R. Peierls, Quantum Theory of Solids. Translated from the English by A. A. Abrikosov, Foreign Literature Publishing House, 259 pp., Moscow, 1956, price 12 rubles 85 kopecks.

conductivity, and these are precisely the questions which the author developed in his time. The question of the thermal conductivity of dielectrics, i.e. of the transfer of energy in a dielectric by thermal vibrations, the author begins by analyzing qualitatively, considering anharmonicity as the cause of phonon scattering. From this qualitative consideration there follows directly the necessity of umklapp processes, i.e. such processes in which, as a result of the interaction of phonons, the total quasi-momentum of the latter is not conserved. At the end of the chapter the author, relying on Boltzmann’s kinetic equation, analyzes the temperature dependence of thermal conductivity at high temperatures and the influence of impurities and of crystal size on thermal conductivity. The author believes that, on the whole, in these questions there is agreement with experiment. In this connection it is perhaps worth noting that, as was shown in recent years by the investigations of Academician A. F. Ioffe and his collaborators, the thermal conductivity in nonmetallic crystals, especially in semiconductors, is in reality of a much more complex character than is provided for by Peierls’ theory.

Chapter 3. Interaction of light with nonconducting crystals. In this chapter infrared absorption, diffraction of X-rays, the thermal factor, coherent and combined scattering of light are considered—optical phenomena not associated with electronic transitions. At the end of the chapter, relying on a mathematical analogy, the scattering of neutrons by a crystal lattice is considered. The exposition in this chapter is based on an expression for the energy of interaction of a crystal with the electromagnetic field, which is assumed to be known and which is then transformed in the proper way in the various cases. Thanks to this, all the phenomena are considered from a single point of view, and the author can touch upon many subtle questions, such as line width, background intensity, etc.

Chapter 4. Electrons in an ideal lattice. In this chapter the author already passes to the study of the properties of metals. Assuming the ions to be fixed, the author studies in detail the motion of electrons in a periodic field. After presenting Bloch’s theorem and clarifying the basic features of the energy spectrum of electrons in a periodic field, the approximations of strongly bound and almost free electrons are considered. Then accelerations caused by electric and magnetic fields are studied; here the formula for the acceleration of an electron by an electric field is derived from the consideration of wave packets, while the formula for acceleration by a magnetic field is given without derivation and is written by analogy. After presenting the fundamentals of the statistics of electrons in metals, heat capacity is considered, a general formula is derived, the role of the density of the statistical weight in heat capacity is shown, and an estimate of this quantity is made in various cases. At the end of the chapter surface problems associated with thermoelectronic and cold emission of electrons are considered.

Chapter 5. Cohesive forces in metals. General ideas on the nature of the metallic bond were briefly formulated already in the first chapter. Here these ideas are developed.

Chapter 6. Transport phenomena. This is perhaps one of the most interesting chapters of the book. First the author proceeds from the assumption that there exists a certain mean free time depending only on the energy, and writes Boltzmann’s equation in the form of a stationarity condition both in the presence of only an electric field and under the joint action of an electric field and a temperature gradient. Next, a generalization is made to the case when the probability of scattering depends on the angle of scattering. This allows the author immediately to obtain an expression for the electric and heat currents, find the coefficient of thermal conductivity and conductivity, and also establish the Wiedemann–Franz law. The author then turns to consideration of the mechanism of electron scattering, and first static obstacles (impurities and lattice imperfections) are considered. It is shown that, generally speaking, as a result of scattering of electrons by static obstacles, phase relations arise between different stationary states. In order to obtain a kinetic equation containing the usual collision term, it is necessary to assume that such phase relations do not exist. Therefore the author introduces, as a first approximation, the assumption that the nature of the scattering potential is such that such phase relations do not exist. The author shows that in the case of static obstacles this will occur if the latter are distributed chaotically over the lattice, which corresponds to the usual idea of the form of the “collision integral” in the kinetic theory of gases. This alone, however, is still not enough. In deriving the kinetic equation it proves necessary to make the assumption that the free path time satisfies the condition \(\tau > \frac{\hbar}{kT}\); if this condition is not fulfilled, then the kinetic equation has no meaning. Thus the author clarifies the limits of applicability of the kinetic theory.

The book under review is a reworking of a course of lectures delivered by the author in 1953 at the summer school of theoretical physics in Les Houches (Grenoble). The book sets forth the contemporary state of the quantum theory of solids, but, unlike many other books devoted to the same subject, the present book is distinguished above all by the fact that it emphasizes the fundamental features of the theory.

