On the Thirtieth Anniversary of Soviet Physics

Theoretical Physics in the USSR over 30 Years

Ya. I. Frenkel

1. Introduction

In former times, the theoretical problems of physics, both of a fundamental and of a mathematical character, were studied, on the one hand, by physicists who had not yet divided themselves into experimentalists and theorists, and on the other, by mathematicians. The multiplication of these problems and their increasing difficulty, both in the mathematical sense and, especially, with respect to their complexity, i.e. their connection with a large number of diverse physical phenomena, led in the end to the emergence of a special discipline intermediate in character between physics as an experimental science and mathematics—namely, theoretical physics. This was aided by the appearance, in the first decade of our century, of such revolutionary and all-embracing theories as the electron theory, the theory of relativity, the theory of light quanta, and the quantum theory of atoms.

The further development of these theories attracted abroad the attention of many young physicists, who gave up any experimental work and devoted themselves entirely to the development of the new theories and to their application to the multitude of various concrete physical phenomena which these theories made it possible to view in an entirely new light.

Representatives of this new specialty perform no experiments and, as a rule, do not know how to perform them (in this respect being the object of good-natured mockery on the part of experimentalists). Their task consists in “bringing order” into the extensive experimental material accumulated by other investigators, in finding the fundamental bases of physical phenomena, revealing their internal mechanism, giving a general mathematical formulation of the laws governing them, and, by means of mathematical analysis, calculating or predicting the course of these phenomena under one or another set of concrete conditions.

It must be emphasized that the solution of all these theoretical problems of physics did not become the monopoly of theoretical physicists. Every major experimental physicist is, in essence, also a theoretician who poses and solves theoretical questions. In exactly the same way, specialists in mathematics still take part in solving those problems of theoretical physics which are reduced mainly to complex mathematical calculations.

In those cases, however, when a question of interest to an experimental physicist has too complex a character, i.e. is connected with phenomena not directly included in the field being studied by him, its solution is usually left to the theoretical physicist. The advantage of the theoretical physicist over the experimentalist consists precisely in the fact that the theoretician, as a rule, is well oriented within a much broader range of physical phenomena belonging to various fields of experimental physics.

The breadth of physical education often also determines the advantage of the theoretical physicist over the mathematician in solving complex physical problems of a mathematical character: not understanding properly the essence of the phenomena under consideration, the mathematician is usually inclined to seek an exact solution where one may confine oneself to a comparatively simple approximate solution obtained as a result of a reasonable, i.e. not essentially altering the matter, simplification of the problem.

Thus, if the negative characteristic of the theoretical physicist is the inability to set up physical experiments, then the positive characteristic is broad encyclopedic knowledge in questions of physics, combined with sufficient mathematical equipment.

Depending on the relation between these two factors, the theoretical physicist may, in his profile, approach either the experimental physicist or the mathematician. The theoreticians of the first, “experimental,” type deal chiefly with various concrete physical or physico-chemical questions that concern experimentalists, with whom they are in close contact. Theoreticians of the second—mathematical—type usually deal with abstract questions, or else with the solution of special mathematical problems connected with phenomena that are quite clear in principle but require an exact quantitative description. This latter field is usually singled out as a special discipline called “mathematical physics.” Its representatives are more mathematicians than physicists; their assignment to one category or the other is of a purely conventional character and depends on what questions, apart from those mentioned above, they deal with—purely mathematical or physical ones.

Having thus established the place of theoretical physics and of its representatives in the general system of physical-mathematical disciplines,

we can return to the subject of the present article—the development of theoretical physics in our country over the last 30 years.

In prerevolutionary Russia, theoretical physics as an independent discipline was poorly represented, which is explained by the very weak development of physics and technology in the prerevolutionary period. The Great October Socialist Revolution, which set the task of transforming an economically and culturally backward country into an advanced industrial state, naturally brought about the rapid development of physics as the foundation of technology and thereby created the most important prerequisite for the emergence of Soviet theoretical physics.

An outstanding role in the creation of theoretical physics in Russia was played at the beginning of the second decade of the twentieth century by P. S. Ehrenfest, who was then working in Petersburg (later—the successor of H. A. Lorentz in the chair of theoretical physics at Leiden).

On the soil of old tsarist Russia, the shoots of the new discipline would probably soon have withered. On the renewed soil of Soviet Russia they took root and began to grow, at first slowly, owing to the extreme smallness of the first contingent of Soviet theoretical physicists, and then, as this contingent grew and as experimental and technical physics developed in our country, ever faster and faster. In what follows we shall dwell on the most important directions of the work of Soviet theoretical physicists.

2. THEORY OF RELATIVITY AND QUANTUM MECHANICS

The first substantial work on Einstein’s theory of relativity and gravitation was done in our country in 1923 by the late A. A. Friedmann, who showed that the cosmological conclusions from this theory are not limited to Einstein’s or Schwarzschild’s closed static world, but have a much more general character, compatible with a monotonic (or periodic) change in the radius of curvature of world space with the passage of time.

Friedmann’s work was of great significance not because it advanced a new cosmological theory, more correct than the preceding ones, but because it undermined the fundamental foundations of the earlier static theories, which had previously been regarded almost as a logical consequence of the general theory of relativity.

Einstein himself at first tried to dispute Friedmann’s conclusions, but in the end was forced to agree with them.

Friedmann’s work may be regarded as a kind of discrediting of cosmological theories based on the general theory of relativity and gravitation, i.e. as proof of the impossibility of deriving from it in an unambiguous way the correct cosmological theory, which from the philosophical point of view represents a very important result.

Another very major work on Einstein’s theory of relativity and gravitation is the work of V. A. Fock, published by him in 1939 and devoted to an approximate solution of the problem of \(n\) bodies (material points). Fock showed that not only the Newtonian law of mutual attraction between these bodies, but also the Newtonian law of motion of each of them under the influence of the others, can be derived from the general equations of the gravitational field, without resorting to any additional principles. It should be recalled that in the original formulation of Einstein’s theory the motion of any body in a given external field was defined as motion along a geodesic line, i.e. along the straightest (or shortest) line in a curved four-dimensional world; this definition therefore represented a generalization of the principle of inertia. Thus Fock succeeded in showing that in the general theory of relativity the principle of inertia has no independent significance and can be entirely reduced to the laws determining the gravitational field as a function of the position and motion of the material bodies that create it—and are moved by it. This result recalls the basic result of the Lorentzian dynamics of an electromagnetic field containing electrons, which consists in the fact that the inertial force of each electron is treated as the resultant of the electromagnetic forces with which the elements of its electric charge act upon one another; here the equation of motion of an electron in a given external field reduces to the equality to zero of the total force which it experiences (i.e. the sum of the external force and the force of electromagnetic “self-action”), which is equivalent to the principle of conservation of the energy and momentum of the field.

