MODERN THEORY OF MAGNETISM*

S. V. Vonsovskii

III. MAGNETISM OF MATTER—FERROMAGNETISM

CONTENTS

1. Phenomenological description of the ferromagnetic state of matter. 11. Theory of the spontaneous magnetization of ferromagnets. 12. Theory of the technical magnetization curve.

The phenomenon of ferromagnetism was discovered in remote antiquity; however, our modern ideas about the causes and laws governing this phenomenon of nature, which is important for practice, were established mainly during the last two or three decades. The wide introduction of ferromagnetic materials into technology also began relatively recently—from the end of the last century.

The basic characteristic of any ferromagnetic substance is the magnetization curve (see § 5), which gives the relation between its magnetic moment (magnetization) and the external magnetic field. The honor of discovering the true magnetization curve of ferromagnets belongs to the well-known Russian physicist A. G. Stoletov (1871),^70 who was the first to take proper account of the influence of the shape of a ferromagnetic specimen on the form of the magnetization curve and who developed a method for recording this curve that eliminates the influence of shape (ballistic measurements of the magnetization of ferromagnetic rings); this method has become firmly established in the modern practice of physical experiment as one of the basic techniques for studying the magnetic properties of ferromagnets. In A. G. Stoletov’s work, curves of magnetic permeability are given for the first time (see Fig. 22), now so well known to physicists and electrical engineers. All subsequent work on the study of magnetization curves of ferromagnets is based on the results of Stoletov’s research. Unfortunately, and to our shame, this circumstance is either

* For Parts I and II see UFN 35, 514 and 36, 30 (1948).

S. V. VONSOVSKII

was either completely passed over in silence, or insufficiently sharply emphasized even in our domestic literature, and therefore the priority of Russian science in the discovery of the magnetization curve of ferromagnets had not been restored until recently.

The study, begun by Stoletov, of the magnetization curves of ferromagnets, the determination of the temperature dependence of their magnetic properties, and the investigation of the connection of the latter with other, nonmagnetic properties of ferromagnetic bodies made it possible to approach the construction of a theory of ferromagnetism. Naturally, at first this theory was of a purely phenomenological—thermodynamic—character. The theory was based on the hypothesis of the so-called “molecular field.” The concept of the existence within a ferromagnetic body of a special “molecular field” was first introduced in 1892 by the Russian physicist B. L. Rozing^71. However, Rozing’s works remained undeservedly unnoticed*), and until recently the concept of the “molecular field” was unjustly associated exclusively with the name of the French physicist P. Weiss^72.

By 1907 the thermodynamic theory of the “molecular field” had in its main features been completed by the works of Weiss and his school and had received qualitative experimental confirmation. Two hypotheses were laid at the foundation of this theory. According to the first of them, it was assumed that the principal characteristic property of the ferromagnetic state of a substance is the presence in it, in a certain range of temperatures (from 0°K to the so-called Curie point), of spontaneous magnetization, independent of the presence of an external magnetizing field. On the other hand, it was known from experiment that in the absence of an external field (if one excludes the secondary phenomenon of magnetic hysteresis) every ferromagnetic body as a whole is demagnetized. This fact forced the formulation of the second basic hypothesis of the theory, according to which every ferromagnetic specimen below the Curie temperature is divided into small regions (domains) possessing homogeneous spontaneous magnetization. In the absence of an external magnetic field, the directions of the magnetization vectors in these regions are distributed over the volume of the specimen in such a way that its resultant magnetic moment is equal to zero. Only under the influence of an external field is such a “zero” distribution of magnetization over the regions disturbed, and the specimen becomes magnetized as a whole.

In accordance with these two hypotheses, the theory of ferromagnetism can be divided into two parts:

*) In his works Rozing attempts to construct a general dynamic theory of magnetism on the basis of the theory of the electromagnetic field. It is clear that at that time modern ideas about the electronic structure of the atom were still lacking. Nevertheless, in a certain sense Rozing introduced the concept of the “molecular field,” calling it a “partial magnetic force,” caused by the “magnetic motion of matter,” which found its full explanation only in the modern quantum theory of ferromagnetism.

1) the theory of spontaneous magnetization, explaining the very nature of ferromagnetism, and

2) the theory of ferromagnetic domains, or the theory of the technical magnetization curve, explaining the behavior of ferromagnetic bodies in an external magnetic field.

It is in this sequence that the theory of ferromagnetism will be presented here.

10. PHENOMENOLOGICAL DESCRIPTION OF THE FERROMAGNETIC STATE OF A SUBSTANCE

a) Phenomenological theory of the “molecular field”

In a phenomenological, thermodynamic description of ferromagnetism, the interaction forces leading to the existence of spontaneous magnetization are taken into account by introducing a certain hypothetical “molecular field,” whose magnitude is connected with the magnetization. To illustrate the theory, let us choose the simplest model of a ferromagnet—a free gas of electron spins (see § 8). The \(N\) electrons of this gas are divided, according to the two possible spin orientations, into \(r\) “right” and \((N-r)=l\) “left” ones. The relative magnetization of the gas “to the right” is equal to

\[ y=\frac{1}{N}(r-l) \]

or

\[ r=\frac{N}{2}(1+y)\quad \text{and}\quad l=\frac{N}{2}(1-y). \tag{10.1} \]

To obtain the magnetic equation of state (see § 4), it is necessary to determine the free energy of the gas \(F\) as a function of \(y\) and to find its minimum. The entropy of the gas \(S(y)\), neglecting the interaction between electrons, is equal to

\[ S(y)=k\ln \frac{N!}{r!\,l!} \tag{10.2} \]

(\(k\) is Boltzmann’s constant). Let us also assume that, in the absence of an external field, the energy of the gas \(U\) does not depend on the magnetization. Then, according to (10.2), (10.1), and (4.14), we obtain:

\[ F(y)=TS(y)\simeq \frac{1}{2}NkT\bigl[(1+y)\ln(1+y)+(1-y)\ln(1-y)\bigr]. \tag{10.3} \]

From the condition for the minimum of (10.3) we find that \(y=0\), i.e., there is no spontaneous magnetization. In order to obtain the possibility of its existence, it is necessary to take into account the dependence of the energy of the ferromagnet \(U\) on \(y\). Following Rosin and Weiss, we postulate this

dependence in the simplest form [using the even character of the function \(U(y)\)]

\[ U=-NA_1y^2, \tag{10.4} \]

where \(A_1>0\) is an as yet unknown, as to its nature, energy of the “molecular field.” In this case the minimum of \(F\) gives:

\[ \frac{4A_1}{kT}\,y=\ln\frac{1+y}{1-y} \quad\text{or}\quad y=\operatorname{th}\frac{\Theta}{T}\,y, \tag{10.5} \]

where

\[ \Theta=\frac{2A_1}{k}. \tag{10.6} \]

Analysis of (10.5) shows that at temperatures below the critical value \(\Theta\) from (10.6), the thermodynamically stable state corresponds to a value of the spontaneous magnetization different from zero. Above this temperature we always have \(y=0\). Thus, if the energy of a ferromagnet depends on the magnetization according to (10.4), then it possesses a spontaneous magnetic moment. As the temperature rises from \(0^\circ\mathrm{K}\) to \(\Theta\), this moment falls according to the law (10.5) from the maximum value \(y=1\) \((y=I_s/I_0\), where \(I_0=N\mu_B\) is absolute saturation, and \(\mu_B\) is the Bohr magneton) to \(y=0\). The physical significance of this conclusion can be established if one determines the order of magnitude of the Curie temperature \(\Theta\), as determined by the energy \(A_1\), from (10.5). It is known from experiment that in typical ferromagnets \(\Theta\sim1000^\circ\). Therefore, for the energy \(A_1\) we obtain from (10.6) a value \(\sim10^{-13}\) erg per atom. Such a magnitude of atomic energy can be determined only by electric forces between electrons, for the maximum value of the magnetic energy between two electrons at atomic distances does not exceed \(10^{-16}\) erg (therefore the “magnetic” molecular field can lead to “ferromagnetism” only with a Curie point \(\sim1^\circ\mathrm{K}\)).

In Fig. 43 the theoretical curve \(y\left(\dfrac{T}{\Theta}\right)\), calculated from (10.5), is shown, and there also are plotted the experimental data for real ferromagnets. From comparison of the curve and the experimental data one sees the general qualitative validity of the theory. For a more detailed comparison, let us give the asymptotic dependences \(y(T)\), respectively at high \((T\sim<\Theta)\) and low \((T\sim0)\) temperatures.

\[ I_s\simeq I_0\sqrt{\frac{3}{\Theta}}\sqrt{\Theta-T} \quad (T\sim<\Theta), \tag{10.7} \]

\[ I_s\simeq I_0\left(1-2e^{-2\Theta/T}\right) \quad (T\sim>\Theta). \tag{10.8} \]

It turns out that formula (10.7) agrees rather well with experiment, whereas (10.8) clearly contradicts it [see below, section в].

With the presence of an external magnetic field, the term \(-\mathbf I\cdot\mathbf H\) \((I=N\mu_B y)\) must be added to the energy (10.4). Then, from the condition of the minimum of the free energy, instead of (10.5) we obtain

\[ y=\operatorname{th}\left(\frac{\Theta}{T}y+\frac{\mu_B H}{kT}\right) =\operatorname{th}\frac{\mu_0}{kT}\left(\frac{k\Theta}{\mu_B}y+H\right). \tag{10.9} \]

From (10.9) one sees the formal possibility of likening the quantity

\[ \frac{k\Theta}{\mu_B}y=H_{\mathrm{mol}} \tag{10.10} \]

to a “molecular field”; at \(T=0^\circ\mathrm K\), \(y_0=1\) or \(I_0=N\mu_B\). Thus, at \(0^\circ\mathrm K\) any weak field magnetizes a ferromagnet along its direction up to saturation, and the magnetization curve

Fig. 43. Temperature dependence of spontaneous magnetization.

has the form of straight lines (\(T=0\)) shown in Fig. 44. With increasing temperature, if \(T\) is not very close to \(\Theta\), in weak fields \((H\ll H_{\mathrm{mol}})\) the magnetization is practically equal to its value from (10.5), smaller than the saturation value \(I_0\). As the field increases, the magnetization increases monotonically and, as \(H\to\infty\), tends to \(I_0\). From (10.9) it is evident that, over a wide range of values of \(H\) and \(T\), the increase of magnetization with field (the true magnetization of a ferromagnet) is so small that it practically plays no role in comparison with the magnitude of the spontaneous magnetization given by (10.5) (the technical saturation of a ferromagnet is \(I_s\)). Thus, analysis of (10.9) allows one to assert that the influence of the “internal” forces on the magnitude of the magnetization may be likened to the influence of a certain “molecular field” \(H_{\mathrm{mol}}\) (10.10). As long as \(y\) is not very small in comparison with unity, which occurs for almost all temperatu-

at temperatures \(T < \Theta\), except in the immediate vicinity of the Curie point, the value \(H_{\mathrm{mol}}\) turns out to be of the order of \(10^7\) oersteds. This value considerably exceeds the external magnetic fields attainable in practice (the largest fields obtained by P. L. Kapitsa\({}^{73}\) did not exceed several hundred thousand oersteds, \(\sim 10^5\) oersteds).

Fig. 44. Idealized magnetization curves of a ferromagnet at various temperatures.

Near \(T = \Theta\) the influence of the external field becomes more active, and the phenomenon of technical saturation gradually disappears (see Fig. 44). Above the Curie point the ferromagnet is transformed into a paramagnet with a linear dependence of \(I\) on \(H\) and with a susceptibility obeying the Curie–Weiss law (5.2)

\[ \chi = \frac{N\mu_B^2}{k\left(T-\dfrac{2A_1}{k}\right)} = \frac{N\mu_B^2/k}{T-\Theta}, \tag{10.11} \]

which is in good qualitative agreement with experiment\({}^{*}\).

The existence of spontaneous magnetization manifests itself not only in the peculiar magnetic behavior of ferromagnets. Experiment shows that the nonmagnetic properties of ferromagnetic bodies also differ by features (anomalies) of the most varied character, especially near the Curie point\({}^{**}\). Thus, for example, the heat capacity

* At \(T\) close to \(\Theta\), experiment gives deviations from (10.11), and, in addition, instead of \(\Theta\) (the ferromagnetic Curie point) in (10.11) one must substitute a somewhat different quantity \(\Theta_p\) (\(\ne \Theta\)) (the paramagnetic Curie point), see Section 6).

** These anomalies have a maximum with a sharp jump on the high-temperature side; by them the Curie point is in fact determined experimentally with precision\({}^{85,78}\).

of ferromagnets has an anomalous temperature course in comparison with other types of solids. In the most general form, the anomaly of the heat capacity may be explained by the fact that in ferromagnets an additional term of the type (10.4), essentially dependent on the magnetization, appears in the free energy. This term also gives an additional (in comparison with non-ferromagnetic bodies) heat capacity \(\Delta c\), which is necessary for the destruction of the spontaneous magnetization when the ferromagnet is heated,

\[ \Delta c=\frac{dU}{dT}=-\frac{A_1}{N\mu_B^2}\,\frac{d\left(I_s^2\right)}{dT}. \]

Using (10.7), we find that at the Curie point the heat capacity of ferromagnetic bodies must undergo a jump of magnitude

\[ \Delta c_{(T=\Theta)}=\frac{3}{2}\,kN=\frac{3}{2}\,R \tag{10.12} \]

(\(R\) is the universal gas constant, if \(N\) is the number of electrons in a mole of spin gas). Figure 45 gives the curve of the specific heat of nickel from the work of Dorfman and Janus[^74]. From this curve one sees the jump in heat capacity at \(T\sim\Theta\), the magnitude of which, when recalculated to the atomic heat capacity, is equal to \(1.08R\). Thus, the general picture of the observed heat-capacity anomaly agrees with the prediction of the theory. It should be noted that small remnants of the heat-capacity anomaly are also present above the Curie point [see Section 6].

Fig. 45. Temperature dependence of the heat capacity of nickel. (After Dorfman and Janus.)

b) Refinement of the theory of the “molecular field” within the framework of the quasiclassical statistical method[^78]

In the theory of the “molecular field” presented in Section a), the interaction between electrons was taken into account too crudely—by the simple introduction of an additional term in the energy (10.4), depending on the magnetization of the body. Such a method of accounting for the interaction is equivalent to taking into consideration only the order in the arrangement of the spins at large distances (“long-range magnetic order”). On the other hand, as will be shown below, the energy \(A_1\) leads to forces acting appreciably only at short distances.

Therefore, the thermodynamically equilibrium state of a ferromagnet must be determined by short-range order between the spins. Here, however, it should be pointed out that, while remaining within the framework of the classical statistical method, there is a danger of obtaining results which may even fail to agree qualitatively with experiment, for ferromagnetism is an essentially quantum phenomenon. Nevertheless, the quasiclassical theory of the “molecular field,” with allowance for order at short distances, is unconditionally useful from an illustrative and qualitative point of view.

For the first time, allowance for “magnetic” order at short distances was made in a paper by the Soviet physicist Stilbans^75, who, using the quasi-chemical method developed in the theory of ordering alloys^76, gave a theoretical explanation of the “residues” of the heat-capacity anomaly at temperatures above the Curie point. These “residues” are caused by the necessity of destroying the order at short distances, which is still partially preserved above the Curie point as well.

The development of this method was continued by Vonsovskii^77,78, who gave a theoretical explanation of the above-mentioned experimental fact of the difference between the ferromagnetic and paramagnetic Curie points and showed that the paramagnetic susceptibility near the Curie point has a finite value, and does not tend to infinity, as follows from (10.11), where short-range order is not taken into account.

Kaner^79 developed a quasiclassical statistical theory of ferromagnetism, using the method of moments, which is also applied in the theory of ordering alloys (Kirkwood and others).

c) A rigorous thermodynamic theory of the ferromagnetic transition

The success of the phenomenological theory of the “molecular field,” despite its plainly simplified character, indicates that a number of its results, which have found good confirmation in experiment, can be obtained from rigorous thermodynamic theory.

As is known, the transition from the ferromagnetic state to the paramagnetic one, according to P. S. Ehrenfest’s classification^80, is a phase transition of the 2nd order. At the Curie point, where the spontaneous magnetization (the long-range “magnetic” order) disappears, the ferromagnet changes its symmetry discontinuously. This jump does not produce a discontinuous change in the first derivatives of the thermodynamic potential \(\Phi\) (entropy, volume, etc.); therefore, for example, at the Curie point there is no latent heat of transformation. However, the second derivatives of \(\Phi\) (heat capacity, coefficient of compressibility, etc.) undergo a jump at \(T=\Theta\).

In his fundamental works on the thermodynamic theory of phase transitions, Landau^81,76 gave a general method for considering transitions of the 2nd order; this method was applied to the case of ferromagnetism by Vonsovskii^82,78.

A second-order phase transition is described thermodynamically by introducing a symmetry parameter of the body, $\xi$. In the case of a ferromagnetic transformation, the role of $\xi$ is played by the square of the relative magnetization,

\[ \xi=y^2=\frac{I_s^2}{I_0^2}. \tag{10.13} \]

Near the Curie point, $y \ll 1$, and the potential $\Phi$, considered as a function of the pressure $p$, temperature $T$, and magnetization $y$, may be represented in the form of a series in increasing powers of the small parameter $\xi$, and one may restrict oneself to the first terms of the expansion:

\[ \Phi(p,T,\xi)=\Phi_0(p,T)+\Phi_1(p,T)\xi+\Phi_2(p,T)\xi^2+\ldots \tag{10.14} \]

It can be shown that near the Curie point

\[ \Phi_1(p,T)\simeq a(T-\Theta), \]

\[ \Phi_2(p,T)\sim b \left( \frac{\partial b}{\partial T}\sim 0,\quad b>0 \right), \]

therefore from the condition of thermodynamic equilibrium for (10.14) we obtain:

\[ y=\frac{I_s}{I_0}=\sqrt{\frac{a}{2b}}\sqrt{\Theta-T}. \tag{10.15} \]

Thus, up to the factor $\sqrt{a/2b}$, which is independent of temperature, (10.15) coincides with the result (10.7) of the approximate theory of the “molecular field.” However, in contrast to (10.7), formula (10.15) has here been obtained under the most general assumptions of a purely thermodynamic type, without any simplifications. It is precisely for this reason that (10.15) should be regarded as a confirmation of the qualitative correctness of the approximate “molecular-field” theory set forth above for temperatures close to the Curie point.

The general thermodynamic theory further makes it possible to obtain the jump in heat capacity at the Curie point and gives a general picture of the diagrams of phase equilibrium between ferromagnet and paramagnet in the planes of various thermodynamic parameters[^78]. In particular, in the phase plane pressure—temperature there is a line of Curie points. In the phase plane temperature—magnetic field, however, there is no such line, since for $H\ne 0$ it is generally meaningless to speak of a ferromagnetic phase transformation, for in this case the long-range “magnetic” order does not disappear. Consequently, on the $T,H$ diagram the Curie point is an isolated point lying on the axis $H=0$[^83].

The thermodynamic theory presented can be refined, according to the scheme indicated by Landau[^84] for the example of the scattering of X-rays in ordering alloys near the temperature of the phase ...

transition, in which he took into account fluctuations of the order parameter at temperatures above the phase-transition point. This method was applied to ferromagnetism by Vonsovskii^77,78, who determined the temperature dependence of heat-capacity anomalies above the Curie point.

11. THEORY OF THE SPONTANEOUS MAGNETIZATION OF FERROMAGNETS

a) From estimates of the orders of magnitude of the energy \(A_1\) in (10.4) (from the values of the Curie points for real ferromagnetic materials), obtained in the phenomenological theory of the “molecular field,” there followed the complete inadequacy of the “magnetic conception” for explaining the causes of the appearance of spontaneous magnetization. The experimentally nonmagnetic nature of the “molecular field” in ferromagnets was proved in the experiments of Ya. G. Dorfman^87 (1927), who measured the deflections of electrons that had passed through magnetized ferromagnetic foil.