In view of the fact that solid-state physics has acquired immediate importance in modern technology, the quantum theory of solids is attracting serious attention not only from theoretical and experimental physicists, but also from broad circles of chemists, engineers, and radio technicians, who in their work necessarily come into contact with various problems of modern solid-state physics. A book devoted to a comprehensive physical analysis of the basic principles of the quantum theory of solids, to the elucidation of the difficulties encountered, and to the analysis of unsolved problems is, of course, of special interest and is, in its way, unique. In order to evaluate this book properly, it must be taken into account that 25–30 years ago, during the years in which the modern quantum theory of solids was being created, the author took the most active part in its development, and many sections of this science are associated with his name. The principal questions that were then developed by Prof. Peierls are the theory of thermal conductivity of dielectrics, the theory of the optical properties of dielectrics and metals, the theory of the electrical conductivity of metals, the theory of galvanomagnetic phenomena in metals, and the theory of the magnetic properties of metals. The author was the first to give a correct explanation of the anomalous sign of the Hall effect in metals and introduced the important concept of scattering processes. After that Prof. Peierls devoted many years to the development of other sections of theoretical physics, and now, twenty years later, in connection with the above-mentioned course of lectures, he has again returned to questions of the quantum theory of solids and has given a new exposition of the basic questions of the theory, taking into account all the achievements of these years and the discussions that have taken place. Naturally, the author has devoted his main attention to those questions whose development was the subject of his own works. Some questions that are being developed in the quantum theory of solids at the present time are touched upon only in passing (for example, the theory of semiconductors), while others are left entirely untouched (for example, cyclotron and spin resonance). But this is rather a merit of the book than a defect, because thanks to this the author was able to concentrate attention on the analysis of the principles of the theory. The style of exposition is also attractive. The author uses a minimum of mathematical apparatus, emphasizes at every stage of the discussion the physical side of the phenomena, and critically analyzes one or another assumption of the models under consideration for physical phenomena in solids. At the same time it cannot be said that the book is written quite easily. It very often contains many extremely subtle arguments which, it is true, eliminate the need for tiresome calculations, but which, in order to be fully understood, require from the reader a sufficiently exacting ability to find his own way in the questions under consideration. In return the reader subsequently receives a far deeper illumination of many things already known to him.

The range of problems considered in the book is very broad. It examines questions of the physics of nonmetallic crystals, including the very interestingly formulated question of the stability of lattices, and the types of bonds in solids. The phenomena of transport in metals, the magnetic properties of metals, and the optical properties of metals, dielectrics, and semiconductors are analyzed in great detail. In conclusion an outline is given of the present state of the theory of superconductivity. Let us now dwell briefly on the contents of the individual chapters.

Chapter 1. Crystalline lattices. General theory. This chapter sets forth the foundations of the general theory of solids, i.e., of crystalline lattices (amorphous bodies are excluded from consideration). After a brief survey of the simplest structures the author turns to the dynamical problem. He analyzes in detail the adiabatic approximation and indicates the limits of its applicability. Speaking of the types of bonds in solids, the author considers briefly electrostatic forces, van der Waals forces, homopolar bonding, overlap, and metallic bonding. Metallic bonding is examined in greater detail in Chapter 5, but already here the author explains very clearly how the motion of electrons in metals leads to the specific features of metallic bonding, as a result of which stable lattices of metals sometimes prove to be rather complicated. The remaining paragraphs of this chapter are devoted to the classical and quantum theory of lattice vibrations. Despite the small size of this section (these paragraphs occupy only 16 pages), an exhaustive exposition of the question is given.

Chapter 2. Crystalline lattices. Applications of the theory. This chapter gives the simplest applications of the previously developed general theory of lattice vibrations. The heat capacity, the influence of the anharmonic term on thermal expansion and heat capacity, the Boltzmann equation, and thermal conductivity are considered. Especially interestingly illuminated in this chapter are the last two questions—the Boltzmann equation and thermal con-

The author then turns to the consideration of the scattering of electrons by lattice vibrations. Here, too, the necessity is clarified of introducing the two assumptions mentioned above. Analyzing the probabilities of transitions caused by lattice vibrations, the author shows that processes with transfer, i.e., processes in which the total quasimomentum of the electrons and phonons is not conserved, also have a place here. This proves especially important at low temperatures. It turns out that at low temperatures collisions without transfer cannot lead to a finite resistance of the metal. Taking transfer processes into account leads to a finite resistance, but the temperature dependence of the resistance thereby obtained differs sharply from that observed experimentally. It is interesting to note that the experimentally observed temperature dependence of the resistance at low temperatures was obtained by Bloch, in whose theory, instead of vibrations of the crystal lattice, vibrations in a continuous isotropic medium were considered. In this case transfer processes are absent and the difficulties mentioned do not arise. Then Prof. Peierls also considers in detail the influence of collisions between electrons on the resistance, which is usually neglected in kinetic theory. It turns out, however, that along this path the problem does not obtain a solution and thus the question of the temperature dependence of the electrical conductivity of metals at low temperatures remains from the theoretical point of view unclear. We emphasize that at high temperatures there is no such difficulty, since in this case taking transfer processes into account does not lead to an anomalous temperature dependence of the electrical conductivity.

