MODERN THEORY OF METALLIC BODIES*)

Ya. I. Frenkel.

§ 1. Reduction of the many-electron problem to a collection of one-electron problems in the case of complex atoms. § 2. Reduction of the many-electron problem to a one-electron problem in the case of crystals; metals and dielectrics. § 3. Critique of the band theories of Bloch–Peierls. § 4. Excited states and partial collectivization of electrons; dielectrics, semiconductors, metals. § 5. Typical metals with a continuous collectivization of electrons. § 6. The simplest model of crystalline bodies and the electrical theory of cohesive forces. § 7. Kinetic theory of repulsive forces in metals. § 8. Improvement of the statistical theory of metals (Gombás and others). § 9. Application of band theory to cohesive forces in metals and alloys (Jones). § 10. Wigner–Seitz cell theory. § 11. Intermetallic alloys. 12. Conclusion.

§ 1. REDUCTION OF THE MANY-ELECTRON PROBLEM TO A COLLECTION OF ONE-ELECTRON PROBLEMS IN THE CASE OF COMPLEX ATOMS.

Quantum mechanics arose, as is known, in connection with the theory of the motion of the simplest mechanical system capable of emitting and absorbing light of a definite oscillation frequency—the harmonic oscillator. In 1913 Bohr extended this theory to more complex mechanical systems—real atoms, or, more precisely, to the electron shells of atoms, leaving aside the question of the structure of the atomic nucleus, which is practically unconnected with the behavior of the electrons and with all those properties of atoms, molecules, and material bodies that are determined by this behavior and for which the only essential characteristics of the nuclei are their charge and mass.

In the case of the problem of the motion of one electron in the Coulomb field of a positive nucleus (the “hydrogen” atom), Bohr’s theory, when relativistic effects were taken into account, at once provided a complete and absolutely accurate quantitative explanation of all observed phenomena (and, in particular, of the spectral regularities).

In the case of complex atoms containing two or more electrons, Bohr’s theory was forced to abandon the search for an exact solu-

*) Revised text of a report delivered on 7/XII-45 in the Chemical Division of the Academy of Sciences of the USSR, as the first Kurnakov lecture.

… and to content oneself with approximate solutions based on reducing the many-electron problem to a set of one-electron ones.

In this approach, the motion of each electron is considered independently of the motion of the others, while their interaction with one another is taken into account approximately, by introducing a certain additional external field that partially screens the Coulomb field of the positive nucleus. In the crudest approximation this screening amounts to reducing the effective charge of the nucleus by an amount that assumes different values for electrons moving in different orbits. In a more exact theory, the action experienced by each of the electrons from the others is characterized by the average value of the force exerted by the latter, during their unperturbed motion along their own orbits, on the former. By choosing these average forces, determined by the orbits of the other electrons and determining the orbit of the given electron, in an appropriate way, one can ensure that the quantized motion of each electron in the corresponding force field leads precisely to the quantized orbits under consideration. This force field, which describes in the best possible way the interaction of the electrons (together with the attraction exerted on them by the positive nucleus), is called “self-consistent.” Using the method of effective electric charges of the nucleus, or the method of the self-consistent field, with allowance for the Pauli principle (according to which at most two electrons may move in identical quantized orbits, provided their spins are opposite), it has been possible, as is well known, to give not only a qualitative but, to a considerable extent, also a quantitative explanation of the chemical, optical, magnetic, and other properties of atoms.

This success is usually regarded as a complete expiation of the “original sin” of the quantum theory of complex atoms, which is associated with replacing the many-electron problem by a set of one-electron problems. In reality, however, this replacement a priori limits the accuracy of those results that can be obtained by means of such a theory, making illusory any attempts to increase the accuracy of the latter by solving more exactly the set of one-electron problems to which it is reduced.

The replacement of Bohr’s theory by modern wave mechanics does not essentially change the situation described above. The insurmountable mathematical difficulties connected with the exact solution of the many-electron problem compel one in this case as well to resort to replacing it by a set of one-electron problems; moreover, the quantized electron orbits of the old theory are replaced by wave functions characterizing the motion of individual electrons. The wave function of the entire electron system is determined by the product of these individual wave functions, or else by a linear combination of such products obtained from one another by permuting electrons among different individual quantized states—

states, and the self-consistent field is chosen in the best way, proceeding from the condition that the probable value of the total energy of the system under consideration be a minimum (the Hartree–Fock method). This condition, as is well known, is satisfied not only by the “normal” state, for which the total energy has an absolute minimum, but also by the excited stationary states of the atom.

The application of the self-consistent-field method to excited states is made difficult by the circumstance that different states of this kind correspond to different self-consistent fields, so that the wave functions characterizing them turn out not to be orthogonal to one another. This circumstance makes any fairly accurate calculation of perturbations caused by external forces and, in particular, the calculation of transition probabilities (between unperturbed states) caused by time-varying forces, extremely unreliable.

§ 2. REDUCTION OF THE MANY-ELECTRON PROBLEM TO A ONE-ELECTRON PROBLEM IN THE CASE OF CRYSTALS; METALS AND DIELECTRICS.

In the case of many-atomic systems, the theory based on the same approximate method of reducing the problem of a system of \(N\) electrons to \(N\) problems concerning the motion of individual electrons in a given—or, more precisely, in a more or less successfully chosen—external field becomes still less accurate than in the case of individual atoms. This applies, in particular, to the modern electron theory of crystals. As applied to monocrystals, the theory has until now been developed on the basis of the assumption, at first sight quite natural, that the potential of the self-consistent electric field possesses in this case three-dimensional periodicity and, consequently, can be expanded in a triple Fourier series in the parameters of the reciprocal crystal lattice. The coefficients of this series, however, are in fact not computed, so that the self-consistent-field method is used only in schematic form. It is assumed, moreover, that in the case of excited states of the crystal as a whole, the motion of the electrons takes place in the very same self-consistent field as in the case of the normal state. This simplification ensures the orthogonality of the wave functions describing the various stationary states of the crystal, but leads to additional errors in the calculation of energy levels and of the probabilities of various transitions caused by external perturbations.

In the case of a many-atomic system, such as a crystal, replacing the interaction of the electrons with one another (and with the positive nuclei) by an external “self-consistent” field leads to incomparably grosser errors than in the case of an individual atom,

Thus the concept of a self-consistent field to a considerable degree loses its meaning and practical usefulness. Indeed, if the motion of each electron is regarded as independent of the motion of the others (taking their interaction with one another into account only by means of a periodic field chosen in one way or another), then the possibility is not excluded that they may accumulate around the same nuclei in numbers exceeding the norm by any amount, and, conversely, that other nuclei may be more or less substantially—or even completely—stripped. In this case the probability of such fluctuations proves to be extremely large, whereas in reality, when the interaction between the electrons is properly taken into account, it must be very small, so that each elementary cell containing one nucleus (in the case of a monatomic crystal) is practically always neutral.

This difficulty is completely ignored in the modern “band” theory of the motion of electrons in crystals, which takes into account only the static (and not dynamic) interaction of electrons, expressed in the fact that, when they are distributed over the various energy levels corresponding to the periodic (supposedly “self-consistent”) field under consideration, they are distributed in pairs (with opposite spin directions) over each of the \(\frac{1}{2}N\) lower levels, while all the remaining, higher levels remain vacant (in the normal state of the crystal).

This scheme is usually realized in two variants. In the first variant, developed by Peierls, all electrons are regarded, in a first approximation, as completely free, so that the action of the periodic field (playing the role of the self-consistent field) reduces to a weak perturbation of their motion. This perturbation leads to the appearance of a series of “zones,” the boundaries between which correspond to such unperturbed motions of the electrons in which the latter undergo selective reflection, analogous to the diffraction of X-rays of the same wavelength. The width of these “forbidden” zones, in the case of electron waves, is somewhat greater than in the case of X-rays (for correspondingly identical wavelength and direction of propagation of the waves), increasing as the energy of the electrons decreases. The set of energy levels enclosed between two adjacent forbidden zones thus forms the so-called “Brillouin zone.” The first of the Brillouin zones, corresponding to the lowest energy level of the electron in an isolated atom, is bounded by a forbidden zone only from the outside (or “above”).