Fock did not confine himself to the first, crudest approximation in solving the equations of the \(n\)-body problem, but also considered corrections to the Newtonian laws of interaction and motion that follow in the second approximation from Einstein’s theory of relativity.

Somewhat later, results completely analogous to Fock’s, and independently of him, were obtained by Einstein himself in collaboration with Hoffmann and Infeld.

A substantial contribution to the development of Einstein’s theory is also the work of the late G. A. Mandel (1930), who showed that by adding to the four dimensions of the space-time world one more, special, fifth dimension, one can achieve a formal unification of the gravitational and electromagnetic fields*). It must be noted that such a unification of gravitational and electromagnetic effects into a “unified field theory” was put forward by Einstein as one of the

*) Somewhat later than Mandel, an analogous theory was developed abroad by Kaluza.

of the most important items in the program for the further development of the theory of relativity.

Much more varied are the works of Soviet theorists in the field of the second theory, under whose banner the development of modern atomic physics took place—namely, the theory of quanta and the quantum theory of atomic phenomena.

These works relate chiefly to the further development of that renewed and deepened form of quantum theory which it received in 1925–1928 after the works of Heisenberg, de Broglie, Schrödinger, and Dirac.

Here one should mention first of all a number of works connected with the further generalization of quantum mechanics in the direction of the general theory of relativity, with the inclusion in Dirac’s equations of terms taking account of gravitational forces (Ivanenko’s work on matrix geometry, subsequently developed by other investigators; the work of Ivanenko and Fock on the “parallel transport” of spinors and, in particular, of the wave function of the Dirac equation, in the sense of the general theory of relativity, 1929).

To the same series of works on the relativistic formulation of the laws of quantum mechanics (within the limits, however, of the special theory of relativity) belong Fock’s works (partly together with Dirac and Podolsky) on the relativistic treatment of the many-electron problem in connection with the application of the principles of quantum mechanics to the electromagnetic field. In this connection it proves possible to derive both the laws of change of this field and the laws of motion of the electrons that create it from a single equation, treating the field and the electrons as two interacting parts of one whole.

As Dirac first showed, the number of electrons interacting with the quantized electromagnetic field may change with the simultaneous appearance or disappearance of positrons. V. A. Fock introduced, for the description of these phenomena, an elegant and convenient mathematical formalism using the concept of functionals—a formalism that may also be applied to other analogous phenomena.

3. QUANTUM THEORY OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI

The most substantial and numerous contributions of Soviet theoretical physicists to the development of quantum theory have been made chiefly in the field of its applications to various concrete problems concerning the properties of elementary particles and of the material bodies formed by them.

The problem of elementary particles is posed in quantum mechanics quite differently than in classical mechanics. First of all, the properties of these particles prove to be hidden in the formalism of the equations describing them much more deeply than in classical theory. Thus, for—

example, the writer of these lines, proceeding from his former relativistic, but not quantum, theory of the rotating electron, proposed in 1928 to describe the waves corresponding to electrons by a system of equations that is a direct generalization of Maxwell’s equations for light waves.

Almost simultaneously Dirac solved the question of electron waves by his famous equation, which contained quantities of a new spinor (“semivector”) type. And in 1937 the French physicist Proca showed that the generalized Maxwell equations describe not electrons, but newly discovered particles—mesons—which have the same charge as electrons, but a greater mass and, most importantly, an integral spin (and therefore obey Bose statistics, not Fermi statistics).

Dirac showed in 1932 that states with negative kinetic energy or mass, in which electrons described by the equation he found can be found, can be interpreted if one assumes that almost all these states are occupied by electrons and are detected, when vacant, in the form of positrons, discovered experimentally by Anderson.

I. E. Tamm, on the basis of Dirac’s theory, was the first to carry out a number of interesting calculations concerning the phenomenon of annihilation, i.e. the mutual destruction of electron–positron pairs with the formation of photons, and also the emergence and disappearance of similar pairs at an intermediate stage in the scattering of light by free electrons. This work contained, in embryonic form, various nonlinear effects connected with Dirac’s theory and the electrodynamics based on it. These include the emergence of electron–positron pairs in collisions of charged particles (Landau and Lifshitz), the scattering of light by positive nuclei through the emergence and disappearance of such pairs (Akhiezer and Pomeranchuk), the scattering of light by light (Akhiezer), and a number of other effects.

Closely connected with these questions are also calculations of effective cross sections for various kinds of collisions between elementary particles.

The computations belonging here, in which young Soviet theorists in particular have recently been excelling, are very complicated and require an excellent command of the mathematical apparatus of quantum mechanics. Some of them, connected with overcoming difficulties of principle (for example, the works of Ivanenko and Sokolov, as well as of Ginzburg, on the problem of eliminating infinite effective cross sections in the scattering of mesons by nuclear particles), are of not only mathematical but also fundamental interest.

Leaving aside a number of works of a purely computational character, we must further note the work, continued by a number of Soviet “quantum theorists,” on the theory of the interaction of elementary particles. A new path in this direction was laid out by Dirac, who

showed that the Coulomb interaction between electrons can be formally described (in calculations by the methods of perturbation theory) as the result of the “throwing” of photons (a photon emitted by one particle and absorbed by another, as it were, produces the interaction between them). I. E. Tamm, and less thoroughly also D. D. Ivanenko (1935), showed that, using this method and replacing photons by electron-neutrino pairs, which, according to Fermi’s theory, are emitted in the $\beta$-decay of radioactive nuclei, one can in principle describe the interaction between nuclear particles. True, the results obtained proved not very satisfactory. Later, however, they were improved by the Japanese theoretician Yukawa, who combined the electron-neutrino pair into a single heavier particle, whose mass he was able to determine approximately from the radius of action of nuclear forces. This hypothetical particle, coinciding in all its characteristics with the meson found later, is capable of spontaneously decaying into an electron and a neutrino. This process is observed in cosmic rays. In the theory of cosmic rays, Soviet quantum theorists also have a number of interesting works to their credit, especially on the theory of showers, where I. E. Tamm and L. D. Landau, as well as D. D. Ivanenko and A. A. Sokolov (1938), and in more recent times (1946) Tamm and Belenky, succeeded in obtaining mathematically exact solutions.

