On the Thirtieth Anniversary of Soviet Physics

Works of Scientists of the USSR on Ferromagnetism

E. I. Kondorskii

Few areas of physics can be named that over the past 30 years have undergone as much development as the theory of ferromagnetism. The causes of this phenomenon have been clarified, theories of magnetization curves and hysteresis have been constructed, and even effects have been calculated (magnetostriction, galvanomagnetic and thermomagnetic phenomena). In the development of the theory of ferromagnetism, the work of our scientists played a significant role. It is enough to point out that almost every article on this question contains references to work carried out in the USSR. In solving very many of the fundamental theoretical questions of ferromagnetism, the initiative belonged to our scientists. A presentation of all the existing works lies far beyond the scope of this article. The author considered it his task to show the main things that have been done in the field of studying and explaining the fundamental phenomena of ferromagnetism by our scientists.

1. Works Devoted to the Causes of Ferromagnetism

The cause of the strong magnetization of ferromagnets, as Weiss first pointed out, should be sought in the interaction between the electrons of neighboring atoms. However, the magnetic forces acting between electrons are too weak to be the cause of ferromagnetism. It was therefore necessary to postulate the existence of a stronger interaction leading to the spontaneous magnetization of individual regions of a body. Weiss succeeded in constructing a formal theory that described well the dependence of the principal properties of ferromagnets on temperature; however, he did not succeed in explaining the nature of spontaneous magnetization. In 1927 Dorfman[^1] (LFTI) carried out experiments on the deflection of β-particles in ferromagnetic bodies and showed that the forces producing spontaneous magnetization cannot be magnetic. In 1928 Frenkel[^2] (LFTI) first noted that spontaneous magnetization may be due to exchange forces arising from the Coulomb interaction between electrons. This same proposition, somewhat later and independently, was laid by Heisenberg at the foundation of the quantum theory of ferromagnetism. Subsequently, theo-

ria was developed in the works of Bloch, Slater, Bethe, Møller, Wigner, and in the work of Fowler and Kapitza[^3]. In the Heisenberg–Bloch theory, the possibility of the existence of excited and polar states was not taken into account. A quantitative development of the polar model in the many-electron treatment of a crystal was first given by Shubin and Vonsovskii[^4] (UPhAN). The authors showed that in the polar model the criterion of ferromagnetism turns out to be more stringent than in the Heisenberg and Bloch theory. In particular, the condition of positivity of the exchange integral in this model is a necessary but not sufficient condition for ferromagnetism. The authors, moreover, considering exchange between \(s\)- and \(d\)-electrons, found the possibility of a new explanation of the “fractionality” of atomic moments. In the works of Weiss, Heisenberg, and Bloch, when determining the most probable states, only the so-called “long-range order” was taken into account, which is characterized, in a quasiclassical treatment, by the total number of spins and by the number of spins oriented along or against the resultant magnetic moment of the region. The influence of how the differently oriented spins are distributed within the region, i.e. the influence of the so-called “short-range order,” was not taken into account in these works. Meanwhile it must play an essential role, since the interaction between electrons rapidly decreases with increasing distance. Assuming that only neighboring spins interact, i.e. assigning the principal role to “short-range order,” Ising calculated the magnetic properties of a chain of spins. He showed that such a chain behaves as a paramagnet obeying, at high temperature, the Curie–Weiss law with Curie point \(\Theta_p > 0\). In passing to a two-dimensional and three-dimensional lattice, the calculations become more complicated. The problem, even in a quasiclassical treatment, cannot be solved exactly. An approximate calculation for a three-dimensional lattice of spins was carried out by Stilbans[^5] (LFTI), who applied to this case the method developed by Guggenheim and Fowler. Vonsovskii[^6], applying the Peierls method, refined and deepened this theory. In Kaner’s work[^7] (LFTI) the three-dimensional case was considered from the quantum-mechanical point of view.

In the works noted it was shown that a three-dimensional lattice of spins, under conditions similar to those in the Heisenberg–Bloch theory, can be ferromagnetic. The inclusion of short-range order, as Vonsovskii showed, explains the cause of the difference in the positions of the paramagnetic and ferromagnetic Curie points and makes it possible to estimate the order of magnitude of this difference.

Recently (1940), Vonsovskii[^8] and Komar[^9] extended the Heisenberg–Bloch theory to the case of binary ferromagnetic alloys and derived the dependence of the Curie temperature on the composition of the alloy. The formula for the Curie temperature, according to Vonsovskii, has the following form:

\[ \Theta = \frac{Z}{2k}\left[A_1 + 2n_2(A_1 - A_{12}) + n_2^2(A_1 + A_2 - 2A_{12})\right], \tag{1} \]

where \(Z\) is the number of nearest neighbors, \(k\) is Boltzmann’s constant, \(n_2\) is the concentration of atoms of type \(B\); \(A_1\), \(A_2\), and \(A_{12}\) are the exchange integrals, respectively, for neighborhoods of the type \(A—A\), \(B—B\), and \(A—B\) (\(A\) and \(B\) are atoms of different components of the alloy).

It follows from the formula that \(\Theta\) depends on the square of the concentration of one of the components of the alloy, which is confirmed for a number of binary alloys. Vonsovsky showed that the Curie temperatures for ordered and disordered solid solutions must be different. This conclusion of the theory is also in good agreement with experimental data.

2. WORKS ON MAGNETIC ANISOTROPY, ON THE STUDY OF THE PROPERTIES AND THEORY OF FERROMAGNETIC CRYSTALS

In the process of spontaneous magnetization, the role of magnetic interaction forces, which are approximately \(10^3\) times smaller than the electric forces, is negligible. However, magnetic forces play an essential role in the process of orienting the magnetic moments of regions of spontaneous magnetization along the direction of an external field. The existence of magnetic interaction, according to modern views, leads to magnetic anisotropy and is the cause of magnetostriction.