The resolution of the “riddle” of ferromagnetism could be given only by quantum mechanics, one of whose consequences is the existence of a dependence of the electrostatic energy of an electron ensemble on its magnetization, caused by the Pauli principle, as we have already seen in the example of the paramagnetism of a free-electron gas (§ 8).

Moreover, in experiments on the gyromagnetic effect^78 it was firmly established that the elementary carriers of the magnetic moment in ferromagnets are electron spins. It therefore became obvious that ferromagnetism is a peculiar form of spin paramagnetism. From comparing this experimentally established fact with the magnetic properties of paramagnets of the alkali-metal type considered above, it follows at once that an explanation of ferromagnetism cannot be given without taking electron interaction into account. Frenkel’s idea^86 amounted to the assertion that the weak paramagnetic properties of a gas of free electrons, predicted by Dorfman^33 and explained by Pauli^50, are by no means general, and that in the presence of a strong electrostatic interaction between electrons the essential dependence of their energy on magnetization may have precisely the opposite character, i.e. may make the magnetized state energetically more favorable.

Frenkel^86, and somewhat later Heisenberg^88, gave the first mathematical formulation of this idea on the basis of a simplified “exchange” model, according to which all the dependence on magnetization conditioned by the Pauli principle manifests itself through the medium of the exchange energy. It is clear that taking into account only the exchange processes, and also rejecting an exact calculation of the energy spectrum even of this simplified model (taking into account only the energetic centers of gravity), could not yield quantitatively any exact...

results. However, the qualitative picture of the phenomenon of ferromagnetism thereby received a complete, principled explanation. The main success of Frenkel’s idea and of its first quantitative formulation consists in the fact that the correct order of magnitude of the energy \(A_1\) in (10.4) was obtained.

The exchange model in its first version is a simple generalization of a problem of quantum chemistry—the hydrogen molecule. One considers a collection of a large number \((N)\) of “hydrogen-like” atoms with one outer \(s\)-electron. In the zeroth approximation the interaction between atoms is completely neglected. The electronic interaction is taken into account according to the scheme of the usual perturbation theory. It is also assumed that the states of the system in which two outer electrons are located on one atom (i.e. polar states) are so energetically unfavorable that they may be disregarded. This assumption makes it possible, in calculating the first approximation, to consider only processes of exchange between electrons of different atoms. The energy correction of the first approximation \(\varepsilon\), which is determined by these processes, is found in the usual way, as the matrix element of the Hamiltonian operator \(H\) of the system (with allowance for interaction), calculated with the aid of the “zero” wave functions \(\psi^0_{h_1\ldots h_r}\) (without allowance for interaction), which, besides the coordinates of the electrons, depend on \(r\) numbers of atoms \(h_i\) with “right” spins \((h_1h_2\ldots h_r)\); thus, we have

\[ \varepsilon(h_1\ldots h_r)=\int \psi^{0*}_{h_i}H\psi^0_{h_i}\,d\tau . \tag{11.1} \]

In computing \(\varepsilon\), however, one makes still another simplification, namely, one calculates the mean value (the energetic center of gravity) of this energy correction for a given number of right spins \(r\), i.e. for a given value of the magnetization of the system.

Most transparently, the calculation of the energetic center of gravity can be carried out in the “vector” form of the electron-spin operators. In this representation the energy operator of the electron system, to within additive constants and neglecting magnetic forces, has the form\({}^{18}\)

\[ H=-2\sum_{qq'} A_{qq'}\sigma_q\sigma_{q'}, \tag{11.2} \]

where \(\sigma_q\) is the operator (vector, more precisely spinor) of the electronic spin of atom \(q\), \(A_{qq'}\) is the integral of electronic exchange between atoms \(q\) and \(q'\), which is approximately written as

\[ A_{qq'}=\int \varphi_q^*(xyz)\varphi_{q'}^*(xyz)\varphi_q(x'y'z')\varphi_{q'}(x'y'z')\times \]

\[ \times [v_{qq'}(|\mathbf r-\mathbf r'|)+G_q(\mathbf r')+G_{q'}(\mathbf r)]\,d\tau d\tau', \tag{11.3} \]

where \(\varphi_q(xyz)\) is the wave function of the \(s\)-electron of an isolated atom, \(G_q(r)\) is the potential energy of the electron in the field of ion \(q\), and \(v_{qq'}(|\mathbf r-\mathbf r'|)\) is the interaction energy of the \(s\)-electrons of atoms \(q\) and \(q'\). In view of the fact that the magnitude of the exchange integral \(A_{qq'}\) falls off sharply with increasing distance between the ions \(q\) and \(q'\), in the sum (10.2) one may restrict oneself to nearest neighbors only and, putting

\[ A_{q,q\pm 1}=A, \tag{11.4} \]

one may write

\[ H=-2A\sum_{\text{(over neighbors)}} \boldsymbol{\sigma}_q\boldsymbol{\sigma}_{q'} . \tag{11.5} \]

The square of the resultant spin vector of the whole crystal is equal to

\[ \left(\sum \boldsymbol{\sigma}_q\right)^2 = \sum_q \sigma_q^2 + \sum_{q\ne q'} \boldsymbol{\sigma}_q\boldsymbol{\sigma}_{q'} = N\sigma(\sigma+1) + \sum_{q\ne q'} \boldsymbol{\sigma}_q\boldsymbol{\sigma}_{q'} = \sigma'(\sigma'+1), \]

where \(\sigma'\) is the spin quantum number (see § 2) of the whole collective of \(N\) electrons, and \(\sigma\) is that of the electrons of one atom. The number of terms in the double sum is \(N(N-1)\). Therefore the mean value of an individual term in the double sum is equal to

\[ \overline{\boldsymbol{\sigma}_q\boldsymbol{\sigma}_{q'}} = \frac{\sigma'(\sigma'+1)-N\sigma(\sigma+1)}{N(N-1)} . \]

The number of terms in the sum (11.5) is equal to \(\frac{1}{2}Nz\), where \(z\) is the number of nearest neighbors. Therefore the mean value of \(H\) is

\[ \overline H = -\frac{zA}{N-1} \left[\sigma'(\sigma'+1)-N\sigma(\sigma+1)\right]. \]

The resultant spin \(\sigma\) of an individual atom is of the order of unity, while \(\sigma'\) is of the order of the magnetization of the system \(m\) (in units of \(\mu_B\)). Therefore, in ferromagnets, to accuracy of order \(\sim 1/N\), we have:

\[ \overline H=-\frac{zA}{N}m^2=-NzA\mu^2 . \tag{11.6} \]

From a comparison of (11.6) and (10.4), the external equivalence of the quantum theory of ferromagnetism and the theory of the “molecular field” is immediately evident; therefore all the results of the latter may be applied to the present theory. In this case, however, the existence of the energy \(A\) is not merely postulated, but is the result of a calculation, which also shows that the value of \(A\) according to (11.3) is sufficient to obtain a Curie point of \(\sim 1000^\circ\), as is indeed observed experimentally.

In its quantitative statements the “exchange” model is, of course, very crude. The only correct quantit—

important conclusions are: the temperature dependence of the paramagnetic susceptibility—the Curie–Weiss law (10.11)—and the temperature behavior of the spontaneous magnetization near the Curie point (10.7). However, the magnitude of the numerical coefficients in these theoretical formulas differs noticeably from experiment. For low temperatures (the region near absolute saturation), the exchange model leads to the incorrect result (10.8).

From the expression for the exchange integral (11.3) it is evident that it may have either sign, since, even if the functions $\varphi_q(\mathbf r)$ are positive, the energies $v$ and $G$ enter under the integral sign with different signs. On the other hand, it is precisely the sign of the exchange integral that determines whether the minimum of the energy $\overline H$ in (11.6) corresponds to a magnetized state with $\gamma=\pm 1$ or, conversely, to a demagnetized state with $\gamma=0$. Therefore one may say that a positive sign of the exchange integral $(A>0)$ is a necessary condition for the appearance of ferromagnetism. However, because of the approximate nature of the exchange model, this criterion cannot have the character of a sufficient condition (see below).

Attempts were made to refine the exchange model in its statistical part (in calculating the phase sum),$^{79}$ but the exact energy spectrum was again replaced by a crude scheme of energy centers of gravity of the bands.

b) The first step toward a genuine refinement of Frenkel’s exchange quantum theory of ferromagnetism was the abandonment of the introduction of energy centers of gravity (11.6) and a more accurate calculation of the structure of the energy spectrum of a system of interacting electrons. This problem can be solved with sufficient accuracy only in the limiting case of low temperatures, when the crystal is close to the state of absolute magnetic saturation. In this case almost all electron spins are directed to one side (for example, “to the right”). Using this circumstance, one can find the exact eigenvalues of the energy operator (11.5). In this case of saturation the energy of the electron ensemble may be represented as the sum of the energies of “elementary excitations” $\varepsilon(\mathbf k_i)$, characterized by quasimomentum vectors $\mathbf k_i$,

\[ E=\operatorname{const}+\sum_{\mathbf k_i} n_{\mathbf k_i}\,\varepsilon(\mathbf k_i), \tag{11.7} \]

where $n_{\mathbf k_i}$ is the number of elementary excitations of quasimomentum $\mathbf k_i$. To these excitations one may associate certain “quasiparticles”—spin waves or ferromagnons, possessing the effective mass

\[ m^*=\frac{\hbar^2}{2d^2\sigma A} \tag{11.8} \]

($d$ is the lattice constant, $\sigma$ is the resultant vector of the spin of the atom,

\(z'\) is the number of nearest neighbors of the lattice sites), \(m^*\) by virtue of (10.6) \((2\sigma z' A \sim k\Theta)\), is determined by the Curie temperature as

\[ m^* \sim \frac{\hbar^2}{d^2 k\Theta}. \tag{11.8} \]

In this refined version of the theory, ferromagnetism is consistently regarded as a cooperative phenomenon, i.e. as the result of strong electrostatic interaction between electrons in the crystalline lattice of a metal. This system of interacting electrons is treated as a single many-electron collective, in which each individual electron has to a considerable extent lost its “free” individuality, and it can no longer in principle be regarded as a particle of an “ideal” gas. But in the limiting case under consideration of states close to saturation (low temperatures), the complex collective motion of the many-electron system turns out to have the character of the motion of an “ideal” gas of “quasiparticles”—ferromagnons. The method of introducing ferromagnons sharply simplifies all the calculations, but at the same time preserves all the specificity of the many-electron treatment of the phenomenon. At the same time, of course, one must always remember that ferromagnons are “quasiparticles” which represent one of the possible approximate representations of the complex real collective motion of the true particles—electrons—which strongly interact with one another.

The most important result of the many-electron theory of spin waves (ferromagnons), developed by Bloch \(^{89}\), Bethe \(^{90}\), Holstein and Primakoff \(^{91}\), and Akhiezer \(^{92}\), is the derivation of the temperature dependence of the spontaneous magnetization near \(0\ \mathrm{K}\), which has the form

\[ I_s = I_0(1 - \alpha T^{3/2}), \tag{11.9} \]

where the coefficient \(\alpha\) depends on the type of lattice and on the exchange integral \(A\). Landau \(^{93}\) believes that, through the experimental determination of the coefficient \(\alpha\), it is at present possible to determine most accurately the magnitude of the exchange energy \(A\), since in the derivation of formula (11.9) a minimum number of arbitrary model assumptions is made.

The generalization of the theory of ferromagnons to the case of higher temperatures encounters great mathematical difficulties, which reduce to the fact that, with increasing temperature, when higher levels of the energy spectrum of the system come into play, the energy of the system can no longer be represented, to a good approximation, as a sum of elementary excitations. In other words, the “gas” of ferromagnons can no longer be regarded as “ideal”*).

*) There have been attempts to consider a “nonideal” gas of ferromagnons \(^{94}\) by formally introducing correction terms into the formulas of the spin-wave theory.

Holstein and Primakoff91, and also Akhiezer92, supplemented the theory of ferromagnons by taking into account the magnetic (dipole) interaction*). This allowance led to the appearance of additional “magnetic” terms in (11.7). In this way it proved possible to obtain a consistent description (in contrast to the phenomenological Weiss one) of the paramagnetic true magnetization of ferromagnets. It turned out that at low temperatures and in not very weak fields \((H \sim 4\pi I_s)\) the susceptibility of such a para-process depends on the field and on the temperature according to the formula**)

\[ \chi_p \sim \frac{T}{\sqrt{H}} . \tag{11.10} \]

As the experiments of Polley95 and, especially, Parfenova96 show, this dependence is indeed realized experimentally in a large number of ferromagnets that have been investigated.

Akhiezer92 was the first to consider in the quantum theory of ferromagnetism the kinetic problem of the interaction of a gas of ferromagnons with a gas of phonons, which represent the thermal vibrations of the crystal lattice of the metal. This enabled him, with the aid of the general scheme of the kinetic equation, to investigate relaxation processes in ferromagnets at low temperatures and to determine the times for the establishment of thermal equilibrium in the gas of ferromagnons \(\tau_s\) and between the gas of ferromagnons and phonons \(\tau_{sf}\). It was found that in the temperature range under consideration \(\tau_s \ll \tau_{sf}\), and therefore the concept of a temperature of the ferromagnon gas, distinct from the temperature of the lattice, is meaningful. Akhiezer also succeeded in calculating the mean lifetime of such a two-temperature state of the ferromagnet.

new waves in the same way as is done in passing from the equation of state of ideal gases to the van der Waals equation. However, the artificial introduction into the theory of a large number of “constants” deprives it of all the advantages of universality and creates the false appearance of success at the expense of an unjustified arbitrary selection of the numerical data of these “constants.”

*) Unfortunately, it has not yet been possible to take into account, in an equally general form, the influence of the so-called spin-orbital magnetic interaction, which, along with the dipole interaction, plays an important role in the phenomenon of magnetic anisotropy (see § 12). However, even without taking this interaction into account, the refinement of the theory considered is of real interest.

**) A certain paradoxical character of the increase of the paramagnetic susceptibility with temperature has a simple visual interpretation. According to (11.9), the number of spins “liberated” from spontaneous magnetization increases with temperature as \(\sim T^{3/2}\), while the susceptibility of the para-process of these electrons, owing to their strong interaction, falls with temperature not according to the classical law \(\sim 1/T\), but has a slower decrease \(\sim 1/\sqrt{T}\), from which (11.10) follows at once.

c) The development of a many-electron*) theory of ferromagnetism must proceed along the path of refining the exchange model that underlies it, and along the path of generalizing the results of this theory to the case of metallic alloys and electronic semiconductors. The point is that Frenkel’s exchange model completely excludes from consideration all processes in which electric charge is transferred. Thereby this model does not make it possible to consider the phenomenon of electrical conductivity and, strictly speaking, is suitable only for describing ferromagnetic dielectrics. In order to make the theory acceptable for ferromagnetic metals, it is necessary to take into account the so-called polar states, in which processes of transfer of electric charge by electrons are allowed. Such a polar model, at least in principle, makes it possible to consider within a unified scheme all the properties of metals and electronic semiconductors. However, the actual realization of such a program encounters substantial mathematical difficulties. It therefore proved useful to construct a simpler model97, 99, which would combine all the principal advantages of the many-electron treatment of ferromagnetism with the simplicity of the one-electron description of the phenomenon of electrical conductivity. This “intermediate” picture is based on the assumption that the principal, leading role in ferromagnetism is played by the inner \(d\)- (or \(f\)-) electrons, while in electrical conductivity it is played by the outer \(s\)-electrons. In this case the \(d\)-electrons are described according to the many-electron exchange scheme, and the \(s\)-electrons according to the one-electron model. However, the exchange interaction between these differently named electrons is taken into account, and precisely in this lies the value of such a treatment**).

The energy of the \(s — d\) exchange is equal to

\[ E_{sd}=-A(\xi)[1+\mu\sigma]; \tag{11.11} \]

here \(\mu\) is the mean atomic magnetic moment of the \(d\)-electrons, \(\sigma=\pm 1\) is the magnetic moment of the \(s\)-electron, and \(A\xi\) is the exchange integral between an \(s\)-electron with quasimomentum \(\xi\) and one (nearest to it) \(d\)-electron. Formula (11.11) may be explained visually as follows: as a result of the \((s-d)\)-exchange interaction, a powerful “quasimagnetic” field acts on the spin of the \(s\)-electron, whose magnitude is of the same order as the “molecular field” (11.6), or even somewhat larger, since \(A(\xi)\) contains the exchange integral between \(s\)- and \(d\)-electro-

*) In the literature there is a large number of papers by American physicists of the schools of Slater, Van Vleck, and also Stoner–Mott and others, in which the problem of ferromagnetism is treated from the standpoint of the one-electron model of metals. These attempts are theoretically unfounded and therefore cannot claim to give a consistent explanation of the essentially cooperative phenomenon of ferromagnetism. Therefore, in this review the description of these works has been omitted; for a more detailed criticism of them see, for example, Ref. 78.

**) See the note at the end of the article, p. 64.

of the same atom, and not of neighboring ones, as is the case in (11.11). Thus, the \((s—d)\)-exchange interaction manifests itself in the “magnetizing” action of the \(d\)-electrons on the \(s\)-electrons. The latter orient their magnetic moments parallel or antiparallel to the magnetization of the \(d\)-electrons, depending on the sign of the exchange integral \(A(\xi)^*\). At the same time it is important that the “quasimagnetic” field (11.11) acting on the \(s\)-electrons depends essentially on the state of the \(s\)-electron (through the dependence of \(A\) in (11.11) on the quasimomentum \(\xi\)). This last circumstance leads to the fact that in ferromagnets the effective mass of the conduction \(s\)-electron proves to depend on the magnitude of the spontaneous magnetization (see below, § 15). Further, the presence of an admixture of magnetization from the \(s\)-electrons immediately gives a natural explanation of the phenomenon of the fractional character of the atomic magnetic moments of ferromagnets\(^{**}\), not only in alloys but also in pure metals. It is known from experiment that if the magnitude of the magnetic saturation of a ferromagnet, extrapolated to \(0^\circ\mathrm{K}\), is divided by the number of atoms per unit volume, we obtain a value that is not a multiple of the elementary magneton \(\mu_B\), but a fractional value of the latter. Thus, for example, in nickel this value is equal to \(0.606\,\mu_B\), in iron \(2.221\,\mu_B\), in cobalt \(1.716\,\mu_B\), and in gadolinium \(7.10\,\mu_B\). (It was precisely this fractional character of the atomic moments that once led to fruitless attempts to introduce a new elementary unit—the Weiss magneton \(\mu_\omega\), which would formally resolve this difficulty of the theory.) The first attempt at a theoretical explanation of the fractional character of atomic magnetic moments was made by Wolff\(^{98}\), who assumed that the atoms of a ferromagnet can be in states of different multiplicity. By an artificial selection of the “proportions” of the different states, Wolff succeeded in fitting the theoretical values of the atomic moments to the experimental data. However, such an adjustment of numerical constants can hardly be regarded as a real explanation of this phenomenon. A more consistent and general explanation of the fractional character of atomic moments is given by the polar model of the metal\(^{99***}\). This explanation follows from it without any additional assumptions, solely from the very supposition of the possibility of polar states. Indeed, if by \(n\) we denote the total number of electrons, and by \(S_0\) the mean value of the so-called “twos” (i.e. such elementary excitations in which, near an atom, two electrons appear which, owing to the Pauli principle,

*) The possibility of such “magnetization” of some electrons by others was first pointed out by Academician L. D. Landau.

**) In the opinion of a number of physicists (Pauli, Pomeranchuk, and others), the absence of an explanation of this phenomenon in the spin-wave theory calls into question even its approximate correspondence to the real phenomenon of ferromagnetism.

***) In a somewhat different form, the application of the polar scheme to the problem of ferromagnetism was also developed by Slater and Geylikman\(^{100}\).

opposite spins), then the magnetization at \(0^\circ\mathrm{K}\) per atom will be equal to

\[ \frac{n-\bar{s}_0}{n}\,\mu_B . \]

Hence there immediately follows the fractional nature of the moment under the sole condition \(\bar{s}_0 > 0\) (it is clear that the greatest value of \(\bar{s}_0\) is equal to \(n/2\)). The \((s-d)\)-exchange model of a ferromagnet, just as the polar one, gives a general explanation of the fractional character of atomic moments\({}^{97}\). From this theory it follows that the atomic moment of a ferromagnet is composed of a part due to \(d\)-electrons and a part due to \(s\)-electrons. In this case both these parts and their sum must, with overwhelming probability, be fractional. It is essential that this conclusion of the \((s-d)\)-exchange theory is also the result not of crude qualitative “adjustments,” but follows naturally from the completely general properties of the theory under consideration.

