Cullity B.D. Introduction to Magnetic Materials. Second Edition (2008)

a Ms ¼ 1714 emu/cm3 or 1.714 MA/m, lsi ¼ 20 1026.

several hundred oersteds or several kA/m, and that an increase in c/a from 1.1 to 1.5 quadruples the coercivity. Increasing c/a beyond 5 produces only moderate increases.

It is of interest to calculate the stresses which would have to be applied to a spherical particle in order to attain the same coercivities. (These are uniaxial stresses, tensile for positive lsi and compressive for negative lsi.) The same value of Ms is assumed, but lsi is arbitrarily taken as 20 1026, which is intermediate between the values for iron and nickel. The results are shown in the central portion of Table 9.1. The required stresses are seen to be very large. For example, a stress of 340,000 lb/in2 ¼ 2350 MPa, which produces the same coercivity as an unstressed nonspherical particle with c/a ¼ 1.1, is above the yield strength of all but the strongest heat-treated steels. The conclusion is that high coercivity is much easier to attain by shape anisotropy than by stress anisotropy.

Finally, in the third part of Table 9.1 are shown values of the uniaxial crystal anisotropy constant K1 which would produce in spherical particles the coercivities listed in the first part. The same value of Ms is assumed. Here the required values of K1 are of the same order of magnitude as those available in existing materials. Thus, barium ferrite has K1 equal to 33 105 ergs/cm3, and cobalt has K1 equal to 54 105 ergs/cm3, leading to calculated coercivities of several thousand oersteds. The fact that Ms for these substances is less than the value of 1714 emu/cm3 assumed in the calculations would make the computed values even higher. The calculated value for barium ferrite, with Ms ¼ 380 emu/cm3,

is 17,000 Oe or 1.35 MN/m and for cobalt, with Ms ¼ 1422 emu/cm3, it is 6300 Oe or 0.5 MA/m. Although the calculations leading to the stress and crystal anisotropy figures in Table 9.1 are based on arbitrarily selected values of Ms and lsi, they serve to fix the order of magnitude of the requirements.

All of the above values apply to an assembly of particles with their easy axes parallel to one another and to the applied field. If the particles are randomly oriented, the calculated coercivities are multiplied by 0.48.

If the crystal structure is cubic, then the rotational processes in an unstressed, spherical particle are much more difficult to calculate. There is now not one easy axis, but three (K1 positive) or four (K1 negative), and the anisotropy constants are generally low. Stoner and Wohlfarth estimate that, if the easy axes of an assembly of particles are aligned with the field, the maximum value of Hci will be about 400 Oe (32 kA/m) for iron and about half as much for nickel.

magnetization change was assumed to occur entirely by unhindered domain wall motion. The dashed lines show the behavior of an assembly of planetary spheroids randomly oriented in space, as calculated by Stoner and Wohlfarth.

9.12.3Remarks

The hysteresis loops of Figs 9.36–9.38 exhibit a remarkable variety of sizes and shapes. They range from square loops to straight lines to curved-and-straight lines without hysteresis. They offer a challenge to the materials designer, who seemingly has at her fingertips the ability to produce a magnetic material tailored to almost any application. However, for particles with a single easy axis, Fig. 9.36 shows that retentivity and coercivity are not independently variable; if one increases, so does the other.

The great difficulty, of course, is to make particles small enough to be single domains, and then imbed them in some controlled manner in a suitable matrix without sacrificing too much of the potential inherent in a single isolated particle. Considerable success in this direction has already been achieved, as we shall see in Chapter 14 on permanent-magnet materials. These materials must have a high coercivity, and magnets with coercivities of 45 kOe (3.6 MA/m) are in production in the early twenty-first century. Thus the even larger values shown in Table 9.1 as theoretically attainable in small particles are of great practical interest, and the processes of magnetization rotation in small particles have been the subject of much experimental and theoretical research.

Throughout this section the assumption has been made that the rotational process in a single-domain particle is one in which all spins in the particle remain parallel to one another during the rotation. This is not always true, as will be shown in Section 11.4 We will find there that the shape-anisotropy coercivities listed in Table 9.1 cannot be obtained in some particles, even in principle.