Returning to questions of nuclear physics investigated or solved by Soviet theorists, we must note the idea expressed by D. D. Ivanenko in 1934 that complex atomic nuclei do not contain electrons, but consist of protons and neutrons. Even earlier—in 1932—Ivanenko put forward the thought that the electrons appearing in $\beta$-transformations do not fly out of nuclei in ready-made form, but arise when the charge of the latter changes, just as photons arise when atoms pass from one state to another with lower energy.

These ideas, developed somewhat later by Heisenberg, constitute the basis of the modern theory of the structure of atomic nuclei.

Of substantial importance for describing the processes of excitation and decay of complex nuclei are the concepts of the temperature of a nucleus, of its heating upon absorption of a proton, neutron, or some other particle, and of subsequent evaporation, expressed by the author in 1936, after the publication of Bohr’s work on the connection between particles in a complex nucleus. This statistical theory also received an interesting development in Landau’s works on the density of levels in the nucleus, which he estimated by applying Fermi statistics to the latter.

Immediately after the publication of communications on the fission of uranium nuclei under the influence of slow neutrons, I (in 1939) developed a theory of the electrocapillary fission of these nuclei, based on Bohr’s idea of complex nuclei as droplets

of a special kind of liquid, and the theory, analogous to Rayleigh’s theory, of the fragmentation of electrically charged drops of a conducting liquid. This theory makes it possible to explain the circumstance that a periodic system of elements breaks off at uranium; moreover, for this theory of electrocapillary fission it was necessary to supplement it with a theory determining, on the basis of Fermi statistics, the normal relation between the charge and the mass of a nucleus, i.e., between the number of protons and neutrons forming it, which ensures its stability with respect to $\beta$-transformations. Similar considerations, in a much more developed form, were advanced somewhat later by Bohr and Wheeler.

The fission of the uranium nucleus or, more precisely, of its light isotope (“actinouranium”), under the influence of the capture of a slow neutron is accompanied, as is known, by the appearance of 2–3 new, faster neutrons. Thus, it turns out that a necessary prerequisite is provided for the development of a chain reaction involving the explosion of a large mass of actinouranium.

In ordinary uranium such an explosion cannot occur, since the atoms of the principal heavy isotope, while capturing neutrons, do not split, but undergo successively two $\beta$-transformations, leading to the formation of plutonium. They capture especially “greedily” neutrons with an energy of about $25\,\mathrm{eV}$. In a very interesting paper, published in 1940, Ya. B. Zeldovich and Yu. B. Khariton showed that a relatively small enrichment of ordinary uranium with its light isotope is sufficient to ensure the possibility of a chain reaction leading to an avalanche-like increase in the number of neutrons.

Among other works by Soviet physicists on the theory of nuclear processes, mention should be made of the work of V. B. Berestetskii and A. B. Migdal on the kinetics of fission of the uranium nucleus, in connection with a critique of the calculations of Bohr and Wheeler, as well as the work of the author, who attempted to explain the asymmetry characteristic of this fission (the inequality of the masses of the two fragments) on the basis of the idea of the tunnel effect.

Among the theoretical achievements of Soviet physicists may also be included the discovery by Rusinov and Yuzefo­vich of the phenomenon of nuclear isomerism or, more precisely, the interpretation of this phenomenon as the result of the transition of a nucleus into an excited metastable state. A number of young theorists, especially Berestetskii and Zavelevich, devoted considerable effort to calculating the probability of electronic conversion in the transition from metastable states to the normal state, i.e., the transfer of the excess energy of the nucleus to one of the nearest electrons of its surrounding shell.

The theory of nuclear processes is at present in an embryonic state. Its development requires a deepening of our knowledge of the nature of the elementary particles of matter, their mutual transformations—

of their appearance and disappearance. These metamorphoses of elementary particles are phenomena of a completely new type, excluded in classical physics, where the principle of conservation of matter is reduced to the principle of invariability of its elementary particles. It should also be mentioned the author’s new work (1945), in which it is shown that, from the point of view of the theory of relativity, complex particles must be treated as material points with internal degrees of freedom and, in principle, cannot be described as an aggregate of simpler particles.

4. QUANTUM THEORY OF METALLIC BODIES

The imperfect and unfinished form of modern quantum theory is quite sufficient as a fundamental basis for a practically irreproachable and exhaustive explanation of all physical phenomena that do not affect the integrity of atomic nuclei. In this respect, modern quantum mechanics plays the same role as Newtonian mechanics, with its law of universal gravitation and laws of motion, does in relation to the problems of celestial mechanics. Despite the entire fundamental grandeur of the reconstruction to which Newton’s theory was subjected by Einstein, when applied to questions of the motion of celestial bodies, in view of their relatively small relative velocities, Einstein’s theory gives results that practically (within the limits of the accuracy of ordinary astronomical observations) do not differ from the results following from Newton’s theory.

Since electrons in atoms and molecules have velocities far from the speed of light, their behavior can be described with accuracy quite sufficient even for spectroscopy by the nonrelativistic quantum mechanics of Heisenberg–Schrödinger, in the extreme case with the introduction of relativistic corrections, which can be borrowed from the approximate form of the Dirac equation and from the equations of quantum electrodynamics.

On the basis of this approximate and, in essence, preliminary form of the laws of quantum mechanics, it is possible, in principle, to give an exhaustive explanation of all physical, physicochemical, and chemical properties of atoms, molecules, and material bodies, and also to establish theoretically the character and regularities of all possible phenomena occurring in them (and not violating the integrity of atomic nuclei).

One should not think that the solution of all these questions is reduced to a purely mathematical problem—the integration of the equations of quantum mechanics for the given aggregate of immutable particles (nuclei, electrons).

With a large number of particles such a problem is practically completely insoluble. The task of the theoretical physicist consists in...

could replace it by a much simpler mathematical problem, admitting an actual solution and at the same time preserving a sufficient degree of similarity to the original in all essential features.

Soviet theoretical physicists have considerable merit in the study, on the basis of quantum mechanics, of objects of this kind, from the simplest atoms to crystalline bodies, covering all possible electrical, magnetic, and optical phenomena that are determined by the interactions of the electric charges of electrons and nuclei.

Here one must first of all mention the works of V. A. Fock on the calculation of the distribution of electrons in complex atoms and the structure of their spectra, based on the application of the self-consistent field method introduced by Hartree and improved by Fock with allowance for the effect of quantum exchange. One may also note several works devoted to problems of “quantum chemistry,” i.e. to the calculation of the structure and properties of the simplest molecules. However, the most substantial and numerous works of Soviet theorists on the application of quantum theory to questions of the structure and properties of matter concern not atoms and molecules, but solid bodies, in particular metals, as well as dielectrics and semiconductors.