The latter also makes it possible to show that, at temperatures close to the Curie point, the magnetization of the \(s\)-electrons is proportional to the magnetization of the \(d\)-electrons and, consequently, in the first approximation has the same temperature dependence as is given, for example, by formula (10.15), and as is confirmed by experiment.

g) The polar theory makes it possible to establish the criterion of a metal and a semiconductor\({}^{99}\) more precisely than this is done in the one-electron approximation. According to the many-electron theory, which takes into account the finiteness of the probability of formation of polar states (= the birth of “pair” quasiparticles), for certain ratios between the various energies in a crystal either the case of a metal or that of a semiconductor may be realized. The second case is simpler, and it admits a comparatively detailed mathematical quantitative treatment. In order for the case of a semiconductor to occur, it is necessary that the energy of two electrons in a “pair” be noticeably greater than the “kinetic” energy of the transition of an electron from atom to atom \(\beta\), i.e. \(U \gg \beta\). Then the lowest states of the system, despite the presence of a certain number of “pairs,” turn out to be insensitive to the accelerating action of an external electric field, i.e. we are dealing with an insulator or semiconductor. In contrast to the one-electron approximation, the spectrum of the system in this theory, also in the case of a dielectric, turns out to be continuous from the very lowest level.

The polar model for the case of a semiconductor may be supplemented by taking account of excited states (“excitons”). Such a refined polar-exciton model can be used for investigating the ferromagnetic properties of semiconductors. Consideration of this problem by the spin-wave method\({}^{101}\) showed that near \(0^\circ\mathrm{K}\) the temperature dependence of the spontaneous magnetization has the form:

\[ I = I_0\left[1-\alpha T^{\frac{3}{2}}-\beta T^{\frac{3}{2}} e^{-\frac{\gamma}{kT}}+e^{-\frac{\Delta E}{kT}}\left(1-\delta T^{\frac{3}{2}}\right)\right], \tag{11.12} \]

where the first term gives the usual dependence (11.9) of the exchange model, the second term gives an additional decrease of the moment due to the admixture of polarized states (“twos”), and, finally, the third term

\[ \Delta I_3=I_0 e^{-\frac{\Delta E}{kT}}\left(1-\delta T^{\frac{3}{2}}\right), \tag{11.13} \]

gives the contribution to the magnetization due to excitons (\(\Delta E\) is the excitation energy of the exciton). From (11.13) it is seen that \(\Delta I_3\) decreases exponentially as the temperature is lowered and is equal to zero at \(0^\circ\text{K}\). Thus, if in a semiconductor ferromagnetism is due only to excited states (the exchange integral is positive only for excitons), then the temperature dependence of the spontaneous magnetic moment must have a form different from the usual curve \(I_s(T)\) of Fig. 43. Namely, as is seen from Fig. 46, in this case there is a second low-temperature Curie point \(\Theta^{(2)}=0^\circ\text{K}\).

Fig. 46. Temperature dependence of the spontaneous magnetization of an “exciton” ferromagnetic semiconductor.

Thus, the theory leads to the possibility of the existence of two fundamentally different types of ferromagnetic semiconductors: a) “normal” ferromagnets, which give saturation at \(0^\circ\text{K}\) and possess a single Curie point on the side of high temperatures, and b) “exciton” ferromagnets, which possess two Curie points, one high-temperature and the other at \(0^\circ\text{K}\).*)

Unfortunately, the very weak experimental study of the magnetic properties of ferromagnetic semiconductors, especially at low temperatures, does not yet make it possible to verify this conclusion experimentally.

d) Ferromagnetism of alloys. At the present time only four pure ferromagnetic metals are known with certainty: iron, nickel, cobalt, and gadolinium. A much more extensive group is formed by numerous alloys and compounds of these metals with one another and with other elements. Some binary and ternary alloys of manganese and chromium are also ferromagnetic,

*) If the ferromagnetism of excitons takes place near \(0^\circ\text{K}\), where the theory is valid, then the low-temperature Curie point is in fact equal to \(0^\circ\text{K}\). If, however, the exciton ferromagnetism begins to decrease sharply at temperatures noticeably above \(0^\circ\text{K}\), then it is more accurate to apply the energy centers of gravity, according to which the low-temperature Curie point may lie above \(0^\circ\text{K}\) [10] (see the dotted curve (2) in Fig. 46). See the note at the end of the article, p. 64.

not containing ferromagnetic components. For technical applications, the study specifically of ferromagnetic alloys is of the greatest interest. However, this study also has substantial theoretical significance. This applies especially to alloys of nonferromagnetic components alone, since their study promises most easily to establish experimentally the necessary conditions for the occurrence of ferromagnetism in a crystal and thereby to test the most essential and interesting question for theory: the necessary and sufficient criterion for the ferromagnetic state.

Although at present, as is clear from what was set forth above, there is as yet no completed quantum theory of ferromagnetism even for pure metals, it is nevertheless possible to take certain preliminary steps toward constructing a theory of ferromagnetic alloys. Here, just as in the case of the theory for pure metals, one has to consider separately the case of high temperatures (the region of the Curie point) and the case of low temperatures. In the first case the matter concerns a generalization of exchange theory, and in the second—the method of spin waves. As Vonsovsky has shown \(^{102*}\), the idea of both these generalizations reduces simply to the fact that it indicates the necessity of taking into account differences in the magnitudes and signs of the exchange integrals taken for different pairs of atoms in the alloy. Thus, for example, in the simplest case of binary alloys it is necessary to consider not one type of exchange integrals between the electrons of nearest-neighbor atoms, but three types. If the atoms of the first component of the alloy are denoted by the index \(a\), and of the second by the index \(b\), then these three types of integrals should be denoted by: \(A_{aa}\), \(A_{ab}\), and \(A_{bb}\)—respectively for nearest neighbors of the type \(a—а\), \(a—b\), and \(b—b\). With the aid of these integrals one may first of all obtain a formula for the spontaneous magnetization of the alloy as a function of temperature and concentration. This formula gives the same temperature dependence for \(I_s\) as the molecular-field theory and the exchange theory, namely:

\[ I_s = N\bar{\mu}_0 \operatorname{th}\frac{zI_s}{2N\bar{\mu}_0 kT} \left[n_a(n_a-n_b\sigma)A_{aa}+2n_an_b(1+\sigma)A_{ab}+\right. \]

\[ \left.+\,n_b(n_b-n_a\sigma)A_{bb}\right], \tag{11.14} \]

where \(\bar{\mu}_0\) is the average magnetic moment per atom of the alloy, \(z\) is the number of nearest neighbors of an atom, \(N\) is the number of atoms in the crystal, \(\sigma\) is the degree of short-range order in the arrangement of the atoms of the alloy components over the lattice sites, and \(n_a\) and \(n_b\) are the concentrations, respectively, of components \(a\) and \(b\). This character of the temperature dependence of \(I_s\) is confirmed in general outline by experiment. It should be noted here that an attempt at a classical generalization of the theory of ferro-

* Analogous ideas, only in a more qualitative form, are developed in the works of Kómar \(^{103}\) and Rudnitskii \(^{104}\).

magnetism in the case of alloys led Bitter \(^{105}\) to the unfounded conclusion of an entirely different form of the dependence \(I_s(T)\) (curves with “humps”), which is never observed experimentally in the case of single-phase alloys.

As for our formula (11.14), it follows from it that the temperature dependence \(I_s(T)\) in an alloy is obtained as completely analogous to that for a pure metal only in the case of a completely disordered alloy \((\sigma=0)\). In this case the whole difference reduces to a different expression for the Curie point \(\Theta\), depending on the exchange integrals and the concentration. Thus, for example, for the pure components \(a\) and \(b\) we have, respectively:

\[ \Theta_a=\frac{z}{2k}A_{aa},\qquad \Theta_b=\frac{z}{2k}A_{bb}, \tag{11.15} \]

and for a disordered binary alloy:

\[ \Theta_{ab\,(\mathrm{n.y})}=\frac{z}{2k}\left(n_a^2 A_{aa}+2n_a n_b A_{ab}+n_b^2 A_{bb}\right). \tag{11.16} \]

Thus, from (11.16) it is seen that the concentration dependence of the ferromagnetic Curie point of a binary disordered alloy has a quadratic course, which, for certain special relations between the magnitudes and signs of the integrals \(A\), may degenerate into a linear course, or in general \(\Theta_{ab\,(\mathrm{n.y})}\) may practically not depend on the concentration of the alloy. It is precisely dependences of this type that are observed experimentally \(^{78}\).

A particularly important consequence of formula (11.16) is that it makes it possible to understand qualitatively the appearance of ferromagnetism in binary alloys made from non-ferromagnetic components. In this case both integrals of the pure components \(A_{aa}\) and \(A_{bb}\) are negative, and, consequently, a necessary condition for ferromagnetism \((A>0)\) is the positivity of the “mixed” exchange integral \(A_{ab}>0\), and, in addition, the inequality

\[ A_{ab}>\frac{n_a}{2n_b}\left|A_{aa}\right|+\frac{n_b}{2n_a}\left|A_{bb}\right|. \tag{11.17} \]

Of course, these criteria are of a formal character, since the question of why precisely they occur in a given alloy remains open within the framework of the theory under consideration. It likewise remains unclear for the time being whether these criteria are determined by a purely geometrical factor, i.e. by a change in the lattice constant with a change in the concentration of the alloy, or whether here an essential role is played by the change in the electron density due to the appearance of “foreign” atoms of another component at an almost unchanged distance between the nodes of the crystalline lattice.

S. V. Vonsovskii

In the case of ordering binary alloys, the expression for the Curie point becomes more complicated; namely, it has the form

\[ \Theta_{ab}=\frac{z}{2k}\left[n_a(n_a-n_b\sigma)A_{aa}+2n_an_b(1+\sigma)A_{ab}+n_b(n_b-n_a\sigma)A_{bb}\right]. \tag{11.18} \]

Analysis of formula (11.18) shows that cases are possible in which the ferromagnetic state arises either with complete order in the arrangement of the atoms, or with complete disorder, or, finally, in a partially ordered state. Experiment also confirms the realization of all these three possibilities. The most detailed comparison of (11.18) with experiment was carried out by Komarov \(^{03,78}\).

In the case of low temperatures it has been possible to show \(^{102}\) that, within the framework of a rigorous calculation by the spin-wave method, for a binary alloy as well the same temperature law is preserved for the spontaneous magnetization (11.9),

\[ I_s=I_0\left[1-\alpha\left(\frac{T}{\Theta_{ab}}\right)^{3/2}\right], \tag{11.19} \]

where \(\alpha\) is a numerical factor \(\sim 10^{-1}\), whose value depends on the type of crystal lattice of the alloy, and \(\Theta_{ab}\) is a quantity of the dimension of temperature, equal to

\[ \Theta_{ab}=\frac{2z}{k}\left\{S_a\left[n_a(n_a-n_b\sigma)A_{aa}+n_an_b(1-\sigma)A_{ab}\right]+\right. \]

\[ \left.+S_b\left[n_an_b(1+\sigma)A_{ab}+n_b(n_b-n_a\sigma)A_{bb}\right]\right\}. \tag{11.20} \]

Here \(S_a\) is the resultant spin of the \(d\)-electrons of atoms of type \(a\), and \(S_b\) is the resultant spin of the \(d\)-electrons of atoms of type \(b\).

From the exposition of these results of the theory of binary ferromagnetic alloys, the need for its further development is clear, since a number of fundamental problems of alloy ferromagnetism (the magnitude of the atomic magnetic moments, the influence of electronic structure on the criterion of the ferromagnetic state, etc.) still remain open to a considerable degree \(^{106}\).

e) Antiferromagnetism and metamagnetism. As already indicated above (§ 5), halide salts of the elements of the iron group and of the neighboring elements chromium and manganese, the sulfates and oxides of these elements, as well as of copper, salts of the rare earths, elements of the palladium and platinum groups, and a number of other compounds, being at high temperatures paramagnets obeying the Curie–Weiss law (10.11), at low temperatures possess peculiar magnetic anomalies (cryomagnetic anomalies). In the main, these anomalies reduce to the fact that at a certain temperature \(\Theta_{a\phi}\) (which may not coincide with the paramagnetic Curie point \(\Theta_p\) in (10.11), and may be either a positive or a negative quantity) the magnetic susceptibility of these substances has a maximum.

However, such a maximum is not obligatory; the curve $\dfrac{1}{\chi}(T)$ may, at low temperatures, bend slightly and tend smoothly to the limiting values $\chi$ at $T = 0^\circ\mathrm{K}$. The opposite cases also occur, when $\dfrac{1}{\chi}(T)$ has a maximum at a certain temperature (see the examples in Fig. 47). At these same temperatures $\Theta_{\mathrm{af}}$ maxima are observed on the temperature curves of the heat capacities of these substances[^107] and on the curves for the magnetocaloric effect[^108]. The latter indicates the presence of a phase transition in these substances at $T = \Theta_{\mathrm{af}}$.

Fig. 47. Reciprocal susceptibility of antiferromagnets as a function of temperature. (After Starr, Bitter, and Kaufmann.)

The dependence of the susceptibility of these substances on the magnitude of the external magnetic field proves to be very complex. Besides the group of substances with a susceptibility that does not depend on the field up to fields of $2 \cdot 10^4$ oersteds (for example, $\mathrm{CuCl_2}$, $\mathrm{FeCl_3}$, $\mathrm{MnF_2}$, etc.), there are substances whose susceptibility decreases with increasing field ($\mathrm{CrCl_3}$, $\mathrm{CuSO_4}$, $\mathrm{NiF_2}$, etc.) or, conversely, increases ($\mathrm{CoCl_2}$, $\mathrm{CoBr_2}$, $\mathrm{NiCl_2}$, etc.). Finally, there is a series of compounds ($\mathrm{FeCl_2}$, $\mathrm{MnBr_2}$, etc.) whose susceptibility resembles that of ferromagnets, i.e. has a maximum at a certain value of the magnetic field.

Upon magnetization reversal, a number of these magnets exhibit the phenomenon of magnetic hysteresis. The magnitude of the residual magnetization is, as a rule, very small (from 0.01 to 1.0 gauss), while the coercive force, on the contrary, reaches hundreds and even thousands of oersteds.

The fact that the magnetic susceptibility of all antiferromagnets and metamagnetics at temperatures above the critical point $\Theta_{\mathrm{af}}$ obeys the Curie–Weiss law makes it possible to suppose that the magnetic properties of these bodies are determined by the exchange interaction of valence electro-

of the atoms in the crystal lattice. In contrast to ferromagnetic crystals, the sign of the exchange integral in these substances is either negative for all atoms that are nearest neighbors in the lattice, or positive for one group of nearest neighbors and negative for the other group. This difference in the sign of the integrals arises from the difference in the distance between the atoms of these two groups, or from the difference in the structure of the electron shell of these atoms. Such a possibility of a difference in the signs of the exchange integrals is precisely most probable in the case of the substances under consideration. As a rule, all these bodies have a crystal lattice in which the paramagnetic atoms are arranged in layers, and in many cases the distance between them is substantially greater than the distance between atoms within a layer.

If the exchange integral is negative for all neighboring pairs of atoms, then, from the point of view of exchange forces, a uniform distribution of right and left spins does not correspond to the minimum of the exchange energy. The minimum of this energy corresponds to such a distribution of spins in which each spin is surrounded only by nearest neighbors oriented antiparallel with respect to it. Consequently, in the case of antiferromagnetism the minimum of the exchange energy is obtained with the simultaneous realization of complete “long-range disorder” of right and left spins (there is no resultant magnetization) and complete “short-range order” in the distribution of antiparallel spins. At \(T = 0^\circ K\), in an antiferromagnetic crystal there is complete order at short distances between antiparallel spins. As the temperature is raised, it begins to be destroyed, and eventually a state with a chaotic distribution of antiparallel spins sets in. By analogy with ferromagnetism, one must suppose that in an antiferromagnet there exist ordered regions of antiparallel spins, similar to regions of spontaneous magnetization.

The transition from the antiferromagnetic state to the paramagnetic one occurs at the temperature \(\Theta_{\mathrm{af}}\), at which thermal motion destroys the regions of spontaneous antiferromagnetism, i.e. the long-range order, while short-range order is preserved only in very small volumes between which no correlation remains. At this temperature one should expect the appearance of maxima of anomalies in the properties of antiferromagnets. The remnants of short-range order should make these maxima diffuse.

The presence of a critical temperature \(\Theta_{\mathrm{af}}\), determined from the maximum of the heat-capacity anomaly or the maximum of the magnetocaloric effect, undoubtedly indicates the cooperative character of the phenomenon, i.e. the existence of a second-order phase transition.

If the exchange integrals for two groups of nearest neighbors have different signs, then the crystal must possess a more complex-

properties. The presence of positive exchange forces leads to the substance’s acquiring, to some extent, properties reminiscent of ferromagnetism. Along with this, the presence of negative exchange forces produces the effect of paramagnetism. Such “cooperation” of ferromagnetic and paramagnetic properties makes the term “metamagnetism” natural.

At present there is no sufficiently satisfactory quantum-mechanical theory of antiferromagnetism and metamagnetism. The existing attempts to construct such a theory (Rudnitskii \(^{109}\), Kaner \(^{110}\)) contain a number of arbitrary assumptions which make it impossible to assess the degree of approximation and, in general, its legitimacy. In addition, there are a number of works developing a quasiclassical theory of antiferromagnetism \(^{111}\).

A calculation according to the quasiclassical theory gives for the initial susceptibility the expression

\[ \chi=\frac{2N\mu_B^2}{kT}\,e^{-\frac{2z'|A|}{kT}}, \tag{11.21} \]

which, in general, qualitatively correctly describes the experimental curves \(\chi(T)\) for some antiferromagnets. At high temperatures

\[ \left(T \gg \frac{z'|A|}{k}\right) \]

we obtain from (11.21)

\[ \chi \sim \frac{N\mu_B^2}{k}\,\frac{1}{T+\frac{2z'|A|}{k}}, \tag{11.22} \]

i.e., the ordinary Curie–Weiss law with a paramagnetic Curie point

\[ \Theta_p=-\frac{2z'|A|}{k}. \]

The antiferromagnetic Curie point corresponds to the maximum on the curve (11.21). Just as in the case of the quasiclassical theory of ferromagnetism, these results can be assigned only an illustrative character.

Landau \(^{112}\) constructed a more rigorous thermodynamic theory, which qualitatively explains the anomaly of the susceptibility of metamagnets at low temperatures. In doing so he proceeds from the following model. The chlorides of chromium, iron, cobalt, and nickel have a layered structure. Between the crystallographic planes dotted with metal atoms there are halide “interlayers.” It is assumed that within the metallic layers there act positive orienting exchange forces, while between different layers there act negative exchange forces, substantially smaller than the positive ones.

Thus, at low temperatures we have spontaneously magnetized layers, but their magnetic moments are oriented in opposite directions, so that the whole crystal as a whole is not

has a macroscopic magnetization, i.e., is not ferromagnetic. If one assumes that the interaction between the different layers is relatively small, then one may expect that in a comparatively weak field there should be observed deviations from the linear law of dependence of magnetization on the field, and even the phenomenon of saturation in weak fields, in which the spontaneous magnetization of all layers will be oriented parallel to the field.

For constructing a quantitative theory it is therefore necessary to take into account three different effects: 1) the saturation magnetization of the individual layers, whose temperature dependence is known from experimental data for ferromagnets; 2) the exchange interaction between the different layers, whose energy in a first approximation may be assumed proportional to the scalar product of the magnetic moments of the layers; 3) the presence of magnetic-anisotropy energy of the layers, which, in view of the symmetry, is proportional to the square of the component of the magnetic moment parallel to the axis of symmetry.