9.13MAGNETIZATION IN LOW FIELDS

The magnetization curves and hysteresis loops of real materials have quite variable shapes. Only in three circumstances, however, do we have algebraic expressions to fit the observed curves:

1.High-field magnetization curves of single crystals, like those of Fig. 7.22.

2.High-field magnetization curves of polycrystalline specimens, to be described in the next section. In both (1) and (2) the magnetization change is by domain rotation.

3.Low-field magnetization curves and hysteresis loops of polycrystalline specimens. These are discussed in this section.

By low fields we mean fields from zero to about one oersted or 80 A/m. Since the Earth’s field amounts to a few tenths of an oersted, precautions have to be taken to ensure that this field does not interfere with the measurements. This range of magnetization was first investigated in 1887 by Lord Rayleigh [Phil. Mag., 23 (1887) p. 225] and is known as the Rayleigh region. In this region, magnetization is believed to change entirely by domain wall motion, except, of course, in specimens composed of single-domain particles.

322 DOMAINS AND THE MAGNETIZATION PROCESS

By means of a suspended-magnet magnetometer, Rayleigh measured the lowfield behavior of iron and steel wire. In extremely low fields, 4 1025 to 4 1022 Oe (3 mA/m to 3 A/m), he found that the permeability m was constant, independent of H, which means that B (or M ) varies linearly and reversibly with H. At higher fields, in the range 0.08–1.2 Oe (6.4–96 A/m), hysteresis appeared, and m was no longer constant but increased linearly with H:

where mi, the initial permeability, and n or h are called the Rayleigh constants. This relation is called the Rayleigh law. (In very low fields, the term nH or hH becomes negligibly small with respect to mi, which explains Rayleigh’s finding of a constant m. Alternatively stated, the observation of linear, reversible magnetization depends on the sensitivity of the measuring apparatus in detecting, or not detecting, the term nH or hH. Rayleigh’s experimental method was extremely sensitive.) Equation 9.49 forms the basis of the standard procedure for finding the initial permeability: experimental values of m are plotted against H and extrapolated to zero field.

If Equation 9.49 is multiplied by H, we have

which is the equation of the normal induction curve in the Rayleigh region. The term miH represents the reversible part, and nH2 or hH2 the irreversible part, of the total change in induction. By substituting (H þ 4pM ) or m0(H þ M ) for B, we can obtain the equivalent expression for M:

where xi is the initial susceptibility.

Rayleigh also showed that the hysteresis loop can be described by two parabolas:

where Hm is the maximum field applied, and where the plus and minus signs apply to the descending and ascending portions of the loop, respectively. Figure 9.39 shows such a loop, together with the initial curve. The slope of the curve leaving each tip is the same as the initial slope of the normal induction curve, namely mi. The hysteresis loss is, from Equation 7.40,

Fig. 9.39 Normal induction curve and hysteresis loop in the Rayleigh region. Dashed lines indicate where slopes are numerically equal to mi.

The loss increases rapidly with Hm but is independent of mi. Hysteresis loss is caused only by irreversible changes in magnetization and therefore depends on n or h; the value of mi affects the inclination of the loop but not its area.

The Rayleigh relations have been confirmed for many materials. The constants mi and n vary over a wide range: 30–100,000 for mi and 0.5–12,000,000 for n, according to Bozorth. Any change in the physical condition of a given material, such as a change in temperature or degree of cold work, usually causes mi and n to change in the same direction; sometimes there is a simple linear relation between the two constants. The maximum value of H, beyond which the Rayleigh relations no longer hold, can only be found by experiment.

The initial permeability mi almost invariably increases with rise in temperature, goes through a maximum just below the Curie point Tc, and decreases abruptly to unity at Tc. As previous noted, this is called the Hopkinson effect, and can be used experimentally to obtain an approximate value for the Curie temperature. This behavior is related to the

fact that the crystal anisotropy constant K and the magnetostriction l both decrease to

p si

zero at or below Tc. Domain wall energy is proportional to K, and wall energy is the main contribution to the hindrance offered to wall motion by inclusions. The hindrance due to microstress depends only on the product lsis. Therefore, whether the hindrances to wall motion are inclusions or microstress or both, they are expected to become less effective as the temperature increases, leading to an increased permeability.

