MAGNETIZATION CURVE AND DOMAINS OF FERROMAGNETS¹)

W. F. Brown, Princeton, U.S.A.

INTRODUCTION

The magnetization curve of a typical ferromagnetic specimen (Fig. 1) may be theoretically divided into three parts: saturation (III), reversible magnetization at high fields (II), and the region of weak fields (I), in which hysteresis is observed. In this article we are interested chiefly in part (I) of the curve, but for a sufficiently clear understanding of its nature it is necessary to dwell briefly on the description of parts (II) and (III).

Fig. 1. Typical magnetization curve

\(H\)—magnetic field; \(J\)—magnetic moment per unit volume, or magnetization; III—saturation; II—slow and reversible magnetization; I—abrupt and, chiefly, irreversible magnetization; \(O\)—demagnetized state. The arrows indicate the direction in which the curve is traversed. At the points \(P\) and \(Q\) the magnetization differs, but the field is the same. The “reversible susceptibility” is represented by the slope \(PP'\) and \(QQ'\).

SATURATION (III)

The large magnetic moment observed in ferromagnetic materials at saturation is attributed by modern theory to the “spins” of atomic electrons. The nature of the forces orienting these spins can be explained only with the aid of quantum mechanics; however, the action of these forces in our case may be taken into account if to each pair of neighboring atoms one assigns the classical potential energy

\[ u = - \frac{1}{2} I \cos \varphi, \tag{1} \]

where \(I\) is a positive constant²), and \(\varphi\) is the angle between the directions of the two magnetic moments (Fig. 2). Since \(u\) has a minimum at \(\varphi = 0\), these forces tend to bring all atomic moments into complete parallelism. At absolute zero temperature nothing opposes—

¹) J. Appl. Physics, 11, 160, 1940. Translated by S. V. [[unclear: surname]].

²) Exchange integral. Translator’s note.

this tendency acts, and the resulting magnetic moment of a unit volume is equal to

\[ J_{0}=n\mu_{0}, \tag{2} \]

where \(n\) is the number of atoms in \(1\ \mathrm{cm}^{3}\), and \(\mu_{0}\) is the magnetic moment per atom.

Thermal motion tends to disturb the complete parallelism of the atomic moments; therefore the “spontaneous” magnetization \(J_s\) decreases with increasing temperature and, finally, disappears at the critical temperature \(\Theta\) (the Curie point). This disappearance will occur when the energy of thermal motion \(kT\) becomes, in order of magnitude, equal to the energy of the orienting forces per atom, or

\[ k\Theta \simeq I. \tag{3} \]

Fig. 2. Energy \(u\) of the orienting forces between the magnetic moments \(\mu_0\) of atoms situated in nearest-neighbor positions

Knowing the value \(\Theta\) (\(=631^\circ\mathrm{K}\) for nickel) and Boltzmann’s constant \(k\) (\(=1.37\cdot10^{-16}\ \mathrm{erg/deg}\)), one can show that the order of magnitude of \(I\) is \(10^{-13}\ \mathrm{erg}\), and the corresponding energy per unit volume is, in order of magnitude, \(In\simeq 10^{-13}\cdot10^{23}=10^{10}\ \mathrm{erg/cm^{3}}\). This value is much greater than the energy density of the interatomic magnetic forces, which in order of magnitude is equal to

\[ \frac{2}{3}\pi J_s^2 \simeq 2\cdot(500)^2 = 5\cdot10^{5}\ \mathrm{erg/cm^{3}}. \]

As is well known, magnetic forces are too small to account for ferromagnetism1.

At room temperature \(J_s\) is only slightly less than the value \(J_0\) at absolute zero. Therefore the problem can be simplified if it is assumed that the atomic moments are completely parallel to one another, but that their magnitude \(\mu\) is somewhat less than \(\mu_0\), so that

\[ J_s=n\mu. \tag{4} \]

The parallel orientation of spins leads to a deformation of the lattice (saturation magnetostriction). The magnitude of the magnetic forces acting between atoms is sufficient to produce such a deformation. However, a calculation made according to the classical theory gives a result that is correct only in order of magnitude. There have as yet been no quantum-mechanical calculations2. Saturation magnetostriction can be calculated from experimental data.

REVERSIBLE MAGNETIZATION AT HIGH FIELDS (II)

The atomic theory of spontaneous magnetization predicts that a ferromagnetic body must be entirely magnetized to saturation \(J_s\), even in the absence of an external magnetic field. Therefore the theory of the magnetization curve for parts (I) and (II) must take into account a number of factors that were neglected in the simplified atomic theory.

One of the most essential factors is the crystalline nature of ferromagnetic materials. An ordinary polycrystalline material consists of grains of single crystals whose crystallographic axes are oriented in different directions. It may be expected that within each grain the spontaneous magnetization is oriented along certain axes of symmetry until a sufficiently strong magnetic field turns the magnetic moments along its own direction. In section \(II\) of the magnetization curve of a polycrystalline specimen, the magnetic moments of individual grains gradually turn from directions determined by crystalline forces into the direction of the external field. The components of the magnetization normal to the direction of the field, for the specimen as a whole, are equal to zero, and the observed magnetization is entirely due to the components parallel to the field.

In single-crystal specimens one can observe a normal component. If the field is applied in the direction in which the magnetization is held by crystalline forces, then section \(II\) of the magnetization curve is absent and saturation is reached in very weak fields. Both of these conclusions are confirmed by experiment: first, a normal component of magnetization is observed, and second, certain directions in a single crystal are “directions of easiest magnetization.”

The theory of section \(II\) developed at present is formal and says nothing about the nature of the atomic forces. It gives formulas for macroscopic effects, deriving them from the requirements of thermodynamics and crystallographic symmetry. Such a theory contains a certain number of constants which are not determined from it and can be obtained only from experimental data. The theoretical determination of the magnitudes of these constants is a task of atomic theory. This part of atomic theory is not yet complete, although the nature of the atomic forces is generally understood;¹ the formal theory is sufficient for our purposes. Let us consider, by way of illustration, two simple cases.

1. Single crystal of cobalt. At room temperature the direction of easiest magnetization coincides with the hexagonal axis. Around this axis the crystal is magnetically isotropic. Therefore the magnetization vector lies in the plane determined by the direction of easiest magnetization and by the—

¹ The cause of magnetic anisotropy is the so-called spin-orbital magnetic forces acting between the magnetic moments of electron spins and the “orbits” of the electrons in the lattice. Bloch and Gentile first showed (Z. Physik, 70, 395, 1931) that a quantum-mechanical treatment of these forces leads to the correct order of magnitude for the magnetic-anisotropy constant. The temperature dependence for cubic crystals, namely iron and nickel, was investigated by van Vleck, and for the case of cobalt by Wonsowski (ZhETF, 8, 1104, 1938).

compared with the field, as shown in Fig. 3, a. This may be regarded as the result of the combined action of the torque \(L_K\), produced by the “anisotropy” forces and tending to turn the magnetization vector into the easy direction, and the torque \(L_H=-HJ_s\sin(\varphi-\vartheta)\), produced by the field and tending to turn the vector in the direction of the field. It is simpler, however, to consider not the torques, but the density of the free energy \(W(\vartheta)\), with which the torque \(L\) acting on the magnetic moment of unit volume is connected by the relation \(L=-\dfrac{\partial W}{\partial \vartheta}\). Stable equilibrium for a given \(H\) corresponds to that value of the angle \(\vartheta\) for which \(W\) is minimal.

Fig. 3. Two simple cases of magnetic anisotropy

a — a cobalt crystal; b — an isotropic material under tension (the magnetostriction is positive). Spontaneous magnetization (\(J_s\)) turns toward the direction of the field (\(H\)) by means of the torque \(L_H\), and toward the direction of easy magnetization by means of the torque \(L_K\), or \(L_\sigma\). The torque \(L_K\) or \(L_\sigma\) may be obtained from the energy function \(W_K\) or \(W_\sigma\).

The part of the free energy \(W\) that depends on the external field is equal to the potential energy of a magnet with moment \(J_s\), directed at an angle \((\varphi-\vartheta)\) to the field \(H\), namely,
\(W_M=-HJ_s\cos(\varphi-\vartheta)\). The part \(W_K\), dependent on the crystalline forces, from symmetry considerations has the form

\[ W_K=K'\sin^2\vartheta+K''\sin^4\vartheta+\ldots \tag{5} \]

Good agreement with the experimental magnetization curves is obtained if one puts \(K'=5.1\cdot10^6\ \mathrm{erg/cm^3}\), \(K''=2.2\cdot10^6\ \mathrm{erg/cm^3}\), and neglects higher powers of \(\sin\vartheta\).

