PHYSICAL THEORY OF THE DOMAIN STRUCTURE OF FERROMAGNETS*)

K. Kittel

CONTENTS

I. Survey of the theory of domain structure . . . . . . . . . . . . . . . . . . . . 453
    1. Introduction. 2. Prerequisites of the domain theory. 3. Origin of domains. 4. Coercive force, hysteresis, and reversible permeability

II. Energy of domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
    1. Exchange energy. 2. Anisotropy energy. 3. Magnetoelastic energy. 4. Magnetostatic energy

III. Bloch wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
    1. Introductory remarks. 2. Estimate of the thickness and energy of a Bloch wall. 3. 180° walls in the [100] plane of iron

IV. Theoretical domain structures . . . . . . . . . . . . . . . . . . . . . . . . . 501
    1. Introduction. 2. Domain configurations with closed flux

V. Experimental study of domains . . . . . . . . . . . . . . . . . . . . . . . . . 511
    1. The method of magnetic powder patterns. 2. Some results obtained by the powder-pattern method

VI. Magnetic properties of small particles . . . . . . . . . . . . . . . . . . . 519
    1. Critical particle sizes at which a single-domain structure appears. 2. Coercive force of small particles

VII. Initial permeability and coercive force . . . . . . . . . . . . . . . . . . 532
    1. General remarks. 2. Theory of inclusions. 3. Deformation theory. 4. Theory of fluctuations of magnetization

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
    Expressions for the anisotropy energy of cubic crystals

Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
    Energy of the magnetic interaction of dipoles in a cubic lattice

Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
    List of symbols

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

*) C. Kittel, Rev. Mod. Phys. 21, 541 (1949).

I. REVIEW OF THE THEORY OF DOMAIN STRUCTURE

I, 1. Introduction

In recent years there has been a great increase in the experimental and theoretical material concerning the origin and behavior of ferromagnetic domains. We now possess the foundations of a theory of domains, which is confirmed in detail by experiments on ferromagnetic single crystals, and less fully in the case of fine ferromagnetic powders. The results obtained confirm the theory in all essential points and thus give confidence that, along this path, more complicated phenomena observed in polycrystalline materials may also be understood, at least qualitatively.

The present article is a complete review of the physical foundations of the theory of domains and of the most important experiments confirming it. In existing books on ferromagnetism the theory of domains is treated insufficiently; for example, in none of these books, with the exception of the recently published Russian book by Vonsovskii and Shur (1948)*), is the fundamental work in this field carried out by Landau and Lifshitz in 1935 discussed.

At the beginning of the present article a brief review is given of the physical foundations of the domain theory. In the following sections the theory is developed in greater detail.

I, 2. Premises of the Theory of Domains

The basic features of ferromagnetism are explained by the following proposition, which follows from experiment: the total magnetization of a specially prepared ferromagnetic specimen can be changed from the initial value zero (in the absence of an external magnetic field) to saturation of the order of 1000 gauss by the application of a field whose intensity may be of the order of 0.01 oersted.

Such a magnetization curve is shown in Fig. 1. Magnetization is defined as the magnetic moment per unit volume. In Fig. 1, however, along the ordinate axis there is plotted not the magnetization, but the magnetic induction, which in the present case is almost equal to the magnetization multiplied by \(4\pi\).

The statement indicated is based on two experimental facts:

*) In addition to the monograph by Vonsovskii and Shur mentioned by the author, see also the review by S. V. Vonsovskii, UFN 35, 514 (1948); 36, 30 (1948); 37, 1, 137 (1949). In this review the reader will also find an extensive bibliography of Soviet work in the field of ferromagnetism, of which only a part has been used by the author of the article. (Translator’s note.)

a) In some cases magnetic saturation can be reached by applying a very weak magnetic field.

b) In a zero (or nearly zero) external field the same specimen may have magnetization equal to zero.

The first of these is remarkable in connection with the fact that, as is known from studies of paramagnetism, when magnetizing a system of free and independent elementary magnetic moments, the application of a field of 0.01 oersted produces an utterly negligible effect.

Fig. 1. Magnetization curve [single crystal of silicon iron. The abscissa axis is approximately the favored axis (Williams and Shockley, 1949).

For example, at room temperature a field of 0.01 oersted increases the magnetization of a paramagnetic salt (such as iron sulfate—FeSO\(_4\)) by an amount of about \(10^{-6}\) gauss, as compared with \(10^3\) gauss in ferromagnetic specimens. As is known, the smallness of the effect in the case of a paramagnetic salt is explained by thermal motion, which counteracts the influence of the magnetic field. In a paramagnetic salt, for every \(10^9\) magnetic moments, a field of 0.01 oersted “orients,” on the average, only one magnetic moment, so that the distribution of magnetic moments over directions remains basically disordered. Such a high degree of chaos is, as has already been said, the result of the predominant role of thermal motion in a system where the magnetic moments of the electrons are free, i.e. noninteracting.

Pierre Weiss (1907) pointed out that the difficulty connected with thermal motion can be largely eliminated by assuming the presence in ferromagnets of a strong internal “molecular” field. We now consider this field to be equivalent to an interaction between electrons that tends to align the magnetic moments parallel to one another.

The magnitude of the Weiss molecular field can be determined without difficulty. At the Curie temperature \(T_c\), the thermal energy of the electron spin, \(kT_c\), must, in order of magnitude, be equal to the energy

\(\mu_B H_{\mathrm{mp}}\) of the magnetic moment of the electron \(\mu_B\) in the acting molecular field \(H_{\mathrm{mp}}\):

\[ kT_c \simeq \mu_B H_{\mathrm{mp}}, \tag{I, 2, 1} \]

whence*)

\[ H_{\mathrm{mp}} \simeq \frac{kT_c}{\mu_B} \simeq 10^{-16}\cdot\frac{10^3}{10^{-20}} \simeq 10^7\ \text{oersted}. \tag{I, 2, 2} \]

The field thus obtained is extremely strong; it is approximately 20 times stronger than any field obtained under laboratory conditions. At temperatures below the Curie point the action of the molecular field exceeds the action of thermal motion, and the specimen becomes ferromagnetic. The orientation of magnetic moments in paramagnetic and ferromagnetic materials is schematically shown in Fig. 2; the value of the magnetic saturation of iron as a function of temperature is presented in Fig. 3.

a)
Paramagnetic salt (or a ferromagnet above the Curie point)

b)
Ferromagnet at a very low temperature

c)
Ferromagnet at a low temperature

d)
Ferromagnet near the Curie point (below it)

Fig. 2. Orientation of the magnetic moments of electrons in paramagnetic and ferromagnetic substances.

It is now known that the molecular field is connected with quantum-mechanical exchange forces; another and better-known manifestation of these forces is the chemical valence bond, although in the case of a chemical bond the exchange forces usually orient the spins of neighboring electrons antiparallel, whereas in the case of ferromagnetism a parallel orientation takes place. Weiss himself made no assumptions about the nature of the molecular field, but he pointed out that the ordinary interaction of the magnetic moments of electrons is too weak to account for the existence of the molecular field. The magnetic field at a lattice site produced by the magnetic moment of an electron situated at a neighboring lattice site is, in order of magnitude,

\[ H \simeq \frac{\mu_B}{r^3} \simeq \frac{10^{-20}}{(2\cdot10^{-8})^3} \simeq 1000\ \text{oersted}. \tag{I, 2, 3} \]

This field is smaller than the acting molecular field \(H_{\mathrm{mp}}\) by approximately \(10^4\) times. The interaction of magnetic moments would lead to

*) The list of symbols is given in Appendix B (p. 541).

to a Curie temperature of about \(0.1^\circ\) K. In dielectrics the situation is completely different, since, in order of magnitude, electric dipole moments are approximately 100 times greater than magnetic dipole moments, which makes the interaction energy greater by \(10^4\) times*). It is therefore not surprising that substances are found which, as a result of the interaction of electric dipoles, are ferroelectrics at room temperature.

We have found out how magnetic saturation is explained by means of a powerful Weiss molecular field. But how, then, shall we explain the fact that a nonzero magnetization is possible at zero external field? At first sight this seems impossible, since one has to suppose that, despite the presence of a molecular field of \(10^7\) oersteds, the magnetic moment of a specimen can be noticeably changed by an external field of \(10^{-2}\) oersted.

Fig. 3. Magnetization of saturation for iron as a function of temperature. At room temperature the saturation value is lower than that occurring at zero degrees Kelvin by 2.0%.

Weiss derived a theory to remove this difficulty, assuming that real specimens consist of a certain number of small regions, called domains, each of which is magnetized to saturation. The direction of magnetization in different domains, however, need not necessarily be the same. A schematic arrangement of domains with zero resultant magnetic moment is shown (for a single crystal) in Fig. 4, \(a\). For polycrystals it was assumed in early investigations that each crystallite may contain one domain and that the resultant magnetic moment may be equal to zero owing to a random distribution of the axes of the grains, as is shown in Fig. 4, \(b\).

The increase of the resultant magnetic moment of a body under the action of an external magnetic field may, in the theory of domains, be regarded as due to two independent processes (as was pointed out by Becker): 1) an increase in the sizes of domains favorably oriented with respect to the field, at the expense of a reduction of unfavorably oriented domains; 2) rotation of the directions of magnetization of the domains toward the direction of the field.

Both of these possibilities, leading to a change in the resultant magnetization, are illustrated by Fig. 5.

) See in this connection the article by V. L. Ginzburg, UFN 39, 490 (1949). (Translator’s note*.)

![Figure 4 diagrams: domain configurations in a monocrystal and a polycrystal.]

Fig. 4. Schematic domain configuration with zero resultant magnetic moment in the case of a monocrystal (a) and a polycrystal (b). For simplicity it is assumed that, in the polycrystalline specimen, each crystallite contains only one domain, which is usually not the case.

![Figure 5 diagrams: principal magnetization processes.]

Specimen not magnetized

$H$

Magnetization in various domains by displacement of boundaries

$H$

Magnetization by rotation

Fig. 5. Principal magnetization processes.

![Figure 6 graph: typical magnetization curve.]

$I$

Rotation of magnetization

Irreversible displacement of the boundary

Reversible displacement of the boundary

$H$

Fig. 6. Typical magnetization curve indicating the regions in which different magnetization processes predominate.

On closer examination it turns out that, in weak fields, the change in magnetization usually occurs through the displacement of boundaries and, consequently, through changes in the sizes of domains. In strong fields the magnetization usually changes as a result of rotation of the direction of magnetization. A typical magnetization curve is shown in Fig. 6, where the regions in which each of the processes predominates are indicated.

Thus we see that Weiss was able to explain the basic properties of ferromagnets with the aid of two assumptions: the existence of a molecular field and the presence of a domain structure. Weiss did not explain either of his assumptions from the standpoint of atomic theory. An explanation of the molecular field from the standpoint of exchange forces was given by Heisenberg in 1926*), and an explanation of the nature of domains from the standpoint of the energy of the magnetic field was given by Landau and Lifshitz in 1935.

We shall now turn to a qualitative consideration of the causes of domain formation.

I, 3. Origin of domains

In this section we shall show that the domain structure is a natural consequence of the existence of different forms of energy in a ferromagnetic body—exchange energy, anisotropy energy, and magnetic energy.

First, however, it is necessary briefly to consider the experimental evidence for the existence of domains.

We have already seen that the assumption of the existence of domains can be made from consideration of the magnetization curve itself. But the existence of a domain structure is demonstrated far more clearly and convincingly by microphotographs of domain boundaries obtained by the method of magnetic powder patterns. This method, first applied by Bitter (1931), yielded in the hands of Williams and his collaborators (1947–1949) numerous and convincing proofs of the existence of domains; moreover, their form and size correspond to theoretical expectations.

The method of powder patterns consists in the following: a drop of a colloidal suspension of a finely divided ferromagnetic substance of the magnetite type is placed on the polished surface of the ferromagnetic crystal under investigation. When observed through a microscope, it is found that the colloidal particles of the suspension concentrate around certain clearly marked lines, which are the boundaries between domains magnetized in different directions. The colloidal particles concentrate—

*) Here, apparently, there is an error, since Heisenberg’s paper belongs to 1928 (Zs. f. Phys. 49, 619 (1928)). Somewhat earlier, analogous considerations in qualitative form were expressed by Ya. I. Frenkel (Zs. f. Phys. 49, 31 (1928)). (Translator’s note.)

are found near domain boundaries because, near these boundaries, there are very strong local magnetic fields, which also attract magnetic particles.

Fig. 7. Simple domain structure in a Si-Fe single crystal (Williams, Bozorth, and Shockley). (\(\times 500\)).

Fig. 8. Complex domain structure in a Si-Fe single crystal (Williams, Bozorth, and Shockley). (\(\times 500\)).

A photograph of a comparatively simple domain structure of iron is shown in Fig. 7 together with an interpretation of the pattern obtained, established both from the photograph itself and on the basis of convincing auxiliary experiments. A more complex type of domain structure is shown in Fig. 8. Structures of this

of a “triple” character arise when the surface of the crystal is slightly inclined with respect to the face of the cube. A detailed explanation of this structure was given by Williams, Bozorth, and Shockley (1949).

We can understand the origin of domains by considering the structures shown in Fig. 9; each of the figures represents a transverse section of a single crystal that is ferromagnetic. In Fig. a) we have a configuration corresponding to saturation, consisting of a single domain; as a result of the formation of magnetic “poles” on the surface of the crystal, this configuration has

Fig. 9. Origin of domains.

a large magnetic energy \({}^{1}/_{8\pi}\int H^2 dV\). For a square cross section the magnetic energy should be of the order of \(I_s^2 \simeq 10^6\) erg/cm\(^3\), where \(I_s\) is the saturation magnetization.

In Fig. b) the magnetic energy is reduced approximately by half as a result of the division of the crystal into 2 domains magnetized in opposite directions. The process of division can be carried further (Fig. c)); in the case of \(N\) domains the value of the magnetic energy is reduced (through a decrease in the spatial extent of the field) approximately to \(1/N\) of the magnetic energy of the configuration indicated in Fig. a).

The process of division can continue until the energy required for the formation of additional boundary layers (or internal faces) separating two oppositely

magnetized domains, will not exceed the decrease in the energy of the magnetic field associated with further subdivision. One may consider that the transition layer does indeed have a definite energy associated with it. In fact, on opposite sides of the layer the magnetization is directed antiparallel, and since the exchange forces tend to maintain a parallel orientation and do not favor an antiparallel one, it is natural to suppose that the formation of the layer is associated with an expenditure of energy. In Section III, after investigating the nature of the layers themselves, we shall calculate this energy and see that it amounts to about 1 erg per \(1\ \mathrm{cm}^2\) of the separating surface. If we then suppose that the number of domains is \(N = 10^3\) per centimeter, then the total surface energy (the energy of the layers) of a crystalline cube with faces of \(1\ \mathrm{cm}\) will be of the order of \(10^3\) ergs, while the magnetic energy will likewise be of the order of \(10^3\) ergs. This situation corresponds approximately to the equilibrium number of domains for the case under consideration.

One can indicate an arrangement of domains for which the magnetic energy is zero (see Fig. 9, g). In this figure the boundaries of the domains, having the form of triangular prisms (the so-called “closure domains”), situated near the outer planes of the crystal, form equal angles of \(45^\circ\) with the directions of magnetization in the domains that they separate. Thus the component of the magnetization normal to the boundary layer is continuous at the boundaries, and no poles are formed in the crystal. Since there are no poles, there is no magnetic field either, and we may speak of a magnetic flux closed within the crystal. Hence the name “closure domains,” since these domains, situated near the surface of the crystal, lead to the formation of a magnetic flux closed within the crystal.

The degree of subdivision of the closed configuration (Fig. 9, d) will depend on the energy required for the formation of closure domains. Hence it is not yet immediately obvious that the optimal closed configuration of type d will necessarily have a lower energy than the optimal configuration of type v; in fact, in different materials, subdivisions close to configurations of both types v and d have been found.

The energy required for the formation of closure domains in uniaxial crystals (such as cobalt) is connected with the so-called energy of crystalline anisotropy. The anisotropy energy accounts for the tendency to direct the magnetization of a domain along certain crystallographic axes. These preferred axes are known as axes of easy magnetization. They are readily detected experimentally, and it is known that, to saturate a specimen along an arbitrary axis, a considerably larger amount of energy is required than in the case of magnetization along one of the axes of easy magnetization. In cobalt

the hexagonal axis of the crystal is the only “easy” axis (the axis of easy magnetization), and cobalt therefore belongs to the uniaxial substances. In iron, which is cubic, the “easy” axes are the edges of the cube; in nickel, which is also cubic, the “easy” axes are the space diagonals of the cube. Fig. 10 shows the magnetization curves of Fe, Ni, and Co in the directions of easy and hard magnetization.

In cobalt, if the principal rectangular domains are magnetized along the easy-magnetization axes, the closure domains will necessarily be magnetized in the “hard” directions. In cubic crystals, such as iron, magnetization is possible along different “easy” axes for both the principal and the closure domains.

Fig. 10. Magnetization curves for single crystals of Fe, Ni, and Co (Honda and Kaya (1926); Kaya (1928)).

The energy expended in this case is associated with magnetostriction: since the closure domains are magnetized along axes different from those of the principal domains, the different domains will, owing to magnetostriction, also elongate along different axes, and in order to combine the different domains together in the crystal it is necessary to overcome elastic forces.

The ordinary structures obtained by the powder-figure method are more complicated than in the simple cases we have analyzed. However, the basic principles are always the same: the domain structure is connected with the possibility of lowering the energy of the system in the transition from a saturated configuration (Fig. 9, a)) with high magnetic energy to a domain configuration (Fig. 9, b) or d)) with lowered energy. Domain structures are essentially caused by the presence of external planes of the crystal, and therefore it is not unexpected that the domain structure changes radically when the surface of the crystal is changed.

A very simple type of domain structure is shown in Fig. 11; this structure was obtained by Williams and Shockley (1949) on a single crystal of silicon iron, cut in the form of a hollow quadrilateral with sides exactly parallel to the crystallographic axes \([001]\) and \([010]\). When the crystal is completely saturated, the domain boundaries are arranged at an angle of \(45^\circ\) (Fig. 11, a); when part of the crystal is magnetized clockwise and part counterclockwise, rectangular boundaries are added (Fig. 11, b). It was found in this case that the magnetization changes by the displacement of the rectangular boundaries; it was also observed that the magnetization curve shown in Fig. 1 is quantitatively explained by displacements of the domain walls.

Fig. 11. Simple domain structures in a single crystal of iron having the form of a rectangular “loop” with sides parallel to the axes \([001]\) and \([010]\).

Domain sizes. An essential feature of the new point of view concerning domains is that the dimensions of domains in a single crystal must be determined to a considerable extent by the dimensions and shape of the crystal itself. Thus the “sizes” of domains are not a constant quantity, but are sensitive to the properties of the crystal under consideration. The volume of some external domain structures may be relatively small (say, \(10^{-6}\ \text{cm}^3\)), whereas the underlying domain structure usually contains larger domains (possibly \(10^{-2}\ \text{cm}^3\)). The limiting size of domains in crystals of suitable shape may, in order of magnitude, be determined by the dimensions of the crystal itself.

Polycrystalline materials. A large part of the present article is directed specifically toward consideration of the domain structure of single crystals. However, the magnetic materials encountered in practice are always polycrystalline, i.e., each specimen consists of a large number of small crystallites. In many materials the orientation of the crystallites is more or less random, so that for some purposes a number of properties of polycrystalline specimens may be obtained by averaging, over directions, the corresponding properties of single crystals. In other materials, particularly those subjected to cold working, the crystallographic axes need not necessarily

may have a random arrangement, and a considerable degree of orientation may occur. For example, in the case of a strip of rolled iron *) the crystallites have a strong tendency to be arranged in such a way that the rolling plane is the \([001]\) plane, while the rolling direction is the \([110]\) direction.

If there is a high degree of orientation, then one may expect from a polycrystalline specimen, with respect to the domain structure, behavior analogous to that of a single crystal.

Fig. 12. Domain structure in a polycrystal of silicon iron (Williams). (×500).

On the other hand, if the orientation of the crystallites is random, there is a high probability that the directions of easy magnetization of adjacent crystallites make a fairly large angle with one another; then one may assume that each crystallite behaves, to a considerable extent, like a single crystal isolated from the neighboring crystallites.

Thus we see that the sizes of domains can in principle be both larger and smaller than the sizes of crystalline grains in polycrystalline materials. In Fig. 12, obtained

*) The orientational anisotropy in a rolled iron strip is beautifully demonstrated by Egger’s microwave-resonance experiment.

Williams has shown a case in which the domain boundaries are practically continuous along more than one grain.

We can also obtain some data on the sizes of domains in polycrystals by measuring the depolarization of polarized neutron beams passing through a ferromagnetic specimen.

I, 4. Coercive force, hysteresis, and reversible permeability

Coercive force*) is probably the most sensitive of the properties of ferromagnets accessible to our control, and at the same time is one of the most important criteria in selecting ferromagnets for practical application. The essential difference between materials for permanent magnets and materials for transformer cores lies precisely in the value of the coercive force, which may vary from 600 oersteds in a loudspeaker magnet (Alnico V alloy) and 20,000 in special highly stable magnets (Fe—Pt) down to a value of 0.5 in power transformers (silicon—iron) or 0.004 in pulse transformers (supermalloy). Thus, the coercive force may vary by a factor of \(5\cdot10^6\).

The task of theory is to interpret the observed values of the coercive force as a function of the physical state of the material. Theory must also indicate methods by which the coercive force can be increased in magnetically hard materials and decreased in magnetically soft ones. Several theories have been proposed (Becker, 1932; Kersten, 1938, 1943; Néel, 1946; Stoner and Wohlfarth, 1947), and some success has been achieved, although the problem is complicated by the usual difficulties in explaining structure-sensitive properties, namely by the difficulty of taking quantitative account of impurities, internal stresses, and other physical factors that are essential in the present question.

When the behavior of the coercive force is understood, we shall also make considerable progress toward understanding hysteresis losses at low frequencies, since the area of the hysteresis loop (Fig. 13) is approximately equal to the product of the saturation induction \(B_s\) by

*) The coercive force \(H_c\) is usually defined with reference to the complete hysteresis cycle (Fig. 13), as the value of the magnetic field corresponding to the point \(B=0\), i.e. the coercive force is the reverse field required to reduce the induction from the saturation value to zero. In theoretical works, however, it is often more convenient to take as the coercive force the field corresponding to the point for which the magnetization \(I\) is zero, i.e. where \(B-H\) is zero. Understood in this sense, the coercive force is denoted by the symbol \({}^{I}H_c\). The difference between \(H_c\) and \({}^{I}H_c\) is significant only in the case of large values of the coercive force, when it is of the order of the saturation magnetization \(I_s\).

coercive force. This means that the energy losses in traversing the hysteresis loop are, to within a factor of 2 to 4, of the order of \(B_s H_c\). We may therefore restrict ourselves to considering only the factor \(H_c\).

