ROTATING ELECTRON.

Ya. I. Frenkel, Leningrad.

§ 1. Introduction.

Electrons are usually treated as small spheres of a certain radius with an electric charge uniformly distributed over their surface or volume. Proceeding from this conception, Abraham as early as 1903 introduced into consideration, alongside the translational motion of the electron, its rotational motion about an axis passing through its center. Such rotational motion was subsequently considered by a number of other authors; however, direct physical significance was first attached to it only in 1922 by A. Compton. Compton arrived at the idea of the rotating electron not on the basis of fundamental considerations, but from an analysis of the mist tracks described by electrons in Wilson’s chamber. In the bends of these curves he found characteristic regularities which, in his opinion, could be explained only if electrons, like artillery shells, possess rapid axial rotation. Compton expressed the idea that such axial rotation is characteristic of all electrons, both outside and inside atoms, and that its speed is in all cases the same, so that all electrons have one and the same “intrinsic” angular momentum. This angular momentum Compton identified with that elementary moment

\[ \frac{h}{2\pi}, \]

which, according to Bohr’s theory, corresponds to the revolution of electrons around the positive nuclei of atoms. A magnetic moment is inevitably connected with a mechanical moment. This applies equally both to the revolution of electrons around nuclei and to their own rotation. Thus, according to Compton, all electrons, both bound and free, possess a quite definite magnetic moment, which must have an essential significance for the magnetic properties of material bodies.

Compton drew no further conclusions from his idea, and it soon died away completely. However, after several years, namely at the end of 1925, it was revived again at the initiative of young

of the Dutch physicists Uhlenbeck and Goudsmit. In doing so, the latter proceeded from a number of new spectroscopic facts, relating chiefly to the anomalous Zeeman effect, as well as to X-ray spectra and their analogy with optical spectra. All attempts to interpret these facts from the point of view of the classical Rutherford–Bohr model of the atom ran up against insurmountable internal contradictions. It was not for nothing that, with Einstein’s light hand, this region of physics—limited, essentially, only to the systematics of empirical regularities—received the somewhat contemptuous name “zoology.” The idea of the rotating electron immediately brought clarity and order into this spectroscopic zoology, removing, as if by magic, a number of contradictions which, as it now turned out, followed from the incompleteness of the old atomic model. Of fundamental significance here was the clarification of those additional forces which the electron experiences in the electric field of the positive nucleus owing to its axial rotation. The clarification of this question is Uhlenbeck’s merit, while the application of the corresponding general principles of the quantum theory of spectra is Goudsmit’s merit. Some discrepancies nevertheless remained; these, however, it proved possible to eliminate, on the one hand, with the aid of the theory of relativity, and on the other—with the aid of the new quantum mechanics.

§ 2. Additional Magnetic Energy of the Electron in the Atom

Let us imagine an electron with charge \(-e\) and mass \(m\), moving around a fixed positive nucleus with charge \(+Ze\). We shall denote the radius vector of the electron (relative to the nucleus) by \(\vec r\), and its velocity by \(\vec v\). The mechanical moment (i.e., the angular momentum) of the electron, or, more precisely, of the system formed by it together with the nucleus, is then expressed by the vector

\[ \vec l = m \vec r \times \vec v^{\,1)} \tag{1} \]

and the magnetic moment by the vector

\[ \vec M = - \frac{e}{2c}\,\vec r \times \vec v . \tag{2} \]

Thus, the two moments—the magnetic and the mechanical—are related to one another by the relation

\[ \vec M = - \frac{e}{2cm}\,\vec l . \tag{3} \]

\(^{1)}\) Here and below, \(\times\) denotes the vector product.

Let us now suppose that the electron, owing to its axial rotation, possesses a mechanical moment \(\vec{i}\) and a magnetic moment \(\vec{\mu}\), and that these moments are connected with one another by the relation

\[ \vec{\mu}=\chi\vec{i}, \tag{4} \]

where \(\chi\) is a coefficient of proportionality, the numerical value of which we shall not consider for the time being.

Under these conditions the electron must experience, from the side of the nucleus, in addition to the Coulomb force of attraction, an additional electromagnetic force, the origin and magnitude of which were determined by Uhlenbeck in the following way.

Instead of the original coordinate system \(S\), in which the nucleus is at rest and the electron moves with velocity \(\vec{v}\), let us introduce another system \(S'\), with respect to which the electron at the moment under consideration is at rest, while the nucleus moves with velocity \(-\vec{v}\). In this system the nucleus must create a magnetic field determined by the Biot–Savart law. Denoting the strength of this field at the point where the electron is located by \(\vec{H}'\), we have

\[ \vec{H}'=-\frac{Ze\,\vec{v}\times\vec{r}}{cr^3} \tag{5} \]

or, according to formula (2),

\[ \vec{H}'=-\frac{2Z\vec{M}}{c^3}. \tag{6} \]

With the help of this formula it is not difficult to calculate the additional force experienced by the electron owing to the presence of its own magnetic moment \((\vec{\mu})\), as well as the couple of forces tending to change the direction of the latter. For what follows, however, it is sufficient to know only the magnitude of the additional energy of the electron, which we can determine from the formula

\[ U=-\vec{\mu}\cdot\vec{H}'\ ^1). \tag{7} \]

Substituting into this formula the preceding expression for \(\vec{H}'\), we obtain

\[ U=+2Z\frac{\vec{\mu}\cdot\vec{M}}{r^3}. \tag{8} \]

If one does not take into account the magnetic moment of the electron, as well as the dependence of its mass on velocity (which follows from the theory of relativity), then its motion around a positive nucleus must

\(^1\) Here and below the dot denotes the sign of scalar multiplication.

proceed along a Keplerian ellipse, whose semiaxes we shall denote by \(a\) and \(b\). In this case, according to Bohr’s theory, the values of \(a\) and \(b\) for stable (or “stationary”) motions are determined by the formulas

\[ a=a_0\frac{n^2}{Z}, \tag{9} \]

\[ b=a\frac{k}{n}, \tag{10} \]

where

\[ a_0=\frac{h^2}{4\pi^2me^2}, \tag{11} \]

is the normal radius (of the one-quantum orbit) of the hydrogen atom, and \(n\) and \(k\) are the principal and azimuthal quantum numbers of the motion under consideration. These numbers have, as is known, the following meaning: \(n\) determines the energy of the electron (or, more precisely, of the electron–nucleus system) according to the formula

\[ W=-\frac{Ze^2}{2a}, \tag{12} \]

and \(k\) is the angular momentum of the electron, i.e. the quantity \(I\) introduced above, according to the formula

\[ I=\frac{h}{2\pi}k. \tag{13} \]

In order to take into account, in a first approximation, the change in the energy of some stationary motion which depends on the magnetic moment of the electron, it is necessary to take the average value of the additional energy \(U\) for the unperturbed motion. Thus to the energy \(W\) we must add the quantity

\[ \overline{U}=-2Z\frac{\overline{\vec M\cdot \vec\mu}}{r^3}, \]

where the bar denotes the average (over time), taken for the normal Keplerian motion. Since in this case the vectors \(\vec M\) and \(\vec\mu\) remain constant both in magnitude and in direction, we must average only the quantity \(\dfrac{1}{r^3}\).