1. The second case, shown in Fig. 3, b, concerns an isotropic material subjected to a uniform and constant tensile stress of magnitude \(\sigma\ \mathrm{dyn/cm^2}\) under the action of an external force. In this case \(W_K\) is replaced by \(W_\sigma\), i.e. by the potential energy of the load producing the elongation1. The saturation magnetostriction \(\lambda_s\) (the relative elongation along the spontaneous magnetization) is assumed positive; in this case the length of the specimen in the direction of tension will be greatest when \(\vartheta=0\) or \(\pi\). Thus, in the absence of a magnetic field, the minimum of \(W\) is reached at \(\vartheta=0\) or \(\pi\) (because in this case the prescribed amount of deformation is obtained under a smaller load than for other values of \(\vartheta\)); consequently, the direction of tension is the direction of easy magnetization. It can be shown that for any \(\vartheta\) the density of the free energy of tension is equal to

\[ W_\sigma=\frac{3}{2}\lambda_s\sigma\sin^2\vartheta . \tag{6} \]

Thus, \(W_\sigma\) has the same form as the first term \(W_K\).

DEMAGNETIZED STATE (0)

Since atomic theory requires the existence of spontaneous magnetization even in a zero external field, it may be assumed that in the demagnetized state the microcrystals are magnetized to saturation; the absence of magnetization over the whole specimen, observed in ordinary measurements, is a consequence of the disordered orientation of the various crystallites of a polycrystalline specimen. However, experiments on single crystals have shown that they too can be demagnetized, and that portion I of the magnetization curve has for them the same form as for ordinary specimens. From this one concludes that even within a single crystal the uniform magnetization required by atomic theory exists only in limited regions, or “domains,” and changes its direction from one easy axis to another at the boundaries between these domains. This violation of the uniformity of the magnetization encounters only slight opposition from the orienting quantum and crystalline forces, since only a comparatively small number of spins at the domain boundaries do not have a parallel orientation.

According to this picture, at the beginning of portion I of the magnetization curve (O, Fig. 1), along each easy direction there is the same volume magnetized to saturation1. At the end of portion I (X, Fig. 1) the whole single crystal, or crystallite, is magnetized along one of the easy axes situated closest to the direction of the field. With a further increase of the field the magnetization vector turns ever closer and closer toward the direction of the field (portion II of the curve), except in the case when the field coincides with one of the directions of easiest magnetization. In the latter case the single crystal is magnetized to saturation at the end of portion I.

Later we shall consider the processes by which a crystal passes from the initial demagnetized state to the final state, when only one easy axis of magnetization is occupied. First we must answer the following questions: 1) What are the direct experimental proofs of the existence of domains, and what information does experiment give about their nature? 2) What are the theoretical grounds for the domain structure of ferromagnets? 3) Can theory explain the observed sizes and shapes of domains?

1. Experimental Proof of the Existence of Domains

Domains are no longer theoretical abstractions: they can be heard and seen.

They can be heard if the secondary winding on a specimen is connected to an amplifier and a telephone, and the magnetizing magnetic field is smoothly increased. The loud crackling heard in the telephone then indicates sudden, discontinuous changes of the magnetic flux in the secondary winding, which may be expected if small pieces of the specimen change, one after another, the direction of their magnetization as the magnetic field increases. This is the well-known Barkhausen effect.

Domains can also be seen by means of experiments which, in their principle, are an improvement of the familiar experiments with “iron filings.” A well-prepared magnetic powder or colloidal solution is applied to the polished surface of a ferromagnetic specimen; the resulting bands (visible under a microscope) not only reveal the presence of domains in a macroscopically demagnetized material, but also provide substantial information about their size, shape, and distribution.

The original crude method of observing the Barkhausen effect (listening to the sound produced when domains are remagnetized) can be replaced by a more delicate procedure, in which the telephone is replaced by a photographic plate, making it possible to obtain quantitative information about the process. This information concerns the regions that discontinuously change the direction of their magnetization; they need not necessarily coincide with the entire volume of the domains, since it may happen that the discontinuous change affects only part of the volume of a domain. Bozorth and Dillinger, on the basis of such experiments, arrived at the following conclusions: 1) the effect is observed both in single crystals and in polycrystals; 2) the sizes of Barkhausen regions do not change noticeably in passing from one material to another, and, depending on their mechanical and thermal treatment, they may be larger or smaller than the grains of the polycrystal; 3) in a wire 60 cm long and 1 mm in diameter, the average volume of a Barkhausen region varies from \(1.2 \cdot 10^{-9}\ \text{cm}^3\) to \(45 \cdot 10^{-9}\ \text{cm}^3\) for a 50% iron–nickel alloy. These experiments leave open the question (which was not raised at the time they were carried out) of a possible dependence of the domain size on the dimensions of the specimen.

2. The Nature of the Domain Structure

The quantum forces described by equation (1) tend to establish a homogeneous spontaneous magnetization throughout the specimen as a whole. The anisotropy forces (5) or (6) can direct this magnetization along certain preferred directions, but they have no tendency to orient the magnetization in different regions of the specimen along different ones of these directions.

If magnetic forces are taken into account, however, the situation becomes different. Magnetic atoms, regarded as tiny current loops,

MAGNETIZATION CURVE AND DOMAINS OF FERROMAGNETS

tend to arrange themselves in such a way as to let as large a magnetic flux as possible pass through them. In Fig. 4, a) the action of this tendency is shown in the case of two atoms. One of them is situated at the point \(O\); the lines of force of its field are shown in the figure. If the second atom is placed at the point \(A\), then it is oriented in the same direction; but if it is placed at the point \(B\), then it is oriented in the opposite direction.

Fig. 4. Influence of magnetic forces

a) Lines of force of an atomic magnet with moment \(\mu\), placed at \(O\). The arrow with a head indicates the direction of the magnetic moment; the circular arrows indicate the direction of the atomic currents that create it. The second atomic magnet at the point \(A\) or \(B\) tends to orient itself as shown in the figure; b) a distribution favorable from the point of view of quantum forces: a uniformly magnetized specimen; c) a distribution favorable from the point of view of magnetic forces: long narrow domains; d) a distribution unfavorable from the point of view of magnetic forces: short broad domains.

In the case of a specimen containing many atoms, homogeneous magnetization [Fig. 4, b)] is magnetically less favorable than the distribution of magnetization shown in Fig. 4, c), in which the flux closes inside the specimen and does not pass out into the external space.

These crude arguments can be made more rigorous if one considers, at some lattice point, the field \(\mathbf{h}\) produced by all the atoms of the other lattice points.

Equilibrium requires that this field coincide with the direction of the magnetic moment \(\mu\) of the atom at the lattice point under consideration. This coincidence is a necessary condition for a minimum of the mutual magnetic energy of all pairs of atoms,

\[ -\frac{1}{2}\sum \mathbf{h}\mu . \]

The overwhelming majority of atoms are situated sufficiently far from the surfaces of the domains; therefore the field \(\mathbf{h}\) is given by the Lorentz formula

\[ \mathbf{h}=\mathbf{H}+\frac{4}{3}\pi \mathbf{J}, \tag{7} \]

where \(\mathbf{H}\) is the macroscopic field, and \(\mathbf{J}\) is the magnetization vector, averaged over the region surrounding the lattice point under consideration. \(\mathbf{J}\) has the same direction as \(\mu\), and magnitude \(J_s\), both in case c) and in case b). The energy density corresponding to this part of \(\mathbf{h}\) is equal to \(-\frac{2}{3}\pi J_s^2\), i.e. to the quantity already mentioned in the discussion of saturation. In case b), \(\mathbf{H}\) is directed opposite to \(\mathbf{J}\)—the demagnetizing field of the poles at the ends of the specimen. In case c), however, \(\mathbf{H}=0\), since there are no poles. Therefore in case c) we have, if not the most favorable distribution of magnetization, at least one better than in case b), from the point of view of magnet-

...fields. On the other hand, \(d)\) is less favorable than \(b)\). The requirements of quantum forces are not satisfied in distribution \(c)\) along the boundaries of the domains. Let us consider a simple cubic lattice with constant \(a\) and assume first that the boundary between oppositely magnetized domains is sharp, as shown in Fig. 5, \(a\), where each arrow represents an atomic magnet. The mutual energy of neighboring atoms on different sides of the boundary, according to equation (1), increases from \(-\dfrac{I}{2}\) to \(+\dfrac{I}{2}\) when two parallel moments are transformed into antiparallel ones. On each unit area of the boundary there are \(\dfrac{1}{a^2}\) such pairs; therefore a sharp boundary leads to an additional energy of magnitude \(\dfrac{I}{a^2}\) per unit area. A gradual transition [Fig. 5, \(b\) and \(c\)] requires less energy. Here the angle \(\vartheta\) between the magnetic moments of the atoms and the direction of easiest magnetization is a slowly varying function of \(x\). Therefore the angle \(\varphi\) between the magnetic moments of neighboring atoms is a small quantity equal to \(a\dfrac{d\vartheta}{dx}\). Consequently, the excess of the energy over its value for parallel orientation is equal to

\[ \frac{1}{2} I(1-\cos\varphi)=\frac{1}{4}I\varphi^2 =\frac{1}{4}Ia^2\left(\frac{d\vartheta}{dx}\right)^2 . \]

Fig. 5. Quantities determining the magnitude of the boundaries between domains \((a—d)\) and of the domains themselves \((e)\)

\(a)\) sharp boundary: too large a quantum energy; \(b)\) gradual transition; \(c)\) graphical representation of \(b\); \(d)\) dependence of the boundary energy per unit area \(\gamma\) on the boundary thickness \(d\); \(T_q\)—quantum energy; \(\gamma_a\)—anisotropy energy; \(\gamma\)—total energy: the minimum is reached at \(\gamma_a=T_q\); \(e)\) dependence of the domain-wall energy \(W_b\) and the surface energy \(W\) on the domain thickness \(w\).

