Modern Theory of Magnetism

S. V. Vonsovskii*)

III. Magnetism of Matter—Ferromagnets (conclusion)

13. Ferromagnets in Alternating Magnetic Fields and Time Effects

The magnetic properties of ferromagnets depend essentially on time. The time dependence manifests itself both in quasistatic magnetization (constant magnetic fields) and in magnetization at a finite rate (alternating fields). This fact is of extremely great practical importance, since in technology ferromagnetic materials find wide application in alternating fields, and the stability of the magnetic properties of these materials over time is also very important. The theoretical study of the time dependence of magnetic characteristics is also of great fundamental interest, because it makes it possible to reveal more deeply the physical mechanism of the processes of technical magnetization of ferromagnetic materials.

Conventionally, time phenomena in ferromagnets may be divided into (a) phenomena of dispersion of magnetic permeability when they are magnetized in alternating fields of different frequency and (b) phenomena of magnetic aftereffect and aging, associated with the magnetic and crystalline structure itself of ferromagnets.

a) Dispersion of Magnetic Permeability

As we have seen, the existence of irreversible magnetization processes in ferromagnets causes the changes in magnetization to lag behind the changes in the magnetic field (magnetic hysteresis), even if these changes are infinitely slow (quasistatic). With increasing rate of change of the field (periodic and aperiodic), the lag between the magnetization \(I\) and the magnetic field \(H\) increases

*) Conclusion. See Usp. Fiz. Nauk 35, 556 (1948); 36, 30 (1949); 37, 1 (1949).

S. V. VONSOvSKII

the phase shift and the lag \(I\) is due not only to one “static” hysteresis, but also to the finite rate of change of \(\mathbf H\). This additional increase of the lag is usually called magnetic viscosity. In the case of very weak fields (the range of applicability of Rayleigh’s formulas), under their quasi-static, slow variation, a ferromagnetic medium can be described phenomenologically by means of ordinary electrodynamics with constant permeabilities \(\mu\) and \(\varepsilon\). However, as the rate of change of \(\mathbf H\) increases, such a description becomes too crude, and therefore the usual system of equations of the electromagnetic field requires generalizations that would also include the phenomenon of magnetic viscosity.

A generalization of the theory of the electromagnetic field for the case of viscous ferromagnetic media was given by V. K. Arkad’ev*) 203,204. Arkad’ev bases his theory on a generalized law of electromagnetic induction, which in differential form has the form (for sinusoidally varying quantities):

\[ -\operatorname{rot}\mathbf E=\rho\mathbf H+\frac{\mu}{4\pi}\frac{\partial\mathbf H}{\partial t}, \tag{13.1} \]

where \(\rho\) is the so-called magnetic conductivity, characterizing the influence of magnetic viscosity on the e.m.f. of induction in a ferromagnet. If the vectors \(\mathbf E\) and \(\mathbf H\) are expressed in complex form

\[ \mathbf E^*=\mathbf E_0 e^{i(\omega t+\delta_1)} \quad \text{and} \quad \mathbf H^*=\mathbf H_0 e^{i(\omega t+\delta_2)}, \]

then

\[ \mathbf H^*=-\frac{i}{\omega}\frac{\partial\mathbf H^*}{\partial t} =-i\frac{T}{2\pi}\frac{\partial\mathbf H^*}{\partial t}, \tag{13.2} \]

where \(\omega\) is the cyclic frequency, and \(T=\dfrac{2\pi}{\omega}\) is the period of the electromagnetic oscillations. Substitution of (13.2) into (13.1) gives

\[ -\operatorname{rot}\mathbf E^* =\frac{\mu-2i\rho T}{4\pi}\frac{\partial\mathbf H^*}{\partial t}. \tag{13.3} \]

Let us introduce the abbreviations

\[ 2\rho T=\rho', \tag{13.4} \]

\[ \mu-i\rho'=\mu'. \tag{13.5} \]

) The numerous works of V. K. Arkad’ev and his extensive school of Russian physicist-magnetologists, which have appeared over the last 35 years (since 1913), have found wide recognition in the Soviet Union and abroad. Most of the works of V. K. Arkad’ev and his school published up to 1936 were used by V. K. Arkad’ev in his well-known two-volume monograph, Electromagnetic Processes in Metals 218. Later works were printed in the special collections Problems of Electrotechnical Metal Science, published by the Department of Technical Sciences of the Academy of Sciences of the USSR (1938), Practical Problems of Electromagnetism (1939), and Problems of Ferromagnetism and Magnetodynamics*, published by the Academy of Sciences of the USSR (1946), as well as in the periodical physics literature.

The quantity \(\mu'\) is called the complex permeability. In the case where \(\mu\) and \(\rho'\) do not depend on time, (13.3) can be written in the form

\[ -\operatorname{rot} \mathbf{E}^{*}=\frac{1}{4\pi}\frac{\partial(\mu' \mathbf{H}^{*})}{\partial t} \tag{13.6} \]

and one may define the complex magnetic induction

\[ \mathbf{B}^{*}=\mu'\mathbf{H}^{*}, \tag{13.7} \]

where

\[ \mathbf{B}^{*}=\mathbf{B}_{0}e^{i(\omega t+\delta_3)}. \tag{13.8} \]

From comparison of (13.5), (13.7), and (13.8) we find expressions for the phase shift between the field and the induction and the relation between the amplitudes of the induction and the field

\[ \begin{gathered} \operatorname{tg}(\delta_3-\delta_2)=\frac{\rho'}{\mu'},\\ B_0=\sqrt{\mu^2+\rho'^2}\,H_0. \end{gathered} \tag{13.9} \]

According to Arkad'ev\(^{203}\), the quantity \(\mu\) is called the conservative (elastic) permeability; it determines the store of “reversible” magnetic energy in the ferromagnetic body

\[ \mu\frac{H^2}{8\pi}; \tag{13.10} \]

which is returned upon its demagnetization; the quantity \(\rho'\) is called the consumptive (viscous) permeability, determining the magnitude of the irreversible losses due to hysteresis:

\[ \rho'\frac{H_0^2}{4}; \tag{13.11} \]

finally, the quantity

\[ r_m=\frac{B_0}{H_0}=\sqrt{\mu^2+\rho'^2} \tag{13.12} \]

is called the total or amplitude permeability\(^*\). A further refinement of the theory was carried out by Arkad'ev\(^{203}\), Bekker\(^{205}\), Hinze, and others, who took into account the dependence of the permeabilities \(\mu,\rho'\) on the strength of the magnetic field. For example, in the case of weak changes of the field one may introduce, following Rayleigh (see § 12):

\[ \mu=\mu_a+\varepsilon H \quad \text{and} \quad \rho'=bH. \tag{13.13} \]

It should be noted that determining the dependence of \(\mu\) and \(\rho'\) on the field and frequency requires the construction of a microscopic theory that takes into account the internal structure of the ferromagnetic body.

\[ \overline{\phantom{xxxxxxxxxxxxxxxxxxxxxxxx}} \]

\(^*\) For a detailed exposition of the theory of magnetic permeability, see the monograph by V. K. Arkad'ev\(^{203}\) (vol. II, § 42).

Experience shows that the permeability of ferromagnets decreases with increasing frequency of the alternating magnetic field. In the region of frequencies of visible and infrared light \((\lambda \sim 30\ \mu\) and \(\nu \sim 10^{13}\ \mathrm{sec}^{-1})\), as Hagen and Rubens\({}^{206}\) showed, ferromagnets lose their characteristic magnetic properties and, in particular, their susceptibility proves at these frequencies to be the same as that of ordinary metals, i.e. \(\mu \sim 1\). It is also known from experiment that in the region of low frequencies (up to \(\nu \sim 10^{8}\ \mathrm{sec}^{-1}\)) the permeability of ferromagnets changes comparatively little. Thus, in the frequency interval from \(10^{8}\ \mathrm{sec}^{-1}\) to \(10^{13}\ \mathrm{sec}^{-1}\) the principal drop of permeability occurs. Arkad’ev\({}^{207,203}\) was the first to point out the inevitability of the existence of such a drop. The experiments of Arkad’ev and his collaborators (Veletskaya\({}^{208}\), Volkovoi\({}^{210}\), Goytannikov\({}^{209}\), Malov\({}^{211}\) and others; for a detailed bibliography see \({}^{203}\) and \({}^{212}\)), as well as of foreign physicists (Hermann, Goldschmidt, Kreilsheimer, Strutt and Knol, and others\({}^{213}\)) on measuring the magnetic permeability of a large number of ferromagnetic materials over a wide interval of frequencies of the external field fully confirmed Arkad’ev’s theoretical assumptions concerning the dispersion of magnetic permeability.

Fig. 62. General scheme of the magnetic spectrum of a ferromagnet (boundaries of broadening of the magnetic-dispersion bands). (After Arkad’ev.)

Figure 62 gives a general schematic picture of the magnetic spectrum of ferromagnets (after Arkad’ev\({}^{212}\)). From this scheme it is seen that, on the long-wave side, there are “steps” on the curve \(\mu(\nu)\) and corresponding diffuse maxima on the curve \(\rho'(\nu)\), corresponding to absorption bands. According to Arkad’ev, the cause of this absorption may be eddy currents and magnetic viscosity. On the short-wave side there is observed magnetic resonance, anomalous dispersion on the curve \(\mu(\nu)\), and a sharp maximum on the curve \(\rho'(\nu)\). On the basis of the general theory he developed for the electromagnetic field in viscous ferromagnetic media, Arkad’ev gave a complete phenomenological description of the magnetic spectra observed experimentally\({}^{212}\).

On the basis of this same electrodynamics of Arkad’ev, Vvedenskii\({}^{214}\), proceeding from the consideration of a single skin effect (without taking magnetic viscosity into account), gave a solution of the problem of the magnetization of a ferromagnetic cylinder in a periodic and an aperiodic magnetic field. Vvedenskii’s investigations were supplemented by Tikhonov\({}^{215}\), who took into account the influence of magnetic viscosity. Divil’kovskii and Filippov\({}^{216}\) calculated the skin effect for the case of a ferromagnetic sphere and gave a new method for determining permeability in alternating fields by measuring the heating of a ferromagnetic sphere under the action of Foucault currents (and partly hysteresis) arising when the sphere is placed in an alternating field. Rytov\({}^{217}\) developed a new method of approximate—

of the calculation of the skin effect by the method of small perturbations. Mash and Enushkov \(^{218}\) used the Divílkovskii method for measuring the permeability of iron in alternating fields with wavelength from \(\lambda=4\ \text{m}\) to \(\lambda=20\ \text{m}\), and found a monotonic increase of \(\mu\) with increasing \(\lambda\). Snoek \(^{213}\) took into account the influence of the demagnetizing action of the surface of a ferromagnetic specimen on the eddy currents in it that arise during magnetization.

Arkadev \(^{203,212}\), as early as 1918, was the first to point out that the cause of the dispersion of magnetic permeability may be “microscopic” eddy currents caused by the remagnetization of regions of spontaneous magnetization. Later \(^{219}\) he gave a qualitative development of this idea.

Landau and Lifshitz \(^{113}\) (1935) were the first to carry out a theoretical calculation of the dispersion of magnetic permeability on the basis of modern detailed conceptions of the mechanism of the processes of technical magnetization. They considered the simplest case of an ideal crystal (without hysteresis) with one axis of easiest magnetization. In this case, for the complex permeability under magnetization along the easy axis, the expression obtained was

\[ \mu'_{\parallel}=1-i\frac{4\pi\mu_0 I_s^2}{\omega\varepsilon\delta d}, \tag{13.14} \]

where \(\mu_0=\dfrac{e}{mc}\), \(\delta\) is the thickness of the boundary layer between regions of spontaneous magnetization, \(d\) is the mean linear dimension of the specimen, \(\varepsilon\) is the dimensionality coefficient of the magnetization \(\sim 10^{-3}I_s\), determining the magnitude of the weak “magnetic” forces in the crystal, and \(\omega=2\pi\nu\) is the angular frequency of the field. For this orientation of the magnetizing field there is no resonance, and we are dealing with pure damping \(\left(\mu=1\ \text{and}\ \rho'\sim\dfrac{1}{\nu}\right)\). As \(\nu\to0\), \(\rho'\to\infty\); this unbounded increase of the viscous permeability is explained by the fact that in deriving (13.14) the phenomenon of magnetic hysteresis and the action of eddy currents were not taken into account.

