THE MODERN THEORY OF MAGNETISM

S. V. Vonsovskii

CONTENTS

Introduction

I. Atomic magnetism. 1. The intrinsic (spin) magnetic moment of the electron. 2. Magnetism of the electron shells of the atom. 3. Magnetism of nucleons (protons and neutrons) and atomic nuclei.

II. Magnetism of matter—weakly magnetic bodies. 4. Phenomenological description of the magnetic properties of matter. 5. Classification of magnetics according to the principal experimental data. 6. Diamagnetism. 7. Magnetic properties of superconductors. 8. Paramagnetism. 9. Magnetic cooling.

III. Magnetism of matter—ferromagnetism. 10. Phenomenological description of the ferromagnetic state of matter. 11. Theory of the spontaneous magnetization of ferromagnets. 12. Theory of the technical magnetization curve. 13. Ferromagnets in alternating magnetic fields. 14. Ferromagnetic materials (soft and hard). 15. Influence of spontaneous magnetization on other properties of ferromagnets. 16. Conclusion.

INTRODUCTION

One of the essential problems of modern physics is the study of the magnetic properties of matter. The importance of this branch of physical science is determined by the fact that, first, all material bodies of the nature surrounding us are, to one degree or another, magnetic and, second, magnetic forces manifest themselves in the interaction of the elementary particles of matter. The range of magnetic phenomena is truly enormous; it extends from the magnetism of the microworld of atomic particles to the magnetism of cosmic bodies.

Man encountered the phenomenon of magnetism already in remote antiquity; even then the discovery of this remarkable property of matter found its first important practical application in the form of the compass.

The unusual universality of the magnetic properties of matter quite naturally gave rise to the desire to understand the physical nature of this phenomenon. The paths by which physical conceptions of the nature of magnetism developed lead from naive hypotheses about the existence in bodies of special magnetic fluids to the idea of molecular magnets.

The discovery, at the beginning of the last century, of the magnetic field of an electric current made it possible at once to put forward the supposition that electric molecular currents are the cause determining the magnetic properties of matter. At that time, however, the atomistic nature of electricity and magnetism had not yet been discovered by direct experiment, and therefore the idea of molecular currents long remained a hypothesis. It found its definitive experimental foundation only with the development of modern atomic physics. One of the essential achievements of the latter was the establishment of the fact that the elementary carriers of magnetism are, first, the elementary particles of matter themselves—electrons, protons, neutrons—which possess their own so-called spin magnetic moment and, second, microscopic currents formed by the motion of electrons in the shell of the atom (orbital magnetic moments). These spin and orbital magnetic moments are precisely those molecular currents which physicists had intuitively guessed at as early as the beginning of the last century. The successes of atomic physics made it possible to create a coherent and unified theoretical conception encompassing all types of magnetic phenomena in matter, and to establish the connection of these phenomena with the electrical, optical, thermal, mechanical, and other properties of matter.

The modern doctrine of magnetism can naturally be divided into two parts: atomic magnetism, i.e. the magnetic properties of isolated elementary particles of matter, atomic nuclei, as well as the electron shells of isolated atoms and molecules, and the magnetism of collectives of atoms and molecules interacting with one another, forming the gaseous, liquid, and solid bodies of the natural world around us, i.e. the magnetism of matter. In the first case we are interested in the magnetic properties of individual elementary carriers of magnetic moment isolated from one another, whereas in the second case the principal interest is the influence of the interaction between these carriers of magnetic moments on their collective magnetic properties.

In the present survey we do not consider the history of the development of the doctrine of magnetism. Here only a brief account is given of the current state of the science of the magnetic properties of matter in a form accessible also to physicists who are not magnetologists. Nevertheless, it should be recalled that in the development of the doctrine of magnetism Russian scientists, both in the pre-revolutionary and in the Soviet periods, have played and continue to play a very large, and in many questions a leading, role. The remarkable works of the Russian scientist A. G. Stoletov on ferromagnetism initiated the modern study of the magnetic properties of ferromagnetic bodies. A major role was played and continues to be played by the works of V. K. Arkadiev, who as early as 1913 gave a generalization of the laws of electrodynamics for ferromagnetic media. The many years of research by V. K. Arkadiev on magnetodynamics have recently acquired

...topical interest in connection with questions of atomic and nuclear magnetism. Of extraordinarily great importance for the development of the doctrine of magnetism were the works of N. S. Akulov, Ya. G. Dorfman, P. L. Kapitsa, I. K. Kikoin, E. I. Kondorskii, L. D. Landau, I. E. Tamm, Ya. I. Frenkel, Ya. S. Shur, R. I. Yanus, and their numerous pupils, as well as of many other Soviet physicists.

We consider it appropriate to note here the fact of the plainly non-objective and grossly tendentious complete suppression of the works of Russian scientists on magnetism that the American physicist Bozorth allowed in his review, published in Reviews of Modern Physics in 1947.

Lack of objectivity is alien to the spirit of Soviet science; but Soviet scientists are also characterized by a high Soviet patriotism, which compels them, in response to the crude attacks of imperialist propaganda in the field of science, to introduce complete and objective clarity, showing, on the concrete material of science itself and without any suppression of the role of foreign scientists, the great and, in many areas, leading role of Soviet physicists in the development of the doctrine of magnetism.

ATOMIC MAGNETISM

1. INTRINSIC (SPIN) MAGNETIC MOMENT OF THE ELECTRON

Among the elementary particles known at the present time, the electron, discovered as early as the end of the last century, has been studied in the greatest detail. For a long time it was believed that the principal characteristics of this particle were its charge and mass, and that magnetic properties were manifested only in the motion of electrons in conductors, in cathode rays, or in the “orbits” of atoms and molecules. However, in the study of the line optical spectra of various atoms, the so-called fine structure of spectral lines¹ was discovered; explanation of its nature made it necessary to refine earlier conceptions of the basic characteristics of the electron. The simplest form of fine structure of spectral lines occurs in the case of hydrogen-like atoms with one valence electron, where it reduces to a double splitting of the lines (doublets). The fact of the appearance of doublets and the magnitude of their splitting proved easy to explain if one assumes that the electron possesses an intrinsic mechanical angular momentum (spin). At the same time it also had to be assumed that this spin angular momentum can have only two orientations, and, moreover, such that with respect to some external field its two possible projections onto the direction of this field are equal in magnitude and opposite in sign, namely:

\[ s^z = \pm \frac{\hbar}{2} \tag{1.1} \]

\(2\pi\hbar = h = 6.624\cdot 10^{-27}\ \mathrm{erg\cdot sec}\) is Planck’s constant). In addition, to the electron spin there corresponds a magnetic moment, the two possible projections of which on the direction of an external magnetic field turned out to be equal to

\[ \mu_s^z = \pm \frac{e\hbar}{2mc} = \pm \frac{e}{mc}\,s^z \tag{1.2} \]

(\(e = 4.8025\cdot 10^{-10}\ \mathrm{CGSE}\) is the charge and \(m = 9.1066\cdot 10^{-28}\ \mathrm{g}\) is the rest mass of the electron, while \(c = 3\cdot 10^{10}\ \mathrm{cm\cdot sec^{-1}}\) is the speed of light), i.e., in absolute value they are equal to the Bohr magneton—the natural atomic unit of magnetic moment:

\[ \mu_B = \frac{e\hbar}{2mc} = 0.9273\cdot 10^{-20}\ \mathrm{erg\cdot gauss^{-1}} . \tag{1.2¹} \]

At first these new fundamental properties of the electron were given a very visual “classical” interpretation. Namely, it was assumed that the electron—a “little ball”—rotates about an axis passing through its center, and that its charge in this rotation creates an intrinsic current, which in turn accounts for the appearance of a magnetic moment. The negative sign of the electron charge determines the antiparallel orientation of the spin and its magnetic moment*).

However, when attempts were made to give such a “classical” explanation of the nature of electron spin, substantial difficulties of principle arose. First, in order to calculate the magnetic moment according to the laws of classical electrodynamics, it was necessary to adopt some definite notions about the structure of the electron (shape, dimensions, charge distribution, etc.). Second, in the classical theory it remained completely incomprehensible why the spin can have only two different orientations (spatial quantization of the spin moment). And, finally, the theory could not give a reasonable explanation of the so-called gyromagnetic anomaly of spin. This anomaly consists in the fact that, as is seen from (1.2), the ratio of the spin magnetic moment to the mechanical moment is equal to

\[ g_s = \frac{\mu_s^z}{s^z} = \frac{e}{mc}, \tag{1.3} \]

whereas according to the classical laws this ratio should be half as large [see below formula (3.4)], i.e., \(\frac{e}{2mc}\). It should be recalled that a similar anomaly was observed in the measurement of the gyromagnetic effect in ferromagnets (see below); it also had to be admitted in order to explain the regularities of the anomalous Zeeman effect (see § 2). These difficulties indicated the inadequacy of the classical interpretation of the nature of the elec-

* A rigorous classical electrodynamics of the “rotating” electron was developed by Ya. I. Frenkel² (1926), and also by I. E. Tamm³.

…of the electron spin and at first forced one simply to regard the existence of spin and of its magnetic moment as an experimental fact that had not yet found an adequate theoretical explanation; moreover, spin was considered as a fourth “internal” degree of freedom of the electron.

Among the vivid and direct experiments proving the existence of the electron spin and of its magnetic moment are experiments with the deflection of atomic beams in an inhomogeneous magnetic field\(^4\) (the Stern–Gerlach experiment\(^*\)). The scheme of the experimental apparatus used in such experiments is shown in Fig. 1.

Fig. 1. Scheme of the experiment on the deflection of molecular beams in an inhomogeneous magnetic field (Stern and Gerlach).

In the electric furnace \(A\) the substance under study is evaporated. From the stream of evaporated molecules or atoms emerging from the furnace through the small aperture \(d\), a narrow and thin beam is selected by means of a series of diaphragms \(bb'\); this beam enters the space between the poles \(BB'\) of an electromagnet and finally falls on the screen \(C\). The shape of the cross sections of the pole pieces \(B\) and \(B'\) of the electromagnet is made such that the magnetic field between them is sharply inhomogeneous along the \(z\)-axis, perpendicular to the direction of the beam \((dH/dz \ne 0)\). Atoms possessing a magnetic moment will be deflected from their initial direction of motion only in such a magnetic field as is already noticeably inhomogeneous over distances of the order of atomic dimensions \((10^{-8}\ \mathrm{cm})\). If this condition is not fulfilled, the atoms in the beam experience no deflection and merely undergo precession about the direction of the magnetic field, while preserving their translational motion unchanged. If the indicated inhomogeneity is present, then on an atom possessing a magnetic moment

\(^*\) It should be noted that, independently of these authors and simultaneously with them, such experiments were also undertaken by the Soviet physicists P. L. Kapitsa and N. N. Semenov\({}^{35}\).

$\mu$, directed at an angle $\vartheta$ to the direction of the field gradient (i.e., to the $z$ axis), will be acted upon by a deflecting force

\[ \mu \frac{dH}{dz}\cos \vartheta . \tag{1.4} \]

This force will affect the motion of the electron over a segment of path of length $l$ (see Fig. 1), where $dH/dz \ne 0$. The magnitude of the deflection $z_{\vartheta}$ of an atom with given $\mu$ and $\vartheta$ at the end of this path, according to the laws of uniformly accelerated motion, is equal to

\[ z_{\vartheta}=\frac{1}{2}\frac{\mu}{M}\left(\frac{dH}{dz}\right)_{\mathrm{av}} t^{2}\cos \vartheta, \tag{1.5} \]

where $M$ is the mass of the atom, and $t$ is the time of flight of the atom through the field, equal to $t=l/v$ (here $v$ is the mean velocity of the atom in the beam, determined by the conditions of thermal equilibrium at the given temperature in the furnace $A$). If all angles $\vartheta$ were equally probable, then on the screen $C$, instead of a narrow image of the slit at the point $O$, there would be obtained a broad band $O'O''$ with upper edge $O'$, corresponding to deflected atoms with magnetic moment parallel to the field ($\vartheta=0$), and with lower edge $O''$, corresponding to atoms with moment antiparallel to the field ($\vartheta=180^\circ$). In fact, an entirely different picture is observed. If the experiment is performed with beams of hydrogen-like atoms, i.e., atoms having one valence electron, then instead of a continuous band from $O'$ to $O''$ we obtain two images of the slit at $O'$ and $O''$. According to the laws of quantum mechanics such atoms in the normal state possess no orbital mechanical or magnetic moments and therefore should not in general be deflected in a magnetic field. If, however, one admits the existence of spin and of its magnetic moment, and also takes into account the quantization rule for them, then the result of the experiment at once becomes understandable. The double splitting of the beam is a consequence of the rule of spatial quantization of the spin magnetic moment, which can have only two possible projections, along ($\vartheta=0$) and against ($\vartheta=180^\circ$) the magnetic field. These experiments also make it possible to determine the magnitude of the projection of the spin magnetic moment on the direction of the external magnetic field, using, with the aid of (1.5), the displacement of the atomic beam $z_{\vartheta}$ measured in the experiment. In Fig. 2 is shown a photomicrograph of the deposit obtained with a beam of sodium vapor. At high temperatures of the furnace the beam consists almost entirely of atoms and gives a “double-humped” curve (the dashed curve in Fig. 2). At lower temperatures the majority of the atoms combine in pairs into molecules $\mathrm{Na}_2$, which have no resultant moment whatever (neither orbital nor spin) and therefore pass through the field without deflection; this circumstance is illustrated by the solid curve in Fig. 2, giving a photomicrograph of the deposit of an undeflected beam of sodium molecules. At present the technique of such experiments has reached such a high—

of perfection, which makes it possible to guarantee an accuracy of measurement of up to \(0.1\)—\(0.2\%\).