For a field \(\vec H = 0\) one may assume that the magnetic moments of the layers are pairwise antiparallel. Therefore all layers may be divided into two groups. This division remains meaningful in the presence of an external field. The expressions for the susceptibility at temperatures above and below the Curie point have the same character as in the classical law (10, 11). However, the theory leads to a sharp anisotropy of the magnetic properties. Unfortunately, up to now there have been no experiments on the study of the magnetic properties of single crystals of antiferromagnets.

12. THEORY OF THE TECHNICAL MAGNETIZATION CURVE

Weiss, on the basis of the hypothesis he had advanced of regions of spontaneous magnetization, attempted to construct a theory of the magnetization curves of ferromagnets. However, his attempts were of a purely qualitative phenomenological character.

The beginning of the modern theory of the technical magnetization curve was laid by the works of Soviet scientists and, first of all, by the well-known works of N. S. Akulov (1928–1931),*) in which he established the universal law of magnetic anisotropy for ferromagnetic crystals and developed applications of this law for explaining magnetization curves, magnetostriction, and a number of effects connected with the magnetic state of a ferromagnet. Akulov’s works predetermined the paths of the further development of the theory of ferromagnetism both in the Soviet Union and abroad. Akulov was the first to show clearly that the whole extensive circle of phenomena associated with the existence of the magnetization curve of a ferromagnet is conditioned not by exchange forces, which ensure only the very po-

) The works of N. S. Akulov and his school are set forth with great completeness in the well-known monograph Ferromagnetism*, ONTI (1939).

the phenomenon of spontaneous magnetization, but by the magnetic interaction between the magnetic moments of electron spins and orbits.

The fundamentally important question of the theoretical justification of the hypothesis of regions of spontaneous magnetization found its complete solution in the works of the Soviet scientists Ya. I. Frenkel and Ya. G. Dorfman, L. D. Landau and E. M. Lifshitz, M. Shirokobokov and others.^78 The theory of magnetic hysteresis was constructed in exactly the same way, in the main, by the works of Soviet scientists and, first of all, of Akulov, Kondorskii, Shur, Yanus, Komar, Zaimovskii, B. G. Livshits, Meskin, and others. A great merit of the major Russian physicist-magnetologist V. K. Arkad’ev, as well as of Akulov, Dorfman, Kondorskii, and Yanus, is the creation by them of a large school of Soviet physicist-magnetologists and the broad introduction of the achievements of the theory of ferromagnetism into industry (magnetic flaw detection, magnetic structural analysis).

a) The principal task of the theory of the technical magnetization curve of ferromagnets consists in calculating this curve and in determining its dependence on various external influences and internal features of the ferromagnetic material itself. First of all, the theory had to provide a justification for the hypothesis of regions of spontaneous magnetization, which predetermine the very existence of the magnetization curve. To solve this question, it is necessary to know those types of interaction between the elementary carriers of magnetic moments which govern all processes of technical magnetization.

Above we have already discussed in detail the description of the exchange forces, which, under known conditions, lead to the existence of spontaneous magnetization. Here we shall be interested only in deviations from the equilibrium value of the energy of these forces (11.6), arising when the homogeneity of spontaneous magnetization is disturbed. According to L. Landau and E. Lifshitz^93,113 this deviation, by symmetry considerations, in the first approximation has the form

\[ \Delta F_{\mathrm{ob}}=\frac{A}{a}\left[(\nabla\alpha_x)^2+(\nabla\alpha_y)^2+(\nabla\alpha_z)^2\right], \tag{12.1} \]

where \(\nabla \alpha_i\) are the gradients of the direction cosines of the vector of spontaneous magnetization with respect to the coordinate axes, \(A\) is the exchange integral, and \(a\) is the lattice constant. This “addition” (12.1) to the exchange energy in a ferromagnet is an essentially positive quantity (\(A>0\)); therefore the exchange forces “oppose” any violation of the homogeneity of the magnetization of the ferromagnet.

From analysis of the magnetization curves of ferromagnetic single crystals it follows that they possess a sharply expressed anisotropy of magnetic properties, reducing essentially to the fact that in ferro-

in a magnetic single crystal there are “axes of easiest magnetization” (see Fig. 48).

The physical cause of this anisotropy was first elucidated by Akulov^114, who showed that the magnetic interaction between electrons is responsible for it. In doing so Akulov, proceeding from symmetry considerations, determined the general form of the dependence of the energy of this magnetic interaction on the orientation of the magnetization in the crystal and on the components of the tensor of elastic deformations of the lattice and of external elastic stresses.

Fig. 48. Magnetization curves of single crystals of iron, nickel, and cobalt for different crystallographic directions.

With possible changes in the orientation of the spontaneous magnetization in a crystal, the equilibrium distances between the lattice sites change. Therefore spontaneous, so-called magnetostrictive, deformations arise. In the particular case of a cubic crystal, in the absence of external stresses, the free energy of the magnetic and elastic interaction will be composed of three expressions (to within sixth powers in the direction cosines of the vector \(\mathbf{I}_s\) and second powers of the tensor of magnetostrictive stresses \(A_{ik}^0\)):

\[ F_{\mathrm{k}}^0 = k_0^0 + k_1^0(\alpha_1^2\alpha_2^2+\alpha_2^2\alpha_3^2+\alpha_3^2\alpha_1^2)+k_2^0\alpha_1^2\alpha_2^2\alpha_3^2 \tag{12.2} \]

— the free energy of the natural magnetic anisotropy of an undeformed crystal (without taking magnetostrictive effects into account), where \(k_1^0\) and \(k_2^0\) are constants of crystallographic magnetic anisotropy;

\[ F_{\mathrm{el}}^0 = \frac{C_1}{2}(A_{11}^0+A_{22}^0+A_{33}^0)^2 + C_2\left(A_{11}^{0\,2}+A_{22}^{0\,2}+A_{33}^{0\,2}\right) + 2C_3\left(A_{12}^{0\,2}+A_{23}^{0\,2}+A_{13}^{0\,2}\right) \tag{12.3} \]

— the free energy of the elastic stresses of the crystal (without taking mag-

finite effects), where \(C_i\) are the elastic moduli of a cubic crystal;

\[ \begin{aligned} F^0_{\mathrm{me}}={}&a_0\left(A^0_{11}+A^0_{22}+A^0_{33}\right) +a_1\left[\left(\alpha_1^2-\frac13\right)A^0_{11} +\left(\alpha_2^2-\frac13\right)A^0_{22}\right.\\ &\left.+\left(\alpha_3^2-\frac13\right)A^0_{33}\right] +2a_2\left(\alpha_1\alpha_2 A^0_{12}+\alpha_2\alpha_3 A^0_{23} +\alpha_1\alpha_3 A^0_{13}\right)\\ &+a_3\left(A^0_{11}+A^0_{22}+A^0_{33}\right)S\\ &+a_4\left[\left(\alpha_1^4+\frac23 S-\frac13\right)A^0_{11} +\left(\alpha_2^4+\frac23 S-\frac13\right)A^0_{22} +\left(\alpha_3^4+\frac23 S-\frac13\right)A^0_{33}\right]\\ &+2a_5\left(\alpha_1\alpha_2\alpha_3^2 A^0_{12} +\alpha_1^2\alpha_2\alpha_3 A^0_{23} +\alpha_1\alpha_2^2\alpha_3 A^0_{13}\right), \end{aligned} \tag{12.4} \]

(where \(S=\alpha_1^2\alpha_2^2+\alpha_2^2\alpha_3^2+\alpha_3^2\alpha_1^2\)) is the free energy of magnetostrictive stresses (here \(a_i\) are the magnetostriction constants, \(i=1,2,3,4,5\)). For an unchanged orientation of the spontaneous magnetization \((\alpha_i=\mathrm{const})\), we find the tensor of magnetostrictive deformations \(A^0_{ik}\) from the conditions for the minimum of the total free energy (12.2)—(12.4).

\[ \frac{\partial}{\partial A^0_{ik}}\left(F^0_{\mathrm{k}}+F^0_{\mathrm{el}}+F^0_{\mathrm{me}}\right)=0 \qquad (i,\ k=1,\ 2,\ 3). \]

Substituting the components \(A^0_{ik}\) found into the expression for the free energy, we obtain:

\[ F^0_{\mathrm{n}} =k_1\left(\alpha_1^2\alpha_2^2+\alpha_2^2\alpha_3^2+\alpha_3^2\alpha_1^2\right) +k_2\alpha_1^2\alpha_2^2\alpha_3^2 +k_3\left(\alpha_1^2\alpha_2^2+\alpha_2^2\alpha_3^2+\alpha_3^2\alpha_1^2\right)^2, \tag{12.5} \]

where

\[ \left. \begin{aligned} k_1={}&k_1^0+\frac{a_1^2}{2C_2}-\frac{a_2^2}{2C_3} +\frac23\,\frac{a_4}{C_2} +\frac76\,\frac{a_1a_4}{C_2} -\frac{3a_0a_3(3C_1-2C_2)}{3C_1+2C_2},\\ k_2={}&k_2^0-\frac{3a_1a_4}{C_2} -\frac{a_4^2}{C_2} -\frac{a_2a_5}{C_3} -\frac{a_5^2}{2C_3},\\ k_3={}&\frac{a_4^2}{2C_2} +\frac{3(3C_1-2C_2)a_3^2}{2(3C_1+2C_2)^2}. \end{aligned} \right\} \tag{12.6} \]

Depending on the ratios of the magnitudes and signs of the resulting constants of magnetic anisotropy \(k_1\), \(k_2\), and \(k_3\), a single crystal has one or another set of axes of easiest magnetization, for which (12.5) has a minimum value; thus, for example, in iron or permalloy \(k_1\) and \(k_2>0\), and therefore the axes of easiest magnetization are axes of type \([100]\), while in nickel \(k_1\) and \(k_2<0\), and therefore these axes are directions of type \([111]\).

Knowing the expressions for the magnetostrictive deformations \(A^0_{ik}\), one can obtain, as Akulov first showed,\(^{119}\) the expression for the relative elongation of a ferromagnet in some direction characterized by the direction cosines \(\beta_1,\beta_2,\beta_3\), for a given orientation of the vector of spontaneous magnetization \((\alpha_1,\alpha_2,\alpha_3)\)

\[ \left(\frac{\delta l}{l}\right)^0_{\alpha_i\beta_i} =\sum_{i,k} A^0_{ik}(\alpha_i)\,\beta_i\beta_k. \]

As a result of elementary calculations, with accuracy up to terms quadratic with respect to \(\alpha_i\), we obtain (for crystals of cubic symmetry):

\[ \left(\frac{\delta l}{l}\right)^0_{\alpha_i} = \frac{3}{2}\lambda_{100} \left(\alpha_1^2\beta_1^2+\alpha_2^2\beta_2^2+\alpha_3^2\beta_3^2-\frac{1}{3}\right) + \]

\[ +3\lambda_{111} (\alpha_1\alpha_2\beta_1\beta_2+\alpha_2\alpha_3\beta_2\beta_3+\alpha_3\alpha_1\beta_3\beta_1), \tag{12.7} \]

where \(\lambda_{100}=-\dfrac{a_1}{3C_2}\) and \(\lambda_{111}=-\dfrac{a_2}{3C_3}\) are the so-called magnetostriction constants of the given crystal, which measure the magnitude of its magnetostriction respectively for the axes \([100]\) and \([111]\) in passing from the demagnetized state of the crystal to its saturation along these axes.

In the presence of external stresses, determined by the tensor \(\sigma_{ik}\), to the free energy (12.2)—(12.4) one must add the term

\[ \sum_{i,k} A_{ik}\sigma_{ik} \tag{12.8} \]

(here \(A_{ik}\) is the strain tensor, caused both by magnetostriction \((A^0_{i,k})\) and by external stresses \((A^\sigma_{i,k})\), i.e. \(A_{ik}=A^0_{i,k}+A^\sigma_{i,k}\)). Again determining the equilibrium values \(A_{ik}\) and substituting them into the expression for the total free energy, we find the additional part of the free energy (12.5) caused by external stresses. In the case of a cubic crystal and homogeneous stresses \((\sigma_{ik}=\sigma\gamma_i\gamma_k\), where \(\sigma\) is the magnitude of the external homogeneous stress, and \(\gamma_i\) are its direction cosines with respect to the crystal axes), this addition to the free energy is equal to (with accuracy up to terms quadratic with respect to \(\alpha_i\))

\[ F_\sigma = -\frac{3}{2}\sigma \left[ \lambda_{100}(\alpha_1^2\gamma_1^2+\alpha_2^2\gamma_2^2+\alpha_3^2\gamma_3^2) + \right. \]

\[ \left. +2\lambda_{111} (\alpha_1\alpha_2\gamma_1\gamma_2+\alpha_2\alpha_3\gamma_2\gamma_3+\alpha_1\alpha_3\gamma_1\gamma_3) \right]. \tag{12.9} \]

The energy (12.9) has an especially substantial significance under strong stresses \((\lambda_s\sigma \gg k)\) or under insignificant natural magnetic anisotropy. In this case the resulting magnetic anisotropy is practically entirely determined by the external stresses. This circumstance can be widely used for the purposes of special treatments of magnetic materials. In particular, for weak anisotropy of magnetostriction \((\lambda_{100}\simeq\lambda_{111}\simeq\lambda_s)\), (12.9) assumes the very simple form

\[ F_\sigma=-\frac{3}{2}\lambda_s\sigma\cos^2\varphi. \tag{12.9′} \]

Although the magnetic interaction is a small correction to the electric exchange forces responsible for spontaneous magnetization, nevertheless they play a decisive role in the entire complex set of phenomena of technical magnetization. Therefore

elucidating the physical nature of the magnetic interaction in ferromagnets has not only theoretical significance, but is also necessary for a clear understanding of the mechanism of those physical processes which determine the whole practical value of the phenomenon of ferromagnetism; for without such a clear understanding of the physical essence of these processes, further progress in the technology of magnetic materials is impossible.

In Akulov’s first works114 the magnetic interaction in ferromagnetic crystals, from the microscopic point of view, was treated in a purely classical manner. A quantum-mechanical treatment of Akulov’s ideas was carried out by a number of authors. Thus, for example, Bloch and Gentile89, Vonsovskii119a, Van Vleck127, and others calculated the magnetic anisotropy of ferromagnetic single crystals; Vonsovskii119b performed a quantum-mechanical calculation of the magnetostriction of ferromagnets.

Holstein and Primakoff91 solved the question of the paramagnetic susceptibility of ferromagnets. Akhiezer92 considered the problem of establishing thermal equilibrium in a gas of ferromagnons and between ferromagnons and the thermal vibrations of the crystal lattice.

At the basis of all these calculations lies allowance for the magnetic interaction between the spin and “orbital” magnetic moments of the electrons participating in ferromagnetism. In the general case the operator of the magnetic energy is composed of three terms

\[ H_{\mathrm{magn}} = u_1 + u_2 + u_3, \tag{12.10} \]

where \(u_1\) is the operator corresponding to the motion of electrons relative to the ions of the lattice—the spin-orbital energy; \(u_2\) is the operator of the magnetic energy arising as a result of the relative motion of the electrons themselves—the orbital energy; \(u_3\) is the operator of the energy of the magnetic interaction of the spin magnetic moments of the electrons—the spin (in first approximation, dipolar) energy.

The effect of the “orbital” interaction \(u_1\) and \(u_2\), which in the case of isolated atoms manifests itself in the formation of the fine structure of spectral lines, leads to the appearance of “internal magnetic fields” of the order of \(10^5\) oersteds. On the other hand, the “equivalent magnetic field” of the anisotropy of ferromagnets, determined by the magnitude of the magnetic field at which saturation is reached in a single crystal along the so-called “hardest directions of magnetization” (see Fig. 48), turns out to be \(\sim 10^2\) oersteds and only in rare cases (cobalt, pyrrhotite) reaches \(10^3\)—\(10^4\) oersteds. The explanation of this discrepancy is that, in contrast to atoms, where the orbital moments differ from zero (with the exception of \(s\)-states), in ferromagnetic crystals, as measurements of the gyromagnetic effect show, the mean orbital magnetic moment over the crystal is practically equal to zero. Therefore, in the first approximation the effect of the orbital energies \(u_1\) and \(u_2\) is also equal to zero.

In this approximation, only the spin part of the magnetic interaction \(u_3\) is important; it is precisely this part that ensures the order of magnitude of the effective anisotropy “fields” observed experimentally.

Despite the absence of a complete quantum treatment of magnetic interaction in ferromagnets, certain successes have nevertheless been achieved in this area. Thus, for example, it has been possible to explain the correct order of magnitude of the magnetic anisotropy constants. In particular, it follows from the theory, without any additional assumptions, that in cubic crystals (iron, nickel) the anisotropy constants should be smaller in absolute value than in the case of hexagonal crystals (cobalt, pyrrhotite). This follows simply from the general fact that, owing to the symmetry properties of cubic crystals, the first approximation for the “dipole” energy \(u_3\) and the second approximation for the orbital energies \(u_1\) and \(u_2\) do not lead to a dependence of the magnetization of the crystal on its orientation relative to the crystallographic axes. To obtain this dependence it is necessary to consider the following approximations, whereas in hexagonal lattices the anisotropy is obtained already in the first case.

Further, quantum theory, in contrast to the classical one, admits a priori the possibility of either sign of the anisotropy constants for one and the same type of crystal lattice.

Fan and Flekov\(^{127}\) succeeded in obtaining the temperature behavior of the magnetic anisotropy constants of cubic crystals, which agrees in general outline with experiment. Vonsovskii\(^{119a}\) succeeded in obtaining the temperature behavior of the anisotropy constant of the hexagonal crystal cobalt and in giving a fundamental explanation of the change of sign of this constant observed experimentally. He also\(^{119b}\) obtained the temperature dependence of the magnetostriction constants, which allows for the possibility of a nonmonotonic change of these constants with temperature, as well as the possibility of a change in their sign. In addition, the calculation of magnetostriction was generalized to the case of binary alloys. Akulov\(^{114}\) developed a theory of the temperature dependence of the constants of magnetic anisotropy and magnetostriction, based on the concept of precessional motions of individual parts of the regions of spontaneous magnetization, caused, according to his hypothesis, by the thermal motion of the lattice.

The constants of magnetic anisotropy, as we shall see in greater detail below, play an essential role in all processes of technical magnetization of ferromagnets. It is therefore natural that researchers show great interest in measuring these constants in various materials. Already in Akulov’s first works\(^{114}\), the principles of experimental methods for determining these constants were indicated (determination of the work of magnetization for the principal crystallographic directions, comparison of experimental and theoretical magnetization curves of single crystals, determination of the normal

component of the magnetization in single crystals; determination of constants from measurements of magnetization curves of polycrystals in the region of very strong fields). The order of magnitude of the constant \(k_1\) ranges from \(10^3\) erg/cm\(^3\) (permalloy) to \(10^7\) erg/cm\(^3\) (cobalt). The sign of the constant may be different (iron \(k_1 > 0\), nickel \(k_1 < 0\)). It is especially important to know the temperature dependence of the anisotropy constants, for to a considerable degree it determines the temperature behavior of the whole aggregate of the “technical” properties of ferromagnets. Experience shows that this dependence is rather sharp, in any case sharper than the temperature dependence of the spontaneous magnetization \(I_s(T)\) (see Fig. 49, where Kirenskii’s data\(^{115}\) are given for \(k_1(T)\) of an iron single crystal, as well as the curve \(I_s(T)\)). The temperature dependence of \(k_1\) for iron was studied in detail by Titov\(^{116}\) and Kirenskii\(^{115}\), for nickel by Bryukhatov and Kirenskii\(^{117}\), and for cobalt by Honda and Masumoto\(^{118}\). In the latter case the constant changes sign at temperatures around \(500^\circ\)K. This has found its theoretical explanation\(^{119}\). Similarly, Tarasov measured the anisotropy constant of important technical iron–silicon alloys\(^{120}\) (transformer steel), and Shubina measured its temperature dependence\(^{121}\), which proved to be similar to the temperature dependence for iron. A detailed investigation of the anisotropy constants of various ferromagnetic alloys was carried out by Al’tgauzen\(^{122}\), Zaimovskii\(^{123}\), Akulov and Puzei\(^{124}\), and other authors\(^{78}\).

