Current State of the Theory of Ferromagnetism

R. M. Bozorth*

About three thousand years have already passed since the discovery of natural magnets and their properties. For centuries ferromagnetism stubbornly resisted the attempts of theoreticians to penetrate this field, and even at the present time theory here lags far behind experiment. In the last 5–10 years, however, the theory of ferromagnetism has achieved considerable success. In the present review the author dwells on those achievements which, in his opinion, are the most essential.

As early as the last quarter of the past century, great successes were achieved in the study of magnetic materials. New data accumulated rapidly, and at the turn of the century there appeared the remarkable book by Ewing², in which all the important experimental and theoretical data of that time were set forth. The form of the magnetization curves for iron, cobalt, and nickel, magnetic saturation and the temperature of magnetic transformations, hysteresis and a number of laws connected with it, the influence of mechanical stresses and magnetostriction, and at the same time the basic methods of measurement—all this was for the first time systematically presented in Ewing’s small book. Although the subsequent 15 years yielded little new information about magnetic materials, it is precisely to this period that many applications of the knowledge obtained to electrical engineering belong, including electrical means of communication. Soon afterward Heusler alloys were obtained (alloys containing no iron but possessing ferromagnetic properties); although these achievements somewhat stimulated the study of theoretical questions of ferromagnetism, there was still little success in this field.

The rapid development of both experimental and theoretical investigations of ferromagnetism began between 1915 and 1920 and continues to the present time. This progress may be illustrated by the gradual improvement of magnetic materials achieved during this period (Table 1). The successes of the last 20 years have resulted from new methods of purifying materials and producing new alloys, as well as from new methods of thermal

* R. M. Bozorth, The Bell System Technical Journal, 15, 63, 1936. Translated by D. I. Penner.

** A little later Dubois’s book appeared, 2a. Translator’s note.

TABLE 1

Limiting values of characteristics of magnetic materials relating to 1915 and 1935

The footnote numbers refer to the bibliography at the end of the article.

processing. Some of the data presented in the table refer only to laboratory specimens, and not to materials used in industry.

The main subject of the present article is the theoretical aspect of ferromagnetism. How are the different values of magnetic permeability, ranging between 1 and 600,000 for different materials, to be explained? Or, if we first pose more fundamental questions, what is an elementary magnetic particle, and why is ferromagnetism inherent in only so few elements?

The Nature of Ferromagnetism

About 100 years ago Ampère proposed the supposition that molecules can behave as magnets thanks to electric

* These substances are free from impurities of other elements, with the exception of oxygen. There is a report by G. Neumann on a substance with an admixture of copper and molybdenum, the initial permeability of which reaches 40,000 (H. Neumann, Arch. techn. Mess., 4, T. 168, 1934).

currents circulating in them. Now, in connection with the development of our knowledge about the structure of the atom, the cause of ferromagnetism can be interpreted in a quite definite way. Strange as it may seem, the elementary magnetic particle was first discovered by spectroscopists. Such a particle is the rotating electron. In order to explain the extensive experimental material obtained in the study of spectra, it proved necessary to revise our conception of the atom. At one time the hypothesis was proposed that the atom consists of a heavy, positively charged nucleus and of electrons moving in circular or elliptical orbits around the nucleus. At present it is assumed that each electron also rotates about its own axis passing through its center. Thus, in the atom electricity circulates both around the nucleus and inside each electron; for the latter motion the term “electron spin”* has been introduced. Each electron in the atom is therefore a little top, possessing a magnetic moment as a result of the motion of the electric charge and an angular momentum as a result of the rotation of the mass. The ratio of the two moments, found by various mutually independent methods, has a definite numerical value. Owing to their motion in orbits, electrons also possess orbital moments, both magnetic and mechanical; their ratio is exactly two times smaller than this ratio for spin.

Fig. 1. Action of a gyroscope (left); the force \(F\) causes rotation \(R\). Gyromagnetic effect (right); the field \(H\) causes rotation \(R\).

Einstein, de Haas, and Barnett^21, by their well-known experiments, directly demonstrated the existence of the magnetic and mechanical moment of the electron and measured their ratio in ferromagnetic substances^22 (Fig. 1). An iron rod, suspended on a thin thread, is instantaneously magnetized; in doing so the rod

* Spin (Engl.)—twisting, rotation.

turns and twists the thread by a small, but nevertheless measurable amount. The rotating electrons, which are responsible for ferromagnetism, are set parallel to the applied field under its action; however, the mechanical moment possessed by these same electrons turns the whole rod in the same way as a top would turn. When the elementary magnets, initially oriented at random, are set approximately parallel to the axis of the rod under the action of the applied field, they acquire a definite angular momentum parallel to this axis. Since every action is balanced by a corresponding reaction, the rod must now in turn recoil with an equal and opposite moment; precisely this moment is manifested in the sudden rotation of the rod and can be calculated from the measured magnitude of the deflection. The sign of this moment indicates that the rotating magnetic particle is negatively charged, while the magnitude of the moment confirms the hypothesis that the elementary carrier of magnetism is the electron rotating about its axis. Thus the change in the state of magnetization is reduced essentially to a change in the orientation of the electronic spins in the atom; it is not connected with a change in the orientation of the whole electron orbit.