For a field direction perpendicular to the axis of easiest magnetization, the theory of Landau and Lifshitz gives

\[ \mu'_{\perp}=1+4\pi\frac{\omega_0^2+i\omega\gamma}{\beta(\omega_0^2-\omega^2-2i\omega\gamma)}, \tag{13.15} \]

where \(\omega_0=\mu_0\dfrac{k}{I_s}\) (\(k\) is the magnetic-anisotropy constant) is the “natural frequency” of displacements of the boundaries between ferromagnetic regions, \(\gamma=\dfrac{\mu_0 k}{I_s^2}\varepsilon\) is the damping decrement of these displacements. The order of magnitude of the natural frequency is \(\omega_0\sim1.5\cdot10^{10}\ \text{sec}^{-1}\) and \(\lambda_0\sim12.6\ \text{cm}\) (if one assumes that the magnetic-anisotropy constant \(k\sim5\cdot10^5\ \text{erg}/\text{cm}^3\), \(I_s\sim10^3\)

and \(\mu_0=\dfrac{e}{mc}\sim 5\cdot 10^7\)); near this frequency anomalous dispersion (resonance) is observed on the \(\mu'(\omega)\) curve, as was shown by Arkad’ev\(^{213}\). At low frequencies \((\omega \ll \omega_0)\), formula (13.15) in the limit gives the static permeability \(\mu_\perp = 1+\dfrac{4\pi I_s^2}{k}\) (see § 12), while at very high frequencies \((\omega_0 \ll \omega)\) \(\mu_\perp \to 1\).

Becker\(^{220}\), using Arkad’ev’s idea, calculated the dependence of permeability on frequency according to the same scheme as Landau and Lifshitz, but for a simpler model (without a detailed account of the distribution of \(I_s\) in the boundary layer). However, Becker takes into account in his calculation inhomogeneities in the material (the finiteness of \(\mu_\parallel\) at \(\omega=0\)) and irreversible displacements of boundaries. This account leads to the fact that in the expression for \(\mu'_{\parallel}\) there appears a natural frequency

\[ \omega_{0\parallel}\cong \frac{c^2}{\sigma \chi_a d^2} \tag{13.16} \]

(\(\sigma\)—specific electrical conductivity, \(d\)—linear dimensions of ferromagnetic regions, \(c\)—speed of light, \(\chi_a\)—static initial susceptibility, which in Landau and Lifshitz was taken to be equal to \(\infty\)). This frequency \(\omega_{0\parallel}\) determines the onset of a noticeable decrease of permeability with frequency. Estimate (13.16) gives for \(\omega_{0\parallel}\) the value \(\sim 2\cdot 10^9\ \mathrm{sec}^{-1}\) (\(\lambda_{0\parallel}\sim 15\ \mathrm{cm}\)), i.e. of the same order as for \(\omega_0\) in (13.15). \(\omega_0\) from (13.15) and \(\omega_{0\parallel}\) depend on the dimensions of the regions of spontaneous magnetization \((d)\), and consequently, by virtue of (12.14), may also depend on the dimensions of the specimens.

Becker also determined the second critical frequency \(\omega'_{0\parallel}\), which is related to (13.16) by the relation

\[ \omega'_{0\parallel}\sim 0.01\omega_{0\parallel}. \]

This second frequency \(\omega'_{0\parallel}\) is due to irreversible displacements and indicates the frequency at which intensive “exclusion” of irreversible displacement processes begins. Thus, one should expect that irreversible processes begin to die out at lower frequencies. This conclusion of the theory is in good agreement with experiment\(^{213}\).

Polivanov\(^{221}\) clearly showed that the observed changes in the magnitude of magnetic permeability are not always true characteristics of the substance. Macroscopic inhomogeneity of magnetization or magnetic viscosity may lead to an apparent dependence \(\mu(\omega)\). The true dependence \(\mu(\omega)\) can be caused only by the existence of regions of spontaneous magnetization. Polivanov investigated in detail a simple model of a ferromagnet with plane-parallel regions and obtained curves \(\mu(\omega)\) and \(\rho'(\omega)\), which qualitatively agree completely with experimental data. It should be noted that Polivanov also pointed out that, in the case of

in calculating \(\mu_t(\omega)\), it is necessary to take into account not only the “microscopic” eddy currents (which Becker took into account), but also ordinary macroscopic ones. Similar calculations were also carried out by Kittel\({}^{222}\), who noted that the permeability dispersion, besides eddy currents, is substantially affected by the shape of the specimens (the demagnetizing factor) and by the energy of magnetic anisotropy. In exactly the same way, the form of the ferromagnetic regions has a substantial influence on the shape of the \(\mu(\omega)\) curves (Polivanov, Kittel, Strett and Knol).

In conclusion, it should be pointed out once more that magnetic spectroscopy, discovered by Arkad’ev, has now acquired a new, timely significance in connection with the development of research on nuclear magnetism (see § 2).

b) Magnetic viscosity

As early as 1881, Ewing\({}^{223}\) discovered the phenomenon of magnetic aftereffect, or magnetic viscosity, which was then studied in detail by Rayleigh\({}^{224}\). At first sight it seemed that the cause of magnetic aftereffect was eddy currents. Indeed, the initial stage of the temporal decay of magnetization, as was shown by the careful investigations of Vvedensky\({}^{225}\), is determined mainly by eddy currents (for example, in thick wires). Telesnin\({}^{226}\) investigated the phenomenon of magnetic aftereffect on sections of the magnetization curve corresponding to maximum permeability. In doing so he developed an original measurement technique which guaranteed the possibility of observing changes in magnetization over very short intervals of time (\(\sim 10^{-6}\) sec.). Mitkevich\({}^{226}\), in a series of works, made a detailed investigation of magnetic aftereffect. She showed that aftereffect phenomena cannot be explained entirely by the retarding action of eddy currents alone. Along with this, magnetic viscosity plays an essential role, its nature being connected with the processes of technical magnetization.

Richter\({}^{228}\) discovered a very sharp dependence of magnetic aftereffect on temperature. Investigating the time properties of a soft magnetic material (carbonyl iron), he found that at a temperature of \(-12^\circ\mathrm{C}\) the decay of magnetization continues for ten minutes, while at \(+100^\circ\mathrm{C}\) the entire magnetic-aftereffect effect is completed in \(10^{-2}\) seconds.

The sharp temperature dependence of magnetic viscosity, with relaxation times the same as for mechanical aftereffect in the same material, makes it possible to suggest that these two phenomena are closely related to one another. When a ferromagnet is magnetized in the region of weak fields (where magnetic viscosity is also observed), we are dealing mainly with processes of displacement of the boundaries between regions of spontaneous magnetization. When the boundaries are displaced, owing to the phenomenon of magnetostriction, internal stresses arise—

their displacement. Their equilibrium distribution is established not immediately, but, owing to mechanical aftereffect, with a finite rate. In more plastic materials, with a longer relaxation time of the mechanical aftereffect, one should also expect a more developed phenomenon of magnetic aftereffect. In this case, the change in the equilibrium conditions of the boundaries between ferromagnetic regions may be influenced by such processes as impurity diffusion, the decomposition of solid solutions, the ordering of atoms in the crystal lattice of alloys, etc.

The phenomenon of magnetic aftereffect can lead not only to a change in the magnitude of the magnetization of a “viscous” ferromagnet with time, but also to a change in the character of the magnetization processes. This is most clearly manifested in the so-called phenomenon of decline of magnetic permeability.

Snoek^229 developed a phenomenological theory of the time decline of permeability (in which he postulates the existence of a relaxation time, without connecting it with the magnetic and mechanical parameters of the crystal), based on the assumption that this phenomenon is connected with elastic aftereffect. After demagnetization, the boundaries between ferromagnetic regions fall into certain “fresh” places of the crystal lattice of the metal. Because of the nonuniformity of magnetization in these boundary layers, appreciable gradients of magnetostrictive stresses arise. As Gorsky^230 showed, in such places of a crystal with large stress gradients there must occur appreciable diffusion of impurities in the crystal. This diffusion causes a rearrangement of the internal stresses, and one may expect that the boundary between ferromagnetic regions will, with time, “press out” for itself a deeper “potential well.” To shift the boundary out of this “well” larger fields are required, which leads to a decrease in permeability with time.

Janus and Drozhzhina^231 investigated the time decline of permeability in a technically important material—silicon iron. They found that refining heat treatment of this material did not weaken the time decline of permeability, in contrast to Snoek’s theory and experiments^230 with carbonyl iron, where an analogous treatment (annealing in hydrogen and then in vacuum) completely destroyed the time decline. In addition, the curves obtained in the experiments of Janus and Drozhzhina do not always coincide with Snoek’s theoretical curves. This indicates that the existing theory of the time decline of permeability requires further refinement.

A strong effect of the time decline of permeability is observed in magnetite. This case, as shown by the experiments of Janus, Shur, Drozhzhina, and V’yukhina^232, is in general covered by Snoek’s theory.

Zavoisky^234 discovered in ferromagnets a phenomenon analogous to magnetic resonance in paramagnetic bodies. In his experiments, ferromagnetic specimens of various materials (nickel, iron-silicon alloys, an iron-nickel-aluminum alloy) were placed in a po-

constant magnetizing field \(H\), on which a weak oscillating magnetic field was superposed in the perpendicular direction, with frequency \(\nu\), corresponding to the centimeter-wave range. By specifying the frequency \(\nu\) and varying the magnitude of the field \(H\), Zavoisky found a characteristic resonance pattern. At the same time, the author believes that only spins not participating in spontaneous magnetization take part in this effect—for example, those that form intermediate layers between ferromagnetic regions. It seems to us more probable that, in this resonance phenomenon discovered by Zavoisky, the participating spins are those that do not take part in the spontaneous magnetization because of its decrease with temperature. It would therefore be highly desirable to continue these experiments by determining the temperature dependence of the resonance curves*).

Recently Snoek\(^{323}\) discovered dispersion of permeability and absorption in magnetic ferrites (general chemical formula \(M\mathrm{Fe}_2\mathrm{O}_4\), where \(M\) is a metal in the form of a divalent ion), which he explains, according to the theory of Landau and Lifshitz\(^{113}\), as a special type of resonance in magnetization processes in the form of pure rotation. In this connection he obtained a formula for the value of the critical frequency \(\omega_0\)

\[ \omega_0 = \frac{3}{2}\,\frac{g I_s}{\chi_a}, \]

where \(\chi_a\) is the initial static susceptibility, and

\[ g = \frac{e}{mc} = 1.76\cdot 10^6 \ \mathrm{CGSE}. \]

§ 14. MAGNETIC MATERIALS

Ferromagnetic materials play an extraordinarily important role in modern technology. They constitute an essential element in the construction of many machines and devices used in industry, transport, and everyday life. If there were no magnetic materials, the present broad development of electrical engineering, radio, measuring instruments, etc., would be impossible. Several million tons of electrical steel and other magnetic materials are produced annually throughout the world. The production of magnetic materials has attained its greatest development during the last 20–30 years. Metallurgists, metal scientists, and physicists all over the world are working on the problems of producing high-quality magnetic metal. As a result of the intensive work of scientists and practicing engineers, great successes have been achieved in the creation of new high-quality magnetic materials. In the Soviet Union, the largest specialists in the development of problems and in the production of magnet—

*) Arkad’ev\(^{203, 212}\) had already pointed out the fundamental possibility of such an effect.

of materials are A. S. Zaimovskii, B. G. Livshits, V. S. Meshkin, D. I. Gabrielyan, A. L. Goldman, who head large schools intensively working on these questions and who have achieved great practical successes in creating a domestic high-quality industry producing magnetic materials. The largest specialists in magnetic materials abroad are Jensen, Bozorth, Goss (America), Brailsford (England), Snoek (Holland), Mishima (Japan), and others.