“Coercive force” in “magnetically soft” (low \(H_c\)) materials may be understood as follows: the principal energy of a given specimen, as a result of local variations in internal stresses, impurities, crystallite sizes, etc., may vary depending on the position of the domain boundaries; these variations are indicated schematically in Fig. 14. In the absence of an external magnetic field the boundary will be located in some minimum position of type \(A\) in Fig. 14. In the presence of a field the boundary will not be able to shift appreciably, as far as the limiting right-hand position \(D\), until the energy has increased to a value that makes it possible for the boundary to pass through point \(B\), corresponding to the maximum of the boundary energy.

Fig. 13. Determination of the coercive force.

The increase in energy may be achieved by reorientation of the internal magnetization \(I_s\) in the applied field \(H\), and the field \(H\) sufficient to reverse approximately one half of the magnetization of the specimen will be the coercive field \(H_c\).

Fig. 14. Change in the energy of a specimen as a function of the position of the domain boundary.

Qualitatively, this picture of the coercive process explains the fact that coercive forces decrease when the amount of impurity decreases (Fig. 15), and also when internal stresses are removed by annealing. Hence it is also clear why alloys containing a precipitated phase are magnetically hard.

While there is no doubt about the correctness of our general picture, the question of the detailed qualitative relation of the coercive

forces with a number of physical factors remains unclear; we shall postpone consideration of this question until Section VII.

The coercive force of one type of magnetically hard material can be understood on the basis of an entirely different scheme; we have in mind materials consisting of very small grains or fine powders, in which each particle is always magnetized to saturation as a single domain. The fact that sufficiently small particles, with a diameter of less than \(10^{-4}\)—\(10^{-5}\) cm, form a single domain is a result of domain theory, which is confirmed by experiment. It can be shown that, in the case of such very small particles, the formation of a domain boundary is energetically unfavorable, since in that case too large a part of the volume of the small particle would be occupied by the transition layer between domains, which does not depend on the dimensions of the particle.

Fig. 15. Effect of a copper addition on the coercive force of iron (Kussmann and Scharnow (1929)).

If a small particle is forced to remain a single domain, then it will be impossible to change and reverse the magnetization by shifting a boundary, which usually requires comparatively small fields. Instead, the magnetization of the particle must rotate as a whole (Fig. 16), i.e., it will change as the result of a process requiring large fields, which depend on the anisotropy energy of the material or on the shape of the particle. The need for large fields is a consequence of the fact that we must rotate the magnetization through an energy barrier corresponding to the direction of difficult magnetization.

Fig. 16. Change of magnetization in very small particles occurs by rotation of the total magnetic moment of the particle.

The coercive force of the smallest iron particles should theoretically be about 250 oersteds, as follows from the magnitude of the crystalline-anisotropy energy, and approximately just such a value was obtained by several observers. Similarly, the high coercive forces of the compounds MnBi \((I H_c > 12\,000)\) and FePt \((I H_c = 20\,000)\) are evidently in agreement with the supposition that the factor hindering rotation is the anisotropy energy.

If small particles have an elongated shape, we may have a large coercive force because of the anisotropy of the energy of the demagnetizing field, even if the crystalline-anisotropy energy is small. This means that the magnetization tends to set itself along the long axis of the particle, and in order to turn the magnetization in the direction of the short axes a strong field must be applied. This apparently explains the high coercive force of a Fe–Co alloy in the form of a fine powder. As is known from measurements on single crystals, this alloy has a low anisotropy energy, so that the anisotropy energy alone cannot explain the experimental data; to explain them it is necessary to invoke the shape effect.

Reversible permeability. The region of fields in which the permeability is reversible is determined by the distance through which the domain boundary can be displaced without passing over a peak on the curve expressing the dependence of the energy of the boundary layer on distance (Fig. 14). An example of such a region of reversible permeability is the region \(CAB\). If the domain boundary leaves this region, it moves irreversibly.

The reversible permeability is determined by the irregularity of the curve of boundary energy as a function of displacement, i.e., it is determined essentially by the same physical conditions as the coercive force. A comparison of the initial permeability \(\mu_0\) and the coercive force \(H_c\) for a large number of magnetic materials is shown in Fig. 17. It can be seen that a high coercive force correlates with a low permeability of the materials, and conversely.

Barkhausen effect. Since many physicists encounter the concept of domain structure only in elementary textbooks, in connection with the discussion of the Barkhausen effect it is appropriate to clarify the widely circulated assertion about the connection between the sizes of Barkhausen jumps and the sizes of domains. The recent experiments of Williams and Shockley (1949) showed with complete clarity that usually there is no direct connection here and that the Barkhausen jumps correspond not to the complete reversal of a domain, but to irregular fluctuations in the motion of the domain boundary (Bloch wall) under the action of the applied magnetic field. A very distinct and prolonged Barkhausen noise was observed during the motion of the boundary of a single domain. This discovery frees us from the difficulties,

which arose in early investigations; for example, the apparent “domain volume” \(10^{-8}\)—\(10^{-9}\ \mathrm{cm}^3\), found from the Barkhausen effect, has no direct relation to the actual volume of the domains, which may be considerably larger.

Ferroelectric (Seignette-electric) domains. Some dielectric crystals, such as Rochelle salt and barium titanate, behave as ferroelectrics,

In the figure: the vertical axis is “initial permeability \(\mu_0\)”; the horizontal axis is “coercive force in oersteds.” The plotted materials include: Supermalloy; 1040 alloy; Sendust; Mumetal; 78 permalloy; hydrogen-containing iron; Hipernik; Hipersil; silicon iron; iron; nickel; steel with 5% chromium; Remalloy; Vicaloy; Alnico I; Alnico V; Vectolite; iron alloy with platinum.

Fig. 17. Correlation between initial permeability and coercive force for a number of magnetic materials.

i.e., regions of spontaneous electric polarization were found in them, analogous to regions of spontaneous magnetization in ferromagnetic crystals*).

In some crystals of barium titanate, ferroelectric domains were found (Matthias and Hippel, 1948), which made it possible to suppose that their formation is determined by electrostatic energy, just as the formation of magnetic domains is determined by magnetostatic energy. However, the electrostriction of barium titanate is of the order of \(10^{-2}\), i.e. considerably greater than the magnetostriction (\(\sim 10^{-5}\)) of ferromagnets. It is therefore possible that in ferroelectric domains a substantial role will

) See A. V. Rzhanov, UFN, 28, 461 (1949); V. L. Ginzburg, UFN 28, 490 (1949); P. W. Forsbergh, Phys. Rev. 76, 1187 (1949).
(Translator’s note.*)

play electrostriction together with considerations concerning the closedness of the lines of electric induction. There is also the possibility of neutralization of charges as a result of the precipitation of ions from the atmosphere.

The saturation polarization in \(\mathrm{BaTiO_3}\) is approximately equal to \(50\,000\) CGSE.

II. DOMAIN ENERGY

The task of this section is to establish quantitative expressions for certain kinds of energy taken into account in the theory of domain structure. The energies with which we shall encounter especially often are the following: exchange energy with density \(f_{\mathrm{ex}}\); anisotropy energy with density \(f_K\); magnetoelastic energy with density \(f_{\mathrm{me}}\), and magnetostatic energy, whose density is \(f_{\mathrm{mag}}\). It is necessary to consider all these kinds of energy: usually the absence of one of them causes noticeable changes in the character of the domain structure. We shall neglect the distinction between energy and free energy, since in ferromagnetics this distinction is essential only near the Curie point.

The most important expressions for the energy densities, obtained in various parts of this section, are given below for a cubic crystal:

Exchange energy:
\[ f_{\mathrm{ex}} = J S^2 \sum_{i>j} \varphi_{ij}^{2}. \]

Anisotropy energy:
\[ f_K = K_1(\alpha_1^2\alpha_2^2+\alpha_2^2\alpha_3^2+\alpha_3^2\alpha_1^2). \]

Magnetoelastic energy:
\[ f_{\mathrm{me}} = \frac{3}{2}\lambda T \sin^2\theta. \]

Magnetic energy:
\[ f_{\mathrm{mag}}=-\frac{1}{2}HI \quad \text{(for the self-energy).} \]

Here \(J\) is the exchange integral; \(\varphi\) is the angle between the directions of neighboring spins \(S\); \(K_1\) is the anisotropy-energy constant; \(\alpha_1,\alpha_2,\alpha_3\) are the direction cosines of the magnetization vector with the crystallographic axes; \(\lambda\) is the isotropic magnetostriction, and \(\theta\) is the angle between the stress \(T\) and the magnetization.

II, 1. Exchange energy

We shall begin the consideration of exchange energy with a preliminary discussion of modern ideas about the nature of ferromagnetism in connection with the electronic structure of magnetic materials. It is believed that almost the entire magnetic moment of ferromagnetic materials is associated with the electron spin, and not with orbital motion—

…of electrons around the nucleus. This conclusion follows from measurements of the gyromagnetic (magneto-mechanical) ratio. The gyromagnetic ratio is the ratio of the magnetic moment to the angular momentum, and theoretically this ratio should be equal to \(\dfrac{e}{mc}\) for spin and to \(\dfrac{e}{2mc}\) for orbital motion.

The experimental observations, summarized by Barnett (1944), are close to \(\dfrac{e}{mc}\), with small but possibly significant deviations. These deviations make it possible to estimate that orbital motion accounts for about 10%, and spin for 90%, of the saturation magnetization. A similar conclusion is confirmed by the results of microwave resonance experiments (Kittel, 1949a). In most cases the effect of orbital motion may be neglected.

Let us now determine how many electron spins participate in the magnetization. We can determine the effective number of Bohr magnetons per magnetic atom from the relation

\[ n_{\mathrm{eff}}= \frac{\text{saturation magnetization}} {(\text{Bohr magneton})\times(\text{number of magnetic atoms per unit volume})}. \]

Some typical values are given in Table I.

Table I

Calculation of the effective number \(n_{\mathrm{eff}}\) of Bohr magnetons per magnetic atom, and also values of the Curie temperature and saturation magnetization

Substance	Magnetization at saturation \(I_s\), room temperature	Magnetization at saturation \(I_s\), \(0^\circ\mathrm{K}\)	\(n_{\mathrm{eff}}\) \((0^\circ\mathrm{K})\)	Density \(g/cm^3\)	Ferromagnetic Curie temperature in \(^\circ\mathrm{K}\)
Fe . . . . . .	1707	1752	2,221	7,86	1043
Co . . . . . .	1400	1446	1,716	8,8	1388
Ni . . . . . .	485	510	0,606	8,85	631
Gd . . . . . .	1090	1980	7,10	7,83	289
Mn Bi . . .	600	675	3,52	9,0	670 *)
Cu\(_2\) Mn Al . .	430	(540)	3,0	1,72 **)	600

The next question is to clarify the relation between the electrons responsible for ferromagnetism and the electrons responsible for the electrical conductivity of metals: are these two phenomena due to the same electrons or to different electrons? It may be supposed that the electrons responsible for

\[ \text{*) Extrapolated.} \]

\[ \text{**) The density refers only to the Mn atoms.} \]

ferromagnetism, make only a small contribution to electrical conductivity. In the iron group of the periodic system, the conduction electrons are taken chiefly from the \(4s\) shell, whereas the ferromagnetic electrons are found in the \(3d\) shell. The \(3d\) shell lies closer to the nucleus and is more strongly bound than the \(4s\) shell.

In this connection it may be noted that there are known materials which are strong ferromagnets but possess very low electrical conductivity. Such compounds as manganese ferrite \(\mathrm{MnO}\cdot\mathrm{Fe}_2\mathrm{O}_3\) and nickel ferrite \(\mathrm{NiO}\cdot\mathrm{Fe}_2\mathrm{O}_3\) have, at room temperature, values of magnetic saturation of the order of 200 or more, while their specific resistivities are of the order of \(10^2\)—\(10^6\ \mathrm{ohm\cdot cm}\). For comparison, let us cite the specific resistivity of iron, equal to \(10^{-5}\ \mathrm{ohm\cdot cm}\).

In magnetic semiconductors, such as ferrites, we are inclined to assume that the \(4s\) electrons and the ferromagnetic \(3d\) electrons are rather more or less attached to definite atoms than freely wander through the crystal. In more metallic materials, the \(4s\) electrons probably make a substantial contribution to the electrical conductivity, while the \(3d\)-electrons remain strongly localized. The results of experiments with polarized neutrons also indicate that electrons with uncompensated spins are localized in the crystal.

The chief difficulty in the model of \(3d\)-electrons fixed at individual atoms is that it does not immediately explain the nonintegral value of the number of magnetons per atom obtained in most materials*). The nonintegral value \(n_{\mathrm{eff}}\) (Table I) is easily explained on the basis of an alternative model of “collective” electronic ferromagnetism, in which the \(3d\)-electrons can move more or less freely throughout the whole crystal.

Neither the atomic nor the collective model, taken separately in their simplest form, gives a complete and consistent explanation of all the numerous phenomena associated with ferromagnetism. In this article we use exclusively the atomic model, not because of firm confidence in its universal applicability, but because on the basis of this model one can determine in a simple way the quantity most important in the theory of domains. This quantity is the exchange energy associated with the gradual change in the directions of the spins observed in the transition layer between domains. The good agreement of the experimental and theoretical values of the surface energy of the transition layer is an important achievement of the atomic model. However, the transition layer can also be treated quantitatively on

*) Van Vleck (1945) showed that a slight modification of the Heisenberg model makes it possible to explain the nonintegral value of the mean number of magnetons.

based on a “collective” model (this is done in the article by Gerring and Kittel now in press).

Exchange energy in the atomic model. We shall now analyze a model in which, at each lattice site of the crystal, there is an atom with total spin quantum number \(S\), where \(2S\) is equal to the number of unpaired spins in the atom and is an integer.

An essential result of the quantum-mechanical treatment of the many-electron problem is that, in the interaction energy between two atoms, there is a term of electrostatic origin that does not arise in the classical treatment. This term tends to orient the electronic spins of the atoms parallel or antiparallel to one another, depending on the algebraic sign of a certain integral \(J\), known as the exchange integral. Usually \(J\) is defined in such a way that, when it is positive, the energy for the parallel orientation of two spins is lower than the energy for their antiparallel orientation by an amount \(2J\) (for spin \(1/2\)).

In the present review we shall assume the existence of an exchange interaction with definite properties, without giving the corresponding proofs, for lack of space. We shall indicate, however, how the exchange integral \(J\) may be approximately related to the Weiss molecular field \(H_{\mathrm{mp}}\). Suppose that the electronic spins of a given atom and of \(z\) of its nearest neighbors are oriented in the same direction. The exchange energy of the selected atom is equal to \(-2zJS^2\). The molecular field is, in essence, defined so that the interaction energy \(-2S\mu_B H_{\mathrm{mp}}\) of the magnetic moment of the atom with the molecular field is equal to the exchange energy:

\[ 2zJS^2 = 2S\mu_B H_{\mathrm{mp}}, \tag{II, 1,1} \]

whence

\[ H_{\mathrm{mp}} = \frac{zSJ}{\mu_B}. \tag{II, 1,2} \]

The effective coupling between spins caused by the exchange effect is equivalent to a potential energy of the form:

\[ V_{ij} = -2J_{ij}S_iS_j, \tag{II, 1,3} \]

where \(J_{ij}\) is the exchange integral of the atoms \(i\) and \(j\); \(S_i\) is the spin angular momentum of atom \(i\), measured in units of \(\frac{h}{2\pi}\). This equation is a fundamental result of quantum theory and serves as the starting point for calculations of the exchange energy of various spin configurations. The equation has been obtained in many places, and we may again refer to Van Vleck’s review (1945).

For many purposes we may, instead of spin matrices, approximately consider classical vectors, and in this sense the equa-

can be written in the form:

\[ w_{\mathrm{ex}}=-\sum_{i>j}2J_{ij}S^2\cos\varphi_{ij}, \tag{II, 1,4} \]

where \(\varphi_{ij}\) is the angle between the directions of the spin vectors, understood in the classical sense; \(w_{\mathrm{ex}}\) is now the exchange energy. The conditions under which formula (II, 1,2) may be used will be discussed in a separate paper; we may summarize its results by saying that the quasiclassical approximation is acceptable when the angles between the directions of neighboring spins are small, as is the case inside the transition layer between domains. In general, in the theory of domains we are interested in the exchange energy only for configurations in which the directions of neighboring spins form small angles with one another.

If we assume that only interactions between nearest neighbors are important for the exchange energy and that these interactions are equal to one another, then

\[ w_{\mathrm{ex}}=-2JS^2\sum_{i>j}\cos\varphi_{ij}. \]

If neighboring spins form a small angle \(\varphi\ll 1\) with one another, we obtain the important result:

\[ \Delta w_{\mathrm{ex}}\simeq JS^2\sum \varphi_{ij}^2, \tag{II, 1,5} \]

and the exchange energy between each pair of spins is

\[ \Delta w_{ij}\simeq JS^2\varphi^2. \tag{II, 1,6} \]

It is often convenient to express (II, 1,5) in another form. Suppose that the direction cosines of the spin at the lattice site \(r_j\) are \(a_j^x, a_j^y, a_j^z\). The direction cosines \(a_i^x, a_i^y, a_i^z\) at a neighboring lattice site may be expanded in a Taylor series:

\[ a_i^x = a_j^x+ \left[ x_{ij}\frac{\partial}{\partial x_{ij}} +y_{ij}\frac{\partial}{\partial y_{ij}} +z_{ij}\frac{\partial}{\partial z_{ij}} \right]a_j^x + \]

\[ +\frac{1}{2} \left[ x_{ij}^2\frac{\partial^2}{\partial x_{ij}^2} +y_{ij}^2\frac{\partial^2}{\partial y_{ij}^2} +z_{ij}^2\frac{\partial^2}{\partial z_{ij}^2} \right]a_j^x+\cdots \tag{II, 1,7} \]

Summing over nearest neighbors in a body-centered cubic lattice with lattice constant \(a\), we have:

\[ \Delta w_{\mathrm{ex}}\simeq -2JS^2a^2\sum_j\left(a_j\cdot\nabla^2 a_j\right). \tag{II, 1,8} \]

The last expression can be put in the form

\[ \Delta w_{\mathrm{ex}}=2JS^2a^2\sum_j \left[ (\nabla a_j^x)^2+(\nabla a_j^y)^2+(\nabla a_j^z)^2 \right], \tag{II, 1,9} \]

using the relation:

\[ \nabla^2(\alpha\cdot\alpha)=0. \tag{II, 1, 10} \]

The density of the exchange energy is equal to (taking into account the presence of two atoms in the elementary cell and taking care not to count the same interaction twice):

\[ f_{\text{ex}}=A\{(\nabla\alpha_1)^2+(\nabla\alpha_2)^2+(\nabla\alpha_3)^2\}, \tag{II, 1, 11} \]

where

\[ A=\frac{2JS^2}{a}. \]

We have now obtained two convenient formulas for the exchange energy, (II, 1, 6) and (II, 1, 11). The next problem is to establish the relation between the exchange integral \(J\), which enters into these formulas, and some experimental quantity strongly dependent on \(J\), such as the Curie temperature or the change of magnetic saturation with temperature. It must be emphasized here that an exact determination of \(J\) from thermal data presupposes the existence of a rigorous statistical theory of ferromagnetism, which we do not at present possess.

The mathematical details of the calculations entering into the statistical theories were compared by Van Vleck (1945, 47). We shall give here the results of two methods. P. R. Weiss (1948), generalizing the Bethe–Peierls method, obtained the following results:

\[ J=0.54\,kT_c \quad \text{(simple cubic lattice, spin } 1/2), \tag{II, 1, 12} \]

\[ J=0.34\,kT_c \quad \text{(body-centered cubic lattice, spin } 1/2), \tag{II, 1, 13a} \]

\[ J=0.15\,kT_c \quad \text{(body-centered cubic lattice, spin } 1). \tag{II, 1, 13b} \]

For iron, taking spin 1,

\[ J=(0.15)(1043)\,k\simeq160\,k. \tag{II, 1, 14} \]

Another method, used by Lifshitz*) (1944), consists in establishing the relation between the experimental value of the constant \(C\) in Bloch’s law

\[ I=I_0\left(1-CT^{3/2}\right) \tag{II, 1, 15} \]

for the temperature dependence of magnetic saturation at low temperatures and the effective exchange integral \(J\). For

*) Some numerical errors in Lifshitz’s work have been corrected here.

for a body-centered cubic lattice and spin \(S\) this relation is as follows (Bloch, 1931; Møller, 1933):

\[ C=\frac{0.0587}{2S}\left(\frac{k}{2SJ}\right)^{3/2}. \tag{II, 1, 16} \]

Now, according to the measurements of Fallot (1936), for iron we have \(C=3.5\cdot 10^{-6}\), and therefore, for \(S=1\),

\[ J=205\,k; \tag{II, 1, 17} \]

this value is in good agreement with formula (II, 1, 14), obtained by an entirely different method.

We shall use the value \(J=205\,k\), obtained from Bloch’s theory, since the picture of spin waves is most closely connected with the behavior of the transition layers between domains. In this picture, as can be shown, the value of \(A\) does not depend on assumptions about the influence of non-nearest neighbors.

Using the indicated value of \(J\), the constant \(A\) in (II, 1, 11) is equal (for \(S=1\)) to:

\[ A=\frac{2JS^2}{a}=\frac{410\,k}{2.86\cdot 10^{-8}}=2.0\cdot 10^{-6}\ \text{erg}/\text{cm}. \tag{II, 1, 18} \]

Fallot’s measurements (1936) indicate that for an Fe alloy with \(4\%\) (by weight) Si, \(A\simeq 1.7\cdot 10^{-6}\ \text{erg}/\text{cm}\). This alloy corresponds approximately to that used by Williams in studying domains by the powder-pattern method. This alloy is more convenient than pure iron, since obtaining crystals of pure iron is more difficult.

II, 2. Anisotropy energy

The anisotropy energy, or, as it is sometimes called, the magnetocrystalline energy of a ferromagnetic crystal, promotes the establishment of magnetization along certain principal crystallographic axes, which are called directions of easy magnetization; directions along which the crystal is most difficult to magnetize are called “hard directions.” It has been established experimentally that the energy required to magnetize a crystal to saturation in a hard direction exceeds, and sometimes considerably exceeds, the energy required to saturate the crystal in directions of easy magnetization. The excess energy required for the hard direction as compared with the easy one is the anisotropy energy.

As an example let us consider the case of cobalt, which is a hexagonal crystal. It was found that the direction of the hexagonal axis is a direction of easy magnetization (at room temperature), whereas all directions in the basal plane, normal to the hexagonal axis, are hard directions.

by them. The magnetization curve of a cobalt crystal is shown in Fig. 10. The energy corresponding to the magnetization curve in the hard direction is equal to \(\int H dI\) per unit volume, which gives an excess energy of about \(5 \cdot 10^6\) erg/cm\(^3\).

