Modern Concepts of Technical Magnetization Processes

S. V. Vonsovsky, Sverdlovsk

A characteristic feature of ferromagnetic bodies, distinguishing them from all other magnetic substances, is the presence in them of spontaneous magnetization, independent of an external magnetic field. Frenkel¹ and Heisenberg², on the basis of quantum-mechanical analysis, showed that the cause of spontaneous magnetization is the specifically quantum part of the electrostatic interaction between electrons in a crystal, the so-called exchange interaction, which has no analogue in classical theory. An essential feature of this interaction is that its energy depends on the magnitude of the resultant magnetization of the crystal. The minimum value of the exchange energy corresponds to complete spontaneous magnetization in ferromagnets and to the absence of the latter in non-ferromagnets. The magnitude of the spontaneous magnetization depends on temperature, since thermal motion disrupts the parallel orientation of the electron spins (the elementary magnetic moments of ferromagnets). At a certain temperature \(\theta\) (the Curie point) the substance loses its ferromagnetic properties. This temperature is determined from the condition \(A \sim k\theta\), where \(A\) (the exchange integral) is the exchange energy per elementary magnet (spin), and \(k\theta\) corresponds to its mean thermal energy at temperature \(\theta\).

It would therefore be natural to expect that every ferromagnetic body should always be spontaneously magnetized to saturation, corresponding to the given temperature. However, as is well known from experiment, for a ferromagnet the most natural state is the unmagnetized state (we exclude permanent magnets, whose residual magnetization is always less than the saturation \(I_s\) and is determined by secondary causes). By placing such an unmagnetized ferromagnet in an external field, we attain saturation in very weak fields—the weaker, the closer the given specimen is to an ideal homogeneous crystal. From this fact alone one may conclude that the absence of resultant magnetization in the whole ferromagnetic specimen without an external field is not proof that it has no spontaneous magnetization. In addition, there exist a number of firmly established experimental facts, such as the magnetocaloric effect, the anomaly of heat capacity

near the Curie point, the very existence of the Curie point, etc., which are predicted by the theory and whose only cause may be spontaneous magnetization.

The apparent contradiction is explained as follows. From the standpoint of exchange interaction it is energetically advantageous to magnetize the whole body to saturation along some arbitrary direction (without an external field), but this is disadvantageous from the standpoint of the other energies of the ferromagnet and, first of all, the energy of the demagnetizing field of the surface. The “struggle” between the exchange and “demagnetizing” forces leads to the ferromagnetic specimen being divided into so-called regions of spontaneous magnetization, and in such a way that the resultant magnetization and the demagnetizing field caused by it are equal to zero throughout the whole specimen. The idea of the existence of these regions was first expressed by Weiss³ in the form of a hypothesis, without any theoretical justification. Later Heisenberg⁴, Bloch⁵, Frenkel and Dorfman⁶, Landau and Lifshitz⁷, and others developed a quantitative theory of the regions of spontaneous magnetization.

In constructing this theory, in addition to the two types of energy already mentioned, one must also take into account the magnetic and magnetoelastic interaction of the electronic spins of the ferromagnet. The presence of these energies is detected experimentally in the phenomena of magnetic anisotropy and magnetostriction. Every ferromagnetic single crystal possesses directions of easy magnetization, along which saturation under magnetization is reached in considerably smaller fields than for all the other “hard” directions (in iron, for example, the tetragonal axes of the crystal are the easy axes). In addition to this natural crystallographic magnetic anisotropy, due purely to the magnetic interaction of spins, we can create an artificial magnetic anisotropy by means of external stresses. Thus, for example, by stretching or compressing a polycrystalline specimen, it can be made sharply magnetically anisotropic, although in its normal state, because of the chaotic distribution of crystallites in it, it did not possess anisotropy. Along the direction of tension or compression we obtain the axis of easiest magnetization. In exactly the same way, the internal stresses in each given small region of a real technical specimen, depending on its orientation relative to the axes of easy magnetization, make one of them the easiest. Along this axis will be directed the magnetization vector of the spontaneous region in the given volume of the specimen. Thus, in a technical material, the following factors chiefly influence the distribution, shape, and dimensions of the spontaneous regions:

a) the crystallographic properties of the specimen,
b) the distribution of internal and external stresses,
c) the shape of the specimen (the demagnetizing field of the surface).