It is not difficult to show\(^1\) that

\[ \overline{\left(\frac{1}{r^3}\right)}=\frac{1}{b^3} \tag{14} \]

\(^1\) Keplerian motion is characterized, first, by the equation of the orbit:

\[ r=\frac{p}{1+\varepsilon\cos\varphi}, \]

where \(\varphi\) is the angle of the radius vector with the direction toward perihelion, \(\varepsilon\) is the eccentricity, \(p\) is the semilatus rectum \(=a(1-\varepsilon^2)=\dfrac{b^2}{a}\), and, second, by the equation \(I=mr^2\dfrac{d\varphi}{dt}=\)

whence it follows that

\[ \overline{U}=\frac{2Z\,\vec M\cdot \vec\mu}{b^3}. \tag{15} \]

Let us denote the angle between the vectors \(\vec M\) and \(\vec\mu\) by \(\theta\); next, in accordance with formulas (3) and (13), put

\[ M=\frac{-e}{2cm}\cdot\frac{hk}{2\pi} \tag{16} \]

and, analogously,

\[ \mu=\frac{-e}{2cm}\cdot\frac{hs}{2\pi}, \tag{17} \]

where \(s\) is a numerical coefficient, the value of which we shall for the time being leave undetermined. If we were to assume that the electron’s intrinsic mechanical moment coincides with the moment of a one-quantum orbit, i.e. is equal to \(\frac{h}{2\pi}\) (Compton’s hypothesis), and that the ratio \(i:\mu\) is equal to the ratio \(\Gamma:M\), then one would have to put \(s=1\).

Using formulas (16) and (17), and also (9) and (10), we can represent the energy \(\overline U\) in the following form

\[ \overline U=\frac{AZ^4}{n^3k^2}\,s\cos\theta, \tag{18} \]

where \(A\) is a universal constant defined by the formula

\[ A=\frac{8\pi^4me^8}{h^4c^2}. \tag{19} \]

§ 3. Comparison of the magnetic energy of the electron with the relativistic correction.

The significance of this result will become clear to us if we compare it with the magnitude of the additional energy arising from the change of the electron mass as a function of its velocity. Namely,

\[ =\text{const.}, \]

expressing the law of conservation of angular momentum. For the mean (in time) value of any function \(f(\varphi)\) we have: \(\overline{f(\varphi)}=\frac{1}{\tau}\int_0^\tau f(\varphi)\,dt\), where \(\tau\) is the period of revolution, or, since \(dt=\frac{mr^2}{\Gamma}\,d\varphi\) and \(\tau=\frac{m\,2\pi ab}{\Gamma}\) (\(\pi ab\) is the area of the ellipse),

\[ \overline{f(\varphi)}=\frac{1}{2\pi ab}\int_0^{2\pi} r^2 f(\varphi)\,d\varphi. \]

In particular we obtain

\[ \overline{\frac{1}{r^3}} = \frac{1}{2\pi ab}\int_0^{2\pi}\frac{1+\varepsilon\cos\varphi}{p}\,d\varphi = \frac{1}{abp} = \frac{1}{b^3}. \]

as Sommerfeld first showed, this “relativistic” correction to the energy of motion, determined by the quantum numbers \(n\) and \(k\), is equal to

\[ -\frac{RhZ^{4}a^{2}}{n^{4}}\left(\frac{n}{k}-\frac{3}{4}\right), \]

where

\[ R=\frac{2\pi^{2}me^{4}}{h^{3}} \]

is the Rydberg constant, and

\[ a=\frac{2\pi e^{2}}{hc} \]

is the fine-structure constant of the hydrogen spectrum.

Substituting these values of \(R\) and \(a\) into the preceding expression, we can represent that part of the relativistic correction which depends on both quantum numbers \(n\) and \(k\) in the following form

\[ \Gamma=-\frac{AZ^{4}}{n^{3}k}, \tag{20} \]

where \(A\) denotes the very same constant as in formula (18). Thus, despite the difference in the causes determining the relativistic and magnetic corrections to the energy of the unperturbed Kepler motion, the magnitude of these corrections turns out, if not absolutely identical, then in any case extremely similar both in numerical value and in the sense of its dependence on the nuclear charge \(Z\) and the principal quantum number \(n\).

Before applying formulas (18) and (20) to the explanation of properties of spectra that have hitherto remained incomprehensible, we think it useful to give a brief derivation of formula (20).

The dependence of the mass of the electron on its velocity is expressed by the formula

\[ m=\frac{m_{0}}{\sqrt{1-\beta^{2}}}, \]

where \(\beta=\frac{v}{c}\). In this case, the kinetic energy \(T\) of the electron is equal to

\[ T=c^{2}(m-m_{0})=m_{0}c^{2}\left(\frac{1}{\sqrt{1-\beta^{2}}}-1\right). \]

Defining the momentum of the electron \(\vec p\) as the derivative of the kinetic energy with respect to velocity (in accordance with the definition of generalized momenta and coordinates in analytical mechanics), we have:

\[ p=\frac{dT}{dv}=\frac{m_{0}v}{\sqrt{1-\beta^{2}}}=mv, \]

which coincides with the usual definition of the quantity of motion, as the product of mass by velocity. Hence we obtain

\[ p^2=\frac{m_0^2 c^2 \beta^2}{1-\beta^2} = m_0^2 c^2\left(\frac{1}{1-\beta^2}-1\right) \]

and further

\[ \frac{1}{\sqrt{1-\beta^2}} = \sqrt{1+\frac{p^2}{m_0^2 c^2}} . \]

Thus, the kinetic energy of the electron, as a function not of its velocity but of its quantity of motion, is expressed by the formula

\[ T=m_0c^2\left(\sqrt{1+\frac{p^2}{m_0^2c^2}}-1\right). \]

If we expand the preceding radical in a series in powers of the ratio \(\dfrac{p^2}{m_0^2c^2}\), then, retaining only the first three terms of this series, we obtain the following approximate expression for \(T\):

\[ T=\frac{p^2}{2m_0}-\frac{p^4}{8m_0^3c^2}. \]

The first term of the right-hand side is the ordinary expression for the kinetic energy (for \(m=m_0\)), and the second is the relativistic correction in approximate form. The mean value of this expression for the unperturbed Keplerian motion is nothing other than the relativistic Sommerfeld correction. To compute it, let us note that the potential energy of the electron with respect to the nucleus is equal to

\[ -\frac{Ze^2}{r}. \]

Thus, if one does not take into account the dependence of mass on velocity, the total energy of the electron (or, more precisely, of the electron–nucleus system) is expressed by the formula

\[ W=\frac{p^2}{2m_0}-\frac{Ze^2}{r}. \]

It follows from this that in the “unperturbed” motion

\[ p^2=2m_0\left(W+\frac{Ze^2}{r}\right). \]

Substituting this value into the preceding expression for the additional kinetic energy, we have

\[ -\frac{p^4}{8m_0^3c^2} = -\frac{1}{2m_0c^2}\left(W+\frac{Ze^2}{r}\right)^2 . \]

Passing to mean values and using the formulas

\[ \overline{\frac{1}{r}}=\frac{1}{a},\qquad \overline{\frac{1}{r^{2}}}=\frac{1}{ab}, \]

and also formulas (10) and (12), we obtain for the desired relativistic correction, i.e. for the additional energy due to the dependence of mass on velocity, the formula

\[ V=-\frac{W^{2}}{2m_{0}c^{2}}\left(4\frac{n}{k}-3\right). \]

The first term on the right-hand side coincides, as is not difficult to verify, with expression (20). As for the second term, which does not depend on \(k\), it is of no essential significance for what follows, and therefore we shall not take it into account.