If we imagine that the transition occurs very smoothly [Fig. 5, c], in a layer of thickness \(d\), then the number of pairs of atoms participating in this transition will grow proportionally to \(d\), while the excess of energy from each pair will decrease as \(1/d^2\). Therefore the total effect will decrease proportionally to \(1/d\). Thus the quantum forces tend to make the transition as gradual as possible. In order of magnitude,

\[ \frac{d\vartheta}{dx}=\frac{\pi}{d}; \]

the number of pairs of atoms per \(1\ \mathrm{cm}^2\) of boundary is equal to

\[ \frac{1}{2}\cdot \frac{1}{a^3}\,d. \]

Therefore the energy per unit area is

\[ \frac{1}{4}Ia^2\frac{\pi^2}{d^2}\cdot \frac{1}{2}\frac{d}{a^3}\simeq \frac{I}{ad}, \]

i.e. it is smaller by a factor \(a/d\) than the energy of an abrupt transition.

But now the forces of anisotropy come into play. Suppose that the anisotropy energy per atom has the form \(A\sin^2\vartheta\); this corresponds to the first term of (5) or (6), with \(A\) equal to \(K'a^3\) in one case and to \(\frac{3}{2}\lambda_s\sigma a^3\) in the other. Therefore the atoms in the transition layer possess an additional energy because their moments are turned relative to the direction of easiest magnetization. If the transition is made more gradual, then the number of atoms participating in it grows proportionally to \(d\), and since in this case there is no compensating decrease in the energy of each atom, the total anisotropy energy per \(1\ \mathrm{cm}^2\) of boundary will grow proportionally to the thickness \(d\) of the boundary. The order of magnitude of this energy is

\[ A(\sin^2\vartheta)_{cp}\left(\frac{1}{a^3}\right)d\simeq A\cdot \frac{d}{a^3}. \]

Thus, the total energy of a unit surface of the boundary between domains consists of a part \(\gamma_q\), proportional to \(1/d\) and due to quantum forces, and a part \(\gamma_a\), proportional to \(d\) and due to the forces of anisotropy. The first part tends to increase the width of the boundary layer, and the second to decrease it. The actual thickness \(d\) must correspond to the minimum of both these parts [(see Fig. 5, d)].

It is easy to show that the function

\[ \gamma(d)=\alpha d+\frac{\beta}{d} \]

has a minimum when the two terms are equal. In this case

\[ d=\left(\frac{\beta}{\alpha}\right)^{\frac{1}{2}} \quad\text{and}\quad \gamma=2(\alpha\beta)^{\frac{1}{2}}. \]

Here

\[ \alpha\simeq \frac{A}{a^3},\qquad \beta\simeq \frac{I}{a}. \]

Therefore, to within numerical factors of order unity, we have

\[ d=a\left(\frac{I}{A}\right)^{\frac{1}{2}}, \qquad \gamma=\frac{2(IA)^{\frac{1}{2}}}{a^2} =\frac{I}{a^2}\frac{2a}{d}. \tag{8} \]

If \(I\simeq 10^{-13}\) and \(A=K'a^3\simeq 10^{-17}\), then we obtain \(d\simeq 100a\), i.e. the transition layer between domains has a width of the order of 100 interatomic-

distances. The total energy per \(1\ \mathrm{cm}^2\) in this case is \(d/2a\), or 50 times smaller than in the case of an abrupt transition.

A more exact analysis only confirms the results just obtained and gives an analytical form of the function \(\vartheta(x)\).

3. Dimensions and shape of the domains

Returning to Fig. 4, c), we can again attempt to determine the thickness \(w\) of the domains themselves. One may expect that it is much greater than the thickness \(d\) of the transition layers between them. Therefore the layers between domains may, with sufficient accuracy, be treated as surfaces of discontinuity with a surface-energy density \(\gamma\), determined by (8). This energy, in turn, tends to reduce the boundary surface as much as possible and, consequently, to reduce the number of domains and increase their thickness. Indeed, there are \(c/w\) boundaries, each of area \(bl\), and therefore the total boundary energy of the specimen is equal to \(\gamma bcl/w\) and decreases as \(w\) increases.

However, thick domains have at their ends regions [Fig. 5, e)] in which the magnetization turns through a right angle to the direction of easiest magnetization and which therefore possess the maximum density of anisotropy energy \(A/a^3\). The two triangular prisms at both ends of a domain together make up a rectangular prism of length \(b\) and square cross-section

\[ \left(\frac{w}{\sqrt{2}}\right)^2=\frac{w^2}{2}. \]

Since the total number of such pairs of prisms is \(c/w\), the total anisotropy energy will be

\[ \frac{A}{a^3}\cdot \frac{bw^2}{2}\cdot \frac{c}{w} = \frac{1}{2}\frac{A}{a^3}bcw; \]

this energy increases as \(w\) increases.

Thus, the energy of the boundary layers tends to make the domains thicker, while the energy of the ends of the domains at the surface of the specimen makes them thinner. The actual thickness is determined by the minimum of the total energy

\[ bc\left(\frac{\frac{1}{2}Aw}{a^3}+\frac{\gamma l}{w}\right), \]

which is a function of the same type as that considered above. The equilibrium value of \(w\) is found by equating the two terms:

\[ w=\left(\frac{2a^3\gamma l}{A}\right)^{\frac{1}{2}} = \left[ \frac{2a^3 l}{A}\cdot \frac{2(IA)^{\frac{1}{2}}}{a^2} \right]^{\frac{1}{2}} = 2(ld)^{\frac{1}{2}}. \tag{9} \]

According to formula (9), the size of the domains depends on the dimensions of the specimen; this result was to be expected, since in the present case we are dealing simultaneously with surface and volume energy. For specimens of linear dimensions \(\sim 1\ \mathrm{cm}\), with \(d \simeq 100a \simeq 10^{-6}\ \mathrm{cm}\), we obtain \(w \simeq 10^{-3}\ \mathrm{cm}\),

1. is the order of magnitude observed in Bitter bands. The volumes observed in the Barkhausen effect, \(10^{-9}\ \text{cm}^3\), do not contradict this, if they are interpreted as portions of domains whose linear dimensions are of the same order of magnitude as the thickness of the domains.

All this calculation\(^{1}\) was undertaken in order to get rid of the large magnetic energy associated with uniform magnetization. The question arises: have we, in the end, obtained an energy small enough to justify our more complicated distribution of magnetization? Substituting (9) into the formula for the total energy and dividing by the volume \(bcl\), we find the average energy per unit volume

\[ W=\frac{2A}{a^3}\left(\frac{d}{l}\right)^{\frac12}\simeq \frac{10^4}{\sqrt{l}}\ \text{erg}/\text{cm}^3 . \tag{10} \]

The magnetic energy of a uniformly magnetized specimen is given approximately by the formula

\[ W_{mag}=\frac12 D J_s^2 \simeq 10^6 D\ \text{erg}/\text{cm}^3, \tag{11} \]

where \(D\) is the demagnetizing factor. For \(l=100\ \text{cm}\), \(W<W_{mag}\) provided that \(D>0.001\). This requires that the diameter of the specimen be greater than \(0.4\ \text{cm}\). Thus the theory is capable of giving a satisfactory description of the domain structure of long thin specimens.

However, in other respects this analysis is incomplete. It was only shown that a certain distribution of magnetic moments into domains leads to a lower value of the energy than some other distributions, but it was not shown that it leads to the very lowest energy. In a complete solution it is necessary to determine the direction of magnetization as a function of position in such a way that this leads to a minimum of the total energy. Mathematically the problem can be formulated in this form, but it leads to differential equations that are too complicated. Moreover, the details of the domain distribution over the whole probability depend very strongly on random factors, such as inhomogeneous stresses and distortions of the lattice due to foreign inclusions, which are not taken into account by the simplified theory.

WEAK FIELDS (1)

We now know that a specimen can be demagnetized and still possess spontaneous magnetization inside domains. By what process can the specimen pass from this state into the state of uniform magnetization within each crystal? In what way can the magnetization change its sign when the field is reduced from the largest value to zero and then to the largest negative value? A detailed theory has been developed only for the simplest cases. Let us first consider these cases, and then discuss their possible generalization to materials with ordinary properties.

\(^{1}\) This calculation was first carried out by L. Landau and E. Lifshitz (Sow. Phys., 8, 153, 1935). Translator’s note.