Among works on the theory of metals I shall permit myself to note, first of all, my own investigations, as chronologically the earliest. Even before the Revolution itself—in the beginning of 1917—I gave a theory of potential jumps in the surface layer of metals, as well as of their surface tension, on the basis of the ideas of Bohr’s quantum theory of atomic structure. In 1925 these ideas were generalized by me to the motion of the so-called free electrons in metals, and it was shown for the first time that the kinetic energy of these electrons, contrary to the ideas of Drude’s classical theory, is practically independent of temperature, but is determined purely by quantum conditions connected with the fact that the character of the orbital motion of valence electrons about individual atoms is partly preserved, being supplemented by their successive transition from some atoms to neighboring ones (in which consists their “liberation,” or, more accurately, collectivization).

In this same work the electrical conductivity of metals was for the first time calculated on the indicated basis; however, this first theory of electrical conductivity, despite the fact that it led to correct results, proved to be essentially incorrect, since it was based on the use of Einstein’s relation between mobility and the diffusion coefficient, which presupposes the applicability of the laws of classical statistics.

Finally, in this work a theory was given of cohesive forces in metals, based on the attraction of the positive ions by the “electron liquid” enveloping and permeating them. The correct and logi-

... a consistent theory of metallic conductivity, based on the idea of the scattering of electron waves by thermal fluctuations of the density of metals in accordance with the principles of Schrödinger’s wave mechanics, was given by me in 1927. In 1928, after the appearance of the works of Fermi, Pauli, and Sommerfeld on the quantum statistics of an electron gas, I modernized my old theory of the double electric layer on the surface of metals, and also developed the theory of a relativistic degenerate gas and showed that its pressure can ensure the equilibrium state of matter in stars, when gravitational forces are taken into account, but only in stars of limited mass, close to the mass of the Sun.

This work found its further development in L. D. Landau, who showed that, with an increase in the mass of a star and an increase in the energy of the electrons, the latter must be captured by protons, turning them into neutrons, which ultimately may lead to the formation of a superdense neutron “core” of the star. Thus it proves possible, to a certain extent, to revive the old theory that the energy radiated by superdense stars of dwarf dimensions is drawn from the energy of gravitational forces*).

Proceeding from consideration of the quantum-mechanical effect of electron exchange in individual atoms and in metals, in 1927 I put forward the now generally accepted idea of the origin of spontaneous magnetization in certain metals. In 1930, together with Ya. G. Dorfman, I gave for the first time an explanation of the fact that ferromagnetic bodies tend to split up into “domains,” i.e. regions with different, on the average mutually compensating, directions of spontaneous magnetization (as a result of the competition between demagnetizing magnetic forces and magnetizing exchange forces).

Two fundamental works on the theory of magnetism were published by L. D. Landau.

In the first of them (1930) he showed, by applying quantum mechanics to the model of an electron gas in metals, that this gas, contrary to the conclusions of classical theory, possesses diamagnetism amounting to one third of the spin paramagnetism found by Pauli. In the second work (together with Lifshitz), Landau gave a complete theory of the splitting of a uniaxial ferromagnet into plane domains parallel to the axis of easy magnetization, taking into account the energy of anisotropy, which had been completely ignored in Dorfman’s and my theory. It should be noted as a remarkable fact, following from both theories and confirmed by experiment, that the dimensions of the domains depend on the dimensions of the whole body. Landau and Lifshitz in their work also extended the theory of magnetization to the case

*) In the case of ordinary stars, radiation is supplied by the energy released in synthetic nuclear reactions (especially the transformation of hydrogen into helium), which take place in the interiors of stars at temperatures of the order of tens of millions of degrees.

variable magnetic fields, calculating the dependence of magnetic susceptibility on the frequency of oscillations.

Of fundamental importance for the development of the doctrine of ferromagnetism and, in particular, of the technical magnetization curve was Akulov’s phenomenological theory (1930) of the phenomena of magnetic anisotropy and magnetostriction (used in part also in the just-mentioned work of Landau and Lifshitz). In the further development of the theory of the technical magnetization curve by means of the displacement of boundaries between domains and the rotation of the vector of spontaneous magnetization, taking into account the influence of internal and external elastic stresses, Soviet physicists S. V. Vonsovskii and, especially, E. I. Kondorskii took a very substantial part.

Of great fundamental importance is the polar model of metals and, in particular, of ferromagnetic metals, developed in 1934–1935 by S. P. Shubin and S. V. Vonsovskii, based on the model representation of “electron pairs” and “holes,” whose presence explains the electrical conductivity of metals and various anomalies in the behavior of ferromagnets. In this theory the transition of a part of the electrons from some atoms (which are transformed into “holes,” i.e., positive ions) to others (which are transformed into “twos,” or negative ions) was assumed to be spontaneous, i.e., not dependent on temperature and associated with a decrease in the total energy of the system.

5. THEORY OF ELECTRONIC SEMICONDUCTORS AND INSULATORS

Entirely analogous ideas were introduced by me somewhat earlier (1932) into the theory of electronic semiconductors, with the sole difference that in this case the transition under consideration, leading to the formation of mobile electrons and positive “holes,” requires an expenditure of energy and therefore occurs only under the influence of thermal motion.

The concept of mobile “holes” was first introduced by me in 1926 in connection with the study of disturbances of regularity in the structure of ionic (salt-like) crystals when the temperature is raised and of the electrical conductivity caused by these disturbances. The idea that the latter is explained by the thermal dissociation of a part of the ions, i.e., by their being torn from the sites of the crystal lattice, was expressed by Joffe in 1924. The quantitative development of the theory of the electrical properties of heteropolar dielectric crystals on the basis of this idea, supplemented by the idea of mobile ionic “holes,” was carried out by me. In view of the large mass of ions, quantum effects in this case play practically no role, whereas in the case of electronic semiconductors their role is very substantial.

Returning to electronic semiconductors, it is necessary first of all to note a number of important works by Soviet theoretical physicists

on questions connected with the passage of an electric current—electronic or hole—through the contact between two semiconductors or between a semiconductor and a metal. This includes, first of all, the works of Blokhintsev, Davydov, and Pekar (1937) on contact resistance and on the effect of current rectification, as well as the works of Pekar and the author on the increase in the conductivity of semiconductors in strong fields (the Poole effect).

At the basis of the theory of Davydov and Pekar lies the notion that the change in contact resistance (and, in particular, an asymmetric change leading to rectification) depends on a change in the concentration of electrons or holes toward an increase or decrease under the influence of an external electric field.