The first systematic works on the study of the properties of ferromagnetic crystals were carried out by Weiss. Weiss did not have at his disposal crystals of the three principal ferromagnetic elements and carried out investigations on magnetite and pyrrhotite. The magnetization curves of iron crystals were measured after Becquerel and, later, Webster, Gerlach, Honda, Kaya, and Masumoto succeeded in obtaining sufficiently large crystals of this metal. Somewhat later, the magnetization curves of nickel crystals were also studied (Sucksmith, Potter, Bradbury, and Kaya), as well as those of cobalt (Kaya, Honda, and Masumoto).

Especially detailed and systematic investigations of the magnetic and electrical properties of crystals of iron, nickel, and cobalt were carried out by scientists of Honda’s school.

The theory of the anisotropy of the magnetic and electrical properties of ferromagnetic crystals was, to a considerable extent, developed by Akulov\(^ {10}\). In 1928 he calculated the energy of a uniformly deformed lattice of dipoles. This work provided the key to explaining the influence exerted by external and internal stresses on the course of magnetization and hysteresis curves.

The part of the energy of a deformed cubic lattice that depends on the components of the strain tensor \(\tau_{ij}\) and on the direction cosines \(s_i\) of the dipole moment, according to Akulov, has the form

\[ U=\sum_{ij}\Phi_{ij}\tau_{ij}+U_c, \tag{2} \]

where

\[ \Phi_{ii}=\frac{Np^2}{a^3}\left(C_0+C_1s_i^2\right);\qquad \Phi_{ij}=\frac{Np^2}{a^3}C_2s_is_j\,(i\ne j) \]

and \(U_c\) is the elastic energy of the lattice, \(N\) is the number of dipoles per unit volume, \(p\) is the dipole moment, \(a\) is the lattice parameter, and \(C_0, C_1, C_2\) are constants having definite numerical values. It follows from (2) that, if the coefficients \(C_k\) are not zero, the spontaneous magnetization of the crystal is always accompanied by a spontaneous deformation characterized by values \(\tau_{ij}\) for which a minimum of \(U_c\) is obtained. Any change in magnetization causes a change in this deformation. The deformation upon magnetization of ferromagnets—magnetostriction—was discovered by Joule. From formula (2) it is easy to obtain the dependence of magnetostriction on the direction of magnetization in a crystal magnetized to saturation. It has the following form:

\[ \lambda = x_0 + x_1 \sum s_i^2 r_i^2 + 2x_2 \sum_{i,j} s_i s_j r_i r_j, \tag{3} \]

where \(x_0, x_1\), and \(x_2\) are parameters having definite numerical values, and \(r_i\) are the cosines determining the direction in which the elongation is measured.

It should be noted that the classical calculation carried out by Akulov, while giving the correct order of magnitude of the parameters \(x\), does not give their true values. They must be determined with the aid of quantum physics. The form of the dependence of \(\lambda\) on \(s\) remains one and the same in the classical and quantum calculations. It may be obtained, independently of the method of calculation, from symmetry considerations.

Akulov showed that formulae similar to (3) must be valid for all even effects in ferromagnetic crystals. With the aid of this proposition, which Akulov called the law of magnetic anisotropy, it proved possible to explain the entire complex picture of the dependence of galvanomagnetic and thermomagnetic, galvano- and thermoelastic effects on the direction of the magnetization vector in crystals of the cubic system magnetized to saturation.

According to modern views, the magnetization of ferromagnets is caused by three simultaneously occurring processes: 1) displacement of the boundaries between individual magnetic phases and the growth of some phases at the expense of others, 2) rotation of the vector of spontaneous magnetization within the phases in the direction of the field, and 3) growth of the absolute value of this vector (i.e., an increase in the “true” magnetization of the phase). Assuming that the initial, steep part of the magnetization curves is due mainly to the first process, and the gently sloping part to the second, Akulov\(^{11}\) for the first time calculated the magnetization curves of crystals of the cubic system along the principal crystallographic directions in the region of the gently sloping part (Fig. 1). The starting formula for the calculation was the dependence of the free energy of the undeformed lattice on the direction of the magnetization vector. This formula has the form\(^{12}\):

\[ U_k = U_0 + 2K(s_1^2 s_2^2 + s_2^2 s_3^2 + s_3^2 s_1^2); \tag{4} \]

E. I. Kondorskii

\(K\) is a constant, usually called the constant of magnetic anisotropy. The free energy of a crystal magnetized to saturation in a magnetic field \(H\), making an angle \(\varphi\) with the direction of magnetization, is equal to:

\[ U = U_k - HI \cos \varphi . \]

With the aid of these formulas it is easy to find the direction of the magnetization vector corresponding to the minimum of \(U\), and hence the dependence between \(H\) and the longitudinal component of the magnetization, i.e., to determine the equation for the magnetization curve for a given direction of \(H\). In 1933 Akulov \({}^{13}\) generalized the previously derived formulas to the case where the crystal is subjected to the action of a uniform tensile or compressive stress, and, in addition, obtained equations for the hysteresis curves of an ideal crystal in which the process of boundary displacement is absent.

Fig. 1. Magnetization curves of single crystals of iron along various axes. Solid curves are calculated; points, crosses, and circles are observational data.

On the basis of the above-mentioned assumptions concerning the sequence of magnetization processes and using formula (3), Akulov derived formulas for the magnetostriction during the magnetization of a crystal [formula (3) gives the magnetostriction of a crystal magnetized to saturation] in the region where the magnetization changes mainly through the process of rotation (Fig. 2). In a similar way he obtained formulas for galvanomagnetic, thermomagnetic, and elastic effects \({}^{14}\) in the process of magnetization.

Finally, on the basis of formula (4), Akulov developed a theory of the magnetic properties of polycrystalline bodies in the region close to saturation, where the magnetic interaction between individual regions is so small that it may be neglected. Assuming that in strong fields the magnetization increases mainly as a result of the process of rotation, and that, owing to the smallness of the magnetic interaction between the crystallites in these fields, the process in each of them proceeds independently (i.e., the magnetic moments of the crystallites turn toward the direction of the external field already independently of one another), Aku-

Akulov\(^ {15}\) derived a formula for the magnetization curve of iron in the region close to saturation.