§ 4. X-ray spectra; “relativistic doublets.”

The relativistic correction was first applied by Sommerfeld to the explanation of the “fine structure” of the lines of the hydrogen spectrum. In Bohr’s original theory, which did not take into account the dependence of mass on velocity, the energies of the different stationary states of the atom depended only on the principal quantum number, according to the formula

\[ W_{n}=-\frac{Rh}{n^{2}}. \]

The latter is obtained from (12) with the aid of formulas (8) and (10), for \(Z=1\), if one introduces into them the Rydberg constant \(R\) (instead of the expression \(\frac{2\pi^{2}me^{4}}{h^{3}}\)). Thus, the frequencies of the various lines of the hydrogen spectrum were determined by the formula

\[ \nu=R\left(\frac{1}{n^{2}}-\frac{1}{n'^{2}}\right), \tag{21} \]

where \(n\) and \(n'\) are integers. When the relativistic correction is taken into account, the energy of different elliptical orbits with one and the same major semiaxis \(a_{n}=a_{0}n^{2}\) turns out to be different depending on their compression, i.e. on the ratio \(\frac{b}{a}=\frac{k}{n}\), according to formula (20). In view of this, instead of a single line corresponding to a change of the principal quantum number \(n'\to n\), there must be obtained a group of “satellite” lines with intervals

\[ \Delta\nu=\frac{A}{h}\left(\frac{1}{n^{3}k}-\frac{1}{n'^{3}k'}\right). \tag{22} \]

This formula, in connection with Sommerfeld’s “selection rule”

\[ k-k'=\pm 1 \tag{23} \]

reproduced exactly the fine structure of the hydrogen spectrum, if the angular quantum number was assigned integer values less than or equal to the principal one. These results were immediately extended to the spectrum of ionized helium \((Z=2)\), and then also to the X-ray spectra. In the latter case it was necessary only to reduce the number \(Z\) (the nuclear charge) by a certain number \(S\), characterizing the “screening” action on the electron under consideration of the other electrons situated in the inner groups or in the same electron group of the atom. The subdivision of the electrons in complex atoms into groups corresponds, as is known, to the successive values of the principal quantum number for their orbits. To the first group, or the \(K\) group, belong 2 electrons moving in one-quantum orbits. For them

\[ W=-Rh\,(Z-1)^2 . \tag{24} \]

The second group, or the \(L\) group, is formed by 8 electrons moving in two-quantum orbits. Their energy, if the relativistic correction is not taken into account, is equal to

\[ W=\frac{Rh\,(Z-S)^2}{2^2}, \tag{25} \]

where the screening constant \(S\) is close to 3. Taking into account that some of these electrons may move in elliptical orbits corresponding to \(k=1\) \(\left(\frac{b}{a}=\frac{1}{2}\right)\), and others in circular orbits \(k=n=2\), and taking account of the corresponding relativistic corrections, we come to the conclusion that the second group may be subdivided into two subgroups, whose energies differ from one another by the amount

\[ \Delta W=\frac{A\,(Z-S)^4}{2^3}\left(\frac{1}{1}-\frac{1}{2}\right)-\frac{1}{16}A\,(Z-S)^4 \tag{26} \]

This difference is found in the spectral interval \(\Delta \nu=\frac{\Delta W}{h}\) between those two lines which are emitted by the atom when an electron passes from the \(L\)-group to the vacant place of the \(K\)-group, in the form of the so-called \(K_{\alpha}\) doublet. And indeed, the width of this doublet for all atoms, from the lightest to uranium \((Z=92)\), is expressed exactly by Sommerfeld’s formula. In view of this circumstance the doublet just mentioned is called “relativistic.”

It would seem that in this circumstance one might see a brilliant confirmation of Sommerfeld’s theory. But at the same time there lies in it an exceedingly great difficulty for that theory. Namely, according to Som—

... to the Sommerfeld “selection principle,” expressed by formula (23), there appears to be a possible transition of the electron to a one-quantum orbit of group \(K\) only from circular orbits of group \(L\). Under this condition the quantum number decreases by one, whereas in the transition from elliptical orbits of group \(K\) (\(n=2,\ k=1\)) it ought to remain unchanged. The impossibility of such transitions, corresponding to the invariance of \(k\), was proved not only in application to the fine structure of the hydrogen spectrum, but also still more clearly in application to the optical spectra of the alkali metals, where different values of the angular quantum number for the final orbit of the valence electron correspond to different series of lines or, what is the same thing, to different, i.e. sufficiently separated, sequences of energy levels or “terms”1. Namely, the terms \(ns\), \(np\), \(nd\), etc. correspond to the values \(k=1\), \(k=2\), \(k=3\). In this case, in complete agreement with Sommerfeld’s rule, only the terms of neighboring sequences combine with one another.

§ 5. Comparison of X-ray spectra with optical spectra; “screened doublets.”

In the case of X-ray spectra, just as in the case of optical spectra, the primary importance belongs not to the lines but to the terms corresponding to the various energy levels of atoms. X-ray terms are readily detected directly in absorption spectra. In this way it was possible to establish that group \(L\) consists not of two subgroups, as we assumed above, but of three; further, that group \(M\) (\(n=3\)) consists not of three (corresponding to the three possible values of the angular quantum number \(k=1,\ 2,\ 3\)), but of five, and so on.

At first sight this circumstance appears extremely puzzling and quite inconsistent with anything. In reality, however, we have the very same thing in the simplest optical spectra, i.e. in the spectra of atoms or ions with one outer (valence) electron. Namely, all the spectral terms of the alkali metals, except for the terms of the \(s\) series, prove to be double. Thus, to the principal quantum number \(n=1\) there corresponds only a single term \(1s\); for \(n=2\) we have three terms, one with the number \(k=1\) (\(2s\)) and two very close terms with \(k=2\) (\(2p_1\) and \(2p_2\)). In exactly the same way, for \(n=3\) we have one term \(s\), 2 terms \(p\), and 2 terms \(d\)—five terms in all, just as in the X-ray group \(M\).

By analyzing the fine structure of the spectra of the alkali metals, Sommerfeld established that, when different terms combine, besides the condition \(k-k'=\pm 1\), one further condition must be satisfied—

...which can be formulated mathematically by assigning to the various terms, alongside the quantum numbers \(n\) and \(k\), a third quantum number \(j\), which he called internal. This internal quantum number serves, first of all, to distinguish the two components of each doublet \((n,k)\) from one another. If, moreover, for the component with the smaller energy (i.e. the larger numerical value of the term\(^1\)) one puts \(j=k-1\), and for the other component \(j=k\) (for \(k=1\) this component is the only one), then the above-mentioned additional condition is expressed as follows:

\[ j-j'=\pm 1 \ \text{or}\ 0. \tag{27} \]

This second selection rule of Sommerfeld, like the first, proves to be fully applicable also to X-ray spectra. The two terms of the group \(L\) which combine with the term \(K\), forming the doublet \(K_{\alpha}\) in the emission spectrum of characteristic X-rays, must, however, be assigned one and the same value of the angular quantum number, namely \(k=2\); as regards the number \(j\), for one of them it is equal to 2, and for the other to 1. The third term of the group \(L\) is characterized by the numbers \(k=1,\ j=1\) and cannot combine with \(K\).