Mathematical expression for the energy of anisotropy. We shall now consider the question of determining the work required to magnetize a cobalt crystal in a direction making an angle \(\theta\) with the hexagonal axis. It is natural to expect that the anisotropy energy density \(f_K\) can be represented by a series of the type

\[ f_K = \sum_n K'_n \sin^{2n}\theta, \tag{II, 2, 1} \]

where odd powers of \(\sin\theta\) are not included for reasons of symmetry, connected with the fact that the directions \(+\theta\) and \(-\theta\) are equivalent in magnetic and crystallographic respects; \(K'_n\) are constants independent of \(\theta\). In fact, for cobalt the experimental results are described very well if one restricts (II, 2, 1) to the first two terms:

\[ f_K = K'_1 \sin^2\theta + K'_2 \sin^4\theta, \tag{II, 2, 2} \]

where, at room temperature,

\[ K'_1 = 4.1 \cdot 10^6\ \text{erg/cm}^3;\qquad K'_2 = 1.0 \cdot 10^6\ \text{erg/cm}^3. \tag{II, 2, 3} \]

It was found that there is no need to include any terms depending on the direction of the projection of the magnetization onto the basal plane, so that, apart from uniaxiality, the specific hexagonal nature of the crystal does not appear in the anisotropy energy.

Iron is a cubic crystal, and the magnetization curve (Fig. 10) shows that the cube edges \([100]\), \([010]\), and \([001]\) are directions of easy magnetization, while the space diagonals (\([111]\) and equivalent axes) are hard directions. The excess work required for magnetization along the \([111]\) axis is, at room temperature, approximately \(1.4 \cdot 10^5\) erg/cm\(^3\).

Writing the anisotropy energy of iron for an arbitrary direction with direction cosines \(\alpha_1, \alpha_2, \alpha_3\) relative to the cube edges, we are guided by the restrictions imposed by cubic symmetry. For example, the expression for the anisotropy energy must contain an even power of each \(\alpha_i\) and must be invariant under mutual interchange of the \(\alpha_i\) among themselves. The simplest combination satisfying the symmetry requirements is \(\alpha_1^2 + \alpha_2^2 + \alpha_3^2\), but this expression is identically equal to unity and does not explain the anisotropy effect. The next combination will be

fourth degree: \(a_1^2a_2^2 + a_2^2a_3^2 + a_3^2a_1^2\), and is sometimes written in the equivalent form \(a_1^4 + a_2^4 + a_3^4\). The equivalence of the two forms follows from the equality:

\[ 1 = (a_1^2 + a_2^2 + a_3^2)^2 = a_1^4 + a_2^4 + a_3^4 + 2(a_1^2a_2^2 + a_2^2a_3^2 + a_3^2a_1^2), \]

whence

\[ a_1^2a_2^2 + a_2^2a_3^2 + a_3^2a_1^2 = \frac{1}{2} - \frac{1}{2}(a_1^4 + a_2^4 + a_3^4). \]

The next term will be of the sixth degree: \(a_1^2a_2^2a_3^2\). These two terms are usually quite sufficient for explaining the experimental data, so that for iron

\[ f_K = K_1(a_1^2a_2^2 + a_2^2a_3^2 + a_3^2a_1^2) + K_2a_1^2a_2^2a_3^2, \tag{II, 2, 4} \]

where at room temperature

\[ K_1 = 4.2 \cdot 10^5\ \text{erg}/\text{cm}^3;\quad K_2 = 1.5 \cdot 10^5\ \text{erg}/\text{cm}^3. \tag{II, 2, 5} \]

Other forms of writing the term with \(K_1\) are given in Appendix A.

The physical nature of the anisotropy energy. Before going more deeply into the experimental data on the anisotropy energy, we wish to discuss the origin of the anisotropy energy from the standpoint of interatomic interaction. The anisotropy energy, in essence, connects the directions of magnetization with the axes of the crystal. From the very beginning we can point out three very important difficulties on the way to understanding the nature of the anisotropy energy: a) the exchange energy by itself does not lead to anisotropy, despite the actually existing geometrical anisotropy of the crystal structure; b) the interaction of magnetic moments leads only to very small values of the anisotropy constants, much smaller than those observed; c) the anisotropy constants, as it turns out, are very sensitive to changes in temperature, and even a change of sign of the constants in passing from low temperatures to high ones is not unusual.

Let us first turn to the exchange energy: the operator of the exchange energy depends only on the angle between spins:

\[ H = -2J\sum S_iS_j \tag{II, 2, 6} \]

and is entirely independent of the angle between the spin and the crystallographic axes. This means that we can rotate the entire system of spins through any angle with respect to the crystal structure without changes in the exchange energy of the system.

The ordinary interaction of the magnetic moments of electrons leads to too small values of the anisotropy. In Appendix B we shall prove that the magnetic dipole interaction gives zero anisotropy for an undeformed cubic lattice; the assumption of spontaneous deformation of a cubic lattice gives very small values of the anisotropy—about \(10^{-}\)

of the observed values in iron and nickel. In uniaxial crystals the magnetic dipole interaction may make a small contribution to the anisotropy energy, but this effect is usually insignificant. The anisotropy energy of a uniaxial MnBi crystal is of the order of \(10^7\) erg/cm\(^3\), whereas the dipole–dipole energy is only of the order of \(I_s^2\), i.e., less than \(10^5\) erg/cm\(^3\). Moreover, the change of sign of the anisotropy constant in the system of face-centered alloys of iron and nickel, as Fig. 18 shows, is incomprehensible from the standpoint of magnetic dipole interaction.

The temperature dependence of the principal anisotropy constants of iron, nickel, cobalt, and the compound MnBi is shown in Fig. 19; it may be seen that this dependence is extremely nonuniform and does not admit of a simple interpretation.

Fig. 18. Anisotropy constant for the face-centered FeNi alloy at room temperature. It should be noted that in the region where the nickel content is close to 70%, the anisotropy is small.

The anisotropy energy, as is now believed (Sommerfeld and Bethe, 1933; Brooks, 1940; Van Vleck, 1947), arises as a result of the combined influence of the spin–orbit interaction and the partial quenching of the orbital angular momentum (as a result of the inhomogeneity of the crystalline electric fields and of the orbital exchange interaction with neighboring spins). In other words, the magnetization of a crystal “follows” the crystal lattice by means of the orbital motion of the electrons; the spin interacts with the orbital motion through spin–orbit coupling, and the orbital motion in turn interacts with the crystal structure through electrostatic fields and as a result of the overlap of the wave functions of neighboring atoms in the lattice.

The theory developed along these lines is very complicated, even in its approximate form. An excellent review of the present state of the theory on this question has been given by Van Vleck (1947).

An important result following from the theory of anisotropy is that the magnetic anisotropy energy will be large for crystals with a low-symmetry lattice of magnetic ions and, conversely, one may expect that the anisotropy energy will be low for crystal lattices of high symmetry. This rule is suggested by the fact that the anisotropy energy

of a cubic crystal manifests itself in a higher approximation than in a uniaxial crystal. The experimental data confirm our hypothesis: the anisotropy energy of cubic crystals Fe and Ni is of the order of \(10^5\) erg/cm\(^3\), whereas it is of the order of \(10^7\) erg/cm\(^3\) for the hexagonal crystals Co and MnBi and, probably, also of the order of \(10^7\) erg/cm\(^3\) for the ordered state of the FePt alloy, in which the Fe atoms form a trigonal lattice.

Fig. 19. Temperature dependence of the principal anisotropy constants in some metals.

The high anisotropy energy is of great interest in connection with the creation of materials with a high coercive force, needed for permanent magnets.

Experimental data on the anisotropy energy. Let us now dwell briefly on a discussion of the experimental material on the anisotropy energy; further data and a description of the measurement methods may be found in the article by Bozorth (1937).

Iron–nickel alloys. The anisotropy constant \(K_1\) at room temperature for the face-centered system (\(\gamma\)-phase) FeNi is shown in Fig. 18. For pure nickel \(K_1 = 3.4 \cdot 10^4\ \mathrm{erg/cm^3}\) and \(K_2 = 5.0 \cdot 10^4\ \mathrm{erg/cm^3}\); for pure iron \(K_1 = 4.2 \cdot 10^5\ \mathrm{erg/cm^3}\) and \(K_2 = 1.5 \cdot 10^5\ \mathrm{erg/cm^3}\).

Iron–cobalt alloys. The results (at room temperature) for the body-centered (\(\alpha\)-phase) FeCo system are given in Fig. 20. For pure Co, which is hexagonal, \(K'_1 = 4.1 \cdot 10^6\ \mathrm{erg/cm^3}\) and \(K'_2 = 1.0 \cdot 10^6\ \mathrm{erg/cm^3}\).

Fig. 20. Anisotropy constant for a body-centered FeCo alloy at room temperature.

Manganese compounds. At room temperature Guillaud (1943) gives \(K'_1 = 8.9 \cdot 10^6\ \mathrm{erg/cm^3}\) and \(K'_2 = 2.7 \cdot 10^6\ \mathrm{erg/cm^3}\) for \(\mathrm{Mn_2Sb}\); for \(\mathrm{Mn_2Sb}\), \(K'_1 = 1.8 \cdot 10^5\ \mathrm{erg/cm^3}\) and \(K'_2 = 0.8 \cdot 10^5\ \mathrm{erg/cm^3}\).

Low values of the anisotropy energy for certain alloys (for example, an alloy containing about \(75\%\) Ni and \(25\%\) Fe, and an alloy containing about \(40\%\) Co and \(60\%\) Fe) are of great interest in connection with the creation of magnetic materials with high permeability. For example, permalloy and related FeNi alloys containing about \(78\%\) Ni have exceptionally high permeabilities; in the case of supermalloy the maximum permeability is of the order of a million.

For alloys with approximately 75% Ni and 25% Fe, the magnetostriction is also very low: this is another requirement for high permeability. The exceptional properties of permalloys are associated with the fact that in them both the anisotropy and the magnetostriction are very small.

II, 3. Magnetoelastic Energy

Magnetoelastic energy is that part of the energy of a crystal which is due to the interaction between the magnetization and the mechanical deformations of the lattice. For an undeformed lattice the magnetoelastic energy is equal to zero.

In the usual consideration of energy relations in ferromagnets, the close physical connection that exists between the anisotropy constants and magnetostriction is not clearly revealed. At the same time it is very important to emphasize that, in the case of an anisotropy energy independent of deformations in the crystal, linear magnetostriction is absent. Magnetostriction arises because the anisotropy energy depends on the deformation in such a way that the stable state of the crystal is deformed with respect to the cubic lattice. Thus, the crystal will deform spontaneously if this lowers the anisotropy energy. We shall now proceed to consider the nature of the interaction between magnetization and deformation in cubic crystals, expressing the energy relations through the experimental constants of magnetostriction. In doing so we reproduce, in simplified form, the usual arguments of Becker and Akulov.

The density of elastic energy in a cubic crystal is expressed (see Love, Theory of Elasticity) as:

\[ f_{\mathrm{el}}=\frac{1}{2}c_{11}\left(e_{xx}^{2}+e_{yy}^{2}+e_{zz}^{2}\right) +\frac{1}{2}c_{44}\left(e_{xy}^{2}+e_{yz}^{2}+e_{zx}^{2}\right)+ \]

\[ +c_{12}\left(e_{yy}e_{zz}+e_{xx}e_{zz}+e_{xx}e_{yy}\right), \qquad (\mathrm{II},\,3,\,1) \]

where \(c_{ij}\) are the elastic moduli and \(e_{ij}\) are the deformations.

For iron (Kimura and Ohno, 1934)

\[ \begin{aligned} c_{11}&=2.41\cdot 10^{12}\ \mathrm{erg}/\mathrm{cm}^{3},\\ c_{12}&=1.46\cdot 10^{12}\ \mathrm{erg}/\mathrm{cm}^{3},\\ c_{44}&=1.12\cdot 10^{12}\ \mathrm{erg}/\mathrm{cm}^{3}. \end{aligned} \qquad \left\} \right. \qquad (\mathrm{II},\,3,\,2) \]

For nickel (Bozorth et al., 1949)

\[ \begin{aligned} c_{11}&=2.50\cdot 10^{12}\ \mathrm{erg}/\mathrm{cm}^{3},\\ c_{12}&=1.60\cdot 10^{12}\ \mathrm{erg}/\mathrm{cm}^{3},\\ c_{44}&=1.185\cdot 10^{12}\ \mathrm{erg}/\mathrm{cm}^{3}. \end{aligned} \]

The anisotropy energy density in an undeformed cubic crystal (Section II, 2), in the first approximation, is equal to:

\[ f_K = K \left(\alpha_1^2 \alpha_2^2+\alpha_2^2 \alpha_3^2+\alpha_3^2 \alpha_1^2\right). \tag{II, 3, 3} \]

Here \(K\) is a constant independent of the direction of the saturation magnetization in the crystal; \(\alpha_1,\alpha_2,\alpha_3\) are the direction cosines of the magnetization with respect to the cube axes. For iron
\(K=4.2\cdot 10^5\ \mathrm{erg}/\mathrm{cm}^3\).

In order to express the dependence of the anisotropy energy on the deformations, let us expand the energy in a Taylor series in the deformations:

\[ f_K=(f_K)_0+\sum_{i\geq j} \left(\frac{\partial f_K}{\partial e_{ij}}\right)_0 e_{ij}+\cdots \tag{II, 3, 4} \]

Here \((f_K)_0\) must satisfy cubic symmetry, but the expressions
\(\left(\frac{\partial f_K}{\partial e_{ij}}\right)_0 e_{ij}\) may have lower symmetry, since these expressions refer to the deformed lattice.

Restricting ourselves to the terms of lowest degree, from symmetry considerations we obtain:

\[ \begin{aligned} \frac{\partial f_K}{\partial e_{xx}}&=B_1\alpha_1^2; & \frac{\partial f_K}{\partial e_{yy}}&=B_1\alpha_2^2;\\ \frac{\partial f_K}{\partial e_{zz}}&=B_1\alpha_3^2; & \frac{\partial f_K}{\partial e_{xy}}&=B_2\alpha_1\alpha_2;\\ \frac{\partial f_K}{\partial e_{yz}}&=B_2\alpha_2\alpha_3; & \frac{\partial f_K}{\partial e_{zx}}&=B_2\alpha_1\alpha_3, \end{aligned} \tag{II, 3, 5} \]

where \(B_1\) and \(B_2\) are constants, which can in principle be computed. The quantities \(B\) are called magnetoelastic coupling constants. For iron, as we shall see later,
\(B_1\simeq -2.9\cdot 10^7\ \mathrm{erg}/\mathrm{cm}^3\),
\(B_2\simeq 3.2\cdot 10^7\ \mathrm{erg}/\mathrm{cm}^3\).

Combining the preceding expressions, we obtain the total energy density, depending both on the deformations and on the direction of the magnetic-saturation vector,

\[ \begin{aligned} f={}&K\left(\alpha_1^2\alpha_2^2+\alpha_2^2\alpha_3^2+\alpha_3^2\alpha_1^2\right) +B_1\left(\alpha_1^2 e_{xx}+\alpha_2^2 e_{yy}+\alpha_3^2 e_{zz}\right)+\\ &+B_2\left(\alpha_1\alpha_2 e_{xy}+\alpha_2\alpha_3 e_{yz}+\alpha_3\alpha_1 e_{zx}\right)+\\ &+\frac{1}{2}c_{11}\left(e_{xx}^2+e_{yy}^2+e_{zz}^2\right)+\\ &+\frac{1}{2}c_{44}\left(e_{xy}^2+e_{yz}^2+e_{zx}^2\right)+\\ &+c_{12}\left(e_{yy}e_{zz}+e_{xx}e_{zz}+e_{xx}e_{yy}\right). \end{aligned} \tag{II, 3, 6} \]

The equilibrium configuration of the crystal magnetized in the direction \(\alpha\) can be found by minimizing \(f\) with respect to

to \(e_{ij}\). The solutions for \(e_{ij}\) can be expressed through the usual magnetostriction constants at saturation \(\lambda_{100}\) and \(\lambda_{111}\).

Moreover, the solutions \(e_{ij}\) of the equations determining the equilibrium state (the equilibrium equations) depend on the direction cosines in such a way that the energy of the equilibrium configuration can be expressed in the form:

\[ f=(K+\Delta K)\left(\alpha_1^2\alpha_2^2+\alpha_2^2\alpha_3^2+\alpha_3^2\alpha_1^2\right), \tag{II, 3, 7} \]

where \(\Delta K\) does not depend on \(\alpha\), but is simply related to the elastic modulus and to the constants of magnetoelastic coupling.

Equilibrium equations. The first problem is to determine the values of \(e_{ij}\) giving the minimum of expression (II, 3, 6) for \(f\):

\[ \left. \begin{aligned} \frac{\partial f}{\partial e_{xx}}&=B_1\alpha_1^2+c_{11}e_{xx}+c_{12}(e_{yy}+e_{zz})=0,\\ \frac{\partial f}{\partial e_{yy}}&=B_1\alpha_2^2+c_{11}e_{yy}+c_{12}(e_{xx}+e_{zz})=0,\\ \frac{\partial f}{\partial e_{zz}}&=B_1\alpha_3^2+c_{11}e_{zz}+c_{12}(e_{xx}+e_{yy})=0,\\ \frac{\partial f}{\partial e_{xy}}&=B_2\alpha_1\alpha_2+c_{44}e_{xy}=0,\\ \frac{\partial f}{\partial e_{xz}}&=B_2\alpha_1\alpha_3+c_{44}e_{xz}=0,\\ \frac{\partial f}{\partial e_{yz}}&=B_2\alpha_2\alpha_3+c_{44}e_{yz}=0. \end{aligned} \right\} \tag{II, 3, 8} \]

Hence

\[ e_{ii}=B_1\,\frac{\left[c_{12}-\alpha_i^2(c_{11}+2c_{12})\right]}{[(c_{11}-c_{12})(c_{11}+2c_{12})]}, \tag{II, 3, 9} \]

\[ e_{ij}=-B_2\,\frac{\alpha_i\alpha_j}{c_{44}}\qquad (i\ne j). \tag{II, 3, 10} \]

Connection with the magnetostriction constants. The ordinary equations of magnetostriction, often used in the analysis of experimental data, in cubic crystals are as follows (Becker and Döring, 1939):

\[ \frac{\delta l}{l} = \frac{3}{2}\lambda_{100} \left(\alpha_1^2\beta_1^2+\alpha_2^2\beta_2^2+\alpha_3^2\beta_3^2-\frac{1}{3}\right) + 3\lambda_{111} (\alpha_1\alpha_2\beta_1\beta_2+\alpha_1\alpha_3\beta_1\beta_3+\alpha_2\alpha_3\beta_2\beta_3), \tag{II, 3, 11} \]

where \(\alpha=(\alpha_1\alpha_2\alpha_3)\) is the unit vector in the direction of magnetization, \(\beta=(\beta_1\beta_2\beta_3)\) is the unit vector in the direction of the measured elongation \(\delta l\); \(\lambda_{100}\) and \(\lambda_{111}\) are the values of the longitudinal magnetostriction at saturation in the directions \([100]\) and \([111]\). We now wish to relate the magnetostrictive constants \(\lambda_{100}\) and \(\lambda_{111}\)

with the magnetoelastic coupling constants \(B_1\) and \(B_2\), which have a more fundamental significance.

In deformations we have:

\[ \frac{\delta l}{l}=\sum_{i\ge j} e_{ij}\beta_i\beta_j; \tag{II, 3, 12} \]

since, by the definition of deformations (see, in particular, the definition of shear deformations used by Love):

\[ \begin{aligned} x'&=(1+e_{xx})x+\frac{1}{2}e_{xy}y+\frac{1}{2}e_{zx}z,\\ y'&=\frac{1}{2}e_{xy}x+(1+e_{yy})y+\frac{1}{2}e_{yz}z,\\ z'&=\frac{1}{2}e_{zx}x+\frac{1}{2}e_{yz}y+(1+e_{zz})z; \end{aligned} \tag{II, 3, 13} \]

whence

\[ \delta(l^2)=2l\cdot\delta l=2l^2\sum e_{ij}\beta_i\beta_j . \tag{II, 3, 14} \]

From this expression there follows directly expression (II, 3, 12).

Substituting the values of \(e_{ij}\) from expressions (II, 3, 9) and (II, 3, 10), we have:

\[ \begin{aligned} \frac{\delta l}{l} &=-\frac{B_1}{c_{11}-c_{12}} (\alpha_x^2\beta_x^2+\alpha_y^2\beta_y^2+\alpha_z^2\beta_z^2)\\ &\quad-\frac{B_2}{c_{44}} (\alpha_x\alpha_y\beta_x\beta_y+\alpha_x\alpha_z\beta_x\beta_z+\alpha_y\alpha_z\beta_y\beta_z)\\ &\quad+\frac{3c_{12}B_1}{(c_{11}-2c_{12})(c_{11}-c_{12})}. \end{aligned} \tag{II, 3, 15} \]

This equation can be written in the form of expression (II, 3, 11), if we set:

\[ \left. \begin{aligned} \lambda_{100}&=-\frac{2}{3}\frac{B_1}{c_{11}-c_{12}},\\ \lambda_{111}&=-\frac{1}{3}\frac{B_2}{c_{44}}, \end{aligned} \right\} \tag{II, 3, 16} \]

and omit the term that does not depend on \(\alpha\) and \(\beta\). Thus we obtain the connection of the magnetostriction constants \(\lambda_{100}\) and \(\lambda_{111}\) with the gradient of the anisotropy energy with respect to the deformations and with the elastic constants of the crystal.

For iron, according to experimental data, \(\lambda_{100}=19.5\cdot10^{-6}\) and \(\lambda_{111}=-18.8\cdot10^{-6}\); using these values, we calculate \(B_1=-2.9\cdot10^7\ \mathrm{erg/cm^3}\), \(B_2=6.4\cdot10^7\ \mathrm{erg/cm^3}\). For nickel \(\lambda_{100}=-46\cdot10^{-6}\) and \(\lambda_{111}=-25\cdot10^{-6}\), and we have \(B_1=6.2\cdot10^7\ \mathrm{erg/cm^3}\), \(B_2=9.0\cdot10^7\ \mathrm{erg/cm^3}\). The values of the various \(\lambda\)'s are taken from Becker and Döring (1939, pp. 277–280).

Connection with the anisotropy energy. We now want to show that the presence of magnetostriction causes the appearance

a noticeable part of the anisotropy energy in the crystal. If experimental determinations of the anisotropy energy could be carried out at constant lattice dimensions, i.e., in a crystal with unchanged deformation, then the existence of magnetostriction should not affect the results of measuring the anisotropy. In practice, however, the anisotropy is measured at constant stresses, so that the lattice may be deformed under the action of magnetoelastic forces.

If, in the general expression for the energy (II, 3, 6), the deformations are eliminated by means of equations (II, 3, 9) and (II, 3, 10), and \(B\) is expressed through \(\lambda\), we finally obtain:

\[ f=(K+\Delta K)(\alpha_1^2\alpha_2^2+\alpha_2^2\alpha_3^2+\alpha_3^2\alpha_1^2), \tag{II, 3, 17} \]

where

\[ \Delta K=-\frac{9}{4}\left[(c_{11}-c_{12})\lambda_{100}^{2}-2c_{44}\lambda_{111}^{2}\right]. \tag{II, 3, 18} \]

For iron \(\Delta K=-7.5\cdot10^2\ \mathrm{erg/cm^3}\), so that here \(\dfrac{\Delta K}{K}\simeq10^{-3}\).

