Bibliography

P. Gombás, Die statistische Theorie des Atoms und ihre Anwendungen, Wien, Springer-Verlag, 1949.

(P. Gombás, Statistical Theory of the Atom and Its Applications. Vienna, Springer, 1949.)

The monograph by Prof. P. Gombás, director of the Physical Institute of the University of Technical Sciences in Budapest, is devoted to an exposition of the statistical treatment of atomic systems on the basis of the work of Thomas and Fermi (1926–1927). The first part (pp. 1–165) is devoted to the general theory; in the second part (pp. 167–356) there follow applications in the form of calculations of various properties of atoms, molecules, and crystals. This book adjoins the valuable monograph by the same author on approximate methods of quantum mechanics, in which the Thomas–Fermi method was considered alongside the ordinary perturbation method, the variational method, the approximate Wentzel–Kramers–Brillouin method, and the self-consistent-field method*).

As is known, in quantum mechanics, both relativistic and nonrelativistic, only the problem of hydrogen, i.e. of one electron in the Coulomb field of the nucleus, is solved exactly; for the solution of the problem of helium and, still more, of other complex atoms, one has to apply approximate methods. In this case the ordinary perturbation method, treating the interaction of the electrons as a small perturbation, proved too crude even in the helium problem and was successfully replaced by the powerful variational method, originating with Ritz. For complex atoms the most accurate results are given by the self-consistent-field method, in which the wave function of the system is represented approximately by a combination of products of individual functions of the separate electrons. However, computations by this method are extremely cumbersome. A further simplification, naturally following from the preceding method (see § 16), consists in abandoning wave functions altogether and passing to the picture of a degenerate gas of electrons filling the cells of phase space corresponding to all momenta up to a certain maximum determined by the condition

\[ \frac{p_m^2}{2m}=e\varphi . \]

Now, on the one hand, the electrostatic potential is determined by the Laplace–Poisson equation

\[ \Delta \varphi=-4\pi e\rho, \]

*) See P. Gombás, Theory and Methods of Solving the Quantum Many-Body Problem (German), Basel, Birkhäuser, 1950.

On the other hand, the density \(\rho\) is directly determined by the volume of phase space occupied by the electrons, i.e., through \(p_m\), or again by the potential \(\varphi\). As a result we obtain the fundamental Thomas–Fermi differential equation

\[ \Delta \varphi=\frac{32\pi^2 e}{3h^3}(2me\varphi)^{3/2}, \]

or, in suitable dimensionless units:

\[ \varphi''=\frac{\varphi^{3/2}}{x^{1/2}}. \]

Solving this equation under the appropriate boundary conditions, one finds \(\varphi\) and, consequently, the density distribution of electrons in the atom \(\rho\), by means of which various quantities can be determined. At first glance it seems incredible that such simple considerations could yield anything more than a preliminary orientation in a small circle of questions. However, the work of many authors, including Dirac, Fermi, and Gombás himself, has demonstrated the remarkable power of the Thomas–Fermi theory, which leads to many results not only for atoms, but also for molecules and crystals, and moreover in excellent agreement with experiment and with an accuracy in many cases not inferior to the results of considerably more painstaking quantum-mechanical calculations by the self-consistent-field method.

Of course, in order to transform the Thomas–Fermi method into a modern, perfected instrument of theory, considerable work had to be done both on its foundation and generalization and on the development of applications.

Let us now proceed to a brief account of the contents of the book. After the exposition of the foundations of Fermi–Dirac statistics (§ 1), the derivation of the exchange energy (per unit volume) is considered:

\[ A=-x_a\rho^{4/3}\qquad (x_a=0.7386\,e^2), \]

arising as a result of the specific quantum repulsion of electrons with parallel spins, as well as the correlation energy (per electron):

\[ w_m=-0.288\,\frac{e^2}{r_s} \]

(\(r_s\) is the radius of a sphere containing one electron), associated with allowance for the specific interaction of electrons of antiparallel spin. The justification of the Thomas–Fermi equation in § 3 is given not only in the usual way, but also by a variational method, which is very convenient for generalizations.

In § 4 approximate solutions of the nonlinear Thomas–Fermi equation are analyzed, both those obtained by machines—integrators—and asymptotic forms suitable at large

\[ \left(\varphi=\frac{144}{x^3}\right) \]

and small

\[ (\varphi=1-1.5880464\,x+4/3\,x^{3/2}\ldots) \]

distances. The virial theorem and the formula for the total energy of the electrons of an atom (in § 6) complete the exposition of the foundations, comparatively well known and accessible from other books, of this part of the theory. We owe the monographic treatment of the subsequent sections to the author.

The question is to generalize the Thomas–Fermi theory by taking into account the following circumstances:

1) the correction to the exchange energy (Dirac), § 2, which makes the decrease of the density at large distances steeper in accordance with the requirements of quantum theory and introduces a finite radius of atoms; 2) the correction for the electrostatic self-interaction of the electrons (Fermi–Amaldi), suitable in the outer regions of atoms and leading to the possibility of treating also negative ions (§ 7); 3) one-

BIBLIOGRAPHY

a time-dependent modification of the Thomas–Fermi method with the aid of both preceding corrections (§ 10); 4) the correction for correlation, with account taken of exchange and exclusion of self-action (§ 11) (Gombás); 5) the correction for kinetic energy by taking account of the nonconstancy of the density according to Weizsäcker; 6) a relativistic generalization, leading, however, to insignificant corrections (§ 14); 7) a generalization to extremely high temperatures (§ 15).