Fig. 49. Temperature dependence of the magnetic anisotropy constant (\(k_1\)) of iron. (After Kirenskii.)

To determine the equilibrium distribution of spontaneous magnetization it is also important to know the energy of the demagnetizing field of surface \((\sigma_m = \operatorname{Div}\mathbf I_s)\) and volume \((\rho_m = \operatorname{div}\mathbf I_s)\) charges. The density of this energy is equal to [see (4.9) and (4.10)]

\[ F_{\text{demag}}=-\frac{1}{2}(\mathbf H\mathbf I). \tag{12.11} \]

The energy density of a ferromagnet relative to the external field \(\mathbf H_e\) is equal to

\[ F_H=-(\mathbf H_e\mathbf I). \tag{12.12} \]

b) Having determined the principal types of interactions in a ferromagnetic crystal, one may pose the question of the character of the distribution in it

*) See the note at the end of the article.

spontaneous magnetization. If only exchange forces were acting, and the ferromagnet were structurally homogeneous, then at first glance it would seem that it should become magnetized to the saturation corresponding to the given temperature, in the direction corresponding to the minimum demagnetizing factor. Such a distribution would correspond both to the minimum of the exchange energy (12.1) and to the minimum of the energy of the demagnetizing field (12.11). However, one can imagine an energetically more favorable distribution of \(I_s\), in which the demagnetizing field is altogether absent. This will be the case if the entire volume of the ferromagnetic specimen is broken up into separate small regions with such a distribution of the orientations of the vectors \(I_s\) in them that the resultant magnetization of the whole specimen as a whole is equal to zero, i.e.

\[ \sum_{\text{(over the specimen)}} I_s v_i = 0, \tag{12.13} \]

where \(v_i\) is the volume of the \(i\)-th region.

In this case there will appear a positive energy associated with the existence of boundaries between ferromagnetic regions. However, experiment shows that, notwithstanding this circumstance, precisely such a demagnetized state is natural for a ferromagnetic specimen; therefore already from this it follows that the energy of the boundaries between regions (surface energy) is less than the energy (volume energy) of the demagnetizing field (12.12). And only when a specimen is magnetized along a direction with zero demagnetizing factor (a homogeneous torus magnetized along its axis) could one expect the absence of regions. In real crystals the matter is complicated by the presence of magnetic anisotropy and of all sorts of structural and chemical inhomogeneities.

The first theoretical quantitative justification of the hypothesis of regions of spontaneous magnetization was given by Frenkel and Dorfman \(^{125}\). They calculated the dimensions of the regions, taking into account only the competition between exchange forces and the demagnetizing action of the surface. In doing so they obtained that the linear dimensions of the regions \(l\) depend on the linear dimensions of the specimen \(L\) according to the formula

\[ l \sim \sqrt{L}. \tag{12.14} \]

In particular, for bodies of average dimensions with \(L \sim 1\ \text{cm}\) they obtained \(l \sim 10^{-2}\ \text{cm}\), i.e. quite macroscopic dimensions. The latter justifies the “classical” thermodynamic approach to the solution of the problem of regions of spontaneous magnetization.

A rigorous quantitative theory of ferromagnetic regions was constructed by L. Landau and E. Lifshitz \(^{113,93}\), who took into account the influence of the energy of magnetic anisotropy. The magnetization in the regions is oriented along the axes of easiest magnetization. In the case of a uniaxial ferromagnetic crystal (cobalt or any material subjected

subjected to strong uniaxial stresses) the domains have the form of plane-parallel layers with surfaces parallel to the axis of easiest magnetization. The equilibrium thickness of the domains \(d\) can be found from a formula of the type (12.14), with a coefficient that is determined by the ratio of the exchange energy to the energy of magnetic anisotropy \(\left(\sim \sqrt[4]{\dfrac{A}{k_1 a}}\right)\). Near the surface the shape of the domains becomes such as to reduce the demagnetizing effect (see Fig. 50), even at the cost of increasing the anisotropy energy in the surface regions, which have the form of triangular prisms and in which the magnetization is directed perpendicular to the axis of easiest magnetization, but parallel to the surface of the specimen.

Fig. 50. Structure of domains of spontaneous magnetization in a ferromagnet with one axis of easiest magnetization (Landau and Lifshitz).

Diagram label: axis of easiest magnetization; \(J_s\).

With a plane, but inclined (to the easy axis) surface of the specimen, the shape of the boundary domains becomes somewhat more complicated. A more detailed calculation by E. Lifshitz\(^{93}\) showed that, as the surface (triangular) domains grow, it becomes energetically favorable for them to split, with the appearance of wedge-shaped domains (see Fig. 51) and surface “magnetic” charges on the contact surfaces of the domains (the latter violates the condition for the disappearance of the demagnetizing field, \(\operatorname{div} \mathbf{J}_s = 0\)).

Fig. 51. “Fine structure” of the domains of spontaneous magnetization of a uniaxial ferromagnet near a surface normal to the axis of easiest magnetization. (According to Lifshitz.)

Diagram label: axis of easiest magnetization; \(J_s\).

Landau and Lifshitz also determined the law of variation of the orientation of the magnetization in the boundary layer between domains, the effective thickness of this layer \(\delta\), and the dependence of the density of the (surface) energy of this layer \(\gamma\) on various magnetic parameters. Namely, for the thickness \(\delta\) the formula obtained was

\[ \delta \sim \sqrt{\frac{A}{k_{\mathrm{eff}}}}, \tag{12.15} \]

where \(k_{\mathrm{eff}}\) is the effective constant of magnetic anisotropy \((\sim \alpha k_1+\beta\lambda_s\sigma)\).

From (12.15) it is seen that the less anisotropic the material, i.e. the smaller \(k_{\mathrm{eff}}\), the thicker the boundary layer, i.e. the less sharply the individual ferromagnetic regions are formed. Moreover, as the temperature rises, \(k_{\mathrm{eff}}\) falls strongly; this also leads to the fact that the boundary loses its sharpness as the Curie point is approached, where the regions disappear altogether. The density of free surface energy proves to be equal to

\[ \gamma \sim \sqrt{k_{\mathrm{eff}}A}. \tag{12.16} \]

Formula (12.16) shows that the less anisotropic the material, the smaller the quantity \(\gamma\). Below we shall see that the character and magnitude of the energy inhomogeneities \(\gamma\) in the material basically determine the magnetization processes in the region of weak magnetic fields. The calculation of Landau and Lifshitz was generalized by Shirobokov\({}^{126}\), who took into account the presence of an external magnetic field. The generalization of this calculation to the case of a crystal with several axes of easiest magnetization was carried out by Shirobokov\({}^{126}\), Lifshitz\({}^{93}\), and Vonsovskii\({}^{128}\). In this case, under certain conditions, there arises the possibility not only of “antiparallel” or 180° neighborhoods between regions (see Figs. 50 and 51), but also of neighborhoods with mutually perpendicular orientation of the magnetization (90° neighborhoods) (Fig. 52).

Fig. 52. Two types of 90° neighborhoods of regions of spontaneous magnetization in a magnetically polyaxial crystal\({}^{78}\).

In real crystals, of course, everything is considerably complicated by their structural inhomogeneities. But in any case one may say that the boundary between regions is arranged in such a way that the increase in the free energy of the ferromagnetic crystal caused by its existence is minimal. Therefore, in the case of antiparallel neighborhoods the boundaries are, as a rule, located in places with a maximum of residual internal stresses, where \(k_{\mathrm{eff}}\sim \alpha k+\beta\lambda\sigma\) is minimal, and consequently, by (12.16), \(\gamma\) is minimal. In the case of mutually perpendicular neighborhoods, the boundary is located in places where the sign of the internal stresses changes (for precisely because of the change of sign the axes of easiest magnetization of neighboring regions are different).

c) Direct experimental proof of the existence of regions of spontaneous magnetization is provided by: 1) the jump-like character of magnetization curves in the region of weak fields (especially near the steepest rise of the curve)—the so-called Barkhausen effect—and 2) inhomogeneities in the distribution of magnetic colloidal particles on the surface of a ferromagnetic crystal (the so-called Akulov–Bitter bands).

Fig. 53. Step-like character of the magnetization curve.

As detailed investigations have shown, the magnetization curve of a ferromagnet has a step-like character (see Fig. 53). The vertical portions of the curve correspond to jump-like changes of magnetization, which occur at a constant external magnetic field because of a rapid change in the orientation of magnetization in individual ferromagnetic regions that were initially magnetized not in the direction of the external magnetic field. This effect was discovered accidentally in 1919 by Barkhausen129. Arkad’ev130 developed a very simple acoustic method for determining these jump-like changes of magnetization. With the aid of an oscillograph one can carry out a detailed study of the jumps78 and determine the average volume of the region remagnetized in a single jump. The volume determined in this way agrees with the theoretical estimate of the dimensions of a ferromagnetic region. By means of special actions on the material (for example, elastic stretching, etc.) one can deliberately increase the magnitude of the jumps, in the limit giving the hysteresis loop an exactly rectangular form (see below).

In 1931 Bitter131, and subsequently Akulov114, showed that a fine ferromagnetic powder, suspended in a liquid, upon settling on a well-polished surface of ferromagnetic

single crystals, figures (bands) of regular form. The dependence of the arrangement of these bands on the magnetic state of the specimen was also shown. Powder deposits on single crystals were likewise investigated by Akulov and Dekhtyarev, Akulov and Raevskii[^132]. Miller and Steinberg[^133] investigated deposits on magnetite crystals, and Akulov and Bazurina[^134] on cobalt crystals. The Akulov–Bitter bands were studied in especially great detail by Elmore[^135], whose data are not only qualitatively but even quantitatively in agreement with theoretical estimates of the dimensions of ferromagnetic domains. In Fig. 54 examples are given of typical deposit patterns on the surfaces of ferromagnetic crystals.

Fig. 54. Pattern of Akulov–Bitter bands on the surface of a single crystal of silicon iron; above—in the presence of a normally applied field, below—without a field. Scale grid: \(14 \mu \times 14 \mu\). The specimen surface is a cube face.

It should be noted that the Akulov–Bitter bands give us information about the structure of domains on the surface of the crystal, but not within its volume.

In addition to the two mentioned methods of direct experimental determination of the magnetic structure of ferromagnets, three further methods have recently been indicated: 1) the method of determining magnetic stray fields at the surface of a ferromagnetic crystal by means of grazing electron beams[^136]*; 2) the method of transmission of ferromagnets by a neutron beam[^137], and finally 3) the method of electron microscopy[^138]. However, as yet there is very little experimental material, which does not allow definite conclusions to be drawn about the limits of applicability of these methods.

d) From the very fact of the existence of regions of spontaneous magnetization it is possible to establish the types of magnetization processes—

* The idea of this experiment was first put forward (1942) by P. I. Lukirskii.

…in ferromagnets. As has already been mentioned, a ferromagnetic body in its natural state, outside a magnetic field, has no resultant magnetization. Therefore

\[ \sum I_s v_i \cos \vartheta_i = 0 \]

(\(v_i\) is the volume of the \(i\)-th region, \(\vartheta_i\) is the angle between the magnetization vector of the \(i\)-th region and any fixed direction in the specimen). If an external magnetic field \(\mathbf H\) is switched on, for example along this fixed direction, then the body begins to become magnetized, i.e., along the direction of \(\mathbf H\) there appears a magnetic moment \(\delta I_H\) of the body different from zero. This moment in the general case consists of two parts:

\[ \delta I_H = I_s \sum_i \cos \vartheta_i \, \delta v_i + I_s \sum_i v_i \delta(\cos \vartheta_i). \tag{12.17} \]

The first term gives in \(\delta I_H\) the contribution obtained from the increase in the volumes of regions with vector \(\mathbf I_s\) in which the direction relative to \(\mathbf H\) is energetically more favorable, at the expense of the volume of regions magnetized in an energetically less favorable way. These processes proceed by displacement of the boundaries between regions and are briefly called displacement processes. The second term on the right-hand side of (12.17) gives in \(\delta I_H\) the contribution obtained from a change in the direction of \(\mathbf I_s\) in the regions \((\delta \cos \vartheta_i!)\). These processes are commonly called rotation processes. Thus the susceptibility of a ferromagnet may be represented as the sum of the susceptibilities of these two types of processes:

\[ \chi = \chi_{\mathrm{см}} + \chi_{\mathrm{вр}} = \left( \frac{dI}{dH} \right)_{\mathrm{см}} + \left( \frac{dI}{dH} \right)_{\mathrm{вр}} . \tag{12.17′} \]

Analysis of the magnetization curves of real materials shows that in the region of weak fields [the initial section of the curve, including the maximum on Stoletov’s curve (Fig. 22) for the permeability] the principal role is played by displacement processes \((\chi_{\mathrm{см}} \gg \chi_{\mathrm{вр}})\). In fields larger than the field corresponding to the maximum on Stoletov’s curve, on the contrary, the principal role is played by rotation processes \((\chi_{\mathrm{вр}} \gg \chi_{\mathrm{см}})\).

Here it should also be mentioned that both these types of processes, in turn, may be reversible and irreversible. The latter type of processes determines the entire phenomenon of magnetic hysteresis [see below, section ж)].

d) Let us first consider reversible magnetization processes, and above all reversible displacement processes. These processes determine important magnetic parameters—the initial and reversible magnetic susceptibility of ferromagnets. The quantitative development of the theory of these processes belongs to E. I. Kondorsky\({}^{139}\).

The question of the real existence of displacement processes at present leaves no doubt. However, in attempts to construct a quantitative theory, substantial difficulties arise.

S. V. VONSOVSKII

The structure of boundary layers between regions and the processes of their displacement depend in a complicated way on the structural state of the ferromagnetic crystal. The lack of sufficiently detailed information about this state, and its sharply “individual” character for each given specimen, make it difficult to construct a quantitative theory of the magnetization curve in weak fields. It is precisely for this reason that the establishment of general regularities, independent of the accidental properties of an individual specimen, is of interest for the development of a quantitative theory.

To each stationary state of a ferromagnet there corresponds a definite distribution of regions of spontaneous magnetization. The position of the boundaries between them is determined from the condition of the minimum of the surface energy \(\gamma\) [see (12.16)] of these boundaries, the magnetoelastic energy of the regions, and the energy of the internal demagnetizing fields.

A displacement of the boundary zone \(S_{kl}\) between neighboring regions, or magnetic “phases” \(k\) and \(l\), can occur if the density of the free energy of the external forces \(F_e\) is different on the two sides of the boundary. If, on a given element \(dS_{kl}\) of the boundary surface, the displacement is equal to \(\delta \mathbf n_{kl}\), then the total work of the external pressure over the whole boundary is equal to

\[ \int_{(S_{kl})}\bigl[(F_e)_l-(F_e)_k\bigr]\,\delta \mathbf n_{kl}\,dS_{kl}. \tag{12.18} \]

Part of the work (12.18) goes to covering the increase of the internal free energy \(F_i\), associated with the forces of magnetoelastic anisotropy, which is equal to

\[ \int_{(S_{kl})}\bigl[(F_i)_k-(F_i)_l\bigr]\,\delta \mathbf n_{kl}\,dS_{kl}. \tag{12.19} \]

The other part of (12.18) goes to compensating the change in the surface energy between the phases

\[ \delta \int_{(S_{kl})}\gamma_{kl}\,dS_{kl}. \tag{12.20} \]

In the general case, the change (12.20) may occur for three reasons: 1) owing to local changes of the surface energy \(\gamma_{kl}\), caused by the displacement of the boundary layer to new places in the crystal,

\[ \int_{(S_{kl})}\frac{\partial \gamma_{kl}}{\partial n}\,\delta \mathbf n\,dS_{kl}; \tag{12.21} \]

2) owing to changes in the magnitude of the surface area \(S_{kl}\), caused by the change of its curvature upon displacement,

\[ \int_{(S_{kl})}\gamma_{kl}\left(\frac{1}{R_1}+\frac{1}{R_2}\right)_{kl}\delta \mathbf n_{kl}\,dS_{kl}; \tag{12.22} \]

3) owing to the change in the magnitude of the surface area caused by deformation of the contour \(\Gamma_{kl}\) bounding it,

\[ \oint_{\Gamma_{kl}}(\gamma_{kl})_{\Gamma_{kl}}(\delta n_{kl})_{\Gamma_{kl}}\,d\Gamma_{kl}. \tag{12.23} \]

Under a reversible displacement of the boundaries, the condition of a minimum of the total free energy of the body \(F\) must be satisfied. The total change \(\delta F\) under displacements of the boundaries between all neighboring magnetic phases must be equal to zero (for arbitrary \(\delta n_{kl}\)). The solution of this variational problem of mixed type is, in general form, very complicated. However, it can be considerably simplified by assuming that the structure of the regions has such a “geometry” that the magnitude of the area of the boundaries does not change appreciably during the process of displacement. Under real conditions this may occur, for example, if the regions have a form close to plane-parallel layers (see Fig. 50). In this case the material must be homogeneous in its structure and, as far as possible, free of inclusions. In the presence of inclusions, the boundaries between magnetic phases, when displaced, are forced to “flow around” the inclusions and may thereby appreciably change the magnitude of their surfaces. Thus, in the case of a homogeneous material, the process of displacement of boundaries between phases is practically completely regulated by local changes in the density of free energy (12.18) and by the boundary energy (12.21) and (12.22). The quantity (12.23) may then be neglected. Conversely, for heterogeneous materials, in the presence of a large number of inclusions, the process of displacement of boundaries between magnetic phases is almost entirely determined by (12.23). The approximation for the “homogeneous” case may be called the “theory of stresses” (Kondorskii), and for the heterogeneous case, the “theory of inclusions” (Kersten).

In the case of homogeneous materials, according to the scheme of the “theory of stresses” and with successive neglect of the quantity (12.23), the minimum of the total free energy \(F\) is attained under the condition that

\[ (F_e)_l-(F_e)_k=(F_i)_k-(F_i)_l+\frac{\partial\gamma_{kl}}{\partial n} +\gamma_{kl}\left(\frac{1}{R_1}+\frac{1}{R_2}\right). \tag{12.24} \]

The equilibrium of the boundary will be stable if the right-hand side of (12.24) increases with increasing external forces; otherwise we pass into the region of irreversible displacements. From (12.24) we find for the complete change in the volume of the \(k\)-th magnetic phase:

\[ \delta v_k=\sum_{l(\ne k)}\int_{(S_{kl})} C_{kl}\,\delta\bigl[(F_e)_l-(F_e)_k\bigr]\,dS_{kl}, \tag{12.25} \]

where the quantity reciprocal to \(C_{kl}\) denotes the gradient of the internal forces, i.e.,

\[ C_{kl}^{-1}=\frac{\partial}{\partial n}\left[(F_i)_k-(F_i)_l +\frac{\partial\gamma_{kl}}{\partial n} +\gamma_{kl}\left(\frac{1}{R_1}+\frac{1}{R_2}\right)\right]. \tag{12.26} \]

Formula (12.25) gives a fundamental solution of the entire problem of reversible displacement of boundaries within the framework of the theory of stresses and with neglect of magnetic stray fields. The main difficulty in applying (12.25) to particular cases lies in determining the gradients (12.26) of the internal forces, which depend on the structural state of the specimen. If the displacement of the boundaries between magnetic phases is caused by an external magnetic field \(\mathbf H\), then

\[ \delta\left[(F_e)_l-(F_e)_k\right]=I_s(h_k-h_l)\delta H, \tag{12.27} \]

where \(h_k\) and \(h_l\) are the cosines of the angles between the vectors \(\mathbf I_s\) in phases \(k\) and \(l\) and the direction of the vector of the change in the magnetic field \(\delta \mathbf H\) (which in the general case need not coincide with the direction of the vector \(\mathbf H\)).