The present state of the technology of manufacturing magnetic materials, and their physical and operating properties, are set forth in a number of monographs and special review articles, to which we refer the reader^235. Here we shall give only a brief description of the physical properties of the most typical magnetic materials.

Depending on their properties and operating conditions, magnetic materials may be divided into two large groups: a) soft magnetic materials and b) hard magnetic materials.

a) Soft magnetic materials

Soft magnetic materials are used in those cases when high magnetic permeability, low coercive force, and small losses upon remagnetization (hysteresis and eddy currents) are required. At present very many different grades of soft materials are known. Typical representatives of this class are pure iron, iron–silicon alloys (transformer and dynamo iron), and iron–nickel alloys (permalloy). Iron is used in all cases when high saturation is required \((4\pi I_s \sim 21\,500\ \text{gauss})\). However, this material has a relatively low electrical resistivity \((\rho \sim 10^{-5}\ \text{ohm}/\text{cm})\) and therefore large eddy-current losses.

Iron–silicon alloys are the cheapest and most widespread material for electrical machines (motors and generators) and transformers. The admixture of silicon (up to \(4\%—5\%\)) noticeably increases the electrical resistivity (compared with pure iron by a factor of 6: \(\rho \sim 6\cdot 10^{-5}\ \text{ohm}/\text{cm}\)) and at the same time neutralizes the harmful action of other undesirable impurities (such as carbon, etc.); the saturation of these alloys decreases, compared with iron, by no more than \(10\%\) \((4\pi I_s \sim 19\,000\ \text{gauss})\).

Iron–nickel alloys possess exceptionally good magnetic properties in the region of weak fields. In the best alloys of this system the initial permeability reaches tens of thousands of units (alloy 1040, mumetal, sendust) and even hundreds of thousands (supermalloy), as against several hundred units for iron–silicon alloys. The maximum permeability of these alloys is also the greatest and reaches hundreds of thousands of units (alloy 1040, 78% permalloy, sendust), and in the record case almost a million (supermalloy \(\mu_{\max}\sim 800\,000\)).

However, these alloys have a lower value of saturation magnetization \((4\pi I_s \sim 10\,000\ \text{gauss})\) than, for example, pure iron or iron–silicon alloys. It should be noted that the materials with the highest value of saturation magnetization are iron–cobalt alloys (permendurs). Thus, for example, the saturation \((4\pi I_s)\) of an alloy with 50% Co reaches 24,500 gauss, i.e., about 15% greater than in pure iron. The reason for such an increase of \(I_s\) in these alloys remains as yet unexplained.

Fig. 63. Initial permeability of ferromagnetic alloys.
(According to Zaimovsky.)

In Fig. 63 a schematic diagram is given of the values of the initial permeability of various technical magnetic alloys[^235]. Table VIII gives data on the most important magnetic characteristics of soft materials used in practice.

Magnetic properties of soft materials depend on their chemical composition, structural state, and thermal and mechanical treatment. As an example, Fig. 64 shows the dependence of the hysteresis loss \(V_h\) and the magnitude of the coercive force \(H_c\) in iron on the amount of carbon and oxygen impurities. Fig. 65 shows the magnetization curves of iron after various thermal and mechanical treatments.

Fig. 64. Dependence of hysteresis losses \((V_h)\) and coercive force \((H_c)\) of iron on carbon \((\%\,C)\) and oxygen \((\%\,O_2)\) impurities.

Fig. 66 illustrates the dependence of the coercive force of an iron–silicon alloy on the size of the polycrystal grain. The question of the influence of grain size on magnetic properties was for a long time con—

Table VIII

Some properties of typical soft magnetic materials (according to data from the world literature)

Material name	Fe	Ni	Co	Si	Cu	Cr	Mo	Al	V	$\mu_0$, gauss/oersted	$\mu_{\max}$, gauss/oersted	$H_c$, oersted	$4\pi I_s$, gauss	$\rho 10^6$, ohm/cm	$\theta^\circ$C	Heat treatment
Iron	99.9	—	—	—	—	—	—	—	—	200	5000	1.0	21 500	10	770	900°C
Silicon iron	96	—	—	4	—	—	—	—	—	450	8000	0.6	19 700	60	630	800°C
Silicon iron	96.7	—	—	3.3	—	—	—	—	—	600	10 000	0.2	20 000	50	700	cold rolled
Hypercil	96.7	—	—	3.3	—	—	—	—	—	1500	40 000	0.1	20 000	50	700	same; 1200° in H₂
Alsifer	85	—	—	9	—	—	—	5	—	30 000	120 000	0.05	10 000	80	500	Casting
45 permalloy	45	45	—	—	—	—	—	—	—	2500	25 000	0.3	16 000	50	440	1050°C
Hypernik	50	50	—	—	—	—	—	—	—	4000	80 000	0.05	16 000	35	500	1200° in H₂
Radiometal	49	47	—	—	3	—	—	—	—	2500	25 000	0.3	15 600	55	—	1050°C
78 permalloy	21	78	—	—	—	—	—	—	—	8000	100 000	0.05	10 000	16	580	1050°C; 600°C
Mo-permalloy (VZÉI)	16	78.5	—	—	—	—	3.8	—	—	12 000	120 000	0.04	8700	60	420	1000°C in H₂
Cr-permalloy	17.7	78.5	—	—	—	3.8	—	—	—	12 000	60 000	—	8000	—	—	—
Mumetal (VZÉI)	17	76	—	—	5	—	3	—	—	15 000	120 000	0.04	7200	60	430	1100° in H₂
Alloy 1040	11	72	—	—	14	—	3	—	—	40 000	100 000	0.02	6000	56	290	1100° in H₂
Supermalloy	15	79	—	—	—	—	5	—	—	100 000	803 000	0.004	8000	60	400	1300° in H₂
Permendur	50	—	50	—	—	—	—	—	—	800	5000	2.0	24 500	7	980	800°C
V-permendur	49	—	49	—	—	—	—	—	2	800	4500	2.0	24 000	26	980	800°C
Powdered permalloy	17	81	—	—	—	—	2	—	—	125	130	—	—	10⁶	480	Pressing, 650°C
Ferrites	—	—	—	—	—	—	—	—	—	$\sim 10^3$	—	$\sim 0.1$–0.01	—	10⁴—10¹³	—	—

method of discussion in studies on magnetic materials. Final clarity on this question was introduced by the work of Meschkin and Pelts ^236, who showed that the improvement of the properties of soft materials with grain growth is a consequence of the decrease in the volume of the specimen occupied by boundary layers between grains, where all kinds of impurities and distortions of the crystal lattice of the alloy, which impair its magnetic properties, are concentrated. Annealing and recrystallization of the material have a substantial effect on its magnetic properties. The study of this question and the successful application of the results of the investigations in technology were carried out by Meschkin ^235a, Zaimovsky ^235, 6, 8, Shur ^249, and others.

Fig. 65. Comparison of the magnetization curves of iron after various treatments.

Experience proves that the properties of soft materials are strongly affected by their crystallographic and magnetic texture. A careful study (Williams ^141, Kondorsky ^149, Shur ^201, and others) of the magnetic properties of single crystals of soft materials showed that this effect is determined by the crystallographic magnetic anisotropy of the initial portion of the magnetization curves and of the coercive force, as well as by the inhomogeneity in the distribution of magnetic phases, caused by external stresses or by the shape of the specimens (the magnetic texture of stresses and of shape). These theoretical propositions form the basis of the modern technology for manufacturing sheet electrical steel. The method of obtaining textured transformer steel by a purely empirical route was developed by Goss ^237. A scientifically substantiated technology for this method was created by Soviet scientists (Goldman ^238, Yakutovich ^249, and others), which made it possible for our industry to manufacture high-

Fig. 66. Influence of grain size in silicon iron on the coercive force.

high-quality sheet material with better properties^238 than those of foreign grades.

At present, two practical methods are known for creating a stable magnetic texture in magnetic materials: 1) cooling specimens in an external magnetic field—thermomagnetic treatment, and 2) cooling specimens while unilateral elastic stresses are applied to them—thermomechanical treatment.

The first detailed experimental investigation and theoretical interpretation of thermomagnetic treatment were carried out by Bozorth and Dillinger^240. Figure 67 shows the deformation of the hysteresis loop of an iron–nickel alloy (65% Ni) after thermomagnetic treatment, and Fig. 68 shows curves of the dependence of maximum permeability on the composition of alloys of the iron–nickel system for various heat treatments.

Fig. 67. Effect of cooling in a magnetic field on the hysteresis loop of 65-permalloy.

Fig. 68. Maximum permeability of iron–nickel alloys under various heat treatments.

According to Bozorth’s theoretical interpretation^240, magnetic cooling creates in the material a strong magnetic texture (by plastic deformation of the material by magnetostrictive stresses in the directions set by the field), which is what causes the sharp increase in magnetic properties in the direction in which the field was applied during cooling.

Work on the further physical study of the mechanism of thermomagnetic treatment of soft materials belongs to Soviet scientists (Shur, Zaimovsky, Yanus, Shubina, Vonsovsky^245, see also^78, § 85); it should be noted that, from the quantitative side, the present state of the theory of magnetic cooling cannot yet be considered complete^78. This “induced” texture is all the greater, the more

the smaller is the natural magnetic anisotropy of the material (therefore, in iron, for example, this effect is negligibly small) and the higher is the Curie point (for the occurrence of plastic deformations caused by magnetostriction upon switching on the magnetic field, high temperatures are required).

Thermomechanical treatment, consisting in the slow cooling of a ferromagnet from temperatures above the Curie point under uniaxial stresses, was discovered by Shur and Khokhlov^244. Analysis of magnetization curves and magnetostriction curves in polycrystalline ferromagnets subjected to such treatment indicates the appearance of magnetic anisotropy (texture). The mechanism of this treatment apparently has the same nature as in the case of thermomagnetic treatment^78. It should be expected that this new method will have fairly broad prospects for technical applications as a means of improving the quality of magnetic materials. The reason for the high magnetic properties of iron–nickel alloys (with a nickel content of ~60–80 atomic percent), as well as of an iron–nickel–aluminum alloy, as was first pointed out by Akulov^241 and then substantiated by Kondorskii^193, consists in the fact that in these materials the constants of magnetic anisotropy \(k\) and the magnetostriction constants \(\lambda_s\) are simultaneously very small and constant. It is known from theory that the smaller \(k\) and \(\lambda_s\), the higher the permeability and the smaller the coercive force (see § 12). Therefore, for equal purity of the material, degree of perfection of the crystal lattice, and crystalline and magnetic texture, a material with smaller \(k\) and \(\lambda_s\) will possess smaller values of \(H_c\) and larger values of \(\mu_a\) and \(\mu_{\max}\).

Fig. 69. Relationship between the magnitude of magnetostriction \((\lambda_s)\) and the initial permeability \((\mu_a)\) for alloys of the iron–nickel system of various compositions.

In Fig. 69 is shown the relationship between the quantities \(\lambda_s\) and \(\mu_a\) and the composition for alloys of the iron–nickel system, from which it is seen that \((\mu_a)_{\max}\) corresponds to a composition with \(\lambda_s \sim 0\).

Zaimovskii^235 and Selisskii^241 carried out a careful investigation of the ternary iron–silicon–aluminum alloy and unambiguously showed that the composition with the best magnetic properties (the Sendust or Alsifer alloy), discovered by Masumoto^243, possesses minimal values of both the constants of magnetic anisotropy and magnetostriction.

These studies indicate clear paths for the experimental search for new soft magnetic materials by means of a systematic study of the course of magnetostriction and magnetic anisotropy in multicomponent ferromagnetic alloys. At the same time this points to the extraordinary importance of theoretical investigations on calculating magnetic anisotropy and magnetostriction in soft magnetic materials.