1. Materials with positive magnetostriction subjected to strong tension

Iron-nickel alloys in a certain range of composition have positive magnetostriction. If a specimen of such a material is subjected to strong longitudinal tension, then its hysteresis loop takes, for example, the rectangular form shown in Fig. 6, a. In this case there is observed

Fig. 6. Magnetic behavior of alloys with positive magnetostriction under the influence of strong longitudinal tension $\sigma$

a—hysteresis loop $(B'ABA'B')$ and initial curve $(OPQB')$; $H_s$ “starting field”; $H_c$—“critical field”; b—formation and propagation of a remagnetization nucleus. The main field $H$ exceeds $H_c$, but is less than $H_s$; the auxiliary coil increases the field to $H_s$ in a small portion of the specimen

100% residual magnetization, which persists up to a certain value of the reverse field, equal to $H_s$. After this field is reached the magnetization changes sign in a single Barkhausen jump $(AB$ or $A'B')$. On the other hand, if the specimen is demagnetized by heating and then a magnetic field is applied to it, the magnetization will reversibly increase or decrease with the field along the curve $OP$ until the field remains smaller than the critical value $H_c$, which is considerably smaller than $H_s$. As soon as $H$ becomes equal to $H_c$, an irreversible jump $PQ$ occurs.

The discontinuous processes $AB$, $A'B'$ have been carefully studied, especially by Sixtus and Tonks. In order to produce the jump $A'B'$, it is not necessary to apply the field $H_s$ along the whole specimen: it is sufficient to have a field of this magnitude in a small part of the specimen, provided that the field in the remaining part exceeds the critical value $H_c$. With the aid of such an arrangement (Fig. 6, b) a nucleus of reverse magnetization is first created at the point where the field is equal to $H_s$, and then it grows until the whole specimen has been remagnetized. The boundary of the remagnetization nucleus moves with a finite velocity $v$, whose order of magnitude is equal to $10^4$ cm/sec. Therefore one may interpret the “starting field” $H_s$ as the field necessary for creating a remagnetization nucleus, and the “critical field” $H_c$ as the field at which a nucleus, once formed, can spread along the whole specimen. If the specimen is demagnetized by heating, then it consists of many domains magnetized in different directions, and not of a single domain magnetized in the opposite direction, as in the case $BA'$. Therefore nuclei always exist, and the field need only reach the value $H_c$ in order that correctly magnetized domains may absorb incorrectly magnetized ones $(PQ)$.

The propagation velocity $v$, found in the experiment of Fig. 6, b, is a linear function of the excess of $H$ over the critical value $H_c$,

\[ v = B(H - H_c). \tag{12} \]

This is easy to explain¹) if one assumes the possibility of displacement of the boundary \(S\) (Fig. 6, \(b\)) and the presence of internal forces opposing this displacement when the field is less than \(H_c\). If the boundary moves forward, this motion changes the magnetic induction, which in turn leads to the appearance of eddy currents. The magnetic field of these eddy currents is oriented in such a direction as to hinder the motion that produced it, and the magnitude of this field is proportional to the velocity \(v\) of motion of the boundary. The effective field acting on the boundary is the resultant of the applied field \(H\) and the field of the eddy currents. It may be written in the form \(H - Cv\), where \(C\) is a constant. If there were no eddy currents, the boundary could move almost instantaneously. But because of them it cannot exceed the velocity at which the effective field is equal to \(H_c\); therefore the actual velocity is determined from the equality

\[ H - Cv = H_c. \tag{13} \]

This formula differs from (12) only in form.

The constant \(B\) or \(C\) can be calculated theoretically in approximate agreement with experiment; it does not depend on the tensile stress \(\sigma\), on which the critical field \(H_c\) depends. The meaning of this critical field can be understood better if one investigates the nature of the reversible change \(OP\).

At the point \(O\) the specimen consists of domains, apparently having approximately the same dimensions and shape as were described above. However, the exact position and shape of the boundaries will be determined by the disordered internal stresses which we previously neglected. Following Kersten, let us imagine an idealized picture of the internal stresses in the form of a periodic alternation of regions of longitudinal tension and compression along the direction \(x\), perpendicular to the axis of the specimen and to the boundaries between domains. If a large external tensile stress \(\sigma_0\) is superposed on such a distribution, then the resultant stress \(\sigma\) will oscillate along the axis \(X\), as is shown in Fig. 7, \(a\). The energy \(\gamma\) per \(1\ \mathrm{cm}^2\) of boundary between domains is given by (8) with \(A = \frac{3}{2}\lambda_s\sigma a^3\). Therefore \(\gamma\) varies depending on the position of the boundary, in view of the fact that \(\sigma\) depends on \(x\). \(\gamma\) has a minimum when the boundary is located at the minimum \(\sigma(x=0)\), as shown in Fig. 7, \(b\). If the boundary is displaced from this position by a distance \(x\) [Fig. 7, \(c\)], then \(\gamma\) increases. The forces acting on the boundary are equivalent, in their action, to a hydrostatic pressure \(p_\gamma\) tending to return the boundary to the position of minimum energy \(x=0\). If \(\gamma(x)\) is regarded as the potential energy of \(1\ \mathrm{cm}^2\) of boundary, expressed as a function of the boundary displacement \(x\), then

\[ p_\gamma = \frac{d\gamma(x)}{dx}; \]

\(p_\gamma = 0\) at the point of minimum \(x=0\) and reaches a maximum at the point \(x_c\), where the slope of the curve \(\sigma(x)\) is greatest.

¹) See, for example, V. K. Arkad’ev, Electromagnetic Processes in Metals, part II, ONTI, 1936.

In a zero field the boundary is at \(x=0\). If a weak field \(H\) is applied in the direction indicated in Fig. 7, c), then it is more advantageous for the magnetized domain \(A\) to tend to grow at the expense of its neighbor \(B\). The potential energy in the given field \(H\) will decrease by \(2J_s H\,dx\) per \(1\ \mathrm{cm}^2\) of boundary when the boundary is displaced by \(dx\). Therefore the action of the field is equivalent

Fig. 7. Reversible and irreversible displacements of the boundary

a) Distribution of stresses obtained as a result of imposing external constant stress \(\sigma_0\) on inhomogeneous internal stresses; b) position of the boundary without a field; c) reversible displacement of the boundary for \(H < H_c\); d) Barkhausen jump for \(H=H_c\); the boundary, having reached \(x_c\), propagates irreversibly through the remnant of region \(B\)

to a hydrostatic pressure \(p_H=2J_sH\), tending to displace the boundary downward, as shown in Fig. 7, c). The equilibrium position of the boundary in the given field is determined from the condition of exact equality of these oppositely directed pressures \(p_\tau\) and \(p_H\) [Fig. 7, c]. Hence we obtain

\[ \frac{d\tau(x)}{dx}=2J_sH. \tag{14} \]

If the field increases slowly, the boundary is gradually displaced, reaching an equilibrium position for each value of \(H\), and returns to its former positions if \(H\) again decreases, until it has reached the point \(x_c\). This gives us the reversible magnetization curve \(OD\) in Fig. 6. But when \(H\) reaches the field \(H_c\) corresponding to the displacement \(x_c\), the equilibrium becomes unstable.

With a very small further displacement there occurs not an increase but a decrease of the opposing pressure \(p_\tau\), and the boundary moves without hindrance, with a velocity limited only by eddy currents, until the domain \(B\) is completely absorbed1. This is what gives us the irreversible jump \(PQ\).

2. Materials with negative magnetostriction, subjected to strong tension

The process of magnetization will have an entirely different character if longitudinal tension is applied to a wire whose saturation magnetostriction \(\lambda_s\) is negative, for example in the case of nickel. In this case the tension orients the magnetization in all domains in a plane perpendicular to the axis of the specimen, but in this plane the directions of magnetization of the domains are distributed chaotically. With respect to the longitudinal field all domains are equivalent, and therefore there is no reason for some of them to absorb others. In this case the whole action of the field reduces to the simultaneous rotation of all domains from a direction with angle \(\vartheta = \frac{\pi}{2}\) to a direction making an angle \(\vartheta < \frac{\pi}{2}\) with the direction of the field\(^{1}\).

The resultant magnetization \(J = J_s \cos \vartheta\) can be found from the condition of minimum total energy. The conclusions of the theory were checked experimentally by Becker and Kersten, who extended the theory of rotation to the case of domains in which the stresses have an internal origin\(^{2}\) and vary from domain to domain. Although this theory of rotation does lead to correct orders of magnitude for many magnetic characteristics, it is nevertheless hardly correct, since the process of rotation plays only a secondary role in the region of weak fields for ordinary ferromagnetic materials\(^{3}\).