For nickel \(\Delta K\simeq2\cdot10^3\ \mathrm{erg/cm^3}\), whence \(\dfrac{\Delta K}{K}\simeq10^{-1}\).

Isotropic magnetostriction. For simplification it is often customary to assume \(\lambda_{100}=\lambda_{111}=\lambda\); this is the case of “isotropic magnetostriction.” For Ni one usually takes \(\lambda=-34\cdot10^{-6}\), for Fe \(\lambda=-7\cdot10^{-6}\), although the assumption of isotropy in neither case gives very good agreement with experimental results.

Expression (II, 3, 11) reduces to

\[ \frac{\delta l}{l}=\frac{3}{2}\lambda\left[(\alpha_1\beta_1+\alpha_2\beta_2+\alpha_3\beta_3)^2-\frac{1}{3}\right], \tag{II, 3, 19} \]

or

\[ \frac{\delta l}{l}=\frac{3}{2}\lambda\left[\cos^2\theta-\frac{1}{3}\right], \tag{II, 3, 20} \]

where \(\theta\) is the angle between the magnetization and the direction along which \(\delta l\) is determined. It can be seen that this expression does not depend on the directions of the crystal axes and therefore is isotropic.

It is very important to calculate the change in anisotropy energy caused by a homogeneous tensile stress \(T\). The components of the stress relative to the crystal axes for a tension with direction cosines \(\gamma_1,\gamma_2,\gamma_3\) are \(P_{ik}=T\gamma_i\gamma_k\), which gives

\[ e_{xy}=-Ts_{44}\gamma_1\gamma_2,\qquad e_{xx}=-T\left[s_{11}\gamma_1^2+s_{12}(\gamma_2^2+\gamma_3^2)\right], \]

where \(s_{ik}\) are the elastic compliance coefficients. The terms in expression (II, 3, 4),

dependent on strain, are as follows:

\[ f_{\mathrm{my}}=-B_1T\left[(s_{11}-s_{12})(\alpha_1^2\gamma_1^2+\alpha_2^2\gamma_2^2+\alpha_3^2\gamma_3^2)\right]- \]
\[ - B_2Ts_{44}(\alpha_1\alpha_2\gamma_1\gamma_2+\alpha_2\alpha_3\gamma_2\gamma_3+\alpha_3\alpha_1\gamma_3\gamma_1). \qquad (\mathrm{II},\,3,\,21) \]

If we now take \(\lambda_{100}=\lambda_{111}=\lambda\), then from (II, 3, 16) we have:
\(B_2(c_{11}-c_{12})=2B_1c_{44}\), so that, using the well-known relation between \(c\) and \(s\) and the relation \(\cos\theta=(\alpha_1\gamma_1+\alpha_2\gamma_2+\alpha_3\gamma_3)\), where \(\theta\) is the angle between the magnetization and the tension, we also obtain:

\[ f_{\mathrm{my}}=\frac{3}{2}\lambda T\sin^2\theta. \qquad (\mathrm{II},\,3,\,22) \]

Terms independent of \(\theta\) have been omitted. We shall use this expression in Section VI, 2.

II, 4. Magnetostatic energy

We shall not here enter into a detailed discussion of the question of magnetic energy, since this would take us too far afield. We shall discuss only a few particular questions directly related to the theory of domains. A more general treatment may be found in the papers of Guggenheim (1936) and Fokker (1939).

Here we shall give the following relations:

1) The energy of interaction of a permanent magnet with a uniform external magnetic field is

\[ f_{\mathrm{mag}}=-\mathbf{I}\cdot\mathbf{H} \qquad (\mathrm{II},\,4,\,1) \]

per unit volume.

2) The self-energy of a permanent magnet in its own field:

\[ f_{\mathrm{mag}}=-\frac{1}{2}\mathbf{I}\cdot\mathbf{H} \qquad (\mathrm{II},\,4,\,2) \]

per unit volume; for an ellipsoid this energy may be written in the form:

\[ f_{\mathrm{mag}}=\frac{1}{2}NI^2 \qquad (\mathrm{II},\,4,\,3) \]

per unit volume, where \(N\) is the demagnetizing factor; for parallel plates with poles of alternating sign (see Fig. 21):

\[ \sigma_{\mathrm{mag}}=0.8525\,I_s^2D \qquad (\mathrm{II},\,4,\,4) \]

per unit surface area, where \(D\) is the width of a plate.

3) The influence of finite anisotropy energy on expression (II, 4, 4), which is applicable only to the case of infinitely large anisotropy energy (completely “frozen” spins),

for the sliding angles of the magnetization vector with the surface is as follows:

\[ \sigma_{\mathrm{mag}}=\left[\frac{2}{1+\mu^{*}}\right]\left(0.8525\, r_s^2 D\right);\qquad \mu^{*}=\frac{1+2r_s^2}{K}. \tag{II, 4, 5} \]

Interaction of a permanent magnet with an external field. It is well known that the interaction energy of a permanent magnetic dipole \(\mu\) with an external field \(\mathbf H\) is equal to \(-\mu H\). The same result is valid for a rigid system of dipoles, so that the density of magnetic energy is equal to:

\[ f_{\mathrm{mag}}=-\mathbf I\cdot \mathbf H. \tag{II, 4, 6} \]

Self-energy of a permanent magnet. When the field against which work is done is not external, but is caused by the magnetization itself, the usual factor \(1/2\) appears. This factor is introduced because, when expression (II, 4, 6) is used to calculate the self-energy, each dipole is counted twice—once as a source of the field and once as a magnet in the field. The correct result is:

\[ f_{\mathrm{mag}}=-\frac{1}{2}\mathbf I\cdot \mathbf H. \tag{II, 4, 7} \]

Fig. 21. Model for calculating the energy of the magnetic field in the case of parallel bands with alternating poles.

Ellipsoidal specimen. If the specimen has the form of an ellipsoid and is magnetized along one of the principal axes, the self-field is equal to:

\[ H=-NI, \tag{II, 4, 8} \]

where \(N\) is the demagnetizing factor. Values of \(N\) in a convenient form are given by Osborn (1945). Expression (II, 4, 7) now takes the form:

\[ f_{\mathrm{mag}}=\frac{1}{2}NI^2. \tag{II, 4, 9} \]

This expression is often used.

Distribution of poles on a plane. Let us consider the energy of the magnetic field in the case of coplanar poles of alternating sign (Fig. 21). Let the plane of the poles be the \(x,y\) plane, with the \(y\)-axis parallel to the axes of the poles; the width of one band is \(D\), and the pole strength per unit surface of the band is \(I\). From (II, 4, 7), the magnetic energy per unit surface is

\[ \sigma=-\frac{1}{2}\int \mathbf H\cdot \mathbf I\,dz. \tag{II, 4, 10} \]

The vertical, or \(z\)-component, of the magnetic field directly under the plane of the poles is given by the Fourier expansion for a “square wave” with amplitude \(-2\pi I\). The approximate solution of Laplace’s equation is as follows:

\[ H_z=\mp 2\pi I\left[\frac{4}{\pi}\sin kx e^{kz}+\text{terms containing overtones }(2n+1)k\right], \tag{II, 4, 11} \]

where \(n=1,2,3,\ldots\)

At first we shall neglect the overtones in this expansion. Then

\[ \sigma=4I^2\int_0^{-\infty} e^{kz}dz\left<|\sin kx|\right>. \]

The mean value of \(|\sin kx|\) is equal to \(\dfrac{2}{\pi}\), so that

\[ \sigma=\frac{8I^2D}{\pi^2}. \tag{II, 4, 12} \]

This expression has been obtained taking into account only first-order terms.

The complete expression, including the effect of the overtones, is obtained by multiplying by \(\sum n^{-3}\), where the summation is over odd integers, and the sum is approximately equal to 1.0517. We finally have:

\[ \sigma_{\mathrm{mag}}=0.8525\, I^2D. \tag{II, 4, 13} \]

In an analogous way we can readily consider the energy associated with an arbitrary periodic distribution of poles on the plane. Let \(\rho(x,y)\) be the surface density of the poles, assumed to be periodic on a rectangle with sides \(2\pi L_x\) and \(2\pi L_y\). Then \(\rho\) can be expanded in a double Fourier series:

\[ \rho(x,y)=\sum_{-\infty}^{\infty}\sum_{-\infty}^{\infty} C_{mn}\exp\{i(m\xi+n\eta)\}, \tag{II, 4, 14} \]

where \(\xi=\dfrac{x}{L_x};\quad \eta=\dfrac{y}{L_y}\), and

\[ C_{mn}=\frac{1}{4\pi^2}\int^{2\pi}\int^{2\pi}\rho(\xi,\eta)e^{-i(m\xi+n\eta)}\,d\xi\,d\eta. \tag{II, 4, 15} \]

The density of the surface energy is equal to:

\[ \sigma_{\mathrm{mag}}=\pi\sum_{-\infty}^{\infty}\sum_{-\infty}^{\infty} C_{mn}C_{-m,-n}P_{mn}, \tag{II, 4, 16} \]

where

\[ P_{mn}=\left[\left(\frac{m}{L_x}\right)^2+\left(\frac{n}{L_y}\right)^2\right]^{\frac12}. \tag{II, 4, 17} \]

Thus we find that the energy of the checkerboard distribution is equal to:

\[ \sigma_{\mathrm{mag}}=0.53 I^{2}D, \tag{II, 4, 18} \]

where \(D\) is the side of each small square.

For a circle of one polarity, inscribed in a square of the opposite polarity,

\[ \sigma=0.374 I^{2}D, \tag{II, 4, 19} \]

where the function \(\rho\) is assumed to be periodic on the square of side \(D\), in which a circle of radius

\[ \frac{D}{(2\pi)^{1/2}} \]

is inscribed.

Correction \(\mu^{*}\). The question of the energy of a pole located on a plane surface is not as simple as it may seem at first sight. The complication is connected with the fact that the spins are not in fact “frozen” along the easy magnetization directions, but can deviate from this direction under the action of the field caused by the presence of the poles. And only for very large values of the spin-anisotropy energy can the spins be regarded as “frozen” along the easy directions.

The corrections to the magnetostatic energy that must be made in this connection depend on the nature of the problem. Various cases were discussed by Lifshitz, Néel, and Shockley. We shall follow Shockley (1948).

Fig. 22. Model for calculating the \(\mu^{*}\)-correction.

Consider the case of parallel plates with alternating-sign poles \(\pm I_s\sin\theta\), where the easy axes make a small angle \(\theta\) with the surface of the crystal, as shown in Fig. 22. The change in the magnetization under the influence of the magnetic field can be expressed with the aid of three permeabilities \(\mu_x\), \(\mu_y\), and \(\mu_z\). Here \(\mu_y \simeq 1\), since the magnetization in the \(y\) direction cannot be appreciably increased; \(\mu_x \simeq \mu_z\) from symmetry considerations.

The first problem consists in finding the field distribution in a medium with permeabilities \((\mu,1,\mu)\) under the condition that the discontinuity of \(H_z\) at the surface must be equal to \(\pm 4\pi I_s\sin\theta\). Suppose that \(\varphi(x,z)\) is the solution of the potential problem for \(\mu=1\). For real-

of the problem we shall assume that the potential is \(A\varphi(x,\alpha z)\) for \(z>0\) and \(A\varphi(x,\beta z)\) for \(z<0\). Then the equality of the surface charges in the two problems leads to

\[ A\alpha\left(\frac{\partial\varphi}{\partial z}\right)_{z=0+} +\mu A\beta\left(\frac{\partial\varphi}{\partial z}\right)_{z=0-} = 2\left(\frac{\partial\varphi}{\partial z}\right)_{z=0+}, \tag{II, 4, 20} \]

or

\[ A=\frac{2}{\alpha+\beta\mu}. \tag{II, 4, 21} \]

The condition \(\operatorname{div}\mathbf{E}=0\) gives the relation

\[ \varphi_{xx}+\beta^2\varphi_{zz}=0, \tag{II, 4, 22} \]

which is satisfied for \(\beta=1\); similarly \(\alpha=1\). Thus,

\[ A=\frac{2}{1+\mu}. \tag{II, 4, 23} \]

Since the potential is proportional to \(A\), the magnetic energy is also proportional to \(A\).

Let us now find the proper value of \(\mu\), i.e. the effective permeability for a small displacement about the easy axis. For cubic and uniaxial crystals \((K>0)\) we have:

\[ f_K \simeq K\varphi^2, \]

where \(\varphi\) is the angle (assumed small) between the magnetization vector and the easy axis. If the magnetic field acts perpendicular to the easy axis,

\[ f_{\mathrm{mag}}=-HI_s\varphi. \tag{II, 4, 24} \]

The total energy \(K\varphi^2-HI_s\varphi\) is minimal if

\[ 2K\varphi-HI_s=0, \tag{II, 4, 25} \]

so that

\[ \varphi=\frac{HI_s}{2K}. \tag{II, 4, 26} \]

Now the magnetization parallel to \(H\) is \(I_s\varphi\), and therefore the magnetic susceptibility is

\[ \chi=\frac{I_s\varphi}{H}=\frac{I_s^2}{2K}, \tag{II, 4, 27} \]

whence

\[ \mu^*=1+\frac{2\pi I_s^2}{K}. \tag{II, 4, 28} \]

The symbol \(\mu^*\) is usually used to denote the effective permeability associated with anisotropy. The values of \(\mu^*\): 46 for Fe and 3.6 for Co.

III. BLOCH WALL

III, 1. Introductory remarks

A “Bloch wall” is the name given to a transition layer that separates adjacent domains magnetized in different directions. This layer was first considered by F. Bloch (1932); the further development of the theory of the transition layer was carried out by Landau and Lifshitz (1935), Lifshitz (1944), Néel (1944, a), and Gerring and Kittel (in press).

The basic idea of the Bloch wall is that the change of spin directions between domains magnetized in different

Fig. 23. Bloch wall.

directions does not occur in a single jump at some atomic plane. The change of direction will rather take place gradually over a length containing many atomic spacings (Fig. 23). The reason for the gradual transition is that, for a given complete change of spin direction, the exchange energy will be lower when the change is distributed over many spins than when it occurs by a jump.

Such behavior can be understood from the expression (II, 1, 6):

\[ w_{\mathrm{ex}} = J S^2 \varphi^2 \tag{III, 1, 1} \]

for the exchange energy between two spins turned through a small angle \(\varphi\) relative to one another; here \(J\) is the exchange integral and \(S\) is the spin angular momentum, expressed in units of \(\frac{h}{2\pi}\).

Let the total required change in angle be \(\varphi_0\); if the change occurs in \(N\) equal steps, then the change in angle between neighboring spins will be \(\frac{\varphi_0}{N}\), and the exchange energy between each pair of neighboring atoms is equal to:

\[ w_{\text{ex}} = JS^2 \left( \frac{\varphi_0}{N} \right)^2 . \tag{III, 1, 2} \]

The total exchange energy of a line of \(N+1\) atoms is equal to:

\[ W_{\text{ex}} = \frac{JS^2 \varphi_0^2}{N}. \tag{III, 1, 3} \]

If the total change in angle between domains is \(\varphi_0 = \pi\), which corresponds to reversal of the direction of magnetization upon passing through the layer, then the exchange energy of a row of atoms for a layer thickness of 100 atoms will be of the order of \(\frac{kT_c}{100}\), as compared with the energy \(kT_c\) for a layer one atom thick.

Since the exchange energy of the layer is inversely proportional to its thickness (III, 1, 3), the layer could extend over a considerable part of the crystal were it not for the restraining influence of the anisotropy energy, which tends to reduce the width of the transition layer. The point is that the spins contained within the layer are strongly deflected from the axes of easy magnetization, so that a definite anisotropy energy is associated with the layer. The magnitude of this energy will be approximately proportional to the thickness of the layer, since the thickness is a measure of the total volume deflected from the easy-magnetization axis.

The real thickness and energy of the transition layer are the result of equilibrium between the competing influences of exchange energy and anisotropy energy: the first of these tends to increase the thickness of the layer, and the second to reduce it.

III, 2. Estimate of the Thickness and Energy of the Bloch Layer

We shall proceed to establish a rough estimate of the thickness and energy of the Bloch layer, postponing for the time being the detailed consideration of particular cases and the comparison of theoretical estimates with experimental results.

Suppose that the layer is parallel to a face of the cube of a simple cubic lattice and separates domains magnetized in opposite directions, as shown in Fig. 24. We wish to determine the thickness of the layer as a function of the number \(N\) of atomic planes contained in it, and also to determine the energy per unit surface \(\sigma_w\).

The energy per unit surface of the layer may, to a good approximation, be represented as the sum of the exchange energy and the energy

anisotropy:

\[ \sigma_w=\sigma_{\text{ex}}+\sigma_{\text{anis}} \tag{III, 2, 1} \]

For each chain of atoms passing through the layer perpendicular to its surface, the exchange energy is approximately given by expression (III, 1, 3). There are \(1/a^2\) such chains per unit area, where \(a\) is the lattice constant; hence

\[ \sigma_{\text{ex}}=\frac{\pi^2 J S^2}{N a^2}. \tag{III, 2, 2} \]

The anisotropy energy will be of the order of the anisotropy constant multiplied by the volume, i.e.

\[ \sigma_{\text{anis}}\simeq KNa \tag{III, 2, 3} \]

Fig. 24. A \(180^\circ\) layer, parallel to a cube face in a cubic crystal, separating domains magnetized in opposite directions parallel to a cube axis.

Thus

\[ \sigma_w\simeq \frac{\pi^2 J S^2}{N a^2}+KNa. \tag{III, 2, 4} \]

This expression is minimal with respect to \(N\) when

\[ \frac{\partial \sigma_w}{\partial N}=0=-\frac{\pi^2 J S^2}{N^2 a^2}+Ka \tag{III, 2, 5} \]

or

\[ N=\left[\frac{\pi^2 J S^2}{K a^3}\right]^{\frac12}. \tag{III, 2, 6} \]

We have obtained the result according to which the thickness of the layer, expressed in atomic distances, is approximately equal to the square root of the ratio of the exchange integral to the anisotropy energy per elementary cell. The order of magnitude for iron is

\[ N\simeq \left[\frac{kT_c}{K a^3}\right]^{\frac12} = \left[\frac{10^{-13}}{10^5\cdot 10^{-23}}\right]^{\frac12} \simeq \]

\[ \simeq 300\ \text{lattice constants}\simeq 1000\ \text{\AA} \]

The total energy of the layer per unit surface is

\[ \sigma_w=2\pi\left[\frac{JKS^2}{a}\right]^{\frac12}; \tag{III, 2, 7} \]

for iron, in order of magnitude:

\[ \sigma_w=\left[\frac{kT_cK}{a}\right]^{\frac12}\simeq \left[\frac{10^{-13}\cdot 10^5}{10^{-8}}\right]^{\frac12} \simeq 1\ \text{erg}/\text{cm}^2 . \]

We see that the contributions of the exchange energy and the anisotropy energy are approximately equal to one another.

In the estimate given, we used the arbitrary assumption that each of the \(N\) atoms of the chain passing through the layer participates equally in the complete change of spin direction. We also used a very rough estimate of the anisotropy energy of the system of spins inside the layer. These assumptions will be discarded in the more rigorous calculations that follow.

III, 3. 180° layer in the (100) plane of iron

Let us now examine in detail the important case of a layer parallel to the (001) plane of iron and separating domains magnetized in opposite directions. The magnetization directions of the domains may be the directions \([100]\) and \([\overline{1}00]\), as shown in Fig. 24. Equivalent solutions for this case were given by Lifshitz (1944) and Néel (1944, a).

We shall assume that the rotation of the spin directions on passing through the layer is such that the spin directions lie in the plane of the layer. This result, for the present particular case, is a consequence of the more general requirement that the normal component of the magnetization remain constant in the layer and that no poles be formed in it. The absence of poles follows from considerations of the minimum magnetostatic energy. Let us note that the magnetostatic energy of a double layer of thickness \(1000\,\text{Å}\), with surface density of poles \(\pm I_s\) per unit surface, is equal to:

\[ \sigma_{\text{mag}}=2\pi I_s^2 d\simeq (10)(10^6)(10^{-5})\simeq 100\ \text{erg}/\text{cm}^2 . \tag{III, 3, 1} \]

This quantity considerably exceeds the layer energy \(\sigma_w\simeq 1\ \text{erg}/\text{cm}^2\), estimated above under the tacit assumption that the changes in spin directions occur in such a way that the normal component of the magnetization remains constant when passing through the layer.

The calculation of the layer characteristics will first be carried out neglecting the magnetoelastic energy; the effect of the magnetoelastic energy contained in the layer will be considered separately.

Let \(\theta\) be the angle between the spin direction and the \(x\)-axis. Then the anisotropy energy density in the \(x,y\) plane, according to (II, 2, 4)

is equal to:

\[ f_K=K(\alpha_1^2\alpha_2^2+\alpha_1^2\alpha_3^2+\alpha_2^2\alpha_3^2)=K\sin^2\theta\cos^2\theta, \tag{III,3,2} \]

since \(\alpha_3=0\).

The density of the exchange energy (see II, 1, 11)

\[ f_{\text{ex}}=A\left[(\nabla\alpha_1)^2+(\nabla\alpha_2)^2+(\nabla\alpha_3)^2\right] \]

takes the form:

\[ f_{\text{ex}}=A\left(\frac{d\theta}{dz}\right)^2, \tag{III,3,3} \]

since \(\alpha_1=\cos\theta,\ \alpha_2=\sin\theta,\ \alpha_3=0\).

The energy of the layer per unit area:

\[ \sigma_w=\int_{-\infty}^{\infty} \left[ K\left(\sin^2\theta\cos^2\theta\right) + A\left(\frac{d\theta}{dz}\right)^2 \right]dz. \tag{III,3,4} \]

Putting \(g(\theta)=K\sin^2\theta\cos^2\theta\) and \(\theta'=\dfrac{d\theta}{dz}\), we can write \(\sigma\) as:

\[ \sigma=\int_{-\infty}^{\infty} \left[g(\theta)+A\theta'^2\right]dz. \tag{III,3,5} \]

The angle \(\theta\) in expression (III, 3, 5) is determined as a function of \(z\) from the requirement that the integral (III, 3, 5) be minimal. We therefore require that the variation \(\delta\sigma_w\) be identically equal to zero for any small variations \(\delta\theta\):

\[ \delta\sigma_w= \int_{-\infty}^{\infty} \left[ g'(\theta)\delta\theta + 2A\theta'\frac{d(\delta\theta)}{dz} \right]dz =0. \tag{III,3,6} \]

Integrating by parts and noting that

\[ \theta'\frac{d}{dz}(\delta\theta) = \frac{d}{dz}(\theta'\delta\theta) - \delta\theta\left(\frac{d\theta'}{dz}\right) \]

and that \(\theta'\delta\theta\) vanishes at both limits, we have:

\[ \delta\sigma_w= \int_{-\infty}^{\infty} \left[ g'(\theta)-2A\frac{d\theta'}{dz} \right]\delta\theta\,dz =0. \tag{III,3,7} \]

This equation can be identically satisfied for all \(z\) only if:

\[ g'(\theta)-2A\frac{d\theta'}{dz}=0. \tag{III,3,8} \]

This is the Euler equation for our problem.