It is also necessary to mention magnetic materials with constant permeability in the initial range of fields (up to 2–3 oersteds), devoid of hysteresis at small amplitudes of the magnetizing field. A material of this type was discovered by Elmen\(^{246}\) in the system of iron–nickel–cobalt alloys (30% Fe, 25% Co, and 45% Ni) and was named perminvar. In Fig. 70,a the curve of Stoletov \(\mu'(H)\) is shown, and in Fig. 70 (b–d) the hysteresis loops for perminvar. From Figs. 70,b, c, d it is seen that even at \(B_{\max}\sim 5000\) gauss the loop has zero residual induction \((B_r=0)\) and coercive force \((H_c=0)\). At larger amplitudes of induction (Fig. 70,d), \(B_r\) and \(H_c\ne 0\). The absence of hysteresis over a wide range of inductions indicates that, in the magnetization of perminvar in weak fields, the dominant role is played by reversible processes of rotation, which proceed intensively already in weak fields. This may be expected with a small magnetic anisotropy, which, indeed, apparently is small in this alloy, as indicated by the large effect of thermomagnetic treatment in this material\(^{240}\).

Fig. 70. Permeability curves and hysteresis loops of perminvar.

Another material, isoperm\(^{235ж}\) (40–45% Ni, 45–50% Fe, and 5–15% Cu), possesses similar properties. It has a constant permeability of \(\sim 50\) gauss/oersted in the range of fields up to 100 oersteds.

These properties are achieved after a complicated combination of thermal and mechanical treatments (cold rolling, quenching, tempering, etc.), as a result of which copper precipitates from the solid solution along certain crystallographic planes. This produces strong internal stresses and creates magnetically single-axis, finely dispersed volumes of the ferromagnetic phase, in which only reversible rotation processes are possible.

Materials with constant permeability can also be obtained by grinding them into small particles isolated from one another, pressed in some insulating nonmagnetic medium. In this case the constancy of the permeability is a consequence of a large internal demagnetizing factor. A rough estimate\(^{247,248}\) shows that the permeability in such a material (if the pressed ferromagnet has a high permeability of the substance) is, in order of magnitude, equal to \(\sim \dfrac{3}{a}\), where \(a\) is the fraction of the volume falling to the insulating material (for example, if \(a \sim 2\%\), then \(\mu \sim 150\)). As ferromagnets for such materials one uses carbonyl iron, mechanically pulverized permalloy, magnetite (Zaimovskii, B. G. Livshits, Al’tgauzen, Sadikov, Rabkin, and others). This powder-like material with extremely low eddy-current losses is used with great success in high-frequency electrical engineering (radiotelephony, etc.).

Alongside powder-like materials, magnetic materials from the class of ferromagnetic semiconductors have recently found broad application. Typical representatives of these materials are ferrites (whence even the new term ferrimagnetics \(^{250}\)). The most detailed investigation of these materials was carried out during the war years by Snoek\(^{253}\) (Holland). These materials have very low electrical conductivity (their specific electrical resistance varies in the range from \(10^{-2}\) to \(10^{+7}\ \text{ohm}/\text{cm}\), as against \(10^{-5}\ \text{ohm}/\text{cm}\), for example, for iron). Therefore eddy-current losses are practically absent in them up to frequencies of \(\sim 1000\) kilocycles/sec. A typical ferrite is, for example, \(\mathrm{CuO \cdot Fe_2O_3}\). This material has low electrical conductivity, but at the same time also low initial permeability. To increase the latter, this material is “diluted” with the nonmagnetic ferrite \(\mathrm{ZnO \cdot Fe_2O_3}\). This greatly lowers the Curie point and shifts into the region of room temperatures the maximum on the curve \(\mu_a(T)\). The latter also leads to the fact that the initial permeability in such complex ferrimagnetics proves to be fairly high (\(\sim 2000\) gauss/oersted).

b) Magnetically hard materials

This group of materials includes ferromagnetic alloys possessing high values of coercive force and residual induction. In engineering these materials are used as perma-

permanent magnets—the sources of a constant magnetic field. Any apparatus in which permanent magnets are used contains an air gap, where the constant magnetic field is produced. Therefore the “working” magnetic characteristics of permanent magnets are determined not only by the magnetic properties of the magnet material itself, but also by its shape (the demagnetizing factor). The working portion of the magnetization curve of a permanent magnet is that part of the descending branch of the hysteresis loop which is called the demagnetization curve and lies in the upper left quadrant of the \((B-H)\) plane (see Fig. 71, a), between the residual induction \(B_r\) and the coercive force \(H_c\).

Fig. 71. “Demagnetization” curves of hard magnetic materials.
\(a\)—demagnetizing portions of hysteresis loops; \(b\)—magnetic-energy curves.

The actual residual induction (of the shape) \(B_d\) is less than \(B_r\); its magnitude is determined by the point of intersection of the demagnetization curve with the shear line [the tangent of whose angle of inclination with respect to the \(B\) axis (\(\operatorname{tg}\alpha\) in Fig. 71, a) is equal to the demagnetizing factor \(N\) of the magnet]. For a given \(N\), obviously, the best material will be the one for which not only the greatest values of \(B_r\) and \(H_c\) are obtained, but also the shape of the demagnetization loop is closest to rectangular (curve 3 in Fig. 71, a).

The quality of a material for a permanent magnet is best characterized by the magnetic-energy curve \((B \cdot H)=f(B)\), calculated for various points of the demagnetization curve (see Fig. 71, b). From these magnetic-energy curves it is evident that the best operating conditions will be achieved in the case when the shear line intersects the curve \(B(H)\) at the point which simultaneously corresponds to \((B \cdot H)_{\max}\).

Hard magnetic materials may conventionally be divided into two large groups: a) steels hardened to martensite,

and b) $\alpha$-alloys (or dispersion-hardening alloys, or ordered alloys with variable structure).

Until 1932, martensitic alloys were the principal magnetic materials for permanent magnets. Along with plain carbon steel ($H_c \sim 60$ oersted and $B_r \sim 9000$ gauss), which has a substantial drawback—appreciable magnetic aging—a large number of alloyed steels were produced. Tungsten, chromium, and molybdenum were used as alloying additions. The best alloyed steel in terms of magnetic properties is cobalt steel ($H_c = 250$ oersted, $B_r \sim 11\,500$ gauss), invented in 1917 (Honda). An especially detailed investigation of these steels was carried out by Minkevič, Stark, and Zaimovskii251, and also by Erachtin252. The disadvantage of cobalt steels is their high cost (because of the scarcity of cobalt).

A new “epoch” in the production of hard magnetic materials was opened by Mishima’s253 discovery in 1931 of a new ternary alloy of the iron–nickel–aluminum system. These alloys are incomparably cheaper than cobalt steel and at the same time possess 2–3 times greater magnetic energy than the latter. With the composition of these alloys within the limits: 11–14% Al, 23–28% Ni, the remainder iron, the coercive force reaches 400–600 oersted, and the residual induction $\sim 6000$–$7000$ gauss.

Fig. 72. Typical demagnetization curves and magnetic energy of magnetically hard materials.

A technological drawback of these alloys is their extraordinary mechanical hardness and brittleness. These alloys cannot be forged or machined by cutting. The only method of obtaining products from iron–nickel–aluminum steel is casting.

After Mishima’s first work (1931), an intensive investigation began of the properties of alloys of the iron–nickel–aluminum system, as well as of more complex alloys of this system with various additions (cobalt, copper, etc.). In Fig. 72 are given typical demagnetization curves and magnetic energy for high-coercivity alloys of this type. In Table IX are given the basic magnetic data for a number of hard magnetic materials. The final picture of the phase diagram of these alloys, outlined in the work of Bradley and Taylor, was obtained in a series of fundamental works by B. G. Liv-

Table IX

Some properties of typical hard magnetic materials according to data in the world literature

Name of material	Chemical composition (weight %)	$H_c$ (oersted)	$B_r$ (gauss)	$(B \cdot H)_{\max} \times 10^{-6}$ (gauss·oersted)	Note
0.65% carbon steel	0.65 C; 0.85 Mn; balance Fe	42	10,000	0.18
1.0% carbon steel	1.0 C; 0.50 Mn; balance Fe	51	9,000	0.20
Tungsten steel	6.0 W; 0.7 C; 0.3 Mn; balance Fe	65	10,500	0.30
1.0% chromium steel	0.9 Cr; 0.60 C; 0.45 Mn; balance Fe	52	10,000	0.23
6.0% chromium steel	6.0 Cr; 1.1 C; 0.40 Mn; balance Fe	74	9,500	0.30
40% cobalt steel	40 Co; 0.7 C; 5 W; 4.25 Cr, balance Fe	242	10,000	1.03
Remalloy	12 Co; 17 Mo (or Cr); balance Fe	250	10,500	1.40
Alni (Mishima steel)	25 Ni; 12 Al; balance Fe	500	7,000	1.40
Alnico 5 (Alcomax)	24 Co; 14 Ni; 8 Al; 3 Cu; balance Fe	650	12,700	5.50
Cunico II	35 Cu; 24 Ni; 41 Co	450	5,300	0.99
Cunife I	60 Cu; 20 Ni; 20 Fe	590	5,700	1.85
Vectolite	30 Fe$_2$O$_3$; 44 Fe$_3$O$_4$; 26 Co$_2$O$_3$	900	1,600	0.50	Specific electrical resistivity 225·10$^6$ ohm/cm
New steel KS (Alnico XII)	35 Co; 18 Ni; 6 Al; 8 Ti; balance Fe	1000	6,100	1.63
Magnico	13.5 Ni; 8 Al; 24 Co; 3 Cu; balance Fe	580	13,300	4.50	Zaimovsky
Vicalloy II	13 V; 35 Fe; 52 Co	450	10,000	3.00

Continuation of Table IX

Name of material	Chemical composition (weight %)	$H_c$ (oersteds)	$B_r$ (gauss)	$(B \cdot H)_{\max} \times 10^{-6}$ (gauss·oersted)	Note
Alnisi	34 Ni; 14 Al; 10 Si; remainder Fe	800	4,200	1.10	Zaimovsky
Silmanal	86.7 Ag; 8.8 Mn; 4.4 Al	590	6,300	0.085
Platinum alloys	77.8 Pt; 22.2 Fe 76.7 Pt; 23.3 Co	1,570 2,700	5,830 4,500	3.07 4.00

[[unclear: beginning of name]]shchits^235г. Meskhin^235а, Zaimovsky^235б, Livshits^235г and their co-workers carried out extensive investigations of the influence of alloying additions on the properties of iron–nickel–aluminum alloys and developed a scientifically substantiated technology for their production.

Komar and Tarasov^198 carried out the most careful and detailed X-ray investigation of these alloys and gave convincing direct proof that the highest magnetic properties are a consequence of the so-called variable structure arising in these alloys at the early stages of ordering processes.

Shur and Shubina^254а and Shur and Shturkin^254б made the most complete study of the magnetization curves and magnetostriction curves of a number of high-coercivity alloys (see Figs. 73A, B, V, G).

Fig. 73A. Magnetization curves and demagnetizing portions of the descending branch of the hysteresis loop of the “alnico” alloy. (According to the data of Shur and Shubina.)

Comparison of these magnetic data with the phase diagram (Livshits) and X-ray data (Komar) makes it possible to make a number of statements about the magnetic structure of high-coercivity alloys. For completeness of the picture, one should also add here the results of thermomagnetic treatment of these alloys. As Oliver and Shedden^255, and also Shur^256, first showed, after cooling the alloys alnico and alni in a constant

in a magnetic field at temperatures above the Curie point, these alloys become magnetically anisotropic.

This treatment gives the greatest effect on magnetically hard alloys of more complex composition \((\mathrm{Fe}—\mathrm{Al}—\mathrm{Ni}—\mathrm{Co}—\mathrm{Cu})\), as

Fig. 73B. Magnetization curves and demagnetizing portions of the descending branch of the hysteresis loop of the alloy vicaloy. (According to Shur and Shubina.)