Let us now consider the question why not every substance is ferromagnetic. In Fig. 2, on the basis of the most recent data, a model of the iron atom is presented. The 26 electrons contained in the iron atom are distributed among four principal shells. Some shells are in turn subdivided into subgroups. The first (inner) shell contains two electrons, the next shell—eight, the third—fourteen, and the last contains two electrons. Since the periodic system of elements begins with the lightest element—hydrogen, the inner shell is filled first of all. When filled, the first four shells contain two, eight, eighteen, and thirty-two electrons (counting from the inner shell outward). Not every shell always already contains its maximum number of electrons before

Fig. 2. Electron shells in the iron atom

is filled next. When, for example, the filling of the fourth shell begins, the third contains only eight electrons instead of eighteen; the subsequent completion of the third shell is precisely connected in the closest way with the phenomenon of ferromagnetism. The magnetic moments, or, more briefly, the spins, of some electrons have one direction, while the spins of the remaining electrons have the opposite direction. These two kinds of spins may conventionally be designated as positive and negative. The figures in Fig. 2 indicate how many electrons in each shell possess positive or negative spins. It should be noted that in iron atoms all shells, with the exception of the third, contain equal numbers of positive and negative spins. The magnetic moments of the electrons in each of these shells on the average compensate one another; consequently the shell is magnetically neutral and cannot exhibit magnetic polarization. In the third, as yet unfilled shell, however, there are five electrons with positive spins and one with a negative spin, as a result of which four spins remain uncompensated; in this consists the polarization of the atom as a whole. If one more proton is added to the nucleus and one electron to one of the outer shells, the iron atom becomes a cobalt atom; if this process is repeated, cobalt is transformed into nickel. In iron the orientation of the electrons added during the completion process, and of their spins, is such that one may conventionally speak of four “excess” spins; for cobalt there are three, and for nickel two such spins. Manganese (the element immediately preceding iron in the periodic system) has an excess of five spins. Excess spins occur only in unfilled shells, in the completion of which heavier atoms arise. Closed shells are magnetically neutral, since the spins on the average compensate one another.

The chemical properties of an element are determined by the electrons of the outer shell. Changes in this shell take place in chemical compounds; ferromagnetism, however, is not connected with the valence electrons.

Exchange Forces

Elements with electrons acquired as a result of the completion of inner shells are found only in certain parts of the periodic system; one of these is the iron group. Since, however, in other places of the periodic system there are also groups of elements whose inner shells have been completed (important here are the parts of the table around palladium, platinum, and the rare earths), a further question arises: why are not all these elements ferromagnetic?

The presence of uncompensated spins in electronic orbits is not yet a sufficient condition for an element to be ferromagnetic; for this it is necessary, in addition, that

the resultant spins in neighboring atoms are parallel. Calculation of the energy of the electrons shows that, in order to obtain the same orientation of the spins in all atoms of some small region, it is necessary that the ratio of the diameter of the atom to the diameter of the electron shell with uncompensated spins have an appropriate value ^23 (Fig. 3). The latter is necessary because the electron spins and charges in neighboring atoms exert an influence on one another,

Fig. 3. Unfilled shells in neighboring atoms. In ferromagnetic substances the ratio \(\dfrac{D}{d}\) has a value greater than 1.5 (Slater)

the magnitude of which depends on their distance. Only under the condition that this interaction, denoted by the term “exchange,” has the proper magnitude can all spins acquire the same orientation^24, or, in other words, can the given substance be ferromagnetic*.

The exchange forces, whose existence has been proved in recent years, tend to arrange the spins parallel, whereas thermal motion, naturally, destroys this ordered arrangement. At a sufficiently high temperature the action of thermal motion predominates, and the substance ceases to be

Fig. 4. Iron-cobalt alloys give the highest values of magnetic saturation

ferromagnetic. This temperature is the Curie point, or “point of magnetic transformation”; for iron it is 770° (or 1043° on the absolute scale). It is clear that the value—

* In their recent work G. Urban, P. Weiss, and F. Trombe^25 found that the rare-earth metal gadolinium is ferromagnetic. The ratio \(D/d\) for \(Gd\) (in its metallic state) is about 3, which agrees with Slater’s rule (Fig. 3). The high value of \(D/d\) indicates that \(Gd\) should have a low Curie point; it has indeed been established that this temperature is equal to 16°.