This formula has the following form:

\[ I=I_s\left(1-\frac{32}{105}\frac{K^2}{I_s^2}\cdot\frac{1}{H^2}\right), \tag{5} \]

where \(K\) is the anisotropy constant and \(I_s\) is the value of the magnetization at saturation. Cherlinsky’s experiments showed that in the field interval

Fig. 2. Magnetostriction of iron single crystals along various axes. Solid curves are calculated; triangles, circles, and squares are observational data.

\(700\)—\(1300\) oersteds, formula (5) is in good agreement with the experimental data. Using this formula, one can determine the magnitude of the anisotropy constant from the course of the magnetization curve of pure polycrystalline bodies. In 1946 Puzey (Moscow State University), using this method, determined the constants \(K\) of the alloys NiCu, NiSn, and NiMo at various temperatures.

In deriving formula (5), Akulov found the mean value of the magnetization, assuming that the axes of the individual crystallites are oriented at random. By means of an analogous method he had previously obtained formulas for the magnetostriction of polycrystalline bodies\(^ {16}\) in the state of saturation, as well as for the parameters characterizing galvanomagnetic and thermomagnetic effects. In deriving the formulas for the magnetostriction of a polycrystalline body, Akulov assumed that,

that its elongation is algebraically composed of the elongations of the individual crystallites. The mechanical action of the crystallites on one another was not taken into account.

Recently, Vladimirsky \(^{17}\), applying an elegant method, obtained formulas for the magnetostriction of a polycrystalline body at saturation with allowance for mechanical interaction.

In parallel with the theoretical work, experimental investigations were carried out in the magnetic laboratory of the Institute of Physics of Moscow State University on the study of the anisotropy constant of even effects in crystals and polycrystalline specimens. In 1932 Akulov and Bryukhatov \(^{18}\) developed a method for determining the anisotropy constant from the measurement of the torque acting on a disk in a magnetic field. With the aid of this method the authors studied the texture of rolled materials. Subsequently Bryukhatov, together with Kirensky, by the same method measured the dependence of the anisotropy constant of nickel on temperature and obtained data which are at present regarded as the most reliable. On the basis of the results of measurements, Bryukhatov and Kirensky \(^{19}\) derived an empirical formula for the dependence of the anisotropy constant on temperature

\[ K_t = K_0 e^{-aT}; \]

where \(K_t\) and \(K_0\) are, respectively, the values of \(K\) at temperature \(T^\circ\) and \(0^\circ\), and \(a\) is a constant. Further, in the same laboratory the influence of elastic stresses on the electrical conductivity, thermoelectromotive force, and magnetostriction of ferromagnets was studied in detail (by Khramov and L’vova \(^{20}\), Fedenev \(^{22}\), Volkov \(^{21}\), Belov \(^{23}\), D’yakov \(^{24}\)). It should be noted that the work on the investigation of thermomagnetic and thermoelastic effects carried out by Volkov presented great experimental difficulties, and the results obtained by him are a very valuable contribution to the field of our knowledge of ferromagnetism.

In 1930 Becker showed that in materials with negative magnetostriction, subjected to a sufficiently strong elastic tension, the axes of easy magnetization concentrate in the plane perpendicular to the stretching force, as a result of which the direction in which this force is applied becomes a direction of difficult magnetization. If the stretching force acts on polycrystalline bodies, then, owing to the indicated concentration of axes, an artificial anisotropy is created—the body becomes, in the magnetic sense, similar to a crystal with one axis of difficult magnetization. For a sufficiently strong tension \(F\), the magnetization curve, as Becker showed, approaches a straight line, and the magnetic susceptibility approaches a certain constant value

\[ \chi = \frac{I_s^2}{3\lambda F}. \tag{6} \]

The validity of the indicated theoretical conclusions and the correctness of formula (6) were confirmed by the experiments of Kersten, who measured the magnetic properties of stretched nickel wires.

In 1938–1939 Grabovsky\(^{25}\) (Institute of Physics, Moscow State University) carried out very detailed investigations of the magnetization curves of nickel wires at various temperatures and tensions. In full agreement with the theoretical conclusions, he showed that at low temperatures, owing to the increase in the anisotropy constant of nickel, considerably greater tensions are required in order to bring the magnetization curve close to a straight line. In this case appreciable deviations from formula (6) are observed.

The foundations of the quantum theory of the magnetic interaction leading to the anisotropy of ferromagnets were developed by Bloch and Gentile. This theory was further developed by Vonsovsky\(^{26}\). He calculated the temperature dependence of the magnetic anisotropy of cobalt crystals and obtained formulae that are in good qualitative agreement with experiment.

According to Vonsovsky, the temperature dependence of the anisotropy constant of cobalt is described by a formula of the following form:

\[ K_1=A\left[\frac{1}{\gamma}\left(1-e^{-\frac{\gamma}{kT}}\right)-\frac{1}{4\beta}\left(1-e^{-\frac{\beta}{kT}}\right)\right], \]

where \(T\) is the absolute temperature, \(k\) is Boltzmann’s constant, and \(A,\beta,\gamma\) are constants connected with the exchange and transfer integrals. For \(\gamma>4\beta\) and \(\gamma>0,\ \beta>0\), at low temperatures \(K_1>0\). At high temperatures

\[ K_1 \approx -\frac{3}{4}\frac{A}{kT}<0, \]

which is in agreement with experiment. Thus Vonsovsky was the first to succeed in explaining the change of sign of the anisotropy constant observed in cobalt when the temperature is raised, which must be noted as a major success of the theory.

