A. Sommerfeld. Atombau und Spektrallinien. 3rd ed., Vieweg und Sohn. Braunschweig, 1922. VIII + 764.

Sommerfeld’s excellent book, already sufficiently appreciated and widely known among us in Russia, has appeared in a third, substantially revised edition.

In terms of content the book has grown considerably: it has been enlarged by 180 pages and contains 125 figures instead of the 109 in the second edition. A new chapter has been added (Die Bandenspektren), and the former chapter on optical series spectra has grown into two. The chapters that have retained their titles have also been supplemented.

Thus, in the first chapter (Vorbereitende Tatsachen, pp. 1–67) a section on the photoelectric effect has been added, serving, among other things, for the reader’s first acquaintance with the concept of quanta (Millikan’s work is presented).

In the second chapter (Das natürliche System der Elemente, pp. 68–134) the doctrine of the structure of the nucleus is set forth in somewhat greater detail, with Rutherford’s work on the artificial transmutation of elements being used; in the section on molecular models (more precisely, in the addition to this section placed at the end of the book) a criticism of Bohr’s hydrogen molecule is given. Aston’s work is presented on the question of isotopy.

In the third chapter (Die Röntgenspektren, pp. 135–236) a section on the structure of crystals has been added. The tables of wavelengths of the lines of X-ray spectra have been compiled anew; new data have been added, and corrections have been introduced into the old ones. The wavelengths of X-ray spectra and the periods of crystals are expressed in Siegbahn units (X-unit = \(10^{-11}\) cm) with two decimal places. For example, \(K_{\alpha}\) for \(Ca\): \(\lambda = 3351,86\ X\).

The fourth chapter (Das Wasserstoffspektrum, pp. 237–309) is devoted entirely to the spectrum of hydrogen. Along the way, the analogy between the hydrogen spectrum and the X-ray spectrum of the \(K\) series is clarified, and an attempt is made to calculate the dimensions of the \(Be\) nucleus, constructed as an inverted molecule of hydrogen (or oxygen), with four positive nuclei (hydrogen) placed on a circular orbit and two electrons at the ends of a line perpendicular to the plane of this orbit and passing through its center. Although the dimensions of such a nucleus turn out to be of the order of \(10^{-12}\) cm instead of \(10^{-13}\) cm, as given by Rutherford from experiments on the scattering of \(\alpha\)-particles, Sommerfeld nevertheless considers it possible to see in this calculation an indication that “the structure of the nucleus is governed by the same quantum laws as the structure of the atom.”

Finally, in the same chapter several pages have been added that are devoted to clarifying the connection between the Weiss magneton and the elementary magnetic moment of Bohr’s atom, which Sommerfeld calls the Bohr magneton. The Bohr magneton is almost exactly five times larger

Weiss’s magneton, and at the present time it still does not appear possible to clarify the misunderstanding that exists here.

Chapter five (Wellentheorie und Quantentheorie, pp. 310–385) corresponds to the sixth chapter of the second edition; in comparison with it, it contains a new section (Die Adiabatenhypothese, pp. 374–386), and differs by a somewhat different arrangement of the material (for example, the question of radiation in a force field and Bohr’s correspondence principle are placed in a separate section, although without substantial additions).

Section six and the supplement to the fourth chapter of the 2nd edition have been developed into an extensive independent sixth chapter of the new edition (Die Serienspektren im allgemeinen, pp. 386–495). In it the selection principle set forth in Chapter 5 already finds wide application. Special sections are devoted to proofs of the series scheme by the method of electron impacts (the work of Franck and Hertz and others) (§ 3), to the concept of the inner quantum number (the multiplicity of orbits corresponding to a given term) (§ 5), and to the anomalous Zeeman effect (§ 7).

Chapter seven (Die Bandenspektren, pp. 506–551) is devoted to band spectra, to which in the second edition only six pages in the appendices had been allotted. After preliminary and historical remarks, the theory of infrared absorption spectra (rotational spectra) is presented; a systematics of the visible band spectra (Cyanbanden) is given; and the many-line spectra of hydrogen and helium are analyzed. The value of the energy moment of the hydrogen molecule obtained from the many-line spectrum provides further evidence of the imperfection of Bohr’s hydrogen molecule, while the many-line spectrum of helium leads to the assumption of the existence of a helium molecule (\(\mathrm{He}_2\)), which, however, can exist only under conditions of electrical excitation of the atom. In the last section of this chapter the question of continuous spectra is touched upon.

Chapter eight (Die Theorie der Feinstruktur, pp. 552–647), devoted to the fine structure of lines, has been considerably supplemented in comparison with the corresponding chapter of the 2nd edition, chiefly with regard to the systematics of X-ray spectra, to which much attention has recently been devoted. Not only is the theory of the structure of \(L\)-doublets expounded, but data are also given concerning doublets in the \(M\)- and \(N\)-series, and the work of Hertz and Wentzel is presented, devoted to clarifying the distinction between so-called regular and irregular doublets (the former include doublets for which \(\Delta\lambda\) does not depend on the atomic number \(Z\); the latter include doublets for which \(\Delta\lambda\) is inversely proportional to the third power of \(Z\)). Finally, a general systematics of X-ray spectra is given, rather complicated, since it distinguishes 21 energy levels:

\[ (K \mid L_1\, L_2\, L_3 \mid M_1\, M_2\, M_3\, M_4\, M_5 \mid N_1\, N_2\, N_3\, N_4\, N_5\, N_6\, N_7 \mid O_1\, O_2\, O_3\, O_4\, O_5), \]

but to the highest degree satisfactory and even “astonishing,” in Sommerfeld’s estimation, for it is developed without any special additional assumptions of a model character, but solely on the basis of the general requirements of quantitative relations and internal symmetry.

The section on mathematical applications (pp. 648–744) is partly relieved by transferring some of the questions treated there into the main text, and partly supplemented by some new notes (Berührungstransformationen, Korrespondenzprinzip, adiabatische Invarianz der Phasenintegrale).

The book makes very extensive use even of the literature of 1921, so that in certain respects it may, here in Russia, somewhat fill the deficiency still unfortunately felt in the current literature.

In conclusion one should mention how Sommerfeld himself evaluates the significance of the two and a half years that have elapsed since the appearance of the first edition. In the preface to the new edition he says: “...I am issuing this edition with a calmer conscience than the first.

At that time much still seemed immature and unreliable. Even now the turbulent fermentation has not yet wholly subsided, but in the course of the years much has already settled and become clear”...

G. Landsberg.