These experiments should be regarded as among the fundamental experiments of all atomic physics, for they made it possible to establish most directly the atomic nature of magnetism.

Fig. 2. Photomicrogram of the “deposit” for a beam of sodium atoms [according to Lious, Zeits. f. Phys. 69, 73 (1930)].

With the development of the consistent atomic theory—quantum mechanics—the question arose of a deeper theoretical explanation of the phenomena of electron spin and its magnetism than was provided by the crude model-based classical theory. Here it should be pointed out, however, that the solution of this problem within the framework of nonrelativistic quantum mechanics, limited to consideration of processes in which the velocities of the particles are small compared with the velocity of light, proved impossible. This is due to the fact that the phenomenon of spin magnetism, like any magnetic phenomenon, belongs to the class of typical relativistic effects, for the consistent explanation of which it is necessary to reckon with the requirements of the theory of relativity.

In 1927 Pauli constructed an approximate semiempirical quantum theory of the electron spin, in which the existence of spin and of its magnetic moment was simply postulated. Then, assuming the possibility of only two orientations of the spin, it followed from the general laws of quantum mechanics that if the magnitude of the spin component in units of \(\hbar\) is \(s^z=\dfrac{1}{2}\), then the magnitude of the spin vector itself in the same units is

\[ |s|=\sqrt{s(s+1)}=\frac{\sqrt{3}}{2}, \tag{1.6} \]

and the magnitude of the vector of the spin magnetic moment is correspondingly

\[ |\mu_s|=\frac{e}{mc}\sqrt{s(s+1)}\,\hbar=\sqrt{3}\,\mu_B . \tag{1.7} \]

The problem of electron spin received its definitive theoretical foundation after Dirac⁵ succeeded in constructing a relativistic quantum theory of the electron. In this theory the need to postulate the existence of spin and of its magnetic moment disappeared. They were obtained automatically from the theory, and precisely in the form required by experiment. Dirac’s theory clearly showed the inadmissibility (from the fundamental point of view) of a visual classical interpretation of spin as the result of the rotation of an electron-sphere. It should be noted that if one nevertheless tries to give a visual illustration of spin, then its appearance must be ascribed to the specifically quantum kinematic properties of the translational motion of the electron. For this reason the spin magnetic moment is often called kinematic, in contrast to a true moment, which is observed in some other particles, for example in the proton or neutron (see below, § 3).

In conclusion of this section we shall make one further remark. From experiments on the deflection of molecular beams in a magnetic field, strictly speaking, it follows only that the atom as a whole possesses one or another value of the magnetic moment. From indirect considerations concerning the character of orbital states, in a number of cases this observed atomic magnetism is ascribed to the electron spin, although this does not follow directly from the experimental result itself. Thus experiments with molecular beams by themselves do not make it possible to separate unambiguously the magnetic effects of the electron spin and of its orbital motion. Therefore at first sight it seems very desirable to carry out an analogous experiment with a beam of free electrons.

However, such an experiment, and any other experiment to determine the magnetic moment of a free electron, as Bohr⁶ showed, is doomed to failure. This is a simple consequence of the uncertainty relation of quantum mechanics. The point is that, since the spin magnetism of the electron has a kinematic character, it cannot be separated from the magnetic effects associated with the translational motion of the electron—a charged particle. It turns out that in any attempt to determine the magnetic moment of the spin, an inevitable uncertainty is introduced into the value of the electron momentum, which causes an uncertainty of the magnetic action due to translational motion, always exceeding the entire magnetic effect of the spin.

There are, however, other ways of observing the electron spin, for example by studying the so-called polarization of electron waves in the double reflection of an electron beam from the surfaces of crystals. This question has not yet found its complete experimental resolution, and therefore the phenomenon of polarization of electron waves cannot be a precision method for measuring the spin of the free electron.⁷

All of the foregoing concerning the magnetic properties of the electron can also be transferred to another elementary particle—the positron, which differs from the electron only in the positive sign of its electric charge. However, this particle, incomparably rarer than the electron, is not a constituent part of the atomic shell and has a short lifetime. Therefore no experiments on the direct detection and measurement of the spin magnetism of positrons have yet been undertaken.

2. MAGNETISM OF THE ELECTRON SHELL OF THE ATOM

After becoming acquainted with the magnetic properties of the electron as an elementary particle, it is natural to turn to the consideration of the magnetism of the simplest collectives of these particles—the electron shells of atoms.

Fig. 3. Elliptical orbit of an electron.

The magnetic properties of the electron shell of an atom are due to three causes: 1) the orbital motion of electrons, 2) electron spins, and 3) the magnetism of the atomic nucleus. We shall dwell first of all on the magnetic properties of the orbital motion of electrons1. If at first we restrict ourselves to a purely classical consideration of the motion of an electron in an orbit, then we immediately obtain a relation between the magnetic and mechanical moments of the orbit. Indeed, the motion of an electron in an elliptical orbit with period of revolution \(T\) is equivalent to a circular current of strength

\[ i=\frac{e}{cT}. \tag{2.1} \]

The magnetic moment of this current is equal to the product of the current strength and the area of the electron “orbit” (Fig. 3)

\[ S=\frac{1}{2}\int_{0}^{2\pi} r^2 d\varphi \tag{2.2} \]

(\(\varphi\) is the angle which the radius vector drawn from the focus makes with the major diameter of the ellipse), i.e.

\[ \mu_{\mathrm{orb}}=iS=\frac{eS}{cT}. \tag{2.3} \]

The angular momentum of the electron \(p_{\varphi}\), in accordance with the law ...

THE MODERN DOCTRINE OF MAGNETISM

conservation of angular momentum (the law of areas), is constant and is, by definition,

\[ p_\varphi = mr^2\frac{d\varphi}{dt}. \tag{2.4} \]

Substituting \(r^2\) from (2.4) into (2.2), we find

\[ S=\frac{p_\varphi}{2m}\int_0^T dt=\frac{p_\varphi T}{2m}. \tag{2.5} \]

Therefore for \(\mu_{\mathrm{orb}}\), by virtue of (2.5) and (2.3), we obtain

\[ \mu_{\mathrm{orb}}=\frac{e}{2mc}\,p_\varphi . \tag{2.6} \]

From the rule of quantization of electronic orbits it follows that

\[ p_\varphi=l\hbar,\qquad (l=1,2,3,\ldots,n) \tag{2.7} \]

(\(l\)—the angular or orbital, \(n\)—the principal quantum numbers), and therefore, from (1.2) and (2.6), we have

\[ \mu_{\mathrm{orb}}=l\frac{e\hbar}{2mc}=l\mu_B . \tag{2.8} \]

Thus, the old semiclassical quantum mechanics asserted that the magnetic moment of the orbital motion of an electron in an atom must be a multiple of the Bohr magneton, while the ratio of this magnetic moment to the mechanical moment of the orbit, according to (2.6), is equal to

\[ g_l=\frac{\mu_{\mathrm{orb}}}{p_\varphi}=\frac{e}{2mc}, \tag{2.9} \]

i.e. two times smaller than the corresponding ratio for spin moments [cf. (1.3)]. Quantum mechanics introduced a certain refinement into this picture, namely, it was shown that the magnitude of the angular-momentum vector for a stationary state of an electron in an atom (which only conditionally, “according to the old habit,” may be called orbital motion) is determined not by formula (2.7), but by another formula

\[ p_\varphi=\sqrt{l(l+1)}\,\hbar . \tag{2.10} \]

The angular or orbital quantum number \(l\) in this case takes the values:

\[ l=0,1,2,\ldots,(n-1). \]

Analogously, for the magnetic moment, instead of (2.8) we obtain

\[ \mu_l=\sqrt{l(l+1)}\,\frac{e\hbar}{2mc}=\sqrt{l(l+1)}\,\mu_B . \tag{2.11} \]

The ratio \(g_l\), however, remains, as is easy to see, the same as that given by (2.9). From (2.10) and (2.11) it is evident that there may be stationar-

states of the atom with \(l=0\), in which there is neither mechanical \((p_\varphi=0)\) nor magnetic \((\mu_l=0)\) moment. These states, with a peculiar “static” distribution of the charge density of the electron cloud of the atomic shell, are commonly called \(s\)-states.

The projections of the vector of mechanical angular momentum \(\mathbf{l}\) (in units of \(\hbar\)), for example, on the direction of an external magnetic field \(\mathbf{H}\), cannot be arbitrary, but only quite definite, forming a discrete set of values (spatial quantization). The magnitudes of these possible projections of the vector \(\mathbf{l}\) (whose absolute value is equal to \(\sqrt{l(l+1)}\,\hbar\)) are determined by the magnetic orbital quantum numbers \(m_l\), which can take \((2l+1)\) values:

\[ m_l=-l,\,-(l-1),\,\ldots,\,-1,\,0,\,1,\,\ldots,\,l-1,\,l . \tag{2.12} \]

Thus, the possible values of the cosine of the angle between the vectors \(\mathbf{l}\) and \(\mathbf{H}\) are determined by the formula

\[ \cos(\mathbf{l},\mathbf{H})=\frac{m_l}{\sqrt{l(l+1)}} . \tag{2.13} \]

Fig. 4. Spatial quantization of the orbital angular momentum of the electron (for \(l=3\)).

Fig. 4 gives a graphical illustration of the spatial quantization of the orbital moment for \(l=3\). The same rule of spatial quantization also holds for the magnetic moment, whose projections, in units of \(\mu_B\), are determined by the quantum numbers \(m_l\), i.e.

\[ (\mu_l)_H=m_l\mu_B . \tag{2.14} \]

Thus, the projections of the orbital magnetic moment remain, in quantum mechanics as well, multiples of the Bohr magneton, while the magnitude of the vector \(\boldsymbol{\mu}_l\) itself is no longer a multiple of the Bohr magneton [because of the appearance of the factor \(\sqrt{l(l+1)}\) in (2.11)].