of the Curie point \(\theta\) (on the absolute temperature scale) is a measure of the exchange forces, which for the time being can be calculated by theory only approximately. In Fig. 4 the Curie points are plotted for elements situated in the periodic system near iron; the curve connecting these points has a maximum near cobalt. Further, let us note that the value of the magnetic saturation depends both on the exchange and on the number of effective electron spins, in other words, on the number of electrons that can be oriented parallel to the field, and on the magnitude of the forces holding them in the state of parallelism. Roughly speaking, the value of the saturation depends on the product of the exchange and the number of uncompensated spins \((S)\) in the atom. If \(\theta\) is taken as a measure of the exchange forces and the products \(\theta \cdot S\) are formed, then the curve shown in Fig. 4 (right) is obtained. The curve shows that the maximum saturation can be obtained in an alloy of iron with cobalt and that manganese, under certain favorable conditions, can be ferromagnetic. Both of these conclusions are confirmed by experimental data. Indeed, alloys of iron and cobalt possess a higher value of saturation than pure iron. Manganese alloys are more magnetic than any other alloys not containing iron, cobalt, or nickel. Heusler alloys, containing manganese, aluminum, and copper, possess a saturation almost equal to the corresponding value for nickel, and numerous other manganese alloys are ferromagnetic to a lesser degree.

Exchange forces have a purely electrostatic origin. They do not, however, constitute electrostatic forces in the classical sense of the word, but are the result of electric charges distributed in a definite manner in space. It is hardly possible to describe them in words; for this purpose there exists a large number of mathematical equations that lead to conclusions following from the premises of quantum mechanics. The fact that in some substances the excess spins in a large group of atoms can easily be arranged in one direction is due to exchange forces. Indeed, if the exchange forces are large, as is the case in ferromagnetic bodies, then the stable state will be that in which the spins are parallel even in the absence of any external field. Under such circumstances, however, the region with parallel orientation of the spins usually does not extend over the whole body as a whole and does not even have visible dimensions; for some incomprehensible reason this region is limited to smaller dimensions. Experiments have established that the volume of such a region is, on average, equal to a cube with an edge of one hundredth of a millimeter. A real ferromagnetic body consists of a large number of such elementary regions, each of which is magnetized to the state of saturation in some direction (i.e., within it the electron spins are parallel). A body is called unmagnetized if the elementary regions are oriented uniformly over

in all directions, so that the magnetization of the whole body as a whole is equal to zero.

Experimental proof of the real existence of these elementary regions is provided by the so-called Barkhausen effect (Fig. 5). If it were possible to magnify many times a small portion of the magnetization curve (for example, the portion shown in Fig. 5), it would become noticeable that the curve consists of separate steps, each of which corresponds to an instantaneous change in the magnetization; after such a jump, the state of magnetization, as the field is increased, does not change until the field reaches a certain, higher value. None of the known instruments makes it possible to carry out such a magnification directly; these jumps can, however, be detected if the body being magnetized is wound with a coil connected to an amplifier, to which in turn a telephone receiver is connected. With a slow increase of the field, series of clicks are heard in the telephone; by quantitative investigation it has been found that, on the average, a click corresponds to the reversal of the magnetic polarization in an elementary region, whose size26 is indicated above. Under favorable conditions the Barkhausen noise can be heard even without an amplifier, by means only of a telephone receiver connected directly to the coil*.

Fig. 5. Instantaneous changes of magnetization causing the Barkhausen effect

It has already been pointed out above that the exchange forces are opposed by the disordering forces of thermal motion. For this reason the value of the magnetic saturation decreases continuously with rising temperature, until, on reaching the Curie point, ferromagnetism disappears. The dependence of magnetic saturation on temperature is shown in Fig. 6, where the saturation at absolute zero and the Curie-point temperature are taken as unity on the ordinate and abscissa axes, respectively. It was found in this way that the data for iron, cobalt, and nickel agree well. The lower curve is theoretical, calculated 30 years ago on the assumption that the elementary magnets, being disoriented by thermal motion, may assume any orientation. If, on the contrary, it is assumed that the spins responsible for ferromagnetism can occupy only two positions (taking into account the presence

* V. K. Arkad’ev27 pointed out the possibility of hearing the noise of the magnetization reversal of iron directly by ear, without the aid of a telephone.

(Translator’s note.)

of the other electrons in the atom), then the upper curve is obtained. If four possible orientations are admitted, the curve computed accordingly is close to the upper curve (Fig. 6), passing only slightly below it; as the number of possible orientations is increased, the corresponding curve approaches the lower curve shown in the same figure. The good agreement between the experimental data and the upper calculated curve is of special interest, since investigators in spectroscopy and atomic theory have independently arrived at the conclusion that each electron in the atom (taking into account the influence of all the rest of the atom) can occupy only a small number of positions.