Multiplying by \(\theta'\) and integrating with respect to \(z\) between \(-\infty\) and \(z\), we find:

\[ g(\theta)=A\left(\frac{d\theta}{dz}\right)^2, \tag{III,3,9} \]

since for \(z=-\infty\), \(\theta'=0\). This solution shows that at each point of the layer the local density of anisotropy energy \(g(\theta)\) is equal to the local density of exchange energy \(A\left(\dfrac{d\theta}{dz}\right)^2\). It follows from this that, in directions with high anisotropy energy, neighboring spins form larger angles with one another than in directions with low anisotropy energy.

From expression (III, 3, 9) we have:

\[ dz=\sqrt{A}\,\frac{d\theta}{(g(\theta))^{1/2}}, \tag{III, 3, 10} \]

so that expression (III, 3, 5) takes the form:

\[ \sigma_w = 2\sqrt{A}\int_{\theta_1}^{\theta_2} (g(\theta))^{1/2}\,d\theta = \]

\[ = 2(KA)^{1/2}\int_{0}^{\pi} |\sin\theta\cos\theta|\,d\theta, \tag{III, 3, 11} \]

which gives as a result:

\[ \sigma_w=2(KA)^{1/2}. \tag{III, 3, 12} \]

Energy of the layer. For iron, according to (II, 1, 18), \(A=2.0\cdot10^{-6}\) erg/cm and \(K_1=4.2\cdot10^5\) erg/cm\(^3\). Thus

\[ \sigma_w(\mathrm{Fe})=1.8\ \text{erg/cm}^2. \tag{III, 3, 13} \]

This result is very important, and we shall therefore make several remarks concerning the plausibility of the value (III, 3, 13) for the energy of a \(180^\circ\) transition layer in the plane \((001)\):

1) We have neglected the anisotropy constant \(K_2\), since the energy term \(K_2\alpha_1^2\alpha_2^2\alpha_3^2\) in the plane \((001)\) is equal to zero.

2) Néel (1944), for the same problem, arrived at the value \(\sigma_w=1.4\) erg/cm\(^2\), using the value of the exchange-interaction constant \(A\) obtained by another method (from the Curie temperature on the basis of the concept of the Weiss field). Any estimate of \(A\) must be regarded as approximate, but we shall assume that, in calculating the energy of the layer, the value obtained from Bloch’s \(T^{3/2}\) law should be used; this conclusion is connected with the fact that the physical situation in spin waves is similar to the situation occurring in a Bloch layer.

3) It will be shown below that the influence of the magnetoelastic energy on the numerical value of the layer energy in iron (III, 3, 13) may be neglected, despite the fact that magnetostriction has a noticeable influence on the thickness of the layer.

4) For a 3.85% SiFe alloy we have approximately \(A=1.7\cdot 10^{-6}\) erg/cm and \(K=2.8\cdot 10^{5}\) erg/cm\(^3\), so that \(\sigma_w=1.4\) erg/cm\(^2\). This value agrees with the rather rough experimental value obtained by Williams, Bozorth, and Shockley (1949) on the basis of observations of domain patterns.

Fig. 25. Change in the direction of the alloys in a 90° Bloch layer.

Layer thickness. The thickness of the layer can be found with the aid of expression (III, 3, 10):

\[ dz=-\left(\frac{A}{K}\right)^{\frac12}\frac{d\theta}{\sin\theta\cos\theta} \tag{III, 3, 14} \]

or

\[ z-z_0=\left(\frac{A}{K}\right)^{\frac12}\ln\left(\frac{\tg\theta}{\tg\theta_0}\right). \tag{III, 3, 15} \]

Taking \(z=0\) at \(\theta_0=45^\circ\), we have

\[ z=\left(\frac{A}{K}\right)^{\frac12}\ln\tg\theta . \tag{III, 3, 16} \]

This equation is shown in Fig. 25 for values \(0<\theta<\frac{\pi}{2}\). The coordinate \(z\) is plotted in characteristic units of length

\[ \left(\frac{A}{K}\right)^{\frac12}, \]

and \(\left(\frac{A}{K}\right)^{\frac12}\) for iron is equal to \(2.3\cdot 10^{-6}\) cm, or 230 Å. A change of angle by \(70^\circ\) occurs for \(\Delta z \cong 3.5\left(\frac{A}{K}\right)^{\frac12}\), or \((\Delta z)_{70^\circ}\cong 800\) Å \(\cong 280\) lattice constants. The angle between neighboring spins will be of the order of \(1/4^\circ\).

The expression (III, 3, 16), however, does not give a finite thickness for a 180° layer, since the layer tends to split into two 90° layers separated by an infinite distance. This difficulty is to some extent fictitious and is eliminated when the influence of magnetostriction is taken into account. A large domain turned through 90° can form between two antiparallel domains only at the cost of considerable magnetoelastic energy. This is connected with the fact that the domain elongates in the direction of its magnetization; consequently, a domain turned through 90° cannot be situated between antiparallel domains without the formation of a system of stresses. As a result, the 90° region between two 90° layers will disappear and both 90° layers will merge into one 180° layer.

Influence of magnetostriction. In solving the problem of the layer, Néel and Lifshitz took into account the influence of magnetoelastic energy. We shall here follow Lifshitz’s treatment*).

As will be shown, the magnetostriction effect can be taken into account by adding to the anisotropy energy density the term:

\[ f_{\mathrm{my}}=\frac{9}{4}(c_{11}-c_{12})\lambda_{100}^{2}\sin^{2}\theta, \tag{III, 3, 17} \]

where \(\lambda_{100}\) is the value of the longitudinal magnetostriction in the \([100]\) direction at saturation; \(c_{11}\) and \(c_{12}\) are elastic moduli. Let us prove this assertion.

Owing to magnetostriction, the part of the crystal magnetized in the direction \(+x\) is deformed as follows (see II, 3, 9):

\[ e_{xx}=-B_1\frac{c_{11}-c_{12}}{(c_{11}-c_{12})(c_{11}+2c_{12})}, \tag{III, 3, 18} \]

\[ e_{yy}=e_{zz}=B_1\frac{c_{12}}{(c_{11}-c_{12})(c_{11}+2c_{12})}. \tag{III, 3, 19} \]

Here \(B_1\) is the magnetoelastic coupling constant, which is related to \(\lambda_{100}\) by the expression (II, 3, 16):

\[ \lambda_{100}=-\frac{2B_1}{3(c_{11}-c_{12})}. \tag{III, 3, 20} \]

The deformation in the transition layer is determined by the deformation of the surrounding domains. The excess magnetoelastic energy of the substance of the layer, in comparison with the magnetoelastic energy of the domains, is equal (from (II, 3, 6)) to:

\[ \Delta f_{\mathrm{my}}=B_1\left[(\alpha_1^2-1)e_{xx}+\alpha_2^2 e_{yy}\right] =B_1\sin^2\theta\,(e_{yy}-e_{xx}). \tag{III, 3, 21} \]

*) Néel’s treatment is to a certain extent erroneous, since he does not distinguish between the anisotropy energy for a lattice of constant dimensions and that for a lattice with constant stresses (see, above (II, 3, 17)); this led him to include in the magnetoelastic energy a term that automatically enters into the experimental value of the anisotropy energy.

Substituting the values of \(B_1\), \(e_{yy}\), and \(e_{xx}\), we obtain:

\[ \Delta f_{\mathrm{mu}}=\frac{9}{4}(c_{11}-c_{12})\lambda_{100}^{2}\sin^{2}\theta . \tag{III, 3, 22} \]

The layer thickness is still obtained from (III, 3, 10), where we must assume that

\[ g(\theta)=K\sin^{2}\theta\cos^{2}\theta+\frac{9}{4}(c_{11}-c_{12})\lambda_{100}^{2}\sin^{2}\theta, \tag{III, 3, 23} \]

whence

\[ -\frac{dz}{\left(\dfrac{A}{K}\right)^{\frac12}} = \frac{d\theta}{\left(\sin^{2}\theta\cos^{2}\theta+P\sin^{2}\theta\right)^{\frac12}}, \tag{III, 3, 24} \]

\[ P=\frac{9}{4}(c_{11}-c_{12})\frac{\lambda_{100}^{2}}{K}. \tag{III, 3, 25} \]

For iron \(P \simeq 2\cdot10^{-3}\), so that in iron the effect of the magnetoelastic energy on the energy of the layer is negligibly small,

Fig. 26. Polar diagram of changes in spin direction in Bloch’s 180° wall in iron. The coordinate \(z\) is perpendicular to the plane of the wall; the plane of the wall is the (100) plane.

The solution of equation (III, 3, 24) is as follows:

\[ \operatorname{sh} z\left(\frac{K(1+P)}{A}\right)^{\frac12} = -\left(\frac{1+P}{P}\right)^{\frac12}\operatorname{ctg}\theta, \tag{III, 3, 26} \]

where \(z\) is measured from the middle of the layer. In the middle part of the layer, for \(P \ll 1\),

\[ \frac{d\theta}{d\left[\dfrac{z}{\left(\dfrac{A}{K}\right)^{1/2}}\right]} \simeq \sqrt{P}, \tag{III, 3, 27} \]

whence it again follows that for \(P=0\) a 90° domain may exist between the two parts of the layer. The changes of the angle are given in Fig. 26 for \(P=2\cdot 10^{-3}\), as is the case for iron.

Fig. 27. Approximate dependence of the energy and thickness of the layer on the energy of crystalline anisotropy. The value of the exchange integral is taken to be approximately the same as for iron (180° layer).

In Fig. 27 we give the approximate dependence of the energy and thickness of the layer on the energy of crystalline anisotropy. The exchange-energy constant is chosen the same as for iron:

\[ A=2\cdot 10^{-6}\ \text{erg}/\text{cm}. \]

IV. THEORETICAL DOMAIN STRUCTURES

IV, 1. Introduction

The sizes and shapes of domains are not a constant property of ferromagnetic materials, but depend on the sizes and orientation of the faces of the crystal, and also on deformations and on the strength of the magnetic field. Domains are formed because, in this process, the magnetic energy is, generally speaking, lowered.

The appearance of domains in a given specimen is determined by the magnitude of the demagnetizing effect that occurs when the specimen consists of a single domain, as compared with the case in which it has a saturated configuration of domains. For example, all the arrangements shown in Fig. 28 are stable. In case a) we have a long, strongly elongated spheroid with the direction of easy magnetization parallel to the axis of the figure; for a sufficiently large axial ratio a single domain will have a lower total energy than a domain structure. Thus the decrease in magnetic energy,

Fig. 28. Stable saturated configurations: a—a long ellipsoid, parallel to the easy axis; b—a hollow rectangle with sides parallel to the easy axes; c—a very small particle.

Fig. 29. Examples of single crystals whose shape favors the formation of a domain structure: a—a hollow rectangular specimen with sides parallel to the easy axis [110] and to its equivalents; b—a crystal of rectangular cross-section with the axes shown in the figure.

accompanying the appearance of a domain structure, in this case may be less than the energy required for the formation of the necessary transition layers (Bloch layers) between domains.

In case b) we have a structure with 4 domains. Here there is no magnetic field, since there are no poles (except for a weak pole at the boundary of a Bloch layer). Poles, in turn, are absent because the normal component of the magnetization is continuous at the diagonal Bloch layers, shown by the dotted lines. Crystals with such a domain structure were obtained by Williams and Shockley (1949).

In the case of Fig. 28, c) we have a very small particle with a diameter of the order (for iron) of 100 Å or less. Here a structure consisting of a single domain is stable, since the number of exchange-

of energy required for the formation of a nonmagnetic configuration will exceed the magnetic energy of a configuration consisting of a single domain. This inequality is valid only in the case of particles of very small dimensions. There is much experimental evidence for the stability of a single-domain structure in small particles. These will be discussed in the next section.

On the other hand, the crystal shapes shown in Fig. 29 are favorable for the formation of the domain structures indicated in the figure, provided that the crystals are large. Let us now consider the optimum thickness \(D\) of the domain structure shown in Fig. 29, b) for a uniaxial crystal having the form of a rectangular cylinder.

The area of the Bloch layer is equal to \(L/D\) per unit of the crystal surface visible from above. The energy of the layer will then be

\[ w_{\text{layer}}=\sigma_w\,\frac{L}{D} \tag{IV, 1, 1} \]

per unit surface, where \(\sigma_w\) is the surface-energy density of the Bloch layer. The energy of the magnetic field associated with parallel charged strips normal to the plane of the drawing, according to (II, 4, 4), is equal to

\[ w_{\text{mag}}=1.7 I_s^2 D \tag{IV, 1, 2} \]

per unit surface, with the surfaces of both bases of the crystal being taken into account. \(I_s\) in (IV, 1, 2) is the magnetic saturation. The dependence of \(w_{\text{layer}}\) and \(w_{\text{mag}}\) on \(D\) is given in Fig. 30. For large \(D\) the energy of the magnetic field predominates, while for small \(D\) the energy of the layer predominates.

Labels in Fig. 30: “Energy \(w\)”; “Magnetic energy”; “Position for which the sum of the energies is minimal”; “Layer energy”; “Domain thickness \(D\).”

Fig. 30. Energy of the domain structure as a function of the domain thickness \(D\).

The total energy per unit surface is equal to

\[ w=w_{\text{layer}}+w_{\text{mag}} \tag{IV, 1, 3} \]

or

\[ w=\sigma_w\,\frac{L}{D}+1.7 I_s^2 D. \tag{IV, 1, 4} \]

The energy will be minimal when

\[ \frac{\partial w}{\partial D} = -\left(\sigma_w\,\frac{L}{D^2}\right)+1.7 I_s^2=0. \tag{IV, 1, 5} \]

or

\[ D=\left[\sigma_w\,\frac{L}{1.7\,I_s^2}\right]^{\frac12}. \tag{IV, 1, 6} \]

Consequently, for crystals with \(L=1\ \mathrm{cm}\), the thickness \(D\) is of the order

\[ D \simeq \left[\frac{2\cdot 1}{1.7(1.7\cdot 10^3)^2}\right]^{\frac12}\simeq 10^{-3}\ \mathrm{cm}. \tag{IV, 1, 7} \]

The domain width \(D\) should not be confused with the thickness of the transition layer \(\delta\), which is of the order of \(10^{-5}\ \mathrm{cm}\).

The energy of the domain structure

\[ w=2[1.7\,I_s^2\sigma_w L]^{\frac12} \tag{IV, 1, 8} \]

will be of the order

\[ w\simeq 2[1.7\cdot(1.7\cdot10^3)^2\cdot2\cdot1]^{\frac12}\simeq 7\cdot10^3\ \mathrm{erg}/\mathrm{cm}^2. \tag{IV, 1, 9} \]

Since for the given crystal \(L=1\ \mathrm{cm}\) was adopted, the energy per unit volume will be of the order of \(7\cdot10^3\ \mathrm{erg}/\mathrm{cm}^3\), whereas the energy density of the magnetic field corresponding to a saturated single-domain structure will be of the order \(I_s^2\simeq10^6\ \mathrm{erg}/\mathrm{cm}^3\). This shows qualitatively that the formation of domains lowers the energy of the system by a very considerable amount.

If we take \(L\simeq10^{-6}\ \mathrm{cm}\), as in a thin film, then the energy density of the domain structure will also be of the order of \(10^6\ \mathrm{erg}/\mathrm{cm}^3\), i.e. of the same order as the density of the magnetic energy of a saturated film; thus, it is clear that dimensions play an important role for the domain structure.

IV, 2. Domain configurations with closed flux

For a rectangular plate one can indicate a domain structure without magnetic poles. Such a structure for a uniaxial crystal, shown in Fig. 31, was first analyzed by Landau and Lifshitz (1935). An analogous structure (Fig. 32, a) was observed by Williams on an iron crystal (unpublished work). Iron, however, is cubic.

The flux is completely closed inside the crystal with the aid of small triangular prisms situated on the upper and lower surfaces. As shown in Fig. 32, b, these “closure domains” transfer the flux of magnetization from one domain to another without the formation of poles. The absence of poles is the result of the continuity, at the prism boundaries, of the normal component of the magnetization.

![Schematic diagram labeled with “easy axes,” dimensions \(D\), \(d\), and \(L\), and alternating arrows indicating domain directions.]

Fig. 31. Domain configuration with a closed flux in a uniaxial crystal.

![Retouched photograph of closure domains with arrows; below it, a schematic labeled \(I_s\), \(Q\), and \(45^\circ\).]

Fig. 32. \(a\)—Retouched photograph of closure domains in an Si–Fe crystal (Williams); \(b\)—closure of the flux in “closure domains,” \(Q\).

K. KITTEL

Let us now turn to the calculation of the optimum domain width and the corresponding energy density.

The energy of the layer per unit surface area of the crystal is approximately equal to

\[ w_{\text{layer}}=\sigma_w \frac{L}{D}. \tag{IV, 2, 1} \]

The magnetic energy is zero, but the anisotropy energy is not zero (if the crystal is uniaxial). The volume enclosed within the closure domains is oriented in the direction of hard magnetization and contains, per unit volume, an energy \(K\), where \(K\) is the anisotropy constant. Per unit area of the crystalline surface on one side of the crystal there is a volume of closure domains, situated on two sides, equal to \(D/2\), so that

\[ w_{\text{anis}}=\frac{KD}{2}. \tag{IV, 2, 2} \]

The layer energy tends to increase the domain width, whereas the anisotropy energy tends to decrease it.

The total energy per unit area is equal to:

\[ w=\sigma_w \frac{L}{D}+\frac{KD}{2}. \tag{IV, 2, 3} \]

This expression is minimal when

\[ \frac{dw}{dD}=-\sigma_w \frac{L}{D^2}+\frac{K}{2}=0. \tag{IV, 2, 4} \]

The condition for the minimum is as follows:

\[ D=\left[2\sigma_w \frac{L}{K}\right]^{\frac12}, \tag{IV, 2, 5} \]

and the corresponding energy per unit area is equal to

\[ w=\left[2\sigma_w LK\right]^{\frac12}. \tag{IV, 2, 6} \]

The energy per unit volume is equal to

\[ f_{\text{domain}}=\left[2\sigma_w \frac{K}{L}\right]^{\frac12}. \tag{IV, 2, 7} \]

Substituting approximate values of the constants for iron and putting \(L=1\ \text{cm}\), we obtain:

\[ D=\left[\frac{2\cdot 2\cdot 1}{4\cdot 10^5}\right]^{\frac12}\approx 3\cdot 10^{-3}\ \text{cm} \tag{IV, 2, 8} \]

and

\[ f=\left[\frac{2\cdot 2\cdot 4\cdot 10^5}{1}\right]^{\frac12}\approx 1.3\cdot 10^3\ \text{erg}/\text{cm}^3. \tag{IV, 2, 9} \]

For these values of the various constants, the energy of the configuration with closed flux (Fig. 31) is lower than for the simple configuration of the type in Fig. 29, b), but as the ratio \(\dfrac{K}{I_s^2}\) increases the closing domains gradually open up, and for \(\dfrac{K}{I_s^2} \gg 1\) the simple configuration is more advantageous (Fig. 29, b)).

Cubic crystals with a positive anisotropy constant. In cubic crystals with anisotropy constant \(K > 0\), the directions of easy magnetization are the edges of the cube. Therefore the direction of magnetization in the closing domains may coincide with one of the easy directions, while the magnetization of the principal rectangular domains (Fig. 31) is directed along another easy direction. In this case it would seem that the domain width will increase until the whole specimen consists of 4 domains. Usually, however, this condition is not fulfilled because of the action of magnetostriction.

With closing domains, generally speaking, magnetostrictive energy is associated. This is the result of the tendency of domains to change slightly in length in the direction of magnetization, so that domains magnetized along different directions cannot be fitted closely to one another without expenditure of elastic energy. Closing domains, for example, may be regarded as compressed by the principal domains, as shown schematically in exaggerated form in Fig. 33.

In the case of closing domains, the deformation caused by the principal domains will be of the order of the longitudinal magnetostriction constant \(\lambda_{100}\). This may be understood from Sec. II, 3. The density of elastic energy of the closing domains is then approximately equal to

\[ f_{\mathrm{el}}=\frac{1}{2}c_{11}\lambda_{100}^{2} \tag{IV, 2, 10} \]

and will be of the order of \(500\ \mathrm{erg}/\mathrm{cm}^{3}\) (for iron).

For cubic crystals of this type the domain width is equal to

\[ D=\left[\frac{4\sigma_w L}{c_{11}\lambda_{100}^{2}}\right]^{\frac{1}{2}}, \tag{IV, 2, 11} \]

and the energy density of the domain structure is

\[ f_{\mathrm{dom}}=\left[\frac{\sigma_w c_{11}\lambda_{100}^{2}}{L}\right]^{\frac{1}{2}} . \tag{IV, 2, 12} \]

Numerically, for iron at \(L = 1\) cm,

\[ D \approx \left[\frac{4 \cdot 2 \cdot 1}{10^3}\right]^{\frac{1}{2}} \approx 0.1\ \text{cm} \tag{IV, 2, 13} \]

and

\[ f_{\text{domain}} \approx \left[\frac{2 \cdot 10^3}{1}\right]^{\frac{1}{2}} \approx 50\ \text{erg}/\text{cm}^3 . \tag{IV, 2, 14} \]

General case. It may be expected that in reality the domain structure near the surface of a crystal will be more complicated than in the simplest cases we have analyzed. For example, Lifshitz showed that the arrangement depicted in Fig. 34 has, under certain conditions, a lower energy than a configuration with triangular prisms. The complete variational problem of finding the domain structure with minimum total energy has not yet been solved; instead, on physical grounds we choose certain domain structures, after which we minimize the energy with respect to one or several characteristic parameters.

Fig. 33. Effect of magnetostriction on closure domains. The dashed lines indicate, on an exaggerated scale, the volume that would be occupied by the closure domain if the compression caused by the other parts of the crystal were eliminated.

Fig. 34. Branching of domains near the surface of a crystal according to Lifshitz. Such a structure may be expected in the case of high anisotropy; structures of this type have been observed experimentally.

Generally speaking, it is natural to assume, to a good approximation, that the total energy consists of two parts: the energy of the Bloch walls \(w_{\text{wall}}\) and the energy \(w_{\text{surf}}\), associated with the outer surfaces of the crystal. Suppose that for lamellar domains

\[ w_{\text{wall}} = \sigma_w \frac{L}{D}; \tag{IV, 2, 15} \]

where \(L\) is some characteristic mean length; further:

\[ w_{\mathrm{surf}}=g(D) \tag{IV, 2, 16} \]

is some function of the plate width. The nature of this dependence is determined by the structure of the surface domains. The energy is minimal for the plate width \(D_0\), when

\[ -\frac{\sigma_w L}{D_0^2}+g'(D_0)=0. \tag{IV, 2, 17} \]

It should be emphasized that the domain structures of most crystals are naturally divided into two classes: the surface domain structure, associated with features of the crystal surface, and the main domain structure, occupying the greater part of the specimen volume. Surface structures are often of very complex form (see, for example, Fig. 8), whereas the main domain structure in single crystals is in most cases apparently quite simple (see, for example, Fig. 7).