This variant of band theory is applied in some cases to all electrons of the crystal, which, in the case of crystals with complex atoms containing a large number of electrons, gives an extremely distorted caricature of reality. Usually, however, the scheme under consideration is applied only to those

electrons which, in the individual atoms, are bound least strongly, i.e. “valence” electrons, whereas the motion of the remaining “inner” electrons is described in the same way as in the case of isolated atoms. Such a “moderate” use of the band scheme with respect to only the outer electrons in some cases—especially in the case of metals with their relatively weakly bound outer electrons—leads to results which, in a number of respects, agree more or less with reality, if not quantitatively, then qualitatively.

The second version of the band theory, developed by Bloch, at first sight appears somewhat less radical than the preceding one; it starts from the approximation of bound electrons. However, in the quantitative development of the theory this binding is taken into account in an obviously incorrect way. Namely, it is assumed that an electron bound to some atom can pass from this atom to another, for example a neighboring one, and become bound to it in exactly the same way, in particular with the very same energy. This assumption would be quite justified in the case of an external electron added to those electrons which make up the normal shell of the atoms under consideration. In Bloch’s theory, however, it is applied not to an external electron, but to one of the “own” electrons forming part of the shell of some atom. Here the circumstance is ignored that this atom, with respect to the electron belonging to it, is a positive ion, whereas other atoms, with respect to that same electron, “foreign” to them, are neutral atoms. Thus, an individual transition of an electron from its “own” atom to a “foreign” one can take place only when the energy is increased by an amount equal to the difference between the ionization energy of the atom and its affinity for the external electron; in such a transition both the binding energy and its character change.

These entirely obvious considerations are completely ignored in Bloch’s theory; at the cost of this neglect it arrives at the idea that electrons which, in isolated atoms, were on one and the same energy level, when these atoms approach one another lose their connection with the individual atoms, as though becoming collectivized by them, and become similar to the free electrons of Peierls’ theory. Their motion in an ideal crystal lattice may then be described as a successive transition from one atom to a neighboring one in an unchanged direction, which, from the point of view of wave mechanics, corresponds to plane waves, speckled with a fine ripple, with a periodicity equal to the periodicity of the lattice itself. In this case each energy level of the electron in the isolated atom is split, in the case of a crystal consisting of \(N\) atoms, into \(N\) levels, the spacing between which is inversely propor-

...proportional to \(N\); all these levels thus form a band of constant (independent of \(N\)) width, or a “zone,” analogous to some extent to the Brillouin zone. To the different energy levels of electrons in isolated atoms there correspond, in the case of a crystal, different Bloch zones, separated from one another by more or less broad intervals—forbidden zones—just as in the case of Peierls’s theory. In this connection the lower zones are regarded as filled with electrons (in the sense of the Pauli principle), whereas the upper zone, corresponding to the valence electrons, may remain unfilled. In the latter case the crystal behaves as a metal; in the opposite case, as a dielectric.

§ 3. CRITIQUE OF THE BLOCH–PEIERLS ZONE THEORIES.

Whereas Peierls’s theory a priori renounces the idea of a connection between the electrons and the individual atoms of the crystal, treating the electrons in the first approximation as free, Bloch’s theory, considering them in the first approximation as bound, subsequently subjects them to a process of “collectivization,” as a result of which they behave in practice in almost the same way as the free electrons of Peierls’s theory.

The essence of both theories lies in the fact that, when atoms—or, more precisely, atomic nuclei—are brought close to one another, the electrons that belonged to definite atoms pass into common use, becoming capable of moving throughout the entire crystal, independently of one another.

This collectivization of the electrons should not be confused with the effect of quantum exchange, when two or more electrons simultaneously exchange their places or orbits. Such an “exchange,” expressed mathematically in the corresponding symmetrization of the wave functions, must occur both for bound and for free or collectivized electrons, reducing, in the latter case, to their permutation among the various quantized energy levels that characterize the motion of each of them in the periodic field of the whole crystal.

The symmetrization, or, more precisely, antisymmetrization of the wave functions (when the spin of the electrons is taken into account) automatically ensures observance of the Pauli principle for the entire aggregate of electrons whose motion is described by these functions. When electrons are assigned to individual atoms, the Pauli principle is usually applied only to those electrons that belong to one and the same atom; the belonging of two electrons to different atoms or nuclei is then equivalent to their being in different quantum states, even if they move along quantum orbits of one and the same kind (but with different centers). When the electrons are collectivized, the Pauli principle must be applied to all the simultaneous...

...precisely, since their states cannot differ by belonging to one nucleus or another.

In an exact solution of the question of the motion of a system of electrons, taking account of the actual interaction between them, it would be possible, alongside the usual method of wave functions of the electrons’ coordinates, to make use of the method of wave functions of their momenta. In this case the potential energy of the electrons, both with respect to the positive nuclei and with respect to one another, would reduce to differential operators in the corresponding momenta (just as the kinetic energy is expressed by a differential operator in the ordinary, coordinate, representation). Peierls’ theory may be regarded as an attempt to approach the solution of the problem of the motion of electrons in a crystal lattice precisely from this side, but as a very crude attempt, one which does not take proper account of the interaction between electrons and quite groundlessly replaces this interaction, together with the action of the positive nuclei, by a periodic field with the symmetry of the crystal lattice.

Bloch’s theory, proceeding from the representation of bound electrons, i.e. from the representation of their motion in configuration space, passes, by means of an illegitimate principle of collectivization, in constructing the wave functions of individual electrons, to a representation of their motion in momentum, or, more accurately, quasi-momentum space, using instead of the electrons’ coordinates the quasi-momenta of the electrons, corresponding to the wave vectors of the electron waves (without taking account of the fine ripple caused by the crystal lattice and possessing the latter’s periodicity).

Both theories are attempts, undertaken with unsuitable means, in the right direction, but without proper account of the concrete features of the problem associated with the strong interaction of the electrons with the nuclei and with one another.

The unsatisfactory character of these theories appears especially clearly when one attempts to apply them to all electrons—both those which, in isolated atoms, occupy outer positions and those which, being in immediate proximity to the nuclei, constitute practically an inalienable property of the latter. It is quite obvious that treating the attraction experienced by these electrons from the corresponding nuclei as a weak perturbation—as is done in Peierls’ theory—cannot lead to any satisfactory results. The situation is no better, however, in Bloch’s theory. The erroneousness of its idea that the transition of an electron from “its own” atom to a “foreign” one is not accompanied by a change of energy, sufficiently crude in the case when outer electrons are in question, becomes quite inadmissible when the inner electrons are collectivized. Thus, for example,

in the collectivization of the electrons forming the very innermost group of the shells of individual atoms (the $K$-group), the wave function of one of these electrons is replaced by a linear combination of this function and analogous functions corresponding to the presence of the very same electron in the $K$-group of all the other atoms. But since each of these groups is filled by a pair of “its own” electrons, the addition to the latter of an extraneous—third—electron is inadmissible not only from the point of view of the energy balance, as in the case of the outer electrons, but also from the point of view of the Pauli principle. Observance of this principle, just like that of the principle of conservation of energy, requires that, when the given electron passes from its atom to a foreign one, one of the electrons belonging to the second atom should pass to the vacated place of the first atom, i.e. that the two electrons exchange places. However, such an “exchange of places” has nothing in common with collectivization in the true sense of the word: it leads only to the “depersonalization” of the electrons, without at all changing the character of their motion in the individual atoms.

§ 4. EXCITED STATES AND PARTIAL COLLECTIVIZATION OF ELECTRONS; DIELECTRICS, SEMICONDUCTORS, METALS.

These shortcomings of the Peierls and Bloch theory can be eliminated if the process of collectivization is restricted to only an insignificant part of all the outer (valence) electrons, linking it with the transition of the collectivized electron from “its own” positive ion, with which it constituted a neutral atom, to a “foreign” neutral atom, which it transforms into a negative ion.