For a long time it was not possible to measure with sufficient accuracy the magnetic properties of iron and nickel crystals in weak fields. The first reliable data in this direction were obtained by Williams in the USA and Kaya in Japan. Williams studied magnetic properties on closed-form specimens cut from crystals of silicon steel. He showed that anisotropy of magnetic properties also occurs in weak fields. According to Williams’s data, the initial susceptibilities in the directions of the axes \([100]\), \([110]\), and \([111]\) are related as \(1: \tfrac{1}{2} : \tfrac{1}{3}\). In his article devoted to the question of anisotropy in weak fields, Kondorsky\(^{27}\) derived a theoretical formula for the initial susceptibility \(\chi_0\) of crystals of the cubic system in various directions:

\[ \chi_0=\chi_1\left(n_1h_1^2+n_2h_2^2+n_3h_3^2\right) +\chi_2\left[n_1n_2\left(h_1^2+h_2^2\right)+n_2n_3\left(h_2^2+h_3^2\right)+n_3n_1\left(h_3^2+h_1^2\right)\right]\ldots, \tag{7} \]

where \(x_1, x_2\) are constants, \(h_1, h_2, h_3\) are the direction cosines of the field vector, \(n_1, n_2, n_3\) are the volume concentrations of the magnetic phases, whose spins are parallel or antiparallel, respectively, to the axes \([100]\), \([010]\), and \([001]\). He showed that the anisotropy of susceptibility observed by Williams can be explained by the fact that, in his specimens, the magnetic moments of most of the regions of spontaneous magnetization were oriented along directions of easy magnetization closest to the axis of the specimen. Such a preferential orientation of magnetic moments (magnetic texture) is often observed in soft specimens. In the presence of such an orientation, formula (7), for \(x_1 \gg x_2\), directly yields the ratio of susceptibilities observed by Williams. In the case where the volumes of regions oriented in different directions are the same, the initial susceptibility of crystals of the cubic system should not depend on the direction of magnetization.

A detailed investigation of the anisotropy of the initial susceptibility of iron crystals and of its dependence on tensile stresses was carried out by Dekhtiar \(^{28}\) (Institute of Physics, Moscow State University). The specimens that he measured had the form of thin strips. Dekhtiar also observed anisotropy of susceptibility. He showed that the initial susceptibility of iron crystals in the directions \([100]\) and \([110]\) increases with increasing tension. At the same time, the shape of the magnetization curves and the value of the initial susceptibility depend substantially: 1) on whether the tension is increased continuously or in jumps, 2) on the sequence in which the tension and the field are applied, and 3) on the state—stressed or free from stress—in which demagnetization was carried out. Dekhtiar showed that the regions of spontaneous magnetization in the crystals, as was to be expected, under the influence of tension are oriented predominantly along certain directions, and that this magnetic texture produced by tension is accompanied by a change in the initial susceptibility.

Measurements of the coercive force of iron crystals in different directions were first made by Ruder and Sykstus and somewhat later by Kaya. However, the data obtained by these authors proved to be very contradictory and could not be regarded as reliable. The first reliable measurements of the coercive force of crystals of the cubic system and, together with this, the first systematic investigations of the anisotropy of the coercive force in these crystals were carried out at UralFTI by Shur \(^{29}\). This author investigated crystals of silicon steel in the form of disks. In preparing the specimens, very careful precautions were taken to avoid the influence of various factors capable of distorting the results (the influence of magnetic fields during the treatment of the crystals, the influence of deviations of the specimens from the correct shape, etc.). As a result of the measurements, Shur obtained curves of the dependence of the coercive force on the direction of magnetization and showed that these curves have a definite period

and that the minimum value of the coercive force occurs when the field is parallel to the axis of easy magnetization lying closest to the plane of the disk. Shur30 further investigated the influence of a magnetic field and of elastic stresses applied during heat treatment on the coercive force of crystals, and showed that such treatment

Fig. 3. Magnitude of the coercive force on single crystals of silicon steel in the (110) plane for various directions. Calculated values; observational data are marked by circles.

substantially changes the character of the anisotropy of the coercive force. In this case the minimum of the coercive force coincides with the direction in which the domains of spontaneous magnetization are oriented. The theoretical explanation of the results obtained by Shur was given by Vonsovskii31 (see below), who derived formulas for the coercive force of single-crystal disks, in good agreement with the experimental data (Fig. 3).

Artificial magnetic anisotropy can be obtained not only under the action of stress, but also after heat treatment in a magnetic field. In the latter case a magnetic texture arises, and the axis of easy magnetization coincides with the direction of the field ...

used in such treatment. Shur and Khokhlov[^32] were the first to show that the same texture arises under so-called thermomechanical treatment in specimens that are annealed in a stretched state.

3. WORKS ON THE STUDY OF THE MAGNETIC STRUCTURE OF FERROMAGNETS

In 1931 Bitter showed that particles of magnetic powder reveal, on the polished surface of iron crystals covered with a magnetic suspension, a series of lines. Akulov and Dekhtyar[^33] in 1931, having carried out analogous experiments, showed that the lines are also revealed in the case when the crystal is demagnetized. This latter circumstance was of very great significance. It showed that in a demagnetized crystal there are sources of strong magnetic fields. These sources could be regions of spontaneous magnetization. Irrespective of whether the discovered lines were situated above the boundaries of regions or above slip lines, the experiments of Bitter, Akulov, and Dekhtyar showed that inside ferromagnetic crystals there are strongly magnetized portions, i.e., Weiss’s hypothesis of spontaneous magnetization is valid. Subsequently it was shown that, when the sign of the field is changed, the lines are displaced and that the direction of the lines is connected in a definite way with the direction of the axes of easy magnetization. From this one could already conclude that these lines are indeed obtained above the boundaries of the regions.

The theory of the magnetic structure of ferromagnets received its development after an expression had been found for the surface energy of regions of spontaneous magnetization. Having introduced the hypothesis of the existence of these regions in a demagnetized ferromagnet, Weiss did not try to seek the reason why the division into regions occurs, or to determine what their dimensions should be. Frenkel and Dorfman[^34] were the first to approach these questions in 1930. These authors pointed out that the volume of the regions should be connected with the magnitude of their surface energy. In 1932 Bloch obtained theoretical formulas for the magnitude of the surface energy and estimated the order of the width of the transition boundary layer between regions of spontaneous magnetization. Bloch attempted to determine their dimensions by means of a peculiar statistical method. Later, however, it was shown that this method is inapplicable to the determination of the dimensions of regions.