Fig. 6. Dependence of the magnetic saturation of iron, cobalt, and nickel on temperature

Influence of the Crystal Structure

To explain the properties of a single crystal, it is necessary to admit the existence of still another kind of force. Owing to the spin of the electrons, and also owing to their orbital motions, each atom may be regarded as a small magnet. These magnets interact with one another in a purely magnetic manner[^28], like a group of ordinary straight magnets. For crystals of a regular system it can easily be established that, owing to the magnetic interatomic forces, certain directions of magnetization are more stable than others. In iron, the most stable direction—

... is an edge of the cube (one of the crystallographic axes); in nickel this direction coincides with a diagonal of the cube (Fig. 7).

A piece of iron usually consists of such small crystalline grains that they are invisible to the naked eye. In recent years, however, methods have been found for determining the size of the crystals of all ordinary metals; at the same time it has proved possible to obtain single crystals

Fig. 7. Magnetic properties and structure of a single crystal of iron and nickel (Beck, Honda and Kaya, Webster)

so large that observations can be made using only one such crystal.

The structure of a single crystal of iron may be represented in the form of a cube with one atom at each vertex and one at the center; the whole crystal is built of such cubes adjoining one another by their faces. For the direction along an edge of the cube (the direction \([100]\)), a magnetization curve has been obtained experimentally, denoted in Fig. 7 by the symbol 100*. The other two curves shown here correspond to the two other principal directions, namely to the diagonal of a side face and to the diagonal of the cube. The discrepancy of the initial portions of the magnetization curves may be neglected; it becomes large only after approximately half-saturation has been reached.

The structure of nickel may likewise be represented as an aggregate of cubes in which, however, the atoms are arranged differently, occupying the vertices and the centers of the side faces of these cubes

* Here the ordinates are the differences \(B - H\) instead of the more customary quantity \(B\), since the former approach the limiting value (saturation).

(Fig. 7). Figure 7 also shows the magnetization curves for nickel corresponding to the same three principal directions; the curves are arranged here in a different order than for iron. For iron the direction \([100]\) is the direction of so-called easy magnetization, while \([111]\) is the direction of most difficult magnetization; in nickel the reverse is the case. The electrostatic forces of exchange arrange the spins parallel to one another, whereas the crystalline forces determine the direction of the axis along which the spins are set by the exchange forces. The exchange forces are so large that they are capable of arranging the spins of an entire group of atoms in one direction—an effect which, in the absence of these forces, could be obtained only

Demagnetized state
Crystallographic axes
Saturated state

Fig. 8. Elementary domains in a single crystal of iron

by the action of an external field of \(10\,000\,000\) oersteds (at room temperature). On the other hand, the crystalline forces are so weak that in order to turn the spins of an entire group of atoms from one common direction into some other direction only \(1000\) oersteds are required. The ratio between these two equivalent fields is therefore equal to the quotient of \(10^7\) divided by \(10^3\), i.e. \(10^4\).

Under the action of the exchange forces and the magnetic crystalline forces, a situation is produced in a single crystal of iron such as that shown in Fig. 8. Even if the crystal is certainly not magnetized, or is demagnetized, it contains small, so-called elementary domains, magnetized to saturation in one of the six equivalent directions of the crystallographic axes. In reality the elementary domains differ noticeably from one another both in size and in shape; conventionally, however, they are shown as squares. Each of the six directions mentioned above, in the absence of an external field, is equally stable and equally probable. The initial result of the action of a magnetic field is a reversal of the direction of magnetization, consisting in a transition from one stable orientation to another, as a result of which the resultant polarization in the direction of the field increases. These changes occur instantaneously and are the cause of the Barkhausen effect; each reversal of an elementary

region is coupled with one step in the magnified magnetization curve shown in Fig. 5; in the telephone it produces a click.

There are, however, still more direct proofs of the real existence of elementary regions in iron. Iron is coated with a colloidal suspension of iron oxide and placed under a microscope giving a magnification of 500 times²⁹. It has been found that the colloidal particles concentrate along lines depending on the crystallographic axes, thereby showing that stray magnetic fields enter the surface of the iron and emerge from it in such a way as though certain portions of the iron were magnetized differently from their neighbors (Fig. 9). This phenomenon is noticeable even in unmagnetized iron, but can never be obtained on non-ferromagnetic bodies.

Fig. 9. Figures on a piece of iron (McKeehan and Elmore); (left) the lines of force of the field enter the iron; (middle part) demagnetized iron; (right) the lines of force emerge from the iron.