3. Ordinary materials

There is no doubt that the two processes described in § 1—reversible and irreversible displacement of the boundaries between domains—are responsible for the magnetization of section (I) of the magnetization curve of ordinary ferromagnetic materials. The boundaries shift reversibly in the region where equilibrium between external and internal forces is possible. Having reached unstable positions, the boundaries move irreversibly with a velocity limited by eddy currents. Both of these processes can be observed on Bitter patterns when a field is applied that has a component along the surface. As the field increases, the distance between some bands increases and between others decreases. This process consists partly of a slow displacement, which can proceed in the reverse direction when the field is decreased, and partly of sudden jumps, after which the boundary does not return to its former position when the field is decreased.

The residual magnetization and the coercive force depend on irreversible jumps. This problem is much more complicated than the simple example considered in § 1, and a quantitative theory is only just beginning to be developed.

\(^{1}\) Moreover, the magnitude \(\vartheta\) depends on the value of the field \(H\) and on the anisotropy forces. Translator’s note.

\(^{2}\) Residual stresses of the material. Translator’s note.

\(^{3}\) This theory may be of interest for the case of so-called hard magnetic alloys, possessing an anomalously large coercive force. Translator’s note.

Reversible processes are much easier to study theoretically, and here many more quantitative results have been obtained. Some properties of ferromagnetic materials in the first approximation depend only on the magnitude of the magnetization \(J\), and not on the method by which this magnitude \(J\) was obtained. For example, these properties are the same both at the point \(P\) and at the point \(Q\) (Fig. 1). This is true for the “reversible permeability,” which is equal to the ratio

\[ \frac{\Delta J}{\Delta H}, \]

measured for small changes \(\Delta H\), made in the reverse direction relative to the increase of the field applied when the initial state was reached. In Fig. 1 this is indicated by the small arrows \(PP'\) and \(QQ'\). This permeability is called “reversible” because such a change is comparatively free of discontinuous Barkhausen jumps. In dealing with properties of this kind, one can obtain sufficiently satisfactory theoretical results by ignoring the irreversible properties and considering a hypothetical reversible specimen, equivalent to the real specimen in the region concerning its reversible properties.

In the ideal specimen of Fig. 7, the function \(\gamma(x)\) can be expanded in a Taylor series

\[ \gamma(x)=\gamma_0+\frac{1}{2}\gamma_2 x^2+\cdots, \tag{15} \]

where the first-order term is absent, and the coefficient of \(x^2\) is positive, since \(\gamma\) has a minimum at \(x=0\). Then (14) gives

\[ \gamma_2 x+\cdots=2J_s H, \tag{16} \]

and, for small reversible displacements \(\delta x\) caused by a change of field \(\delta H\),

\[ (\gamma_2+\cdots)\delta x=2J_s\delta H, \tag{17} \]

where the omitted terms are small if \(x\) is small. Therefore, for not too large \(x\),

\[ \delta x=k\cdot 2J_s\delta H, \tag{18} \]

where \(k\) is approximately constant. If the total surface of the boundaries between domains per unit volume is equal to \(S\), then the change of magnetization is

\[ \delta J=2J_s\cdot S\delta x=4kSJ_s^2\delta H. \tag{19} \]

Following Kondorsky, let us suppose that, owing to the disorder of the internal stresses, the shape of the domains is irregular and equation (19) is valid at a constant mean value of \(k\) throughout the entire magnetization process.

The surface \(S\) depends on the relative volumes \(v_A\) and \(v_B\) occupied by domains of the two types \(A\) and \(B\) [Fig. 7, c)]. It vanishes when one of these domains disappears (saturation in some direction). One may expect that \(S\) is maximal when the volumes are equal (the demagnetized state). Kondorsky proposes the relation

\[ S=s\cdot v_A\cdot v_B,\quad s=\mathrm{const.}, \tag{20} \]

which, although it cannot be regarded as exact, has a simple form and satisfies the conditions indicated above. Since the magnetization is given by the formula \(J=J_s(v_A-v_B)\), and since \(v_A+v_B=1\), we have

\[ v_A=\frac{1+\frac{J}{J_s}}{2}, \qquad v_B=\frac{1-\frac{J}{J_s}}{2}. \]

From (19) and (20) we arrive at the relation between the reciprocal susceptibility \(\dfrac{\delta J}{\delta H}\) and the magnetization \(J\):

\[ \frac{\delta J}{\delta H} = k s J_s^2 \frac{1-\frac{J^2}{J_s^2}}{J_s^2}. \tag{21} \]

In the same way, in a more general case, Kondorskii was able to obtain formulas for the reciprocal susceptibility of crystals of various types. Unfortunately, there are as yet no experimental data that could verify the regularity of Kondorskii’s assumptions. However, the calculation can be generalized to the case of a polycrystalline material, using the concept of an “equivalent reversible specimen.”

For a reversible specimen, \(J\) is a single-valued function of \(H\), which can be found by integrating (21). For the specimen shown in Fig. 7, or for a cobalt single crystal, this gives

\[ J=J_s \operatorname{th} k s J_s H. \tag{22} \]

If \(H\) makes an angle \(\vartheta\) with the direction of easiest magnetization, then \(H\) must be replaced by its component \(H\cos\vartheta\) along this direction, and the entire expression must be multiplied by \(\cos\vartheta\), in order to obtain the component of the magnetization in the direction of the field. In the case of a polycrystalline specimen this result must be averaged over all orientations of the crystals. As a result one obtains an expression \(J=f(H)\) for the magnetization curve of a fictitious reversible polycrystalline cobalt specimen. The derivative \(\chi=f'(H)\) is equal to the susceptibility. These expressions may be regarded as parametric equations determining \(\chi\) as a function of \(J\). According to the hypothesis of the equivalent reversible specimen, the reciprocal susceptibility \(\chi_r\) of the actual specimen is the same function of \(J\), although the parameter \(H\) no longer has a simple physical meaning.

In Fig. 8 the predictions of the theory (curve 2) are compared with experimental data for polycrystalline cobalt. Although the agreement is not brilliant, curve 2 coincides with the experimental points incomparably better than curve 1, calculated by the formula proposed many years ago by Gans. Gans’s formula corresponds to the limiting case—a crystal with several axes of easiest magnetization, distributed uniformly over the sphere of unit radius. It agrees well with experiment in the case of polycrystalline iron and nickel.

Curve 3 was calculated according to another theory, in which the symmetry of the crystal is taken into account, but a different mechanism of magnetization is assumed. The practical coincidence of curves 2 and 3 and their sharp difference from curve 1

show that the crystallographic nature of the material has a more substantial influence on its magnetic properties than does one or another mechanism of magnetization. From this one may draw two conclusions. The first of them is somewhat discouraging, namely, that agreement between theory and experiment is not necessarily proof of the correctness of the mechanism assumed by the theory.

Fig. 8. Reciprocal susceptibility \(\chi_a\) of a cobalt polycrystal. Experimental points (M. Samuyel): \(o\)—initial curve; \(\bullet\)—“ideal,” or “hysteresis-free,” curve; \(\triangle\)—descending branch of the hysteresis loop; \(\times\)—ascending branch. Theoretical curves: \(1\)—displacement of boundaries; directions of easiest magnetization distributed isotropically; \(2\)—displacement of boundaries; correct crystallographic symmetry. Artificial model with constant internal forces, but with correct crystallographic symmetry; \(\chi_0\)—initial susceptibility.

Fig. 9. Change of magnetostriction \(\lambda\) with magnetization for a polycrystalline nickel at \(6^\circ\). Circles represent experimental values (D. Kirkham). The curve is calculated from a formula which may be derived either from the theory of boundary displacement or by a statistical method. \(\lambda_\infty\)—saturation magnetostriction.

of the mechanism which the theory assumes. The second conclusion is more encouraging—knowledge of the exact mechanism is not necessary for an approximate calculation of magnetic properties.

A large number of properties can be calculated quite satisfactorily with the aid of a theory in which no definite mechanism of magnetization is adopted, but only the given structure is taken into account in the most general form. Suppose that we investigate a ferromagnetic specimen by means of a large number of test holes (peep-holes), which are small in comparison with the domains and are located sufficiently far from one another, so that what we see in one of them does not especially help us to guess what we shall see in another hole. Averaging over all the test holes, we shall obtain a good representation of the average magnetization, magnetostriction, and, in general, of the specimen as a whole. If the distribution of domains is determined mainly by random internal stresses, then statistical methods may be used to determine the relative fraction of the various axes of easiest magnetization and thereby to compute certain magnetic properties. Historically, this method more

older than Kondorskii’s method. All the results obtained by the old method can be obtained if Kondorskii’s assumptions are combined with the hypothesis of an equivalent reverse sample. Fig. 9 shows how well this theory predicts the dependence of magnetostriction on magnetization in polycrystalline nickel. Of course, such good agreement is an exception. In most cases there are small systematic deviations, which indicate that a number of factors have not been taken into account in the theory.

CONCLUSION

An attempt has been made here to give a general characterization of the methods used for studying very complex phenomena. The basic ideas have been set forth and simple relations have been obtained, without any claim to completeness or accuracy. It seems to us, however, that the fundamental propositions of the theory are correct. Although a mathematical theory of the hysteresis loop has not yet been constructed, the existence of domains has been proved.