Analogous relations also hold for the terms of the group \(M\); when an electron jumps from the group \(M\) to a vacant place in the group \(L\) or \(K\), the atom emits only those lines for which conditions (24) and (27) are fulfilled. The correspondence between the optical and X-ray terms is evident from the following table, where they are arranged in order of increasing energy (decreasing term).

Table 1.

	\(K_{1,1}\)	\(L_{1,1}\)	\(L_{2,1}\)	\(L_{2,2}\)	\(M_{1,1}\)	\(M_{2,1}\)	\(M_{2,2}\)	\(M_{3,2}\)	\(M_{3,3}\)
X-ray term	\(K_{1,1}\)	\(L_{1,1}\)	\(L_{2,1}\)	\(L_{2,2}\)	\(M_{1,1}\)	\(M_{2,1}\)	\(M_{2,2}\)	\(M_{3,2}\)	\(M_{3,3}\)
Optical term	\(1s_1\)	\(2s_1\)	\(2p_1\)	\(2p_2\)	\(3s_1\)	\(3p_1\)	\(3p_2\)	\(3d_1\)	\(3d_2\)
\(n\)	1	2	2	2	3	3	3	3	3
\(k\)	1	1	2	2	1	2	2	3	3
\(j\)	1	1	1	2	1	1	2	2	3

From the point of view of the systematics of spectral terms (i.e. from a purely “zoological” point of view) this result leaves nothing to be desired. However, from the physical point of view it represents something highly unsatisfactory and paradoxical. First, the physical meaning of the internal quantum number remains unclear. Secondly, the “relativistic correction,” calculated

\(^1\) The energies of the various stationary states of the atom are expressed by negative numbers.

for two neighboring values of the number \(k\), in fact corresponds to two neighboring values of the number \(j\), with one and the same \(k\). Finally, thirdly, it turns out that the interval between two terms corresponding to neighboring values of the number \(k\), with one and the same \(j\), for example, \(L_{1,1}\), \(L_{2,1}\), is completely independent of the atomic number—just as if, for the same values of \(Z\), \(n\), and \(k\), these terms differed from one another only by the values of the screening constant \(S\) in formula (25) and others like it. Were it not for the relativistic correction, the origin of these “screening doublets” would be directly explained by the difference in eccentricity of the electron orbits of the group under consideration. Thus, for example, in the case of the \(L\) group we have circular orbits with \(k=2\) and elliptical ones with \(k=1\). Since the latter pass much closer to the nucleus (situated at their focus) than the former, the screening action of the remaining electrons must be smaller for them than for the former. In other words, the quantity \(Z-S\) for \(L_{1,1}\) (\(k=1\)) must be larger than for the term \(L_{2,1}\) (\(k=2\))—which is indeed the case.

Entirely analogous relations were discovered by Millikan and Bowen in the case of optical terms. In doing so they compared the spectra of atoms or ions having one outer electron (for example, the spectrum of lithium, the spectrum of ionized beryllium, doubly ionized boron, etc.). In all cases they found the same two types of “doublets” that are observed in X-ray spectra, namely “relativistic” doublets, the distance between which is expressed by formulas of the form (26), but which correspond to one and the same value of the number \(k\) (i.e. \(p_{1}-p_{2}\), \(d_{1}-d_{2}\), etc.), and “screening” doublets, which correspond to neighboring values of the number \(k\), but the distance between which does not depend on the degree of ionization (i.e. on the effective charge of the nucleus).

§ 6. Explanation of doublets in X-ray and optical spectra.

All attempts to explain these glaring contradictions remained, until recently, completely fruitless. They, however, vanish instantly in the light of the new idea—the rotating electron. Indeed, we saw above that the additional energy \(U\), caused by the electron’s own rotation, i.e. by its magnetic moment, is expressed in almost the same way as the relativistic correction \(V\), i.e. the additional energy caused by the dependence of mass on velocity [cf. formulas (18) and (20)].

As a consequence of this circumstance, the relativistic correction may at least partly be masked (i.e. diminished) by the magnetic correction or, conversely, be simulated (i.e. increased) by it.

it). The former occurs in the case of “screened,” and the latter in the case of “relativistic” doublets.

Such is the essence of the explanation proposed by Uhlenbeck and Goudsmit. In order to make this explanation more precise, however, it proves necessary to make the following assumptions:

a. The axis of rotation, i.e. the mechanical or magnetic moment of the electron, can, for \(k>1\), assume only two opposite directions, perpendicular to the plane of the orbit. This corresponds to a splitting of terms characterized by one and the same value of the number \(k\) into pairs or doublets. The only exception is provided by the terms \(s\) (\(k=1\)), for which only one of the indicated directions is possible, namely that for which the intrinsic magnetic moment of the electron \(\vec{\mu}\) is directed in the same direction as the magnetic moment of its orbit \(\vec{M}\) (\(\cos\theta=+1\) in formula (18); for \(k>1\), \(\cos\theta=\pm1\)).

b. The coefficient \(s\) in formula (17) is equal to \(\frac{1}{2}\). This means that the intrinsic magnetic moment of the electron (\(\mu\)) is equal to one half of the Bohr magneton.

c. The angular quantum number takes not integral but fractional values

\[ k' = k-\frac{1}{2}=\frac{1}{2},\ \frac{3}{2},\ \frac{5}{2}\ldots, \]

so that

\[ I=\frac{hk'}{2\pi} \]

and

\[ M=\frac{e}{2cm_0}\,\frac{hk'}{2\pi}. \]

Let us denote the magnitude of the relativistic and magnetic correction for the above fractional values of the angular quantum number by \(V'\) and \(U'\). In the case of a nucleus with effective charge \((Z-S)e\), their sum is expressed by the formula

\[ V' + U' = -\frac{A(Z-S)^4}{n^3} \left( \frac{1}{k'} \mp \frac{s}{k'^2} \right), \tag{29} \]

where the upper sign corresponds to the same, and the lower to the opposite, direction of the vectors \(\vec{\mu}\) and \(\vec{M}\).

If the number \(k\) is very large in comparison with 1, then one may put, with accuracy up to quantities of the second order of smallness relative to \(\frac{1}{k}\),

\[ \frac{1}{k'}= \frac{1}{k-\frac{1}{2}} \simeq \frac{1}{k}+\frac{1}{2k^2} \]

and

\[ \frac{1}{k'} \overset{s}{\asymp} \frac{1}{k^2}. \]

We have, therefore,

\[ \frac{1}{k'} \mp \frac{s}{k^2}\overset{s}{\asymp}\frac{1}{k}+\frac{1}{k^2}\left(\frac{1}{2}\mp s\right). \tag{29'} \]

Thus, for \(s=\frac{1}{2}\), the sum \(V' + U'\) approximately coincides with the magnitude of the relativistic correction

\[ V=-\frac{A(Z-S)^4}{n^3 k}, \]

corresponding to the integer angular quantum number \(k=k'+\frac{1}{2}\) (the upper sign in formula 29′) or \(k-1=k'-\frac{1}{2}\) (the lower sign) \(^{1}\).