Let us now consider in more detail what processes take place during changes of magnetization. A large part of these changes is connected with the reorientation of spins in the elementary regions, with the transition from one direction of easy magnetization to another (Fig. 10). These changes correspond to the long central part of the magnetization curve. It is obvious, however, that this process is in general completed before saturation is reached. Suppose that all the elementary regions are polarized parallel to that direction of easy magnetization which is closest to the direction of the external field; then the only way of further increasing the magnetization is the turning of the electron spins in each elementary region from their stable position toward the direction of the field. Sometimes this phenomenon is somewhat

inaccurately called “rotation of elementary domains”; it takes place in fields of the order of 10 to 100 oersteds. From Fig. 7 it is clear that the beginning of these processes corresponds to the portion where the curves turn sharply away from their vertical part.

Fig. 10. With increasing magnetic-field strength, the elementary regions first change their direction instantaneously, and then smoothly.
a—state of demagnetization, b—instantaneous overturning of elementary regions is completed (bend in the magnetization curve), c—state of saturation; elementary regions after rotation in a strong field

Fig. 11. The vectors depict the differences \(B-H\) for iron, growing in magnitude as the magnetic field \(H\) is strengthened. At first the vector \(B-H\) is parallel to \(H\) (1); then, as \(B-H\) grows, the vectors deviate from \(H\) (2); finally, in high fields the vector \(B-H\) is again parallel to \(H\) (3)

Only in the case when the field applied to the single crystal coincides with the direction of easiest magnetization does the above-described process not take place. When the external field has the direction of hardest magnetization, the process of rotation of the elementary regions begins at field strengths lower than in any other cases.

The same picture explains yet another important property of a single crystal; it appears if the field applied to the single crystal is not parallel to any principal axis. Let, for example, the external field make an angle of \(30^\circ\) with some axis of a cubic crystal of iron, as indicated in Fig. 11 by the long arrow. As the field increases, starting from zero, the magnetic polarization in magnitude and direction is represented by the other arrows. At first the polarization is parallel to the external field. But as the field is increased it deviates, approaching the direction of easy magnetization, until, finally, saturation is reached in this direction. In ...

further increase of the polarizing field turns it toward the direction of the field, and ultimately saturation occurs in this direction. In agreement with experiment, the theory predicts the \(3^\circ\) direction and magnitude of the deviation of \(B\) from \(H\) for any given value of \(B\).

Both paths of change of magnetization described above for single crystals—namely, the instantaneous turning to new directions of the easy magnetization and the gradual rotation of the elementary regions—are equally well applicable to ordinary polycrystalline bodies, whose properties are those of a single crystal averaged over all directions. As a result the body behaves as an isotropic one, and the course of \(B\) is parallel to \(H\).

The last remark, however, requires some qualification: magnetic materials used in engineering are not always isotropic; in other words, the crystallographic axes are not always distributed uniformly in all directions. It has long been known that, when a metallic strip is rolled, its crystals tend to occupy definite positions corresponding to the direction and plane of rolling. Even after annealing and recrystallization these special orientations continue to exist in some metals right up to the melting point. Since the magnetic properties of a single crystal depend on the direction of the crystallographic axes, metal sheets consisting of crystals with special orientations cannot have identical magnetic properties in all directions. Several years ago this phenomenon was detected in iron, nickel, and iron–nickel alloys[^31]. Still later an alloy of iron and silicon appeared on the market[^32], in which the permeability assumes very different values in different directions. Parallel to the direction of rolling this material, in high fields (\(B = 10\,000\) gauss), has a permeability of 4000, whereas perpendicular to the direction of rolling the permeability is only 400. As established by X-ray analysis[^33], the orientation of the crystals in this material is such that in the majority of them one axis is deviated by only a few degrees from the direction of rolling. Thus the direction of rolling coincides with the direction of easy magnetization.

In considering above the properties of a single crystal, we passed over their properties in very weak fields; this is due chiefly to the fact that obtaining accurate data on single crystals involves great difficulties. The process taking place in single crystals and polycrystals in such fields must differ from both magnetization processes considered earlier, since in the magnetization of an ordinary polycrystalline body there is no discontinuity; in other words, the Barkhausen effect is absent here, and, moreover, these fields are not strong enough to turn the elementary regions appreciably against the crystalline forces away from the direction of easy magnetization. Knowing the relation between the intens—

...by the field strength and by the angular displacement for strong fields, it was possible to calculate that, with the same mechanism of change of magnetization in very weak fields, the highest value of the initial permeability for iron would be 20 instead of many thousands. The process taking place in weak fields has until recently been considered repeatedly. At present, apparently, a satisfactory explanation has been found. Magnetization in weak fields is now reduced to the displacement of the boundaries of elementary regions (Fig. 12); the transition

Fig. 12. Magnetization in very weak fields consists in an insignificant displacement of the boundaries of elementary regions

Fig. 13. Three types of changes of magnetization: 1—displacement of boundaries, 2—instantaneous change of orientation, 3—slow change of orientation

of a region having a size of several atomic diameters (its size, calculated from exchange forces, is about 30 atomic diameters) proceeds in such a way as to increase the elementary region magnetized in the direction of the field at the expense of another elementary region with a less favorable orientation. Such an increase can be only insignificant in comparison with the linear dimensions of the elementary region, being limited by the stresses present in every body.