Fig. 73V. Curves of longitudinal \((a)\) and transverse \((b)\) magnetostriction of the alloy “alnico” before and after thermomagnetic treatment. (According to Shur and Shturkin.)

this was shown by Kijzer257, Jones et al.258, and Zaimovskii259. The latter called an alloy subjected to such treatment “magnico.” In Fig. 74 a comparison is given of the demagnetization and magnetic-energy curves of an alloy of the “magnico” type. From the curves it is seen that in this

In this case a record value of the maximum magnetic energy is obtained, \((\mathbf{H}\cdot \mathbf{B})_{\max}\sim 5\cdot 10^6\) gauss·oersted, when the samples are magnetized along the direction of the field applied during the thermomagnetic treatment. The magnetization curve of “magnico” (Fig. 73A) and of vicalloy (Fig. 73B), as well as the curves of longitudinal and transverse magnetostriction (Fig. 73V and G), indicate a pronounced magnetic texture in these materials after the corresponding treatments. In addition, from the form of the initial magnetization curve there is every reason to suppose that the process of magnetization in these materials proceeds by rotation of the vectors of spontaneous magnetization \(^{254,78}\).

Fig. 73G. Curves of longitudinal (a) and transverse (b) magnetostriction of vicalloy alloy before and after tempering (according to Shur and Shturkin).

Along with alloys based on the iron–nickel–aluminum system, iron–nickel–copper and iron–cobalt–vanadium (vicalloy) alloys have recently become widespread (see, for example, Fig. 73B and G). These alloys have more favorable plastic properties, which make it possible to subject them to mechanical working (rolling, cutting, etc.).

Fig. 74. Influence of thermomagnetic treatment on the magnetic properties of a high-coercivity alloy (alnico).

A material possessing special magnetic properties is made by pressing and calcining, in various proportions, powdered iron oxide \((\mathrm{Fe}_3\mathrm{O}_4)\) and cobalt ferrite \(\mathrm{CoO}\cdot \mathrm{Fe}_2\mathrm{O}_4\). After thermomagnetic treatment this alloy has \(B_r \cong 4000\) gauss, \(H_c\sim 600\) oersted and \((\mathbf{B}\cdot \mathbf{H})_{\max}\sim 1.3\cdot 10^6\) gauss·oersted; at the same time this material has a very low density \((3.55\ \mathrm{g/cm^3})\) and high electrical resistivity. In Fig. 75 the hysteresis loop of this material is presented.

Some alloys made of nonferromagnetic components possess a very large coercive force (an alloy of composition $\mathrm{Ag}_5\mathrm{MnAl}^{260}$, $H_c \sim 5500$ oersteds), as do alloys of platinum with iron and cobalt (see Table IX).

15. THE INFLUENCE OF SPONTANEOUS MAGNETIZATION ON THE NONMAGNETIC PROPERTIES OF FERROMAGNETICS

The existence of spontaneous magnetization $I_s$ in ferromagnetics, as already mentioned (see § 10), is manifested not only in their peculiar magnetic behavior, but in general all the properties of ferromagnetics, owing to the presence of $I_s$, differ to one degree or another from the properties of nonferromagnetic bodies. In particular, above (§ 10) the so-called ferromagnetic anomaly of heat capacity near the temperature of the ferromagnetic transformation was already considered. It is essential to note that all these features characterizing ferromagnetism appear independently of the external magnetic state of the ferromagnetic substance (i.e. they occur both in the demagnetized and in the magnetized state). Ferromagnetic anomalies are manifested most vividly in thermal, mechanical (elastic), electrical, galvanomagnetic, and optical properties. We shall dwell briefly below on a description of some of them.

Fig. 75. Hysteresis loop of cobalt ferrite $(\mathrm{CoOFe}_2\mathrm{O}_4) + (\mathrm{Fe}_3\mathrm{O}_4)$ (after thermomagnetic treatment).

a) Thermal properties

Changes in thermal energy accompanying the processes of magnetization and remagnetization of ferromagnetics may be both reversible and irreversible.

From general thermodynamic considerations it follows that in a ferromagnetic substance, under its adiabatic magnetization, there is a reversible change in temperature (the magnetocaloric effect). This effect was discovered by Weiss$^{261}$. The change in free energy under true magnetization is equal to

\[ dF_0 = H\,dI_s, \]

therefore the entropy density corresponding to this part of the thermodynamic potential is equal to

\[ S_0 = -\left(\frac{\partial F}{\partial T}\right)_H = -H\left(\frac{\partial I_s}{\partial T}\right)_H . \tag{15.1} \]

The free energy of magnetic anisotropy, according to (12.5), gives for the entropy the expression

\[ S_{\text{anis}}=-\left(\frac{\partial F_{\text{anis}}}{\partial T}\right)_H =-\left(\frac{\partial k}{\partial T}\right)_H \sum \alpha_i^2 \alpha_k^2 . \tag{15.2} \]

On the other hand, the change in entropy is determined by the equality

\[ \Delta S=C_{p,H}\frac{\Delta T}{T}. \tag{15.3} \]

Therefore, choosing \(H, p, \alpha_i\) as the independent variables, from (15.1)—(15.3) for strong fields (where \(I\sim I_s\)) we find

\[ \Delta T=-\frac{T}{C_{p,H}}\left(\frac{\partial I}{\partial T}\right)_H\Delta H -\frac{T}{C_{p,H}}\left(\frac{\partial k}{\partial T}\right)_H \Delta \sum \alpha_i^2\alpha_k^2 = \Delta T_w+\Delta T_{\text{anis}} . \tag{15.4} \]

Weiss investigated the magnetocaloric effect at temperatures close to the Curie point \((\Delta T_w)\). This investigation for the first time made it possible to determine the temperature dependence of the spontaneous magnetization. Indeed, from (15.4) one can obtain the dependence of \(\Delta T_w\) on \(I\), namely for \(T<\Theta_f\)

\[ \Delta T_w=\frac{\mathfrak{N}}{C_{I,p}}\left(I^2-I_s^2\right), \tag{15.5} \]

where \(I\) is the measured resultant magnetization, and \(I_s\) is its spontaneous part \((I>I_s)\). In Fig. 76 are shown experimental curves \(\Delta T_w(I^2)_{T=\mathrm{const}}\) from Potter’s data \(^{262}\) for iron. Extrapolation of these curves to the \(I^2\)-axis gives the values \(I_s(T)\).

Fig. 76. Determination of the spontaneous magnetization from measurements of the magnetocaloric effect: \(\Delta T=f(I^2)\).

The quantity \(\Delta T_{\text{anis}}\) was predicted and discovered by Akulov and Kirensky \(^{263}\). These authors found a temperature change of a single-crystal disk of a ferromagnet during its rotation in a strong constant magnetic field in the region of low temperatures (where true magnetization is practically absent). Fig. 77 presents the results of measurements of \(\Delta T_{\text{anis}}\) carried out by Akulov and Kirensky \(^{263}\) on a nickel single crystal. Vonsovskii \(^{119a}\) predicted the possibility of observing changes in the heat capacity for different orientations of the saturation magnetization in

ferromagnetic single crystals, owing to the existence of a term of type (15.2) in the expression for the entropy.

A large number of works, beginning with Warburg \(^{264}\) (1881), have been devoted to the study of the irreversible heat changes \(Q_{irr}\) caused by hysteresis during remagnetization of ferromagnets. The quantity \(Q_{irr}\) is determined by the area of the hysteresis loop

\[ Q_{irr}=-\oint H\,dI . \tag{15.6} \]

The experiments made it possible to separate the observed total thermal effect into an irreversible part, connected with irreversible displacement processes, and a reversible part, connected with reversible rotation processes (Akulov and Kirenskii).

Fig. 77. Curve of the magnetocaloric effect \(\Delta T_{\text{aniz}}\) in a single crystal of nickel (magnetic field lies in the plane \((110)\). (According to Akulov and Kirenskii.)

b) Magnetostriction

It is known from experiment that magnetization curves can change their shape very sharply if a ferromagnet is subjected to the action of external stresses.

In Fig. 78, as an example, magnetization curves are given for specimens subjected to the action of various uniaxial tensile or compressive loads. In the case of permalloy (Fig. 78a), tension leads to an increase of the permeability in weak fields and to a more rapid attainment of saturation.

In nickel (Fig. 78b) tension produces the opposite effect. In iron, tension increases the permeability in weak fields and decreases it in stronger fields (Fig. 78c). Such a dependence of the form of the magnetization curves of ferromagnets on external stresses is a consequence of the phenomenon of magnetostriction, i.e. of the dependence of the shape and volume of a ferromagnet on its magnetization. This phenomenon was discovered long ago by Joule (1842) and was theoretically investigated by Maxwell and Helmholtz. The magnetostrictive effect can be obtained at once from general thermodynamic considerations.

Fig. 78a. Influence of uniaxial elastic tension \((\sigma)\) on the magnetization curve of 68-permalloy (positive magnetostriction, \(\lambda_s>0\)).

In particular, from the fundamental thermodynamic equation (4.17) it follows that

\[ \left(\frac{\partial I}{\partial p}\right)_{H,S} = -\left(\frac{\partial v}{\partial H}\right)_{p,S}, \tag{15.7} \]

i.e., that the dependence of the magnetization on pressure is connected with the dependence of the volume of the body on the field strength.

Equation (15.7) is easily transformed for the case of uniaxial stresses \(\sigma\left(=-\frac{p}{S}\right)\) and changes of the linear dimensions \(l\left(=\frac{v}{S}\right)\) of the body (linear magnetostriction, \(S\)—cross section of the specimen). Namely,

\[ \left(\frac{\partial I}{\partial \sigma}\right)_{H,S} = \left(\frac{\partial l}{\partial H}\right)_{p,S}. \tag{15.8} \]

The different influence of tension and compression on permalloy, nickel, and iron (Fig. 78) is connected with the different signs of the magnetostrictive effect in these ferromagnets.

Fig. 78b. Influence of unilateral elastic tension \((\sigma=+2\ \mathrm{kg/mm^2})\) and compression \((\sigma=-0.65\ \mathrm{kg/mm^2})\) on the magnetization curve of nickel (magnetostriction is negative, \(\lambda_s<0\)).

Fig. 78c. Influence of unilateral elastic tension \((\sigma>0)\) on the magnetization curve of iron (magnetostriction changes sign).

From Fig. 79 it is seen that in nickel the magnetostriction \(\left(\frac{\Delta l}{l_0}\right)\) is negative at all fields, in permalloy it is positive, while in iron it is positive in weak fields and negative in strong fields.

The beginning of the modern study of the phenomenon of magnetostriction was laid by the works of Akulov \(^{265,114}\) (1928), who for the first time gave the correct explanation of this phenomenon in ferromagnetic crystals.

and indicated ways for further conscious study of this important phenomenon.

According to Akulov’s theory, in a ferromagnet, when it is cooled below the Curie temperature, spontaneous deformations arise (spontaneous magnetostriction*), connected with the fact that, upon the appearance of spontaneous magnetization, i.e. a parallel orientation of the electron spins, the conditions of equilibrium between the lattice points change in the crystal\({}^{196}\), and its deformation (magnetostrictive deformation) occurs (see § 12). Such deformation takes place in every region of spontaneous magnetization. These deformations, as we have seen, are anisotropic (see § 12) and, for example, in cubic crystals are characterized in first approximation by two constants \(\lambda_{100}\) and \(\lambda_{111}\) (see 12.7). If the ferromagnetic crystal as a whole is unmagnetized, then the spontaneous magnetostriction does not manifest itself; it can be detected only in processes of technical magnetization. The magnetostriction curves (Fig. 79) are precisely the result of measuring changes in the length of a ferromagnet along the direction of the magnetizing field when the distribution of spontaneous magnetization in it changes. In the region of the process of rotation, proceeding from Akulov’s anisotropy law (12.7), one can calculate the magnetostriction curves of single crystals.

Fig. 79. Magnetostriction curves of polycrystalline iron, nickel, and 68-permalloy.