From a quantitative point of view this result is, of course, not entirely satisfactory, since the equality \(V' + U' = V\) holds only for large values of \(k'\). Here, however, the failure is explained not by the incompleteness of the mechanical model of the atom, but by the incorrectness of those mechanical principles which we applied to this model. If the calculation of the additional relativistic-magnetic energy of the electron is carried out by means of the new quantum mechanics, then, as Heisenberg and Jordan have shown, complete agreement between theory and experiment is obtained. At the same time, the requirement (c) of replacing integer values of the angular quantum number by half-integer ones also follows directly from the new mechanical theory. We give here only its result, which reduces to the following.

The relativistic correction for \(k=\frac{1}{2}\) is partially compensated by the magnetic correction, so that their sum coincides with the value of \(V\) for \(k=1\).

For \(k=\frac{3}{2}\) we have two opposite directions of the magnetic moment of the electron. One of them—the one opposite to the magnetic moment of the orbit—gives for the sum \(V+U\) a value equal to the value of \(V\) for \(k=1\); the other corresponds to the relativistic correction for \(k=2\). Analogous relations are obtained for \(k=\frac{5}{2}, \frac{7}{2}\), etc. They are represented graphically in the following scheme (borrowed by me from the article by Uhlenbeck and Goudsmit \(^{2}\)):

\(^{1}\) For

\[ \frac{1}{k-1}\overset{1}{\asymp}\frac{1}{k}+\frac{1}{k^2} \]

\(^{2}\) Nature, 20 February 1926.

After all that has been said, this scheme requires no explanation. It gives an exhaustive quantitative explanation of the structure of optical and X-ray spectra (if one takes into account the difference in the screening constants for orbits corresponding to different values of \(k'\)). At the same time it determines the physical meaning of Sommerfeld’s “inner” quantum number \(j\). Namely, the latter characterizes the total magnetic moment of the atom, since this moment depends on the electron under consideration, i.e. on the geometrical or, if one wishes, algebraic sum of both moments of the electron \(\vec{\mu}\) and \(\vec{M}\).

Putting

\[ M=\frac{e}{2cm}\frac{h}{2\pi}k' \]

and

\[ \mu=\pm\frac{e}{2cm}\frac{h}{4\pi}, \]

we obtain

\[ M+\mu=\frac{e}{2cm}\frac{h}{2\pi}j, \tag{30} \]

where

\[ j=k'\pm\frac{1}{2} \tag{31} \]

(cf. Table 1; \(k'=k-1\)).

Let us note that the preceding results still give no grounds for judging the magnitude of the mechanical moment of the electron \(i\).

If we assume that the ratio \(\dfrac{\mu}{i}\) coincides with the ratio \(\dfrac{M}{I}\), i.e. that

\[ i=\frac{1}{2}\frac{h}{2\pi}, \]

then for the total mechanical moment of the atom we obtain the formula

\[ I+i=\frac{h}{2\pi}j, \tag{32} \]

completely analogous to (30). The study of the magnetic properties of atoms shows, however, that in reality the ratio \(\dfrac{\mu}{i}\) is twice as large as \(\dfrac{M}{I}\).

§ 7. The Zeeman Effect.

The intrinsic magnetic moment of the electron, like the magnetic moment of its orbit, which manifests itself indirectly in the absence of an external magnetic field, must evidently manifest itself directly in the presence of such a field. And indeed, it is observed very distinctly in the anomalous Zeeman effect. Until recently, however, it remained, so to speak, unrecognized. Namely, the additional magnetic moment of the atom, whose existence had been inferred by Pauli and Landé from an analysis of the magnetic splitting of spectral terms, was attributed by them not to the “light-emitting electron” itself, but to that “atomic core” (Atomrumpf) around which this electron revolves and which, with respect to it, plays the role of a positive nucleus. In doing so, it proved necessary to assume that, contrary to the basic requirements of electrodynamics, the ratio of the magnetic moment of the atomic core to its mechanical moment is twice as large as for the valence electron—or, more precisely, for the orbit of the latter. This circumstance followed from the following considerations.

According to Larmor’s theorem, a system of electrons revolving in an entirely arbitrary manner about some fixed center (nucleus) acquires, in a homogeneous magnetic field of not too great intensity \(\vec H\), an additional precessional motion, consisting in a uniform rotation of the entire system as a whole about an axis parallel to \(\vec H\), with angular velocity

\[ \vec\Omega=\frac{e}{2cm}\vec H . \tag{33} \]

The quantity \(-\dfrac{e}{2cm}\) is nothing other than the ratio of the magnetic moment of each electron (more precisely, of the electronic orbit) \(\vec M\) to the mechanical moment \(\vec I\). Thus formula (33) is equivalent to the following:

\[ \vec I\cdot \vec\Omega=-\vec M\cdot \vec H . \tag{34} \]

The right-hand side of this equality is nothing other than the magnetic energy of the electron under consideration. As for the left-hand side, it is, as is easily verified, equal to the additional kinetic energy of the electron caused by the precessional rotation of its orbit. Indeed, denoting the linear vel—

the electron in its normal motion by \(\vec v\), and in the perturbed motion by \(\vec v'\), we have

\[ \Delta {1\over 2}m\vec v^{\,2} = {1\over 2}m\vec v'^{\,2}-{1\over 2}m\vec v^{\,2} = {1\over 2}m\left(\vec v+\vec v'\right)\cdot\left(\vec v'-\vec v\right) = \]

\[ = {1\over 2}m\left(\vec v+\vec v'\right)\cdot\left(\vec\Omega\times\vec r\right). \]

If \(\vec H\) is not too large [and this assumption underlies formula (33), which ceases to be valid for extremely strong fields], then in this formula one may replace \(\dfrac{\vec v+\vec v'}{2}\) by \(\vec v\), which gives

\[ \Delta {1\over 2}mv^2 = m\vec v\cdot\left(\vec\Omega\times\vec r\right) = \vec\Omega\cdot\left(\vec r\times m\vec v\right) = \vec\Omega\cdot\vec I, \]

which is the definition of \(\vec I\).

Thus formula (34) expresses the essentially self-evident fact that the magnetic energy of the electron is nothing other than the additional energy of its perturbed motion (the potential energy does not change under precession).

Denoting the magnetic moment of the “atomic core” by \(\vec\mu\), and the mechanical moment by \(\vec i\), we can evidently write

\[ \vec i\cdot\vec\omega=-\vec\mu\cdot\vec H, \tag{35} \]

where

\[ \vec\omega=-{\mu\over i}\vec H \tag{36} \]

is the angular velocity of precession of this core. Since the latter consists of electrons entirely similar to the valence “light-emitting” electron, we ought to have expected that \(\dfrac{\mu}{i}=\dfrac{M}{I}\), i.e. \(\omega=\Omega\).