If the atomic shell consists of several electrons, then the orbital quantum number \(L\) of the total angular momentum \(\mathbf{L}\) has, for example in the case of two electrons with quantum numbers \(l_1\) and \(l_2\), the following possible values:

\[ L=l_1+l_2,\quad l_1+l_2-1,\ldots,\quad l_1-l_2;\qquad (l_1>l_2). \]

The magnitudes of the vector \(\mathbf{L}\) and of the total magnetic moment \(\boldsymbol{\mu}_L\) are respectively equal to:

\[ |\mathbf{L}|=\sqrt{L(L+1)}\,\hbar \quad\text{and}\quad |\mu_L|=\sqrt{L(L+1)}\,\mu_B . \tag{2.15} \]

The projections of these vectors on the direction of the external field \(\mathbf{H}\) are quantized

are made in the same way as in the case of a single electron. Analogous rules of addition also apply to the total spin mechanical moment \(\mathbf{S}\) and to the corresponding magnetic moment \(\boldsymbol{\mu}_S\), whose magnitudes are equal to

\[ |\mathbf{S}|=\sqrt{S(S+1)}\,\hbar \quad\text{and}\quad |\boldsymbol{\mu}_S|=\sqrt{S(S+1)}\,\mu_B; \tag{2.16} \]

the projections of these vectors on the direction of the magnetic field \(\mathbf{H}\) are likewise multiples of the quantities \(\hbar\) and \(\mu_B\) and are determined by the total spin magnetic quantum number \(m_S=S, S-1,\ldots,-S\). The total angular momentum \(\mathbf{J}\) of the electron shell of an atom is the vector sum of the resultant orbital moment \(\mathbf{L}\) and the resultant spin moment \(\mathbf{S}\) (Russell–Saunders coupling rule1)

\[ \mathbf{J}=\mathbf{L}+\mathbf{S}. \tag{2.17} \]

The total angular quantum number \(J\) assumes the following values:

\[ J=L+S,\quad L+S-1,\ldots,\quad L-S, \]

if \(L>S\) (altogether \(2S+1\) values),

\[ J=S+L,\quad S+L-1,\ldots,\quad S-L \]

if \(L<S\) (altogether \(2L+1\) values). The magnitude of the vector \(\mathbf{J}\) is then equal to

\[ |\mathbf{J}|=\sqrt{J(J+1)}\,\hbar. \tag{2.18} \]

The projections of the vector \(\mathbf{J}\) on the direction of the external field have, just as those of the vectors \(\mathbf{L}\) and \(\mathbf{S}\), only integral values in units of \(\hbar\), and are determined by the resultant magnetic quantum numbers \(m_J\), which may have \((2J+1)\) different values:

\[ m_J=J,J-1,\ldots,0,-1,\ldots,-J. \tag{2.18'} \]

In an atom with one electron, if \(l=0\), there is only one value of the total angular quantum number \(j=s=\frac{1}{2}\); if \(l>0\), then there are two values \(j=l+\frac{1}{2},\ l-\frac{1}{2}\). Thus, \(j\) is equal to an odd half-integer number \(\left(\frac{1}{2},\frac{3}{2},\frac{5}{2},\text{ etc.}\right)\). In Fig. 5, as an example, a graphical picture is given of the addition of the vectors \(\mathbf{l}\) and \(\mathbf{s}\) for the case \(l=2,\ s=\frac{1}{2}\) and \(j=\frac{3}{2},\ \frac{5}{2}\).

The resultant magnetic moment of the atomic shell \(\boldsymbol{\mu}_J\), owing to the gyromagnetic anomaly of the spin \((g_s=2g_l)\), will not coincide in direction with the resultant mechanical moment \(\mathbf{J}\). This is shown graphically in Fig. 6. In the chosen scale the length of the vector \(\boldsymbol{\mu}_L\) is taken equal to the length of the vector \(\mathbf{L}\); therefore the length of the vector \(\boldsymbol{\mu}_S\) must be equal to twice the length of the vector \(\mathbf{S}\). Owing to the negative—

of the electron charge, as was already mentioned above, the directions of \(\mu_L, \mu_S\) are respectively antiparallel to the directions of \(\mathbf L\) and \(\mathbf S\). The resultant magnetic moment \(\mu\) makes with the vector \(\mathbf J\) an angle different from \(180^\circ\). The vectors \(\mathbf L\) and \(\mathbf S\), speaking in classical language, precess about the direction of the vector \(\mathbf J\), and therefore \(\mu_L\) and \(\mu_S\) must also precess about \(\mathbf J\). If each of these vectors is resolved into two components: one parallel to \(\mathbf J\) and one perpendicular to it, then the values of the perpendicular components of each of the vectors \((\mu_L)_\perp\) and \((\mu_S)_\perp\), averaged over the time of rotation, will be equal to zero, since they continuously change their direction.

Fig. 5. Addition of the orbital \((l)\) and spin \((s)\) angular momenta of an electron \((l=2,\ s=1/2)\).

Fig. 6. Addition of the mechanical and magnetic moments of the electron shell of an atom.

Therefore the effective magnetic moment of the electron shell of an atom will be equal to the sum of the projections \((\mu_L)\) and \((\mu_S)\) parallel to the vector \(\mathbf J\), i.e.

\[ \mu_J=\mu_L\cos(\mathbf L,\mathbf J)+\mu_S\cos(\mathbf S,\mathbf J), \tag{2.19} \]

where \(\cos(\mathbf L,\mathbf J)\) and \(\cos(\mathbf S,\mathbf J)\) are respectively the cosines of the angles between the vectors \(\mathbf L\) and \(\mathbf J\), \(\mathbf S\) and \(\mathbf J\). Applying the usual trigonometric formulas to the triangle formed by the vectors \(\mathbf L,\mathbf S\) and \(\mathbf J\), we obtain

\[ \cos(\mathbf L,\mathbf J)= \frac{L(L+1)+J(J+1)-S(S-1)} {2\sqrt{L(L+1)}\sqrt{J(J+1)}}, \]

\[ \cos(\mathbf S,\mathbf J)= \frac{S(S+1)+J(J+1)-L(L-1)} {2\sqrt{L(L+1)}\sqrt{J(J+1)}}. \tag{2.19'} \]

Substituting these values of the cosines, as well as the values of \(\mu_L\) from (2.15) and \(\mu_S\) from (2.16), into (2.19), we find

\[ \mu_J=\left[1+\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}\right]\sqrt{J(J+1)}\,\mu_B= \]

\[ = g_J\sqrt{J(J+1)}\,\mu_B , \tag{2.20} \]

where

\[ g_J=1+\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)} , \tag{2.21} \]

the so-called Landé factor. If \(L=0\), then \(J=S\) and \(g_J=g_S=2\), i.e., in the case of spin angular momentum the Landé factor is equal to two. If, conversely, \(S=0\), then \(J=L\) and \(g_J=g_L=1\), i.e., for a purely orbital angular momentum the Landé factor is equal to unity. Thus, for spin the Landé factor is twice as large as for orbital angular momentum—this is a direct result of the gyromagnetic anomaly of spin.

Fig. 7. Diagram of the normal Zeeman effect.

Zeeman effect\(^1\). One of the most direct manifestations of the magnetic properties of the atom is the Zeeman effect (1896), which consists in the splitting of the spectral lines of atomic spectra when the radiating atoms are placed in a magnetic field. In the laboratory reproduction of this phenomenon the light source (a sodium flame, mercury arc, etc.) is placed between the poles of an electromagnet, and the radiation itself is directed into a spectroscope of high resolving power. In this way one can investigate the spectral composition of the light emitted both parallel to the magnetic field (longitudinal effect) and perpendicular to it (transverse effect) (see Fig. 7). If the field is “strong” (see below), then we have the so-called normal Zeeman effect (or Paschen–Back effect). In the normal longitudinal effect, instead of the single spectral line of frequency \(\nu_0\) observed in the absence of the field, as a rule not

polarized one observes two lines symmetrically displaced relative to $\nu_0$: a line with the lower frequency $\nu_2$ and one with the higher frequency $\nu_1$ ($\nu_0-\nu_2=\nu_1-\nu_0$). Both these lines prove to be circularly polarized (Fig. 7). In the normal transverse effect three lines are observed: the undisplaced line $\nu_0$, linearly polarized along the field, and two lines $\nu_1$ and $\nu_2$, displaced in the same way as in the longitudinal effect, but, in contrast to it, linearly polarized perpendicular to the direction of the magnetic field (see Fig. 7).

In the case of “weak” fields this phenomenon becomes considerably more complicated: additional lines appear. An explanation of this “anomalous” Zeeman effect will be given below.

The normal effect can be quite fully explained with the aid of the elementary classical electron theory (Lorentz). Let us suppose, for simplicity of calculation, that the electron in the atom moves with angular velocity $\omega_0$ in a circular orbit of radius $r$, the plane of which is normal to the vector of the magnetic field $\mathbf{H}$. In the absence of a magnetic field, the electron is then acted upon by the centripetal force

\[ F_0=m\omega_0^2 r . \]

If the magnetic field is switched on, then in the very process of switching it on the magnetic flux through the area of the orbit will vary with time. Therefore there arises an additional induced electric field directed tangentially to the orbit. This additional field causes a change in the velocity of motion of the electron along the orbit. At the same time, on an electron moving in a magnetic field there will act a Lorentz force directed along the radius. Its magnitude and direction will be such as to ensure that the radius of the orbit remains unchanged. Therefore the switching on of the magnetic field will lead only to an increase or decrease of the angular velocity of the electron, depending on the direction of its motion relative to the direction of the magnetic-field vector.

If the changed value of the angular velocity of the electron is equal to $\omega_1$, then the additional radial Lorentz force is equal to

\[ F_H=\pm \frac{1}{c}He\omega_1 r . \]

The plus sign corresponds to $\omega_1>\omega_0$ and the minus sign to $\omega_1<\omega_0$. The total centripetal force is equal to the sum of $F_0$ and $F_H$, i.e.

\[ m\omega_1^2 r=m\omega_0^2 r \pm \frac{1}{c}He\omega_1 r . \]

Hence we find approximately (noting that in the atom $\omega_0\sim 10^{16}\ \mathrm{sec}^{-1}$, while $eH/cm\sim 10^{12}\ \mathrm{sec}^{-1}$ even for superstrong magnetic fields in the experiments of P. L. Kapitza) that

\[ \omega_1=\omega_0\pm \frac{eH}{2mc}. \tag{2.22} \]

The quantity

\[ \omega_H = 2\pi \nu_H = \frac{eH}{2mc} \tag{2.23} \]

is called the Larmor frequency; it determines the magnitude of the influence of the magnetic field on the orbital motion of an electron in an atom. If the magnetic field is not normal to the plane of the orbit, the effect of the field is nevertheless determined by the quantity \(\omega_H\), which in this general case is the angular velocity of the Larmor precession of the electronic orbit about the direction of the magnetic field (Fig. 8). The additional energy of the electron, caused by this additional angular velocity, is, by virtue of (2.11) and (2.13),

\[ \Delta E_H = \mu_l H \cos \theta = m_l \frac{e\hbar}{2mc} H = m_l \omega_H \hbar . \tag{2.24} \]

This entire effect of Larmor precession is a special case of Lenz’s induction rule (it is easy to see that the additional motion of the electron, caused by switching on the magnetic field, creates its own magnetic field antiparallel to the former) and underlies the universal phenomenon of diamagnetism inherent in all atoms (see below, § 6).

We shall use the result obtained to explain the normal Zeeman effect. Let the magnetic field be directed along the \(x\)-axis (Fig. 7). At the light source \(O\), the atomic orbits have all possible orientations. Under the influence of the magnetic field they begin to precess about the \(x\)-axis.

Fig. 8. Precession of the electronic orbit (\(l\)) about the magnetic field (\(H\)).

Because of the transverse character of light waves, along a given direction only waves from electron-acceleration components perpendicular to this direction propagate. Therefore, in longitudinal observation (along the \(x\)-axis), light comes only from the components of the electronic motion lying in the \((y,z)\) plane, i.e. from the projections of the electron orbits onto this plane. Owing to the chaotic orientations of the electron orbits of the different atoms of the source, for one half of the number of the projections mentioned the angular velocity will be \(+\omega_0\), and for the other half \(-\omega_0\). Therefore, for \(H \ne 0\), the resulting angular velocities in absolute value will be respectively equal to \((\omega_0+\omega_H)\) and

\((\omega_0-\omega_H)\), and consequently the linear frequencies of the two shifted spectral lines will be

\[ \nu_1=\nu_0+\frac{eH}{4\pi mc}\quad \text{and}\quad \nu_2=\nu_0-\frac{eH}{4\pi mc}. \tag{2.25} \]

Both these lines, in complete agreement with experiment, are shifted relative to the initial line \(\nu_0\) by the same amount \(\Delta\nu=\dfrac{He}{4\pi mc}\) and are circularly polarized in opposite directions.

When observing across the field (along the \(z\)-axis), three lines are observed. The unshifted line \(\nu_0\) is obtained from the component of the electron motion along the \(x\)-axis, on which the magnetic field does not act (the Lorentz force is zero if \(\mathbf H \parallel \mathbf V\)). Therefore the unshifted line \(\nu_0\) is linearly polarized along the \(x\)-axis, as is indeed observed. The component of the motion along the \(y\)-axis, perpendicular to the field \(\mathbf H\) (light from the component along the \(z\)-axis does not propagate along the \(z\)-axis), changes its frequency by the amount \(\omega_H\) and will give two shifted lines \(\nu_1\) and \(\nu_2\), linearly polarized along the \(y\)-axis.

By comparing the theoretical and experimental values for \(\Delta\nu\), it was possible to determine the specific charge of the electron, which proved to be \(e/m=1.757\cdot 10^7\) CGSM, in excellent agreement with the data for the specific charge obtained by experiments on the deflection of cathode rays in electric and magnetic fields. Moreover, from the direction of the circular polarization of the shifted lines in the longitudinal effect, it was possible for the first time to show experimentally that the source of the emitted light in the atom is precisely the negative electric charge, i.e. the electron.

Let us now clarify the refinements that quantum mechanics introduced into the explanation of this magnetic phenomenon.