LITERATURE

I. Monographs

1. E. C. Stoner, Magnetism and Matter, Methuen & Co., Ltd., London, New York, 1934.
2. F. Bitter, Introduction to Ferromagnetism, McGraw-Hill, New York, 1937.
3. R. Becker (editor), Probleme der technischen Magnetisierungskurve. (For a translation of the article by K. Sixtus and W. Döring see Uspekhi Fizicheskikh Nauk, 22, 58, 63, 78, 1939. See also the recently published monograph: R. Becker and W. Döring, Ferromagnetismus. Translator’s note.)
4. L. F. Bates, Modern Magnetism, Cambridge, and Macmillan, New-York, 1939.
5. R. H. Fowler, Statistical Mechanics, second edition, Macmillan, New-York, 1936.

II. References on special questions

The first several references under each heading refer to specific pages (indicated in parentheses) of the books cited in Section I. The remaining references pertain to the periodical literature. The articles indicated in them are cited either because they are of general significance, or because they contain data that cannot be found in the books of Section I. In the latter case, in the cited articles it is necessary to take into account references to earlier works by the same or other authors.

A. Spontaneous magnetization

1. (114—129, 350—367, 434—435, 437); 2. (29—40, 126—143);
2. (233—241); 5. (477—501).
3. L. Pauling, Phys. Rev., 54, 899, 1938.

B. Saturation magnetostriction and magnetic anisotropy

1) Crystals

1. (385—401); 2. (176—178, 194—213, 246—255); 3. (150—163, 298—303);
2. (150—163, 298—303); 5. (503—517).
3. R. M. Bozorth, J. Appl. Physics, 8, 575, 1937.
4. H. J. Williams, Phys. Rev., 52, 1004, 1937.
5. J. H. Van Vleck, Phys. Rev., 52, 1178, 1937.

2) Anisotropy produced by stresses
2. (243—246); 4. (305—307).
10. R. Becker, Physik. Z., 33, 905, 1932.
11. L. W. McKeehan, Phys. Rev., 53, 301, 1938.

C. Domains—experiment

1) Surface poles
2. (55—66); 4. (320—321).
12. W. C. Elmore, Phys. Rev., 53, 757, 1938; 56, 210, 1939.

2) The ordinary Barkhausen effect
1. (404, 411—413); 2. (21, 290—291); 4. (321—326).

3) Large Barkhausen jumps
1. (413—415); 3. (9—25); 4. (326—329).
13. C. W. Heaps, Phys. Rev., 50, 176, 1936.

4) Magnetization curves of alloys under tension
3. (3—4); 4. (324).
14. M. Kersten, Z. techn. Physik, 19, 546, 1938 or Physik. Z., 39, 860, 1938.

5) Magnetization curves of nickel under tension
2. (245—246); 3. (2—3); 4. (306—307); reference 10.

D. Domains—theory

1) Nature, form, and dimensions of domains
2. (185—192, 293—299); first reference 12.
15. F. H. Reinard, Phys. Rev., 55, 312, 1938.

2) Propagation of Barkhausen jumps
3. (30—41); 4. (329).
16. W. Döring and H. Haake, Z. techn. Physik, 19, 551, 1938 or Physik. Z., 39, 865, 1938.
17. R. Becker, Z. techn. Physik, 19, 542, 1938 or Physik. Z., 39, 856, 1938.
18. H. Schlechtweg, Ann. Physik, 35, 657, 1939.

3) Reversible and irreversible displacement of boundaries
1. (409—411); 3. (42—72); references 10, 14.
19. W. F. Brown, Phys. Rev., 55, 568, 1939.

4) Theory of rotation
reference 10.

5) Statistical theory
1. (390—392); 2. (255—260); 5. (506—507, 517—521); reference 19.

Addendum by the translator S. V. Vonsovskii

1. THE FORM AND DIMENSIONS OF DOMAINS IN A FERROMAGNETIC CRYSTAL WITH SEVERAL AXES OF EASIEST MAGNETIZATION

a) In Brown’s article a general description is given of the domain structure of a ferromagnetic crystal with one axis of easiest magnetization. The extension of this description to the case of a crystal with several axes of easiest magnetization encounters a number of difficulties and is not trivial. Most technical ferromagnetic materials (iron, nickel, and their alloys) have three (iron) or four (nickel) axes of easiest magnetization; therefore such a generalization is of undoubted interest for the theory of the technical magnetization curve.

The number of domains with different orientation of the spontaneous magnetization, or, as it is commonly said, the number of magnetic phases, is equal to twice the number of axes of easiest magnetization. For definiteness let us dwell on the case

for the case of iron and all alloys having three such axes, coinciding with the tetragonal axes \([100]\), \([010]\), and \([001]\) of the cubic lattice. In this case the number of magnetic phases is equal to six. Between these six phases there are two types of adjacency: 1) \(180^\circ\) adjacencies between phases with magnetization parallel and antiparallel to one and the same

Fig. 10. \(a\)—\(180^\circ\) adjacency, \(b\)—\(90^\circ\) adjacency of the 1st type,
\(c\)—\(90^\circ\) adjacency of the 2nd type

easy axis (Fig. 10, \(a\)); 2) \(90^\circ\) adjacencies between phases belonging to two mutually perpendicular axes of easiest magnetization. Each of the \(90^\circ\) adjacencies, depending on the geometrical arrangement of the adjoining magnetic phases, may in turn be of two kinds: 2, \(a\)) \(90^\circ\) adjacencies with rotation of the magnetization \(J_s\) through \(90^\circ\) (in passing from one domain to the neighboring one) about the easy axis perpendicular to the domain boundary (Fig. 10, \(b\)); we shall call them \(90^\circ\) adjacencies of the first type; 2, \(b\)) \(90^\circ\) adjacencies with rotation of the magnetization through \(90^\circ\) in the plane in which the vectors \(J_s\) of the neighboring domains lie (Fig. 10, \(c\)); we shall call them \(90^\circ\) adjacencies of the second type.

An investigation similar to that set forth in Brown’s paper shows that \(180^\circ\) adjacency in an unstressed three-axial crystal is energetically unfavorable. With gradual rotation of the spins in the boundary layer between neighboring domains [Fig. 5, \(b\)], after a rotation through \(90^\circ\) the spins are oriented along an easy axis and tend to form a \(90^\circ\) adjacency of the first type. If, nevertheless, a \(180^\circ\) adjacency is artificially “imposed” on the crystal, then in the middle of the boundary layer there will be a discontinuity in the orientation of \(J_s\), equal to the change of the angle \(\hat{\theta}\) (Fig. 5) by \(90^\circ\) (from \(45^\circ\) to \(135^\circ\)) between two neighboring atomic planes. The boundary energy of such a layer with a “discontinuity” is equal (to within factors \(\sim 1\)) to the sum of the energy \(\gamma\) [from formula (8)] and the discontinuity energy

\[ \frac{I}{a^2} \]

[Fig. 5, \(a\)]. The magnetostrictive stress arising in a \(90^\circ\) adjacency leads to a lower energy than the \(180^\circ\) boundary.

Therefore, the natural adjacency in an ideal, undeformed three-axial crystal is a \(90^\circ\) adjacency of the first type. The width and energy of the corresponding boundary layer have a form completely analogous to (8) (the difference will be only in numerical factors \(\sim 1\)).

Somewhat more complicated is the question of a 90° neighborhood of the second type. The distribution of the orientations \(\mathbf{J}_s\) in the boundary layer must not only satisfy the condition of a minimum of the sum of the exchange and magnetic-anisotropy energies, but must also be such that no volume density of “magnetic” charges arises in the boundary layer, which would create a magnetic field and thereby increase the energy of the crystal. Therefore the distribution \(\mathbf{J}_s\) must satisfy the condition \(\operatorname{div}\mathbf{J}_s=0\) (\(\operatorname{div}\mathbf{J}_s\) is equal to the volume density of “magnetic” charges). The distribution of Fig. 5, b) satisfies this requirement. Indeed, if we denote by \(X\) the axis perpendicular to the boundary layer, and choose the other two axes \(Y\) and \(Z\) in the plane of the layer, then the whole distribution will be a function only of \(x\); \(J_{sx}=0\), and therefore

\[ \operatorname{div}\mathbf{J}_s=\frac{\partial J_{sy}(x)}{\partial y}+\frac{\partial J_{sz}(x)}{\partial z}=0. \]

In order that \(\operatorname{div}\mathbf{J}_s=0\) also in the case of a 90° neighborhood of the second type, it must have the form shown in Fig. 11. First, the boundary must be inclined at an angle of 45 and 135° to the directions \(\mathbf{J}_s\) in the neighboring domains, and the whole distribution must depend only on the coordinate \(x'\). Secondly, the rotation of the vector \(\mathbf{J}_s\) in passing from domain \(I\) to domain \(II\) must occur in such a way that the projection of \(\mathbf{J}_s\) on the axis \(X'\), \(J_{sx'}\), remains constant and equal to \(J_s\cos45^\circ=\dfrac{J_s}{\sqrt{2}}\). This can be accomplished if the vector \(\mathbf{J}_s\) is made to rotate along the generatrix of a cone whose height is the direction parallel to the axis \(X'\), and whose angle at the vertex is 45° (Fig. 11). In this case

Fig. 11

Fig. 12

\[ \operatorname{div}\mathbf{J}_s=\frac{\partial J_{sy'}(x')}{\partial y'}+\frac{\partial J_{sz'}(x')}{\partial z'}=0. \]

As calculation has shown, the thickness of the boundary layer \(d\) and its energy \(\gamma\) in the case of a 90° neighborhood of the second type are, to within a factor \(\sim 1\), the same as in the case of a 90° neighborhood of the first type, i.e. analogous to expressions (8).

b) Knowing the character of the boundaries between domains, one can draw a number of conclusions about the form of the domains in a triaxial ferromagnetic crystal. Let us consider several specimens of the simplest form.