In reality, however, from the analysis of the complex or “anomalous” Zeeman effect it follows that

\[ \omega=2\Omega, \tag{37} \]

whence we must conclude that

\[ {\mu\over i}=2{M\over I}. \tag{38} \]

Without going into the details of this analysis, for which we are indebted chiefly to Lande, let us briefly recall only its most important results.

In the case of the simple or “normal” Zeeman effect, each energy level is split into several magnetic levels, according to the formula

\[ \Delta W=\frac{h}{2\pi}\Omega\cdot m, \tag{39} \]

where \(m\) is the so-called magnetic quantum number, determining the angle \(\theta\) formed by the moment of the atom (or electron) with the direction \(\vec H\), according to the formula

\[ \cos\theta=\frac{m}{k}; \tag{40} \]

the number \(m\) takes all integral values in the range from \(-k\) to \(+k\), if by \(k\) one understands the ordinary “angular” quantum number \((k=1,2,3\ldots)\), which corresponds to the presumed absence of an intrinsic magnetic moment in the electron. In this case only those levels combine with one another for which

\[ m-m'=\pm 1 \quad \text{or} \quad 0. \tag{41} \]

As a result of such combinations one obtains the normal “Zeeman triplets,” consisting of the undisplaced spectral line of frequency \(\nu\) and of its two components of frequencies

\[ \nu+\frac{\Omega}{2\pi} \quad \text{and} \quad \nu-\frac{\Omega}{2\pi} \]

(\(\frac{\Omega}{2\pi}\) is the frequency of the Larmor precession).

Such a “normal Zeeman effect” is observed only in the so-called singlet terms, which have no fine structure, and also in hydrogen, if one does not take into account the fine structure of its spectral lines.

In the case of the alkali metals and, in general, of more complex systems with one outer electron, which in the absence of a magnetic field possess doublet terms (i.e. double energy levels), the magnetic decomposition of these terms is determined, in comparatively weak fields, by the following formula

\[ \Delta W=\frac{h}{2\pi}g\Omega m'. \tag{42} \]

Here the magnetic quantum number \(m'\) assumes not integral, but half-integral (or semi-integral) values \(\pm \frac{1}{2}, \pm \frac{3}{2}\) within the limits of ...

await \(j-\dfrac{1}{2}\) and \(j+\left(l-\dfrac{1}{2}\right)\). As for the multiplier \(g\), its values lie between 1 and 2, being determined as a function of \(j\) and \(k'\left(=k-\dfrac{1}{2}\right)\) by the following formula

\[ g=1+\frac{\left(j+\dfrac{1}{2}\right)\left(j-\dfrac{1}{2}\right)+1-k'^2}{2\left(j+\dfrac{1}{2}\right)\left(j-\dfrac{1}{2}\right)}, \]

which was first established purely empirically by Landé.

The latter proposed that the product \(g\Omega\) be regarded as the “anomalous” precession velocity of the valence electron together with the atomic core. The origin of such a precession was explained by Pauli as the result of a kind of compromise between the electron, or more precisely the electronic orbit, which tends to precess with velocity \(\Omega\), and the “core,” which—for an unknown reason—tends to precess with velocity \(2\Omega\). This compromise solution is realized, however, only as long as the coupling between the electron and the atomic core, determined by their mutual magnetic energy \(U_i\), remains sufficiently strong in comparison with the action which they experience from the external field and which is determined by the corresponding “external” energies \(-\vec{\mu}\cdot\vec{H}\) and \(-\vec{M}\cdot\vec{H}\). As for the magnitude of the energy \(U_i\), Landé and Pauli showed that it may be regarded as proportional to the cosine of the angle between the vectors \(\vec{\mu}\) and \(\vec{M}\), and inversely proportional to the cube of the radius of the electronic orbit.

With a gradual strengthening of the external field there occurs a peculiar change in the type of the Zeeman phenomenon, which ends with its transformation into the simplest “normal” type, determined by formula (39). This effect, discovered by Paschen and Back, is explained from the point of view of Landé and Pauli very simply, namely, by the breaking of the bond between the valence electron and the atomic core, which in a very strong external field precess independently of one another with the velocities characteristic of each of them,

\[ \Omega=\frac{e}{2cm}H \]

and

\[ 2\Omega=\frac{e}{cm}H. \]

In this case the magnetic energy of the atom reduces to the sum of the corresponding “external” energies \(-\vec{\mu}\cdot\vec{H}\) and \(-\vec{M}\cdot\vec{H}\), i.e.

\[ \Delta W=-(\vec{\mu}+\vec{M})\cdot\vec{H}=+(j+2i)\cdot\Omega. \]

§ 8. Explanation of the complex Zeeman effect; properties of the rotating electron.

The results set forth in the preceding section acquire a new and, moreover, much simpler and clearer meaning from the point of view of the theory of the rotating electron.

The additional magnetic moment of the atom, revealed in the complex Zeeman effect, must merely be ascribed not to the core of the atom, but to the valence electron itself.

At once there disappears the glaring contradiction with the fundamental laws of electrodynamics which arises in the case when the “anomalous” precessional velocity \(\omega = 2\Omega\) is ascribed to the “core” of the atom, i.e. to the system of electrons, each of which has a “normal” precessional velocity \(\Omega\). From the new point of view the core of the atom, i.e. the electron shell of a monovalent ion of an alkali metal, has neither mechanical nor magnetic moment, which is fully in accord with its similarity to the electron shell of the neighboring “noble” element, i.e. an inert gas: the latter, as is known, are diamagnetic. The role of the core reduces merely to screening the nucleus, i.e. to diminishing its effective charge.

The validity of this interpretation of the complex Zeeman effect is made most evident by the remarkable fact that also in the case of hydrogen atoms, whose “core” consists of a single “bare” nucleus (proton), a phenomenon is observed analogous or, more likely, identical to the Paschen–Back effect. In strong magnetic fields the “fine structure” of the hydrogen lines is blurred and completely disappears, giving way to normal Zeeman triplets. This phenomenon, discovered by Erochin, has until recently been subjected to strong, though wholly unfounded, doubt. From the point of view of the former theory (Sommerfeld’s), which explained the “fine structure” of the hydrogen terms by a relativistic correction alone, it was, of course, quite incomprehensible. From the new point of view—Uhlenbeck and Goudsmit’s—it is self-evident. Moreover, it should, it seems to me, be regarded as the simplest and fundamental experimental justification of the theory of the rotating electron.

The fact that the ratio \(\frac{\mu}{i}\) is twice as large as the corresponding ratio for electronic orbits presents no fundamental difficulties for this theory. Moreover, it follows directly from the general laws of electrodynamics if the electron is treated as a ball with a uniformly distributed surface charge.

In fact, the magnetic moment of such a small sphere is expressed by the formula

\[ \vec{\mu}=-\frac{ea^{2}}{3c}\,\vec{o}, \tag{45} \]

where \(-e\) is its charge, \(a\) its radius, and \(\vec{o}\) the angular velocity.