All these generalizations of the Thomas–Fermi equation are introduced, as a rule, by adding to the principal energy of the electron gas

\[ E = E_k + E_p,\qquad E_k = \varkappa_k \int \rho^{5/3}\,d\tau; \]

\[ E_p = \frac{1}{2}e^2 \iint \frac{\rho(r)\rho(r')\,d\tau\,d\tau'}{|r-r'|} - e\int V_k \rho\,d\tau, \]

where \(V_k\) is the external potential of new terms corresponding to the exchange energy, the correlation energy, etc., and by subsequent variation of the expression obtained. As a result, a picture arises of an impressive generalization of the original equation. The graph of the density, computed with all these corrections, from which the role of the first and second corrections can be clearly seen, convincingly illustrates the accuracy of the general course of the density calculations, which, however, are still not able to convey the characteristic bends in the course of the quantum density \(\rho\).

In the general part, the application of perturbation theory to the statistical model (§ 17), the nonstatic treatment of the electron gas (§ 20), and certain other questions are also considered.

In the second part, devoted to applications, atoms (§§ 21–29) and crystals (§§ 34–35) are considered in two principal chapters. Here, too, the applications of statistical theory to molecules (§§ 31–33), a theory of which is only beginning to develop and presents difficulties because of the absence of spherical symmetry, are set out more briefly. In the concluding sections (§§ 36, 37) the behavior of matter at high pressures is analyzed and a relation between pressure and density is derived, applied in a curious way to the investigation of the inner layers of the earth.

As for atoms and metals, here statistical theory is reaping its greatest successes. In § 21 the derivation of the formation of electron shells is presented; moreover, as is known, statistical theory, in striking agreement with Mendeleev’s periodic system, points to the elements hydrogen (\(Z=1\)), boron (\(Z=5\)), scandium (\(\mathrm{Sc}, Z=21\)) and cesium (\(Z=55\) instead of \(Z=58\)) as the places where the filling of the corresponding \(s\)-, \(p\)-, \(d\)-, \(f\), … shells begins. Let us note that the ability of the statistical model to describe shell formation has prompted us to undertake a similar investigation for atomic nuclei, where, as is known, the works of Soviet and foreign authors in recent years have undoubtedly shown the presence of periodicities in many properties (spins, magnetic moments, quadrupole moments, etc.).*

Next, the conditions for the last energy position (§ 22), and calculations of terms (x-ray and optical) by means of a combined method, which uses the Thomas–Fermi potential in the Schrödinger equation (§ 24), are set out. The agreement of Rosetti’s statistical calculations for \(M\)-terms (\(3d\)-states) gave excellent agreement with experiment; for example, for uranium the value in rydbergs is equal to 259, whereas empirically we have 261.2, and the theoretical calculation without account of screen-

* See D. Ivanenko and V. Rodichev, DAN 70, 605 (1950); D. Ivanenko and A. Sokolov, DAN 74, 33 (1950).

... would give 940. In §§ 28 and 29 questions are considered concerning the calculation of the radii of atoms and ions, in good agreement with Goldschmidt’s empirical values, as well as the related problem of calculating the diamagnetic susceptibility by the well-known Langevin formula,

\[ \chi=-\frac{e^2}{6mc^2}\,\overline{r^2}, \quad \text{where } \overline{r^2}=\int \rho r^2\,d\tau \]

with the aid of the Thomas–Fermi density \(\rho\).

In § 29 an important problem is examined: the statistical determination of the form factor for the scattering of Roentgen rays and electrons and, finally, in § 30, the Bloch theory of energy losses in the passage of charged particles through matter.

Of interest are the paragraphs devoted to ionic crystals and metals, based in particular on the works of the author of the book. Prof. Gombás, giving the theory of these crystals and deriving the values of the binding energies, coefficients of compressibility, etc., emphasizes the absence of any new empirical constants.

On the whole, in this first monograph in world literature devoted to the Thomas–Fermi method, Gombás unfolds before the reader a fascinating broad picture of the gradual generalization of the method and the diversity of its applications. Let us stress that the book is written in a very clear style; each paragraph is provided with a brief summarizing introduction; the book contains a large number of drawings and tables of solutions of the Thomas–Fermi equation; the literature on the subject is given in exhaustive form. Due place is given to well-known Soviet works on the self-consistent field (Fock and his collaborators), on the theory of metals (Frenkel and others). Particular interest is aroused by the presentation of the works of Gombás himself and of his collaborators, printed until recently in Hungarian editions that were difficult to obtain.

Congratulating the author on an undoubted success, we consider it necessary to express the insistent wish that the works of scholars of the countries of people’s democracy should as soon as possible appear in Russian translation and, for their part, serve the great cause of the further cultural rapprochement of our peoples.

D. Ivanenko