In the region of weak fields, where the principal role is played by processes of displacement of the boundaries between regions of spontaneous magnetization, the calculation of magnetostriction curves presents great difficulties. Akulov\({}^{266}\) and then Heisenberg\({}^{267}\) developed a scheme for the statistical calculation (statistics of regions of spontaneous magnetization) of changes in the concentrations of various magnetic phases, which they successfully applied to the calculation of magnetostriction curves. This method was further developed by Akulov and Kondorskii\({}^{268}\), who, along with determining the magnetostriction curves of undeformed crystals, took into account the influence of external elastic stresses on these curves**,

* This phenomenon is sometimes called (not very aptly) “thermostriction.”

** These calculations were experimentally confirmed by Dzhirenchin\({}^{276}\).

as well as an explanation of the anomalies of the elastic properties of ferromagnetic crystals) (the phenomenon of mechanostriction*, or the \(\Delta E\)-effect—an anomalous decrease of Young’s modulus in ferromagnets when they are magnetized). Vladimirskii\(^{269}\) applied Akulov’s statistical method to the calculation of magnetostriction curves of polycrystals.

Along with the investigation of magnetostriction curves, the study of the dependence of the saturation magnetostriction \(\lambda_s\) on the direction of the vector \(I_s\) in a single crystal is of great interest. In Fig. 80 a comparison is given of the theoretical curve \(\lambda_s(\varphi)\), calculated by Akulov from formula (12.7), with experimental data for a nickel single crystal [with

Fig. 80. Comparison of the theoretical curve \(\lambda_s(\varphi)\)* (dashed line) (Akulov) with experimental data.

Fig. 81. Hysteresis loop of the magnetostriction of nickel. (According to Shur’s data.)

magnetization in the plane (100)], according to Honda et al.\(^{270}\); the agreement is very good. Titov\(^{271}\) made an analogous comparison for iron single crystals.

The magnetostriction of ferromagnets exhibits the phenomenon of hysteresis. Fig. 81 shows a typical hysteresis loop for magnetostriction (nickel).

The magnetostriction constants, just like the magnetic-anisotropy constants, depend very strongly on temperature. According to the classical calculation of Akulov\(^{265}\) and Becker\(^{272}\), the magnetostriction constants should depend on temperature in the same way as \(I_s^2\). However, experiment in a number of cases (Shturkin\(^{273}\), Kaya and Takaki\(^{274}\), Dyakov\(^{275}\)) does not confirm so simple a dependence. A quantum calculation\(^{1996}\)

*) Experimentally verified by Bychkov\(^{281}\).

gives a more complicated temperature dependence of \(\lambda_s(T)\), which is qualitatively confirmed by experiment.

The magnetostriction constants of ferromagnetic alloys depend very intricately on composition. Fig. 82 presents such a dependence on composition of the constants \(\lambda_{100}\) and \(\lambda_{111}\) for alloys of the iron–nickel system.

Fig. 82. Dependence of the values of the magnetostriction constants \((\lambda_{100}, \lambda_{111})\) on the alloy composition of the iron–nickel system.

From Fig. 82 it is seen that the rule of simple additivity is not fulfilled in this case. The existing quantum theory of magnetostriction \(^{196}\) makes it possible to obtain the dependence of the constants \(\lambda_s\) on the composition and degree of order of an alloy. In the first approximation, for disordered alloys there is a quadratic dependence on the concentration of the components.

Shur and Khokhlov \(^{277,78}\) were the first to emphasize distinctly the importance of studying magnetostriction curves for obtaining and determining the magnetic texture of ferromagnetic materials. These curves are among the most sensitive indicators of the character of the distribution of the concentration of magnetic phases in a ferromagnet. Indeed, for example, if the material is completely magnetically textured, i.e. it contains only two magnetic phases (magnetized antiparallel), then the magnetostriction curve taken along the direction of magnetization of these phases coincides with the abscissa axis. Conversely, a curve taken along a direction perpendicular to the vector \(I_s\) in these two magnetic phases gives a magnetostriction curve with the maximum saturation value \(\lambda_s\).

Besides linear magnetostriction, ferromagnets also exhibit volume magnetostriction (change in the volume of a ferromagnet upon its magnetization). Just like linear magnetostriction, this effect occurs both under true and under technical magnetization of ferromagnets. In addition, as Becker \(^{278}\) showed theoretically and Kornetskii \(^{279}\) confirmed experimentally, volume magnetostriction also depends on the shape of the magnetized ferromagnetic specimen. A detailed theoretical investigation of volume magnetostriction was given by Simonenko \(^{280}\), who, using Akulov’s law of anisotropy, without any additional assumptions introduced by Becker \(^{278}\), obtained the course of volume magnetostriction for all portions of the magnetization curve.

In a number of ferromagnetic alloys, the magnetostriction of the paraprocess (true magnetostriction) has an anomalously large value even at low temperatures (far from the Curie point), as does the true magnetization \(^{282}\).

A systematic study of the magnetoelastic properties of these alloys was undertaken by Belov\(^{283}\), who for the first time discovered a change in the saturation magnetization under elastic tension in ferromagnetic substances (alloys 64% Fe—36% Ni and 44% Fe—56% Pt) (Fig. 83*).

The existence of spontaneous magnetostriction is also manifested in anomalies of the thermal expansion of ferromagnets. Ferromagnets with negative magnetostriction have somewhat smaller dimensions below the Curie point, and those with positive magnetostriction—somewhat larger dimensions than would correspond to the volumetric thermal expansion. Near the Curie point, where the spontaneous magnetization and the accompanying spontaneous deformation disappear, heating produces either a decrease in the resultant expansion (for \(\lambda>0\)) or an increase (for \(\lambda<0\)). At the Curie point there is observed a sharp minimum (\(\lambda>0\)) or maximum (\(\lambda<0\)) of the thermal coefficient of expansion \(\alpha\). In Fig. 84, as an illustration, experimental curves of the temperature

Fig. 83. Temperature dependence of the change in spontaneous magnetization under the action of uniaxial tensions. (After Belov.)

*) Belov’s result can be interpreted on the basis of simple thermodynamic relations (not given by the author). Indeed, from (15.8) it follows that for \(I=I_s\)

\[ \left(\frac{\partial I_s}{\partial \sigma}\right)_{H,T} = \left(\frac{\partial l}{\partial H}\right)_{\sigma,T} \]

(\(\sigma\)—uniaxial tension). Near the Curie point, to a first approximation, one may assume that \(I_s\) is a function of \(\dfrac{T}{\Theta}\) [cf., for example, (10.7)]; therefore

\[ \left(\frac{\partial I_s}{\partial \sigma}\right)_{H,T} = \frac{\partial I_s}{\partial \left(\dfrac{T}{\Theta}\right)} \frac{\partial \left(\dfrac{T}{\Theta}\right)}{\partial \sigma} = -\frac{T}{\Theta} \left(\frac{\partial I_s}{\partial T}\right)_{\sigma} \frac{\partial \Theta}{\partial \sigma}. \]

From this formula there immediately follow the conclusions of Belov’s work concerning the connection between the change in the saturation \(I_s\) under tension and \(\dfrac{\partial I_s}{\partial T}\), and with the sign and magnitude of the shift of the Curie point under tension (under the condition that \(I_0\), i.e. the saturation at \(0^\circ\) K, does not depend on \(\sigma\)).

dependence of this coefficient \(\alpha(T)\) for alloys of the iron–nickel system with different signs of magnetostriction.

This anomaly of the coefficient of thermal expansion \(\alpha\) finds an important technical application in obtaining materials with a prescribed temperature course of \(\alpha(T)\)—the problem of the so-called “invars.” In particular, it may happen that the “anomalous” ferromagnetic part of the coefficient \(\alpha\) in some temperature interval can exactly compensate the usual “nonmagnetic” part of the coefficient. Such an alloy was first discovered by Guillaume\(^{284}\) (1897) in the iron–nickel system (35% Ni). These alloys were studied in detail by Masumoto\(^{284}\).

Fig. 84. Experimental curves of the temperature dependence of the coefficient of thermal expansion in alloys of the iron–nickel system.

Akulov\(^{114}\), on the basis of his theory of even effects, related the magnitude of the ferromagnetic part of the coefficient of thermal expansion to the ferromagnetic part of the heat capacity. The development of this theory and the experimental proof of the ferromagnetic nature of invar-type alloys belong to Belov\(^{285}\).

As a consequence of the phenomenon of magnetostriction in ferromagnets, mechanical oscillations arise when they are periodically magnetized. On the other hand, when external oscillations are imposed on a ferromagnet, then in it, again by virtue of magnetostriction, reversible and irreversible displacements of the boundaries between regions of spontaneous magnetization begin to occur, which can change the entire character of the damping of mechanical oscillations. Magnetostrictive oscillations are beginning to find wide application in technology\(^{286}\).

c) Electrical, galvanomagnetic, thermoelectric, thermomagnetic, and optical properties of ferromagnets

In the case of ferromagnets, the enumerated properties have a specific character. First, they all have an anomalous temperature course; in most cases, at the Curie point there is a sharp maximum or minimum (jump) of the temperature coefficient of the corresponding quantity (electrical conductivity, thermoe.m.f., thermo-

conductivity, etc.). Secondly, the magnitude of these effects depends on the orientation of the spontaneous magnetization in the crystal and on the distribution of the concentrations of the various magnetic phases throughout the volume of the ferromagnet. The first type of anomaly is connected with the very existence of spontaneous magnetization and therefore is determined mainly by exchange forces. The second type of anomaly, however, is a consequence of the processes of technical magnetization, i.e., it is determined by magnetic interaction in the ferromagnetic crystal.

Experience shows that the temperature dependence of the electrical resistivity of ferromagnets \(\rho(T)\) is not like such a dependence for nonferromagnetic metals. In Fig. 85 this is illustrated by comparing the \(\rho(T)\) curves for nickel and for the element palladium, which is similar to it according to the Mendeleev table. The scale of the curves is different and is chosen so that above the Curie point, where both metals are not ferromagnetic, the curves coincide. Below the Curie point the curves diverge sharply; the curve \(\rho(T)\) for nickel lies below the curve for palladium. The negative value of the difference of the ordinates of these curves \(\Delta \rho / \rho_0\) (\(\rho_0\) is the electrical resistivity at \(0^\circ\mathrm{C}\)) for \(T \leq \Theta\) is the “anomalous” decrease of the electrical resistance of nickel in the ferromagnetic state \((T < \Theta)\), connected with spontaneous magnetization. The experiments of Gerlach \(^{287}\) et al. showed that, at least for temperatures close to the Curie point \((T \lesssim \Theta)\), there is a simple relation

Fig. 85. Comparison of the temperature dependence of the specific electrical resistivity of nickel and palladium.

\[ \frac{\Delta \rho}{\rho_0}=a I_s^2 . \tag{15.9} \]

Gerlach \(^{287}\) and Englert \(^{288}\) indicated a method for determining the magnitude and the temperature dependence of the spontaneous magnetization by measuring the quantity \(\Delta \rho\). The point is that at temperatures near the Curie point \((T \lesssim \Theta)\) and above it there occurs true magnetization, which likewise causes a decrease of the resistance proportional to \(I^2\)*). However, the type of the curves \(\Delta \rho=f(I^2)\) above and below the Curie point, as is clear from Fig. 86, is essentially different.

For \(T>\Theta\) (the paramagnetic region) the curves (isotherms) \(\Delta \rho(I^2)\) pass through the origin with the angular coefficient \(\left|\Delta \rho/\Delta I^2\right|\), poss—

*) See the note at the end of the article, p. 179.

increasing with temperature. For \(T<\Theta\) (the ferromagnetic region) the curves intersect the \(I^2\) axis at \(I^2\ne 0\). Thus, for \(T<\Theta\), the isothermal decrease of the electrical resistance due to the true magnetization turns out to be proportional to the difference between the square of the resulting magnetization \(I^2\) and a certain initial value, whose magnitude increases as the temperature is lowered. This initial value, according to Gerlach and Englert, is the spontaneous magnetization. This conclusion is fully confirmed by the coincidence of the values of \(I_s(T)\) obtained from the curves of Fig. 86 with the values of \(I_s(T)\) obtained from measurements of the magnetocaloric effect (see Fig. 76).