This formula is obtained from the definition of \(\vec{\mu}\) as the geometrical sum of the moments of the various elements of the electron—\(de=-\eta ds\) \(\left(-\eta=-\frac{e}{4\pi a^{2}}\right.\) is the surface charge density, \(ds\) is an element of the surface) with respect to its center. Denoting the radius-vector of the element under consideration by \(r\), we have:

\[ d\vec{\mu}=-\frac{de}{2c}\,\vec{r}\times(\vec{o}\times\vec{r}) =\frac{de}{2c}\left\{\vec{o}\,r^{2}-\vec{r}(\vec{o}\cdot\vec{r})\right\} \]

and, consequently,

\[ \vec{\mu}=-\frac{e}{2c}\left[\vec{o}\,a^{2}-\overline{\vec{a}(\vec{o}\cdot\vec{a})}\right], \]

where the bar denotes the mean over all possible directions of the radius-vector \(\vec{a}\). From considerations of symmetry it is clear that this mean is a vector parallel to \(\vec{o}\) and, consequently, coincides with the mean value of its projection on the direction \(\vec{o}\), i.e. with \(oa^{2}\overline{\cos^{2}\theta}\), where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{o}\). Noting that \(\overline{\cos^{2}\theta}=1/3\), we arrive at formula (45).

As for the mechanical moment, i.e. the moment of momentum of the rotating electron, it is determined by means of the concept of electromagnetic momentum, as a quantity distributed in space with volume density \(\vec{G}=\frac{1}{4\pi c}\vec{E}\times\vec{H}\), where \(\vec{E}\) is the electric, and \(\vec{H}\) the magnetic field strength. Inside the spherical electron, \(E\), and consequently also \(G\), vanish. Outside, \(\vec{E}\) coincides with the field of a point charge \(e\), concentrated at its center, and the magnetic field with the field of an elementary magnet with moment \(\mu\), likewise concentrated at the center of the electron. We therefore have

\[ \vec{E}=\frac{e\vec{r}_{o}}{r^{2}} \quad\text{and}\quad \vec{H}=\frac{3\vec{r}_{o}(\vec{\mu}\cdot\vec{r}_{o})-\vec{\mu}}{r^{3}}, \]

where \(\vec{r}_{o}=\frac{\vec{r}}{r}\) is the unit vector characterizing the direction of \(\vec{r}\). Thus

\[ \vec{G}=-\frac{e}{4\pi c r^{5}}\,\vec{\mu}\times\vec{r}_{o}. \]

The angular momentum is obtained from this by the formula

\[ \vec i=\int_{r>a} \vec r \times \vec G\,dV, \]

where \(dV\) is an element of volume of the external space. Taking into account that the mean value of the vector \(\vec r_0\times(\vec\mu\times\vec r_0)\) for all possible directions of \(\vec r_0\) is equal, according to the preceding, to \(\frac{2}{3}\vec\mu\), we obtain

\[ \vec i=-\frac{2}{3}\frac{e\vec\mu}{4\pi c}\int_a^\infty \frac{4\pi r^2}{r^4}\,dr =-\frac{2}{3}\frac{e\vec\mu}{ca}. \]

Since the electromagnetic mass of a spherical electron is connected with its radius by the formula

\[ m=\frac{2}{3}\frac{e^2}{c^2a}, \tag{46} \]

we finally arrive at the formula

\[ \mu=-\frac{e}{cm}\vec i, \tag{47} \]

i.e. the ratio \(\frac{\mu}{i}\) turns out to be twice as large as the ratio \(\frac{M}{I}\), in complete agreement with the experimental facts.

One should not, however, attach too much significance to this agreement. First of all, it holds only in the case of a spherical electron and disappears if we assume that the charge of the electron is uniformly distributed throughout its volume. Further, putting \(m\simeq 9\cdot10^{-28}\), \(e\simeq 4.7\cdot10^{-10}\), and \(\mu\simeq 10^{-20}\) (the Bohr magneton), we obtain for the radius of the electron the known value \(a\simeq 2\cdot10^{-13}\ \mathrm{cm}\), and for its angular velocity [from (45)] \(\omega\simeq 10^{25}\). Thus the linear velocity of the elements of the electron at its equator, \(a\omega\), turns out to be approximately \(10^{12}\ \mathrm{cm/sec}\), i.e. almost one hundred times greater than the velocity of light \(c\). These difficulties are not, of course, serious. We arrive, however, at a serious difficulty—more precisely, a contradiction—if we take into account that additional mass of the electron which it must have by virtue of its rotation. Without entering into exact calculations, let us note that this mass is equal (at least approximately) to the energy of the magnetic field it creates,

\[ \frac{1}{8\pi}\int H^2\,dV, \]

divided by \(c^2\). In order of magnitude it is therefore determined by the expression

\[ \frac{\mu^2}{a^3c^2}. \]

Substituting here the preceding values—

tion for \(\mu\) and \(a\), we obtain the quantity \(m' \lesssim 10^{-21}\), which is a million times greater than the fundamental electromagnetic mass of the electron determined by formula (46). This formula, however, serves not to determine the mass of the electron (which is known from experiment), but to calculate its radius. One might imagine that the mass of the electron depends in reality not on its electric field, but on its magnetic field. In that case, for determining its radius we would have the formula

\[ m \lesssim \frac{\mu^2}{a^3 c^2}, \tag{48} \]

from which one obtains \(a \lesssim 10^{-11}\,\mathrm{cm}\)—a figure that is clearly absurd.

§ 9. THE ELECTRIC MOMENT OF THE MAGNETIC ELECTRON AND THE RELATIVISTIC THEORY OF ITS MOTION.

Without entering into hypotheses about the structure of the electron, we shall regard relation (47) as an experimental fact following from the analysis of the complex Zeeman effect. On the other hand, the analysis of spectra in the absence of a magnetic field has led us to the conclusion that the intrinsic magnetic moment of the electron is equal to one half of the Bohr magneton. Hence it would seem to follow necessarily that the mechanical moment of the electron \(i\) is not one half of the elementary quantity \(\dfrac{h}{2\pi}\), as was supposed at the end of § 6, but only one quarter of it.

This result in itself presents nothing unusual. However, a number of considerations indicate that in reality \(i=\dfrac{1}{2}\dfrac{h}{2\pi}\) and, consequently, \(\mu\) is equal to an entire Bohr magneton.

These considerations are based on investigating that change which the mechanical moments of the electron \(\vec{i}\) and \(\vec{J}\) undergo in the absence of an external magnetic field. In this case their geometric sum \(\vec{i}+\vec{J}\) must remain constant, i.e. we must have the equality

\[ \frac{d\vec{i}}{dt}+\frac{d\vec{J}}{dt}=0. \tag{49} \]

The derivatives \(\dfrac{d\vec{i}}{dt}\) and \(\dfrac{d\vec{J}}{dt}\) are, evidently, equal to the moments of the couples \(\vec{q}\) and \(\vec{Q}\), tending to turn the axis of the electron and the axis of the orbit and causing them to precess with one and the same velocity about the unchanged direction \(\vec{i}+\vec{J}\).

Considering the electron in that coordinate system \(S\), in which it is instantaneously at rest, we would obtain for the torque of the couple acting on it the formula

\[ \vec q=\vec \mu \times \vec H', \]

and for the corresponding additional force

\[ \vec f=(\vec \mu\nabla)\vec H', \]

where \(\vec H'\) is determined by formula (5). It is not difficult to see that the moment of this force

\[ \vec r\times \vec f=\vec Q \]

has little in common with \(\vec q\), and that the sum \(\vec q+\vec Q\) is by no means equal to zero, as would follow from (49).