¹) A rectangular parallelepiped with edges parallel to the easy axes (Fig. 12). In this case there is no need to divide the specimen into small domains. It is sufficient to distribute the magnetization over five domains (Fig. 12) in order that the conditions \(\operatorname{div}\mathbf J_s=0\) and \(J_{sn}=0\) (\(J_{sn}\) is equal to the normal component of \(\mathbf J_s\) to the surface of the specimen) be satisfied. The appearance of the intermediate domain \(V\) is needed in order to eliminate the 180° adjacency between domains \(I\) and \(III\), for the two 90° boundaries \(I—V\) and \(III—V\) are energetically more favorable than one 180° boundary \(I—III\).

The choice of the distribution shown in Fig. 12 is, of course, not unique. It may be replaced, for example, by a distribution rotated through 90° about one of the easy axes. In passing from one distribution to another there will be a small change in the energy of the crystal. Therefore the realization of one or another of these energetically almost equivalent distributions will depend substantially on the conditions and methods of obtaining the initial demagnetized state. As a general conclusion it must be stated that in an ideal crystal of the form under consideration there should be a tendency for the domains to grow to a size of the order of the dimensions of the specimen itself.

2) Let us now consider a specimen in which at least one face is not parallel to any of the easy axes. In this more general case the condition \(\operatorname{div}\mathbf J_s=0\) can no longer be satisfied by dividing the specimen into a small number of large domains. Here there occurs a distribution of magnetization analogous to that obtained for cobalt by Landau and Lifshitz². Inside the crystal (Fig. 13) there are regular plane-parallel domains \(I, II, III, IV, \ldots\), magnetized along and against the axes \([001]\) and \([010]\). Domains \(I, III, \ldots\), on approaching the surfaces of the specimen \(aa'\) and \(bb'\), “sharpen.” Between these sharpened ends there are surface domains \(V, VI, VII, VIII, \ldots\), having, in the plane of the drawing, a cross-section in the form of a trapezoid. These domains form 90° adjacencies of the first type with domains \(II, IV, \ldots\) and 90°¹) adjacencies of the second type with domains \(I, III, \ldots\). The equilibrium value of the thickness \(d\) of the domains is determined from the condition of a minimum of the free energy of the given distribution, which consists of two parts: a) the magnetic-anisotropy energy \(E_s\) of the surface domains \(V, VI, \ldots\), magnetized in directions not coinciding with the axes of easiest magnetization, and b) the energy \(E_i\) of the boundary layers between the domains. If by \(l_1, l_2\), and \(l_3\) we denote the dimensions of the specimen along the three axes, and by \(\vartheta_1, \vartheta_2, \varphi_1, \varphi_2\) and \(\psi_1, \psi_2\), respectively, the angles between the axes \([001]\), \([010]\), \([100]\) and the lines of intersection of the faces of the specimen with the corresponding section (Fig. 13), then it is easy to show that

Fig. 13

¹) Here, of course, 90° must be regarded simply as an abbreviated notation, for in fact the angle between the vectors \(\mathbf J_s\) of these adjacent domains is not equal to 90°.

the equilibrium thickness of the domains \(d\) is equal to

\[ d=2\sqrt{l_2l_3}\sqrt[4]{\frac{I}{ak}}\,[\,l_1(\sin\psi_1\cos^2\psi_1+\sin\psi_2\cos^2\psi_2)+ \]
\[ {}+l_2(\sin\vartheta_1\cos^2\vartheta_1+\sin\vartheta_2\cos^2\vartheta_2)+ \]
\[ {}+l_3(\sin\varphi_1\cos^2\varphi_1+\sin\varphi_2\cos^2\varphi_2)\,]. \tag{1'} \]

If all the angles tend to \(0\) or \(\frac{\pi}{2}\), then \(d\to\infty\). This means that, in passing to a specimen in the form of a rectangular parallelepiped with edges along the easy axes, the domains become commensurable with the dimensions of the specimen.

Since the dimensions \(l_i\) and all the angles \(\vartheta_i\), \(\varphi_i\), and \(\psi_i\) may change from section to section of the specimen and even from point to point, the domain thickness will also be a function of the coordinates of the specimen. The fact that the factor \(\sqrt{l_2l_3}\) contains only the two dimensions of the specimen belonging to the section in which the regular domains lie is a mathematical expression of the “physical ambiguity” of the given distribution \(\mathbf J_s\).

The replacement of the “laminar” distribution of domains by any other—for example, by a distribution of \(\mathbf J_s\) in the form of prismatic domains—is energetically unfavorable. Thus one may say that in a crystal with three easy axes the domain structure has a “laminar” character. The system of plane-parallel domains forms \(90^\circ\) neighborhoods of the first type. Each pair of nearest antiparallel magnetized domains (separated by a domain magnetized at an angle \(+90^\circ\)) forms a closed magnetic chain together with the surface domains, with which they form \(90^\circ\) neighborhoods of the second type.

c) In passing to a real crystal it is necessary to take into account the presence of internal inhomogeneities, residual strains, foreign inclusions, etc. Therefore the plane-parallel form of domains can no longer correspond to the minimum of the free energy of the crystal. The positions of the plane boundaries between domains passing through the whole crystal are not connected with the distribution of stresses in the crystal. Therefore the division of the crystal into plane domains should not depend on internal stresses. But this clearly contradicts experiment, which indicates the dependence of all technical magnetic properties of a ferromagnet on internal stresses. In particular, the magnitude of the initial permeability, which is connected with the position of the boundaries between domains, depends strongly on them. If in a real crystal the boundaries were plane independently of stresses, then the crystal would be magnetized to saturation in vanishingly small fields under any treatment of the material. Moreover, the existence of residual magnetization, which appears in all real specimens after the first magnetization to saturation, would be completely incomprehensible. From the point of view of the ideal model, in the absence of a magnetic field spontaneous division into plane domains should occur. Therefore, in all probability, in real ferromagnets, in which

there are always internal stresses, somehow distributed over the volume of the crystal; the boundary zones pass through places with a minimum value of the stresses, or through places where the stress changes sign (the boundary of regions of compression and tension). Such a distribution of the boundaries between domains corresponds to a minimum of their energy \(\gamma\).

In the case of a polycrystalline material, the presence of large homogeneous stresses inside a volume containing several domains can make the material magnetically uniaxial and thereby make \(180^\circ\) neighborhoods energetically favorable\(^1\). For this it is necessary that the signs of the stresses \(\sigma_i\) and of the magnetostriction \(\lambda\) coincide, and that their products \(\lambda\sigma_i\) be considerably greater than the constant of magnetic anisotropy \(K\). If this condition is satisfied and \(\sigma_i\) is homogeneous, then the polyaxial material becomes magnetically textured, which can considerably increase its magnetic properties.