If an atom is placed in a magnetic field \(\mathbf H\), which is relatively weak, then the coupling between the vectors \(\mathbf L\) and \(\mathbf S\) is not “broken,” and the resultant vector \(\mathbf J\) will precess about the direction of the magnetic field. The additional energy \(\Delta E_H\), caused by the action of the magnetic field on the resultant atomic magnetic moment \(\mu_J\), will, by (2.20), be equal to

\[ \Delta E_H=\mu_J H\cos(\mathbf J,\mathbf H) = g_J \mu_B H \sqrt{J(J+1)}\cos(\mathbf J,\mathbf H). \tag{2.26} \]

But the projection of the vector \(\mathbf J\) on the direction \(\mathbf H\), i.e. \(\sqrt{J(J+1)}\cos(\mathbf J,\mathbf H)\), according to \((2.18')\), is equal to the magnetic quantum number \(m_J\). Thus,

\[ \Delta E_H=m_J g_J \mu_B H. \tag{2.27} \]

Consequently, each atomic level with energy \(E_n\) is split in an external magnetic field into \((2J+1)\) levels with energies \(E_n+(\Delta E_H)_n\). The number \((2J+1)\) gives us the magnitude of the so-called mul-

multiplicity of the level. The frequencies emitted by an atom in a magnetic field will be equal to

\[ \nu_{ik}+\Delta\nu= \frac{[E_i+(\Delta E_H)_i]-[E_k+(\Delta E_H)_k]}{h}; \]

taking into account that \(h\nu_{ik}=E_i-E_k\) and (2.27) and (2.23), we find

\[ \Delta\nu=\left[(m_J)_i(g_J)_i-(m_J)_k(g_J)_k\right]\nu_H. \tag{2.28} \]

As an illustration, let us apply formula (2.28) to explain the anomalous Zeeman effect of the yellow doublet of the principal series of the sodium spectrum (the \(D\)-lines), consisting of two lines. Table I gives the values of \(L\), \(S\), \(J\), \(g_J\), \(m_J\), and \(g_Jm_J\) for the three unperturbed energy levels, transitions between which give the two \(D\)-lines,

Table I

States	\(L\)	\(S\)	\(J\)	\(g_J\)	\(m_J\)	\(g_Jm_J\)
\(3^2S_{1/2}\)	0	\(1/2\)	\(1/2\)	2	\(1/2,\ -1/2\)	\(1,\ -1\)
\(3^2P_{1/2}\)	1	\(1/2\)	\(1/2\)	\(2/3\)	\(1/2,\ -1/2\)	\(2/3,\ -1/3\)
\(3^2P_{3/2}\)	1	\(1/2\)	\(3/2\)	\(4/3\)	\(3/2,\ 1/2,\ -1/2,\ -3/2\)	\(2,\ 2/3,\ -2/3,\ -2\)

and Fig. 9 gives a graphical picture of the splitting of the energy levels and shows the Zeeman components for the transverse effect. Lines denoted by the letter \(\sigma\) are polarized perpendicular to the magnetic field, while those denoted by the letter \(\pi\) are polarized parallel to the field. The polarization of these lines can be predicted theoretically; the theory also yields selection rules, which allow only part of the possible transitions between the various levels, provided that the magnetic quantum number remains unchanged in the transition (\(\Delta m_J=0\), \(\pi\)-components) or changes by one unit (\(\Delta m_J=\pm1\), \(\sigma\)-components). In the longitudinal effect only the \(\sigma\)-components (\(\Delta m_J=\pm1\)) remain, and they turn out to be circularly polarized.

If the external magnetic field becomes strong and “breaks” the coupling between the vectors \(\mathbf L\) and \(\mathbf S\), then in this case the quantum number \(J\) loses its meaning, and the vectors \(\mathbf L\) and \(\mathbf S\) precess independently about the magnetic field. The magnitude of the projections of these vectors on the direction of the field is determined by the orbital and spin magnetic quantum numbers \(m_L\) and \(m_S\). The additional energy in the magnetic field, instead of (2.27), will have the form (\(g_L=1\) and \(g_S=2\)):

\[ \Delta E_H=\Delta E_L+\Delta E_S=(m_L+2m_S)\mu_B H \]

and therefore, by virtue of (2.23) and (1.21),

\[ \Delta\nu=(\Delta m_L+2\Delta m_S)\nu_H. \tag{2.29} \]

The selection rules in this case state: \(\Delta m_L=0,\pm1,\ \Delta m_s=0\); therefore (2.29) gives only three possibilities

\[ \Delta\nu=0,\pm\nu_H, \]

i.e., the classical normal Zeeman effect. It should be noted that the magneto-optical Zeeman phenomenon, throughout the entire development of electronic physics up to the present day, has invariably served as a powerful experimental criterion for the correctness of the development of its theoretical interpretations.

Fig. 9. Scheme of the levels and transitions of the anomalous Zeeman effect for the principal doublet of the \(D\)-line of the line spectrum of the sodium atom.

Recently Stern\(^8\) proposed a new method for the exact determination of atomic magnetic moments, which reduces to balancing the action of gravity on a very long molecular beam by means of a magnetic force directed against gravity (and produced, for example, by the inhomogeneous magnetic field of a current flowing through a conductor stretched exactly beneath the molecular beam parallel to its direction of motion). It is easy to see that the condition of equality of the forces, referred to one gram-molecule of the substance investigated in the beam, has the form

\[ N\mu \frac{\partial H}{\partial z}=NMg, \]

where \(N\) is Avogadro’s number, \(M\) is the mass of the atom, \(g\) is the acceleration of gravity, and \(z\) is the vertical direction. Since \(g\), \(N\), \(M\), and \(\partial H/\partial z\) can be known with great accuracy, it becomes possible to determine very accurately the value of the molar magneton \(N\mu = Ne\hbar/2mc\) (if, for example, the resultant magnetic moment of the atom is produced only by the spin of a single valence electron). On the other hand, Faraday’s number \(F = Ne\) and the speed of light \(c\) are also known very accurately; therefore this method makes it possible to determine precisely the ratio \(\hbar/m\).

In conclusion of this paragraph we also note that, besides studying the magnetic properties of atoms in isolated states, where the interaction between different atoms or molecules is known to be small (molecular beams, atomic spectra), one may raise the question of studying the magnetic moments of the electron shells of atoms in solids and liquids. Although in this case, strictly speaking, we can speak only of the magnetism of a collective of atoms in the condensed phase, nevertheless some information about the “individual” magnetic properties of separate atoms can be obtained in this case as well. In particular, the already well-known experiments on the gyromagnetic effect in ferromagnetics (see below §§ 10 and 11) made it possible to establish that in practically all known ferromagnetic bodies the carriers of magnetism are electron spins. Using the method of magnetic resonance (see in more detail § 3), it proves possible to determine not only the Landé \(g\)-factors, but also the mechanical and magnetic moments of atoms in solids and liquids (Arkadev, Dorfman, Rabi, Zavoisky, Gorter, and others).

Thus, for example, Zavoisky\(^{9}\) determined, in paramagnetic salts \((\mathrm{MnSO_4\cdot 3H_2O}\) and \(\mathrm{CuCl_2\cdot 2H_2O})\), the value of the spin of the ions \(\mathrm{Mn}^{++}\) and \(\mathrm{Cu}^{++}\), which turned out to be respectively equal to \(\sqrt{\frac{5}{2}\left(\frac{4}{2}+1\right)}\,\hbar\) and \(\sqrt{\frac{1}{2}\left(\frac{1}{2}+1\right)}\,\hbar\). The Landé factor in the first case turned out to be equal to 1.96, and consequently the magnetic moment is \(\mu = 4.90\,\mu_B\). Zavoisky’s experiments are the first direct proof of the spatial quantization of the spin of an atom in a solid.

3. MAGNETISM OF NUCLEONS (PROTONS AND NEUTRONS) AND ATOMIC NUCLEI

Heavy elementary particles—protons and neutrons (they are now commonly called by the general term nucleons), as well as the atomic nuclei composed of them*)—possess their own magnetic moments. Therefore one may speak of “nuclear magnetism.” However, the magnitude of the magnetic moments of nucleons and of complex nuclei is, on the average, a thousand times smaller than the spin or orbital magnetic moment of the electron shell of an atom. For this reason nuclear magnetism mani—

*) The idea of the proton-neutron structure of atoms belongs to the Soviet scientist D. D. Ivanenko.

manifests itself in much more subtle phenomena of the atomic world (for example, the hyperfine structure of spectral lines) and for its detection and study requires the application of complex experimental methods.

Nevertheless, advances in the development of the modern technique of physical experiment are so great that it has proved possible not only to detect, but also to measure with great accuracy the magnetic moments of individual nucleons and of complex nuclei.

By analogy with the magnetic properties of the electron, at first sight it would seem that the proton—a positively charged elementary particle—whose spin is equal to \(\sqrt{\frac{1}{2}\left(\frac{1}{2}+1\right)}\,\hbar\), should possess a spin magnetic moment

\[ \frac{e\hbar}{Mc}\sqrt{\frac{1}{2}\left(\frac{1}{2}+1\right)}, \]

where \(M\) is the mass of the proton, 1836.5 times greater than the mass of the electron. Thus, in nuclear magnetism the role of the elementary magnetic moment should be played by the value of the projection of the proton magnetic moment on the direction of the external magnetic field, equal to

\[ \mu_{\text{nucl}}=\frac{e\hbar}{2Mc}=\frac{1}{1836.5}\,\mu_B . \tag{3.1} \]

\(\mu_{\text{nucl}}\) is called the nuclear magneton, which is 1836.5 times smaller than the Bohr magneton; in this lies the cause of the smallness of nuclear magnetism in comparison with the magnetism of the electron shell.

As for the neutron—a particle devoid of electric charge—it seemed natural to assume that it also has no magnetic moment.

In reality, experiment gave an entirely different picture. Measurements showed that the magnetic moment of the proton \(\mu_p\) is almost three times greater than the nuclear magneton, namely

\[ \mu_p=(2.7896 \pm 0.0008)\,\mu_{\text{nucl}}, \tag{3.2} \]

and the magnetic moment of the neutron turned out to be different from zero and equal to

\[ \mu_n=-(1.9103 \pm 0.0013)\,\mu_{\text{nucl}}. \tag{3.3} \]

Such a discrepancy between “natural” assumptions and experiment, when one attempts to explain the nature of nuclear magnetism, from the very first steps leads to a difficulty that did not arise in considering the magnetic properties of the electron shell[^10]. The modern theory of the atomic nucleus is still at the initial stage of its development and is very imperfect; nevertheless, it makes it possible to understand qualitatively the reason for the apparent divergence between our theoretical assumptions about the magnitude of the magnetic moments of nucleons and the experimental data for them. The point is that, independently of the electromagnetic interaction between the electric charges of nucleons, powerful nuclear forces act between nucleons.

In exactly the same way as the electromagnetic interaction of charges can be represented as the result of an “exchange” (i.e., mutual emission and subsequent absorption) of quanta of electromagnetic radiation—photons—between the interacting charges, the specific nuclear interaction of nucleons can be represented as an “exchange” of special particles—mesons—between the interacting nucleons*).

Unlike photons, whose rest mass is zero and which have no electric charge, mesons have a finite rest mass (about two hundred electron masses) and a positive or negative elementary electric charge). One may say that, as a result of the continuous emission and absorption of mesons, nucleons create around themselves a kind of meson field, in the same way that electric charges create around themselves an electromagnetic field as a result of the analogous emission and absorption of photons. In this process, for example, a proton, emitting a positively charged meson or absorbing a negatively charged meson, turns into a neutron. Conversely, a neutron, absorbing a positively charged meson or emitting a negatively charged meson, turns into a proton. Thus, nucleons, which are observed experimentally either as protons or as neutrons, undergo continuous transformations into one another, while nevertheless spending most of the time—protons in the proton state, neutrons in the neutron state. Since the process of meson exchange does not occur instantaneously but requires some time, in the nucleus mesons with their spin moments, almost ten times greater than those of the nucleons, constantly appear for relatively short intervals; and therefore the resulting magnetic moment of the proton turns out to be greater than the nuclear magneton, while in the neutron, solely due to the meson field, a negative magnetic moment appears. Indeed, from a comparison of the experimental data (3.2) and (3.3), it is evident that the difference of the absolute values of the magnetic moments of the proton and the neutron is very close to one nuclear magneton, and that the signs of the moments are opposite*).

When one proceeds to the consideration of the magnetic properties of more complex nuclei, the picture also becomes more complicated than is the case for the electron shell. Table II gives the values of the spins and magnetic moments of the atomic nuclei of a number of isotopes.

*) The idea that the nuclear interaction is carried by particles was first put forward by I. E. Tamm and D. D. Ivanenko (1934).