On the basis of general considerations\(^{289,294}\) of the quantum theory of metals, one may assert that the ferromagnetism of transition metals is due mainly to \(d\)-electrons, while the conductivity is due to \(s\)-electrons. The outer \(s\)-electrons of ferromagnetic atoms behave in the crystal, apparently, just as in other metals. Their degeneracy temperature is a quantity of order \(10^4\), whereas the Curie point in typical ferromagnets is a quantity \(\sim 10^3\). It may be assumed that all anomalies of the electrical conductivity of ferromagnets are caused either by the \(d\)-electrons alone, or by their distorting influence on the \(s\)-electrons.

Fig. 86. Determination of the spontaneous magnetization from measurements of the change in electrical resistance upon magnetization

Let us first consider the first possibility, i.e. let us not take into account the outer \(s\)-electrons. For this it is necessary to turn to the polar model (see § 11\(^{99,289}\)), which admits, alongside exchange processes, also processes of transfer of electrons from one atom to another, even if there is already one electron there. Polar states may be energetically more favorable if the corresponding width of the energy band is so large that its minimum levels lie below the minimum levels of the nonpolar band. Polar states are characterized by the number of quasiparticles, “twos”; the mean value of the number of these twos, \(\bar{s}\), also plays the role of an effective number of conduction electrons (\(s\)-electrons) in this model. The free energy of a ferromagnet in the polar theory depends on the number of twos, on the number of ferromagnons, and on temperature. The equilibrium values of the effective number of conduction electrons and of the spontaneous magnetization are found from the conditions for a minimum of the total free energy. Approximately, the free energy may be represented in the form of the sum of two terms—the term \(\Phi_1(\bar{s})\), conventionally called

free energy of the conduction electrons, which depends on the number of the latter \(\bar s\), and the term \(\Phi_2(\bar s,\bar m)\), which may be called the free energy of ferromagnetism and which depends on the spontaneous magnetization \(\bar m\), the number of conduction electrons, and the usual thermodynamic parameters (temperature, pressure), on which, of course, the first term also depends. Above the Curie point the second term is equal to zero, and the equilibrium value of the number of conduction electrons is determined from the minimum conditions for the first term

\[ \Phi'_1(\bar s_0)=0,\qquad \Phi''_1(\bar s_0)>0. \tag{15.10} \]

Below the Curie point the conditions for the minimum of the free energy are

\[ \Phi'_1(\bar s)+\Phi'_2(\bar s,\bar m)=0. \tag{15.11} \]

For temperatures close to the Curie point, one may assume that the number of conduction electrons differs little from its equilibrium value above the Curie point, and therefore the minimum condition (15.10) may be expanded in powers of the difference \(\Delta s=\bar s-\bar s_0\), and then, using the known expression \(\Phi_2\) (10.14) for temperatures close to the Curie point, it is easy to obtain that

\[ \Delta s=-\frac{ak\Theta}{\Phi''_1(\bar s_0)}\,\bar m^2, \tag{15.12} \]

where \(\alpha>0\) is a constant determined from (10.15), and \(k\) is Boltzmann’s constant. Thus, with decreasing temperature and with the growth of the spontaneous magnetization, the number of conduction electrons also increases, \(\Delta s>0\), and consequently the resistance will decrease, i.e.

\[ \Delta\rho\sim -A\bar m^2 \qquad (A>0), \tag{15.13} \]

as required by Gerlach’s empirical law (15.9).

However, against such an interpretation one may object that all this in general plays no role, since the main contribution to conductivity is made by the outer electrons.

To answer this question it is useful to recall what determines the temperature dependence of the electrical resistance in a metal. In ordinary bodies the resistance at high temperatures is proportional to the temperature for two reasons: a) the resistance is caused only by collisions with phonons; the number of such collisions at ordinary temperatures is proportional to the temperature, b) all other factors which a priori may affect the resistance, in particular the distribution of electrons over velocities, do not depend on temperature (for example, because of the high temperature of degeneracy of the gas).

If it is assumed that such a “scheme” also holds in ferromagnetic metals, then in order to explain the ferromagnetic “anomalies” it is necessary to accept that, in addition to collisions with phonons, there may also be other processes that give rise to resistance. In particular, the energy of the electrons may pass directly into the ferromag-

the kinetic energy of electronic exchange, etc. Such a representation was developed by Bethe\(^{90}\), who indicated a possible mechanism for transitions of this type. Bethe proceeds from the fact that conduction electrons are distinct from the electrons of ferromagnetism. Then the mechanism of energy transfer can be due to exchange between two types of electrons. An \(s\)-electron is scattered in the “alloy” of right and left spins of the \(d\)-electrons. From this point of view one might obtain a resistance by considering the atoms immobile (i.e., in general disregarding the existence of phonons). The resistance would then be due to exchange of \(s\)- and \(d\)-electrons.

Consideration of the problem of exchange between an \(s\)-electron and a system of \(d\)-electrons showed\(^{289,290}\) that the electrical resistance in such a model is absent: when an external electric field is applied, the current increases without bound.

It is clear, of course, that in considering this problem one cannot regard the distribution of the “lower” \(d\)-electrons as given. Indeed, the exchange processes themselves, which are of interest to us here, change not only the state of the external electron but also the states of the internal ones. Therefore the only correct treatment here will be an exact consideration of a system of \(N+1\) electrons, taking into account all possible interaction forces between them. In such a system one can distinguish three types of transitions\(^{290}\): transfer and two classes of exchange (\(s-d\) and \(d-d\)). The latter two types do not give transitions with current, whereas the first does. Thus the stationary states of the system possess a current different from zero. Such a system is qualitatively in no way different from the electron of the ordinary one-electron theory of metals. Upon application of an electric field the “kinetic” energy will increase, while the remaining energies do not change. There will be no compensating processes. Thus, in order to obtain a finite electrical resistance in this scheme, it is again necessary to introduce phonons.

Thus, to explain the anomalies of the electrical resistance of a ferromagnet, it is necessary to know how the distribution of electrons over velocities changes. Such a hypothesis finds its full experimental confirmation in the anomaly of the heat capacity of the electrons of transition metals and, in a more striking form, in ferromagnetic metals. It is only necessary to assume that the anomalies are somehow connected with a redistribution not only of \(d\)- but also of \(s\)-conduction electrons.

The effect considered above of the change in the number of conduction electrons under the assumption that they coincide with the \(d\)-electrons of ferromagnetism (the polar model) gives an example of this general proposition.

It is possible, however, to carry out an analogous calculation\(^{294}\), starting from the model already mentioned above (§ 11), in which the \(s\)- and \(d\)-electrons are treated differently, and not within a single polar scheme. In this case, for a first orientation one may assume that the \(d\)-electrons have no polar states. Then, as we have seen, the action of \((s-d)\)-exchange is equivalent to the action of a powerful quasi-magnetic molecular field (analogous to the field of Rozing-Weiss) from the \(d\)-electrons on

spins of the outer electrons. It turns out that the magnitude of this energy of \((s-d)\)-exchange depends substantially on the state (energy or quasi-momenta) of the \(s\)-electron (see 11.11). On the other hand, this energy also depends on the spontaneous magnetization. And since the magnitude of the latter—especially near the Curie point—depends substantially on temperature, a noticeable redistribution of electrons over quasi-momenta, as well as a change in the magnitude of their effective mass, must occur in the “gas” of \(s\)-electrons of a ferromagnet. As a result of simple calculations\(^{294}\) one can readily obtain that near the Curie point

\[ \Delta \rho \sim - B \left(\overline{m}_d + \overline{m}_s\right)^2, \tag{15.14} \]

where the constant \(B \sim 1\), and \(\overline{m}_d + \overline{m}_s\) is the total spontaneous magnetization of the \(d\)- and \(s\)-electrons, i.e., again Gerlach’s law (15.9). Thus, the model of the \((s-d)\)-exchange interaction, just like the polar model, leads in a nontrivial way to the correct result. Nevertheless, the problem of a more detailed study of electrical resistance is not removed from the agenda of the theory of metals.

Mott\(^{291}\) made an attempt to construct a theory of the electrical conductivity of transition metals and, in particular, of ferromagnets. In doing so he adopts a one-electron treatment for the \(s\)- and \(d\)-electrons. This is a weak point of the theory, for a one-electron treatment cannot give a consistent explanation of the phenomenon of ferromagnetism.

He explains the fact of the large magnitude of the resistance in transition metals, compared with simple ones, by the additional possibility of transitions of \(s\)-electrons into the \(d\)-band. Mott explains the “exclusion” of \(d\)-electrons from conduction by the large effective mass of these electrons. Wilson\(^{292}\) solved this problem more rigorously, though for nonferromagnetic transition metals. For the case of high temperatures he obtained an additional term in the resistance, increasing linearly with temperature. At low temperatures the resistance decreases to zero according to the exponential law \(\sim e^{-\Theta/T}\). However, Potter’s recent experiments\(^{293}\) did not confirm these theoretical conclusions.

Apparently, the discrepancy between theory and experiment is due to the inadequacy of the one-electron treatment of the whole problem.

Vonsovskii\(^{294}\) calculated the electrical conductivity of ferromagnets on the basis of the more general theory mentioned above, which takes into account the interaction between “outer” and “inner” electrons. The calculation was carried out for the case of low temperatures. Collisions between electrons and “ferromagnons” (spin waves) are considered. The role of the energy of the perturbation leading to transitions (with emission or absorption of “ferromagnons”) is played by the term in the exchange energy that takes into account the deviation of the spontaneous magnetization

from saturation. In this case there is a conservation law for the quasipulse of the electron and the ferromagnon in the acts of emission and absorption of the latter. Solving the usual kinetic equation for the given problem and assuming that a Bose distribution holds for ferromagnons, we obtain that the part of the electrical resistance specific to ferromagnets at low temperatures (the spontaneous magnetization is very close to absolute saturation) depends on the temperature according to the law \(\sim T^3\).

One can make the mechanism of interaction between electrons and ferromagnons taken into account in this calculation somewhat more concrete. The point is that the emission of one ferromagnon is equivalent to an increase of the magnetization, and absorption—to a decrease. One may imagine such an interaction between electrons, for example spin–orbit interaction, in which the spin of the whole system is not an integral of the motion and need not be conserved in each act of interaction. Thus, among the indicated transitions leading to emission or absorption of a magnon, there are also possible those in each of which the spin of the \(s\)-electron does not change.

Fig. 87. Dependence of the relative change in the specific electrical conductivity of pure polycrystalline nickel on longitudinal \((H \parallel i)\) and transverse \((H \perp i)\) magnetization.

But such transitions are probably also possible in which the total spin of the system is conserved. For example, in processes of \((s-d)\)-exchange. In this case, in each act of emission or absorption of a ferromagnon the spin of the \(s\)-electron must correspondingly “flip over.” Here there are two possibilities:
a) the exchange energy of the \(s\)-electron \(A\) is large in comparison with the mean thermal energy, \(A \gg kT\); but then transitions of \(s\)-electrons with spin reversal are inessential because of the impossibility of satisfying the energy conservation law, for at low temperatures the number of ferromagnons with energy appreciably exceeding \(\sim kT\) is negligibly small;
b) the exchange energy of the \(s\)-electron is small in comparison with \(kT\), \(A \ll kT\). In this case transitions with spin reversal differ in no way from transitions without spin reversal.

One may therefore think that the law found above, \(\sim T^3\), is a general (universal) result of the theory and does not depend on particular model assumptions.

We now turn to the consideration of galvanomagnetic phenomena in ferromagnets, namely, the effect of a change in electrical resistance in a magnetic field (the Thomson–Goldhammer phenomenon) and the Hall effect.