The component of the resultant magnetization \(I_{\parallel}\) along the field in the general case may be written as follows⁸:

\[ I_{\parallel}=\sum_i V_i \cos \vartheta_i, \]

where \(V_i\) is the relative volume \(\left(\sum_i V_i=1\right)\) of the spontaneous region numbered \(i\), and \(\vartheta_i\) is the angle between the spontaneous magnetization of this region and the direction of the external field. \(I_{\parallel}\) can change its magnitude in two ways:

1. Process of rotation: \(V_i=\mathrm{const}\), the angles \(\vartheta_i\) change; the spontaneous magnetization in the regions turns toward the direction of the external field.

2. Process of boundary displacement (inversion): \(\vartheta_i=\mathrm{const}\), the volumes \(V_i\) change in such a way that the regions situated more favorably energetically with respect to the field grow at the expense of their less favorably oriented neighbors.

The process of rotation, as experiment shows, takes place in the region of medium and high fields (\(\gtrsim 100\) gauss). It is reversible in the sense that, when the sign of the change of the field is reversed, the rotation proceeds in the opposite direction. The process of boundary displacement in very weak fields (or, more precisely, for very small changes of the field) also proceeds reversibly. But after some critical field \(H_0\) has been reached, displacement of the boundary can proceed even without any further increase of the field, with a finite velocity, i.e. the process of devouring neighboring regions occurs by a jump (the Barkhausen effect). The magnitude of the critical field \(H_0\) is determined from the following considerations. The boundaries between spontaneous regions have a finite thickness, forming a kind of transition layer. The direction of magnetization in this layer changes continuously from the direction \(I_s\) in one region to its direction in the neighboring one. The thickness of the boundary layer and the law of variation of the direction of magnetization within it are determined from the minimum of the total energy of this layer, which is treated as a certain surface energy \(\gamma\). Owing to the presence of inhomogeneities in any real crystal, the energy \(\gamma\) depends on the coordinates. Therefore, when the boundary layer moves during inversion, its energy changes. In the initial state the boundary is in an energy well; for its displacement work is required, which is supplied by the external field. Such a quasi-elastic displacement of the boundary continues until we reach the top of the energy barrier. After this, the motion of the boundary can proceed by itself, without any further increase of the external field, up to the next potential barrier or until the neighboring region is completely devoured—a Barkhausen jump occurs. The height of the potential barrier is determined by the gradient of the surface energy; the magnitude of the critical field is determined by the same quantity.

Irreversible phenomena are manifested most vividly in the magnetization of specimens that have previously been magnetized to saturation (hysteresis). By reducing the field \(H\) that produces saturation in some direction, we shall at first descend along the basic—