For displacement of the boundaries between magnetic phases caused by a change in a homogeneous stress \(\delta\sigma\), we have:

\[ \delta\left[(F_e)_l-(F_e)_k\right]= \begin{cases} \dfrac{3}{2}\lambda_{100}\left(\beta_k^2-\beta_l^2\right)\delta\sigma, & \text{axes of easiest magnetization of type }[100],\\[6pt] \dfrac{3}{2}\lambda_{111}\left(\beta_k^2-\beta_l^2\right)\delta\sigma, & \text{axes of easiest magnetization of type }[111]. \end{cases} \tag{12.28} \]

Here \(\beta_k\) and \(\beta_l\) are the cosines of the angles between the direction of the stress \(\delta\sigma\) and the magnetizations of phases \(k\) and \(l\). Under the simultaneous action of a magnetic field and stresses, the reversible change in the volume of the \(k\)-th magnetic phase is equal to

\[ \delta n_k=\sum_{l(\ne k)}\int_{(S_{kl})} C_{ik}\left[ I_s(h_k-h_l)\delta H+\frac{3}{2}\lambda_s\left(\beta_k^2-\beta_l^2\right)\delta\sigma \right]\,dS_{kl}, \tag{12.29} \]

where \(\lambda_s=\lambda_{100}\) in the case of iron, and \(\lambda_s=\lambda_{111}\) in the case of nickel.

From (12.29) it is evident that, for small reversible displacements of the boundaries, the effects of the magnetic field and of stresses are additive. However, as soon as irreversible displacements come into play, this additivity disappears. In particular, \(\delta n_k\) will depend on the sequence in which the field and stresses are applied, on the manner in which they are applied and varied, etc. (see below).

On the basis of the general relation (12.29), Kondorskii obtained formulas for the reversible susceptibility \(\chi_r\) due to reversible processes of displacement of the boundaries between regions of spontaneous magnetization. This quantity is additively composed of reversible displacements of \(180^\circ\)-boundaries \(\chi_{\parallel r}\) and \(90^\circ\)-boundaries \(\chi_{\perp r}\), with

\[ \chi_{\parallel r}=\chi_1\sum_i \left[ 1-\left(\frac{n_i-\bar n_i}{n_i+\bar n_i}\right) \right](n_i+\bar n_i)^2 h_i^2. \tag{12.30} \]

If antiparallelly magnetized regions are grouped into blocks, then (12.30) has a somewhat different form:

\[ \varkappa_{\parallel r} = \varkappa_1 \sum_i \left[ 1- \left( \frac{n_i-\overline{n_i}}{n_i+\overline{n_i}} \right)^2 \right] (n_i+\overline{n_i})\,h_i^2, \tag{12.30'} \]

\[ \varkappa_{\perp r} = \varkappa_2 \sum_i \left[ (n_i+\overline{n_i})(n_k+\overline{n_k})(h_i^2+h_k^2) - 2(n_i-\overline{n_i})(n_k+\overline{n_k})h_i h_k \right]. \tag{12.31} \]

(where the factors \(\varkappa_1\) and \(\varkappa_2\) are determined through the gradients of the internal forces (12.26) and, consequently, depend on the structural properties of the material; \(n_i\) is the concentration of the \(i\)-th magnetic phase with the positive direction of magnetization, and \(\overline{n_i}\) is the same for the phase with the negative direction of the vector \(I_s\)). From (12.30)—(12.31) it is seen that the reversible susceptibility depends: 1) on the structural properties of the lattice, through the constants \(\varkappa_1\) and \(\varkappa_2\), 2) on the magnetic structure, i.e. on the ratio of the concentrations \(n_i\) of the various magnetic phases\(^*\), 3) on the orientation of the magnetization in the crystal—the anisotropy \(\varkappa_r\), due to its dependence on the direction cosines \(h\).

It follows from Kondorskii’s theory that, for a uniform distribution of magnetic phases (all \(n_i\) identical), the anisotropy \(\varkappa_r\) is absent. In the case of a sharply expressed magnetic texture (the concentrations \(n_i\) differ strongly from one another), however, one may expect a considerable anisotropy of \(\varkappa_r\). In particular, Kondorskii\({}^{140}\) thus explained the results of Williams’s experiments\({}^{141}\), who found that the initial magnetic susceptibilities in single crystals of silicon iron having the form of closed flat frames with sides parallel to the principal crystallographic axes \([100]\), \([110]\), and \([111]\) are related respectively as \(\varkappa_{ar[100]}:\varkappa_{ar[110]}:\varkappa_{ar[111]}=6:3:2\). In this case it is only necessary to assume that, owing to their flat shape, the samples possess a sharp anisotropy of the demagnetizing factor, which leads to a strongly expressed magnetic texture.

By generalizing Kondorskii’s calculation, one can obtain\({}^{142}\) a dependence of the magnetic susceptibility on external elastic stresses which is qualitatively confirmed by experiments, though as yet very few in number.

Owing to the sharp structural dependence of the initial permeability (\(\mu_a=4\pi\varkappa_a\)), the theory in its present state cannot determine numerical values of \(\varkappa_{ar}\). In real materials these values vary within wide limits—from several units or tens (for example, in nickel \(\mu_a\sim 30\)) to tens of thousands (for example, the iron–silicon–aluminum alloy Sendust: \(\mu_a\sim 35000\)). From the theory,

\(^*\) With a sharply nonuniform distribution of these concentrations, we are dealing with the phenomenon of magnetic texture.

S. V. Vonsovskii

However, one should make the general qualitative assertion that the less magnetically anisotropic the material is, the purer it is, and the less distorted its crystal lattice is, the higher its initial permeability. It follows from the theory that the structurally sensitive quantities \(\chi_1\) and \(\chi_2\), which enter expressions (12.30)—(12.31), vary, respectively, inversely proportionally to the inhomogeneities of the boundary energy \(\gamma\)

\[ \left(\chi_{\parallel}\sim \frac{1}{\Delta\gamma}\right) \]

or to the product of the magnetostriction constant and the mean amplitude of internal stresses

\[ \left(\chi_{\perp}\sim \frac{1}{\lambda_2\sigma_i}\right). \]

Therefore, other conditions being equal, in order to obtain a material with the maximum value of \(\mu_a\), it is necessary to seek a reduction of magnetostrictive deformations. For example, this can be achieved by additions to the metal or alloy of certain supplementary elements. Experience shows that, indeed, alloys with small magnetostriction possess such properties (for example, alloy 1040 of the Fe—Ni—Cu—Mo system\(^{143}\), sendust alloy of the Fe—Si—Al system\(^{144}\)). Detailed investigations of the influence of the magnitude of magnetostriction on the magnitude of the initial susceptibility were carried out by Zaimovskii\(^{145}\) and others. The temperature dependence of the initial susceptibility is highly peculiar. Near the Curie point the curve \(\chi_a(T)\) has a sharp maximum (the Hopkinson effect) (see Fig. 55). A detailed investigation of this phenomenon was carried out by a number of authors: Zaimovskii\(^{146}\), Honda and Nishina\(^{147}\), Kirkham\(^{149}\), Dekhtyar and Androshin\(^{148}\), Drozhzhina and Shur\(^{150}\). However, the present state of the theory of the technical magnetization curve does not yet make it possible to give a quantitative explanation of this dependence.

Kersten\(^{151}\), developing the above-mentioned theory of “inclusions,” also obtained a formula for the initial susceptibility, which has the form

\[ \chi_a(T)=C\,\frac{I_s^2(T)}{\sqrt{k_1(T)}}. \tag{12.32} \]

Thus, the temperature dependence of \(\chi_a\) enters only through the spontaneous magnetization \(I_s\) and the anisotropy constant \(k_1\). However, the constancy of the factor \(C\) with temperature and its magnitude remain, within the framework of this theory, still entirely undetermined.

As Gans\(^{152}\) showed, the reversible susceptibility of ferromagnets (which is defined as the limit of the ratio of the change in magnetization of the specimen \(\Delta I\) to the change in the magnetic field \(-\Delta H\) as it decreases and as \(|\Delta H|\to 0\)) in many cases proved to be a universal function of the magnetization. In parametric form this dependence has the form

\[ \left. \begin{aligned} \frac{\chi_r}{\chi_{ar}} &= \frac{1}{x^2}-\frac{1}{\operatorname{sh}^2 x},\\ \frac{I}{I_s} &= \operatorname{cth} x-\frac{1}{x}. \end{aligned} \right\} \tag{12.33} \]

In Fig. 56 such a theoretical curve (12.33) is shown for soft iron, together with experimental points. Recently, however, sharp deviations from the law (12.33) have been found. This was observed, for example, by Samuel153 for cobalt, and by Goldschmidt154 and Shubina78 for silicon iron. Kondorskii’s general theory139 makes it possible to understand the nature of these deviations, which are connected with the phenomenon of magnetic texture. Braun155, on the basis of conclusions from Kondorskii’s theory, showed that the empirical relation (12.33) can be fundamentally

Fig. 55. Temperature dependence of the initial susceptibility of nickel. (According to Kirkham.)

Fig. 56. Reciprocal susceptibility. (According to Gauss.) Theoretical curve (12.33) and experimental points for soft iron.

justified only under very special assumptions about the character of the internal stresses in the material and its magnetic texture.

Kondorskii47,156 was the first to note that the initial susceptibility may fail to coincide, as is usually assumed, with the initial reversible susceptibility \((\chi_a > \chi_{ar})\). Even in an arbitrarily weak field an irreversible displacement of the boundaries may occur. Studying the magnetization of ferromagnets (an iron–nickel alloy with 15% Ni) subjected to tension, as a function of the character of the increase of the magnetic field, Kondorskii found that the inequality \(\chi_a > \chi_{ar}\) occurs in specimens demagnetized without load and then subjected to tension and magnetized by a smooth increase of the field. This happens because here the boundaries in the initial state are in unstable equilibrium, and irreversible magnetization must begin in weaker fields than when the specimen has been demagnetized under load and, as a result, regions stable under the given conditions have been formed.

Changes in the initial portion of the magnetization curve may be expected in all cases when, by one means or another, ...

change in the distribution of the magnetic phases of the initial (for example, demagnetized) magnetic state of the body.

In particular, as Dekhtyar[^157] showed, the initial susceptibility of single crystals of meteoritic iron depends substantially on the sequence of applying elastic stresses and switching on the magnetic field, and also on the conditions of demagnetization of the crystals. Drozhzhina and Shur[^158] carried out a comprehensive study of the influence of demagnetization conditions, of the sequence of application and removal of elastic loading, and of switching on the magnetic field, on the initial portion of the magnetization curves of polycrystalline specimens of iron, nickel, and transformer steel. They noted that the increase in susceptibility caused by an “elastic wave” during loading or unloading of the specimen has, as its upper limit, the susceptibility of the so-called hysteresis-free (ideal) magnetization curve.

The magnitude of the magnetization in weak magnetic fields depends substantially on the temperature prehistory of the specimen and on the moment at which the magnetic field is switched on. This effect is explained by: 1) the temperature hysteresis of structural transformations in the crystal lattice of heterogeneous ferromagnetic materials and 2) the temperature magnetic hysteresis inherent in the very processes of technical magnetization.

The phenomenon of an ambiguous temperature dependence of magnetization, for a given value of the magnetic field, in the initial portion of the curve \(I(H)\), has long been used, for example, in the so-called temperature aging of permanent magnets (successive heating and cooling of residually magnetized specimens), or in obtaining hysteresis-free (ideal) magnetization curves. However, the beginning of the modern study of this phenomenon is associated with the works of Drozhzhina and Shur[^150], who established, in qualitative form, the connection between temperature magnetic hysteresis and the processes of technical magnetization.

e) Reversible processes of rotation. The completion of displacement processes in ferromagnetic single crystals must lead to the technical saturation of the latter along one of the axes of easiest magnetization closest to the direction of the magnetizing field. A further increase of the field causes the process of rotation of the vector \(I_s\), which ends when the vectors \(I_s\) and \(H\) become parallel to one another. Such a sharp division of the magnetization curve into two different regions, differing in their nature, is, of course, only relative in character. One can only assert that, as already indicated, in the region of weak fields (below the maximum on the Stoletov curve, Fig. 22) \(\chi_{\mathrm{cm}} \gg \chi_{\mathrm{вр}}\), while in medium fields (after the maximum on the Stoletov curve up to saturation) \(\chi_{\mathrm{вр}} \gg \chi_{\mathrm{cm}}\). This delimitation of the magnetization processes is observed in its sharpest form on the magnetization curves of ferromagnetic single crystals (see Fig. 48).

The law of magnetic anisotropy of Akulov \(^{114}\) makes it possible to calculate theoretically the magnetization curve for any single crystal and for any direction in it in the region of fields where \(\chi_{\mathrm{во}} \gg \chi_{\mathrm{см.}}\). The calculation of these curves reduces to determining the minimum of the total free energy of the single crystal, consisting of the energy of magnetic anisotropy (12.5) and the energy relative to the external field (12.12)\(^*\). As an example, we shall present the calculation of the magnetization curve for a single crystal of cubic symmetry having the form of a wire whose axis coincides with the direction \([110]\) (the diagonal of a cube face), as is shown in Fig. 57. According to (12.5) and (12.12), the density of the free energy of the crystal is equal to

\[ F = k_1\alpha_1^2\alpha_2^2 - HI_s \cos \vartheta . \tag{12.34} \]

Fig. 57.

In the state corresponding to the kink on the magnetization curve, there are only two magnetic phases with magnetizations \(I_s\) parallel to the axes \([100]\) and \([010]\), which make the same angle of \(45^\circ\) with the axis of the wire and the direction of the field \(\mathbf H\). Since the position of \(I_s\) relative to \(\mathbf H\) in both phases is exactly the same, the whole calculation may be carried out with one phase, assuming conventionally that all the magnetization is directed along the axis \([100]\). Let us denote by \(\vartheta\) the angle between \(I_s\) and \(\mathbf H\) (see Fig. 57); then

\[ \alpha_1 = \cos(45^\circ - \vartheta) = \frac{1}{\sqrt{2}}(\cos \vartheta + \sin \vartheta), \]

\[ \alpha_2 = \cos(45^\circ + \vartheta) = \frac{1}{\sqrt{2}}(\cos \vartheta - \sin \vartheta), \]

\[ \alpha_3 = 0 \]

and, introducing the relative magnetization \(j = \cos \vartheta = \dfrac{I}{I_s}\), we obtain from (12.34)

\[ F(j) = \frac{1}{4}k_1(2j^2 - 1)^2 - HI_s j . \]

The condition of thermodynamic equilibrium \(\dfrac{\partial F}{\partial j} = 0\) gives:

\[ H = \frac{2k_1}{I_s}(2j^2 - 1)j . \tag{12.35} \]

\(^*\) Strictly speaking, to this one must also add the energy of the demagnetizing field \(F_H\), determined by the shape of the specimen \(^{159}\). But for specimens in the form of long wires or closed frames [Williams \(^{141}\)] taking account of the energy \(E_H\) is immaterial.

(12.35) and is the equation of the magnetization curve for the process of rotation of a cubic crystal for the axis [110]. Graphically this curve is shown in Fig. 58. At \(H=0\) (if displacement processes are completely excluded) \(j=\dfrac{1}{\sqrt{2}}\), i.e. at the kink point on the experimental curve \(I\) must be equal to \(I_s\sqrt{2}\), which is indeed observed in reality (see Fig. 48). The saturation field \(H_{s[110]}=\dfrac{2k_1}{I_s}\). For example, in the case of iron \((k_1\sim 4\cdot 10^5,\ I_s\sim 1.7\cdot 10^3)\), \(H_{s[110]}\sim 470\) oersted, which also agrees with experimental data. Similar agreement between theory and experiment is observed in other cases.

Fig. 58. Theoretical magnetization curve of a ferromagnetic cubic crystal along the [110] axis, calculated on the assumption that all magnetization proceeds by the rotation process.

Akulov \(^{114}\) also calculated magnetization curves of polycrystals (for fields where \(\chi_{\mathrm{rot}}\gg \chi_{\mathrm{disp}}\)) by simple averaging of the data for single crystals for a specified distribution of the directions of the axes of the individual crystallites of the polycrystalline specimen. These calculations were confirmed by the experiments of Zhigadlo and Sidel’nikova \(^{160}\). However, in these calculations a difficulty arises because of the impossibility of accurately taking into account the magnetic interaction between the individual crystallites (see below), which, above all, blurs the sharp boundary between the sections of the curve for the rotation and displacement processes (delay in the completion of displacement processes and simultaneous “untying” of the intensive rotation of the vectors \(I_s\)). Therefore, the use of the results of calculating the curves \(I(H)\) for single crystals in the case of polycrystalline specimens makes sense mainly at sufficiently high fields [see below, section ж].

A very convenient magnetic quantity for the study of ferromagnetic single crystals in the region of intensive rotation processes is the component of the magnetization normal (to the field), \(I_\perp\). These measurements, usually carried out with the aid of single-crystal disks, depend to a much lesser degree on the demagnetizing factor of the specimen than do measurements of the curves \(I_{\parallel}(H)\). Therefore it is precisely measurements of \(I_\perp\) that can be successfully used for the most accurate determination of the constants of magnetic anisotropy.

When a single-crystal disk is placed in an external uniform field \(H\) (see Fig. 59), parallel to its surface, on the vec-

tor \(I_s\) (with the disk fixed) will be acted upon by a torque from the external field

\[ t_H=-HI_s\sin\vartheta, \]

and, by virtue of the magnetic anisotropy, there will arise still another torque

\[ t_k=-\frac{dF_k}{d\varphi}. \]

In the state of equilibrium \(t_H=t_k\), and, consequently, for the orientation of the disk shown in Fig. 59, we shall have the curve for the normal component of the magnetization in the plane of the disk

\[ I_\perp=I_s\sin(\varphi_1-\varphi)=\frac{k_1}{2H}\sin4\varphi. \]

Akulov\(^{114}\) was the first to calculate the curves \(I_\perp(H,\varphi_1)\). The method of measuring \(I_\perp\) or \(t_H\) found wide practical application. Akulov and Bryukhatov\(^{114}\) developed a special type of torsion magnetometer for determining \(I_\perp\), and also for determining, from the curves \(t_H(\varphi,H)\), the crystallographic texture in disks cut from polycrystalline specimens. These investigations led to the creation of a special practical method of magnetic control of the texture of products (chiefly of sheet material)—the so-called magneto-textural analysis. The development of this method is due chiefly to the work of Soviet scientists Akulov and Bryukhatov\(^{114}\), Aksyonov and Grigorov\(^{61}\), Titov\(^{162}\), Volkov\(^{163}\), and others.

Fig. 59. For the calculation of the normal component of magnetization in a single-crystal disk.

A more detailed investigation of the curves \(I_\perp(\varphi,H)\) showed the presence of a number of deviations from the formulas originally obtained by Akulov. These deviations, however, as shown by the analysis and measurements of Tarasova\(^{16}\), Bozort and Williams\(^{105}\), Kirensky\(^{106}\), and Shubina\(^{67}\), can be explained by the same Akulov theory, if only one takes into account the influence of the demagnetizing factor of the disk and more accurately allows for the energy of magnetic anisotropy. Therefore, the study of the curves \(I_\perp\) and their practical use constitute brilliant evidence of the fruitfulness and accuracy of the theory of magnetic anisotropy, created first and foremost by the works of Akulov\(^{114}\).

As already indicated above, the magnetic anisotropy of ferromagnetic crystals can depend substantially on external and internal stresses. In particular, under very strong stresses \((\lambda\sigma \gg k_1)\), the latter practically alone set the “tone” of the entire aniso-

crystal. It is precisely for this reason that the magnetization curves in the region of the rotation process are also very sensitive to stresses. Akulov\(^{114}\), Gans\(^{168}\), and others calculated curves for single crystals with allowance for external stresses. Akulov and Kirensky\(^{169}\) performed a general calculation of the magnetization curves of polycrystals near saturation for a diffuse distribution of stresses and under homogeneous elastic deformation. These calculations make it possible to determine the constants of magnetostriction and magnetic anisotropy, as well as the magnitude of internal stresses, from measurements on polycrystals. Similar work was carried out by Yanshin\(^{170}\).