**) There are all grounds to believe that there exist mesons without charge (neutretto), but their existence has not yet been proved by direct experiments.

***) It is quite possible that the processes of such quasi-splitting of nucleons proceed in a more complicated way. For example, the “emitted” mesons may still decay into an electron (positron) and a neutrino. But the qualitative picture is not changed by this.

Table II

Values of the spin quantum number \((I)\) and of the projections of the magnetic moment for the atomic nuclei of certain isotopes

Nucleus	\(I\)	\(\mu_I\) (in units of \(\mu_{\mathrm{nuc}}\))	Nucleus	\(I\)	\(\mu_I\) (in units of \(\mu_{\mathrm{nuc}}\))
\({}^{1}_{0}n\)	\(1/2\)	\(-1,9103 \pm 0,0012\)	\({}^{40}_{19}K\)	\(4\)	\(-1,230\)
\({}^{1}_{1}H\)	\(1/2\)	\(+2,7896 \pm 0,0008\)	\({}^{69}_{31}Ga\)	\(3/2\)	\(?\,2,11\)
\({}^{2}_{1}H\)	\(1\)	\(+0,85647 \pm 0,0003\)	\({}^{71}_{31}Ga\)	\(3/2\)	\(?\,2,69\)
\({}^{6}_{3}Li\)	\(1\)	\(+0,8813 \pm 0,0005\)	\({}^{85}_{37}Rb\)	\(5/2\)	\(+0,10127\)
\({}^{7}_{3}Li\)	\(3/2\)	\(+3,2532 \pm 0,0015\)	\({}^{87}_{37}Rb\)	\(3/2\)	\(+2,733 \pm 0,009\)
\({}^{9}_{4}Be\)	\(3/2\)	\(-1,176 \pm 0,005\)	\({}^{113}_{49}In\)	\(9/2\)	\(+0,998 \times \mu_I({}^{115}_{49}In)\)
\({}^{10}_{5}B\)	\(1\)	\(+0,598 \pm 0,003\)	\({}^{115}_{49}In\)	\(9/2\)	\(+5,43 \pm 0,03\)
\({}^{11}_{5}B\)	\(3/2\)	\(+2,686 \pm 0,008\)	\({}^{133}_{55}Cs\)	\(7/2\)	\(+2,558 \pm 0,007\)
\({}^{13}_{6}C\)	\(1/2\)	\(+0,701 \pm 0,002\)	\({}^{135}_{56}Ba\)	\(3/2\)	\(+0,837 \pm 0,023\)
\({}^{14}_{7}N\)	\(1\)	\(+0,403 \pm 0,002\)	\({}^{137}_{56}Ba\)	\(3/2\)	\(+0,936 \pm 0,003\)
\({}^{15}_{7}N\)	\(1/2\)	\(?\,0,280 \pm 0,003\)
\({}^{19}_{8}F\)	\(1/2\)	\(+2,625 \pm 0,003\)	\({}^{4}_{2}He\)	\(0\)	\(0\)
\({}^{23}_{11}Na\)	\(3/2\)	\(+2,215 \pm 0,002\)	\({}^{12}_{6}C\)	\(0\)	\(0\)
\({}^{27}_{13}Al\)	\(5/2\)	\(+3,630 \pm 0,010\)	\({}^{16}_{8}O\)	\(0\)	\(0\)
\({}^{35}_{17}Cl\)	\(5/2\)	\(+1,368 \pm 0,005\)	\({}^{32}_{16}S\)	\(0\)	\(0\)
\({}^{37}_{17}Cl\)	\(5/2\)	\(+1,136 \pm 0,005\)	\({}^{80}_{34}Se\)	\(0\)	\(0\)
\({}^{39}_{19}K\)	\(3/2\)	\(+0,391 \pm 0,002\)	\({}^{20}_{10}Ne\)	\(0\)	\(0\)

From Table II it is evident that for nuclear spin there is a simple rule of additivity, whereas the magnetic moments of nuclei are plainly nonadditive. Even in the simplest complex nucleus—the deuteron (the nucleus of the heavy hydrogen atom), consisting of one proton and one neutron, with spin quantum number equal to unity (which indicates a parallel orientation of the mechanical moments of the proton and neutron in the deuteron nucleus)—the resultant magnetic moment is not equal

exactly to the algebraic sum of the magnetic moments of the proton and the neutron. From Table II we obtain

\[ \mu_{\mathrm{H}_2}-(\mu_p+\mu_n)=-0.0228\,\mu_{\mathrm{nucl}}. \]

The accuracy of the measurements guarantees the correctness of quantities above 0.0013, i.e. less than 5% of the observed difference.

The fact that, for example, in the case of light nuclei the spin quantum numbers do not exceed \(3/2\), permits the assumption that nucleons, like the electrons of the atomic shell, form closed “shells” with zero value of spin and magnetic moment. Examples of nuclei with such closed “shells” are given at the end of Table II. The spin of the nucleus is the result of the addition of the spins of only a small number of nucleons not included in the closed shells *). Thus, for example, one may consider that in the nuclei \(\mathrm{He}_2^4\), \(\mathrm{C}_6^{12}\), \(\mathrm{O}_8^{16}\), \(\mathrm{Ne}_{10}^{20}\), etc., with \(I=0\), all nucleons enter into closed shells, while in the nucleus \(\mathrm{N}_7^{14}\) 6 neutrons and 6 protons form closed shells, and the seventh neutron and proton give the resultant nuclear spin, equal to their sum \(I=1\). However, the corresponding magnetic moment of this nucleus is by no means equal to that for the deuteron, where \(I\) is also equal to unity, but amounts to only \(0.403\,\mu_{\mathrm{nucl}}\).

This absence of additivity of magnetic moments in complex nuclei is explained by Frenkel \({}^{11}\), for example, by the same cause as the anomalous magnitude of the magnetic moments of individual nucleons, i.e. by the meson field of nucleons in nuclei, and also, possibly, by relativistic effects in the interaction of nucleons.

In the case of heavier nuclei the values of the spin quantum number may reach more considerable magnitudes. For example, for the indium isotopes \(\mathrm{In}^{113}\) and \(\mathrm{In}^{115}\), \(I=9/2\), and the magnetic moments are correspondingly large: \(5.50\,\mu_{\mathrm{nucl}}\) and \(5.49\,\mu_{\mathrm{nucl}}\).

The comparatively small magnitude of the spin and magnetic moment of atomic nuclei, as Frenkel \({}^{11}\) pointed out, permits one to draw an analogy between nuclear magnetism and the paramagnetism of alkali metals, where likewise, as a consequence of the requirements of the Pauli principle (closed “shells”), the resultant spin of the collective of conduction electrons in the absence of excitation (at \(0^\circ\mathrm{K}\) and at \(H=0\)) is always equal to zero. Frenkel compares the appearance of spin in nuclei with the presence of a resultant spin and magnetic moment different from zero in atoms of transition elements with incomplete inner shells of the electron shell, or even with the spontaneous magnetization of ferromagnetic bodies.

Let us now turn to the experimental methods for determining the spin and the spin magnetic moment of nucleons and complex atomic nuclei. Here, first of all, one should point out the phenomenon

*) In contrast to the case of an atomic electron shell, all these arguments in the case of nuclei, because of the very strong interaction between nucleons, are of a very qualitative character.

of the hyperfine structure (structure) of spectral lines of atomic spectra, first discovered by the Russian physicists Dobretzov and Terenin\(^{12}\) as early as 1928. They showed that individual lines, even in the fine structure (§ 2) of the spectrum, are in fact a set of several different lines with very close frequency values. The sodium doublet (the \(D\)-line) studied by these scientists has a distance between its two fine-structure lines of \(6\) Å (on the wavelength scale), while the distances between the individual lines of the hyperfine splitting are \(0.0021\) Å (for the line with \(\lambda = 5890\) Å) and \(0.023\) Å (for the line with \(\lambda = 5896\) Å), i.e., quantities hundreds and thousands of times smaller than \(6\) Å.

Further study showed\(^{13}\) that two types of hyperfine structure can be distinguished. In the first case all lines of the spectrum have the same number of components. The cause of the appearance of hyperfine structure here is the presence of two or more stable isotopes of the given element,* since the difference in the masses of the nuclei of the isotopes slightly changes the energies of the stationary states of the electron shell of the atoms.

The second type is characterized by the presence of different numbers of components in different lines of the spectrum, which cannot be explained by isotopic displacement, all the more so since this type of hyperfine structure is also observed in the spectra of atoms possessing only a single stable isotope (for example, the case of Bi). In this case the hyperfine structure can be explained only if one takes into account the existence of the nuclear spin and the corresponding nuclear magnetic moment; owing to their interaction with the moments of the electron shell, splitting of the energy levels of the atomic shell occurs. Thus formula (2.17) must be refined, and the total angular momentum of the atom \(\mathbf{F}\) is equal to the sum of the total angular momentum of the electron shell \(\mathbf{J}\) and the resultant moment of the nucleus \(\mathbf{I}\):

\[ \mathbf{F}=\mathbf{J}+\mathbf{I}. \tag{3.4} \]

According to the rules of spatial quantization, the total quantum number of the mechanical moment of the atom \(F\) takes the following possible values:

\[ F=J+I,\quad J+I-1,\ldots,(J-I). \tag{3.5} \]

Thus, the energy levels of the electrons split into \((2J+1)\) (if \(I>J\)) or \(2I+1\) (if \(J>I\)) sublevels (multiplet) with somewhat different energy, which is what gives rise to the hyperfine structure. In Fig. 10 two vector models (analogous to Fig. 5) are given for the addition of the mechanical and magnetic moments of the electron shell and the nucleus of the atom for the cases: (a) when the electron

* By such an “isotopic” displacement of all the lines of the Balmer series in the spectrum of hydrogen, the heavy isotope of hydrogen—deuterium—was discovered.

the shell has orbital and spin moments; (b) when the electron shell has only a spin moment (\(S\)-state with \(L=0\)). From Fig. 10 it is again seen that, owing to the gyromagnetic anomaly of the spin, the resultant magnetic moment of the atom \(\mu_F\) is not parallel to the resultant mechanical moment \(\mathbf F\), about which it precesses. For given values of the angular quantum numbers

Fig. 10. Vector model of an atom with a nuclear moment.
a) case of an electron shell possessing nonzero orbital \((L)\) and spin \((S)\) moments; b) case of an electron shell with one electron in an \(S\)-state \((L=0)\) (the scale for \(\mu_I\) is taken evidently enlarged by \(\sim 10^3\) times).

of the shell \(J\) and of the nucleus \(I\), the angular quantum number \(F\) of the whole atom, according to (3.5), has a number of possible values, each of which corresponds to its own value of the energy. One may say that the electron shell creates at the center of the nucleus an effective magnetic field \(\mathbf H_{\mathrm{el}}\), parallel to the vector \(\mathbf J\). Therefore the energy of interaction of the nuclear magnetic moment \(\mu_I\) in this field is equal to

\[ \Delta E_{\mathrm{s.t.s.}}=\mu_I H_{\mathrm{el}}\cos(\mathbf I,\mathbf J), \]

where

\[ \cos(\mathbf I,\mathbf J)= \frac{F(F+1)-I(I+1)-J(J+1)} {2\sqrt{I(I+1)J(J+1)}};\qquad \mu_I=g_I\sqrt{I(I+1)}\,\mu_{\mathrm{nucl}}. \]

Thus,

\[ \Delta E_{\mathrm{s.t.s.}}= \frac{\mu_{\mathrm{nucl}}H_{\mathrm{el}}g_I}{2\sqrt{J(J+1)}} \,[F(F+1)+I(I+1)-J(J+1)], \tag{3.6} \]

where \(F\) is determined, for given \(I\) and \(J\), by the series of values (3.5).

The energy difference (3.6) for given \(I\) and \(J\) gives the magnitude of the splitting of the hyperfine structure. At the same time it is easy to see that the rela-

…the ratios of the distances between a series of sublevels of the hyperfine structure, characterized by the values of the quantum numbers \(F, F+1, F+2,\ldots\), will depend only on the quantum number \(F\); namely, these ratios are equal to \((F+1):(F+2):(F+3):\ldots\), as follows from (3.6). The same also holds for the ratios of these intervals expressed in terms of the wave numbers (or frequencies) of the hyperfine-structure components, i.e.

\[ \Delta\nu_1:\Delta\nu_2:\Delta\nu_3:\ldots=(F+1):(F+2):(F+3):\ldots . \]

In addition to this interval rule, one can also obtain an intensity rule, which gives the values of the ratios of the intensities of the various hyperfine-structure components as a function of the quantum number \(F\). With the aid of these two rules it is possible, on the basis of experimental spectroscopic data, to determine the value of the spin quantum number of the nucleus \(I^{13}\).