Interesting investigations were also carried out on questions connected with the internal properties of semiconductors, the dependence of their conductivity on the concentration of impurities (Frenkel, 1940), on the theory of various galvanomagnetic effects and, in particular, on the theory of the peculiar effect discovered by Kikoin and Noskov, consisting in the appearance of an electromotive force under one-sided illumination of a semiconductor plate placed in a magnetic field parallel to its plane. A qualitative explanation of this phenomenon was given by me, and the quantitative theory was developed by Landau.

Landau, together with Kompaneets, gave a statistical or, more precisely, kinetic theory of the non-Maxwellian distribution of velocities among electrons in a semiconductor under the influence of a strong electric field, which brings about an increase in the mean energy of the electrons.

A completely analogous theory for electrons in the plasma of a gas discharge was developed by the Dutch physicist Druyvesteyn and then worked out in a more perfected form by Davydov. The latter proposed a very interesting theory of the electrical breakdown of gases, based on the idea of stepwise ionization of atoms, i.e. ionization of atoms previously excited (at least twice) by electron impact. In view of the fact that the breakdown of solid crystalline dielectrics is externally very similar to the breakdown of gases, it is necessary to admit, if Davydov’s ideas are correct, that in dielectric crystals as well atoms can not only be ionized, but also pass into an excited state without the corresponding electron being torn away from them.

Investigating this question in 1931, I showed that the state of excitation can in this case move from one atom to another, like an electron or a hole, i.e. travel throughout the entire crystal, like an elementary particle, in accordance with the general laws of quantum mechanics. These pseudoparticles were called by me excitons. At the same time it was shown that the absorption of light in crystals reduces to the formation of excitons, and the “selection rules” were found,

determining the spectra of light absorption in dielectric crystals at low temperatures, and elucidated the mechanism by which excitons are transformed into phonons, i.e. into the energy of thermal motion. In a later work (1936) I further showed the inapplicability of Wilson’s scheme of electronic levels in semiconductors to the description of the normal and, in part, excited states of dielectric crystals. Unfortunately, this circumstance is still ignored by many physicists, which often leads them, in interpreting photoelectric phenomena in dielectrics and semiconductors, to various difficulties that are essentially fictitious.

Interesting schemes of electronic levels in ionic crystals were proposed by the late P. S. Tartakovsky. In an extremely interesting work published in 1933, I. E. Tamm showed that, alongside the usual levels corresponding to the motion of electrons inside a crystal, there exist additional “surface” levels corresponding to the motion of electrons along the surface of the crystal or, if one prefers, to their being bound to its surface.

Of fundamental importance is also the work of I. E. Tamm and S. P. Shubin on the external photoelectric effect in metals (1931). In this work it was shown that the photoeffect consists of two component parts—the surface part, which is due to the jump of the potential at the metal surface, and the internal part, which depends on the periodic oscillations of the potential inside the metal.

Among other works connected with optical phenomena in crystals (especially dielectrics), one should mention the works of Mandelstam and Leontovich and, in particular, of I. E. Tamm (1929) on the theory of the combination scattering of light in crystals, discovered in 1928 by Mandelstam and Landsberg.

In this connection I. E. Tamm was the first to give a consistent quantum-mechanical theory of this process, establishing the dependence of the Stokes and anti-Stokes components on temperature, and introduced into physics the concept of quanta of sound, or “phonons.”

Of very great interest is the work of I. E. Tamm and I. M. Frank on the theory of the effect discovered by Cherenkov in the laboratory of S. I. Vavilov and under his direction—a distinctive radiation emitted by a transparent body (solid or liquid) when an electron moves in it with a velocity exceeding the velocity of propagation of light in that body (the analogue of a Mach shock wave in acoustics). The theory of Tamm and Frank was further developed in a number of works by V. L. Ginzburg.

One should also mention the works of M. A. Leontovich on the theory of the scattering of light by a nonuniformly heated body, and also by the surface of a liquid, and, in particular, the works of Blokhintsev and Antonov-Romanovsky on the theory of the phosphorescence of crystals. The latter author proved experimentally that the luminescence of phosphors is governed by a recombination mechanism, which, depending on

of the concentration of positive centers, can be both bimolecular and monomolecular in character. In addition, Antonov-Romanovsky experimentally substantiated the idea that thermal electrons in a crystal have a much lower mobility than electrons possessing large energies (for example, those torn out by light in the first period of their existence). This low mobility is explained by the process of “self-arrest,” or sticking, of electrons in an ideal crystal lattice, the regularity of whose structure is disturbed by the presence of these very electrons, which cause a deformation of the lattice near those atoms through which they pass and at which, as a result of this deformation, they are delayed.

This question was first considered by L. D. Landau, who associated it with the phenomenon of activation necessary, in his opinion, for the process of “binding” an electron, and excluding the possibility of this process at low temperatures, i.e., precisely when the tendency toward such binding, due to the accompanying decrease in energy, proves to be especially strongly expressed. From the statistical point of view this question was considered somewhat later by me in connection with various phenomena in semiconductors and, in particular, dielectrics, which can be explained from this point of view. In doing so I pointed out that analogous “binding” should take place in the case of mobile positive holes, and also in the case of excitons (the liquidation of which by converting their energy into the energy of photons apparently proves possible only upon transition through such a “bound” state). Finally, quite recently S. I. Pekar gave a rigorous quantum-mechanical theory of the binding of electrons in ionic lattices, for which the accompanying decrease in energy is especially great. He succeeded in actually calculating this energy and in applying the results obtained to the quantitative explanation of many interesting optical properties of ionic crystals.

Among other investigations of a theoretical character in physical optics, one should mention the interesting works of S. I. Vavilov and his collaborators (especially Levshin) on the luminescence of solutions, in which various aspects of this phenomenon were theoretically elucidated and experimentally verified, in particular the question of the quenching of fluorescence under the influence of various factors and of its polarization as a function of the age of the solution (more precisely, of the solvent).

Of great interest are the works of Vlasov and Fursov on the theory of light absorption in gases in connection with the interaction between atoms, and the resulting form of the spectral lines in the emission and absorption spectra of gases. These works apparently represent the most successful solution of this question.