As an illustration, let us consider the calculation of magnetization curves in the region of the rotation process under very strong stresses \((\lambda_s\sigma \gg k_1)^{171}\). In this case the free energy of the crystal, by virtue of (12.9′) and (12.12), is equal to

\[ F=-\frac{3}{2}\lambda_s\sigma\cos^2\vartheta-HI_s\cos\vartheta, \tag{12.36} \]

where \(\vartheta\) is the angle between the vectors \(\mathbf{H}\) and \(\mathbf{I}_s\). The stresses in this case are uniaxial and parallel to the vector \(\mathbf{H}\). From the condition of thermodynamic equilibrium

\[ \frac{\partial F}{\partial \vartheta}=0 \]

we find from (12.36):

\[ I=I_s\cos\vartheta=-\frac{I_s^2}{3\lambda_s\sigma}\cdot H \]

and, consequently,

\[ \chi=-\frac{I_s^2}{3\lambda_s\sigma}. \tag{12.37} \]

Fig. 60. Influence of uniaxial elastic tensile stress \(\sigma\) on the magnetization curves of polycrystalline nickel. (After Becker and Kersten.)

From (12.37) it follows that, under very strong tensile stresses \((\sigma>0)\), in materials with negative magnetostriction \((\lambda_s<0)\) (for example, nickel), the magnetization curves have the form of straight lines, whose slope depends on \(I_s\), \(\lambda_s\), and \(\sigma\) and is determined by (12.37). Measurements of such curves on nickel were carried out by Becker and Kersten\(^{171}\). These measurements (see Fig. 60) showed the complete validity of formula (12.37). Grabovsky\(^{172}\) investigated the magnetization curves of stretched nickel at low temperatures with the aim of determining the course of these curves upon a sharp increase of the constants of crystallographic magnetic anisotropy. It turned out that, in contrast to room temperatures, at \(-183^\circ\mathrm{C}\) the rectilinear

the magnetization curves of strongly stretched nickel ($\sigma \sim 17\ \text{kg}/\text{mm}^2$) have a noticeable bend in the initial portion. This corresponds to the fact that at low temperatures the condition $\lambda_s\sigma \gg k_1$ is violated, and in weak fields, along with rotation processes, displacement processes begin to play a noticeable role.

Gorelik$^{173}$ and Lyubina$^{174}$ investigated the influence of stresses on magnetization curves in the region of the rotation process under the superposition of two mutually perpendicular magnetic fields. These investigations are of substantial importance for practical application (in the design of highly sensitive magnetometers).

In addition to the cases considered of magnetization of ferromagnets by means of the rotation process, one should mention one more case, which may occur in materials possessing strong internal stresses $\sigma_i$ with a disordered distribution of their orientations (for example, hard magnetic materials of the type of steels quenched to martensite). It may be assumed that in small volumes inside the specimen the $\sigma_i$ are homogeneous, while along the boundaries of these volumes (where the orientation of the vectors $\sigma_i$ changes) there are “peaks” of stresses or a “network” of inclusions, which create a very high potential barrier that retards displacement processes during magnetization of the material. If the average sizes of these “isolated” volumes are commensurate with the equilibrium sizes of ferromagnetic domains, then it is natural to suppose that in each such volume there will fit one domain, the boundaries of which will be rigidly “pinned” at the “peaks” of stresses or at the “network” of inclusions. Owing to this latter circumstance, rotation processes attain appreciable development in the region of weak fields with practically complete inhibition of displacement processes. Therefore in such materials the initial susceptibility will be determined entirely by rotation processes, and not by displacement, as is the case in soft materials. As was already mentioned above, the general solution of the problem of calculating the magnetization curve was given by Akulov and Kirenskii$^{165}$, and also by Becker$^{175}$. At the present time our information on the magnitude and distribution of internal stresses in real materials is very scanty. Therefore, conversely, magnetic measurements on materials with strong internal stresses are being used in attempts to determine the magnitude and dispersion of $\sigma_i$. The most convenient quantity for this determination is the initial susceptibility, which has the form

\[ \chi_a=\frac{I_s^2}{a\lambda_s}\left(\overline{\frac{1}{\sigma_i}}\right), \tag{12.38} \]

where $a$ is a numerical factor of order unity, and $\left(\overline{\dfrac{1}{\sigma_i}}\right)$ is the mean value of the reciprocal of the amplitude of the internal stresses in a polycrystalline material. Formula (12.38) is also the basis of the magnetic method for determining internal stresses. Experimen-

Much work has been devoted to the experimental determination of \(\sigma_i\) from magnetic measurements by Akulov\(^{114}\), Kersten\(^{176}\), Tissin\(^{177}\), Förster and Stambke\(^{178}\). It should be noted that the theoretical substantiation of this practically important method requires further refinement by taking into account the magnetic interaction between regions, the inevitable “admixture” of displacement processes, and the influence of the magnetic texture.

In constructing a theory of the technical magnetization curve, the greatest difficulty is the allowance for the substantial influence of the structure of the material on the processes of magnetization. These difficulties consist chiefly in the fact that the mentioned influence of structure has a highly individual character from specimen to specimen, whereas theory strives above all to establish certain universal regularities of the phenomenon. From this point of view it is very interesting to investigate the course of the magnetization curve of ferromagnets in the limiting case of very strong fields, where the magnetization can be represented in the following universal form (“law of approach to saturation”):

\[ I = I_s \left( 1 - \frac{a_1}{H} - \frac{a_2}{H^2} - \frac{a_3}{H^3} - \cdots \right). \tag{12.39} \]

The works of Akulov\(^{114}\) laid the foundation for the modern theory of the law of approach to saturation. If it is assumed that in the region of very high fields magnetization is effected only by means of the process of rotation, then the term with \(a_2\) in (12.39) is wholly determined by the energy of crystallographic anisotropy \(F_k\) and the energy of elastic stresses \(F_\sigma\). As Akulov showed,

\[ a_2 = \frac{1}{2 I_s^2}\, \overline{\left[\nabla (F_k + F_\sigma)\right]^2}_{\vartheta \to 0}, \tag{12.40} \]

where the bar means that the mean value of the square of the gradient of the energy \((F_k + F_\sigma)\) over the volume of the specimen must be taken, and \(\vartheta \to 0\) indicates that the entire calculation is carried out in very strong fields, when the angle between the vectors \(\mathbf H\) and \(\mathbf I_s\) throughout the volume of the specimen may be considered a small quantity. From (12.40), after calculation, one can obtain a relation between the quantity \(a_2\), determined from experiment, and the constants of magnetic anisotropy and magnetostriction. This relation underlies the above-mentioned method for determining the constants of magnetic anisotropy. Numerous experimental investigations\(^{78}\) have shown the validity of Akulov’s theory, since the anisotropy constants thus determined from measurements on polycrystals coincide with the constants measured in experiments with single crystals. Holstein and Primakov\(^{179}\) refined the formula for \(a_2\), taking into account in a general form the magnetic interaction between the crystallites of a polycrystal. This further improved the agreement between theory and experiment.

The term with \(a_1\) in (12.39), as shown by the experiments of Polli\(^{180}\), is determined by plastic deformations of the crystal. Brown\(^{181}\) developed a theory for

of this case. He showed that sharply inhomogeneous local deformations (dislocations) disturb the homogeneous distribution of electron spins in a volume much larger than that occupied by these deformations themselves, which makes it possible to detect deviations of the magnetization from saturation caused by this reason. Brown’s formula, in contrast to (12.39), has the form

\[ I=I_s\left[1-\frac{a_1'}{H^{1/2}}-\frac{a_1''}{H}-\frac{a_1'''}{H^{3/2}}-\frac{a_2}{H^2}-\ldots\right], \tag{12.41} \]

where the term with \(a_1'\) corresponds to point localization of stresses, \(a_1''\) to linear localization, and \(a_1'''\) to surface localization. Parfenov\({}^{96}\) investigated the law of approach to saturation in a large number of magnetic materials, both soft and hard, and found the presence of the terms with \(a_1'\) and \(a_1'''\) predicted by Brown. However, Brown’s theory is still of a very preliminary character and requires further refinement.

g) Magnetic hysteresis. Irreversible changes in the magnetization of ferromagnets during their magnetization and remagnetization lead to the phenomenon of magnetic hysteresis.

Kondorskii\({}^{183}\) clearly indicated three main mechanisms of hysteresis.

1. Hysteresis caused by irreversible rotation processes (in the absence of remagnetization nuclei).

2. Hysteresis caused by a delay in the growth of remagnetization nuclei.

3. Hysteresis caused by a delay in the displacement of boundaries between regions of spontaneous magnetization.

Let us consider these three mechanisms of hysteresis in somewhat more detail from the theoretical and experimental points of view.

If in a ferromagnetic material the possibility is excluded of the occurrence of remagnetization nuclei (i.e., volumes with spontaneous magnetization of the opposite direction relative to the previous orientation of the saturation magnetization of the specimen), then displacement processes are altogether excluded in it, and remagnetization can take place only by means of the process of rotation of the vectors \(I_s\). Such remagnetization of a ferromagnet was first treated theoretically by Akulov\({}^{114}\). Akulov’s theory can be clearly illustrated by the case already considered above of magnetization of a single-crystal wire with an axis parallel to the crystallographic axis [110]. The equation of the magnetization curve is given by formula (12.35). From Fig. 58, which gives the graph of this curve, it is seen that its segment \(EOF\) is thermodynamically unstable

\[ \left(\frac{\partial^2(F_k+F_H)}{\partial j^2}<0\right); \]

the segments \(EDC\) and \(HGF\) are stable and correspond to two minima of the free energy \((F_k+F_H)\), separated by a potential barrier. The fields corresponding to the boundaries of this

of a two-valued interval, at which the energy barrier vanishes together with one of the minima \((F_k + F_H)\), are determined from the condition

\[ \frac{dH}{dj}=0. \]

By virtue of (12.35), this gives for the coercive force

\[ H_{c[110]}=\pm \frac{4}{\sqrt{6}}\frac{k_1}{I_s}. \]

In the general case, for an arbitrary orientation of the field, in the case of polycrystalline specimens and upon the application of external stresses \((k_{\mathrm{eff}}\sim k_1+\alpha \lambda_s\sigma)\), the expression for \(H_c\) is given, in order of magnitude, by the formula

\[ H_c \sim \frac{k_{\mathrm{eff}}}{I_s}. \tag{12.42} \]

For example, for pure iron \((k\sim 5\cdot 10^5,\ I_s\sim 1.7\cdot 10^3)\), \(H_c\sim 300\) oersteds. Already from this estimate it is clear that the coercive forces of real soft materials, for which experiment gives values not exceeding 1–10 oersteds, are not connected with this mechanism of hysteresis.

Akulov^114 developed a theory of hysteresis losses in rotating magnetic fields, proceeding from ideas of remagnetization as a pure process of rotation. Bryukhatov^189 investigated the relation between the coercive force and the anisotropy constants of various ferromagnetic materials. Mes’kin and Somin^190 showed experimentally that in real hard ferromagnetic materials the magnitude of the coercive force cannot be explained by the simple effect of irreversible rotation processes.

If in a specimen the magnetic anisotropy is vanishingly small \((k+\alpha\lambda\sigma\sim 0)\) and at the same time the possibility of the appearance of remagnetization nuclei is excluded, then the process of rotation will be determined by the anisotropy of the demagnetizing factor. For example, in the case of a specimen in the form of an elongated ellipsoid of revolution with a demagnetizing factor along the axis of revolution \(N_1\) and along any direction in the plane perpendicular to this axis \(N_2\) (with \(N_1 \ll N_2\)), the coercive force proves to be equal to

\[ H_c=|N_2-N_1|I_s. \]

In the case of a very long specimen, \(N_1\sim 0,\ N_2\sim 2\pi\), and, consequently, for a material with \(I_s\sim 10^3\), \(H_c\sim 5\cdot 10^3\) oersteds; such values of coercive forces are not observed in experiment at all. Nevertheless, there are real cases in which hysteresis may be caused by irreversible rotation of the vectors \(\mathbf{I}_s\). These cases are the following: 1) high-coercivity heterophase ferromagnetic materials, 2) thin ferromagnetic films, 3) colloidal ferromagnetic particles in an isolated state (aerosols) or coagulated micro-impurities in nonferromagnetic materials. In all materials of this kind the coercive force indeed has anomalously large values (hundreds of oersteds), however the experimental

the data still remain very few in number and unsystematic, and therefore we do not possess sufficiently well-founded proof that, in these materials, remagnetization is effected by means of the rotation process (see, however, below on high-coercivity alloys).

The second mechanism of hysteresis, caused by retardation of the growth of nuclei, occurs to one degree or another in all ferromagnets, and the reality of its existence has undoubtedly been proved by experiment. In its pure form this mechanism of hysteresis can be very clearly studied (and not only studied, but also controlled) in specimens with the so-called rectangular hysteresis loop. Such specimens can be prepared artificially, for example, from a polycrystalline material (usually in the form of wire) with a high yield point, by subjecting it to very strong unilateral external tensile stresses. In this case \((\lambda_s \sigma_e > k)\) the entire magnetic anisotropy of the specimen is practically determined by the stresses, and the axis of the stretched wire becomes (for \(\lambda_s > 0\)) the axis of easiest magnetization. In Fig. 61, as an example,\(^{184}\) hysteresis loops are given for a wire made of an iron–nickel alloy under various tensile stresses, illustrating the process of formation of a rectangular hysteresis loop.

Fig. 61. Formation of a rectangular loop in iron–nickel wire (14% Ni).

With the rectangular form of the loop, the process of remagnetization is carried out by a single jump, whose mechanism consists in the fact that at some place in the specimen a nucleus of remagnetization is created; at a certain magnitude of the magnetic field (the starting field) it begins to grow with finite velocity, and in the end absorbs the entire volume of the specimen. These large jumps of remagnetization have been studied with great care by many investigators (Preissach\(^{184}\), Sixtus and Tonks\(^{185}\), Steinberg\(^{186}\), Miroshnichenko\(^{187}\), and others\(^{78}\)). The importance of these investigations lies in the fact that they, first of all, gave direct proof of the real existence of processes of displacement of boundaries between ferromagnetic regions and, secondly, made it possible to follow in detail the kinetics of the remagnetization process. In these experiments the existence of two characteristic values of the magnetic field was discovered, which determine the kinetics of remagnetization. One of these values—the “starting field” \(H_s\)—determines the beginning of the remagnetiza-

tion; it is needed for the formation in some small region of the specimen of a nucleus of reversal, of dimensions that make its further growth energetically more favorable than its disappearance (“evaporation”!). However, after the starting field \(H_s\) has once been reached, and the nucleus has begun to grow with a finite velocity, this growth can continue in a weaker, so-called critical field \(H_0(<H_s)\). This field is needed in order that the boundary of the reversing region, in its motion, may overcome all the potential barriers arising from inhomogeneities of the specimen material.

The theory of the starting field was developed in detail by Döring\(^{188}\), who, proceeding from general ideas about the processes of technical magnetization (Akulov), obtained theoretical conditions for the growth of nuclei of reversal. In particular, he obtained the formula for the starting field

\[ H_s = H_0 + a\,\frac{\gamma}{I_s}\,\frac{1}{d}, \tag{12.43} \]

where \(a\) is a numerical constant \(\sim 1\), \(\gamma\) is the density of the free energy of the boundary zone of the nucleus of reversal, and \(d\) is the diameter of the transverse section of the nucleus in cm (the form of the nucleus is assumed to be that of an elongated ellipsoid of revolution). In experiments with large discontinuities of reversal it has been possible to “freeze” nuclei in the process of their growth and then, by etching the wires, to extract them from the bulk of the material, thereby obtaining direct proof of their real existence and the possibility of a quantitative verification of the theory. Formula (12.43) is of especially substantial importance because it gives the simplest way of determining the magnitude of the boundary energy \(\gamma\). Indeed, the quantities \(H_s\), \(H_0\), \(I_s\), and \(d\) are determined from experiment; therefore the quantity \(\gamma\) can also be found, to within the estimate of the numerical factor \(a(\sim 1)\), and compared with its theoretical formula according to the theory of Landau and Lifshitz\(^{113}\) (12.16). Thus, for example, for an iron–nickel wire, from (12.43) we obtain \(\gamma \sim 2.7\ \mathrm{erg/cm^2}\), while from formula (12.16) \(\gamma \sim 2.1\ \mathrm{erg/cm^2}\).

Taking into account the approximate character of these relations, the agreement obtained is very good. The kinetics of the growth of nuclei was studied in detail (Miroshnichenko\(^{187}\) and Haaque\(^{191}\)), and experiment fully confirmed the predictions of the theory.

In the whole theory of the starting field, one admittedly very important question remains unclear—how the very nucleation of the reversal region takes place before the beginning of its growth. Here one can only make the following assumptions. First, it may turn out that in a ferromagnet, even at saturation, there remain small portions of the former regions of spontaneous magnetization with direction \(I_s\), opposite to the direction of the magnetic field causing reversal. In this case it must be assumed that the critical field \(H_0\)

of such regions exceeds the magnitude of the field that brings the specimen to preliminary saturation.

Secondly, the demagnetizing field created by internal inhomogeneities of the specimen, foreign inclusions in the metal, or voids can have a substantial influence on the process of nucleus formation. These fields may make the internal field in small volumes of the specimen near such distortions opposite in relation to the external field. Finally, a third cause of the appearance of reversal-magnetization nuclei may be thermal fluctuations. However, calculation of the probability of this process for the creation of a nucleus of critical size (possessing the ability for further growth) shows that this cause can hardly play a substantial role. Kondorskii^192 showed that the area of the hysteresis loop and the magnitude of the coercive force, if the hysteresis is partly due to the second principal cause (delay in the growth of nuclei) or the first (irreversible rotations), may depend on the shape of the specimen. If there is no such dependence, this means that the rectangular loop is due mainly to the delay of boundary-displacement processes. Kondorskii explains the independence of the coercive force from shape in the majority of polycrystalline ferromagnets either by the cause just mentioned, or by the fact that, owing to structural inhomogeneities and differences in the directions of the axes of easy magnetization in separate regions of the specimen, the coercive force depends no longer on the shape of the ferromagnet, but on the shape of the individual regions that are sufficiently homogeneous and have a single direction of easiest magnetization.

A general formula for the critical field \(H_0\) can be obtained from the conditions for the maximum of the difference of the free energies (12.24). In the case of \(180^\circ\) displacements \((F_l)_k-(F_l)_l\sim 0\), and if one further assumes that the curvature of the boundaries between the domains is small \(\left(\dfrac{1}{r_1}+\dfrac{1}{r_2}\sim 0\right)\), then, in order of magnitude, \(H_0\), by virtue of (12.24) and (12.27), will be equal to

\[ H_0=\frac{1}{2I_s}\left(\frac{\overline{\partial\gamma}}{\partial n}\right)_{\max}, \tag{12.44} \]

where the bar denotes the mean value of \(\left(\dfrac{\partial\gamma}{\partial n}\right)_{\max}\) over the boundary surface. If, in addition to local changes of the energy \(\gamma\), a noticeable increase in the area of their surface takes place upon displacement of the boundaries (for example, during the growth of nuclei or in flowing around obstacles), then in (12.44), instead of \(\gamma\), one must substitute the product \(\gamma \bar S\), where \(\bar S\) is the mean value of the area of the boundary surface, and divide the entire expression by \(\bar S\). Thus, the critical field according to the “inclusion theory” (for relatively weak local inhomogeneities \(\partial\gamma/\partial n\)) is given by the formula

\[ H_0=\frac{\bar{\gamma}}{2I_s\bar S}\left(\frac{\partial S}{\partial n}\right)_{\max}. \tag{12.45} \]

Kondorsky^192 showed that, in cases where

\[ \left(\frac{\partial \overline{S}}{\partial n}\right)_{\max} \ll \left(\frac{\partial \gamma}{\partial n}\right)_{\max}, \]

the quantity \(H_0\) is determined mainly by the gradients of internal stresses. Indeed, from formula (12.16) for \(\gamma\) it follows that

\[ \frac{\partial \gamma}{\partial n} \sim \lambda_s \delta \frac{\partial \sigma}{\partial n}, \]

and therefore

\[ H_0 \sim \frac{\lambda_s \delta}{I_s}\left(\frac{\partial \overline{\sigma}}{\partial n}\right)_{\max}. \tag{12.46} \]

In a more general case formula (12.46) may, according to Kersten^191, be written in the form

\[ H_0 = p_0 \frac{\lambda_s \overline{\Delta \sigma}}{I_s}, \tag{12.47} \]

where \(\overline{\Delta \sigma}\) is the mean value of the fluctuations of internal stresses, and the factor \(p_0\) depends on the ratio of the thickness \(\delta\) of the boundary layer between ferromagnetic regions to the mean length \(l\) of the “wave” of internal stresses; in the case \(l \gg \delta\), \(p_0 \sim \frac{\delta}{l}\), while for \(l \ll \delta\), \(p_0 \sim \frac{l}{\delta}\), and therefore the largest value of \(H_0\) is to be expected when \(l \sim \delta\).