curve, the magnetization vector leaves the direction of the field and turns toward the nearest axis of easy magnetization. As one approaches small fields, the demagnetization curve will pass above the main curve. At \(H = 0\) the magnetization vector in each crystallite will be directed along the nearest easy axis, and the so-called residual magnetization appears. To demagnetize the specimen it is necessary to apply a reverse field. At a certain value \(H_C\) of this field, which is called the coercive force, the specimen is completely demagnetized. With a further increase of the reverse field we can magnetize the specimen in the opposite direction up to saturation and then return again to the initial state, describing a hysteresis loop. In the state of saturation we have no spontaneous regions, since the entire material is magnetized to saturation. The process of rotation likewise cannot create them. It is therefore necessary to assume that there exist and can arise nuclei of remagnetization. These may be either remnants of the former regions, surrounded by very high energy barriers inaccessible even to large external fields, or they may be formed as a result of thermal fluctuations. Each of these nuclei begins to grow if the reverse field reaches for it some definite value. This field is called the “start” field \(H_S\); its magnitude is connected with the value of the critical field and with the form of the nucleus. So far we do not have any reasonably satisfactory theory of the origin of remagnetization nuclei. Apparently, this nucleation occurs at the places of greatest lattice distortions, which play the role of free magneto-catalytic surfaces, for, as can be shown, within a homogeneous and uniformly magnetized material the appearance of remagnetization nuclei is very improbable. In real technical materials the observed coercive force is the result of averaging the start fields \(H_S\) over the whole specimen. Likewise, the field corresponding to the steep rise on the main curve is the statistical mean of the critical fields of the individual regions of the specimen. Therefore the study of the elementary processes of magnetization and the verification of the basic theoretical conclusions on ordinary technical material are very inconvenient because of secondary statistical factors. A convenient object for studying the elementary processes of technical magnetization could be single crystals. Here, however, we encounter another difficulty. All ferromagnetic single crystals obtained up to now have very small dimensions. Therefore, when working with them one cannot avoid the strong demagnetizing action of the surface, which greatly distorts the magnetization curve, especially in the initial part, in the region of the process of inversion and hysteresis; moreover, the calculation of the distortion presents insurmountable mathematical difficulties. There is, however, a possibility of bypassing this difficulty and obtaining a homogeneous material suitable for the detailed study of the elementary processes of technical magnetization. As we have already indicated, external stresses

can create in the material one axis of easiest magnetization. Therefore, by strongly elastically stretching a wire (magnetostriction \(>0\)), we can make it, in the magnetic sense, similar to a uniaxial single crystal—the axis of the wire will be the axis of easiest magnetization. By remagnetizing such a wire, we obtain a rectangular hysteresis loop. The inversion process is accomplished by a single Barkhausen jump through the entire wire. The wire can be made as long as desired and thereby the demagnetizing effect of the end surfaces can be reduced to a minimum.

The content of the translated articles placed below consists in the description of experiments and the presentation of the theory of these experiments on such elastically stretched wires.

In the first article, Sixtus gives a review of the results of this author’s brilliant experiments, begun in 1931 jointly with Tonks. In these experiments the kinetics of remagnetization was studied for the first time, and the displacement of boundaries between antiparallel magnetized regions was directly observed for the first time. The velocity of motion of the boundary proved to be directly proportional to the difference between the field in which the experiment is performed and the critical field \(H_0\). Moreover, these experiments made it possible to establish the very concepts of the critical field \(H_0\) and the starting field \(H_S\), and to clarify their dependence on external stresses and on the properties of the samples studied. Also very important for understanding the processes of technical magnetization, and in particular for clarifying the kinetics of growth of already formed nuclei, are Sixtus’s later experiments with the freezing of large nuclei. The theoretical analysis of precisely these experiments, carried out by Döring \(^{9}\) (see the second of the proposed articles), made it possible for the first time to give a direct experimental verification of the predictions of theory. In particular, Döring obtained an estimate of the magnitude of the energy of the boundary surface between antiparallel spontaneous regions.

In one of their subsequent works, Döring and Haake \(^{10}\) showed experimentally that the dependence of \(\gamma\) on stresses predicted by Bloch \(^{5}\) is in good agreement with observations.

LITERATURE

1. J. Frenkel, Z. Physik, 49, 31, 1928.
2. W. Heisenberg, Z. Physik, 49, 619, 1928.
3. P. Weiss, J. de Physique (4), 6, 601, 1907.
4. W. Heisenberg, Z. Physik, 69, 287, 1931.
5. F. Bloch, Z. Physik, 74, 295, 1932.
6. J. Frenkel a. J. Dorfman, Nature, 126, 274, 1930.
7. L. Landau a. E. Lifschitz, Sow. Phys., 8, 153, 1935.
8. See, for example, the introductory article by R. Becker in the collection Probleme der Technischen Magnetisierungskurve, J. Springer, 1938.
9. W. Döring, Z. Physik, 108, 137, 1938.
10. W. Döring u. H. Haake, Z. Physik, 39, 865, 1938.