Plastic deformation decreases both Rayleigh constants. Figure 9.40 shows three m, H curves, linear in accordance with Equation 9.49; the material is a reasonably high-purity iron, annealed in hydrogen to reduce the interstitial content (carbon and nitrogen in interstitial solid solution) to 23 ppm C and 20 ppm N (1 ppm ¼ 1 part per million ¼ 1024%). The upper curve is for an annealed specimen and the other two for specimens prestrained in tension. These curves show the great sensitivity of the low-field magnetic behavior to a small amount of plastic deformation. For some materials the m, H curve bends downward as H approaches zero. Measured values of m at very low fields are then somewhat less than the value of mi found by extrapolating the main linear portion of the curve.

The domain size is important because it determines the domain wall area per unit volume. If that is known, then the average distance a wall moves can be calculated from

324 DOMAINS AND THE MAGNETIZATION PROCESS

Fig. 9.40 Permeability m vs field strength in the Rayleigh region for iron with varying degrees of cold work. Numbers on curves are degree of plastic elongation, in percent. [R. M. Rusak, Ph.D. thesis, University of Notre Dame, South Bend, Indiana.]

the observed change in magnetization. This distance may be compared to the scale of the structural irregularities in the specimen. As an example of such a calculation we will take the annealed iron specimen whose m, H curve appears in Fig. 9.40. At H ¼ 0.1 Oe, m ¼ 617, B ¼ 62 gauss, and M ¼ 5.0 emu/cm3, or H ¼ 8 A/m, m ¼ 775 1026 (mr ¼ 617), B ¼ 62 1024 T and M ¼ 4900 A/m. When unit area of a single 1808 wall is moved a distance x by a field parallel to the Ms vector in one of the adjacent domains, the magnetization of that region changes in the direction of the field by an amount 2Msx. If the spacing of the walls is d, then the number of walls per unit length is 1/d, and

If the field is almost normal, instead of parallel, to the Ms vector, then the factor 2Ms in this expression, resolved parallel to the field, is almost zero. When we consider the motion of both 1808 and 908 walls, variously oriented with respect to the applied field and assumed to be present in equal numbers, the factor 2Ms in Equation 9.54 is reduced to a value of roughly (3Ms/4). Then,

For the annealed iron specimen referred to, d was estimated as 3 mm from Bitter patterns.

distance a domain wall moves in a field of 0.1 Oe or 8 A/m is less than half of the domain

˚

wall thickness, estimated at 300–400 A in iron.

Such calculations suggest that the conventional view of wall motion, as in Fig. 9.34, wherein a wall moves reversibly from an energy minimum to a position of maximum energy gradient and then jumps to a new position, is grossly out of scale in the low-field region. Instead, it appears that in the demagnetized specimen many walls are already

poised at metastable positions, ready to move irreversibly when even slight fields are applied, because n is nonzero in this region. The above calculation assumes that all walls in the specimen move when a field is applied. An alternative view is that many of the walls are pinned so that they do not move at all in these weak fields; the remainder would then have to move several times the distance calculated above in order to produce the observed magnetization. It is interesting to translate into wall motion the results of W. B. Elwood [Rev. Sci. Instrum., 5 (1934) p. 300], whose measurements of magnetization are possibly the most sensitive ever made. With a special galvanometer operated in a special way, he was able to detect an induction B of 2 1024 gauss or 2 1028 T in a specimen of compressed iron powder. If the domain wall spacing in his specimen was 3 mm, as above,

tance is not only smaller than the wall thickness but is less than the diameter of an atom, by a factor of 1024. Another alternative is that only a fraction 1024 of the walls were

˚

unpinned, and these moved an average distance of 4 A. In either case, the notion of wall motion becomes rather strained, and must be replaced with the image of the spins in the walls being rotated through very small angles by the field.