Thus, in the magnetization of an ordinary, well-annealed ferromagnet, three processes take place, corresponding to the three well-known parts of the magnetization curve (Fig. 13): the growth of one elementary region at the expense of a neighboring one in the initial part of the curve, the instantaneous overturning of elementary regions in the middle part (with large energy losses), and the gradual or smooth rotation of elementary regions in the upper part. The last two processes appear at high values of induction; the first process takes place only in weak fields after demagnetization.

The Influence of Mechanical Stresses

The picture drawn above of changes in magnetization applies to materials free from any substantial mechanical stresses. Mechanical stresses can exert a very considerable influence on the state of magnetization; thus, for example, a tension of 350 kg/cm² under certain conditions

can change the induction by more than 10,000 gauss,³⁴ almost from zero magnetization to saturation. This phenomenon is well illustrated by the data of Fig. 14 for permalloys 65 and 85 (iron–nickel alloys containing, respectively, 65 and 85% Ni). For

Fig. 14. Influence of mechanical stresses on magnetization (Buckley and McKeehan)

permalloy 65 the result of tensile stress is an increase of magnetization in all fields; for permalloy 85 the phenomenon proceeds in the opposite way. In each case the influence of compression is opposite

Fig. 15. Magnetostriction in iron, nickel, and in two iron–nickel alloys (permalloy)

to the influence of tension. For ordinary iron, tension causes an increase of magnetization in weak fields and a decrease of it in strong fields.

The influence of mechanical stresses on magnetization corresponds to the inverse phenomenon, namely the influence of magnetization on

the length of a ferromagnetic body. An iron rod, when magnetized, lengthens slightly. Here we are dealing with one example of a large class of phenomena encompassing all ferromagnetic bodies and known under the general name of magnetostriction. Fig. 15 illustrates the change in the length of rods made of nickel, iron, and two alloys, with a change in the field \(H\) (left) or in the relative difference \(B-H\) (right). When magnetic saturation is reached, the limiting value of magnetostriction is also reached, the so-called saturation magnetostriction. Its values for some iron-nickel alloys are shown in Fig. 16. Let us note here that alloys containing less than \(81\%\) Ni lengthen when magnetized, whereas alloys with a higher Ni content shorten. There is a definite relation between magnetostriction and the influence of mechanical stresses on magnetization;

Fig. 16. Saturation magnetostriction in various permalloys (McKeehan and Cioffi, Schulze)

the general rule here is as follows: if magnetostriction is positive (lengthening upon magnetization), then the result of stress is an increase in magnetization, and conversely (Figs. 14, 15, and 16).

What can theory say about magnetostriction and the influence of mechanical stresses on magnetic properties? Fig. 17 depicts the arrangement of atoms in an iron crystal; here it is assumed that each atom, owing to the presence of spin and orbital motion of the electrons, possesses a definite magnetic moment. Thanks to this assumption it is possible to calculate the average magnitude of the magnetic forces opposed by the elastic forces that determine the strength of the crystal. For iron it has been found by calculation\(^5\) that equilibrium is established with a small elongation in the direction of magnetization and a shortening perpendicular to this direction, so that the volume remains practically unchanged. The calculated value of the magnetostriction agrees with the experimental data both in sign and in order of magnitude. For nickel this agreement is less satisfactory. But in any case

the theory predicts the correct qualitative relation between magnetostriction and the change in magnetization caused by mechanical stresses.

Thus magnetostriction and the effect of mechanical stresses on magnets are reciprocal phenomena, caused by the very same kinds of magnetic interatomic forces with which the difference in magnetic properties along different directions in the crystal is associated. Just as in a stress-free crystal the direction of easy magnetization is determined by the crystal structure, so, in the presence of a mechanical stress of sufficient magnitude, the direction of easy magnetization is determined by the latter. Fig. 18 shows that the elementary regions

Fig. 17. Magnetic interatomic forces produce a small elongation in iron (magnetostriction)

Fig. 18. Orientation of elementary regions determined by crystalline forces and mechanical stresses; \(H = 0\). \(a\)—iron free of stress, \(b\)—iron under stress, \(c\)—nickel under stress, \(d\)—crystallographic axes, \(e\)—direction of stress

are arranged parallel to the crystallographic axes in stress-free iron, whereas under a sufficiently large stress the magnetic polarization becomes parallel to the direction of this stress (and perpendicular to it, if nickel is in question). When the stress reaches a value from 700 to 2000 \(kg/cm^2\), its effect begins to predominate over the effect of the crystal structure, and then the direction of magnetization is determined chiefly by the stress. Calculation also leads to the conclusion that in bodies with positive magnetostriction the magnetization increases under tension. These considerations qualitatively explain the increase of permeability in permalloy 65 (which has positive magnetostriction) and its decrease in permalloy 85 (with negative magnetostriction). The theory, however, is still quite powerless to predict the quantitative value of the effect.