Among the above-mentioned works on the theory of crystals there also belong a number of works by I. Ya. Pomeranchuk and A. Akhiezer on the quantum theo-

magnetic and, in particular, thermal properties of crystals at low temperatures. In particular, Pomeranchuk succeeded in explaining certain anomalies in the thermal properties of ferromagnetic bodies by using the concept of “magnons,” i.e., particles associated with spin waves, which had been investigated earlier by Bloch in connection with the theory of ferromagnetism. Further, Pomeranchuk showed that, in order to explain the finite thermal conductivity of crystals, it is necessary to consider anharmonic effects associated not only with terms of the third order (as had previously been assumed), but also of the fourth order of smallness. Pomeranchuk also produced the first theory of the scattering and diffraction of slow neutrons in crystals, taking into account additional effects caused by the presence of randomly distributed isotopes.

The latter circumstance had somewhat earlier been taken into account by I. M. Lifshitz in his theory of the scattering of infrared rays by ionic crystals. The presence of isotopes manifests itself in this case in the appearance, alongside the fastest resonant vibrations (residual rays), of vibrations of lower frequencies (corresponding to finite wavelengths). The work under consideration is interesting not only for its results (which also concern the influence of foreign impurities), but also from the methodological point of view; I. M. Lifshitz developed an extremely convenient and elegant matrix method for calculating various phenomena connected with the vibrations of the crystal lattice. Somewhat earlier the same author had published a number of other interesting works on disturbances of regularity in the structure of real crystals and solid solutions, as well as on various phenomena caused by these disturbances, especially in the scattering of X-rays.

6. THEORY OF THE LIQUID STATE AT HIGH AND LOW TEMPERATURES

One of the fundamental areas of physics that was set in motion by Soviet theoreticians is the theory of the liquid state. Earlier theories proceeded from the approximation of the liquid state to the gaseous state. In 1926 I first developed a kinetic theory of liquids on the basis of the idea of the similarity of liquids, at temperatures close to the crystallization temperature, to the properties of the corresponding solids. In doing so I used a model of a liquid which combines, in the sense of the character of thermal motion, the properties of solids, on the one hand, and of gases, on the other (a combination of oscillatory motion about certain equilibrium positions with jumps from one position to neighboring ones). Starting from this representation, one can construct a quantitative theory of the (ordinary) viscosity of liquids, making it possible to explain its experimentally observed dependence on temperature and on pressure.

This theory was successfully developed further, albeit in a somewhat formal manner, in the direction of a molecular-kinetic explanation of the mechanical, electrical, and in part optical properties of liquids (with anisotropic molecules) associated with various kinds of relaxation phenomena. In addition, the kinetic theory of liquids made it possible to explain the discontinuous character of the process of melting and crystallization, proceeding from the fundamental continuity between solid (crystalline) and liquid (amorphous) states, with the instability or partial metastability of a number of intermediate states (similarly to what occurs in van der Waals’ theory). From the point of view of the modern kinetic theory of liquids, their “second,” or “bulk,” viscosity is of great interest; it manifests itself very sharply in connection with the fact that the absorption of sound and, in particular, ultrasonic vibrations in liquids is usually many times, and sometimes even hundreds and thousands of times, greater than that which follows from taking into account the ordinary viscosity of liquids according to the Navier–Stokes theory. From the formal point of view this fact is very simply explained by the circumstance that liquids possess viscosity of two types, of which one (ordinary viscosity) is connected with tangential stresses arising under changes of shape (i.e., in ordinary flow), while the other is connected with normal stresses caused by changes of volume (density). The latter circumstance, i.e., the presence in liquids of bulk viscosity, had until very recently been completely ignored.

A semi-phenomenological theory of its origin in connection with various relaxation processes in liquids was given in 1935 by M. A. Leontovich and S. L. Mandelstam, and also by me and Yu. A. Obraztsov.

The theories mentioned above take no account at all of quantum effects, which must appear at sufficiently low temperatures. The only liquid that does not freeze (at ordinary pressure) even at absolute zero temperature is liquid helium. Experimental investigations by a number of physicists and, in particular, P. L. Kapitsa revealed in helium at temperatures below \(2.19^\circ\mathrm{K}\) the peculiar phenomenon of superfluidity, which L. D. Landau was able to explain by applying to this liquid a new quantum hydrodynamics, based on the idea that at each point of a liquid stream two different velocities may coexist—that of “dead,” or degenerate, helium and that of “living,” non-degenerate helium—in connection with the idea of the existence of a certain “gap,” i.e., a discontinuity, in the energy spectrum of this substance. These paradoxical ideas led Landau to predict the existence in liquid helium II, alongside the ordinary sound vibrations connected with pressure oscillations, of vibrations of a second kind, connected mainly with temperature (or entropy) oscillations. This theory, brought into a more concrete form by E. M. Lifshi-

which was then brilliantly confirmed experimentally by Peshkov. Despite these successes, Landau’s conception is in a number of respects controversial and, apparently, may be replaced by a less paradoxical conception based on the consistent application to helium of Bose statistics (to which its atoms must obey). The latter conception was developed as early as 1937 by London. However, London did not succeed in taking account of the interaction between the atoms of liquid helium, which he treated as an ideal gas, as a result of which the program he outlined remained unfulfilled. This program has recently (in 1946) been carried out by the Soviet mathematician N. N. Bogolyubov, who showed that the property of superfluidity of helium can be explained on the basis of Bose statistics if the predominance of repulsive forces between the atoms over attractive forces is taken into account. This, among other things, could explain the absence of superfluidity in liquid hydrogen, whose molecules are characterized by a comparatively strong cohesion with one another. The phenomenon of superfluidity of liquid helium is outwardly very similar to the phenomenon of superconductivity observed in a number of metals and alloys at very low temperatures. Although a satisfactory molecular theory of superconductivity still does not exist, L. D. Landau succeeded in showing that the transition of a superconducting alloy into the ordinary state (with normal electrical resistance) under the influence of a magnetic field takes place through a series of intermediate states, which are characterized by an alternation of superconducting layers with layers of finite resistance gradually thickening (at their expense). The existence of this layered structure, reminiscent of the layered structure of uniaxial ferromagnets, was confirmed experimentally by A. I. Shalnikov.

7. MECHANICAL PROPERTIES OF SOLIDS

The most backward field of modern theoretical physics is, strange as it may seem, the field of phenomena that would seem to be the most ordinary and qualitatively comprehensible, namely phenomena connected with the mechanical properties of solids—crystalline and polycrystalline. These include, for example, the problem of the plasticity of crystals, into which clarity has still not been introduced, despite a number of attempts.

One of such attempts, relating to the linear model of a crystal, is the work of the author together with T. A. Kontorova (1937), in which the problem of the propagation of dislocations, which according to Taylor underlie plastic deformation, was solved for the first time. Of interest is Kontorova’s work on intercrystalline interlayers between twinned regions, entirely analogous to the theory of transition zones between neighboring domains, i.e. regions of spontaneous magnetization in ferromagnets.