For the first time, a consistent and rigorous theory of the magnetic structure of a homogeneous ferromagnet was developed by Landau and Lifshitz[^35]. By means of an elegant method, these authors obtained a formula for the surface energy of a plane boundary and found the law according to which the direction of the magnetic moment changes in passing from one region to another (within the boundary zone). Landau and Lifshitz found

formula determining the sizes of the regions, and showed that these sizes depend on the dimensions of the ferromagnet. On the basis of these works it proved possible to give an analysis of the patterns of powder deposition observed on the polished surface of crystals. The chief significance of these works, as well as of Bloch’s work, consists, however, in the fact that, based on their results, it was possible to develop the theory of magnetization curves in the region of maximum permeability values, where the main role in magnetization is played by the process of displacement of boundaries.

The theory of the magnetic structure of pure uniaxial ferromagnets was detailed in the work of Shirokobokov (GIFTI) ^36. Shirokobokov considered the change in magnetic structure under the action of an external field and gave a fully rigorous theory of the magnetization curve of an ideal crystal with one axis of easy magnetization. In particular, he gave a rigorous derivation of the formula for the magnetization curve of a cobalt crystal obtained in 1931 by Heisenberg. On the other hand, Shirokobokov’s work showed that the purest crystals at our disposal are still very far from ideal.

A detailed theory of the magnetic structure for a triaxial crystal was given by Vonsovskii ^37. Vonsovskii considered how homogeneous elastic stresses affect the magnetic structure, and showed that under the action of stress the boundary layers between regions are displaced. The regions that are energetically less favorable decrease in volume.

4. WORKS ON THE THEORY OF THE MAGNETIZATION PROCESS IN THE REGION OF WEAK AND MEDIUM INDUCTIONS AND ON THE THEORY OF HYSTERESIS

The physical essence of the magnetization process in the region of the initial and steep part of the curve remained unknown until 1930–1931. In 1930, Sixtus and Tonks showed that reversal of magnetization in a wire homogeneous with respect to its magnetic properties occurs similarly to a phase transformation. The nucleus of a region magnetized in the direction of the field increases in volume at the expense of the surrounding, oppositely magnetized medium. In this process the boundary between the new and old phases moves along the wire with a definite velocity. The experiments of Sixtus and Tonks showed that, along with the process of rotation of the spontaneous-magnetization vector from the axis of easy magnetization toward the field direction, there occurs a process of displacement of the boundaries between regions, and this latter process plays the main role in weak fields. Considering the displacement of boundaries between regions whose moments make angles of 90° (boundaries of the 2nd type), Becker was the first to give a theoretical formula relating the initial permeability to the magnitude of the internal stresses, and explained the reasons for the dependence of the course of the magnetization curves in weak fields on the structure of the material.

The magnitude of the surface energy plays no substantial role in the process of displacement of boundaries of the second type. However, it plays an important role in the process of displacement of boundaries between regions with oppositely directed magnetic moments (boundaries of the first type). Bloch, using the formula he had obtained for the surface energy, showed that, in the presence within a ferromagnet of inhomogeneities leading to local changes in the exchange integral, displacement of boundaries of the first type can occur only when the external field reaches a definite value. At these field values irreversible displacements of boundaries of the first type must occur. The question of the nature of irreversible changes in magnetization was analyzed in detail by Kondorskii1. Kondorskii showed that in real ferromagnets with internal stresses the energy of the boundary layer must depend substantially on the magnitude of these stresses. He was the first to note that inhomogeneous internal stresses are a much more probable cause of the hindrance to boundary displacement than inhomogeneities leading to a change in the exchange integral, which are unlikely to exist in soft magnetic materials. From consideration of the conditions for the motion of boundaries in a ferromagnet with inhomogeneous stresses, the author obtained a formula for the critical field at which one region of spontaneous magnetization can absorb a neighboring one, and showed that the magnitude of this field must be proportional to the magnitude of the gradient of the internal stresses. This made it possible to explain the influence of heat treatment on the magnitude of the coercive force and the connection, long established in physical metallurgy, between the coercive force and the dispersion of the structure. With the aid of the formula obtained, it was also possible to explain the decrease in the coercive force of materials with positive magnetostriction when stretched along the direction in which the field is applied.

In order to verify his conclusions, Kondorskii2 carried out an experimental investigation of the action of stresses on the coercive force and of the influence of the initial state on the initial and reversible susceptibility. This investigation showed that the values of the reversible susceptibility in the case of strong tension did not depend on the state in which demagnetization had been carried out (under tension or free from stress). Conversely, the values of the irreversible susceptibility, characterizing irreversible increments of magnetization, depended substantially on the initial magnetic structure. If demagnetization was carried out in the free state and the specimen was then subjected to tension, an unstable magnetic structure is obtained, in which the boundaries between regions are located in places where, under another method of demagnetization, they would not have been. Therefore, after such artificial demagnetization, displacement of boundaries of the first type begins in the weakest fields, and the susceptibility increases the more, the greater the applied tension;

Obtaining high values of permeability in hard materials, with an artificially created magnetic structure, showed the correctness of the ideas underlying the theory.

Analyzing the causes of hysteresis in ferromagnets, the author^40 noted that hysteresis is a consequence of magnetic anisotropy—crystallographic or caused by stresses—and may be due to:

1) The absence, after magnetization to saturation, of nuclei of oppositely magnetized phases, as a result of which remagnetization cannot occur by displacement of boundaries and only rotation of the vector of spontaneous magnetization is possible. Hysteresis of this kind should be ascribed to an ideal ferromagnet.

2) A delay in the growth of nuclei of the new phase owing to the increase, during this growth, of the surface area and surface energy of the boundary layer separating the nucleus from the surrounding medium.