2. THE THEORY OF THE COERCIVE FORCE OF BLOCH—KONDORSKY

The modern ideas set forth by Brown concerning the processes of technical magnetization in the region of weak fields make it possible to construct not only a theory of reversible phenomena in weak fields, but also of irreversible ones. The basis of the theory of irreversible phenomena is the equilibrium equation (14) between the hydrostatic pressure of the external magnetic field

\[ p_H = 2J_s H \]

and the internal pressure

\[ p_\gamma = \frac{d\gamma(x)}{dx}. \]

The reversible displacement of the boundary, described by equation (14), will occur until the boundary, having left its equilibrium position \(x = 0\), corresponding to

\[ \frac{d\gamma}{dx}=0 \]

(Fig. 14), reaches the position \(x_A\), which corresponds to the nearest maximum

\[ p_\gamma = \left(\frac{d\gamma(x)}{dx}\right)_{\max}. \]

At this point the field reaches the value

\[ H'_0 = \frac{1}{2J_s}\left(\frac{d\gamma}{dx}\right)_{\max}, \]

which plays the role of a critical field. After the field \(H'_0\) has been reached, the boundary may continue its displacement already without further increase of the external field. This spontaneous displacement will continue either until the neighboring domain is completely absorbed, or to the point \(x_B\) (Fig. 14), beginning from which the internal pressure \(p_\gamma\) increases again. With a further increase of \(H(>H'_0)\), reversible displacement of the boundary begins again, which may pass into an irreversible jump after reaching at \(x_C\) the next, higher maximum of the pressure \(p_\gamma\). Irrever-

Fig. 14

\(^1\) See, for example, Ya. Shirobokov, ZhETF, 11, 1941.

the possibility of the transition of the boundary from \(x_A'\) to \(x_B\) follows from the following: if, after reaching the point \(x_C\), the magnitude of the magnetic field is decreased, then at \(H=H_0'\) the boundary will remain at the point \(x_B\), and with a further decrease of the field it will not make the jump to \(x_A\), but will be displaced back to the point \(x_D\), which corresponds to the nearest maximum of \(p_x\) (under reverse displacement the roles of maxima and minima are interchanged). After this the boundary, without any further decrease of the field, may pass by a jump to the point \(x_E\). Only from this position can the boundary return to the initial point \(x=0\), but, generally speaking, at a different value of the magnetic field. From this consideration, which is a schematic illustration of the mechanism of technical magnetization, it is seen that the cause of irreversibility is the inhomogeneity of the surface energy \(\gamma\). The more homogeneous the energy \(\gamma\), i.e. the smaller its gradients, the smaller are the values of the critical fields \(H_0\), and the narrower is the interval of fields in which irreversible phenomena occur in ferromagnetics.

Bloch\(^3\) assumed that the change in the energy \(\gamma\) upon displacement of the boundaries is caused by changes in the exchange integral due to inhomogeneities of the lattice of a real crystal. Bloch did not take into account the possibility of changes in the magnetoelastic part of the energy \(\gamma\) upon displacement of the boundaries. Since experiment indicates a direct connection between coercive force, critical field, and elastic stresses, Kondorskii\(^4\) considers it more probable that the main part of the change in the energy \(\gamma\) upon displacement of a boundary is caused by changes in the magnetoelastic energy.

The general expression for the critical field according to (14) has the form:

\[ H_0=\frac{1}{2J_s}\left(\frac{d\gamma}{dx}\right)_{\max}, \tag{2′} \]

where \(\gamma\), according to (8), is equal to

\[ \gamma=\frac{2}{a^2}\sqrt{IA}. \tag{3′} \]

The magnetoelastic energy \(A=\frac{3}{2}\lambda\sigma\), where \(\sigma\) denotes internal stresses,\(^a\) and \(\lambda\) is the saturation magnetostriction. Assuming that in (2′) only \(\sigma\) depends on \(x\), we find:

\[ \frac{d\gamma}{dx}=\frac{3}{2}\lambda d\,\frac{d\sigma}{dx}, \]

where \(d\) is the boundary thickness, determined from (8). Consequently, from (2′)

\[ H_0=\frac{3\lambda d}{4J_s}\left(\frac{d\sigma}{dx}\right)_{\max}. \tag{4′} \]

If the mean distance between two successive minima of \(\sigma(x)\) is large in comparison with the boundary thickness \(d\), then for simplicity one may suppose that \(\sigma(x)\) has the form shown in Fig. 15. In this case

\[ \frac{d\sigma}{dx}=\frac{2\Delta\sigma}{l}, \]

where \(\Delta\sigma\) is the mean magnitude of the stress inhomogeneities, and \(l\) —

the average distance between neighboring minima. Therefore from (4′) we have:

\[ H_0=\frac{3}{2}\frac{\lambda \Delta\sigma}{J_s}\frac{d}{l}\quad (d\ll l). \tag{5′} \]

In the opposite case \((l\ll d)\) the stress distribution will

Fig. 15

Fig. 16

have the form shown in Fig. 16. As Kersten\(^5\) showed,

\[ H_0 \simeq \frac{1}{2}\frac{\lambda\Delta\sigma}{J_s}\frac{l}{d}\quad (d\gg l). \tag{6′} \]

In the general case (5′) and (6′) can be written in the form

\[ H_0=p_0\frac{\lambda\Delta\sigma}{J_s}, \tag{7′} \]

where \(0\le p_0\le 1\) is a numerical factor whose magnitude is determined by the distribution of internal stresses.

In Fig. 17 a graph is presented of the dependence of \(H_0\) on the ratio \(\frac{l}{d}\). For \(\frac{l}{d}\ll 1\), \(p_0\sim \frac{l}{d}\), while for \(\frac{l}{d}\gg 1\), \(p_0\sim \frac{d}{l}\); at \(\frac{l}{d}\sim 1\), \(p_0\sim 1\), and \(H_0\) reaches its maximum value.

Fig. 17

Fig. 18

*) As was indicated in Broun’s paper, the coercive force in a stretched wire with a rectangular loop (Fig. 6) coincides not with the critical

by the field \(H_0\), but with the starting field \(H_s\), which is determined by the conditions for nucleation of regions of magnetization reversal. However, in a normal ferromagnet, as Kersten\(^3\) indicated, the coercive force \(H_c\) is determined by the mean value of the critical field \(H_0\). According to (7′),

\[ H_c=\frac{3}{2}\rho_c \frac{\Delta \varepsilon}{J_s}, \tag{8′} \]

where \(\frac{3}{2}\rho_c\) takes into account the distribution of stresses and critical fields \(H_0\) over the whole specimen.

Formula (8′) is confirmed by experiment. In Fig. 18\(^5\) the experimental dependence of \(H_c\) on \(\Delta \varepsilon\) is given for two specimens (\(a\) and \(l\)) of nickel. The difference between the specimens is due to the difference in the distribution of stresses.

3. DEPENDENCE OF FERROMAGNETISM ON PARTICLE SIZE

The appearance of a domain structure in ferromagnets is the result of competition between the magnetizing action of the exchange forces and the demagnetizing action of the surface of the body. When a ferromagnet is divided into domains, a positive surface energy \(\gamma\) of the boundary layers arises. The appearance of this substantially positive energy in specimens of ordinary size is more advantageous than if there existed a positive energy of the demagnetizing field of the surface. If the dimensions of a ferromagnet are reduced, then at some critical value of these dimensions it may turn out that the appearance of the surface energy \(\gamma\) is thermodynamically disadvantageous\(^1\). Therefore, with further reduction of the dimensions of a ferromagnet one should expect the appearance of spontaneous magnetization of the whole specimen as a whole. However, if the dimensions of the body are reduced still further, the ferromagnetic state may possibly disappear altogether.

Recently Haul and Schoon\(^6\) found that aerosols of \(\gamma\)—\(Fe_2O_3\) with linear dimensions \(\sim 16\,\text{\AA}\) are weakly paramagnetic. When the dimensions are increased to \(30\)—\(40\,\text{\AA}\), their susceptibility increases strongly and becomes dependent on the field. The authors come to the conclusion that in the range of dimensions from 30 to \(40\,\text{\AA}\) lies the lower limit for the size of regions of spontaneous magnetization in isolated form.

Baiher and Vinkel\(^7\), Vinkel and Haul\(^8\) investigated aerosols of nickel and iron. In the case of nickel, particles of dimensions \(35 \times 60 \times 210\,\text{\AA}\) spontaneously stuck together into chains, which under the action of a field oriented themselves along it and, upon magnetization reversal, rotated through \(180^\circ\). The sticking together took place under careful shielding from the earth’s magnetic field (the field was in any case less than 0.004 oersted) and disappeared on heating above the Curie point.

Elmore\(^9\) and Hines\(^ {10}\) carried out similar investigations with ferromagnetic colloids. The magnetization curve obtained with such colloids coincides with the Langevin curve for a paramagnetic gas, for which

\(^1\) See Ya. I. Frenkel and Ya. G. Dorfman, Nature, 1930.

the role of elementary magnets is played by individual spontaneously magnetized colloidal particles.

It must be pointed out, however, that as yet there is no complete experimental and theoretical solution of the question of the dependence of ferromagnetism on particle size.

References

1. S. V. Vonsovskii (in press).
2. L. Landau and E. Lifshitz, Sow. Phys., 8, 153, 1935.
3. F. Bloch, Z. Physik, 74, 295, 1932.
4. E. I. Kondorskii, JETP, 7, 1117, 1937; see also Sow. Phys., 11, 597, 1937.
5. M. Kersten, in the collection edited by R. Becker, Probleme der techn. Magnetisierungskurven, J. 1938, p. 42.
6. W. C. Elmore, Phys. Rev., 54, 1092, 1938.
7. C. W. Heaps, Phys. Rev., 57, 528, 1940.

1. Unless, along the way, the pressure \(p_\tau\) again increases, which can occur if domain \(B\) spans several “periods” of \(\sigma(x)\). Translator’s note. ↩↩↩↩

2. A calculation of magnetostriction according to quantum theory was recently carried out by S. V. Vonsovsky, ZhETF, 10, 762, 1940. ↩