Fig. 35. Powder pattern on the hexagonal face of a cobalt single crystal (Williams).

The width of the main structure is determined by the surface-energy density of the surface structure \(g(D)\) and by the wall energy \(\sigma_w\). The nature and dimensions of the surface structure depend on the relative values of the magnetic energy, the anisotropy energy, and the magnetoelastic energy at the surface of the crystal.

In cobalt the anisotropy energy is predominant, and as a result the flux does not close completely. A typical domain pattern obtained on the hexagonal face of a cobalt crystal is shown in Fig. 35. L. H. Germer (1942) confirmed, by the electron-scattering method, the existence of strong local magnetic fields of the order of 10,000 oersteds precisely on the hexagonal face of a cobalt crystal. Electrons with an energy of 30 kilovolts, scattered from the hexagonal face, produced a very complex pattern on a photographic plate (Fig. 36). Crystals of iron and nickel do not give a similar picture, which indicates the weakness of the magnetic fields, as was to be expected for a domain structure with flux,

closed inside the crystal. The result obtained for cobalt, of course, indicates that in this case the flux is not closed inside the crystal. This conclusion is in agreement with theoretical predictions.

In the limiting case of very high anisotropy energy one may expect that the domains will be plate-like throughout the entire extent

Fig. 36. a—Apparatus for detecting surface magnetic fields by observing electron scattering; b—result obtained in the scattering of electrons from the hexagonal plane of a cobalt single crystal (Germer, 1942).

and that there will be no surface domains. In the other limiting case, when the anisotropy and magnetostriction are equal to zero, any semblance of a discrete domain structure disappears, and the requirement of flux closure is not taken into account. The exception is the case when the direction of magnetization changes over distances of the order of \(10^{-5}\ \mathrm{cm}\)—in this case there must also

the exchange energy must be taken into account. Two possible domain structures of this type are shown in Fig. 37. The domains grow in such a way that each domain occupies a large part of the volume of the crystal. Such a limiting case may be expected when the thickness of the Bloch layer becomes comparable with the dimensions of the crystal. For a thickness of \(10^{-2}\ \mathrm{cm}\), the anisotropy energy should be of the order of only \(1\ \mathrm{erg}/\mathrm{cm}^3\). It is quite possible that we approach such a situation in a thin tape of supermalloy (Bozorth, 1948).

Fig. 37. Theoretical possibilities for the domain structure in the limiting case of zero anisotropy.

We have not discussed in this section the question of the domain structure in the presence of an external magnetic field. Calculations in this case must be carried out in a manner analogous to that used above. The reader may find details in the articles by Néel (1944c) and Kholodenko (1947). Experimental confirmation of the field-dependence of the domain width predicted by the theory was given by Bates and Neale (1949) for silicon iron (the field was applied in the direction \([011]\)).

V. EXPERIMENTAL STUDY OF DOMAINS

In the preceding sections, the theoretical questions connected with the structure of ferromagnetic domains have been considered in detail. We now dwell on the experimental foundations of the theory.

Experimental data on the domain structure can be obtained from various sources. The most important of these is the method of magnetic powder patterns. The method of powder patterns was proposed by Bitter, Elmore, and others, and was widely developed in the work of Williams, Bozorth, and Shockley (1949) and of Williams and Shockley (1949). The principle of the method will be considered in Section V, 1, and the experimental results are given in Section V, 2. The reader may find details in the papers of Williams and his collaborators.

Other experiments that give information on the domain structure are the following:

a) Scattering of electron beams (Germer, 1942; Marton, 1948).

b) Depolarization of beams of polarized neutrons (Berdzhi, Hughes, and Wallace, 1948).

c) Domains in stressed wires (Sixtus and Tonks, 1933).

d) Dependence of magnetostriction and magnetic resistance on the applied field and stresses (Bozorth, 1946).

V, 1. Method of magnetic powder figures

The method of magnetic powder figures consists in applying a thin layer of liquid to a carefully prepared surface of a ferromagnetic specimen. The liquid contains a colloidal suspension of fine ferromagnetic powder. Usually magnetite is taken, with particle sizes of the order of one micron.

We now wish to show that, as a result of strong changes of the magnetic field at the boundaries of the Bloch layer, a strong local concentration of particles in the colloidal film is to be expected.

In the presence of an inhomogeneous magnetic field, a force acts on the magnetic particles, attracting them to those points of the surface of the specimen where the magnetic-field intensity is greatest. In fact this occurs as if the particles were in thermal equilibrium, so that the distribution of the density of particles in the liquid is determined by the Boltzmann formula. This means that the particle density \(p(H)\) at a point with field intensity \(H\) is related to the density \(p(0)\) at a point with zero field by the relation

\[ p(H,\theta)=p(0)\exp\left(\frac{\mu H\cos\theta}{kT}\right), \tag{V, 1, 1} \]

where \(\theta\) is the angle of the magnetic moment \(\mu\) of the particle with the field \(H\). Here we have assumed that the particle has a constant magnetic moment; we have also neglected the interaction between particles. Averaging \(p(H,\theta)\) over all angles \(\theta\), we obtain the mean value:

\[ p(H)=\frac{1}{4\pi}\int_{0}^{\pi}\exp\left(\frac{\mu H\cos\theta}{kT}\right)2\pi\sin\theta\,d\theta = \frac{\operatorname{sh}\frac{\mu H}{kT}}{\frac{\mu H}{kT}}. \tag{V, 1, 2} \]

A graph of the function \(p(H)=\dfrac{\operatorname{sh}x}{x}\), where \(x=\dfrac{\mu H}{kT}\), is given in Fig. 38. It can be seen that the particle density increases rapidly when \(x\) is of the order of 3 and greater, i.e. when \(\mu H>3kT\). Taking, for a rough estimate, the particle saturation magnetization \(I_s\) equal to 400 and the particle volume equal to \(10^{-12}\ \mathrm{cm}^3\), in order to obtain a noticeable change in density we must have a field change

\[ H>\frac{3kT}{\mu} = \frac{3(1.4\cdot 10^{-16})(3\cdot 10^{3})}{(4\cdot 10^{2})(10^{-12})} = 3\cdot 10^{-4}\ \text{oersted}. \]

For a particle volume of \(10^{-15}\ \mathrm{cm}^3\), corresponding to a diameter of approximately \(0.1\) micron, the changes in the magnetic field must exceed \(0.3\) oersted. The changes of the field at the surface of the crystal greatly exceed both \(0.0003\) and \(0.3\) oersted. On a crystal surface with a closed domain configuration (Fig. 31), there must exist local fields along the lines of intersection of the Bloch layers with the surface. Since the change of spin directions inside a Bloch layer occurs in the plane of the layer, it is evident that, if we cut the layers—as we must do at the surface of the crystal—the spins in the layer will have a component of magnetization normal to the surface of the crystal*). Thus, wherever Bloch layers emerge at the surface of the specimen, there will be lines of north or south poles. These lines of free poles create a magnetic field quite sufficient for the formation of dense lines of colloidal particles.

Fig. 38. The function \(\dfrac{\operatorname{sh} x}{x}\), determining the particle density as a function of \(x=\dfrac{\mu H}{kT}\), where \(\mu\) is the magnetic moment of one particle.

As a very rough estimate we may assume that, in iron, the perpendicular intersection of a layer with the surface of the crystal gives a pole with density \(I_s\) on a strip about \(10^{-5}\ \mathrm{cm}\) wide. The corresponding linear density of poles is

\[ g = 1700 \cdot 10^{-5} \simeq 0.02\ \text{gauss}/\mathrm{cm}. \]

At a distance of one micron from the layer this leads to the formation of a field

\[ H = 2g/r = 2(0.02)/10^{-4} = 400\ \text{oersted}. \]

This estimate shows that the field intensity created by the edge of a Bloch layer exceeds by several orders of magnitude the inten—

*) See Fig. 23.

ness necessary for the formation of dense lines of colloidal particles.

In fact, our estimate must be corrected by taking into account the \(\mu^{*}\)-effect considered in Section II, 4. This means that we must take into account the fact that the line of poles lies on a permeable material with permeability

\[ \mu^{*}=1+2\pi I_s^2/K . \]

The solution of the boundary-value problem for the field produced by a line of poles on a homogeneous permeable half-space shows that the field in air differs from the field in free space by the factor \(2/(1+\mu^{*})\), which for iron is equal to \(0.047\). The corrected field strength at a distance of one micron is approximately equal to 20 oersteds, which is also still quite sufficient for the formation of dense lines of colloidal particles, as is in fact observed.

The theory of line formation was also developed (Kittel, 1949 b) for the case of colloidal particles that are not permanent magnets, but permeable spheres, as occurs in the majority of cases.

A concentration of particles should also be observed in the absence of a domain configuration with closed flux, when the domains themselves emerge on the surface of the crystal. Such a situation should occur in materials with high anisotropy energy and is in fact usually observed in cobalt and on the (111) plane in iron.

We have shown that strong local concentrations of particles arise on the surface of a ferromagnetic specimen. These concentrations, with appropriate illumination of the specimen, can be observed under a microscope.

V, 2. Some results obtained by the powder-pattern method

Let us now discuss some results obtained in the study of powder patterns. We shall consider the following three questions:

1) Determination of the direction of magnetization by the scratch method (Williams, 1947).

2) The relation between the change in magnetization and the displacement of boundaries (Williams and Shockley, 1949).

3) Dagger-shaped domains around cavities and crystalline defects.

The scratch method. In the absence of a magnetic field, the directions of magnetization of the domains are directed along the easy axes of magnetization of the crystal. In an iron crystal there are three mutually

perpendicular axes of easy magnetization, directed along the edges of the cube.

In interpreting powder figures, we want to know the directions of magnetization in all parts of the figure. In some important cases we can determine the axis of magnetization by the so-called scratch method. The direction of magnetization can then be determined from observations of the expansion or contraction of a domain under the action of an external magnetic field parallel to its axis.

The scratch method can be understood at least from the example of the (001) surface of an iron crystal. The easy axes are parallel to the axes [100]

Magnetization perpendicular to the scratch

Magnetization parallel to the scratch

Fig. 39. Determination of the direction of magnetization by the scratch method.

and [010]. If we make a thin scratch on the surface, parallel to the direction [100], then colloidal precipitate from domains parallel to [010] will collect in the scratch, but the scratch will not collect colloidal particles from domains parallel to [100]. Thus, colloidal particles accumulate in the scratch when the magnetization crosses the scratch, and do not accumulate in it when the magnetization is parallel to the scratch. This is explained by Fig. 39. When the magnetization is perpendicular to the scratch, a leakage flux is produced which attracts the colloidal particles, whereas when the magnetization is parallel to the scratch, there is no leakage.

Domain figures demonstrating this effect are given in Fig. 40, obtained by Williams (1947).

Relation between changes in magnetization and displacement of boundaries. In Fig. 11 we gave a sketch of a single-domain structure found by Williams and Shockley (1949) in an iron single crystal (containing a small amount of silicon). Reversal of the magnetization in the circuit shown in Fig. 11, a, occurs by the formation and movement through the crystal of a Bloch layer having the form of a quadrilateral (see

Fig. 40. Diagram of a powder pattern on the (100) plane with scratches parallel to the [010] axis. In regions where the scratches are not visible, the magnetization is parallel to the [010] axis; in regions where the scratches are visible, the magnetization is directed along the [001] axis.

Fig. 11, b). The quadrilateral increases or decreases in size in accordance with the relative magnitude of the fluxes clockwise or counterclockwise.

If the layer just mentioned passes through the crystal as shown in Fig. 11, b, then one may expect a linear dependence between the change in magnetization and the displacement of the layer relative to the sides of the crystal. This conclusion is confirmed by measurements, the results of which are given in Fig. 41.

This experiment may be regarded as one of the most fundamental experiments in the field of ferromagnetism, since

it proves the existence of a direct dependence between the flux and the displacement of the layer.

Domain structure around voids and inclusions. Néel (1944, b) pointed out that the presence of voids or inclusions in magnetic materials should lead to the appearance of considerable magnetostatic energy associated with the formation of north and south poles on the opposite sides of the inclusion.

Fig. 41. a) Dependence of the magnetization on the displacement of the 180° Bloch layer; b) patterns showing this layer in three positions (Williams and Shockley (1949)).

On the graph: vertical axis \(B-H\); horizontal axis “Distance in mm”; curve points \(a\), \(b\), \(c\).

He further assumed that around such an inclusion or cavity a local domain structure of the type shown in Fig. 42 would form, which would lead to a decrease in the total energy. Néel’s structure leads to a distribution of poles along the curved portions of the domain boundaries. The elongation of the domains causes a decrease in the magnetostatic energy and an increase in the energy of the layers; in the structure realized experimentally, the sum of the two energies is minimal.

Williams (1947) found powder patterns of the expected type around holes in silicon-iron crystals.

Fig. 42. Domain structure around voids: a) structure predicted from energy considerations by Néel (1944, b); b) and c) figures observed experimentally by Williams (1947).

Measurement of the ratio of the length of Néel domains to their width makes it possible to determine experimentally the surface-energy density of the Bloch layer \(\sigma_w\). Values estimated in this way give the correct order of magnitude.

VI. MAGNETIC PROPERTIES OF SMALL PARTICLES

If a specimen is cooled from a temperature lying above the Curie point, in the absence of a field, then in large ferromagnetic crystals the demagnetized state is stable. In the demagnetized state the domains are oriented so that the magnetic flux is enclosed almost completely within the specimen, whose total magnetic moment is approximately equal to zero.

As the size of the specimen is reduced, the specific weight of the various energy terms in the total domain energy changes, with surface energies becoming more important than volume energies. The energy of the transition layer (Bloch layer) between domains is a surface energy, while the energy of the magnetic field (the proper magnetostatic energy) is a volume energy.

With decreasing size there comes a point at which the absence of transition layers is energetically most favorable; as a result the specimen becomes a single domain and acts as a permanent magnet. This was first predicted by Frenkel and Dorfman (1930), although in their note the surface energy of the boundary layer between domains was overestimated by approximately a factor of 50, which led to excessively large values of the critical dimensions sufficient for the formation of a single-domain structure. Improved and corrected calculations were published by Kittel (1946), and then by Néel (1947) and Stoner and Wohlfarth (1947, 1948).

Experimental proof of the existence of permanent magnetization in small ferromagnetic particles was first given by Elmore (1938, 1941), although the existence of the effect had been suspected earlier, for example by Antik and Kubyshkina (1934), on the basis of certain considerations connected with the magnitude of the coercive force.

The general agreement of experiment with theory in the field of small particles is not as complete as one might wish, but in many cases it is quite clear that the theory is on the right track. The question of complications connected with the clumping of particles has not yet been discussed sufficiently.

The single-domain structure of small particles may be of great practical interest from the standpoint of materials for permanent magnets. This is connected with the very high values of the coercive force in single-domain structures.

Analogously, a single-domain structure may be expected in very thin films (Kittel, 1946); experimental proof of this proposition is given in the work of Drigo and Pizzo (1948), devoted to the Barkhausen effect in thin films of Fe, Ni, and Co. They found that the Barkhausen effect disappears when the film thickness is reduced to \(10^{-5}\) cm, which is in complete agreement with the theoretical values of the critical thickness for the appearance of a single-domain structure.

VI, 1. Critical particle sizes at which a single-domain structure appears

Let us consider a small spherical ferromagnetic particle of radius \(R\); for definiteness we shall regard the particle as a single crystal. First of all we are interested in the critical dimensions of the particle for which the energy of the single-domain configuration is lower than the energy of a domain configuration close to the configuration with closed flux.

The energy density of the saturated single-domain configuration is the magnetostatic energy density, equal for a sphere to: (see (II, 4,3))

\[ f=\frac{1}{2}NI_s^2=\frac{2\pi I_s^2}{3}. \tag{VI, 1,1} \]

The numerical value for iron is approximately \(6\cdot 10^6\) erg/cm\(^3\). For a sphere of radius \(R\) the energy

\[ w=fV=\frac{1}{2}\left(\frac{4\pi}{3}\right)^2 R^3 I_s^2 \tag{VI, 1,2} \]

and is approximately equal to \(24\cdot 10^6\) erg for \(R=1\) cm and \(20\cdot 10^{-12}\) erg for \(R=10^{-6}\) cm.

We must now consider the energy of simple domain configurations. If the anisotropy is small, it is natural, as the most acceptable, to choose the configuration with closed flux shown in Fig. 43, \(a\); if, however, the anisotropy energy is very large, one may expect the pattern shown in Fig. 43, \(b\) for a cubic crystal and in Fig. 43, \(c\) for a uniaxial crystal.

Let us now examine each of these three cases separately.

Low anisotropy. In this case the energy is mainly exchange energy. Consider the spins on a ring of radius \(r\). On the ring there are

\[ \frac{2\pi r}{a} \]

spins, where \(a\) is the length of the elementary cell. The total change of angle in one revolution is \(2\pi\), so that the angle \(\varphi\) between successive spins will be equal to

\[ \varphi=\frac{a}{r}. \tag{VI, 1,3} \]

and, according to (II, 1, 5), for \(S=1\)

\[ w_{\text{ring}}=\frac{1}{2}J\left(\frac{a}{r}\right)^2\frac{2\pi r}{a}=\frac{\pi Ja}{r}. \tag{VI, 1, 4} \]

Now let us consider a sphere consisting of circular cylinders (Fig. 44), each one elementary cell thick. The number of cells in a cylinder will be

\[ \left(\frac{2}{a}\right)(R^2-r^2)^{\frac12}, \]

so that

\[ w_{\text{cyl}}=\frac{2\pi J(R^2-r^2)^{\frac12}}{r}, \tag{VI, 1, 5} \]

\[ w_{\text{sph}}=\frac{2\pi J}{a}\int_a^R \frac{(R^2-r^2)^{\frac12}}{r}\,dr \simeq \frac{2\pi JR}{a}\left[\ln\frac{2R}{a}-1\right], \tag{VI, 1, 6} \]

or per unit volume

\[ f_{\text{vol}}=\frac{3}{2}\,\frac{J}{aR^2}\left[\ln\frac{2R}{a}-1\right]. \tag{VI, 1, 7} \]

This energy density depends on the size of the sphere. Substituting

Fig. 43. Simple domain configurations in a small sphere:
a) case of low anisotropy; b) case of high anisotropy in a cubic crystal; c) case of high anisotropy in a uniaxial crystal.

Fig. 44. Division of a sphere into circular cylindrical layers.

for iron the value

\[ \frac{J}{a}=2\cdot 10^{-6}\ \text{erg} \]

from (II, 1, 18), we have:

\[ \text{a) } R=1\ \text{cm}\qquad w_{\text{vol}}=1.3\cdot 10^{-4}\ \text{erg}, \]

\[ \text{b) } R=10^{-6}\ \text{cm}\qquad w_{\text{vol}}=23\cdot 10^{-12}\ \text{erg} \]

and

\[ f_{\text{vol}}=0.8\cdot 10^7\ \text{erg}/\text{cm}^3. \]

The values of \(w\) can be compared with the values calculated from (VI, 1, 2) for a saturated configuration. We see that a configuration with closed flux has a considerably lower

energy when the radius is equal to 1 cm, while when the radius is equal to \(10^{-6}\) cm, the saturated configuration has the lower energy.

Thus, for sufficiently small particles the saturated configuration has a lower energy than the configuration with a closed flux. The critical radius \(R_c\) is determined from

\[ \frac{1}{2}\left(\frac{4\pi}{3}\right)^2 R_c^3 I_s^2 = \frac{\pi J R_c}{a}\left[\ln\frac{2R_c}{a}-1\right] \tag{VI, 1, 8} \]

and is approximately inversely proportional to the magnetic saturation. The value of the critical radius for iron is \(\sim 10^{-6}\) cm.

An expression practically equivalent to (VI, 1, 8) was first proposed by Néel (1947, a).

High anisotropy, cubic crystal. In the preceding calculations the anisotropy energy associated with the configuration corresponding to a closed flux was assumed to be negligibly small in comparison with the exchange energy; this is the case if the critical radii are considerably smaller than the thickness of the Bloch wall, since the anisotropy energy and the exchange energy in a Bloch wall are equal, but if a change of spin directions is forced to take place over a distance smaller than the wall thickness, the exchange energy dominates.

If, however, the anisotropy energy is high, it is possible that the critical radius will appreciably exceed the wall thickness. When this condition is fulfilled, the critical radius may be calculated using the model shown in Fig. 43, \(a\). Here the energy essentially coincides with the energy of the wall and is equal to

\[ w_{\text{wall}} = 2\sigma_w \pi R^2 . \tag{VI, 1, 9} \]

For an iron particle with \(R = 10^{-6}\) cm we have \(w_{\text{wall}}\simeq 14\cdot 10^{-12}\) erg, and the value of the critical radius is about \(0.7\cdot 10^{-6}\) cm, since, according to (VI, 1, 9) and (VI, 1, 2),

\[ R_c=\frac{9}{4\pi}\frac{\sigma_w}{I_s^2}. \tag{VI, 1, 10} \]

This value of the critical radius is smaller than the wall thickness and, consequently, the calculations are not valid. The calculations will be applicable if the anisotropy energy is increased by a factor of 10 or more.

High anisotropy, uniaxial crystal. For the model shown in Fig. 43, \(c\), the energy balance is approximately given by the equality:

\[ \frac{1}{2}\cdot\frac{1}{2}\left(\frac{4\pi}{3}\right)^2 R_c^3 I_s^2 + \pi R_c^2\sigma_w = \frac{1}{2}\left(\frac{4\pi}{3}\right)^2 R^3 I_s^2, \tag{VI, 1, 11} \]

which leads to

\[ R_c=\frac{9\sigma_w}{4\pi I_s^2}. \tag{VI, 1, 12} \]

This expression is identical with (VI, 1, 10). For MnBi the estimate gives:

\[ R_c \simeq \frac{20}{600^3} \simeq 4 \cdot 10^{-5}\ \text{cm}, \]

whereas the thickness of the layer is of the order of \(\delta \simeq 2 \cdot 10^{-6}\ \text{cm}\). Thus, the assumption adopted for the calculations, according to which

\[ \frac{\delta}{R_c} \ll 1, \]

is fulfilled in this case.

Analogous calculations for thin wires and thin films, as well as for small particles, were given by Kittel (1946).

VI, 2. Coercive force of small particles.

We have just seen that, in the case of sufficiently small particles, the formation of domain boundaries is energetically unfavorable. In the absence of domain boundaries, the change in magnetization cannot be carried out by the “easy” process of boundary displacement (Fig. 5), but must occur exclusively as a result of the “difficult” process of rotation of the total magnetic moment of the particle (Fig. 16).