3) A delay in the displacement of boundaries between different magnetic phases (regions) owing to the presence of inhomogeneities in the substance.

The author showed that if hysteresis in a ferromagnet is caused in part by the second cause or arises from the first cause, then the area of the hysteresis loop and the magnitude of the coercive force must depend on the shape of the specimen. From this he concluded that, first, in pure crystals and uniaxial materials one should expect a dependence of the hysteresis loops on shape (since in these materials hysteresis must, to a considerable degree, be due to the second cause), and, second, that the independence of the coercive force from shape in most polycrystalline ferromagnets should be explained either by the fact that, in the main, their hysteresis arises not from the first two causes, or by the fact that, owing to the noncoincidence of the directions of easy magnetization in the individual parts of the specimen, the coercive force depends no longer on its shape as a whole, but on the shape of the individual regions. In both cases, the independence of the coercive force from the shape of the specimen is a consequence of the inhomogeneity of the material.

Developing the ideas indicated above, Kondorskii^41 gave a theory of reversible changes in magnetization occurring through boundary displacement, and derived formulas for the reversible permeability of crystals of the cubic system and of polycrystalline bodies. This theory received its further development in the work of Brown (USA).

In studying the magnetic properties of crystals, the specimens in most cases have the form of thin wires, strips, or disks. In these cases the shape of the specimen affects the position of the vector of spontaneous magnetization.

The first theory of the process of remagnetization of single-crystal disks was given by Vonsovskii^31. Vonsovskii showed that the character of the dependence of the coercive force on the direction of magnetization is substantially affected by the orientation of the normal to the disk relative to the cry-

crystallographic axes. In this case, as the author showed, two methods of calculation are possible, based on two models. In the first, approximate, model it is assumed that the magnetic moments of the regions remain parallel to the plane of the disk during the process of remagnetization and are oriented in the direction corresponding to the energy minimum. In the second, more exact, model it is assumed that the magnetic moments of the regions are distributed over all axes of easy magnetization, but in such a way that the resultant magnetization vector lies in the indicated direction. With the aid of these models, Wonsowski calculated formulas for the coercive force of single-crystal disks and gave a theoretical explanation of the curves obtained in Shur’s experiments.

Developing his theory, Wonsowski^42 gave an explanation of the influence of the magnetic field and of elastic stresses applied during thermal treatment on the magnetization curves and on the anisotropy of the coercive force of crystals. Wonsowski also analyzed in detail the question of the influence of elastic stresses on the magnitude of the reversible susceptibility of crystals and polycrystalline bodies. He showed^43 that, in isotropic polycrystalline specimens subjected to tension, the reversible susceptibility must at first increase somewhat as the latter grows, and then decrease. This conclusion of the theory is in full agreement with experimental data^39.

5. WORKS ON THE THEORY OF POLYCRYSTALLINE FERROMAGNETS

The first attempt to derive formulas for the magnetization and hysteresis curves of polycrystalline ferromagnets was made by Weiss. In addition to the natural assumption of a uniform distribution of the axes of easy magnetization of the individual crystallites in all directions, which followed from the isotropy of the body, Weiss was forced to adopt certain artificial and, as became clear later, generally incorrect assumptions. They may be formulated as follows:

1) The magnetic moments of the regions of spontaneous magnetization change their orientation independently of one another only under the action of the external field.

2) Reversal of the sign of the magnetic moments that make obtuse angles with the direction of the field occurs when the component of this field in the direction of the moment reaches a certain critical value. This value may be many times smaller than the value of the field that appreciably deflects the moment from the direction of easy magnetization. On the basis of these assumptions, Weiss derived formulas for the magnetization and hysteresis curves. The curves constructed with their aid resembled the actual ones only in their external appearance and, even with a suitable choice of the arbitrary parameter, deviated noticeably from the latter.

The chief obstacle to the development of a theory of magnetization curves in the case of polycrystalline bodies is the difficulty of taking into account the magnetic interaction between individual regions of the body. It is easy to show that in a specimen with equiaxed grains the magnetic field inside the majority of the grains (crystallites), owing to this interaction, differs very greatly from the external field. Therefore, the only consistent theory of a polycrystalline ferromagnet can be a theory in which magnetic interaction is taken into account. The outlines of such a theory were first sketched in the works of Kondorskii. The author ^44 first considered the case of a polycrystalline ferromagnet in which the magnetic interaction between individual crystallites is known to be small. This is a ferromagnet with grains strongly elongated in one direction. Using the expression derived earlier for the critical field, Kondorskii showed that, for the magnetization and hysteresis curves of such a ferromagnet, formulas similar to the Weiss formulas should be valid.

Fig. 4. Hysteresis losses of cold-drawn nickel wire. The solid curve is calculated; circles and triangles are observational data.

This conclusion the author of the present article verified experimentally on ferromagnetic wires subjected beforehand to cold drawing, which made the crystallites of the specimens strongly elongated. The measurements showed that the theoretical formulas indicated above and, in particular, the formulas for residual magnetization, coercive force, and loss curves, describe well the actual properties of these wires (Fig. 4). The formulas cease to be valid as soon as the length of the uniformly magnetized regions inside the body decreases (for example, as a result of annealing, which makes the grains from elongated ones equiaxed, or as a result of the action of elastic tension, which brings the magnetic moments out of their former directions).

Having verified the principal conclusions of the theory in this special case, the author proceeded to the general case of a polycrystalline ferromagnet with uniaxial crystallites of arbitrary shape, between which magnetic interaction exists. He showed ^45 that, in the process of magnetization, these crystallites can be divided into definite groups. For some groups the intensity of the local field remains unchanged, while for others the irreversible part of the magnetization does. This circumstance greatly facilitates the derivation of general formulas for

dependence of the average magnetization on the intensity of the external field, i.e., formulas for the magnetization and hysteresis curves.