9.14 MAGNETIZATION IN HIGH FIELDS

Between the low-field Raylegh region and the high-field region near saturation there exists a large section of the magnetization curve, comprising most of the change of magnetization between zero and saturation. The main processes occurring here are large Barkhausen jumps, and the shape of this portion of the magnetization curve varies widely from one kind of specimen to another. It is not possible to express M as a simple function of H in this intermediate region.

In the high-field region, on the other hand, domain rotation is the predominant effect, and changes in magnetization with field are relatively small. The relation between M and H in this region is called the “law of approach” to saturation and is usually written as

The term xH represents the field-induced increase in the spontaneous magnetization of the domains, or forced magnetization; this term is usually small at temperatures well below the Curie point and may often be neglected. Constant a is generally interpreted as due to inclusions and/or microstress, and b as due to crystal anisotropy.

There are both practical and theoretical difficulties with this equation. One practical difficulty is in deciding over what range of fields it should be applied. The upper field limit depends on the maximum field available, which may range from a few kilo-oersteds (less than 1 T) to several hundred kilo-oersteds (tens of tesla) in a superconducting or Bitter magnet, or even higher in pulsed fields. The lower field limit is at the discretion of the experimentalist. It should be chosen high enough so that all domain wall motion is complete, and only magnetization rotation is occurring, but there is no clear criterion to determine when this condition is reached. The values of a and b may depend quite strongly on the lower field limit chosen for fitting the equation. Another practical problem is that changes in magnetization with field at high field are small (except near the Curie point), and therefore hard to measure accurately. Some workers have elected to directly measure

326 DOMAINS AND THE MAGNETIZATION PROCESS

changes in magnetization with field, rather than the absolute magnetization, to minimize this problem.

The theoretical difficulties are of three kinds. First, a 1/H term leads to an infinite energy of magnetization at infinite field, which means the a/H term is unphysical or that it only applies over some limited range of fields. Second, the physical significance of the a/H term has been variously interpreted, with no consensus. Third, 1/H and 1/H2 terms are not the only possibilities: 1/pH and 1/H3/2 have been proposed on various theoretical grounds.

We may conclude that understanding the approach to saturation is in a distinctly unsatisfactory state. However, this has no significant consequences in the engineering applications of magnetic materials.

9.15SHAPES OF HYSTERESIS LOOPS

So far we have examined the shapes of major hysteresis loops of assemblies of singledomain particles and the minor loops, in the Rayleigh region, of polycrystalline specimens. Now we will consider the shapes of major loops of polycrystalline specimens.

Two extreme kinds are sketched in Fig. 9.41. Both apply to a material with a strongly developed uniaxial anisotropy, due, for example, to stress, uniaxial crystal anisotropy together with preferred orientation, or other causes. In Fig. 9.41a the applied field is parallel to the easy axis. The domain walls are predominantly 1808 walls parallel to the easy axis, as idealized, for example, in Fig. 8.21c. Magnetization reversal, from saturation in one direction to saturation in the other, occurs almost entirely by motion of these walls, in the form of large Barkhausen jumps occurring at a field equal to the coercivity Hc. The magnitude of Hc depends on the extent to which crystal imperfections impede the wall motion or on the mechanism discussed in the next paragraph. The result is a “square loop,” with vertical or almost vertical sides and retentivity Br almost equal to the saturation induction Bs.

In many square-loop materials, the loop shape is not a real characteristic of the material but rather an artifact, due to the usual method of measuring the loop. Normally the saturated specimen is exposed to a constantly increasing reverse field. Essentially no change in B is

Fig. 9.41 Hysteresis loops of uniaxial materials: (a) field parallel to easy axis; (b) field perpendicular to easy axis.