If the distance between the “transplanted” electron and the “positive hole” left by it is sufficiently large, and if the total number of such electrons and holes is small in comparison with the total number of atoms, then each transplanted electron must behave among the surrounding neutral atoms practically in the same way as a foreign electron introduced into the crystal from outside. At the same time it must be collectivized by these neutral atoms—in precisely the sense in which this concept is understood in Bloch’s theory, where, however, it is applied in an illegal manner based on identifying the excited state of an electron that has attached itself to a “foreign” atom with its normal state in its own atom.

Let us note that, in an analogous manner, positive holes are also collectivized: when each hole is replaced by an electron that previously belonged to one of the atoms surrounding it, this hole passes to the place of the latter, moving in such “relay-like”

in this way throughout the whole crystal; it behaves like a collectivized electron with a positive charge (a “positron”).

This picture of partial collectivization, associated with the expenditure of energy on the transfer of electrons from positive ions to neutral atoms, is usually applied to crystals of dielectrics and electronic semiconductors. Being in such an “excited” state, the crystal acquires the ability to conduct an electric current by the displacement of collectivized electrons and holes in opposite directions, whereas in the normal state it possesses insulating properties, like a gas. In the case of true dielectrics, the energy necessary for the transfer of electrons and, consequently, for their collectivization is of the order of several volts; it can be obtained at the expense of the energy of quanta of visible or ultraviolet light, whereas heating to several thousand degrees is not very effective. In the case of electronic semiconductors, characterized by a relatively small value of the “collectivization energy”—of the order of several hundredths of a volt—a considerable degree of collectivization can already be achieved at room temperatures*). However, in this case as well, as in the preceding one, the electrical conductivity of the crystal must tend to zero at temperatures lowered to absolute zero.

The applicability of the above conception to metallic bodies, whose electrical conductivity does not decrease with lowering temperature but, on the contrary, increases, seems at first sight to be completely excluded. However, Slater¹ (in America), as well as Shubin and Vonsovskii² (in the USSR), attempted to extend it to this case, proceeding from the assumption—which has so far not been either proved or refuted—that the expenditure of energy on the transfer of electrons can be compensated owing to the splitting of the energy levels of the collectivized electrons when the lower boundary of the band formed by them is sufficiently lowered (in the sense of the theory of Bloch or Peierls).

The consideration of the influence exerted by collectivized electrons and holes on one another leads to an analogous result. Namely, it is not difficult to show that with an increase in the number—or, more precisely, the concentration—of collectivized atoms (holes), the energy necessary for their further collectivization decreases; in some cases it may even become negative. Under such conditions the collectivization of electrons must be established spontaneously and be preserved at absolute zero temperature at some definite level. The corresponding bodies are, obviously, metals.

*) Let us recall that the energy of thermal motion at room temperature corresponds to 0.03 volt.

§ 5. TYPICAL METALS WITH COMPLETE COLLECTIVIZATION OF ELECTRONS.

The possibility of explaining the metallic state in the spirit of the concept set forth above is indisputable. However, the question remains open whether, in this case, the degree of collectivization proves to be sufficiently small for each collectivized electron or positive hole to be regarded as “lost” among a sea of neutral atoms—for example, in the alkali metals; this question is undoubtedly answered in the negative. In this case we are dealing with complete collectivization of all valence electrons, so that the notion of displaced electrons and positive holes loses its meaning together with the notion of neutral atoms.

Such a typical metal should be conceived much more simply: namely, as an aggregate of positive ions immersed in a gas or liquid formed by collectivized electrons. This conception, developed by me as early as 1925,[^3] does not, however, lend itself to adequate mathematical formulation. The motion of the collectivized electrons was initially described by me as a combination of rotation around one of the atoms along a certain quantum orbit with transition from this atom to one of its neighbors. Multiple repetition of this combined motion ensures the displacement of each electron throughout the entire volume of the metal. The motion of the electron may then be likened to the motion of a point on the circumference of a rolling wheel: the rotation of this point about the center of the wheel corresponds to the rotation of the electron around the center of the atom of which it is a temporary guest, while the motion of the center of the wheel corresponds to the displacement of the electron from one atom to another.

The main difficulty in this representation consists in establishing such a correlation in the motion of the various electrons as would make it possible for each of them to carry out the described motion without substantial interference from the other electrons. Strict fulfillment of this requirement leads to the conception of the completely coordinated motion of all electrons, or at least of separate electron chains, closed or else extending through the whole metal from one point of the surface to another. Such a conception is apparently applicable to metals in the superconducting state*); in the ordinary state, however, in which

*) This view of the nature of superconductivity was developed by me in two notes published in 1933 and 1935. In the first note[^4] the electrostatic interaction between conduction electrons was supplemented by an electromagnetic one, which, in the presence of a resultant current, is capable of stabilizing their motion at sufficiently low temperatures.[^6] In the second note[^5] it was shown that a superconductor should be treated as a superdielectric (or superdiamagnetic), i.e. as a body whose electrical conductivity is zero, since its magnetic permeability is zero and its dielectric constant is infinite.

they possess finite electrical resistance, the coordination in the motion of the various electrons must have a limited character, not excluding temporal and local disturbances, which may be interpreted as the result of their “collisions” with one another because of the right to “visit” one and the same atom.

The problem of electron correlation remained unresolved also in the further development of the electron theory of metals on the basis of wave mechanics, i.e., with the introduction of the conception of “electron waves” propagating in a metal in all directions. This conception, introduced by me in 1927. ^7 *) made it possible, however, to specify the influence of those “disturbances” which electrons experience in their motion in a metallic body owing to the inhomogeneity of the medium formed by this body—inhomogeneity due partly to foreign impurities and, chiefly, at not too low temperatures, to density fluctuations associated with thermal motion. Just as in the propagation of light waves in a liquid or solid medium, these fluctuations cause scattering of electron waves, i.e., attenuation by a factor \(e^{\mu x}\) in traversing a path \(x\). In this case the quantity reciprocal to the scattering coefficient \(\mu\) can be defined, from the corpuscular point of view, as the mean free path of electrons in a metallic body. Knowing this length as a function of temperature, it is not difficult to calculate the electrical conductivity of a metal. In doing so the theory agrees with experimental data both as regards the temperature dependence of electrical conductivity and as regards its absolute magnitude.

The further development of the electron theory of metals on the basis of wave mechanics, proposed in the already considered above works of Bloch and Peierls, did not lead to any substantial improvement of it either from the fundamental side or in the sense of agreement of its results with experimental data. It may even be said that in some respects this agreement became worse. Namely, the theories of Bloch and Peierls apply, strictly speaking, only to crystalline bodies (more precisely, to single crystals). The conceptions of electron “zones,” characteristic of these theories, fall away in passing from crystalline bodies to molten ones, i.e., to liquid metals. Meanwhile, in most cases the melting of a metal is accompanied only by a comparatively small change in its electrical properties (and also in density and in the energy of the cohesive forces; see below). Namely, the electrical conductivity of a metal upon melting usually falls several-fold, increasing only in the case of such metals whose volume thereby decreases (for example, antimony and bismuth). The character of the temperature dependence above the melting temperature also changes somewhat. These changes are not, however, substantial—

*) In this work I also introduced, in embryonic form, the conception of zones, later developed in greater detail by Peierls and Brillouin.

...constant. In particular, the abrupt increase in electrical conductivity upon melting may be explained by an increase in the compressibility of the metal (associated with an increase in its specific volume) and the consequent increase in the magnitude of density fluctuations.

It follows from this that emphasizing the periodicity of the electric field in solid metals and the use of the resulting band structure of the electron energy levels—which constitutes the distinctive feature of the theories of Bloch and Peierls—can by no means be regarded as an advantage of these theories. A comparison of the properties of solid and liquid metals shows that band effects associated with the crystalline structure are of secondary importance and that, for understanding the most essential properties of metals, they may be ignored altogether.