Thus, from Kondorsky’s theory, for the critical field of soft materials devoid of noticeable inclusions, two important conclusions follow: 1) the critical field increases proportionally to the increase in the mean amplitude of internal stresses \(\overline{\Delta \sigma}\), and 2) \(H_0\) has its largest value when the dispersion of these stresses is comparable with the thickness of the boundary layers between regions of spontaneous magnetization (\(l \sim \delta\)). An enormous amount of experimental material fully confirms these two principal qualitative conclusions of the theory. Unfortunately, a quantitative comparison of theory with experiment cannot be carried out with proper effect because of the pronounced structural sensitivity of the phenomenon of magnetic hysteresis.

For materials with a large number of inclusions, (12.45) applies. This formula, in a more explicit form for the particular case of a regular distribution of spherical inclusions of approximately identical size (cementite grains in carbon steel may serve as an example), takes the form:

\[ H_0 = p' \frac{k_{\text{eff}}}{I_s}\beta^n, \tag{12.48} \]

where \(k_{\text{eff}}\) is the magnetic-anisotropy constant, \(\beta\) is the concentration of impurities, \(n\) is an exponent \(\left(\sim 1,\ \frac{2}{3},\ \frac{4}{3}, \text{ etc.}\right)\), and the factor \(p'\) depends on the ratio of the thickness of the boundary layer \(\delta\) and \(d\), the diameter of the inclusions; for \(\delta \ll d\), \(p' \sim \frac{\delta}{d}\), while for \(\delta \gg d\), \(p' \sim \frac{d}{\delta}\). Thus, \(H_0\) increases with increasing concentration of inclusions and has its greatest value at a definite dispersion of them (when \(\delta \sim d\)). From (12.48)

it follows that the temperature dependence of \(H_0\) is determined mainly by that for \(k_{\mathrm{eff}}\) and \(I_s\) (in \(p'\), \(k_{\mathrm{eff}}\) enters through \(\delta\)), which can be used in testing the theory by means of measurements of the temperature dependence of the coercive force in such materials. However, as yet we do not have systematic experiments to test the theory of inclusions*).

Kondorskii\(^{195}\) constructed a theory of magnetization curves and hysteresis loops for polycrystalline materials. The main difficulty in constructing such a theory is connected with taking into account the magnetic interaction inside a ferromagnetic specimen. Nevertheless Kondorskii, considering limiting cases that are amenable to calculation (a polycrystal with elongated grains and with plane-parallel grains), gave theoretical asymptotic formulas for zero magnetization curves, for maximum hysteresis loops, and for partial-cycle loops in the case of ideal magnetization curves.

Kondorskii was the first to obtain a theoretical formula for the dependence of the hysteresis losses \(W\) on the amplitude of the magnetizing field \(H_0\)

\[ W = 4 I_s H_0 \left(1 - \frac{H_0}{H}\right), \tag{12.49} \]

which agrees well with experiment. The theoretical curves for the dependence of the residual magnetization on the amplitude of the magnetizing field, calculated by Kondorskii, also agree well with experiment. Kondorskii obtained a general formula relating the magnetization curve and the remagnetization curve of a polycrystalline material. If the magnetization curve has the form \(I = f(H)\), then the remagnetization curve can be found at once from the formula

\[ \frac{I_m - I}{2} = f\left(\frac{H_m - H}{2}\right), \tag{12.50} \]

where \(H_m\) is the amplitude of the magnetizing field, and \(I_m\) is the corresponding maximum magnetization, from which the remagnetization process begins. For example, if this result is applied to the case of very weak fields, where, according to Rayleigh, the magnetization curve has the form

\[ I = \chi_a H + bH^2, \tag{12.51} \]

then for the Rayleigh hysteresis loop we immediately find

\[ I = I_m - \chi_a (H_m - H) - \frac{b}{2}(H_m - H)^2. \tag{12.52} \]

Previously these formulas had been regarded as independent. Kondorskii’s theoretical formulas have been confirmed experimentally on a large number of varied soft magnetic materials by experiments of the author himself, and also by Popov and Chernikova\(^{196}\).

*) See the note at the end of the article, p. 64.

Of especially great practical importance are heterogeneous ferromagnetic alloys. From the point of view of obtaining magnetically hard materials, phase transformations may provisionally be divided into three groups[^78]: a) martensitic transformations; b) decomposition of solid solutions; and c) ordering phenomena in an alloy. According to the theoretical considerations set forth above, as a result of the first two types of transformations one obtains such magnetic materials in which the nature of the coercive force is explained either by inhomogeneities of internal stresses or by foreign inclusions in the material. However, neither the “stress” theory nor the “inclusion” theory can explain the nature of magnetic hysteresis in the so-called high-coercivity alloys. Experience shows that in all these alloys high magnetic properties are obtained as a result of phase transformations of the third type, i.e. as a result of processes of ordering of the atoms of these alloys, often with simultaneous decomposition of solid solutions. This general theoretical conclusion was obtained on the basis of numerous experimental works, among which the principal role is played by the works of the Soviet scientists Zaimovskii[^197], Kondorskii[^193], Komar[^198], Livshits B. G.[^199], Mes’kin[^200], Shur[^78], Gabrielian and their collaborators.

Unfortunately, there is still no sufficiently clear picture of the processes of technical magnetization of high-coercivity alloys. One can only say that careful X-ray investigation of these materials (especially the experiments of Komar and Tarasov[^198]) indicates the presence in the alloy of a so-called variable structure, corresponding to the initial stages of ordering of the atoms of the components. With such a variable structure the material breaks up into a finely dispersed aggregate of separate small volumes in the form of thin plates of different phases. Each such plate possesses spontaneous magnetization and sharp magnetic anisotropy (one axis of easiest magnetization). These plates are, to a certain extent, magnetically isolated; displacement processes are almost completely excluded, and the entire mechanism of technical magnetization proceeds by the process of rotation. However, as yet we do not have a sufficient quantity of experimental data that would directly prove the existence of precisely such a mechanism for obtaining high coercive force in these alloys.

The phenomenon of magnetic hysteresis exhibits strong anisotropy. A careful experimental investigation of this question on monocrystalline disks of silicon iron, carried out by Shur[^201], showed that the magnitude of the coercive force of a single crystal depends on crystallographic directions. In the case of a disk, the minimum values of $H_c$ are obtained for those directions in the plane of the disk for which the value of the magnetic-anisotropy energy is minimal. A simple theoretical analysis[^202] of these experiments showed that the observed anisotropy $H_c$ in single crystals of ferromagnets

is determined mainly by the influence of the shape of the specimen (the demagnetizing action of the surfaces of the bases of the disks).

The magnetic hysteresis of ferromagnetic bodies depends substantially on temperature. With increasing temperature the coercive force, as a rule, decreases. In exactly the same way the hysteresis losses decrease with temperature. Modern theory can give only a qualitative explanation of this decrease with temperature of the magnetic parameters characterizing the hysteresis loop^78.

(To be concluded in the next issue.)

LITERATURE CITED FOR PART III

§ 10

1. A. G. Stoletov, Collected Works, vol. I, GTTI (1939).
2. B. L. Rozing, ZhRFKhO (physical section) 24, 105 (1892); 28, 59 (1896); 42, 71 (1910).
3. P. Weiss, Journ. de Phys. et de Radium (4) 6, 661 (1907).
4. P. L. Kapitsa, Proc. Roy. Soc. 131, 243 (1931).
5. Ya. G. Dorfman and R. I. Yanus, Zeits. f. Phys. 54, 277 (1929).
6. L. S. Stilbans, ZhETF 9, 432 (1939).
7. See, for example, L. D. Landau and E. M. Lifshitz, Statistical Physics, ONTI (1940).
8. S. V. Vonsovskii, DAN 27, 550 (1940); Izv. AN SSSR, ser. fiz. 11, 485 (1947).
9. S. V. Vonsovskii and Ya. S. Shur, Ferromagnetism, Gostekhizdat (1948).
10. F. Kainer, ZhETF 10, 67 (1940).
11. P. S. Ehrenfest, Proc. Kon. Akad. Amsterdam, 36, 153 (1933).
12. L. D. Landau, ZhETF 7, 19 (1937); see also E. M. Lifshitz, Journ. of Phys. 6, 61 (1942).
13. S. V. Vonsovskii, Izv. AN SSSR, ser. fiz. 11, 485 (1947); see also V. L. Ginzburg, ZhETF 17 (1937).
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15. L. D. Landau, ZhETF 7, 1232 (1937).
16. See, e.g., V. Gerlakh, UFN 24, 368 (1940).

§ 11

1. Ya. I. Frenkel, Zeits. f. Phys. 49, 31 (1928).
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13. Wolf, Zeits. f. Phys. 70, 519 (1931).
14. S. P. Shubin and S. V. Vonsovskii, Sov. Phys. 7, 292 (1935); 10, 348 (1936); see also S. V. Vonsovskii, Izv. AN SSSR, ser. fiz. 12 (1948).

1. B. T. Geilikman, ZhETF 13, 168 (1943); see also J. C. Slater, Phys. Rev. 52, 198 (1937).
2. E. N. Agafonova, Dissertation, Sverdlovsk (1948).
3. S. V. Vonsovskii, DAN 26, 564 (1940); ZhTF 18, 131 (1948).
4. A. P. Komar, Izv. AN SSSR 11, 497 (1947).
5. V. E. Rudnitskii, ZhETF 10, 63 (1940).
6. F. Bitter, Phys. Rev. 54, 79 (1938).
7. Ya. G. Dorfman, Sov. Phys. 3, 399 (1933).
8. Trapeznikova and Shubnikov, Sov. Phys. 7, 66, 255 (1935).
9. Milyutin and Shalt, DAN 24, 679 (1939).
10. V. E. Rudnitskii, ZhETF 12, 542 (1942).
11. F. Kaner, ZhETF 10, 83, 407 (1940).
12. See, for example, Dressnandt, Zeits. f. Phys. 115, 369 (1940); J. H. VanVleck, Journ. Chem. Phys. 9, 85 (1941).
13. L. D. Landau, Sov. Phys. 4, 675 (1933).

§ 12

1. L. D. Landau and E. M. Lifshitz, Sov. Phys. 8, 153 (1935).
2. N. S. Akulov, Ferromagnetism, ONTI (1939).
3. L. V. Kirenskii, ZhETF 7, 879 (1937).
4. E. F. Titov, Sov. Phys. 10, 337 (1936).
5. N. L. Bryukhatov and L. V. Kirenskii, ZhETF 8, 198 (1938).
6. K. Honda, H. Masumoto, Sc. Rep. Tôhoku Univ. 20, 323 (1931).
   119(a). S. V. Vonsovskii, ZhETF 8, 1104 (1938).
   119(b). S. V. Vonsovskii, ZhETF 10, 762 (1940).
7. L. P. Tarasov, Phys. Rev. 56, 1245 (1939).
8. L. A. Shubina, Izv. AN SSSR, ser. fizich. 11, 527 (1947).
9. O. N. Altgauzen, ZhETF 8, 1014 (1938).
10. A. S. Zeitsovskii, Qualitative Steel, No. 6, 35 (1935).
11. N. S. Akulov and I. M. Puzei, Izv. AN SSSR 11, 533 (1947).
12. Ya. I. Frenkel and Ya. G. Dorfman, Nature 126, 274 (1930).
13. M. Shirobokov, DAN 24, 426 (1939); ZhETF 15, 57 (1945), see also L. Kholodenko, ZhETF 17, 698 (1947).
14. J. H. VanVleck, Phys. Rev. 52, 1178 (1937); see also Van Peyp, Physica 5, 465 (1938).
15. S. V. Vonsovskii, UFN 26, 64 (1944).
16. Barkhausen, Phys. Zeits. 20, 401 (1919).
17. V. K. Arkad’ev, Electricity 8, 255 (1927); DAN, p. 277 (1927).
18. F. Bitter, Phys. Rev. 38, 1903 (1931).
19. N. S. Akulov and M. V. Dekhtyar, Ann. d. Phys. 15, 750 (1932); N. S. Akulov and S. Raevskii, Ann. d. Phys. 20, 113 (1934).
20. N. I. Miller and D. S. Shteinberg, Technical Physics 1, 205 (1934).
21. N. S. Akulov and I. A. Bazurina, ZhETF 8, 745 (1938).
22. I. S. Elmore, Phys. Rev. 53, 757 (1938); 62, 486 (1942).
23. Germer, Phys. Rev. (A) 62, 295 (1944).
24. O. Halpern and T. Holstein, Phys. Rev. 59, 960 (1941); F. Bloch, Hamermesh and Staub, Phys. Rev. 69, 47 (1943).
25. L. Marton, Phys. Rev. 73, 1475 (L) (1948).
26. E. I. Kondorskii, DAN 19, 397, 401 (1938).
27. E. I. Kondorskii, DAN 18, 325 (1938).
28. H. Williams, Phys. Rev. 52, 747, 1004 (1937).
29. S. V. Vonsovskii, ZhETF 17, 1094 (1947).
30. O. V. Auwers u. Neumann, Wiss. Veröff. Siemens-Werke 14, Hf. 2, 93 (1935).
31. H. Masumoto, Honda; Anniv. Vol. Sci. Rep. Tôhoku Univ., p. 389 (1936).

1. A. S. Zaimovskii, Soft Magnetic Materials, Gosenergoizdat (1941).
2. A. S. Zaimovskii, Bulletin of VEI No. 2, 1 (1941).
3. K. Honda and H. Nishina, Zeits. f. Phys. 103, 728 (1936).
4. M. V. Dekhtyar and N. Andryushin, ZhETF 10, 1402 (1940).
5. Kirkham, Phys. Rev. 52, 1162 (1937).
6. V. I. Drozhzhina and Ya. S. Shur, Izv. AN SSSR, ser. fizich. 11, 539 (1947).
7. M. Kersten, Phys. Zeits. 44, 63 (1938).
8. R. Gans, Ann. d. Phys. 27, 1 (1908); 29, 301 (1909).
9. M. Samuel, Ann. d. Phys. 86, 798 (1928).
10. Goldschmidt, Phys. Zeits. 31, 1059 (1930).
11. W. Brown, Phys. Rev. 55, 568 (1939).
12. E. I. Kondorskii, DAN 20, 117 (1938).
13. M. V. Dekhtyar, ZhETF 8, 1124 (1938); 9, 438 (1939).
14. V. I. Drozhzhina and Ya. S. Shur, ZhETF 11, 116 (1941).
15. H. Schlechtweg, Ann. d. Phys. 27, 573 (1936).
16. A. Zhigadlo and S. Sidelnikov, Uchenye zapiski MGU, issue 2 (1934).
17. G. I. Aksenov and K. V. Grigorov, Kachestvennaya stal’, No. 10, 44 (1935) and No. 2, 19 (1938).
18. E. Titov, ZhETF 5, 817 (1935); ZhTF 7, 2084 (1937).
19. D. Volkov, ZhETF 5, 952 (1935); G. Akimov and L. Pevzner, ZhTF 4, 1935 (1934).
20. L. P. Tarasov, Phys. Rev. 56, 1224 (1939).
21. R. Bozorth and H. Williams, Phys. Rev. 59, 827 (1941).
22. L. V. Kirenskii, Izv. AN SSSR 12, 327 (1948).
23. L. A. Shubina, DAN 57, 455 (1947).
24. R. Gans, Ann. d. Phys. 24, 680 (1935).
25. N. S. Akulov and L. V. Kirenskii, ZhETF 9, 1145 (1939).
26. I. Yanshin, ZhETF 10, 786 (1940).
27. R. Becker and M. Kersten, Zeits. f. Phys. 69, 660 (1930).
28. M. Grabovskii, ZhETF 9, 180 (1939); Izv. AN SSSR, ser. fizich. 11, 553 (1947).
29. G. S. Gorelik, Izv. AN SSSR, ser. fizich. 8, 172 (1944); G. S. Gorelik, K. Goronin and Zhukov, DAN 44, 257 (1944).
30. A. G. Lyubina, Dissertation (Gorky, 1942).
31. R. Becker, Wiss. Veröff. Siemens-Werke 11, 1 (1932).
32. M. Kersten, Zeits. f. Phys. 82, 723 (1933).
33. G. Thissen, Ann. d. Phys. 38, 153 (1940).
34. Förster and Sambre, Zeits. f. Metallkunde 33, 97 (1941).
35. T. Holstein and H. Primakoff, Phys. Rev. 59, 388 (1941).
36. R. Becker and H. Polley, Ann. d. Phys. 37, 534 (1940).
37. W. Brown, Phys. Rev. 58, 736 (1940); 60, 139 (1941).
38. N. S. Akulov, Zeits. f. Phys. 81, 790 (1933); A. A. Baskakov and N. Bryukhatov, ZhETF 9, 984 (1939).
39. E. I. Kondorskii, Problems of Ferromagnetism and Magnetodynamics, Izv. AN SSSR (1946).
40. Preisach, Ann. d. Phys. 3, 737 (1929).
41. Sixtus, UFN 22, 63 (1939).
42. D. S. Shteinberg, Sov. Phys. 7, 155 (1935); 2, 227 (1932).
43. F. D. Miroshnichenko, Sov. Phys. 10, 540 (1940).
44. V. Döring, UFN 22, 78 (1939).
45. N. L. Bryukhatov, ZhETF 4, 933 (1934); N. L. Bryukhatov and D. R. Fedenev, ZhETF 4, 920 (1934).
46. V. S. Mes’kin and B. Somin, Zeits. f. Phys. 98, 610 (1936).
47. Haake, Zeits. f. Phys. 113, 218 (1939).

1. E. I. Kondorskii, ZhETF 10, 420 (1940).
2. E. I. Kondorskii, ZhETF 7, 1117 (1937).
3. M. Kersten, Grundl. Theor. ferrom. Hysterese u. d. Koerzitivkraft, Leipzig (1944).
4. E. I. Kondorskii, DAN 30, 598 (1941).
5. N. P. Popov, Dissertation, Moscow State University (1935); N. P. Popov and Chernikova, Journal of Phys. 10 (1946).
6. A. S. Zaimovskii and V. V. Usov, Metals and Alloys in Electrical Engineering, Gosenergoizdat (1941).
7. A. P. Komar and D. M. Tarasov, ZhTF 10, 1745 (1940); A. P. Komar and N. V. Volkenshtein, ZhETF 11, 711 (1941).
8. B. G. Livshits, High-Coercivity Alloys, Metallurgizdat (1945).
9. V. S. Mes'kin, Ferromagnetic Alloys, ONTI (1937).
10. Ya. S. Shur, ZhTF 8, 1817 (1938).
11. S. V. Vonsovskii, ZhTF 8, 1805 (1938); 9, 1151 (1939).

PROOFREADING NOTES

To p. 16. Of course, such a division of the electrons in a crystal into two classes is highly conditional. Without it, however, it is hardly possible to calculate anything. Taking into account the interaction between \(s\)- and \(d\)-electrons leads to the fact that both these types of electrons participate both in electrical conductivity and in ferromagnetism, but to different degrees.

To p. 19. In addition, there may be a mixed case, i.e., the spontaneous moment is created by both normal and excited states. In this case the temperature dependence \(I_s(T)\) will have the form of the dotted curve (3) in Fig. 46.

To p. 59. In a recently published work by E. I. Kondorskii (DAN 63, 507 (1948)), a critique is made of the theory of inclusions (according to Kersten). Kondorskii has shown that the principal role in the delay of boundary layers between magnetic phases may be played by an internal demagnetizing field arising from magnetic “charges” on the surfaces of inclusions when the boundary is displaced from them.