It should also be noted that there are several works by the same author, jointly with me, on the statistical theory of strength, as well as those of Aleksandrov and Zhurkov.

More serious successes were achieved, in particular, by Soviet physicists on the question of the technical strength of solids, i.e. on the smallness of this strength in comparison with the theoretical strength that follows from the electronic theory of interparticle forces. Here one should especially note the classical works of A. F. Ioffe, who succeeded in explaining the deficit of strength by surface cracks, although the mechanism of their occurrence remains, incidentally, not yet fully clarified, despite the encouraging work in this respect of A. V. Stepanov, who connects their occurrence with preliminary plastic deformations.

In recent years Stepanov has been trying to develop a new theory of the strength of solids, connected with the concept of nuclei of fracture, which can reversibly grow and shrink until they reach a certain critical size, after which irreversible destruction of the body sets in. In the case of crystalline bodies the character of this destruction depends in an essential way on their anisotropy.

Garber, independently of Stepanov, arrived at analogous ideas with regard to the process of twinning of crystals. Recently the foundations of a quantitative theory of these processes have been laid by I. V. Obreimov and E. M. Lifshitz.

8. GENERAL PRINCIPLES OF THERMODYNAMICS AND STATISTICS

Alongside the special questions indicated above, Soviet theorists have a number of important works in the field of general questions of the statistical and thermodynamic theory of material bodies. Among the works belonging here one must first of all mention the works of L. D. Landau on the general thermodynamic theory of phase transformations (1936). These works are perhaps the greatest step forward in the thermodynamics of heterogeneous systems since the classical investigations of Gibbs. Landau’s theory, however, applies only to transformations of the second kind, connected with Curie points (i.e. peaks of the heat capacity and of the coefficient of thermal expansion), in particular to transformations characterized by the disappearance of long-range order in the arrangement of particles, and hardly concerns transformations of the first kind, connected with a discontinuous change of entropy and volume. Moreover, Landau does not take fluctuations into account (with the exception of one paper on superstructural X-ray lines in binary alloys near Curie points). This circumstance substantially limits the physical significance of his conclusions, since a thermodynamic theory that ignores fluctuations,

fundamentally excludes both unstable and metastable states, whereas the latter are realized and observed in reality. It leads, among other things, to a physically not entirely justified opposition between crystalline and liquid bodies, despite the preservation in the latter of short-range order, which may be treated as fluctuations of long-range order or of the anisotropy characteristic of crystals.

In 1946 N. N. Bogolyubov published two fundamental works on the foundations of statistical mechanics, connected with the development of the idea of introducing a sequence of distribution functions for the particles of a molecular system—one at a time, two at a time, three at a time, etc. This idea had been expressed earlier as well by various authors, but had not received any significant mathematical development. Bogolyubov gave a general method for constructing this sequence of functions both for a statistically equilibrium state (usually described by the Gibbs distribution) and for a nonequilibrium state (for which no general theory had hitherto existed at all), and indicated some approximate methods for solving the corresponding equations for the previously investigated cases of rarefied gases and dilute electrolyte solutions. I am convinced, however, that Bogolyubov’s method will also be applicable to more complicated condensed systems, in particular to liquids*).

It should also be noted that B. I. Davydov wrote several works devoted to the question of the increase of entropy of molecular systems, in which he attempts to reduce this increase to the intervention of the experimenter in measuring entropy (in accordance with Heisenberg’s uncertainty principle). Davydov’s considerations are undoubtedly interesting; however, to me personally they appear very doubtful.

Mention has already been made above of Davydov’s works on statistical kinetics, i.e. the statistical theory of irreversible processes connected with the approach to the equilibrium state. Some works of L. D. Landau also pertain to this same question, as do those of L. E. Gurevich, who published an entire book on the “foundations of physical kinetics.”

Also of great fundamental interest are the works of the young theoretician N. S. Krylov, who died prematurely in 1947, collected in his doctoral dissertation (but, unfortunately, still unpublished), on a new general theory of irreversible processes, which makes no use at all of the concept of entropy and confines itself to the study of the “dispersion” over time of the points representing the state of the molecular system under consideration in configuration space.

*) Apparently, an analogous method, independently of Bogolyubov, was developed somewhat later by Born (in England). It should be noted that this method may be regarded as a generalization of the self-consistent-field method.

§. PHYSICAL CHEMISTRY, GEOPHYSICS, AND ASTROPHYSICS

Among works in theoretical physics one could include a number of interesting theoretical investigations by A. N. Frumkin, B. V. Deryagin, P. A. Rebinder, V. G. Levich, and others on various questions of molecular physics, especially in the field of capillary and electrocapillary phenomena, as well as works by N. N. Semenov and Ya. B. Zel’dovich on the theory of the propagation of combustion and explosions.

However, these investigations were carried out under the banner of physical chemistry or chemical physics and are reviewed in the corresponding articles, in view of which we can only touch upon them very briefly in this survey. A. N. Frumkin is responsible for fundamental works on the theory of the electrocapillary curve and of the diffuse ionic layer, which determines the phenomena of electrocapillarity, on the formation of vapor bubbles (during boiling) or gas bubbles during electrolytic evolution, and on the jump of electric potential in electrolyte solutions.

Further, A. N. Frumkin gave a thermodynamic substantiation of the phenomenon of the disjoining action of thin liquid layers, discovered and experimentally studied by B. V. Deryagin, who on its basis constructed a quantitative theory of the stabilization and coagulation of colloids.

P. A. Rebinder clarified, experimentally and theoretically, the question of the lowering of hardness under the influence of adsorption of surface-active substances and established a fact very important for the theoretical understanding of the structure of solids: the existence in them of microporosity (probably connected with a mosaic structure).

V. G. Levich gave an interesting solution to the question of the surface viscosity of adsorbed layers, and also of the calming action of thin oil films on the undulation of a water surface.

Among works on the theory of surface phenomena I shall also note my own 1924 work on the theory of the phenomena of adsorption and condensation of molecular beams on the surface of solids, in which a quantitative explanation was given of the critical temperature of condensation; its existence was discovered by Knudsen and studied by Semenov and Chariton.

The various fields of science, and in particular of physics, cannot be sharply delimited. They also pass continuously into one another, just as the head passes into the torso and the torso into the arms and legs. Therefore the dismemberment of physics into theoretical and experimental physics, and the singling out from it of questions of a physico-chemical and even a “purely” chemical character, are connected with very painful and ugly amputations. Following, however, the generally accepted classification of the sciences, we shall not consider the further development of Soviet theoretical physics in the direction of physical chemistry and chemistry.