Fig. 9.42 Re-entrant hysteresis loop.

observed until the point A in Fig. 9.42 is reached; then, when the coercivity Hc1 is exceeded, a large change in B, practically equal to 2Bs, is observed as the induction changes along the dashed line. However, feedback circuits can be used which will sense the beginning of this large change in B and quickly reduce the magnitude of the reverse field. Ideally, such a circuit will provide only enough reverse field, at any level of B, to cause B to slowly decrease toward zero. In such a case, an irregular line crossing the H-axis at the “true” coercivity Hc2 would be traced out, the irregularities reflecting irregular impediments to wall motion. Re-entrant loops of this kind have been observed in a number of soft magnetic materials. Two interpretations of what occurs at point A are possible:

1.The specimen is truly saturated, in the sense of consisting of a single domain. If the magnetization is to reverse by wall motion, the reversal can be initiated only if one or more small reverse domains are nucleated. This is a relatively difficult process, requiring the field Hc1.

2.The specimen is not truly saturated. Some small reverse domains persist, but their walls are so strongly pinned that the field Hc1 is required to free them.

Interpretations 1 and 2 are known as the nucleation model and the pinning model of coercivity; the play an important part in discussions of coercivity mechanisms of permanent magnet materials. In either case, mobile walls suddenly appear at point A; these walls can then be made to keep on moving by a field weaker than the field required to initiate their motion.

The other extreme kind of loop is shown in Fig. 9.41b. Here the applied field is at right angles to the easy axis. The change of B with H is nearly linear over most of its range, which is an advantage for some applications but is obviously obtained at the cost of decreased permeability. The retentivity and coercivity both approach zero.

Between these extremes of square loops and almost linear loops lie those for specimens with more or less randomly oriented easy axes. Certain parameters of such loops can sometimes be calculated. Consider, for example, a uniaxial material like cobalt in which each grain has a single easy axis and in which the grains are randomly oriented. Then Fig. 9.43 illustrates several states of magnetization. The arrangements of Ms vectors in space is represented by a set of vectors drawn from a common origin, each vector representing a group of domains. The ideal demagnetized state is shown at point O. When a positive field is applied, domains magnetized in the minus direction are eliminated first, by 1808 wall motion, leading to the distribution shown at point B. Further increase in field rotates vectors into the state of saturation shown at C. When the field is now removed, the

328 DOMAINS AND THE MAGNETIZATION PROCESS

Fig. 9.43 Domain arrangements for various states of magnetization. [S. Chikazumi, Physics of Ferromagnetism, Oxford University Press (1997).]

domain vectors fall back to the easy direction in each grain nearest to the þH direction. Because the easy axes are assumed to be randomly distributed, the domain vectors are then uniformly spread over one half of a sphere, as indicated at D. If Ms in any one domain makes an angle u with the þH direction, the magnetization of that domain is Mscos u, and the retentivity Mr of the specimen as a whole is given by the average of Mscos u over all domains. This average is found by the method of Fig. 3.2b to be

Thus Mr/Ms, which is usually called the remanence ratio, is 0.5. S. Chikazumi [Physics of Ferromagnetism, 2nd edn, Oxford University Press (1997)] has shown how this ratio can be calculated for other kinds of materials, with the following results, where random grain orientation is assumed throughout:

1.Cubic crystal anisotropy, K1 positive (three k100l easy axes). Mr/Ms ¼ 0.83.

2.Cubic crystal anisotropy, K1 negative (four k111l easy axes). Mr/Ms ¼ 0.87.

However, Chikazumi points out that such calculations ignore the free poles that form on most grain boundaries. These free poles set up demagnetizing fields which can cause the actual value of Mr/Ms to be substantially lower than the calculated value.

Returning to Fig. 9.43, we note that the effect of applying a negative field to the remanent state is to first reverse the domain magnetizations pointing in the þH direction, leading to state E at the coercivity point. We note also that states E, O, and E0 all have M ¼ 0 but quite different domain distributions; of these, E and E0 are unstable in the sense that an applied field of one sign or the other is necessary to maintain these distributions.

Although not a hysteresis loop, it is convenient to mention here the ideal or anhysteretic (¼ without hysteresis) magnetization curve. A point on this curve is obtained by subjecting the specimen to a constant unidirectional field H1 together with an alternating field of