Since we have excluded the possibility of motion of boundaries, we can obtain a considerable increase in the coercive force as a result of increasing the effective anisotropy of the specimen, i.e. by making as difficult as possible the rotation of the magnetization of the domain as a whole. This was first pointed out by Kittel (1946). In order to reverse the direction of magnetization in a small particle, it is necessary that the magnetic energy acquired by the particle in an external magnetic field be greater than the internal energy that tends to prevent rotation of the domain direction. The effective internal anisotropy will be high if the magnetocrystalline anisotropy energy is high (Kittel, 1946), or if the particle has an elongated shape (Néel, 1947, b), or, finally, if some anisotropic strain is created (Stoner and Wohlfarth, 1943). Of the high observed values of the coercive force (for example, 12,000 oersteds in MnBi and 20,000 oersteds in FePt), the first of these causes is probably responsible.

Let us now consider the coercive force in each of these three cases separately. In all cases it is assumed that a single-domain configuration is present. First we shall analyze the case of an isolated particle, and then consider the changes that arise when the particles are densely packed. Values of the coercive force of small Fe, Co, and Ni particles, calculated by various methods, are given in Table II.

Coercive force associated with magnetocrystalline anisotropy. The anisotropy energy density of a uniaxial crystal, according to (II, 2, 2), in the first approximation is equal to:

\[ f_K = K'_1 \sin^2 \theta, \tag{VI, 2, 1} \]

where \(\theta\) is the angle between the crystal axis and the direction of magnetization.

The density of the magnetic energy for a magnetic field \(H_0\), parallel to the crystal axis, is equal to:

\[ f_{\mathrm{mag}}=H_0 I_s \cos\theta, \tag{VI, 2, 2} \]

where the choice of sign corresponds to the direction of the field, opposite to the projection of the magnetization onto the crystal axis (Fig. 45). The total energy is equal to

\[ f=K'_1\sin^2\theta+ H_0 I_s\cos\theta \tag{VI, 2, 3} \]

and will be minimal with respect to \(\theta\) when

\[ \frac{\partial f}{\partial\theta}=0 = 2K'_1\sin\theta\cos\theta- H_0 I_s\sin\theta \tag{VI, 2, 4} \]

or

\[ H_0=\frac{2K'_1}{I_s}\cos\theta. \tag{VI, 2, 5} \]

Table II

Maximum coercive force of small particles, due to various causes. (It is assumed that there is complete orientation; the packing-effect correction is neglected.) \(T=2\cdot10^{10}\) dyn/cm

	Anisotropy \(2K_1/I_s\)	Shape \(2\pi I_s\)	Internal deformation \(3\lambda T/I_s\)
Fe	500	10 700	600
Co	6000	8 800	600
Ni	135	3 150	4000

This expression is maximal (and, consequently, equal to the coercive force) when \(\theta=0\). Therefore

\[ H_c=\frac{2K'_1}{I_s}. \tag{VI, 2, 6} \]

This result is also applicable to the case of a cubic crystal if the field is applied along the direction \([001]\). For small \(\theta\) the anisotropy energy according to equation (A, 3) is equal to

\[ f_K\simeq K_1\theta^2. \tag{VI, 2, 7} \]

The total energy is

\[ f=K_1\theta^2+H_0 I_s\cos\theta. \tag{VI, 2, 8} \]

This expression will be minimal with respect to \(\theta\) when

\[ \frac{\partial f}{\partial\theta}=0 = 2K_1\theta-H_0 I\sin\theta. \]

Hence, for \(\theta\to0\), we find the coercive force

\[ H_c=\frac{2K_1}{I_s}. \tag{VI, 2, 9} \]

In real specimens of powder materials the grains usually have random orientation. For this case Néel (1947, c) showed that in cubic crystals having random orienta-

...orientation, the coercive force for the mean hysteresis loop at \(K>0\) is

\[ \langle H_c\rangle_{\mathrm{av}}=0.64\,\frac{K}{I_s}. \tag{VI, 2, 10} \]

This gives, for iron, about 100 oersteds.

If we apply the same expression to cobalt, which is uniaxial, we obtain \(\langle H_c\rangle_{\mathrm{av}}\simeq 2500\) oersteds.

Coercive force associated with anisotropy of particle shape. Suppose that the particle has the shape of a prolate spheroid, and let us restrict ourselves to consideration of the case in which the applied magnetic field \(H_0\) is parallel to the long axis of the spheroid and opposite to the initial direction of magnetization. Let \(N_0\) denote the demagnetizing factor of the prolate spheroid in the direction of the principal axis, and let \(N_t\) be the demagnetizing factor in any direction at right angles to the long axis. We have \(N_0\leq N_t\); for a long circular cylinder \(N_0=0\) and \(N_t=2\pi\); for a sphere \(N_0=N_t=\dfrac{4\pi}{3}\).

Fig. 45. Geometric picture that must be taken into account in calculating, in a uniaxial crystal, the coercive force due to magnetocrystalline anisotropy.

The energy

\[ f_{\mathrm{mag}}=\frac{1}{2} I_s^2\left(N_0\cos^2\theta+\right. \]
\[ \left.+\,N_t\sin^2\theta\right)+H_0 I_s\cos\theta, \tag{VI, 2, 11} \]

where the first term is the energy of the demagnetizing field, and the last is the magnetic energy associated with the applied field \(H_0\). The energy will be minimal with respect to \(\theta\) when

\[ \frac{df}{d\theta}=0=I_s^2(N_t-N_0)\cos\theta\sin\theta-H_0I_s\sin\theta. \]

The coercive force corresponding to \(\theta=0\) is equal to

\[ H_c=(N_t-N_0)I_s. \tag{VI, 2, 12} \]

In Fig. 46, according to (VI, 2, 12), values are given of the coercive force for iron \((I_s=1700)\) as a function of the axial ratio of the prolate spheroid.

The coercive force is maximal for the limiting case of a long circular cylinder; in this case

\[ H_c=2\pi I_s=\frac{B_s}{2}. \tag{VI, 2, 13} \]

The maximum theoretical values of the coercive force \(H_c\), associated with the “shape effect,” are given below for various materials. These values do not take into account the interaction effect, which will be discussed below.

Substance	Maximum \(H_c\) (oersted)
Fe	10 700
Co	8 800
Ni	3 150

The character of the magnetization curve when the shape effect dominates is shown in Fig. 46, where the curves are given for a field applied parallel, perpendicular, and at an angle of \(45^\circ\) to the long axis of the particle.

Fig. 46. Coercive force in an iron particle, due to the shape effect, as a function of the particle axial ratio.

Stoner and Wohlfarth (1948) showed that, for a random orientation of the particle axes, the mean coercive force is expressed as:

\[ < H_c >_{\mathrm{av}} = 0.48 (N_t - N_0) I_s, \tag{VI, 2, 14} \]

so that, for a random orientation, the values given above must be reduced by approximately one half.

Néel (1947, b) analyzed the case in which the axial ratio \(\frac{c}{a}\) of an elongated spheroid is approximately equal to unity, i.e.

\[ \frac{c}{a} = 1 + \varepsilon,\quad \varepsilon \ll 1. \]

Here, transforming the analytical expression for the demagnetizing factors, we obtain:

\[ N_t - N_0 \simeq \frac{8\pi \varepsilon}{5}. \tag{VI, 2, 15} \]

Then, according to (VI, 2, 12):

\[ H_c \simeq \frac{8\pi \varepsilon I}{5}. \tag{VI, 2, 16} \]

As a result of graphical calculations, Néel found that if the directions of the principal axes are distributed chaotically, the average coercive force is reduced by a factor of 0.48, so that

\[ \langle H_c\rangle_{\mathrm{av}} \simeq 0.48\,\frac{8\pi}{5}\,\varepsilon I_s. \tag{VI, 2, 17} \]

Hence, for iron,

\[ \langle H_c\rangle_{\mathrm{av}} = 4100\,\varepsilon . \]

Coercive force associated with longitudinal stresses. The density of magnetoelastic energy (for isotropic magnetostriction), according to (II, 3, 22), is

\[ f_{\mathrm{mu}}=\frac{3}{2}\lambda T\sin^2\theta, \tag{VI, 2, 18} \]

where \(\lambda\) is the magnetostriction constant and \(T\) is the applied stress. The density of the total energy in a field \(H_0\), applied parallel to the stress, is

\[ f=\frac{3}{2}\lambda T\sin^2\theta+HI_s\cos\theta. \tag{VI, 2, 19} \]

This expression is minimal with respect to \(\theta\) when

\[ \frac{\partial f}{\partial\theta}=0=3\lambda T\cos\theta\sin\theta-HI_s\sin\theta, \]

or, for \(\theta=0\),

\[ H_c=\frac{3\lambda T}{I_s}. \tag{VI, 2, 20} \]

This result was obtained by Stoner and Wohlfarth.

Fig. 47. Magnetization curves for elongated particles in a field parallel, perpendicular, and situated at an angle of \(45^\circ\) to the long axis of the particle.

If the stress is due to internal deformations, a reasonable upper value of the stresses, according to the authors cited, is \(200\ \mathrm{kg/cm^2}\), i.e. \(2\cdot 10^{10}\ \mathrm{dyn/cm^2}\). The maximum coercive forces for iron, nickel, and cobalt are respectively 600, 4000, and 600.

Dependence of the coercive force on particle size. From experimental data it follows that the coercive force increases smoothly as the size of the particle decreases. Changes in the coercive force probably occur even more

more smoothly than might have been expected because of the scatter of particle sizes in any sample. The experimental results of Guillaud (1943), obtained with a fine powder of the compound MnBi,

Fig. 48. Comparison of Guillaud’s data for MnBi with the theoretical lower limit for the coercive force \(H_c\) as a function of the particle diameter \(d\).

are shown in Fig. 48; as can be seen, when the diameters change from 3 to 100 microns there is a considerable increase in the coercive force \(H_c\).

Fig. 49. Initial stage of magnetization reversal by the formation of a flat transition layer of thickness \(\delta\).

A rough theoretical explanation of these changes was given by Kittel (1948), using a model (Fig. 49) that applies to spherical particles of highly anisotropic materials.

Let us consider a small sphere magnetized to saturation, and suppose that, when a field \(H_c\) is applied, a domain layer is formed, as shown in Fig. 49. In this case the energy balance is approximately expressed by the equation

\[ \sigma \frac{\pi p^2}{4} = H_c I_s \frac{\pi p^3 \delta}{8} + \frac{1}{2} I_s^2 V \frac{\delta}{2d}. \tag{VI, 2, 21} \]

Here \(\sigma\) is the surface-energy density of the Bloch wall, \(p\) is the diameter of the layer, \(\delta\) is the thickness of the layer, and \(V\) is the volume of the sphere. The term on the left-hand side of this equation is an approximate expression for the energy of formation of the layer; the first term on the right-hand side approximately represents the magnetic energy of the substance of the layer in the applied field \(H_c\), and the second term on the right is a rough estimate of the change in the sphere’s own magnetic energy, based on dimensional considerations. Using the geometrical relation \(p^2 = 4\delta d\), we may write expression (VI, 2, 21) in the form

\[ \frac{H_c}{H_c^\infty}=1-\frac{d}{d_0}, \tag{VI, 2, 22} \]

where \(H_c^\infty=\frac{2\sigma}{\delta I_s}\) and \(d_0=\frac{24\sigma}{I_s^2}\). Further, from the theory of the Bloch wall

\[ \sigma \sim \left(\frac{K k T_c}{a}\right)^{1/2} \]

and

\[ \delta \sim \left(\frac{k T_c}{K a}\right)^{1/2}, \]

so that, in agreement with the value obtained from rotation of domains,

\[ H_c^\infty \sim \frac{2K}{I_s}. \]

For MnBi we obtain \(H_c^\infty = 40\,000\) and \(d_0 = 7\cdot 10^{-4}\ \text{cm}\).

In view of the crudeness of the estimate of the sphere’s own energy, the above estimate of the value of \(d_0\) is very inaccurate. The theoretical curve shown in Fig. 48 was constructed using the value \(H_c^\infty = 20\,000\) and \(d_0 = 9\cdot 10^{-4}\ \text{cm}\).

Fig. 50. Approximate upper limit of the particle diameter for which a single-domain configuration appears.

The present theory gives a lower limit of the coercive force as a function of particle size; it is a lower limit because, without detailed calculations, it is not clear that the case shown in Fig. 49 actually corresponds to the maximum energy barrier for formation of the layer. In this model the particle diameter for \(H_c = H_c^\infty/2\) is equal to

\[ D=\frac{d_0}{2}=\frac{12\sigma}{I_s^2}. \tag{VI, 2, 23} \]

The graph of this equation is given in Fig. 50. The values calculated for Fe, Co, Ni, and MnBi, owing to the approximate nature of the calculations, are shown by short vertical lines. The particle diameter calculated from (VI, 2, 23) may be regarded as an approximate upper limit of the diameter for which the single-domain structure is clearly expressed. It is assumed here that the anisotropy energy is high, so that the ratios obtained are in fact inapplicable to nickel and iron.

Dependence of the coercive force on the packing density. The interaction of the magnetic moments of particles in a specimen made of pressed powder leads in some cases to a decrease in the coercive force. When this effect occurs, it is the greater the denser the “packing.” The results obtained by Weil (1947) with fine powders of the FeCo alloy are shown in Fig. 51. It may be expected that the effect will be greatest for elongated particles whose coercive force is determined mainly by the shape effect considered above.

Fig. 51. Influence of packing density on the coercive force in magnets pressed from fine powders containing 70% Fe and 30% Co (according to Weil (1947). The packing factor is the fraction of the magnet volume filled by magnetic material.

For a square lattice of infinitely thin circular cylinders, Shokley and Kittel showed in an unpublished work that the coercive force is equal to:

\[ H = (1 - \beta p) 2\pi I_s, \qquad (\mathrm{VI},\,2,\,24) \]

where \(\beta\) is a coefficient equal to 1.1 for cubic “packing” and 1.0 for dense hexagonal “packing.” The calculations are too long and therefore are not given here.

Dependence of the coercive force on temperature. When the coercive force of a fine powder is determined by magnetocrystalline anisotropy, the temperature changes of the coercive force will be determined by the temperature changes of

\[ \frac{K}{I_s}. \]

If the coercive force is determined by shape anisotropy, then its temperature changes will be determined by the temperature changes of the magnetic saturation \(I_s\).

The results of calculations for these two types of temperature dependence in the case of a fine iron powder are given in Fig. 52.

The results of experimental measurements by Weil and Marfure (1947) on fine nickel particles are given in Fig. 53. The increase

Fig. 52. Theoretical temperature dependence of the coercive force of a fine iron powder for two different models. The Curie temperature is \(770^\circ\mathrm{C}\).

Fig. 53. Experimental dependence of the coercive force of nickel on temperature. Comparison of massive nickel and fine nickel powder (Weil and Marfure (1947)).

of the coercive force with decreasing temperature is very marked, but not as rapid as might have been expected if the coercive force were due exclusively to crystalline anisotropy.

VII. INITIAL PERMEABILITY AND COERCIVE FORCE

VII, 1. General remarks

Initial permeability and coercive force are properties that depend strongly on the structure of the material, i.e., they may change considerably as a result of small changes in the metallurgical treatment and chemical composition of the material. On the contrary, density and magnetic saturation usually do not depend on the structure. Our knowledge in the field of structure-sensitive properties is, in general, not especially complete. This is explained by the difficulties of obtaining reliable information about the true physical state of the material (for the physical dimensions of impurities, or deformed centers, or any other cause of structural sensitivity are very small, often of the order of a micron or less). Furthermore, it is difficult to carry out control experiments with a single impurity or deformation center.

Fig. 54. Displacement of a domain boundary as a result of applying a magnetic field.

In practice, the very existence in a massive material of a non-infinite initial permeability and of a coercive force different from zero testifies to the imperfection and inhomogeneity of the specimen. In an ideal specimen, the boundary layer separating two oppositely magnetized domains should move easily upon application of an extremely small external field \(H\) (Fig. 54).

The essential physical problem of coercive force can be reduced to the problem of determining the critical magnetic field \(H_0\) necessary for the displacement of a transition layer separating oppositely magnetized domains. Theory must also indicate the mechanism by virtue of which the energy of the specimen will change more or less irregularly with a change in the position of the Bloch layer. If the energy changes irregularly, then positions with minimum energy will be found, and the layers will naturally occupy these positions. The layers may be displaced from these positions as a result of applying a magnetic field, which exerts on them a pressure tending to displace them in such a way as to increa—

change the magnetization in the direction of the field. The initial permeability is a measure of the internal “restoring” force which, for small displacements, tends to return the layer to its initial position. The coercive force is a measure of the maximum “restoring” force acting on the layer. The coercive force indicates to us the field strength necessary for carrying the layer over the highest energy hump from one potential-energy well into another potential well.

Let us formally relate all the energy changes accompanying the motion of the layer to changes in the energy of the layer itself. Suppose that, in order to displace the layer by a distance \(\Delta x\), the energy of the layer per unit area \(\sigma_w\) must increase by \(\Delta\sigma_w\). The required energy is obtained by reversing the magnetic moment of the volume under consideration \(I_s\Delta x\) in a magnetic field \(H'\), just sufficient to cause the displacement \(\Delta x\). Then

\[ 2H'I_s\Delta x=\Delta\sigma_w, \tag{VII, 1, 1} \]

where the left-hand side represents the decrease in the magnetic energy of the system, caused by the change in the direction of the magnetic moment \(I_s\Delta x\) from an orientation antiparallel to \(H'\) to an orientation parallel to \(H'\). The magnetic energy is transformed into the surface energy of the boundary layer. The effective pressure exerted by the field will be \(2H'I_s\).

The critical field \(H_0\) for displacing the boundary over the entire length of the domain will be determined by the greatest local obstacle encountered along the path of this boundary.

Thus,

\[ H_0=\frac{1}{2I_s}\left(\frac{d\sigma_w}{dx}\right)_{\max}. \tag{VII, 1, 2} \]

This expression gives the order of magnitude of the coercive force. The problem is now reduced to estimating \(\left(\dfrac{d\sigma_w}{dx}\right)_{\max}\). There are three principal mechanisms which are considered in connection with the theory of the coercive force. F. Bloch was the first to suggest that nonuniform internal strains may determine the resistance to the motion of the boundary. This idea was developed by Kondorskii (1937) and Kersten (1938). A theory taking into account the influence of aggregates or inclusions of foreign atoms was developed by Kersten (1943). A substantial criticism and generalization of these two theories was carried out by Néel (1946), who emphasized the role of the demagnetizing energy associated with changes in magnetization caused by internal strains and inclusions.

The experimental work now under way will probably make it possible in the near future to gain a better understanding of

mechanism of the coercive force. After this it will be possible to develop a theory of the coercive force that is more substantiated experimentally than any of those existing at the present time.

VII.2. Theory of Inclusions

As an example of the calculations used in the theory of the coercive force, we shall give here a brief estimate of the influence of nonmagnetic inclusions on the energy of a layer. The coercive force in this model will arise because a layer intersecting several inclusions will have a smaller area, and therefore a smaller energy, in comparison with a layer not intersecting inclusions.

Fig. 55. Model for calculating the coercive force in the theory of inclusions.

We use an extremely simplified model in which the inclusions are taken to be spheres of diameter \(d\), arranged in the form of a cubic lattice with lattice constant \(s\) (Fig. 55). When the boundary intersects a sphere, the energy of the layer is lowered by an amount corresponding to the energy of the surface of the layer that is eliminated or “closed” by the inclusion.

Consideration of Fig. 55 shows that the energy of the layer for \(x < d/2\) is equal to

\[ \sigma(x)=\sigma_0 \frac{s^2-\pi\left[\frac{d^2}{4}-x^2\right]}{s^2}. \tag{VII, 2, 1} \]

Taking the derivative,

\[ \frac{d\sigma}{dx}=\frac{2\sigma_0\pi x}{s^2}, \tag{VII, 2, 2} \]

we obtain:

\[ \left(\frac{d\sigma}{dx}\right)_{\max}=\frac{\sigma_0\pi d}{s^2}. \tag{VII, 2, 3} \]

Combining expressions (VII, 1, 2) and (VII, 2, 3), we find the coercive force:

\[ H_c=\frac{\pi}{2}\frac{\sigma_w}{I_s}\frac{d}{s^2}. \tag{VII, 2, 4} \]

PHYSICAL THEORY OF THE STRUCTURE OF FERROMAGNETS

Let us now introduce the relative volume of the inclusions

\[ \alpha=\frac{\pi d^{3}}{6s^{3}} \tag{VII, 2, 5} \]

and the quantity \(\delta\), equal to the half-width of the boundary layer; \(\delta\) is of the order of \(\sqrt{\frac{z_w}{K}}\), where \(K\) is the anisotropy-energy constant. Then

\[ H_c \simeq \frac{K}{I_s}\frac{\delta}{d}\,\alpha^{\frac{2}{3}} . \tag{VII, 2, 6} \]

This expression was derived under the assumption that the diameter \(d\) of the inclusion is considerably greater than the thickness \(\delta\) of the layer. In a rough approximation we may assume that \(H_c\) has its maximum value when \(\delta=d\), i.e.

Fig. 56. Coercive force of iron with heterogeneous copper inclusions as a function of the excess of copper over 0.5% Cu, dissolved at \(600^\circ\mathrm{C}\).

\[ (H_c)_{\max} \simeq \frac{K}{I_s}\,\alpha^{\frac{2}{3}} . \tag{VII, 2, 7} \]

Kersten gives:

\[ (H_c)_{\max} \simeq 2.5\,\frac{K}{I_s}\,\alpha^{\frac{2}{3}} . \tag{VII, 2, 8} \]

These values are shown in Fig. 56 and compared with the experimental values of \(H_c\) for iron with an addition of copper.

Initial permeability in the model with inclusions. The initial susceptibility \(\chi_0\) is equal to

\[ \chi_0=\frac{dI}{dx}\Big/ \frac{dH}{dx}. \tag{VII, 2, 9} \]

The change in magnetization \(\Delta I\), associated with a displacement \(\Delta x\) of the boundary...

... boundary between oppositely magnetized domains, is equal to

\[ \Delta l=\frac{2I_s\beta \Delta x}{s}, \tag{VII, 2, 10} \]

where \(\beta=\dfrac{s}{h}\), and \(h\) is the mean domain thickness.

From (VII, 1, 1) and (VII, 2, 2):

\[ H=\frac{1}{2I_s}\frac{d\sigma}{dx} =\pi\frac{\sigma}{I_s}\frac{x}{s^2} \tag{VII, 2, 11} \]

or

\[ dH=\pi\frac{\sigma}{I_s}\frac{dx}{s^2} =2\pi\frac{K}{I_s}\frac{\delta}{s^2}\,dx. \tag{VII, 2, 12} \]

This gives

\[ \chi_0=\frac{\beta I_s^2 s}{2\pi K\delta}. \tag{VII, 2, 13} \]

Taking into account the \(90^\circ\) and \(180^\circ\) layers, with \(\mu_0\gg 1\) and \(d\gg\delta\), we have

\[ \mu_0\simeq 4\pi\chi_0= \frac{1.6\,\beta I_s^2 d}{k\delta^{1/3}}. \tag{VII, 2, 14} \]

VII, 3. Deformation theory

In the presence of a stress \(T\), the surface-energy density of the Bloch wall, as can be shown (see the argument contained in Sections II, 3 and III, 3), is approximately equal to:

\[ \sigma_w=2[A(K+\lambda T)]^{1/2}, \tag{VII, 3, 1} \]

where we neglect a numerical factor of order unity. Here \(\lambda\) is the magnetostriction at saturation. The only new feature in this expression is the explicit allowance for the anisotropy of strains by means of the term \(\lambda T\).