Alongside definitely directed homogeneous stresses, such as arise, for example, when a wire is stretched along its length, one often encounters randomly directed (heterogeneous) stresses, varying in magnitude, sign, and direction from point to point throughout the body. Such stresses appear during cold working, during phase transformations, etc. In such bodies the direction of magnetization

in some elementary region is caused by a local stress; the stability of this direction will be the greater, the more considerable the stress. Thus one can explain why it is more difficult to change the magnetization in a body that has undergone heavy cold working. The hardness of a metal is also caused by these same internal stresses; hence follows the well-known correspondence between magnetic and mechanical hardness.

The relation between internal stress and permeability is illustrated by the data \(^{35}\) given in Fig. 19. The value

Fig. 19. Increase of magnetic permeability owing to the reduction of internal stresses by annealing (Dillinger and Houorth)

of the permeability for a series of strips of permalloy 70, initially subjected to cold rolling, increases as the annealing temperature is raised. X-ray analysis data for these same specimens (the point in question is the angular width of the reflected beam of X-rays) establish the magnitude of the existing internal stresses; these same data indicate a gradual decrease of internal stresses with increasing annealing temperature, the most rapid change in each case occurring between 400 and 600°. In this interval, as the microscope shows, recrystallization takes place.

Developing these propositions, one may arrive at the idea that, as a good material for permanent magnets, one should take a material with very intense internal stresses. The magnitude of the internal stresses in a good permanent magnet, directly determined with the aid of X-rays, confirms this assumption (Fig. 20). Here the internal stress is measured directly by the width of the reflected X-ray beams. For comparison with the material of permanent magnets, our diagram gives the curves of other materials with smaller internal stresses. As the magnetic material here there is taken an alloy of iron, nickel, and aluminum, subjected to hardening

precipitation hardening (Ausscheidungshärtung, precipitation hardening)—a method that has been increasingly used during the last three or four years in the treatment of certain materials. This method is often applicable when, in the stable state at room temperature, an alloy contains two phases (Fig. 21), while at higher temperatures one phase dissolves in the other, forming a solid solution. A body heated to a high temperature is rapidly cooled, and then heated again to \(700^\circ\); at this temperature the second phase slowly precipitates in a very finely divided form. After precipitation of the optimum amount, the body is cooled to room temperature, after which no further changes occur. Each precipitated submicroscopic particle is a center of stresses, and it is the presence of these extraordinarily large internal stresses that accounts for the good quality of the permanent magnet.

Fig. 20. The width of the beam of reflected X-rays indicates the magnitude of internal stresses. Along the ordinate is plotted the intensity of the X-rays reflected from a metallic surface; along the abscissa, the angle of reflection.
\(I\)—permalloy hardened by quenching, \(II\)—Fe + Ni + Al alloy hardened by precipitation, \(III\)—permalloy subjected to heavy cold working.

Fig. 21. Precipitation hardening of an alloy for a permanent magnet (for example, an alloy of iron, nickel, and aluminum).

We shall now turn to the other extreme, i.e., to the case in which easy magnetization is required. It is known that complete annealing

and a homogeneous structure of the body favor magnetization. There exist, however, at least two further kinds of stresses that are not eliminated by annealing. One of them is connected with the presence of nonmetallic impurities that hinder the attainment of the proper arrangement of atoms in the metal or in the alloy. At the present time it has been found that, by hot treatment of iron in an atmosphere of hydrogen at a temperature of about \(1500^\circ\), nonmetallic impurities are removed to a considerable extent and the so-called chemical stresses are greatly reduced. It has been established that as a result of such treatment the maximum permeability increases from \(10\,000\) to \(340\,000^{12}\) (Fig. 22), and at the same time the mechanical hardness is greatly reduced.

Fig. 22. Permeability curves of ordinary iron and of iron purified by heat treatment in an atmosphere of hydrogen at \(1500^\circ\) (Chioffi)

After the elimination of both chemical stresses and stresses caused by cold working, it is necessary to take into account still another kind of residual stress, connected with magnetostriction. These stresses are usually randomly directed, since they are associated with chaotically oriented elementary regions; however, by means of the method indicated below they can be given an arrangement favorable to magnetization in one desired direction, to the detriment of ease of magnetization in perpendicular directions. Such a method is hot treatment in the presence of a magnetic field. Without entering into more detailed explanations, we shall in what follows present experimental data obtained during the last two years.