§ 6. THE SIMPLEST MODEL OF CRYSTALLINE BODIES AND THE ELECTRICAL THEORY OF COHESIVE FORCES.

One of the most essential properties of metallic bodies is their interparticle cohesion and the mechanical and thermal properties associated with it. The key to understanding these properties is the conception, considered in the preceding section, of the collectivization of electrons in metals.

In the early works of Haber^8 and Thomson^9 (in 1916–1925), which treated a metallic crystal as a heteropolar compound of the NaCl type, in which the role of negative ions is played by collectivized electrons (forming a lattice similar to the lattice of positive ions), the cohesive forces were calculated according to the same scheme that had first been applied by Born to crystals of salt-like compounds. In this approach the attractive forces were reduced to electrostatic forces between positive ions and electrons, while the nature of the repulsive forces remained unexplained. Under such conditions, the calculation of the normal distance between neighboring atoms (i.e., the density of the metal) proved impossible, and for the calculation of its energy or compressibility one had to use empirical constants. A further shortcoming of the Haber–Thomson theory was the unjustified assumption of a regular arrangement of collectivized electrons at the nodes of a crystal lattice. If no forces other than electrical ones are taken into account, such an arrangement is unstable and, consequently, cannot persist for long. Moreover, it is entirely unclear how it could arise in the process of formation of a solid metal by condensation of a metallic vapor.

In my first work (1925), a new conception of collectivized electrons was given, which partially eliminated the shortcomings indicated above. In this conception, the valence electrons, upon condensation of a metallic vapor, do not pass into any equi-

...equilibrium positions, but they preserve their quantized rotational motion, complicating it only by (more or less coordinated) transitions from some atoms to neighboring ones. It is precisely in this that the phenomenon of collectivization consists. Application of Clausius’ virial theorem to the Coulomb forces acting between electrons and positive ions shows that the collectivization of electrons must be accompanied by an increase in their kinetic energy by an amount equal to the energy liberated in this collectivization, i.e., in the transition of vapor into a liquid or solid metal.

Thus, in contrast to Drude’s classical theory, it turned out that free electrons in metallic bodies retain, at absolute zero temperature, a kinetic energy different from zero and, moreover, even somewhat greater than that which they possess in isolated atoms.

Abstracting from the rotational component of the motion of the collectivized electrons and considering only their translational motion (from atom to atom), one may, following Drude, liken them to an ideal gas enclosed within an impermeable envelope formed by the surface of the metal*). In this case the electrons, owing to their motion, exert on this “envelope” the same pressure as the particles of an ordinary gas possessing, on the average, the same kinetic energy. The pressure of the electron gas also represents those repulsive forces by which the electric forces of attraction between the positive ions and the collectivized electrons passing between them are balanced. Let us note that these forces are somewhat weakened, but by no means compensated, by the mutual repulsion between like-charged particles, in view of the fact that the nearest distances between the latter are smaller than the nearest distance between oppositely charged particles, i.e., between positive ions, on the one hand, and collectivized electrons, on the other.

The collectivization of electrons occurs spontaneously when the atoms of a metallic vapor are brought sufficiently close together (to distances comparable with the sizes of the orbits of the outer valence electrons), for the reason that it is accompanied by a decrease in the potential energy of the Coulomb forces, only half compensated by the increase in the kinetic energy of the electrons (according to the virial theorem). The decrease in the total energy of the aggregate of metallic atoms upon condensation of the vapor formed by them into a liquid or solid body is nothing other than the binding energy between these

*) Near this surface the electrons experience forces directed inward; moreover, the work that must be expended to overcome them is greater than the maximum kinetic energy of the electrons at absolute zero temperature. As I showed as early as 1917, the additional potential energy of the electrons at the surface of a liquid metal determines the surface tension of the latter.

atoms, i.e. a measure of the work of the corresponding cohesive forces (taking into account the repulsive forces of inertial origin caused by the motion of the electrons).

We thus see that the collectivization of electrons is at the same time a consequence of the rapprochement of the atoms of a metallic vapor and the cause of this rapprochement, or rather of the preservation of their proximity at certain stable distances, at which the electrical forces of attraction are balanced by the inertial forces of repulsion.

From this point of view, nonmetallic bodies differ from metallic ones in that the bond between their atoms is established either without collectivization of electrons, at the expense of the mutual polarization of the atoms (“dispersion” forces), or by means of pairwise sharing of electrons between neighboring atoms (a homopolar bond), or, finally, in the case of salt-like compounds, by the transfer of electrons from metallic atoms to metalloidal ones. In the first case the bond between atoms proves weak; in the second and third it is as strong as in the case of metals. At the same time, however, it has an essentially different character, expressed in the difference between the crystalline structure of metallic and nonmetallic bodies.

For the most typical metals, compactly packed structures are characteristic, in which each atom is surrounded by twelve neighbors. These structures are realized in the form of a face-centered cubic lattice or a hexagonal lattice with a quite definite ratio of axes. Such structures follow directly from the idea of a solid metal as an aggregate of positive ions floating in a negative liquid formed by collectivized electrons. Pressing them toward one another, this liquid acts on them in the same way as an external pressure on an aggregate of identical small spheres, which then arrange themselves in the most compact manner, i.e. so that each small sphere is in contact with twelve neighboring small spheres.

The question of why a number of metallic bodies and, in particular, the alkali metals crystallize in a less compact manner—in the form of a body-centered cubic lattice, in which each atom is surrounded by 8 neighbors—remains unresolved to this day. In this respect, the various refinements introduced into the electronic theory of metals on the basis of wave mechanics have proved completely useless.

§ 7. KINETIC THEORY OF REPULSIVE FORCES IN METALS.

In the preceding section it was indicated that in a solid metallic body at absolute zero temperature and in the absence of external pressure the electrical forces of attraction between ions and

collectivized electrons are balanced by inertial forces caused by the (translational) motion of the latter and reducible to the pressure of the gas formed by them.

In order that this equilibrium have a stable character, the repulsive forces must, with increasing distance \(R\) between neighboring atoms, decrease faster than the attractive forces, so that for \(R<R_0\), where \(R_0\) is the equilibrium distance, the repulsive forces predominate, and for \(R>R_0\) the attractive forces predominate.

It is not difficult to show, from the most general considerations, that this condition is fulfilled.

In the adiabatic expansion of an ordinary gas, the latter cools (and in adiabatic compression it heats up). This means that the average kinetic energy of its particles decreases (or increases). In the case of a monatomic gas, the relation between the kinetic energy \(E\) and the volume of the gas \(V\) (in adiabatic processes) is expressed by the well-known formula \(E V^{2/3}=\mathrm{const}\) *). This formula remains valid also for an electron gas at absolute zero temperature. At the same time, however, while in the case of an ordinary gas the energy is a measure of the absolute temperature, in the case of an electron gas it is an athermal quantity.

Since the volume of a metal is proportional to the cube of the distance between neighboring atoms, the preceding formula for the kinetic energy of the electron gas may be rewritten in the form \(E=A/R^2\), where \(A\) is a constant. As for the potential energy of the electric forces \(U\), according to Coulomb’s law its dependence on \(R\) is determined by the formula \(U=-\frac{B}{R}\). Here the coefficient of proportionality \(B\) is of the order \(e^2N\), where \(e\) is the charge of the electrons and \(N\) is their total number.