In conclusion to the chapter, the limits of applicability of the assumptions made are analyzed. In particular, it is shown that the condition \(\tau > \dfrac{\hbar}{kT}\) is fulfilled very poorly for metals. However, the author does not consider it correct on this basis to draw the conclusion that the entire theory of transport phenomena rests on a very shaky foundation. The author refers to considerations by Academician Landau, which lead to the conclusion that the mentioned restriction must be replaced by another, namely \(\tau > \dfrac{\hbar}{\mu}\), where \(\mu\) is the chemical potential of the electron gas. Since in metals \(\mu \gg kT\), the latter condition proves to be quite satisfied. Nevertheless, the considerations leading to the latter criterion are to a considerable extent qualitative, and the question as a whole remains insufficiently clear. The impression is created that, taking into account another difficulty with transfer processes, the generally accepted methods of kinetic theory are inapplicable to metals (that they are inapplicable to semiconductors raises no doubts). However, this is the reviewer’s opinion. The author does not draw such a conclusion.

Chapter 7. Magnetic properties of metals. This chapter considers both questions of the equilibrium of electrons in a magnetic field (paramagnetism, diamagnetism) and questions of kinetics (Hall effect, resistance in a magnetic field). In considering the diamagnetism of free electrons, the author does not restrict himself to calculating only the stationary susceptibility, but, elegantly using Poisson’s summation formula, also obtains oscillations of the susceptibility (the van Alphen–de Haas effect).

Chapter 8. Ferromagnetism. In this chapter the results of the modern theory of ferromagnetism are briefly considered. After a short discussion of the Weiss model, the theory of spin waves is presented in an elementary and clear manner—first in the one-dimensional case, then in the three-dimensional case. The difficulties of the theory of spin waves are noted, as is the contradiction between the predictions of the spin-wave theory for heat capacity of ferromagnets and the results of experiment. Therefore the author also considers ferromagnetism from the standpoint of the model of collectivized electrons, investigated by Bloch, Stoner, and others. The author then briefly considers the scattering of neutrons in ferromagnets and shows what information about the structure of domains the study of neutron scattering can provide. In conclusion, the questions of magnetization curves and antiferromagnetism are briefly considered. The importance of taking into account the so-called indirect exchange for a correct explanation of antiferromagnetism is clarified.

Chapter 9. Interaction of light with electrons in solid bodies. In this chapter the author again returns to optical phenomena in solids, but unlike Chapter 3, processes connected with optical transitions of electrons are studied here. At first the classical theory is presented, valid at not too high frequencies. Then transitions between energy bands are considered, and in connection with this Skinner’s work on the emission spectra of soft X-rays is discussed, which makes it possible to obtain a picture of the distribution of energy levels in a band. The photoelectric effect is discussed qualitatively; the question of volume and surface effects is analyzed. A paragraph on the interaction of light with nonconducting crystals is written very interestingly, where, in particular, a picture is drawn of the exciton mechanism of light absorption.

Chapter 10. Semiconductors and luminescence. The chapter on semiconductors considers the following questions: what semiconductors are, the calculation of the number of charge carriers, electrical conductivity and the Hall effect in semiconductors, the spatial charge and rectifying contacts, the behavior of nonequilibrium electrons, and the mechanism of luminescence. Although all these questions are presented in Prof. Peierls’s characteristically exceptionally clear form, here, nevertheless, there is lacking that profound analysis of the physical premises and difficulties of the theory with which the chapters devoted to metals were replete.

The last chapter, Chapter 11, is devoted to superconductivity. After a concise but clear survey of the principal experimental facts, the foundations of the Fröhlich—Bardeen theory are set forth, and the action of a magnetic field is analyzed. In conclusion the author gives a critique of the existing theory of superconductivity, notes the difficulties it contains, but at the same time considers that the Fröhlich—Bardeen theory possesses many encouraging features.

We think that the above survey of the book’s contents gives a clear idea of the richness of its content and of the high level of discussion of the questions touched upon in it. Undoubtedly, this book will be studied with interest by everyone interested in the modern quantum theory of the solid state. Anyone who studies this book will, without doubt, read it with profit.

A. Samoilovich