However, this is insufficient for determining the Landé factor or the nuclear magnetic moments themselves. For this purpose it is necessary to use the Zeeman effect, i.e. to investigate the splitting of the hyperfine-structure components of spectral lines in external magnetic fields. Here, just as in the case of the electron shell, one must distinguish two limiting cases—weak and strong fields.

a) Weak field. Weak fields are those in whose presence the vectors of the mechanical moments of the nucleus \(\mathbf I\) and of the shell \(\mathbf J\) are still strongly coupled. Therefore their sum \(\mathbf F\) behaves as a whole in an external magnetic field. The quantum numbers for the projections of the vector \(\mathbf F\) on the field direction are then

\[ m_F=F,F-1,\ldots,-F+1,-F. \tag{3.7} \]

The possible transitions between levels are determined by the selection rules

\[ \Delta m_F=0,\ \pm 1. \tag{3.8} \]

In Fig. 11, as an example, the splitting of the hyperfine structure of the \({}^{2}S_{1/2}\)-term (i.e. of the level with \(L=0,\ S=\frac{1}{2}\)) in a weak magnetic field is given, where it is assumed that \(I=\frac{3}{2}\). Of the six possible \(\pi\)-components (see § 2), corresponding to transitions with \(\Delta m_F=\pm1\), four are actually observed, since the transitions \(1\to0\), \(0\to1\) and \(1\to-1\), \(-1\to0\) merge pairwise into one line (in Fig. 11,b it is shown that these lines have doubled intensity). Transitions with \(\Delta m_F=0\) correspond to three \(\sigma\)-components of the hyperfine structure.

The change in the energy of a level with quantum number \(F\) in an external magnetic field \(H\), by analogy with (2.27), is equal to

\[ \Delta E_H=m_F g_F \mu_B H. \tag{3.9} \]

The value of the Landé factor \(g_F\) of the entire atom can be expressed in terms of the Landé factors of the shell \(g_J\) and of the nucleus \(g_I\). Indeed, the magnetic moment of the entire atom \(\sqrt{F(F+1)}\,g_F\mu_B\) is composed of the magnetic moment of the shell \(\sqrt{J(J+1)}\,g_J\mu_B\) and the magnetic moment of the nucleus \(\sqrt{I(I+1)}\,g_I\,\dfrac{\mu_B}{1836.5}\) [see (3.1)], i.e.

Fig. 11. Splitting of the hyperfine structure of the term \(^{2}S_{1/2}\) in a weak magnetic field for \(I=3/2\).
a) level and transition diagram; b) intensities and positions of the Zeeman \(\pi\)- and \(\sigma\)-components.

\[ \sqrt{F(F+1)}\,g_F\mu_B = \left\{ \sqrt{J(J+1)}\,g_J\cos(\mathbf{J},\mathbf{F}) + \sqrt{I(I+1)}\,g_I \cos(\mathbf{I},\mathbf{F})\, \frac{1}{1836.5} \right\}\mu_B . \]

Replacing \(\cos(\mathbf{J},\mathbf{F})\) and \(\cos(\mathbf{I},\mathbf{F})\), by analogy with (2.19′), by their quantum-mechanical values, we find:

\[ g = g_J \frac{F(F+1)+J(J+1)-I(I+1)}{2F(F+1)} + \frac{g_I}{1836.5}\, \frac{F(F+1)+I(I+1)-J(J+1)}{2F(F+1)} . \tag{3.10} \]

If in (3.10) the second term on the right-hand side is neglected because of its smallness

multiplier \(\dfrac{1}{1836.5}\), then in the particular case of the level \({}^{2}S_{1/2}\) (Fig. 11)

\[ g_{F=I+1/2}=\frac{g_J}{2I+1}, \qquad g_{F=I-1/2}=\frac{-g_J}{2I+1}. \tag{3.11} \]

Thus, the magnitudes of the splitting (3.9) of both hyperfine-structure levels are, in this approximation, the same, while their sequence with respect to the values of \(m_F\) is reversed, owing to the difference of signs in (3.11) for \(F=2\) and \(F=1\). Consequently, measurement of \(\Delta\nu_H\) makes it possible to determine \(I\).

The entire splitting structure lies symmetrically with respect to the frequency of the hyperfine-structure component \(\nu_0\) at \(H=0\) (see Fig. 11, b). The number of \(\pi\)- and \(\sigma\)-components unambiguously gives the magnitude of the nuclear spin quantum number. For example, for \(I=1\) there are three \(\pi\)-components \((\Delta m_F=\pm1)\) and two \(\sigma\)-components \((\Delta m_F=0)\).

b) Strong field. By a strong field we shall again mean (see § 2) such a field in which the coupling between the vectors \(\mathbf I\) and \(\mathbf J\) is broken. Both these vectors precess independently of one another around the direction of the magnetic field \(\mathbf H\). In this case, instead of one common quantum number \(m_F\), two separate quantum numbers \(m_I\) and \(m_J\) appear, and it no longer makes sense to speak of hyperfine-structure sublevels determined by given values of \(F\) (see Fig. 11, a), or of their splitting. Therefore the energy of the Zeeman levels should be reckoned not from the levels of the initial hyperfine structure, but from the energy center of gravity \(E_S\) of the whole multiplet system. Thus, for each of the Zeeman levels (for given \(I\) and \(J\)) we have

\[ E=E_S+g_J m_J\mu_B H+g_I m_I\frac{\mu_B}{1836.5}\,H+A m_I m_J . \tag{3.12} \]

The second term on the right-hand side of (3.12) is the energy of the magnetic moment of the shell, and the third is the energy of the magnetic moment of the nucleus in the external field. The last term gives the average energy of the interaction between the nucleus and the shell.

The coefficient \(A\) depends on the quantum numbers \(I\) and \(J\), and in the particular case of the \({}^{2}S_{1/2}\) state has the form: \(A=\dfrac{h\nu_0}{I+1/2}\). Although in the case of strong fields the “rigid” coupling of the vectors \(\mathbf I\) and \(\mathbf J\) is completely “broken” and they precess independently around the direction of the field, nevertheless a magnetic interaction takes place between the nucleus and the shell, which is determined by the mean value of the cosine of the angle between the vectors \(\mathbf I\) and \(\mathbf J\):

\[ \sqrt{I(I+1)\,J(J+1)}\cos(\mathbf{IJ}) = \]

\[ = \sqrt{I(I+1)\,J(J+1)}\cos(\mathbf{IH})\cos(\mathbf{JH})=m_I m_J . \]

A scheme of the splitting of the energy levels of the same term \({}^{2}S_{1/2}\) in a strong field for \(I=\dfrac{3}{2}\) is shown in Fig. 12. The selection rules in this case read: \(\Delta m_I=\pm 1,\ \Delta m_J=0\) or \(\Delta m_I=0,\ \Delta m_J=\pm 1\). As will be seen below, for determining the magnetic moments of nuclei use is made of transitions corresponding to the first group of selection rules and shown in Fig. 12.

Fig. 12. Splitting of the hyperfine structure of the term \({}^{2}S_{1/2}\) in a strong magnetic field \((I=3/2)\) and transitions with \(\Delta m_I=\pm 1\) and \(\Delta m_J=0\).

Formula (3.12) for \(\Delta m_J=0\) gives the following values of the Zeeman frequencies:

\[ \nu=\frac{\nu_0}{I+\tfrac{1}{2}}\,m_J\Delta m_I+ \frac{\mu_B}{1836.5}\,\frac{Hg_I\Delta m_I}{h}. \tag{3.13} \]

For \(\Delta m_I=\pm 1\) and \(m_J=\pm \dfrac{1}{2}\) we have

\[ \nu=\frac{\nu_0}{2I+1}\pm \frac{\mu_B}{1836.5}\,\frac{Hg_I}{h}. \tag{3.14} \]

In contrast to the weak field, here the interaction term between the nuclear moment and the external field also plays an essential role. (For example, for \(H=6000\) oersteds and \(g_I=2\) the second term in (3.14) is equal to \(\sim 1\cdot 10^7\ \mathrm{sec}^{-1}\), while the first for \(I=\dfrac{3}{2}\) is \(\sim 20\cdot 10^7\ \mathrm{sec}^{-1}\).)

It follows from (3.14) that the frequencies of transitions with \(\Delta m_J=0,\ \Delta m_I=\pm 1\) in strong fields tend to two values differing from each other by

\[ \frac{2\mu_B}{1836.5}\,\frac{Hg_I}{h}, \]

from which the desired quantity \(g_I\) can be determined. The arithmetic mean of the frequencies (3.14) gives the value \(\dfrac{\nu_0}{2I+1}\) and, consequently, for a known value of the frequency

$\nu_0$ components of the hyperfine structure (without a magnetic field it is possible to determine the spin quantum number $I$ of the nucleus).

The theory makes it possible to obtain formulas also for the intermediate case of medium fields, which is very important for the interpretation of experimental results.

The simplest case is an atom in which the electron shell has no mechanical or magnetic moment at all (${}^1S_0$ level). In this case all the magnetism of the atom is determined by the small magnetic effect of the nucleus. In an external magnetic field the ${}^1S_0$ level is split into $2I+1$ sublevels, equally spaced from one another. The magnitude of this splitting, independently of the magnitude of the field $H$, is determined by the energy

\[ \Delta E = \mu_I H \cos(\mathbf{I}, \mathbf{H}) = \mu_I H \frac{m_I}{\sqrt{I(I+1)}} . \tag{3.15} \]

The selection rules in this case are: $\Delta m_I = \pm 1$ and, consequently, the transition frequency is equal to

\[ \nu_H = \frac{\mu_I H}{h\sqrt{I(I+1)}} = \frac{g_I H}{h}\,\mu_{\text{яд}} . \tag{3.16} \]

The order of magnitude of (3.16) for $g_I \sim 2$ and $H = 1000$ oersteds is $\sim 10^6\ \text{sec}^{-1}$, and the corresponding wavelength is $\lambda \sim 10^2$ meters. This frequency, in classical terminology, is simply the Larmor frequency of the nuclear spin.

Thus, measurement of the frequency (3.16) makes it possible to determine the $g_I$-factor of the nucleus. Unfortunately, there are few such simple cases. And if they do occur, it is chiefly for isotopes with an even atomic number and an even mass number, for which the spin and magnetic moment of the nucleus are absent (see, for example, Table II).

A convenient object for measurements is provided by molecules whose ground electronic state is devoid of magnetic moment (${}^1\Sigma_0$ state). In this case the magnetism of the molecule is due to the magnetic moments of the nuclei and, possibly, to the magnetic moment of the rotational states of the molecule; the latter, however, may be absent in normal states.

At first, experimental methods for determining nuclear magnetic moments developed in the direction of improving the technique of deflecting molecular beams in an inhomogeneous magnetic field.^14 However, real progress in this matter was achieved only with the development of the so-called magnetic resonance method. As early as 1922, Einstein and Ehrenfest^15,16 pointed out that a change in the orientation of the magnetic moments of atoms under the action of a magnetic field should be accompanied by the emission of electromagnetic waves in the radio-frequency region. On the basis of these considerations Ya. G. Dorfman^17 predicted the “photomagnetic effect,” which consists in a change of the magnetic state of paramagnets or ferromagnets under

under the action of radio-frequency irradiation. Dorfman also pointed out that, apparently, the selective absorption of radio waves in ferromagnets, discovered and studied in detail by V. K. Arkad'ev^18 and his school, is at least partly explained by the photomagnetic effect.

Majorana and then Gorter^19 considered theoretically the question of the influence of radio-frequency magnetic fields on atoms situated in a magnetic field. However, the idea of using radio-frequency fields for measuring the magnetic moments of atoms in atomic beams acquired real existence in the well-known works of Rabi^14. In very recent times the further development of this method has made it possible to determine nuclear moments not only in molecular beams, but also in the condensed phase of matter^9, ^15, ^20.

The magnetic resonance method amounts to producing resonance between the frequency of precession of the nuclear magnetic moment about the direction of a constant magnetic field and the frequency of a simultaneously applied radio-frequency magnetic field. In this case what is directly investigated are transitions of the atom between the levels of the hyperfine structure of a given multiplet, and not transitions between levels of the hyperfine structure of different multiplets, which are observed in ordinary spectroscopic investigation. As we have seen, the wavelengths corresponding to transitions between the Zeeman levels of the hyperfine structure of one multiplet lie in the interval \(1—10^4\) cm. This radiation has the character of magnetic dipole radiation. Rabi’s idea consists in observing this radiation not in the “natural” state, when its intensity is vanishingly small (because of the small probability of magnetic dipole radiation), but when it is artificially induced by an external alternating magnetic field, whose frequency is chosen to be in resonance with the frequency of the transitions between the Zeeman levels of the hyperfine structure. In the simplest case the frequency of precession of the nuclear moment about the direction of the external magnetic field is determined by formula (3.16). Determining it from the resonance observed in experiment, one finds the value of the nuclear Landé factor \(g_I\).