The derivative of the energy with respect to the volume, taken with a minus sign, is the pressure due to the corresponding forces. Putting \(V=\gamma R^3\), where \(\gamma\) is a numerical coefficient of order 1, we obtain for the kinetic pressure of the electron gas the positive value \(\frac{2}{3}\frac{A}{\gamma R^5}\), and for the potential (electric) pressure—the negative value \(-\frac{B}{3\gamma R^4}\), characterizing the magnitude of the cohesive forces. Their algebraic sum becomes zero for a value of \(R\) equal to \(R_0=\frac{2A}{B}\), which corresponds to the equilibrium state—

*) This formula is derived as follows. Equating the external work \(p\,dV\), performed by the gas when its volume is increased by \(dV\), to the decrease \(-dE\) of its kinetic energy, and expressing the gas pressure through its energy and volume by the formula known from the kinetic theory of gases,

\[ pV=\frac{1}{3}NmV^2=\frac{2}{3}E, \]

we obtain:

\[ -dE=\frac{2}{3}\frac{E}{V}\,dV, \]

i.e.

\[ d\lg E=-\frac{2}{3}\,d\lg V, \]

whence it follows that

\[ EV^{2/3}=\mathrm{const}. \]

to the state. The stability of this state follows from the fact that the positive pressure (repulsion) changes with \(R\) more rapidly than the negative pressure (attraction). The same result is obtained by considering the total energy of the metal \(W=E+U\), which at \(R=R_0\) attains a minimum value equal to

\[ -\frac{B}{2R_0}, \]

i.e. \(\frac{1}{2}U_0\).

Hence, incidentally, it follows that in the state of equilibrium the kinetic energy is equal to one half of the potential energy taken with the opposite sign—in agreement with the virial theorem.

The preceding considerations are sufficient for an understanding of the nature of the forces which bind atoms in a metallic body and ensure its stable equilibrium; however, they are insufficient for a quantitative determination of these forces, and likewise of the normal volume characterized by the equilibrium value of the interatomic distance \(R_0\). To calculate this distance and the quantities connected with it—the compressibility of the metal, its strength, energy of evaporation, etc.—it is necessary to know the magnitude of the coefficient \(A\) in the formula

\[ E=\frac{A}{R^2} \]

for the kinetic energy of the electrons. This problem was first solved by Fermi in 1927 by applying Pauli’s principle to the free (or collectivized) electrons in metallic bodies. The result obtained by Fermi was applied by me in 1928.¹⁰ for an approximate calculation of the dimensions, compressibility, and other properties of metals at ordinary temperatures, at which the thermal energy of the electrons may be neglected in comparison with their athermal energy, i.e. at temperatures practically indistinguishable from zero.

Under such conditions the Fermi distribution reduces to pairwise occupation by electrons of the lower \(N/2\) energy levels corresponding to their free motion inside the metallic body (\(N\) is the total number of electrons). In this case the maximum velocity of the electrons \(v_{\max}\) corresponds to the minimum de Broglie wavelength \(\lambda_{\min}\), approximately equal to twice the distance between neighboring electrons

\[ 2\left(\frac{V}{N}\right)^{1/3}, \]

or, more exactly,

\[ \left(\frac{8\pi}{3}\frac{V}{N}\right)^{1/3}. \]

Putting

\[ mv=\frac{h}{\lambda}, \]

where \(h\) is Planck’s constant, we obtain for the maximum value of the electron energy at \(T=0\),

\[ \varepsilon_{\max}=\frac{(mv_{\max})^2}{2m}, \]

the expression

\[ \varepsilon_{\max}=\frac{h^2}{2m}\left(\frac{3}{8\pi}\frac{N}{V}\right)^{2/3}. \]

The mean value of the energy of one electron is \(3/5\) of the preceding value; thus the total energy of all the free (or collectivized) electrons \(E\) can be represented in the form

\(A/V^{2/3}\), as we did above, with the constant \(A\) turning out to be equal to \(\dfrac{3}{10}\dfrac{h^2}{m}\left(\dfrac{3N'}{8\pi V}\right)^{2/3}N\). Using this value and the approximate value of the constant \(B\) in the expression for the energy of the Coulomb forces indicated above \((B \simeq e^2N)\), in 1928 I calculated the interatomic distances for a number of typical metals, as well as their compressibility coefficients and the magnitude of their tensile strength, obtaining satisfactory agreement with experimental data both with respect to the relations existing among these quantities and with respect to their absolute values.

§ 8. IMPROVEMENT OF THE STATISTICAL THEORY OF METALS

(GOMBÁS AND OTHERS)

In the following years a large number of works appeared whose aim was a more exact calculation of the energy of metallic bodies and of the quantities connected with it.

Some of these works (for example, those of Gombás\(^{11}\)) are, in essence, merely an attempt to refine the electron theory set forth in the two preceding sections, not connected with any new considerations or conceptions. Thus, for example, in my original theory (1925 and 1928) I assumed, for simplicity, that the “negative liquid” formed by the free electrons is distributed throughout the whole volume of the metal uniformly, i.e. with constant density, except, perhaps, for that part of this volume which is enclosed inside the positive ions (the latter assumption leads to additional repulsive forces, whose energy is inversely proportional to the cube of the distance). Ten years later Gombás arrived at the same conclusions, applying Fermi’s statistical equation to the problem of the spatial distribution of free electrons, taking into account the “exchange effect” (which practically reduces to excluding from the volume negative charge acting on a given electron that part of this charge which is due to this very same electron). In calculating the electrical energy of the metal, Gombás determined not only the order of magnitude of the coefficient \(B(\simeq e^2N)\), but its exact value, identifying the potential energy of one of the positive ions with respect to all the surrounding charges with its potential energy with respect to a sphere of negative charge of equal magnitude, at the center of which this ion is located (since the action it experiences from the other ions is compensated by the action of the corresponding negative spheres).*

In one of his articles Gombás attempted to apply Fermi’s statistical method not only to the valence electrons, but also to

* In my first work (1925) the same result was obtained by a more complicated route.

internal ones, determining the energy of formation of the crystalline lattice of a metal as the difference between the energy of the latter and the sum of the energies of the atoms composing it, taken separately. It turned out that the density of the electronic fluid inside the ions (i.e., at small distances from the atomic nuclei) practically does not change upon condensation of the metallic vapor, i.e., consequently, that the corresponding “internal” electrons are practically not affected by the process of collectivization of the “outer” (valence) electrons, which are distributed almost uniformly in the spaces between the positive ions, forming as it were a disperse medium in which these ions are suspended, like colloidal particles of the dispersed phase.

Somewhat earlier than Gombás, an analogous method was applied by Jensen¹² to calculate the energy of formation of heteropolar (ionic) crystals of the NaCl type; in doing so Jensen assumed that the distribution of the negative charge in the crystal lattice can be represented as the sum of charges formed by the electron shells of ions of both signs, with preservation of the spherical symmetry of these shells (with respect to the corresponding atomic nuclei), but with the electron density changing in the radial direction as a function of the lattice constant. Proceeding from the condition of a maximum of the total (i.e., potential and kinetic) energy, Jensen determined, for a number of heteropolar crystals, the normal dimensions and the energy of formation in satisfactory agreement with experimental data.

§ 9. APPLICATION OF THE ZONE THEORY TO COHESIVE FORCES IN METALS AND ALLOYS (JONES).

The advantage of the statistical method in comparison with more rigorous methods of wave mechanics consists chiefly in its simplicity, which in a number of cases makes it possible, with its aid, to obtain results not inferior in accuracy to those that can be obtained on the basis of a more exact quantum-mechanical formulation of the corresponding problem, when one must necessarily be content with a roughly approximate solution of it, ignoring the effects of correlation in the motion of different electrons (in the statistical theory these effects disappear)*. However, some characteristic wave effects, manifested in the motion of individual electrons, cannot be taken into account when this motion is described by the statistical method (which is based on the use of classical mechanics in connection with Fermi statistics).

It is therefore natural that the question of cohesive forces in solid metals was treated by a number of authors with the aid of the same methods of quantum

* Since, by taking into account the interaction of the electrons with one another, we are thereby not considering the motion of each of them separately.

mechanics, which prove necessary for understanding and calculating the electrical conductivity of metallic bodies, as well as their other, more “subtle” properties—electrical, magnetic, galvanomagnetic, optical, etc. At the same time, however, there inevitably arise, in one form or another, all those difficulties of which we spoke above in connection with the band theories of Bloch and Peierls, and, above all, the difficulty of correctly taking into account the interaction between electrons and the correlations in their motion.