The resolution of this contradiction proves possible only from the point of view of the theory of relativity. The latter makes it possible to consider the motion of the electron with respect to the coordinate system \(S\), connected with the nucleus at rest. In this system the latter, obviously, produces no magnetic field. The additional action which it exerts on the electron must, therefore, be determined by some additional electric property of the electron, distinct from its electric charge and connected with its magnetic moment in approximately the same way as the magnetic field \(\vec H'\) in the system \(S'\) is connected with the electric field of the nucleus \(\vec E\) in the system \(S\).

It is easy to convince oneself that this property is the electric or dipole moment \(\vec p\), which must be found in the electron from the point of view of any system relative to which it is moving.

We are accustomed to think that a magnetic or electric moment represents invariant properties of material particles, i.e. that the magnitude of these moments does not depend on the choice of the coordinate system to which the motion of these particles is referred. This notion is fundamentally erroneous. The electric and magnetic moments are two properties connected with one another and capable, so to speak, of passing into one another, exactly as takes place in the case of the electric and magnetic fields.

According to the theory of relativity, the three-dimensional vectors \(\vec H\) and \(\vec E\) form a four-dimensional tensor of the electromagnetic field with 16 compo-

components \(H_{\alpha\beta}=-H_{\beta\alpha}\;(\alpha,\beta=1,2,3,4)\), defined through the components of the vectors \(\vec H\) and \(\vec E\) by the formulas

\[ H_{23}=H_1,\quad H_{31}=H_2,\quad H_{12}=H_3;\quad H_{14}=-iE_1,\quad H_{24}=-iE_2, \]
\[ H_{34}=-iE_3, \tag{50} \]

where \(i=\sqrt{-1}\) \((H_{\alpha\alpha}=0)\).

In an analogous way, the vectors \(\mu\) and \(p\) jointly determine the antisymmetric tensor of the electromagnetic moment \(\mu_{\alpha\beta}=-\mu_{\beta\alpha}\), where

\[ \mu_{23}=\mu_1,\quad \mu_{31}=\mu_2,\quad \mu_{12}=\mu_3;\quad \mu_{14}=+ip_1,\quad \mu_{24}=+ip_2,\quad \mu_{34}=+ip_3 \tag{50'} \]

For what follows, we need not go more deeply into the investigation of this question. We need only establish those formulas by which the vectors \(\vec\mu\) and \(\vec p\) are transformed in passing from one coordinate system \((S')\) to another \((S)\). These formulas are identical with the corresponding formulas for \(\vec H\) and \(-\vec E\). If in \(S\), \(H=0\), then, in the first approximation, for smallness of the velocity \(v\) in comparison with the velocity \(c\), we must have for the system \(S'\), which moves with respect to \(S\) with velocity \(\vec v\),

\[ \vec H'=-\frac{\vec v}{c}\times \vec E \tag{51} \]

and \(\vec E'\simeq \vec E\). The preceding formula is identical with formula (5), if one takes into account that in the case under consideration

\[ \vec E=Ze\frac{\vec r}{r^3}. \]

Let us now imagine that in the system \(S\) the electric field \(E\) is equal to zero. Then in the system \(S'\) we would have

\[ \vec E'=+\frac{\vec v}{c}\times \vec H \quad\text{and}\quad \vec H'\simeq \vec H. \]

From the point of view of the system \(S'\), where the electron is at rest, it has a definite magnetic moment \(\vec\mu'\). We shall suppose that in this system it has no electric moment, i.e. that \(p'=0\). Passing to the system \(S\), we then obtain \(\mu\simeq \mu'\) and

\[ \vec p=\frac{\vec v}{c}\times\vec\mu. \tag{52} \]

Thus, with respect to the coordinate system in which the positive nucleus is at rest, the electron behaves as an electric dipole

with moment (52). Let us note that the latter is perpendicular to the velocity of the electron. If the vector \(\vec{\mu}\) is parallel to the axis of the orbit, i.e. to its magnetic moment \(\vec{M}\), then the vector \(\vec{p}\) is parallel to the radius vector \(\vec{r}\). The question of the origin of the vector \(\vec{p}\) is of no essential importance for us. Assuming that the magnetic moment of the electron is due to its rotation, one may reduce the electric moment \(\vec{p}\) to a definite redistribution of the electron charge, obtained in passing from the system \(S'\), where this distribution has radial symmetry, to the system \(S^1\).)

An electric dipole with moment \(\vec{p}\) has, in an external electric field \(\vec{E}\), the potential energy

\[ U=-\vec{p}\cdot\vec{E}. \tag{53} \]

Substituting here the preceding value of \(\vec{p}\) and \(\vec{E}=Ze\,\dfrac{\vec{r}}{r^3}\), we obtain

\[ U=-\frac{Ze}{r^3}\left(\frac{\vec{v}}{c}\times\vec{\mu}\right)\vec{r}, \]

or, in view of the identity \(\vec{r}\cdot(\vec{v}\times\vec{\mu})=\vec{\mu}(\vec{r}\times\vec{v})\),

\[ U=-\frac{Ze}{cr^3}(\vec{r}\times\vec{v})\vec{\mu} =2Z'\frac{\vec{M}\cdot\vec{\mu}}{r^3}, \]

i.e. the formula (8) established above.

The difference between the old and the new method of treating the motion of the electron is obtained in calculating the couple and the force acting on it. The moment of the former, from the new point of view, is equal to

\[ \vec{q}=\vec{p}\times\vec{E}, \tag{54} \]

i.e., consequently, it differs from the previous expression \(\vec{\mu}\times\vec{H}'\). In fact, with the aid of formula (51) the latter is reduced to the form

\[ \vec{q}=\vec{\mu}\times\left(\vec{E}\times\frac{\vec{v}}{c}\right), \]

whereas according to (52) and (54)

\[ \vec{q}=\left(\frac{\vec{v}}{c}\times\vec{\mu}\right)\times\vec{E}. \]

For the force \(\vec{f}\) we further have

\[ \vec{f}=(\vec{p}\nabla)\vec{E} =\frac{Ze}{r^3}(\vec{p}\nabla)\vec{r} +Ze\vec{r}\left(\vec{p}\cdot\nabla\frac{1}{r^3}\right), \]

\(^1\) See my Lehrbuch der Elektrodynamik 1, pp. 294–296 (Springer, 1926).

with respect to the magnetic moment of atoms. Thus, for example, the magnetic moment of the hydrogen atom (as well as of other more complex atoms with one external electron), measured by the method of Stern and Gerlach, is equal to one Bohr magneton, whereas according to the preceding it should have been equal to \(3/2\) of this unit. Analogous difficulties arise in the interpretation of the Zeeman phenomenon in strong fields, i.e., when reducing formula (44) to the form (39) with integral \(m\).

These difficulties disappear if one passes from classical mechanics to quantum mechanics.

We do not, however, have the opportunity here to enter into a consideration of this question.

1. Let us recall that by a spectral “term” is meant the absolute magnitude of an energy level divided by \(h\). ↩