Bibliography

F. Seitz, The Modern Theory of Solids, translated from the English under the editorship of G. S. Zhdanov. Gostekhizdat, Moscow, 1950, price 36 rubles.

The theory of solids is one of the most extensive and important chapters of modern physics. In addition to the great fundamental interest presented by the study of solids, solid-state physics provides a basis for the development of a whole series of related branches of science and technology. It is enough to mention, for example, such fields of knowledge that are essentially connected with it as metallurgy, ultra-high-frequency radio engineering, the study of catalysis and corrosion, for the full breadth and importance of the problems facing solid-state physics to become clear. It is no accident, therefore, that at the present time it is experiencing rapid growth. An ever-increasing number of workers in related fields is coming face to face with problems belonging to the domain of solid-state physics (in particular, with electronic processes in crystals). Meanwhile, there are not so very many books on the theory of solids, and works intended not only for theoretical physicists are almost entirely absent. One may therefore only welcome the translation of Seitz’s well-known book, which has long since gained popularity among physicists.

It should be noted, however, that the title of the book is considerably broader than its actual content. In essence, it deals mainly with electronic phenomena in solids; the theories of heat capacity and phase transitions are covered only in two chapters of a semi-popular character, while such questions as the theory of the mechanical properties of crystals, for example, are not touched upon at all. In addition, owing to the rapid development of solid-state theory during the war and especially in the postwar period, Seitz’s book, published in 1940, is now already obsolete to a considerable degree. This applies above all to the theory of ionic crystals and semiconductors. After the work of a number of Soviet scientists (A. F. Ioffe, V. E. Lashkarev, S. I. Pekar, F. F. Volkenshtein, E. I. Adirovich, and others), the corresponding sections of the book under review make an entirely archaic impression. In the theory of metals as well it has fallen rather far behind the present day (the works of N. N. Bogolyubov and S. V. Tyablikov, S. V. Vonsovsky). We note, incidentally, that Seitz in general cites very few Soviet authors, as a result of which, from a book devoted to the theory of solids, the investigations of A. F. Ioffe and his school on semiconductor physics, of S. V. Vonsovsky on the polar model of the metal, of L. D. Landau and I. Ya. Pomeranchuk on the influence of electron interaction on the electrical conductivity of metals, and others have been omitted. The number of such examples could easily be increased, but what has been said is sufficient to show the incompleteness and limitations of the book under review. This is its first major shortcoming. It is all the more unpleasant because the insertion into the text of the corresponding corrections and additions would have required very serious reworking of the entire book, for it is too much of a...

connected whole. Therefore the editor had to leave the book in an “untouched” state, supplying it only with small, though very numerous, notes.

Zeitz’s book would more properly have been called not “The Modern Theory of Solids,” but “The Band Theory of Metals,” for its main content is precisely a systematic exposition of the Bloch–Peierls band theory and its application to a number of concrete problems. Within the framework of this problem the book is written masterfully; the exposition of the theory of free electrons in a metal may, in our opinion, be considered classical. At the same time, the author’s striving to stay as close as possible to experiment is also attractive: the whole book is saturated with experimental curves and tables; a careful comparison of theory with experiment is steadily pursued. Unfortunately, however, the author quite inadequately emphasizes all the crudeness and the purely qualitative character of band theory. Only occasionally does he make a few timid remarks about the possibility of applying the band model to $d$-shell electrons, continuing, however, to use it even when it is plainly inapplicable—for example, in the theory of ferromagnetism; in the rest of the book he tries to extract quantitative conclusions from band theory, being little troubled by the circumstance that some of his calculations have the character of rather “uncovered” constants fitted to experimental data. (Let us note in parentheses that such a “method of calculation” is quite popular in certain foreign circles, chiefly among American physicists.) At the same time it is well known—this has been shown in a number of works by Soviet scholars—that the band theory of metals is only a very schematic, rough representation of reality. The reason for this is that the interaction of electrons in band theory is taken into account in an extremely crude and inadequate way, whereas in metals it is by no means small. Therefore the conclusions of band theory have only a purely qualitative character; moreover, what calls for explanation is not the discrepancy of its quantitative predictions with experiment (this is self-evident), but the agreement of them, if such agreement is obtained for some reason! In forgetting this circumstance lies the second shortcoming of the book.

Let us note, incidentally, that Zeitz was altogether “unlucky” with the many-electron problem: the method of spin waves in the theory of ferromagnetism, which is still the best-known example of a “many-electron” approach to the question, is presented quite unsatisfactorily, without even saying that spin waves obey Bose statistics, as a result of which, incidentally, the analogy drawn in the book between a spin wave and an exciton is made very superficial.