The application of these theories (or, more precisely, Peierls’ theory) to the problem of the structure of metallic bodies—in particular intermetallic alloys—and of the cohesive forces determining this structure was undertaken chiefly by Jones[^13]. The initial principle of his theory was the agreement of the number of valence (collectivized) electrons with the volume of the first Brillouin zone (separated from the next by a sufficiently large energy discontinuity), in which these electrons must be placed in accordance with the Pauli principle and the requirement of minimum energy. It is precisely the latter requirement that leads to the necessity of placing the electrons inside the first zone, since their penetration into the second is associated with a stepwise increase in energy.

The principles set forth enabled Jones to give a theoretical interpretation of the well-known Hume-Rothery rule, which connects the crystal structure of alloys of various types with the number of electrons per elementary cell of the crystal lattice. However, the extension of Jones’ theory to dilute solid solutions of metals of different valence encountered an insurmountable difficulty, not of a mathematical but of a fundamental nature. Imagine, for example, that divalent zinc atoms are embedded in the lattice of a monovalent metal, say silver (according to the substitution principle). According to the basic principles of Peierls’ band theory, both valence electrons of each Zn atom, just like the valence electrons of Ag atoms, must enter the “common pot” of the alloy, i.e. undergo collectivization. Since, however, the zinc ion must then possess a doubled positive charge compared with the copper ion, such collectivization proves impossible (according to Mott, the excess charge of the impurity ion must be partially screened by a nearby electron, which thus in fact remains in a bound state).

We shall return below to the question of the theory and structure of alloys. Here it is necessary to note the circumstance that, in application to pure metals, Jones’ theory made it possible—though not without some strain*—to explain the structure of such an intermediate element as bismuth, whereas the structure of typical metallic ele-

* Connected with replacing the first Brillouin zone by a zone of higher order, characterized by a large magnitude of the energy jump at its boundary.

ments and, in particular, of the alkali metals with their body-centered cubic lattices (instead of the lattices of the “close-packed” type expected on the basis of the elementary theory of lattices), received no explanation.

§ 10. THE THEORY OF WIGNER–SEITZ CELLS.

Of more serious interest than Jones’s theory is the theory of Wigner and Seitz[^14]. Strictly speaking, this theory altogether rejects the idea of electron collectivization and replaces it by the idea that each atom in the crystalline lattice of a metal has the normal complement of valence electrons. The difference between such a “lattice” atom and an isolated atom (in a metallic vapor) consists in the fact that in the latter case the valence electrons may be at an arbitrarily large distance from the nucleus, whereas in the former they must remain inside a certain polyhedral cell, separating the “sphere of influence” of this nucleus from the corresponding spheres of influence of neighboring nuclei. Replacing the polyhedron by an equal-volume sphere, Wigner and Seitz introduce, as the boundary condition for the valence electron “assigned” to the corresponding atom, the vanishing of the derivative of the electron wave function in the radial direction at the surface of the sphere (instead of the usual condition \(\psi = 0\) for \(r = \infty\)). By this condition they attempt to express the circumstance that an electron “assigned” to a given atom, i.e. lying in its sphere of influence, can in reality pass into the sphere of influence of a neighboring atom, replacing the electron which had previously been there. If it is assumed that such a “substitution,” or more precisely “permutation,” of the two electrons has no fundamental significance (owing to the indistinguishability of electrons), and if, accordingly, one considers the wave functions \(\psi_i\) of the electrons assigned to various atoms \(i\) as the values of the wave function \(\psi\) of one (any one) of them in the corresponding regions (cells), then the Wigner–Seitz boundary condition ensures the continuity of the function \(\psi\) throughout the whole volume of the crystal, and the function \(\psi\) itself describes the motion of an electron as if it could move throughout this entire volume, i.e. were in a collectivized state. In fact, however, as we have just seen, such motion is possible only with the proper correlation in the motion of different electrons, ensuring the preservation in each cell of an unchanged (normal) number of electrons.

In the Wigner–Seitz theory this correlation is taken into account in a very incomplete way, namely purely statistically—by describing the entire aggregate of valence electrons with the aid of a wave function \(\Psi\), composed of products of “local” wave functions \(\psi_i\), for the individual electrons, symmetrized in accordance

in accordance with the requirements of Fermi–Pauli–Dirac statistics. In this, the mutual repulsion of the electrons is taken into account only indirectly, in determining the local functions \(\psi_i\), which is carried out under the assumption that the force experienced by each electron from the others is compensated by the action of the corresponding positive ions, so that the resultant field acting on the given electron reduces to the field of the nucleus to which it is taken to belong (i.e., just as in the case of an isolated atom).

Such a model of a metallic body—which in essence differs in no way from the model of some metalloid (for example, inert) element in the solid state—is incompatible with the existence of ordinary electrical conductivity, associated with more or less uncoordinated transitions of individual electrons from one cell to another and, consequently, with an anomalous increase in the number of electrons in some cells at the expense of others, i.e. with the formation of negative ions and positive holes (cf. § 4)*).

However, if the number of such polar (i.e., negatively or positively charged) cells is small in comparison with their total number, then this circumstance cannot have any appreciable effect on the magnitude of the cohesive forces and on the structure of the metal, so that the conclusions of the Wigner–Seitz theory which pertain here remain valid when this circumstance is disregarded.

These conclusions for monovalent metals—lithium and sodium—if one starts from the structure which they actually possess, are in good agreement with experiment both with respect to the value of the specific volume (i.e., the interatomic distance \(R_0\)) and with respect to the condensation energy. However, the question why they have precisely this structure and not some other remains unresolved; for this it would be necessary to carry out analogous calculations, starting from other possible types of lattice, and to show that for all of them the energy of the metals considered would turn out to be greater than in the case of the body-centered cubic lattice observed experimentally. In view of their complexity, however, such calculations have not been carried out by anyone up to now.

Naturally, the question arises: do the results of calculations of this kind justify the labor that must be expended in carrying them out. In my opinion, this question should be answered in the negative. The accuracy of the results increases to an incomparably smaller degree than the complexity and tediousness of the calculations, and is, moreover, in principle limited by the inevitable inaccuracy of the initial approximations (connected, for example, with reducing the many-electron problem to a one-electron one). An increase in the accuracy of the calculations has

*) It does not, however, exclude the possibility of the existence of “superconductivity,” associated with a coordinated “chain” displacement of electrons.

makes sense only if it makes it possible to explain such fundamentally important aspects of the phenomenon under consideration as cannot be explained by less exact calculations. In this, of course, much depends on what questions are to be regarded as fundamentally important and what as inessential.

As was already noted in § 3, the melting of metallic bodies is accompanied by a comparatively small increase in volume (about \(10\%\), which corresponds to an increase in interatomic distances of only \(3\%\)) and by an insignificant increase in internal energy (the latent heat of fusion of metals amounts to several percent of the latent heat of vaporization). If, consequently, in calculating the energy, density, and other properties connected with them of metallic bodies one is satisfied with an accuracy of the order of several percent, then there is no need to take into account the distinction between the solid and liquid states of these bodies. Statistical theory, even in the crudest form in which it was first developed by me, gives, under such conditions, a quite sufficient approximation to reality.

§ 11. INTERMETALLIC ALLOYS.

In § 9, in considering Jones’s theory, we already briefly touched upon the question of intermetallic alloys. This question, despite its fundamental importance for metal physics, is at present in a very poorly developed state. In the case of alloys of stoichiometric composition \((A_mB_n,\) where \(m\) and \(n\) are small integers), which are chemical compounds between the corresponding metallic elements, we so far have only Jones’s theory and the as yet unpublished work of Smirnov*), who, like Jones, proceeds from the band theory of Peierls (in the approximation of nearly free electrons) and of Bloch (in the approximation of strongly bound electrons). Unlike Jones, however, Smirnov, developing an idea expressed by me as early as 1937 and sketched before him by Rudnitskii \(^{15}\), and also, independently of him, by the Japanese physicist Muto \(^{16}\), generalized the Brillouin-zone theory to the case of a binary alloy of the type \(A_mB_n\) (or one close to it), introducing, along with the principal zones, which correspond to the most intense reflections of X-rays, additional zones corresponding (in the case of a fully ordered alloy) to “superstructure” X-ray lines. On this basis it proves possible not only to determine the type of crystal structure of the alloy (this problem, considered by Jones, Smirnov, and Muto, was not in fact dealt with by them), but also to calculate—with the limited degree of accuracy allowed by the band method—the energy

*) Doctoral dissertation, Leningrad, 1946.

formation of an alloy, the lattice constant, and also the electrical conductivity and various other electrical and galvanomagnetic coefficients.