For the justification of this method, the solution of the question of what will happen to an already spatially quantized beam if it is passed again through another magnetic field is of great importance. Will spatial “requantization” take place, or will the atoms still merely precess about the direction of the field, retaining their previous state? This question was examined by a number of investigators^25, who showed that if the transition from one field to another takes place adiabatically, then the atoms precessing in the first field will likewise precess in the second. However, in the case of a nonadiabatic mode of transition, i.e., when the time during which the field changes direction is of the same order as, or less than, the period of Larmor precession, a reorientation of a certain number of atoms will occur. The results of all

experiments using the magnetic resonance method confirm this conclusion.

Fig. 13 shows the basic scheme of an experimental setup for determining \(g_I\) by the magnetic resonance method in molecular beams. A narrow beam of molecules or atoms, selected by diaphragms and emerging from the furnace \(O\), which is at constant temperature, passes between the pole pieces of magnet \(A\) and enters the space between magnets \(A\) and \(B\). In the middle of this space there is another diaphragm with slit \(S\), and beyond the interpole space of magnet \(B\) there is a receiver \(D\) (photographic plate, ionization chamber, etc.). The aperture of the furnace \(O\), the slit \(S\), and the receiver \(D\) lie on one straight line, which coincides with the direction of the beam in the absence of a deflecting magnetic field. Magnets \(A\) and \(B\) produce sharply inhomogeneous magnetic fields along the \(z\) axis \((\partial H/\partial z \ne 0)\)—up to \(10^5\) oersted/cm. These fields are directed parallel to one another, and their gradients are antiparallel, as shown in Fig. 13 by arrows. Molecules with a nonzero magnetic moment will be deflected in the direction of the field gradient if the projection of the moment on the \(z\) axis is positive \((\mu_z > 0)\), and against the gradient if the projection of the moment on this axis is negative \((\mu_z < 0)\). Molecules that leave the furnace \(O\) along the direction \(OSD\) (with angle \(\alpha = 0\)), if their moment is not vanishingly small and their velocity \(v\) is not too large, will be deflected away from the line \(OSD\) to one side or the other and will not enter slit \(S\). However, molecules that leave the furnace at other angles to the line \(OSD\) \((\alpha \ne 0)\) may, precisely as a result of deflection in the inhomogeneous magnetic field, enter slit \(S\). In general, for a molecule with given \(\mu_z\) and \(v\), one can find the initial value of the angle \(\alpha_0\) at which the molecule will pass through slit \(S\) (see Fig. 13). According to (1.4), the force deflecting molecules in an inhomogeneous field is equal to \(\sim \mu_z(\partial H/\partial z)\). The beam of molecules, def—

Fig. 13. Basic scheme of an apparatus for determining nuclear magnetic moments by the magnetic resonance method.

deflected from their original direction by the field of magnet \(A\) and having passed through the slit \(S\), would not, in the absence of magnet \(B\), fall on the receiver \(D\), but would be displaced relative to it by the distance

\[ d_A=\mu_z\left(\frac{dH}{dz}\right)_A \frac{\beta_A}{2Mv^2}, \]

where \(M\) is the mass of the molecule, and \(\beta_A\) is a factor depending on the geometry of the apparatus. In view of the fact that the field gradient of magnet \(B\) is antiparallel to the field gradient of magnet \(A\), the molecular beam undergoes, in the field of magnet \(B\), a deflection in the direction opposite to the deflection in the field of magnet \(A\). If, when the molecule passes from the field of magnet \(A\) into the field of magnet \(B\), \(\mu_z\) does not change, then the displacement of the beam relative to the receiver \(D\), caused by the field of magnet \(B\), will be equal to

\[ d_B=\mu_z\left(\frac{dH}{dz}\right)_B \frac{\beta_B}{2Mv^2} \]

and opposite in sign to \(d_A\). Therefore, if the absolute values of the deflections \(d_A\) and \(d_B\) are equal, the molecules of the beam are “focused” on the receiver \(D\). This focusing for a given \(\mu_z\) does not depend on the velocity of the molecules and requires only that the equality

\[ \beta_A\left(\frac{dH}{dz}\right)_A=\beta_B\left(\frac{dH}{dz}\right)_B, \]

be satisfied, which is easily achieved by the corresponding change in the dimensions of the apparatus and in the magnitudes of the gradients of the magnetic fields. Experimentally it was found that the number of molecules reaching the receiver when the fields are switched on is practically the same as in their absence.

Let us now place in the space between the slit \(S\) and magnet \(B\) a small magnet \(C\) (see Fig. 13), producing a homogeneous magnetic field \(H_0\), parallel to the fields \(H_A\) and \(H_B\). In this same space we create, in a direction perpendicular to \(H_0\), a radio-frequency alternating magnetic field \(H_1\) by means of two parallel wires placed between the pole pieces of magnet \(C\). Molecules with a given value of \(\mu_z\), entering the magnetic field \(H_0\), begin to precess about its direction with the Larmor frequency (3.16). If the frequency of the field \(H_1\) coincides with one of the allowed frequencies of transitions between the Zeeman terms of the hyperfine structure (according to the selection rules for magnetic-dipole radiation indicated above), then the corresponding transitions can take place. As a result of such a transition induced by the field \(H_1\), the component \(\mu_z\) of the magnetic moment of the molecule will change and, upon entering the inhomogeneous magnetic field \(H_B\), the molecule will undergo a deflection \(d'_B\) different (greater or smaller) than the deflection \(d_B\). The equality with the deflection \(d_A\), which existed at \(H_1=0\), will be violated, and thereby the number of particles reaching the receiver \(D\) will decrease in comparison with the number reaching it at \(H_1=0\).

Keeping the field \(H_0\) fixed and investigating the dependence of the intensity of the molecular beam reaching the receiver \(D\) on the frequency \(f\) of the magnetic

of the field \(H\), we find such a frequency \(f_{\min}\) at which the intensity proves to be the smallest (“resonance minimum”). According to what was set out above, the frequency \(f_{\min}\) coincides with one of the Zeeman frequencies of the hyperfine structure of the molecular spectrum at the given field \(H_0\).

Conversely, one may fix the frequency \(f\) of the magnetic field \(H_1\) and vary the intensity of the constant field \(H_0\), achieving resonance by changing the Larmor frequencies of the molecule until they coincide with the external frequency \(f\). Fig. 14 shows the typical form of a resonance minimum obtained for a beam of \(K^{39}\) atoms by varying the frequency \(f\) of the radio-frequency field. Fig. 15 shows an analogous resonance minimum\({}^{14}\), obtained for a beam of lithium chloride molecules LiCl in the state \({}^{1}\Sigma_0\), by choosing the intensity of the constant field \(H_0\) at a given frequency of the radio-frequency field \(H_1\). Substituting in (3.16), instead of \(\nu\), the value \(f_{\min}\) found from experiment and solving (3.16) for \(g_I\), we find

\[ g_I=\frac{4\pi Mc}{e}\frac{f_{\min}}{H_0}. \tag{3.17} \]

Fig. 14. Dependence of the intensity of the atomic beam (potassium atoms \(K^{39}\)) entering the detector on the frequency \(f\) of the alternating magnetic field \(H_1\) [according to the data of Kusch, Millman, and Rabi—Phys. Rev. 57, 765 (1940)].

Table III\({}^{14}\) gives the ratios of the experimental values \(f_{\min}/H_0\), obtained in various experiments (different \(H_0\), \(f\), and types of molecules) in determining the nuclear Landé factor for the nucleus of the lithium isotope \(Li^7\). The results of the measurements differ by less than \(0.5\%\), which testifies to the high accuracy of the method.

The measurements of the magnetic moments of the proton and deuteron\({}^{14}\) were carried out most thoroughly, but in principle they do not differ from the scheme described above. Recently, experiments have been performed to determine the magnetic moments of the proton and deuteron by means of the magnetic radio-frequency method, but no longer in molecular beams; rather, in solid or liquid specimens\({}^{20,21}\).

Of particular interest is the problem of determining the magnetic moment of the neutron \(\mu_n\). Direct measurement of the magnetic moment in a beam of free neutrons is associated with great technical difficulties because of the impossibility of obtaining a narrow, sharply defined beam. Therefore another method of deter-

moment $\mu_n$. A beam of neutrons, passing through some substance, undergoes scattering because of the interaction of the neutrons with the nuclei of the atoms of the substance (“nuclear scattering”). In the case of fast neutrons (whose energy is considerably higher than $kT$) this type of scattering is dominant. For slow neutrons (with thermal velocities), alongside nuclear scattering, a coequal role begins to be played by scattering caused by the interaction of the magnetic moment of the neutron with the magnetic field of the atomic shell. This “magnetic” scattering gives the greatest effect if the magnetic moments of the atoms of the substance are oriented in one direction, which, for example, occurs in ferromagnets. If by $\sigma_0$ we denote the effective cross section for nuclear scattering, and by $p$ the ratio of the effective cross section of magnetic scattering to the nuclear one, then, as was shown by calculations $^{22,23}$, the total cross section for the neutron is equal to $\sigma_0(1+p)$ or $\sigma_0(1-p)$, depending on whether the neutron spin is parallel

Fig. 15. Dependence of the intensity of the molecular beam entering the receiver (LiCl molecules predominantly in the state ${}^1\Sigma_0$) on the strength of the constant magnetic field $H_0$ [according to the data of Rabi, Millman, Kusch, and Zacharias — Phys. Rev. 55, 526 (1939)].

Table III

Ratio of the frequency of the radio-frequency magnetic field corresponding to the resonance minimum to the strength of the constant magnetic field $H_0$ in determining the Landé factor $g_I$ of the $Li^7$ nucleus, obtained with molecular beams of various lithium compounds

Type of molecules	$f_{\min}$ in $10^6\ \mathrm{sec}^{-1}$	$H_0$ in oersteds	$f_{\min}/H_0$
LiCl	5.611	3399	1651
LiCl	6.587	3992	1650
LiCl	2.113	1278	1654
$Li^7—Li^7$	3.084	1879	1652
LiF	5.621	3401	1653

($m_I=+\dfrac{1}{2}$) to the direction of magnetization of the scattering substance

(directed along some axis \(z\)) or antiparallel \(\left(m_I=-\frac{1}{2}\right)\). It is assumed here that the electron shell acts on the neutron as a magnetic dipole*). The value of \(p\) in this case, as experiments have shown, turned out to be approximately equal to \(0.1\).

Let us now suppose that an intense unpolarized beam of neutrons, directed along the \(x\)-axis, is incident at right angles on a plane-parallel ferromagnetic plate \(P\) of thickness \(x_1\), magnetized along the \(z\)-axis (Fig. 16). Let the intensity of the incident beam be \(I_0\). Then, by virtue of the fact that in an unpolarized beam one half of the neutrons have \(m_I=\frac{1}{2}\) and the other half \(m_I=-\frac{1}{2}\), the intensity of the beam after passing through the magnetized plate will be equal to

Fig. 16. Schematic diagram of an apparatus for measuring the magnetic moment of the neutron.

\[ I_{\mathrm{m}}=\frac{1}{2}I_0\left[e^{-n x_1\sigma_0(1+p)}+e^{-n x_1\sigma_0(1-p)}\right]\sim \]

\[ \sim I_0 e^{-n x_1\sigma_0}\left(1+n^2\sigma_0^2 x_1^2 p^2\right), \tag{3.18} \]

where \(n\) is the number of scattering centers per unit volume, and it is assumed that \(p\ll 1\). If the plate \(P\) were unmagnetized, then the intensity of the beam transmitted through it would be

\[ I_{\mathrm{nm}}=I_0 e^{-n x_1\sigma_0}. \tag{3.19} \]

Thus, magnetic scattering somewhat increases the transparency of the magnetized plate \(P\) for neutrons \(\left(I_{\mathrm{nm}}<I_{\mathrm{m}}\right)\). The magnetized plate \(P\) plays the role of a polarizer, since the number of neutrons \(N_{1/2}\) with \(m_I=\frac{1}{2}\) after passage through the plate \(P\) becomes—

* A detailed and exhaustive analysis of this interaction and criticism of some incorrect conclusions\({}^{22}\) are given by A. B. Migdal\({}^{23}\).