It was shown that between the magnetization curves and the hysteresis curves for partial cycles there is a definite general correspondence, described by the following formulas, which are valid so long as

Fig. 5. Induction curves of steel (a) and permalloy (b) (calculated); circles—observational data.

the maximum value of the field intensity does not exceed the coercive force of the principal cycle

\[ I' = I_m - 2F\left(\frac{H_m - H'}{2}\right), \]

where \(I'\) is the value of the magnetization on a partial cycle at an arbitrary point,

\[ I_m \]

is the maximum value of the magnetization for the given cycle,

\(F\) is a function representing the magnetization curve in the interval \((0, I_m)\),

\(H_m\) is the maximum value of \(H\) for the given cycle, and

\(H'\) is the field intensity at an arbitrary point.

The formulas derived, as was shown\({}^{46}\), are in good quantitative agreement with the curves of real polycrystalline ferromagnets in the region of small inductions, for which the theory is valid (Fig. 5). From the equations obtained the author further derived theoretical formulas for hysteresis-free (ideal) curves, determining stable states, and determined the quantities on which the form of these curves depends. Finally, he showed the relation in which the coercive force of polycrystalline ferromagnets stands to the magnitude of the critical field at which occurs

displacement of boundaries of the first type, and on what factors the magnitude of the maximum permeability of ferromagnetic bodies depends. Recently, Poptsov and Chernikova^46, on the basis of extensive experimental material, have shown that the theoretical formulas for the hysteresis curves of polycrystalline ferromagnets are in very good agreement with experimental data (Fig. 6).

6. WORKS ON THE STUDY OF THE MAGNETIC PROPERTIES OF FERROMAGNETS IN APERIODIC AND PERIODIC FIELDS

Owing to the existence of irreversible processes, even under infinitely slow cyclic changes of the field there occurs a lag of the induction—hysteresis. However, the angle characterizing the phase shift between the induction and the field under infinitely slow changes tends to zero as the magnitude of the field is decreased. This follows directly from Rayleigh’s empirical formula for hysteresis curves. Therefore electromagnetic phenomena occurring in ferromagnets in very weak, infinitely slowly varying fields are described by Maxwell’s equations with constant coefficients \(\mu\) and \(\varepsilon\). If the change of the field does not take place infinitely slowly, then, in addition to the lag due to static hysteresis, there is a lag resulting from the finite rate of change of the magnetic field. This phenomenon, called magnetic viscosity, from the point of view of modern ideas arises because the rate of motion of the boundaries between regions of spontaneous magnetization is, for one reason or another, slowed down, and the magnetization corresponding to the new value of the field is established only after a certain time has elapsed following the establishment of the field.

Fig. 6. Partial hysteresis cycles for an iron–nickel alloy with 45% Ni. Curves II, III, and IV A are calculated; circles and points are observational data.

The behavior of ferromagnetic bodies with strongly pronounced magnetic viscosity in periodic and aperiodic fields, even very weak ones, acquires a specific character.

Works on the study of magnetization processes in alternating and aperiodic fields are to a considerable extent associated with the name of Arkad’ev^47, who, together with his collaborators, carried out research in this field for more than thirty years. To describe phenomena in ferromagnetic media, Arkad’ev proposed introducing into Maxwell’s equations an additional term \(\rho H\), characterizing the influ-

effect of magnetic viscosity on the electromotive force of induction. He showed that Maxwell’s equations, supplemented by this term, yield solutions that well describe the behavior of ferromagnets in weak alternating and aperiodic fields. Arkadiev further indicated methods by which, while preserving the linearity of the equations, they could be applied to an approximate description of phenomena occurring in stronger fields, where static hysteresis plays an essential role and where the magnetic permeability depends on the field. On the basis of his theory, Arkadiev^48 was the first to point out that in high-frequency alternating fields there must exist a region within which a decrease of permeability takes place.

In Arkadiev’s laboratory, investigations were carried out of the magnetic properties of ferromagnets in alternating fields with frequencies varying over a wide range. Arkadiev’s principal conclusion concerning the decrease of magnetic permeability in a certain frequency interval was fully confirmed.

In recent times the phenomenon of magnetic viscosity has been the subject of detailed investigations both in our country and abroad. Very careful measurements were made by Goytannikov and Veletskaya^49,50. Measurements of the frequency dependence of the coefficients $\mu$ and $\rho$ were made by Volkova^51, who showed that in the wavelength interval from 70 to 120 m, $\mu$ and $\rho$ are approximately constant. The frequency characteristics were then investigated by Mash and Enushkov. In 1937–1939 the phenomenon of magnetic viscosity was studied in detail by Telesnin^52,53, who was the first to investigate viscosity on the steep portions of the hysteresis loop.

In 1935 Landau and Lifshitz^54 were the first to analyze theoretically the process of displacement of the boundaries between regions of spontaneous magnetization in alternating-frequency fields and showed that, with increasing frequency, a decrease in permeability is possible. In 1938 Becker gave a semiquantitative theory of the drop in magnetic permeability at various frequencies. In 1940–1941 Polivanov^55 gave a general theory of the influence of magnetic structure on the drop in magnetic permeability and on skin-effect phenomena.

Owing to the skin-effect phenomenon, the average values of the induction in ferromagnetic bodies of finite dimensions are not equal to the values at the surface. For the same reason, the magnetic flux in ferromagnetic bodies placed in a constant magnetic field becomes established, after some time has elapsed, even in the absence of magnetic viscosity. In 1921–1923 Vvedensky^56 solved the problem of magnetizing a cylinder in aperiodic and periodic fields. He constructed curves by means of which it was possible to determine the value of the flux in a cylindrical specimen of any size at any moment of time. The problem of magnetizing a cylinder and a plate with allowance for magnetic viscosity was solved by Tikhonov^57.

Arkadiev^58 developed a general scheme for calculating the magnetic characteristics of ferromagnets in alternating and aperiodic fields

and constructed curves with the aid of which, from quantities determined experimentally, one can obtain the true values of the coefficients \(\mu\) and \(\rho\). Arkad’ev also derived formulas by means of which it was possible to calculate the electrical resistance of ferromagnetic conductors under alternating current. This calculation for iron wires was made by Antik[^59], who constructed theoretical curves for the dependence of the resistance of these wires on the current strength and showed that the curves coincide very closely with the experimental curves obtained by Ermolaev[^60].