The third and principal shortcoming of the book consists, it seems to us, in the not entirely successful arrangement of the material. Thus, after the very successful first chapter, devoted to the empirical classification of solids and a summary of a number of experimental facts, there follows a chapter entitled “Classical Theory of Ionic Crystals,” in which, however, the results of quantum-mechanical calculations concerning van der Waals forces are substantially used. In the third chapter the specific heat of simple solids is examined, and electronic specific heat is also discussed; the latter, however, is calculated only in the following chapter, devoted to the theory of free electrons (Sommerfeld’s model). And only after all this—in Chapter 5—are set out the elements of quantum mechanics necessary for understanding the preceding material (including the Pauli principle)! It should also be noted that, in contrast to the masterfully written first four chapters, the fifth chapter gives the impression of a collection of fragmentary items of information from quantum mechanics, from which it is still impossible to form any coherent idea of it. The reader not familiar with quantum mechanics will understand neither the second nor the fourth chapter, while the fifth chapter will help him little in this respect; for the prepared

...the reader does not need it at all. Therefore it seems to us altogether superfluous.

The next two chapters are devoted to an exposition of several approximate methods for solving the many-body problem and to their application to molecular theory. Here the variational method and approximation originating from one-electron functions are considered (on its basis, in the end, the Fock equations are obtained); the results of solutions of the Fock equations for various atoms are reviewed. The quality of the exposition here is very high; rather complex mathematical questions are analyzed very clearly. It should be emphasized, however, that the author not only insufficiently stresses the necessity (and not merely the “possibility”) of representing the complete wave function in the form of a determinant (in the approximation based on one-electron functions). These two chapters serve as, in effect, an introduction to the following, eighth, chapter, where the foundations of band theory are finally set out. The “nearly free” electron method is presented excellently; this cannot be said of the method of “nearly bound” electrons: it should have been pointed out that the procedure usually employed in it is nothing other than the solution of the problem by the variational method, and that the choice of atomic functions as the basis system is justified by the assumption of sufficient separation of the atoms. It should also be noted that there is a significant shortcoming in § 66, devoted to the study of complete wave functions for a system of electrons in a solid: the main defects of the Bloch and Heitler–London methods are not indicated, namely that in the first of them the “polar” states, corresponding to an anomalously large accumulation of electrons at one atom, are assigned too great a weight, whereas in the second method these states are excluded altogether (which also does not permit this method to be used for considering questions of the electrical conductivity of metals).

The ninth chapter is devoted to an exposition of one approximate method for solving the one-electron problem in a crystal—the so-called Wigner–Seitz “cell method”; exchange corrections to it are also considered for the cases of completely free and strongly bound electrons. The exposition is good, but, taking into account what has already been said above about the purely qualitative character of band theory, attempts at any exact calculations within its framework (and the cell method itself was invented for this purpose) seem to us rather pointless.

In the next chapter the theory of cohesive forces in metals, ionic and molecular crystals is considered. The main part of the chapter is devoted precisely to metals and is based chiefly on the cell method; the results of calculations for lithium and sodium are fairly well brought into agreement with experiment; however, as has already been said above, there is no reason to attach too much importance to this circumstance.

There follow two short but well-written chapters on the theory of work function and on excited states of the electron system in a solid. Especially good is this last chapter, which contains an exposition of Frenkel’s exciton theory.

The next, thirteenth, chapter is, in the author’s words, the central one in the book: it is devoted to consideration of the electronic structure of all five types of solids indicated in the first chapter. Here (chiefly qualitatively) the character of the bands in metals, solid solutions, ionic and valence crystals, and semiconductors is considered. As in the whole book, no particular objections can be raised to the quality of the exposition in the sense of its clarity and distinctness; it seems to us, however, that for metals the consideration of all the properties of the crystal as a whole on the basis of the band model is in general not very fruitful. In this chapter the “sterile” character of many of the foundations of band theory stands out especially vividly. Consider, for example, such a phrase: “From the fact that silver

considerably more weakly colored in comparison with copper, we may conclude that the difference between the $d$- and $s$-bands is greater for this metal. Apparently, this difference again decreases for gold, since gold too is colored” (p. 450). A particularly unpleasant impression is produced by the section devoted to transition metals, whose properties, following Mott’s example, are “interpreted” on the basis of ideas about the electron $d$-band (although Zeitz himself recognizes that band theory is inapplicable in this case).

As for ionic crystals and semiconductors, where the band approximation can indeed be justified (though not in the Bloch sense), the corresponding sections of the book under review are so hopelessly obsolete that it is better not to mention them at all.