The main interest of these works, however, lies not in such calculations, which lead to numerical results of very doubtful value, but in elucidating the influence exerted on the various properties of an alloy by a decrease in the degree of order in the arrangement of the atoms of both kinds that form it over the corresponding (and non-corresponding) sites of the crystal lattice (on the assumption that the latter remains unchanged). The influence of such “disordering” on the electrical resistance of an alloy had in part been considered earlier as well (for example, by Nordheim), namely in that part in which this disordering causes additional scattering of electrons, imparting to the alloy the properties of a turbid medium with respect to electron waves—in addition to that “turbidity” which is due to thermal fluctuations of density (see § 5)*).

Alongside this direct influence of disordering, however, it exerts on the electrical resistance and, in particular, on the galvanomagnetic properties (Hall effect, additional resistance in a magnetic field) an indirect influence, depending on the constant weakening of the boundaries between additional Brillouin zones, corresponding to the weakening of superstructure X-ray lines in the X-ray diagrams of the alloy. By this indirect influence of disordering it proves possible to explain, at least qualitatively, the anomalous change of the Hall effect with decreasing degree of order (for example, when the temperature of the alloy is raised, especially near the Curie temperature)**).

Another very interesting point in Smirnov’s work is the indication of the possibility of stoichiometric intermetallic compounds which, with complete ordering of the atoms of both components, would be dielectrics and whose electrical conductivity should increase sharply as they become disordered, or else when the composition deviates from the stoichiometric one. This result was obtained by Smirnov for alloys with a body-centered cubic lattice, on the assumption of equal valence of both metallic elements and of a small difference between their ionization potentials.

This paradoxical conclusion is confirmed by a number of recently discovered experimental facts, to which attention was drawn

*) The additional “turbidity,” depending on the irregular distribution of impurities, causes, in the case of weak solutions, an additional resistance determined by Matthiessen’s rule and theoretically explained by me as early as 1928.

**) In the experiments of A. P. Komar, the influence of the degree of order on galvanomagnetic properties was investigated at constant temperature by comparing alloys quenched by rapid cooling from different initial temperatures.

A. F. Ioffe*). These facts, however, do not pertain to such alloys for which they should be expected according to Smirnov’s theory, but to alloys between certain typical metals and elements with a weakly expressed metallic character (for example, lead and sulfur, thallium and sulfur, zinc and antimony).

I think that the absence of conductivity—or, more precisely, its very small value—in alloys of this kind is explained not at all by the considerations of band theory from which Smirnov proceeded (and which are connected with the position of the surface of maximum kinetic energy, according to Fermi, relative to the boundary of the first additional Brillouin zone), but by much simpler and more evident considerations connected with the relative character of metallic and metalloidal properties in the interaction of different atoms with one another in molecules or crystals.

It can be shown that, when the two components of an alloy \(AB\) are unlike, the density of the electron liquid near atoms of one kind must be somewhat greater than near atoms of the other kind, just as occurs in a more sharply expressed form in the case of salt-like compounds such as NaCl, in which the metalloid atoms expropriate the valence electrons from the metal atoms. In the case of intermetallic alloys of metals, different atoms behave in such a way as if some of them had an excess negative charge and others a positive charge—that is, in other words, as if some of them, together with the electron liquid surrounding them, constituted negative ions, and others positive ions. The charge of these “ions” in the case of alloys between metallic elements of the same valence may amount to a small fraction of an elementary charge, i.e., of the charge of an electron or proton**).

Under such conditions, the possibility seems quite natural that a fully ordered stoichiometric alloy of two metals may possess the same electrical and mechanical properties as an ionic dielectric. The conditions for the realization of this possibility, however, remain unclear, owing to the present lack of a satisfactory quantitative theory of the phenomenon of collectivization of electrons in solids, one which would properly take into account the interaction of electrons with one another and the correlation in their motion.

The inadequacy in this respect of the modern electron theory of metallic bodies and, in particular, of the various versions of band theory has already been noted by us more than once above, and cannot be eliminated by refining and complicating this theory within the fram—

*) In a report at the session of the Division of Physical and Mathematical Sciences of the Academy of Sciences of the USSR in Leningrad, in December 1945.

**) According to calculations by Yu. P. Bulashevich, Candidate’s dissertation, Leningrad, 1939.

as its basic initial assumptions, connected with reducing the many-electron problem to a one-electron one. This circumstance is illustrated by the inability of the modern theory to explain the phenomenon of superconductivity*), and also to give quantitative criteria for solving the fundamental question of whether some element (not to mention chemical compounds) belongs to the class of conductors or dielectrics. Thus, for example, the question of why diamond is a dielectric and not a conductor, whereas the elements homologous to it—silicon, germanium, and, in particular, tin and lead—have more or less sharply expressed metallic properties, remains open at the present time. What is beyond doubt is only the circumstance that, in the case of dielectric elements—for example, that same diamond—the bond between atoms has not a metallic but a homopolar character, i.e. is due not to complete collectivization of all the outer electrons, but to their pairwise sharing between neighboring atoms, leading to the formation of directed “chemical” bonds. It is also beyond doubt that this character is preserved, to a greater or lesser degree, also in the case of elements with a clearly expressed metallic character, so that the “metallic bond” is not a special kind of interatomic bond essentially different from the covalent one, but only a certain modification of the latter, connected with the ability of individual electrons to pass from some atoms (or, more precisely, pairs of atoms, which they bind to one another) to other atoms (or pairs of atoms) more or less independently of the remaining electrons. The factors responsible for the loosening of the covalent bond, corresponding to its “metallization,” have not yet been clarified. It is clear, however, that the most important of them is the magnitude of the binding of the valence electrons in the isolated atoms of the element under consideration: the weaker this binding (measured by the magnitude of the ionization energy), the greater the degree of metallization when atoms combine into a solid or liquid aggregate**).

In the limiting case of the most typical metals—for example, the alkali metals—the covalent bond is modified to the greatest degree, losing its usual character and reducing to the typical “metallic bond,” which is effected by the attraction of the negative fluid formed by collectivized electrons to the positive ions floating in it.

*) In any case, quantitatively.

**) Many years ago Herzfeld expressed the idea that metallic elements can be characterized by an anomalously large polarizability of their atoms, leading, according to the Lorentz formula
\[ (\varepsilon - 1)(\varepsilon + 2)=\frac{3}{4\pi}\alpha N \]
to an infinitely large (or negative) value of the dielectric constant in the solid state. This point of view is in full agreement with ours, since the polarizability of an atom is the greater, the weaker the binding of its valence electrons.

The fact that the latter are arranged not compactly, but in the form of a body-centered cubic lattice, indicates, however, that even in this limiting case the matter is not so simple. In the opinion of A. F. Ioffe, this characteristic structure of the alkali elements is explained by the fact that the spins of the electrons in neighboring atoms tend to orient themselves in opposite directions—which would be impossible if each of them were surrounded by twelve neighbors. As for the indicated tendency, which characterizes antiferromagnetic substances, it may be explained in the same way as the tendency toward identical orientation of spins in ferromagnetic substances, proceeding from a consideration of the magnitude or, more precisely, the sign of the “exchange energy” corresponding to the interchange of the two electrons binding two neighboring atoms in the crystal lattice.

This point of view is fully consistent with the conception of the chemical or covalent character of the interatomic bond even in the case of the most typical metallic bodies*).