On the basis of the theory it was also possible to explain the dependence of the resistance of iron wires on tension, which had been studied experimentally by Sadikov[^61].

The theory of the skin effect in ferromagnets has recently been developed in the works of Polivanov[^62], who proposed formulas convenient for practical calculations.

7. WORKS IN THE FIELD OF OBTAINING MAGNETIC MATERIALS

The past 30 years have been marked by major successes in improving the properties of magnetic materials and in discovering materials with special properties. As a result of the joint work of metallurgists and physicists, the average losses in transformer steel were significantly reduced and its magnetic permeability increased. In 1923 a new alloy—permalloy—was discovered, with an exceptionally high value of magnetic permeability. In 1927 cobalt steel was obtained, whose coercive force exceeded by a factor of three the coercive force of tungsten and chromium steels known up to that time. Finally, in 1932 alloys of iron, nickel, and aluminum were discovered, whose coercive force reached 450–500 oersteds, i.e., exceeded the coercive force of tungsten steel by a factor of 7.

Work on improving the magnetic properties of materials, and the related investigations of the influence of structure on magnetic properties, were begun in the USSR at the Institute of Steel (Leningrad) by Mes’kin[^63] and at the All-Union Electrotechnical Institute (Moscow) by Zaimovskii[^64]. Mes’kin studied in detail the influence of various treatments on the properties of steels and the influence of grain size on the magnetic permeability and coercive force of iron. According to the data of Mes’kin and Pel’ts, in full accordance with modern conceptions, as the sizes of iron crystallites increase, the coercive force of this material decreases, while the permeability increases.

The initiative in studying the magnetic properties of Soviet transformer steel belongs to Zaimovskii. Beginning in 1930, this author carried out at the All-Union Electrotechnical Institute an investigation of the influence of various treatments on the properties of this steel and elucidated ways of improving its properties. Since 1933, work has been under way at the All-Union Electrotechnical Institute on mastering the technology.

and in obtaining special grades of permalloy, as well as magnetic alloys with high coercive force. Work on obtaining and studying the magnetic properties of high-coercivity alloys was carried out simultaneously in Leningrad by Meshcheryakov and at the TsZL of the ATE plant by Livshits.

As a result of these works, at the present time the technology of special magnetic materials has been mastered and alloys are being obtained which, in quality, are not inferior to foreign specimens.

In 1938–1940, Zaimovsky \(^{65}\) carried out a detailed study of the magnetization curves and coercive force of transformer steel and iron-nickel alloys of various compositions under various heat treatments. In addition, he made a systematic study of the temperature dependence of magnetic permeability and coercive force. This author established that the highest value of permeability in iron-nickel alloys corresponds to compositions lying in the interval between the alloy with the minimum value of magnetostriction and the minimum value of the anisotropy constant. Studying the temperature dependence of the coercive force of permalloy, he discovered an increase in its magnitude with increasing temperature near the Curie point. This previously unknown, very interesting phenomenon has not yet received an explanation. Zaimovsky showed that, on the basis of modern ideas about the process of magnetization, one can give a simple qualitative explanation of the temperature dependence of magnetic permeability near the Curie point. In addition, he discovered an “anomalous” dependence of magnetic permeability on temperature, not amenable to such an explanation. In 1939–1940, Selissky \(^{66}\) (VEI) studied the permeability, coercive force, and magnetostriction of alloys of iron, silicon, and aluminum. He showed that in this case as well the maximum permeability corresponds to the composition at which the magnetostriction and the anisotropy constant have minimum values.

A systematic investigation of the phase diagram of iron-nickel-aluminum alloys, and of the influence of composition and heat treatment on their magnetic properties, was carried out by Livshits \(^{67}\). On the basis of careful measurements he definitively proved the correctness of the diagram proposed by Bradley and Taylor and corrected it quantitatively. Studying the influence of tempering on coercive force, Lifshits found new evidence for the correctness of the view that the high coercive force of these alloys is the result of dispersion hardening.

During the years of the Great Patriotic War, Soviet researchers achieved significant successes in obtaining new magnetic alloys. Zaimovsky and Livshits developed new iron-nickel-cobalt-aluminum alloys with high values of residual induction and coercive force. These quantities are of especially great importance in the magnico alloy obtained in Zaimovsky’s laboratory.

This alloy undergoes special treatment in a magnetic field. Gabrielyan at TsNIIchermet has obtained new alloys with high values of initial and maximum permeability.

8. MAGNETIC FLAW DETECTION

Magnetic methods for quality control of metals are at present widely used in industry. Of especially great technical importance is the magnetic-suspension method, used for detecting cracks and hairline cracks. In the present article it is not possible to dwell in detail on all the work carried out and the achievements of Soviet researchers in this field. Systematic scientific research on magnetic flaw detection was conducted at the Scientific Research Institute of Physics of Moscow State University (Arkadev, Akulov, Dekhtyar, Kondorsky), at the Ural Branch of the Academy of Sciences (Yanus, Mikheev, Fakidov, Grigorov, Khalileev, Shur, Vonsovsky), at VIAM (Akimov, Shraiber, Zhigadlo, Kubyshkina, Rozhdestvensky), and at TsNIITMash (Akulov, Eremin, Sitolaev). Of great importance for the development of magnetic methods of materials testing, as is now recognized, is their theoretical substantiation. The first systematic works in this direction were those of Arkadev^68^ and Yanus^69,70^. These works may be considered the beginning of the construction of the theory of magnetic flaw detection.

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2

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3

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4

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5

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6

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7

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8

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2. R. I. Yanus, ZhETF 8, 307 (1938); 15, 3 (1945).

3. R. I. Yanus, Magnetic Defectoscopy (1946).