I consider it necessary, however, to add a few words about the works of Soviet theorists in the field of disciplines connected with appli-

…by the application of general physical theories to special phenomena—on the Earth and in the heavens, i.e., to geophysics and astrophysics.

As for geophysics, here one should first of all note Kibel’s work on the theoretical forecasting of weather, proceeding from the approximate methods he developed for solving the equations of the hydrodynamics of the atmosphere. Recently, in connection with the establishment of a number of centers of atmospheric activity, Kibel’s methods have made it possible to create long-term forecasts for an entire season. Further, one must point to the works of V. V. Shuleikin, which relate partly to the same subject, and also to the problem of the motion of the Earth’s poles and, in particular, to the physics of the sea (including several very interesting and original works on the mechanics of the motion of fish and other inhabitants of fresh and marine waters).

Soviet theorists have recently obtained a number of important results in the field of turbulence and, in particular, atmospheric turbulence (the works of Obukhov on these questions should be especially noted).

Finally, I must mention the new theory of the phenomena of atmospheric electricity, which I have developed over the last three years, proceeding from the consideration of clouds as an aerosol suspended in an ionized atmosphere. Closely connected with this is the new conception of the origin of terrestrial magnetism as the result of the self-excitation of electric currents in the liquid metallic core of the Earth (regarded as a special kind of “turbogenerator”). This conception was developed by me on the basis of the analogous conception which L. E. Gurevich and Lebedinsky developed for explaining the magnetic field of sunspots.

I have already mentioned above some works (my own and Landau’s) relating to the problem of the structure of stars. To this should be added the extremely interesting new works of O. Yu. Schmidt on the origin of binary stars, as well as of the planetary system, by the “capture” of some celestial bodies (in particular, meteors) by others (for example, growing planets).

10. MATHEMATICAL PHYSICS

In conclusion to this article we shall briefly consider the development of theoretical physics in the USSR in the direction of mathematics, or “mathematical physics.” The latter, as a rule, consists in solving, at a high mathematical level, problems—particularly of a technical character—that are quite clear from the qualitative side but require an exact quantitative solution.

The largest works in this practically very important direction, which so far has still not received the development it deserves among us, belong to V. A. Fock and G. A. Grinberg.

In 1927, developing ideas expressed by N. N. Semyonov and A. F. Walter, Fock gave an exhaustive theory of thermal electri—

of dielectric breakdown of dielectrics (i.e., electrical breakdown caused by their self-heating). Subsequently V. A. Fock developed the theory of breakdown in cables (two- and multicore) and gave a theory of the skin effect in a ring, making use for this purpose of toroidal functions. V. A. Fock discovered and corrected an error in Sommerfeld’s well-known theory of the propagation of radio waves along the earth’s surface. He is responsible for brilliant works of a mathematical character on the most varied questions of physics and technology—from the quantum mechanics of many-electron systems to the theory of photometry or the theory of logging.

In recent years (1945–1946) V. A. Fock has been engaged, in particular, with the theory of the propagation of radio waves along the earth’s surface. In this connection he succeeded for the first time in solving the very difficult problem, posed already by Poincaré, of the diffraction of radio waves by the surface of the terrestrial sphere, as well as by various local irregularities of this surface.

G. A. Grinberg opened up a number of new paths in mathematical physics, systematically developing and applying to the solution of concrete electrostatic and magnetostatic problems powerful new mathematical methods based on the use, on the one hand, of integral equations that connect the electric and magnetic field with the surface charges of dielectrics or magnetizing bodies immersed in it, and, on the other hand, of the theory of integral transformations of the Laplace-transform type that underlie operational calculus (1937). Similar methods have recently been applied by him with exceptional success to the solution of problems on the propagation of radio waves along the earth’s surface under an arbitrary variation of electrical conductivity and dielectric constant with height or depth, and moreover not only for a “flat” earth, but also for the real spherical Earth (i.e., under far more general assumptions than in Sommerfeld’s theory). A particularly interesting and elegant approximate method, based on the use of S. M. Rytov’s work on the theory of the skin effect, was applied by him to the solution of problems on the “coastal refraction” of radio waves, i.e., their propagation along the surface of the sea (ideally conducting), bounded by arbitrary shorelines (with a finite conductivity of the earth’s surface).

Alongside the theory of the electromagnetic field, G. A. Grinberg developed with great success the mathematical theory of various electronic and ionic devices, or, more precisely, of the oscillatory processes taking place in them, overcoming a number of difficulties before which the greatest mathematicians, both in our country and abroad, had failed. In particular, he was the first to give (jointly with Lukoshkin) a theory of the split magnetron (1934), of the cylindrical diode with a magnetic field, and of the initial regimes in a plane diode. Very interesting work on this question was also done by I. M. Lifshits.

Of this series of works, special mention should be made of the work (1942) on the focusing of cathode rays in constant electromagnetic fields, in which the question is solved of choosing the field that ensures the desired type of focusing. This work represents an entirely new and fresh page in electron optics and will undoubtedly be of great practical importance.

Finally, G. A. Grinberg, together with his collaborators (Kantorovich and Lebedev), solved the problem of the development in time of the thermal breakdown of capacitors, investigated and solved the very difficult problem of short-circuiting in an oil-filled cable, and a number of other cable problems.

All these investigations of G. A. Grinberg are of very substantial importance for electrical engineering and radio engineering.

It is also necessary to note a number of interesting works by the students and collaborators of G. A. Grinberg—N. N. Lebedev and M. O. Kantorovich—on the theory of diffraction of radio waves by means of methods developed by Grinberg. The investigations of Vvedensky, Mandelstam, and Papaleksi on the propagation of radio waves along the earth’s surface also belong to this same question; in them some new results were obtained by using the old, cumbersome methods of Sommerfeld, Watson, and other foreign investigators (1940–1941).

Works important for physics and technology on the theory of nonlinear oscillations were carried out by L. I. Mandelstam and his collaborators (in particular, A. A. Andronov), as well as by N. M. Krylov and N. N. Bogolyubov (whose monograph has been translated into English).

Finally, it is necessary to point to the great role played in the development of mathematical physics in our country, and also in the creation within it of a school of applied mathematics and mathematical physics, by the works of A. N. Krylov and, in particular, by his fundamental work “On Certain Differential Equations of Mathematical Physics Having Applications to Technical Problems.”