If an annealed specimen of permalloy 65 is held for several minutes at a temperature of \(650^\circ\), while at the same time subjecting it to the action of a magnetic field of 10 oersteds, then the maximum permeability increases, beginning from approximately \(20\,000\), to \(600\,000\) and higher (cf. Fig. 23). This material gives record values

maximum permeability, the lowest coercive force, and the smallest hysteresis losses at high values of induction. It is interesting to compare this material with the most permeable material known in 1900, which was then iron with a maximum permeability not reaching 3000.

Fig. 23. Permeability curves of permalloy 65 after hot treatment in a hydrogen atmosphere and additional hot treatment in a magnetic field (Dillinger and Bozorth)

Until now we have considered only the effect of stress on the orientation of elementary regions in medium and high fields. However, stresses also affect the initial permeability. It has already been said above that in very weak fields the change in magnetization is associated with the displacement of the boundaries of the elementary regions, i.e., with the growth of elementary regions parallel to the field at the expense of neighboring regions with a less favorable orientation. This growth is, understandably, hindered by stress. The initial permeability is related to the internal stress and other magnetic properties by the following equation ^37:

\[ \mu_0 = \frac{0.018(B-H)_s^2}{\left(\frac{\Delta l}{l}\right)_s \cdot \sigma_i}, \]

where \(\mu_a\) is the initial permeability, \((B-H)_s\) and \(\left(\frac{\Delta l}{l}\right)_s\) are the induction and magnetostriction of iron at saturation, and \(\sigma_i\) is the mean value of the internal stress, expressed in dynes per square centimeter.

Even if a body contains no internal stresses due to impurities, insufficient annealing, etc., it usually contains stresses due to magnetostriction itself, which cause some elementary domains to grow at the expense of others (Fig. 24). In this case the stress appearing in the preceding equation is equal to Young’s modulus \(E\) multiplied by the magnetostrictive elongation

\[ \sigma_i = E \left( \frac{\Delta l}{l} \right)_s, \]

and we arrive at the following equation:

\[ \mu_0 = \frac{0.018 (B - H)_s^2} {\left( \dfrac{\Delta l}{l} \right)_s^2 E}. \]

Fig. 24. Magnetostriction in the shaded domain acts like pressure of the bar, preventing any further change in magnetization.

Labels in the figure: “without stress”; “under compression”; \(H\).

This equation gives the theoretical upper limit for \(\mu_0\). These limits, as well as the highest experimental values for iron–nickel alloys, are shown in Fig. 25. It is evident from this

Labels in the graph: \(\mu_0\); “composition of permalloy—percent Ni content”; “Highest experimental values of \(\mu_0\)”; and the theoretical expression

\[ \mu_0 = \frac{(B-H)_{\max}^2} {18\pi E(\Delta l/l)_{\max}^2}, \]

“calculated upper limit for \(\mu_0\).”

Fig. 25. Comparison of the theoretical upper limit of the initial permeability (Kersten) with the highest initial permeability measured in iron–nickel alloys (Arnold and Elmen, Schulze).

why the permalloy alloy, which has the highest initial permeability, is very close to alloys with zero magnetostriction.

The influence of stresses must now be briefly summarized. The cause of these effects lies in the magnetic interaction between neighboring atoms. The magnetic interaction is balanced by elastic (electrostatic) interatomic forces. The balancing of these two forces leads to a change in the shape of a body when it is magnetized (magnetostriction), and also to a change in the magnetization caused by it. A uniform rectilinear stress can either facilitate or hinder magnetization; the phenomenon depends on magnetostriction, and the character of this dependence can be established qualitatively, but not quantitatively. More difficult is the magnetization of a material with randomly directed local stresses, since the latter impede changes in magnetization; as the intensity of these stresses increases, the difficulty of magnetizing or demagnetizing the body increases. The influence of local stresses on the initial permeability can be successfully calculated, whereas other magnetic properties, such as, for example, the maxi-

TABLE 2

Brief data on the carriers of magnetic properties

mal permeability, for the time being can be determined only qualitatively.*

In conclusion we give the following table (Table 2).

REFERENCES

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1. K. Honda, Magnetic Properties of Matter, Syokuwabo, Tokyo, 1928.

3a. V. K. Arkadiev, Electromagnetic Processes in Metals. Moscow, Part I, 1935; Part II, 1936.

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* In the present article the author considers exclusively the phenomena of the magnetostatics of matter, i.e. ferromagnetic properties in a constant field. At present we possess extensive material concerning the phenomena of magnetization in alternating fields, in periodic as well as instantaneously arising or disappearing ones. This field of phenomena, united under the general name of magnetodynamics, has been examined in detail in a number of studies; see 3a. (Translator’s note.)

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