…less than the number of neutrons \(N_{-1/2}\) with \(m_I=-\dfrac{1}{2}\), namely, from (3.18) it follows that

\[ \frac{N_{1/2}}{N_{-1/2}} \sim 1-2 n x_1 \zeta_0 p, \tag{3.20} \]

i.e., the neutron beam becomes partially polarized. If now this partially polarized neutron beam is passed through a second ferromagnetic plate \(A\) (Fig. 16) of thickness \(x_2\), then it will play the role of an analyzer. If the magnetization of the plate \(A\) is the same as that of the polarizer \(P\) (along the \(z\)-axis), then the transmission of the beam through it is equivalent to increasing the thickness of the scattering substance, both for nuclear and for magnetic scattering, i.e.

\[ I_{M(1+2)}=I_0 e^{-n\sigma_0(x_1+x_2)} \left[1+n^2\zeta_0^2(x_1+x_2)^2p^2\right]. \tag{3.21} \]

If, however, the analyzer \(A\) is magnetized antiparallel to the polarizer \(P\) (opposite to the \(z\)-axis), then the action of the plate \(A\) for nuclear scattering is still equivalent to an increase in thickness, while for magnetic scattering the thickness \(x_2\) must enter into (3.21) with the opposite sign (with respect to \(x_1\)), i.e.

\[ I_{M(1-2)}=I_0 e^{-n\sigma_0(x_1+x_2)} \left[1+n^2\zeta_0^2(x_1-x_2)^2p^2\right]. \tag{3.22} \]

When the method of magnetic resonance is used to determine the magnetic moment of the neutron\({}^{24}\), in the space between the polarizer \(P\) and the analyzer \(A\) (Fig. 16) a constant magnetic field \(H_0\), directed along the \(z\)-axis, is produced, together with an alternating radio-frequency field \(H_1\) perpendicular to it, directed along the \(y\)-axis. The neutrons in the partially polarized beam of intensity \(I_M\) that has passed through the polarizer \(P\) will precess about the direction of the field \(H_0\). The number of “Zeeman levels” for neutrons is only two (for \(m_I=\pm \dfrac{1}{2}\)), and only a single transition between them is possible.

Under the action of the alternating magnetic field \(H_1\) with frequency

\[ f=\frac{\mu_n H_0}{h\sqrt{\frac{1}{2}\left(\frac{1}{2}+1\right)}} \]

[see (3.16)] in a beam of neutrons moving in the field \(H_0\), transitions between the two “Zeeman levels” will be induced. The number of these transitions is determined by the strength of the magnetic field \(H_1\) and by the probability \(W_{1/2,-1/2}\) of such a transition. As a result, some fraction of the neutrons in the beam will change the value of \(m_I\) to the opposite one, and the numbers \(N_{+1/2}\) and \(N_{-1/2}\) will be changed. Because of subsequent scattering in the analyzer \(A\) (Fig. 16), the intensity \(I'\) of the neutron beam reaching the receiver \(D\) will be less than their intensity \(I_{(H_1=0)}\) that occurred when \(H_1=0\), i.e. \(I'=I_{(H_1=0)}-\Delta I\). The probability that a neutron during the time \(t\) will pass from the state with

\(m_I=+\dfrac{1}{2}\) into the state with \(m_I=-\dfrac{1}{2}\), is equal to\({}^{25}\)

\[ W'_{1/2,-1/2} = \frac{f^2\sin^2\vartheta}{f^2+\nu_H^2-2f\nu_H\cos\vartheta} \sin^2\pi t\left[f^2+\nu_H^2-2f\nu_H\cos\vartheta\right], \tag{3.23} \]

where \(\operatorname{tg}\vartheta=H_1/H_0\), \(f\) is the frequency of the field \(H_1\), and \(\nu_H\) is the Larmor frequency of the neutron in the field \(H_0\), given by (3.16). From (3.23) it is seen that the probability \(W'_{1/2,-1/2}\) depends on time, i.e., on the velocity of the neutron with which it passes through the region in which the field \(H_1\) acts. Since the neutron beam is very inhomogeneous in velocities, the square of the sine in (3.23) may be replaced by its mean value, i.e. \(1/2\), and, using the fact that in experiments always \(H_1 \ll H_0\), formula (3.23) may be approximately replaced by the simpler one

\[ \overline{W}_{1/2,-1/2}^{\,t} \approx \frac{1}{2}\, \frac{1}{1+\left[2\dfrac{H_0}{H_1}\left(1-\dfrac{\nu_H}{f}\right)\right]^2}. \tag{3.24} \]

From (3.24) it is immediately seen that the probability \(\overline{W}_{1/2,-1/2}^{\,t}\) has a maximum when the frequency of the field \(H_1\) coincides with the Larmor frequency of the neutron \((f=\nu_H)\). The resonance will be the sharper, the smaller the ratio \(H_1/H_0\).

The scheme of the experimental apparatus for determining the magnetic moment of the neutron by the resonance method is shown in Fig. 17\({}^{14}\).

Fig. 17. Schematic of the experimental apparatus for determining the magnetic moment of the neutron.

A beam of neutrons is obtained either as a result of a nuclear reaction (for example, \(\mathrm{Be}^9+\mathrm{D}^2=\mathrm{B}^{10}+n^1\)) by irradiating some target with a beam of accelerated particles (protons, deuterons) in an accelerator \(A\) (cyclotron, synchrotron), or from a nuclear reactor. The paraffin block \(B\) (Fig. 17) is used to slow the neutrons down to thermal velocities. Then, with the aid of a cadmium tube \(Cd\), which plays the role of a diaphragm, a narrow beam of slow neutrons is selected. This beam passes through the polarizer \(C\), magnetized in a plane perpendicular to the beam and to the plane of the drawing (along the \(z\)-axis), and enters a strong-

...a constant field \(H_0\) (\(\sim 600\) oersted), directed along the \(z\)-axis and producing precession of the neutron moments. Here, too, the neutrons are acted upon by a radio-frequency alternating field \(H_1\) (\(\sim 10\) oersted), produced by a solenoid. The beam then passes through analyzer \(D\) and finally enters receiver \(E\), in the form of a chamber filled with boron trifluoride (\(\mathrm{BF}_3\)) and shielded from the entry of “extraneous” neutrons; in this chamber the intensity of the neutron beam that has passed through the entire apparatus is measured. Fig. 18 shows a typical curve of the resonance minimum obtained on such an installation.

These experiments were also used to determine the sign of the neutron magnetic moment,\(^{24}\) which, as was to be expected, turned out to be negative.

Dorfman\(^{26}\) rightly pointed out that the method of magnetic resonance proposed by him and now so widely used makes it possible to determine only the nuclear Landé factor \(g_I\). In order to find the magnetic moment of the nucleus, it is necessary additionally to determine the nuclear spin, for example from spectral measurements. In the same work Dorfman proposed a new modification of the magnetic-resonance method, which makes it possible to determine at once both the magnetic moment and the spin of the nucleus. The principle of the method consists in using measurements of the additional magnetic susceptibility \(\chi_n\), caused by the orientation of nuclear spins in a constant magnetic field (\(\chi_n\) is of the order of \(10^{-10}—10^{-13}\)). Under ordinary conditions, nuclear magnetism is masked by the magnetism of the electrons, which greatly exceeds it. However, if perpendicular to a strong constant magnetic field \(H_0\) one applies a weak alternating field \(H_1\) and chooses its frequency in resonance with the Larmor frequency \(\nu_H\) of the nuclear spins, then the contribution of the nuclei to the paramagnetic susceptibility of the substance can be eliminated. Dorfman proposed the following schematic arrangement for the experimental setup.\(^{26}\)

Fig. 18. Curve of the resonance minimum for neutrons.

The substance under study is placed in an ampoule \(AB\), which is arranged symmetrically between the poles of an electromagnet \(NS\) (Fig. 19). The ends of this ampoule (\(A\) and \(B\)) are in the same nonuniform field. By virtue of the symmetry of the field, the ampoule is in equilibrium. If, however, at end \(A\) one creates, for example, a radio-frequency field \(H_1\),

perpendicular to \(H_0\), then the nuclear moments will begin to precess about the direction of the field \(H_1\) and will leave the magnetization along \(H_0\). Consequently, now a force

\[ f=\chi_n v H_0 \frac{dH_0}{dz} \]

(where \(v\) is the volume of the part of the ampoule located in the field \(H_1\)) will act on the end of ampoule \(A\), tending to move the ampoule to the left. This force can in principle be measured (for example, with the aid of sensitive torsion balances). Its maximum value will occur at resonance:

\[ f_{\max}=\nu_H. \]

Measuring the frequency \(H_1\) at resonance \((f_{\max})\), we find \(\chi_n\) and \(\nu_H\), which are connected with the desired quantities \(g_I\) and \(I\), namely, by (3.16) we have

\[ f_{\max}=\frac{g_I H}{h}\,\mu_{\text{яд}} \]

and by the formula for the paramagnetic susceptibility (see below, § 8),

\[ \chi=\frac{4N\mu_{\text{яд}}^{2}}{3kT} I(I+1), \]

where \(N\) is the number of atomic nuclei of the given kind per unit volume, \(k\) is Boltzmann’s constant, and \(T\) is the absolute temperature. From these formulas one can simultaneously and directly calculate \(I\) and \(g_I\), and thereby also \(\mu_I\), using measurements in a single apparatus.

Fig. 19. Scheme of Dorfman’s experiment for determining nuclear magnetic moments.

(To be continued in the next issue)

LITERATURE CITED FOR PART I

§ 1

1. See, for example, E. V. Shpolsky, Atomic Physics, Gostekhizdat (1944).
2. Ya. I. Frenkel, Electrodynamics, vol. 1, GTTI (1934), chs. VII and X.
3. I. E. Tamm, Zeits. f. Physik, 55, 199 (1929).
4. W. Gerlach und O. Stern, Ann. d. Physik, 74, 673 (1924).
5. See, for example, Ya. I. Frenkel, Wave Mechanics, vol. II, GTTI (1935), ch. VI.

1. W. Heisenberg, Physical Foundations of Quantum Mechanics, GTTI (1932), pp. 122–124.
2. L. D. Landau, DAN 26, 436 (1940); A. A. Sokolov, JETP 16, 3 (1946).

§ 2

1. O. Stern, Phys. Rev. 51, 852 (1937).
2. Z. Zavoisky, DAN 57, 887 (1947); see also Ya. G. Dorfman, Izv. Acad. Sci. USSR, Phys. Ser. 11, 598 (1947).

§ 3

1. See, for example, I. E. Tamm and S. Altshuler, DAN 1, 455 (1934); N. Kemmer, W. Heitler and H. Fröhlich, Proc. Roy. Soc. 166, 154 (1938).
2. Ya. I. Frenkel, Izv. Acad. Sci. USSR, Phys. Ser. 11, 593 (1947).
3. L. Dobretsov and A. Terenin, Naturwiss. 16, 656 (1928).
4. See, for example, S. E. Frisch, Atomic Nuclei and Spectra, GTTI (1934), or his Atomic Spectra, GTTI (1933).
5. See, for example, J. Kellogg and S. Millman, UFN 34, 72 (1948), or N. Korfeman, Naturwiss. 29, 563, 581 (1941).
6. Ya. G. Dorfman, Izv. Acad. Sci. USSR, Phys. Ser. 11, 598 (1947).
7. A. Einstein und P. Erenfest, Zeits. f. Physik 11, 21 (1922).
8. Ya. G. Dorfman, Zeits. f. Physik 17, 98 (1923).
9. V. K. Arkad’ev, Magnetic Spectroscopy, Moscow (1924); Electromagnetic Processes in Metals, vol. II, ONTI (1936).
10. E. Majorana, Nuovo Cim. 9, 43 (1932); C. J. Gorter, Physica 3, 995 (1936).
11. E. M. Purcell, H. C. Torrey and R. V. Pound, Phys. Rev. 69, 37 (1946); F. Bloch, W. Hansen and M. Packard, Phys. Rev. 69, 127 (1946); 70, 474 (1946); F. Bloch, Phys. Rev. 70, 460 (1946).
12. W. Arnold and A. Roberts, Phys. Rev. 70, 878 (1947).
13. F. Bloch, Phys. Rev. 50, 259 (1936), 51, 994 (1937).
14. A. B. Migdal, DAN 20, 555 (1938); JETP 10, 5 (1940).
15. L. Alvarez and F. Bloch, Phys. Rev. 57, 111 (1940).
16. J. Schwinger, Phys. Rev. 51, 648 (1937); I. Rabi, Phys. Rev. 51, 652 (1937); F. Bloch and I. Rabi, Rev. Mod. Phys. 17, 237 (1945).
17. Ya. G. Dorfman, DAN 57, 769 (1947).

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