Fig. 57. Displacement of a wall in the presence of sinusoidal stresses.

Suppose that \(T\), as a function of \(x\), varies as follows:

\[ T=T_0+\Delta T\sin\frac{2\pi x}{l}. \tag{VII, 3, 2} \]

as is shown in Fig. 57. Then

\[ \frac{d\sigma_w}{dx} =\lambda\left[\frac{A}{K+\lambda T}\right]^{1/2}\frac{dT}{dx} =\frac{2\pi\lambda\Delta T}{l} \left[\frac{A}{K+\lambda T}\right]^{1/2} \cos\frac{2\pi x}{l}. \tag{VII, 3, 3} \]

The minimum value of this expression is

\[ \left(\frac{d\sigma}{dx}\right)_{\max} = \frac{2\pi\lambda \Delta T}{l} \left[ \frac{A}{K+\lambda T} \right]^{\frac12} \tag{VII, 3, 4} \]

or, approximately,

\[ \left(\frac{d\sigma_w}{dx}\right)_{\max} \simeq 2\pi\lambda \Delta T\,\frac{\delta}{l}, \tag{VII, 3, 5} \]

where \(\delta\) is the thickness of the Bloch layer and \(l\) is the length over which the stress changes appreciably.

From (VII, 1, 2) and (VII, 3, 5)

\[ H_c \simeq \pi\,\frac{\lambda\Delta T}{I_s}\,\frac{\delta}{l}. \tag{VII, 3, 6} \]

Experimental results showing the dependence of \(H_c\) on the magnitude of the internal stress \(\Delta T\) are given in Fig. 58.

These schematic calculations are sufficient to give an idea of the dependence of the coercive force on the magnitude of the stress variation that causes a change in the energy of the layer. It may be expected that this mechanism will play an essential role in materials with high magnetostriction (as, for example, in nickel). The reader will find details in the papers of Kondorskii (1937) and Kersten (1938).

Fig. 58. Coercive force \(H_c\) of nickel as a function of the mean internal stress \(T_i\):
\(a\) — recrystallized wire, \(b\) — strongly stretched wire (Kersten (1938)). The value of \(T_i\) is determined from other independent magnetic measurements.

VII, 4. Theory of Magnetization Fluctuations

Néel (1944, b, 1946) pointed out that the inclusion theory and the deformation theory are erroneous in a number of respects. First, the assumption of a regular arrangement of inhomogeneities—for example, the assumption that the inclusions form a cubic lattice—leads to a considerable overestimate of the coercive force for real materials, in which the inhomogeneities are distributed more or less randomly. Second, the assumption of the “rigidity” of the domain layers also leads to excessively large values of the coercive force. If, however, the necessary corrections taking these remarks into account are introduced into the calculations, then, according to Néel, the theory gives—

the maximum coercive force is of the order of magnitude of 1 oersted. This value is much smaller than the coercive force of many magnetic materials.

Néel drew attention to the fact that the magnetic energy associated with inclusions or with changes in stresses may be considerably greater than that arising under the same conditions of deformation.

Let us consider, for example, two positions of a layer, shown in Fig. 59. In position a) the magnetic energy of an inclusion having the form of a sphere of radius \(a\), according to (II, 4, 3), is equal to

\[ w_a=\frac{1}{2}\,\frac{4\pi}{3}\,\frac{4\pi a^3}{3}\,I_s^2 . \tag{VII, 4, 1} \]

At the same time in position b), according to the calculations contained in Néel’s work (1944), the magnetic energy is equal to:

Fig. 59. Diagram illustrating the dependence of the magnetic energy of an inclusion on the position of the boundary layer.

\[ w_b=0.46\,w_a . \tag{VII, 4, 2} \]

The difference of the energies \(w_a\) and \(w_b\) is a measure of the field stress which must be applied in order to move the layer from position a) to position b).

As a result of the development of this basic idea, Néel in a 1946 paper obtained the following expressions:

\[ \text{iron:}\qquad H_c=2.1\,v+360\,v'\ \text{oersted}, \tag{VII, 4, 3} \]

\[ \text{nickel:}\qquad H_c=330\,v+97\,v'\ \text{oersted}. \tag{VII, 4, 4} \]

Here \(v'\) is the part of the volume occupied by the inclusion, and \(v\) is the part of the volume subjected to irregular internal stresses, the magnitude of which is \(30\ \text{kg}/\text{cm}^2\).

Let us also note that the work of Williams and Shockley with iron crystals having a simple domain structure shows that the interaction of the boundaries of the main domains with lancet-shaped domains around crystalline inhomogeneities (section V, 2, 3)

exerts a substantial influence on the coercive force in such crystals. Further work of this kind will probably provide a more reliable physical basis for understanding the mechanism of the coercive force.

APPENDIX A

Expressions for the anisotropy energy of cubic crystals

Expression (II, 2, 4)

\[ f_K = K_1\left(\alpha_1^2 \alpha_2^2 + \alpha_1^2 \alpha_3^2 + \alpha_2^2 \alpha_3^2\right), \tag{A, 1} \]

which depends on the direction cosines \(\alpha_1\), \(\alpha_2\), and \(\alpha_3\) of the magnetization with the crystal axes, can also be written in a somewhat different form.

Thus, if \(c_3 = 0\), then in the plane \((001)\)

\[ f_K = K_1 \sin^2 \theta \cos^2 \theta = \frac{1}{4} K_1 \sin 2\theta . \tag{A, 2} \]

For small \(\alpha_2\) and \(\alpha_3\)

\[ f_K \simeq K_1 \theta^2, \tag{A, 3} \]

where, as in (A, 2), \(\theta\) is the angle between \(\alpha\) and the axis \([100]\).

In the general case (Néel, 1944):

\[ f_K = K_1\left(\sin^2 \theta - \frac{7}{8}\sin^4 \theta - \frac{1}{8}\sin^4 \theta \cos 4\varphi\right), \tag{A, 4} \]

if a side of the cube is chosen as the polar axis from which the angle \(\theta\) is measured. If the space diagonal is chosen as the polar axis,

\[ f_K = K_1\left(\frac{1}{3}\cos^4\theta + \frac{1}{4}\sin^4\theta - \frac{\sqrt{2}}{3}\cos\theta \sin^3\theta \cos 3\varphi\right). \tag{A, 5} \]

Finally, choosing the diagonal of a face as the polar axis, we have:

\[ f_K = \frac{1}{4} K_1 \left[(1 - 4\sin^2\theta + 4\sin^4\theta) + \right. \]

\[ \left. {} + (6\sin^2\theta - 4\sin^4\theta)\sin^2\varphi - 3\sin^4\theta \sin^4\varphi \right]. \tag{A, 6} \]

The azimuthal angle \(\varphi\) is measured around an edge of the cube.

APPENDIX B

Energy of the magnetic interaction of dipoles in a cubic lattice

We wish to prove here the well-known fact that the magnetic interaction between dipoles in an infinite, undeformed cubic lattice does not lead to the appearance of anisotropy energy. The energy of the magnetic interaction

between any two dipoles is equal to

\[ V_{ij}=\frac{\mu_i\mu_j}{r^3}-\frac{3(\mu_i r_{ij})(\mu_j r_{ij})}{r^5}. \tag{B, 1} \]

Therefore the density of the total energy of dipole interaction in a saturated simple cubic lattice is equal to

\[ f=\frac{N\mu^2}{2a^3}\sum_{lmn}'\left[ \frac{1}{(l^2+m^2+n^2)^{3/2}} -\frac{3(la_1+ma_2+na_3)^2}{(l^2+m^2+n^2)^{5/2}} \right], \tag{B, 2} \]

where \(N\) is the number of dipoles per unit volume, \(\alpha\) is the direction of magnetization, and \(r=(x,y,z)=(la,ma,na)\) is the radius vector of a dipole at the lattice site \((l,m,n)\) relative to the dipole located at the origin. The first term in the sum entering into (B, 2) is completely independent of the direction of magnetization and thus cannot give anisotropy. The second term, using the identity \(\alpha_1^2=1-\alpha_2^2-\alpha_3^2\), can be rewritten in the form:

\[ -3\sum_{lmn}'\frac{l^2+\alpha_2^2(m^2-l^2)+\alpha_3^2(n^2-l^2)} {(l^2+m^2+n^2)^{5/2}}, \tag{B, 3} \]

where terms containing products \(lm\), \(ln\), etc., have been omitted, since these terms in the sum obviously give zero. Further, from symmetry considerations,

\[ \sum_{lmn}'\frac{l^2}{(l^2+m^2+n^2)^{5/2}} = \sum_{lmn}'\frac{m^2}{(l^2+m^2+n^2)^{5/2}} = \sum_{lmn}'\frac{n^2}{(l^2+m^2+n^2)^{5/2}} \tag{B, 4} \]

and, consequently, in (B, 3) the terms containing \(\alpha_2\) and \(\alpha_3\) are equal to zero, while the remaining term does not depend on the direction of magnetization. Thus, in a simple cubic lattice anisotropy is absent; the same result is obtained for face-centered and body-centered lattices. One can also verify that \(f=0\).

However, if we admit that the lattice can deform spontaneously, then the magnetic dipole interaction will cause magnetostriction, and this, for the reasons we indicated earlier, will lead to the appearance of a certain, albeit small, anisotropy. As Becker (1930) showed, the magnetostriction caused by dipole interaction, in the case of iron, amounts to only about one fifth of the observed value; in the case of nickel, the magnetostriction calculated in this way even has the opposite sign from that observed experimentally. We thus see that in a regular lattice the magnetic dipole interaction explains neither the observed anisotropy nor the observed magnetostriction.

According to Becker’s calculations, the constants of magnetoelastic coupling \((11,3,5)\) for the body-centered lattice are equal to

\[ B_1=-6Sl_s^2;\qquad B_2=4Sl_s^2, \tag{B, 5} \]

and for a face-centered lattice

\[ B_1=-3SI_s^2;\qquad B_2=2SI_s^2, \tag{B, 6} \]

where

\[ S=\frac{3}{2}\sum'\left\{\frac{1}{(l^2+m^2+n^2)^{3/2}}-\frac{5l^4}{(l^2+m^2+n^2)^{7/2}}\right\}. \tag{B, 7} \]

For a body-centered lattice \(S=0.4\), and for a face-centered lattice \(S=0.6\). Therefore for iron, whose lattice is body-centered, we have:

Calculations: \(B_1=-0.7\cdot10^7\ \text{erg}/\text{cm}^3;\quad B_2=0.5\cdot10^7\ \text{erg}/\text{cm}^3.\)

Observations: \(B_1=-2.9\cdot10^7\ \text{erg}/\text{cm}^3;\quad B_2=6.4\cdot10^7\ \text{erg}/\text{cm}^3.\)

For nickel, whose lattice is face-centered:

Calculations: \(B_1=-0.04\cdot10^7\ \text{erg}/\text{cm}^3;\quad B_2=0.03\cdot10^7\ \text{erg}/\text{cm}^3.\)

Observations: \(B_1=6.2\cdot10^7\ \text{erg}/\text{cm}^3;\quad B_2=9.0\cdot10^7\ \text{erg}/\text{cm}^3.\)

This comparison of the calculated and observed values indicates that taking account of the magnetic dipole interaction is quite insufficient for explaining the experimental data.

APPENDIX B

List of Notation

\(B=H+4\pi I\) — magnetic induction,
\(H\) — magnetic-field strength,
\(H_m\) — effective molecular field,
\(I\) — magnetization \(=\dfrac{\text{magnetic moment}}{\text{volume}}\),
\(I_s\) — magnetization at saturation (magnetic saturation),
\(I_0\) — magnetization at saturation at \(0^\circ\text{K}\),
\(K\) — anisotropy constant,
\(J\) — exchange integral,
\(T_c\) — Curie temperature,
\(k=1.38\cdot10^{-16}\dfrac{\text{erg}}{\text{deg}}\) — Boltzmann constant,
\(\alpha;\ (\alpha_1,\alpha_2,\alpha_3)=(\alpha_x,\alpha_y,\alpha_z)\) — vectors of the magnetization and its direction cosines with respect to the crystal axes,
\(\lambda_{111},\lambda_{100}\) — magnetostriction constants at saturation,
\(c_{11},c_{12},c_{44}\) — elastic moduli of a cubic crystal,
\(e_{ij}\) — strain components,
\(f\) — energy density \(=\dfrac{\text{energy}}{\text{volume}}\),
\(f_{\text{ex}}\) — exchange-energy density,
\(f_{\text{anis}}=f_K\) — anisotropy-energy density,
\(f_{\text{me}}\) — magnetoelastic-energy density,
\(f_{\text{mag}}\) — magnetic-energy density,
\(f_{\text{el}}\) — elastic-energy density,
\(B_1,B_2\) — magnetoelastic-coupling constants,
\(N\) — number of atoms per unit volume,
\(a\) — lattice constant,
\(\varepsilon_w\) — energy of a Bloch layer referred to unit area,
\(\varphi\) — angle between the directions of neighboring spins,

$n_{\mathrm{eff}}$ — effective number of Bohr magnetons per atom,
$A$ — exchange-energy constant,

$\mu_B = 0.927 \cdot 10^{-20}$ erg/oersted — Bohr magneton,
$T$ — stress; absolute temperature,
$g$ — linear density of poles,
$S$ — spin quantum number,
$\chi = \dfrac{I}{H}$ — susceptibility of a unit volume,

$\mu = 1 + 4\pi\chi$ — permeability,
$\mu^{*}$ — effective permeability of a single domain in a field perpendicular to the direction of magnetization in the domain (see Section II, 4).

LITERATURE *)

I. Antik and T. Kubyshkina, On losses in the hysteresis of liquid ferromagnetics (amalgams), Uchenye Zapiski MGU 2, 143—149 (1934).

S. V. Vonsovskii and Ya. S. Shur, Ferromagnetism. GTTI (1948).

E. Kondorskii, On the nature of the coercive force and irreversible change of magnetization, Physik. Zeits. Sow. 11, 597—620 (1937).

L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetics, Physik. Zeits. Sow. 8, 153—169 (1935).

E. Lifshitz, On the magnetic structure of iron, J. of Phys. 8, 337—346 (1944).

Ya. Frenkel and Ya. Dorfman, Spontaneous and induced magnetization in ferromagnetic bodies, Nature 126, 274—275 (1930).

L. Kholodenko, On the domain structure of ferromagnetics in the presence of a magnetic field, ZhETF 17, 698—707 (1947).

L. F. Bates and F. E. Neale, A quantitative examination of recent ideas on domain structures, Physica 15, 220—224 (1949).

S. J. Barnett, New researches on magnetization by rotation and the gyromagnetic ratios of ferromagnetic substances, Proc. Am. Acad. Sci. 75, 109—129 (1944).

R. Becker, Zur Theorie der Magnetisierungskurve, Zeits. f. Physik 62, 253—269 (1930).

R. Becker, Elastische Spannungen und magnetische Eigenschaften, Physik. Zeits. 33, 905—913 (1932).

R. Becker and W. Döring, Ferromagnetismus (Verlag Julius Springer, Berlin; reprinted I. W. Edwards, Ann. Arbor, 1939).

G. Bethe and A. Sommerfeld, Electronic Theory of Metals (1938).

F. Bitter, On inhomogeneities in the magnetization of ferromagnetic materials, Phys. Rev. 38, 1903—1905 (1931).

F. Bloch, Zur Theorie des Ferromagnetismus, Zeits. f. Physik 61, 206—219 (1930).

F. Bloch, Zur Theorie des Austauschproblems und der Remanenzerscheinung der Ferromagnetika, Zeits. f. Physik 74, 295—335 (1932).

R. M. Bozorth, Directional ferromagnetic properties of metals, J. Appl. Phys. 8, 575—588 (1937).

*) Different articles by the same author that appeared in one and the same year are distinguished by a letter appended to the year (for example, 1944b).

R. M. Bozorth, Magneto-resistance and domain theory of ironnickel alloys, Phys. Rev. 70, 923 (1946).

R. M. Bozorth, On magnetic remanence, Zeits. f. Physik 124, 519—527 (1948).

Bozorth, Mason, McSkimin and Walker, Elastic constants and internal loss of single nickel crystals, Phys. Rev. 75, 1954 (1949).

H. Brooks, Ferromagnetic anisotropy and the itinerant electron model, Phys. Rev. 58, 909—918 (1940).

Burgy, Hughes and Wallace, Double transmission and depolarization of neutrons, Phys. Rev. 74, 1207 (1948).

A. Drigo and M. Pizzo, Particolari aspetti della magnetizzazione di sottili pellicole ferromagnetiche, Il Nuovo Cimento 5, 1—11 (1948).

W. C. Elmore, The magnetization of ferromagnetic colloids, Phys. Rev. 54, 1092—1095 (1938).

W. C. Elmore, Theory of the optical and magnetic properties of ferromagnetic suspensions, Phys. Rev. 60, 593—596 (1941).

M. Fallot, Ferromagnétisme des alliages de fer, Annales de Physique 6, 305—387 (1936).

A. D. Fokker, Remark on the fundamental relations of thermomagnetics, Physica 6, 791—796 (1939).

L. H. Germer, Stray magnetic fields from cobalt, Phys. Rev. 62, 295 (1942).

E. A. Guggenheim, On magnetic and electrostatic energy, Proc. Roy. Soc. A155, 49—70 (1936).

C. Guillaud, Ferromagnétisme des alliages binaires de manganèse, Thesis, Strasbourg, 1943.

C. Herring and C. Kittel, Energy of a Bloch walle on the band picture, Phys. Rev. (to be submitted).

H. Honda and S. Kaya, Magnetization of single crystals of iron, Sci Rep. Tôhoku Univ. 15, 721—753 (1926).

S. Kaya, On the magnetization of single crystals of nickel, Sci. Rep. Tôhoku Univ. 17, 639—663 (1928a).

S. Kaya, On the magnetization of single crystals of cobalt, Sci. Rep. Tôhoku Univ. 17, 1137—1177 (1928b).

M. Kersten, Probleme der technischen Magnetisierungskurve, Zur Deutung der Koerzitivkraft, edited by R. Becker (Verlag Julius Springer, Berlin; reprinted by J. W. Edwards, Ann. Arbor, 42—72 (1938).

M. Kersten, Grundlagen einer Theorie der ferromagnetischen Hysterese und der Koerzitivkraft (S. Hirzel, Leipzig; reprinted by J. W. Edwards, Ann. Arbor, 1943).

R. Kimura and K. Ohno, On the elastic constants of single crystals of iron, Sci. Rep. Tôhoku Univ. 23, 359—364 (1934).

C. Kittel, Theory of the structure of ferromagnetic domains in films and small particles, Phys. Rev. 70, 965—971 (1946).

C. Kittel, Domain theory and the dependence of the coercive force of fine ferromagnetic powders on particle size, Phys. Rev. 73, 810 (1948).

C. Kittel, Gyromagnetic ratios and splitting factors of ferromagnetic substances, Phys. Rev. 76, 743 (1949a).

C. Kittel, Theory of the formation of powder patterns on ferromagnetic crystals. Phys. Rev. 76, 1527 (1949b).

A. Kussmann and B. Scharnow, Zeits. f. Physik 54, 1—15 (1929). Über die Koerzitivkraft. I. Teil. Koerzitivkraft und mechanische Härte.

L. Marton, Ferromagnetic domain observation, Phys. Rev. 73, 1475 (1948).

B. Matthias and A. von Hippel, Domain structure and dielectric response of barium titanate single crystals, Phys. Rev. 73, 1378—1381 (1948).

C. Moller, Zur Theorie der Austauschproblems und des Ferromagnetismus bei tiefen Temperaturen, Zeits. f. Physik 82, 559—567 (1933).

L. Néel, Quelques propriétés des parois de domaines élémentaires ferromagnétiques, Cahiers de Physique 25, 1—20 (1944a).

L. Néel, Effet des cavités et des inclusions sur le champ coercitif, Cahiers de Physique 25, 21—24 (1944b).

L. Néel, Les lois de l’aimantation et de la subdivision en domaines élémentaires d’un monocristal de fer, J. de phys. et rad. 5, 241—251, 265—276 (1944c).

L. Néel, Bases d’une nouvelle théorie générale du champ coercitif, Annales Univ. Grenoble 22, 299—343 (1946).

L. Néel, Propriétés d’un ferromagnétique cubique en grains fins, Comptes Rendus (Paris) 224, 1488—1490 (1947a).

L. Néel, Le champ coercitif d’une poudre ferromagnétique cubique à grains anisotropes, Comptes Rendus (Paris) 224, 1550—1551 (1947b).

L. Néel, Théorie de l’anisotropie à aimants traités à chaud dans un champ magnétique, Comptes Rendus (Paris) 225, 109—111 (1947c).

J. A. Osborn, Demagnetizing factors of the general ellipsoid, Phys. Rev. 67, 351—357 (1945).

W. Shockley, Energy calculations for domains, Phys. Rev. 73, 1246 (1948).

K. J. Sixtus and L. Tonks, Propagation of large Barkhausen discontinuities, IV, Regions of reversed magnetization, Phys. Rev. 43, 931—940 (1933).

E. C. Stoner and E. P. Wohlfarth, Interpretation of high coercivity in ferromagnetic materials, Nature 160, 650 (1947).

E. C. Stoner and E. P. Wohlfarth, A mechanism of magnetic hysteresis in heterogeneous alloys, Phil. Trans. A240, 599—644 (1948).

J. H. Van Vleck, A survey of the theory of ferromagnetism, Rev. Mod. Phys. 17, 27—47 (1945).

J. H. Van Vleck, Quelques aspects de la théorie du magnétisme, Annales de l’Institut Henri Poincaré 10, 57—190 (1947).

L. Weil, Variation du champ coercitif en fonction de la densité de poudres ferromagnétiques agglomérées, Comptes Rendus (Paris) 225, 229—230 (1947).

L. Weil and S. Marfoure, Variation thermique du champ coercitif du nickel aggloméré, J. de phys. et rad. 8, 358—361 (1947).

P. Weiss, L’hypothèse du champ moléculaire et la propriété ferromagnétique, J. de Phys. 6, 661—690 (1907).

P. R. Weiss, The application of the Bethe-Peierls method to ferromagnetism, Phys. Rev. 74, 1493—1504 (1948).

H. J. Williams, Direction of domain magnetization in powder patterns, Phys. Rev. 71, 646 (1947).

Williams, Bozorth and Shockley, Magnetic domain patterns on single crystals of silicon iron, Phys. Rev. 75, 155—178 (1949).

H. J. Williams and W. Shockley, A simple domain structure in an iron crystal showing a direct correlation with the magnetization, Phys. Rev. 75, 178—183 (1949).

W. A. Yager, Phys. Rev. (to be submitted).