The next chapter considers questions connected with the motion of nuclei in the lattice. First the adiabatic approximation is introduced (a short but very successful paragraph), then a qualitative theory of phase transitions and an elementary theory of ordering processes in alloys are given, and the problem of the free rotation of molecules in crystals is also considered. In general, this chapter, in its content, stands somewhat aside from the main direction of the book, is rather superficial in character, and can serve only as an introduction to the study of the questions touched upon in it.

Chapter 15 is devoted to the theory of electrical conductivity. It considers the electrical conductivity of metals (within the framework of the photon-electron scheme), ionic conductivity, and photoconductivity. As for the first question, here, if one does not count the rather vacuous paragraph on superconductivity, no special claims can be made against the author, apart from those already noted above (not a word is said about the work of L. D. Landau and I. Ya. Pomeranchuk, who clarified the role of electron interactions in the resistance of metals at low temperatures; for the “explanation” of anomalies in the electrical conductivity of transition metals an ad hoc $d$-band scheme is used). The quality of the exposition is quite satisfactory; it is especially pleasant to note the comparatively moderate use of a complex mathematical apparatus. The section on ionic conductivity is also good. On the other hand, the exposition of the question of photoconductivity, like almost everything in this book relating to nonmetallic crystals, is very obsolete.

In Chapter 16 the magnetic properties of solids are considered. Writing the Hamiltonian in the presence of a magnetic field and deriving the general thermodynamic formula for magnetic susceptibility, the author considers the orbital diamagnetism of free and nearly bound electrons, as well as spin paramagnetism. From the methodological point of view the exposition raises no objections; however, the author’s attempts to obtain some quantitative results by “fitting constants in a series” cannot but cause bewilderment. We shall also note the extremely obsolete exposition of the question of the magnetic susceptibility of nonferromagnetic metals at low temperatures; this phenomenon, as is known, has recently been thoroughly investigated by Soviet scientists.

Next the problem of the effective field in a magnet is considered, after which the author passes to the theory of ferromagnetism. Weiss’s phenomenological scheme and Heisenberg’s theory are presented well; the theory of spin waves, however, as was already said above, is presented very poorly.

The last, Chapter 17, is devoted to the optical properties of solids. Its first two paragraphs, concluding the classical theory of absorption and dispersion, do not differ in their content from an ordinary university physics course; next a quantum formulation of optical properties is given, which is applied to metals and ionic crystals. Then the problem of determining the number of $F$-centers in alkali-halide crystals from the form of impurity absorption bands is considered.

Next comes a paragraph devoted to the infrared spectrum of ionic crystals; unfortunately, this question is treated here far from completely—for example, there is not a word about the scattering of light by a solid. The chapter ends with a paragraph entitled “Special Questions,” in which certain phenomena, sometimes very important ones, not included in the main text of the book are mentioned extremely briefly: the photoelectric effect in metals, the width of absorption and emission bands, fluorescence of crystals, photolysis. On average, about two pages of rather general discussion are allotted to each of these sections, and it therefore makes no sense to discuss this paragraph in greater detail. As a whole, the entire chapter gives the impression of having been compiled according to some accidental criterion; its comparatively small content in no way corresponds to the richness of optical phenomena in solids.

In conclusion, we note one more major shortcoming of the book: it does not discuss the theory of crystal symmetry. Yet this is absolutely necessary for understanding a number of processes in crystals, and in fact the author is repeatedly compelled to refer to it. This may create considerable inconvenience for a reader not familiar with the theory of symmetry.

To sum up, one may say that Seitz’s book is by no means a modern course in the theory of the solid state, or even in the theory of electronic processes in it. It is a well-written textbook on the band theory of metals, interesting both for the theorist and for the experimentalist, on whose bookshelf it must always stand, and quite suitable for acquaintance with the subject, if the reader is already somewhat familiar with quantum mechanics. However, the theory of nonmetallic crystals, as well as questions connected with the limits of applicability of band theory and with going beyond those limits, have not received proper treatment in the book under review.

Believing that the translation of Seitz’s book will prove useful, we must emphasize the urgent need for a Soviet book on the theory of the solid state, reflecting its present state and relying on the major achievements of Soviet science.

The translation of the book has been done well, apart from certain inaccurate expressions. Thus, on p. 440 we read: “The mean energy of this wave function is...,” whereas it should have been said: “The mean energy in the state described by this wave function is....” It should be noted that the editor has done a very large and useful piece of work, supplying the book with notes and compiling, together with Prof. V. I. Iveronova, a bibliography of the postwar Soviet works on the physics of the solid state (more than 500 titles). The book’s external production is quite satisfactory, but the misprints that occur rather often are unpleasantly striking (for example, in formula numbers).

V. Bonch-Bruevich