The contraposition of the homopolar and metallic bonds in the case of elementary substances is just as incorrect as the very widespread present-day contraposition of the homopolar and heteropolar bonds in the case of chemical compounds. The notion that in a molecule or crystal of the NaCl type the atoms of the metal completely give up their valence electron to the atom of the metalloid, forming a compound of purely ionic character, is inaccurate. In reality, as Kossel pointed out in 1916, the valence electrons of both atoms are shared by them, while, however, the center of gravity of the system of shared electrons is displaced toward the “stronger” partner, which is the metalloid atom. In the case where the two partners are identical (for example in the molecule H$_2$ or O$_2$), the shared electrons are distributed symmetrically, in accordance with the usual conception of the purely homopolar character of the covalent chemical bond. When the two atoms are not identical, this bond is “polarized,” acquiring a more or less pronounced ionic character, but remaining essentially covalent, i.e. conditioned by the sharing of the valence electrons of neighboring atoms, just like the ordinary homopolar bond. Thus, polarity is a certain property of the covalent bond, disappearing in the limiting case of identical atoms and by no means capable of serving as a basis for opposing the homopolar bond to the heteropolar one, which should be regarded—

*) Similar considerations were expressed by me in a report on the theory of antiferromagnetism in Kiev in 1936. In all probability, however, they have no direct bearing on the question of the crystal structure of the alkali metals. Indeed, iron, which is the prototype of a ferromagnetic substance, possesses the very same structure of a body-centered cubic lattice.

considered as a covalent bond with a sharply expressed polar character.

In exactly the same sense, the “metallic bond” should be regarded as a covalent bond with a sharply expressed metallic character, conditioned by the ability of the electrons forming it to pass more or less independently of one another from one pair of atoms to another (decreasing it by half in the first case and increasing it by the same amount in the second).

In intermetallic alloys the covalent bond is complicated not only by this metallization, but also by polarization, i.e. by a bond of the heteropolar type; moreover, as we saw above, this polarization may be brought about through metallization, leading to the formation of intermetallic alloys of stoichiometric composition possessing sharply expressed dielectric properties.

These complications are manifested not only in the electrical, but also in the magnetic and mechanical properties of metallic elements and alloys. However, the theory of metals will not be able to give a satisfactory explanation of all these complications until it is based on the self-consistent-field method, i.e. on the reduction of the many-electron problem to a set of one-electron ones.

§ 12. CONCLUSION.

The more complex the system under consideration, the more simplified, of necessity, its theoretical description must be. It is impossible to demand from the theoretical description of a complex atom—and still more of a molecule or a crystal—results of as high an accuracy as from the theory of the simplest hydrogen atom. Such a demand, moreover, is not only impossible to fulfill but, in essence, also “worthless.” Physical theory may set itself two kinds of tasks: fundamental and practical. Fundamental tasks amount to the explanation and prediction of physical phenomena, while practical ones amount to the calculation of those quantities by which these phenomena are characterized, with the aim of making their technical use more convenient.

The exact calculation of the constants characterizing the simplest physical systems is of essential importance as a test of the correctness of the basic principles of the theory. After, however, it has brilliantly withstood this test, there is no sense in subjecting it to new tests in application to more complex systems. The most ideal theory cannot withstand such a test because of the practically insurmountable mathematical difficulties that it inevitably encounters in application to complex systems. In this case all that is required of the theory is a correct interpretation of the general character of the quantities and regularities pertaining to such a system. In this respect the theoretical physicist is like a caricaturist, who must reproduce

original not in all its details—like a photographic apparatus—but to simplify and schematize it in such a way as to reveal and emphasize its most characteristic features. Photographic accuracy can—and should—be demanded only of the theoretical description of the simplest systems. A good theory of complex systems must be only a good caricature of these systems, exaggerating those of their properties which are most typical and deliberately ignoring all the other inessential properties.

Approaching the assessment of the modern electron theory of metals from this point of view, we see that, with respect to a number of properties of metallic bodies, the theory which I gave as early as 1925–1929 is sufficiently satisfactory, and that its further mathematical development on the basis of quantum mechanics and statistics cannot lead to any substantial improvement of it. A good caricature of some person cannot be substantially improved by a more careful and accurate depiction of the uncharacteristic details of his face and figure.

The band theory of Bloch–Peierls, the statistical theory of Gombás–Jensen, and Wigner–Seitz’s “cellular” theory make it possible to obtain somewhat more accurate results, a somewhat more complete resemblance of the caricature or “model” to the original, but only to a very limited degree and, moreover, at the cost of a completely inadequate expenditure of labor, and, above all, of the obvious ignoring of other features of the original, connected with the collectivization of electrons, the type of structure, the character of the bond, mechanical properties, etc., with respect to which these complex theories prove to be just as helpless as the elementary theory.

The latter, therefore, needs further development in an entirely different direction, which until now has remained without due attention and is connected with a more exact allowance for the interaction of electrons and the correlation in their motion. In this connection, the task of theory should consist not in a more accurate calculation of various quantities characterizing metallic bodies, but in a more correct understanding of the distinctive properties of these bodies in comparison with nonmetallic bodies. In other words, the theory should be developed further not so much in a quantitative as in a qualitative direction. It is far more essential to understand the nature of the connection between the electrical conductivity of metals and their plasticity (the nature of which still remains entirely unclear) than to calculate more accurately than has been done until now the magnitude of the electrical conductivity or the energy of the cohesive forces.

It is far more interesting and important to explain the difference between diamond and lead with respect to their electrical properties, or the difference between the ordinary metallic state and the superconducting state, than to try to refine the calculation of the galvanomagnetic or thermoelectric coefficients of various metallic bodies.

Thus matters stand in the approach to the modern theory of metals from a fundamental point of view*). As for the practical approach to this theory, here the further path of its development may be illustrated by comparison with the most recent development of hydrodynamics and the theory of heat transfer as applied to the complex problems of modern technology. In both cases the development of the theory proceeded along the path of creating semi-empirical methods of calculation, not ignoring experiment, but incorporating its results into theoretical investigation by using various theoretical “criteria,” or dimensionless numbers—Reynolds, Prandtl, Nusselt, and so on—the relations among which are established experimentally. The most recent development of the theory of the internal structure of stars has followed this path. I think that the development of the theory of metals, as a practical tool for metal science and metallurgy, should proceed along the same path.

References

1. J. C. Slater, Phys. Rev., 36, 57 (1930).
2. S. P. Shubin and S. V. Vonsovsky, Pros. Roy. Soc., 145, 159 (1934).
3. Ya. I. Frenkel, ZhRFKhO, phys. part, 1924; see also: Zs. f. Phys. 26, 117 (1924).
4. J. Frenkel, Phys. Rev., 43, 907 (1933).
5. J. Frenkel, Nature, 133, 730 (1934).
6. W. Band, Phys. Rev., 69, 41 (1945).
7. J. Frenkel, Congress International de Physique à Como (1927); see also: Zs. f. Phys., 47, 819 (1928) Frenkel u. Mirolubow, Zs. f. Phys., 48, 835 (1928).
8. F. Haber, Ber. Berl. Akad. (1919) S. 506 u. 909.
9. J. J. Thomson, Phil. Mag., 43, 736 and 44, 657 (1922).
10. J. Frenkel, Zs. f. Phys., 49, 31 (1928); see also Wave Mechanics, part I, p. 304.
11. P. Gombas, Zs. f. Phys., 93, 378 (1935); 94, 473 (1935); 99, 749 (1936); 100, 599 (1936); 104, 81, 543 (1937).
12. H. Jensen, Zs. f. Phys., 77, 722 (1932).
13. H. Jones, Pros. Roy. Sos., 144, 225 (1934), 147, 396 (1934); see also Mott a. Jones: Theory of Metals and Alloys (1936).
14. E. Wigner u. F. Seitz, Phys. Rev., 43, 804 (1933).
15. V. Rudnitskii, ZhETF (1940).
16. T. Mito, Sc. Pap. Inst. Phys. Chem. Res., Tokyo, 34, 377 (1938).

*) We have a completely analogous situation in modern quantum chemistry, even with respect to simple molecules. The methods used in it for calculating bond energies, interatomic distances, and so on are of a roughly approximate character, sufficient, however, for a fundamental clarification of the corresponding phenomena and regularities.