Experiment shows that, when a ferromagnet is magnetized in an external magnetic field, its electrical resistance changes in magnitude. Here one may be interested either in the change of resistance when the current and the field are parallel (the longitudinal effect \(\Delta\rho_{\parallel}\)), or in the change of resistance for a mutually perpendicular orientation of current and field (the transverse effect \(\Delta\rho_{\perp}\)). In Fig. 87 typical curves \(\Delta\rho_{\parallel}(H)\) and \(\Delta\rho_{\perp}(H)\) are given, according to the data of Englert \(^{288}\)). From these curves it is seen that the longitudinal effect in the region of technical magnetization has a positive sign, while the transverse effect has a negative one. In the region of strong fields, where the para-process takes place, both curves \(\Delta\rho_{\parallel}(H)\) and \(\Delta\rho_{\perp}(H)\) show practically the same decrease in electrical resistivity, associated with the growth of the true magnetization (see Fig. 87). Akulov \(^{296}\), on the basis of his universal law of magnetic anisotropy, showed that the formulae he obtained for calculating the magnetostriction of ferromagnetic crystals in the region of the rotation process can be applied to the calculation of any so-called even effects* (not depending on the sign of the field or of the magnetization)—galvanomagnetic, galvanoelastic, thermomagnetic, thermoelastic, and so on. In the region of weak fields, where magnetization is effected by displacement processes, Akulov successfully applied the static method developed by him and already mentioned. If, in the anisotropy law, one restricts oneself to terms of the expansion no higher than biquadratic (with respect to the direction cosines of the spontaneous magnetization), then, as Akulov \(^{114}\) showed, two general rules for even effects can be formulated.

1) In the region of the displacement process, the change in the magnitude of (elastic) even effects under small deformations is proportional, in the first approximation, to the first power of the external stresses, and under magnetization—to the square of the magnetization. The ratio of the constant of each elastic effect to the constant of the corresponding magnetic effect is a quantity (approximately) constant for all effects and equal to the product of the initial susceptibility of the material by the magnetostriction constant of saturation, measured along the axis of easy magnetization.

2) The sum of the results of measurements of any even effect along three arbitrary mutually perpendicular directions (in the absence of the para-process) is equal to zero. If, however, the para-process takes place, then this sum increases in proportion to the change in the square of the resulting magnetization (\(\sim \delta I^2\)).

) It should be recalled that the first thorough investigation of the quantitative aspect of this phenomenon belongs to the Russian physicist Goldhammer (carried out as early as 1887 and published in the Uchenye Zapiski of Moscow University and in Wiedem. Annal.*).

An experimental substantiation of the theory of even effects was given almost exclusively by Soviet physicists, chiefly by Akulov’s school. Shteinberg and Miroshnichenko\({}^{297}\) investigated the influence of the orientation of spontaneous magnetization on the resistance of nickel, iron, and iron–nickel alloys. At the same time they investigated the influence of tension and torsion on this effect. The same authors showed, in agreement with the theory, that displacements of 180° boundaries do not affect the change in the resistance of ferromagnets.

A detailed experimental investigation by Khramova and Lvova\({}^{298}\) of the thermomagnetic and thermoelastic effect is in agreement with Akulov’s theory. Fedenev,\({}^{299}\) studying the change in resistance in iron and nickel under the action of a magnetic field and weak elastic stresses, confirmed the above-mentioned rules of even effects. Belov\({}^{300}\) carried out an analogous verification of the theory under the simultaneous action of field and stresses on the thermoelectromotive force in ferromagnets. Akulov and Annaev\({}^{301}\) showed that the anisotropy of the thermoe.m.f. in iron crystals is described by two constants (\(\alpha_{100}\) and \(\alpha_{111}\)). Fedenev and Uskov\({}^{302}\) found that a linear relation between the change in electrical resistance and the square of the magnetization holds at all temperatures. Volkov,\({}^{303}\) studying thermoelastic and thermomagnetic effects, discovered a new possibility of investigating internal stresses by their influence on these effects. Further studies in this direction were carried out by Belov and Volkov\({}^{304}\) for the case of the galvanic-elastic effect. Fedenev and Vampilov\({}^{305}\) and Belov\({}^{306}\) investigated the influence of elastic stresses on even effects. Belov\({}^{307}\) also investigated the question of the influence of elastic stresses on the hysteresis of even effects.

Akulov, Volkov, and Belov\({}^{308}\) found that when an iron wire, through which an electric current had previously been passed, is stretched, an electric potential difference arises at its ends. The appearance of this potential difference occurs as a result of a change in the residual circular magnetization (created by the current) under the influence of tension.

Kapitza\({}^{309}\) investigated the change in resistance \(\frac{\Delta \rho}{\rho}\) in ferromagnets in very strong magnetic fields. Akulov\({}^{310}\) also gave a theory of the anisotropy of “odd” effects (Hall, Kerr, Nernst, etc.).

In the case of Heusler alloys,\({}^{311}\) highly coercive alloys, and silicon iron, as the experiments of Shur and Drozhzhina\({}^{312}\) and Bates\({}^{313}\) showed, a deviation from the second rule of even effects is observed for the Thomson effect (\(\Delta \rho_{\parallel}\) and \(\Delta \rho_{\perp}\) have the same sign in the region of technical magnetization). These deviations are not connected with the para-process, but are apparently determined\({}^{314}\) by the fact that in these materials an essential role is played by terms of higher order with respect to the direction cosines in Akulov’s anisotropy law, which were not taken into account in the derivation of the second rule of even effects.

Khalileev^295 investigated changes in the electrical resistivity of natural magnetite in a magnetic field at low temperatures (from \(80^\circ\) to \(120^\circ\mathrm{K}\)). In doing so he showed that the electrical resistivity of magnetite decreases in a magnetic field. The relative decrease in electrical resistivity \(\Delta\rho/\rho\) has a sharply pronounced maximum at \(T=111.4^\circ\mathrm{K}\), reaching, in a field of 9000 oersteds, \(8\%\) of the entire initial value. This result indicates that at a temperature of \(111.4^\circ\mathrm{K}\) magnetite undergoes some second-order phase transition, as is indicated by the sharp jump on the \(I(T)\) curves obtained by Weiss and Forrer*). Shur et al.^312 were the first to point out clearly that the study of the curves

\[ \frac{\Delta\rho}{\rho}=f(H) \]

of the Thomson–Goldhammer effect is one of the most convenient and sensitive means of determining the magnetic texture of ferromagnets.

The Hall effect in ferromagnets has an entirely special character. As Pugh^315 first discovered and Kikoin^316 finally established, the magnitude of the Hall potential difference \(E\) in ferromagnets is determined not by the magnitude of the magnetic-field intensity, but by the magnitude of the magnetization of the specimen \(\mathbf I\).

\[ E=R[\mathbf I\mathbf j], \tag{15.15} \]

where \(\mathbf j\) is the density of the electric current.

The Hall constant \(R\) in Kikoin’s formula (15.15) has an anomalously large magnitude in comparison with non-ferromagnetic metals.

Kikoin^316 also showed that, in the paramagnetic region at \(T>\Theta\), the Hall potential difference is, in the main, determined by formula (15.15). Kikoin, using data^317 on measurements of the magnetic susceptibility and Hall effect of palladium, as well as his own measurements^316 for copper–nickel and palladium–nickel alloys at temperatures above the Curie point, showed that in the general case the Hall constant (referred to the magnetic-field intensity) is equal to the sum of two terms

\[ R_p=R_0+R_I, \tag{15.16} \]

where \(R_0\) is the “classical” Hall constant, independent of temperature, and \(R_I=\dfrac{C}{T-\Theta_p}\) is that part of the Hall constant which is determined by the paramagnetic susceptibility \(\chi_p=\dfrac{C}{T-\Theta_p}\). Komar and Volkenshtein^318 investigated the Hall effect in the binary alloy \(\mathrm{Ni}_3\mathrm{Mn}\) at various degrees of ordering. In this alloy, for the Hall potential difference, Kikoin’s formula (15.15) also holds. The most interesting result of the measurements of Komar and Volkenshtein is the dependence, obtained by them, of the constant \(R\) in (15.15) on the degree of long-range order in the alloy. This

*) See the note at the end of the article, p. 179.

dependence makes it possible to conclude that \(R\) is a function of the spontaneous magnetization, namely \(R \sim I_s^2\). The same conclusion can also be obtained from Kikoin’s data \(^{316}\) for the temperature dependence \(R(T)\) in nickel, if it is compared with that for \(I_s^2(T)\).

Thus, two problems arise before the theory of the Hall effect in ferromagnetics: 1) to explain the dependence of the Hall constant on the spontaneous magnetization, and 2) to explain why, in Kikoin’s formula (15.15), the vector \(\mathbf{I}\) appears instead of \(\mathbf{H}\). The existing attempts \(^{319}\) to construct an electronic theory of the Hall effect in ferromagnetics have not yet been crowned with success. Thermoelectric phenomena were investigated by Dorfman, Janus, and Kikoin, as well as by Dorfman, Janus, Grigorov, and Chernikovsky \(^{320}\).

Among the anomalies of optical effects in ferromagnetics, one should mention the Faraday effect (rotation of the plane of polarization of light as it passes through a substance) and the magneto-optical Kerr effect (an analogous rotation upon reflection of light from the surface of a magnetic ferromagnetic).

The classical electronic theory of magneto-optical effects (Goldhammer, Voigt, Drude, Lorentz) did not make it possible to understand the mechanisms of these effects. There is only one quantum-mechanical work, by Hulme \(^{321}\), which explained the dependence of Faraday rotation in ferromagnetics on the basis of the exchange theory of ferromagnetism. However, his calculation does not take into account an essential circumstance—the absorption of light, whose inclusion may have a substantial influence on the result of the entire calculation.

16. CONCLUSION

In the present survey we have not set ourselves the task of giving an encyclopedic exposition of all the material that can be attributed to magnetism in general, which constitutes an extensive branch of physical science. We have pursued a more modest aim—to note the most important questions in the modern physical treatment of the phenomena of magnetism in atomic particles and their macroscopic aggregates.

The survey has not at all touched upon such an important problem as, for example, terrestrial magnetism and the magnetism of celestial bodies. This question attracted the attention of Russian scientists in the prerevolutionary epoch (Lebedev), and of Soviet physicists (Frenkel and others), and may constitute the subject of a wholly independent large survey.

Likewise, in the present article questions of the theory and practice of magnetic measurements, intensively developed in our country in the Soviet Union by Shramkov, Yanovsky, Arkadiev, Janus, and others, have not been covered. Questions of geophysical applications of magnetism (Kalashnikov, Kondorsky) have also remained untouched.

The methods of magnetic flaw detection and magnetic structural analysis (Aku-

…lov, Arkad'ev, Grigorov, Dekhtyar, Eremin, Zhigadlo, Kondorskii, Mikheev, Khalileev, Shur, Yanus, and many others). These questions likewise go far beyond the scope of the present review.

In conclusion, it may be stated that the modern doctrine of magnetism embraces an enormous range of problems and is an important part of modern physics. Like every genuine science, the theory of magnetism is organically connected with the urgent practical demands of all modern technology.

The concrete material set forth in this review, illustrating the development of the doctrine of magnetism, despite its incompleteness, once again clearly and indisputably convinces us that to Russian, Soviet science belongs the principal, decisive role in resolving the majority of the nodal questions of the science of magnetism. All the more discordant, therefore, must sound any unscrupulous and anti-scientific attempts to belittle or even completely ignore the obvious fact of the enormous creative role of Soviet scientists in the field of the doctrine of magnetic phenomena on the part of certain scientists of the capitalist world. (See the note on this page.)

NOTES ADDED IN PROOF

To p. 169. Such a quadratic dependence was first discovered by the Russian scientist Goldhammer (Uchen. Zap. Moskovskogo in-ta, issue 8, 1889).

To p. 177. Apparently, this phase transition reduces to the phenomenon of the so-called “electronic ordering”; for more detail see.^255

To p. 179. In addition to the work of Bozorth mentioned at the beginning of our review, there has recently appeared another review work on ferromagnetism by the Englishman Stoner (in the collection Reports on Progress in Physics, 11, 43–112 (1949), London), where the works of Soviet authors are likewise ignored.

CITED LITERATURE

(conclusion)

§ 13

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§ 14

1. Monographs on magnetic materials:
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§ 15

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