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

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Fig. 10.4 Possible atom arrangements in a square lattice.

the direction of the local magnetization such that the two tend to be parallel. (The exact physical nature of this interaction is not specified; presumably it is related, like magnetocrystalline anisotropy, to spin–orbit coupling.) Thus, if a saturating field is applied at a high temperature, the magnetization will be everywhere in one direction and diffusion will occur until there is a preferred orientation of like-atom pairs parallel to the magnetization and the field. On cooling to room temperature, this directional order will be frozen in; the domains that form when the annealing field is removed will then have their local magnetization bound to the axis of the directional order. The anisotropy energy, for random polycrystals, is then of the same form as for other uniaxial anisotropies:

where Ku is the anisotropy constant and u is the angle between the direction of magnetization and the direction of the annealing field.

The value of Ku for polycrystals may be determined by measuring the area between magnetization curves measured in the easy and hard directions, or by the analysis of torque curves, if due allowance is made for the possible contribution of preferred orientation to the observed anisotropy. For single crystals, Ku may be found from torque curves, if these are Fourier analysed to separate the field-induced anisotropy from the crystal anisotropy. The values of Ku so found are not particularly large. They are typically

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magnetic annealing are also capable of long-range ordering is a complication that must be kept in mind in the experimental study of magnetic annealing.

3.When magnetic annealing is carried out at constant temperature rather than by continuous cooling, Ku is found to decrease as the field-annealing temperature is increased. The higher the temperature, the greater is the randomizing effect of thermal energy and the smaller the degree of directional order.

As noted above, the anisotropy induced by field annealing is not particularly strong. This suggests that the directional order is not very strong, certainly not as pronounced as that shown in Fig. 10.4c. If it were, such properties as lattice parameter and elastic constants should be detectibly different in directions parallel and perpendicular to the directionalorder axis in a single crystal; such differences have not been observed. Nor has X-ray or neutron diffraction ever furnished any direct evidence for the existence of directional order. Calculations indicate that the degree of directional order attained in practice is something like 1% of the theoretical maximum.

The behavior of field-annealed single crystals is more complex than that of polycrystals, and Equation 10.1 is not adequate to describe all the phenomena observed. Whereas the annealing field may be made parallel to any crystallographic direction that the experimenter chooses, the axis of directional order is determined by the structure of the crystal. If directional order is due to an interaction between the local magnetization and the axis of nearest-neighbor pairs, as is usually assumed, then the ordering axis must be a crystallographic direction on which nearest neighbors are located, i.e., a direction of close-packing for the structure involved. This direction is the face diagonal k110l for the face-centered cubic structure and the body diagonal k111l for the body-centered cubic structure. If the local magnetization, which we can label as Ms since in a domain the magnetization is saturated, is not parallel to a close-packed direction, directional order can still occur and create an easy axis, but this axis may or may not be parallel to Ms. For example, suppose the annealing field is parallel to [001] in an fcc crystal. Then directional order is equally favored along

k l ¯ ¯

four of the six 110 directions, namely, [011], [101], [ 011], and [101], which are all equally

8 ¯

inclined at 45 to Ms, but not along the [110] and [110] directions, which are at right angles to Ms. As a result, an easy axis is created parallel to Ms. If the annealing field is along [001] in a bcc crystal, then all four k111l directions are equally inclined to [001] and no anisotropy should result, because none of the four close-packed directions is favored over the others.

The experimental results on single crystals are as follows. When the annealing field is parallel to a simple crystallographic direction like k100l, k110l, or k111l in a cubic crystal, the resulting easy axis is parallel to the annealing field direction, but the value of Ku is different for each of these directions. When the annealing field is parallel to k100l in a bcc crystal, Ku is observed to be small, but not zero as predicted by theory. Presumably, directional order of second-nearest neighbors, which lie on k100l axes, is taking place. Or maybe the annealing field was not exactly parallel to k100l. When the annealing field is not parallel to one of these simple directions, an easy axis results, but it is not parallel to the annealing field direction; in fact, it can deviate from it by as much as 208. For a fuller account of single-crystal behavior and a summary of directionalorder theory, the reader should consult S. Chikazumi [Physics of Ferromagnetism, Oxford University Press (1997)].

Magnetic annealing has also been observed in solid solutions of cobalt ferrite and iron ferrite (called cobalt-substituted magnetite) and in some other mixed ferrites containing cobalt. The effect depends on, but is not caused by, the presence of metal-ion vacancies

in the lattice. (Such vacancies can be created by heating the ferrite in oxygen.) The Co2þ ion in a cubic spinel occupies an octahedral site (Table 6.2, Fig. 6.2) lying on a k111l axis of the crystal, and, for reasons described by Chikazumi, this ion creates a local uniaxial anisotropy parallel to that k111l axis on which it resides. In a non-field-annealed crystal all four k111l axes will be equally populated by Co2þ ions and the local anisotropies will cancel out, leaving only the cubic crystal anisotropy. In the presence of an annealing field, however, the Co2þ ions will tend to occupy the k111l axis closest to the field, producing an observable and rather large uniaxial anisotropy. Directional order of Co2þ –Co2þ pairs is thought to be a contributing factor. The role of vacancies is to insure enough diffusion of the metal ions for substantial ordering to take place.

It is important to realize that directional order, whether of like-atom pairs in an alloy or Co2þ ions in a cobalt-containing ferrite, is not due to the field applied during annealing, but to the local magnetization. All that the applied field does is to saturate the specimen, thereby making the direction of spontaneous magnetization uniform throughout. It follows that a material that responds to magnetic annealing will undergo local self-magnetic-annealing if it is heated, in the demagnetized state and in the absence of a field, to a temperature where substantial diffusion is possible. Directional ordering will then take place in each domain, parallel to the Ms vector of that domain, and in each domain wall, parallel to the local spin direction at any place inside the wall. [Strictly, the directional ordering will take place along the close-packed direction(s) nearest to Ms in the domains and in the domain wall.] This directional order, differing in orientation from one domain to another, is then frozen in when the specimen is cooled to room temperature. The result is that domain walls tend to be stabilized in the positions they occupied during the anneal.

The effect differs in kind for 908 and 1808 walls. When a 908 wall is moved a distance x by an applied field, it sweeps out a certain volume. In the volume swept out, the Ms vector is now at 908 to the axis of directional order. The potential energy of the system, which is just the anisotropy energy, must therefore increase from 0 to Ku per unit volume swept out, according to Equation 10.1. The potential energy then varies with x as shown by the dashed line of Fig. 10.7a. The slope of this curve is the force on the wall which the field must exert to move the wall. This force soon becomes constant, as shown in Fig. 10.7b, and so does the restoring force tending to move the wall back to its original position. A 1808 wall, on the other hand, separates domains in which the directional order is identical. But the directional order axes within the wall itself differ from the axis in the adjoining domains. The energy of the system therefore increases as the wall is moved from its original position, but soon becomes constant when the wall is wholly within one of the original domains. The force to move the wall, and the balancing restoring force, increases to a maximum and then returns to zero. Each kind of wall is stabilized by the potential well built up around it by directional order, but the shape of the well depends on the type of wall.

These considerations explain the constricted or “wasp-waisted” hysteresis loops found in self-magnetically annealed alloys and ferrites. Such a loop was first observed in Perminvar and is sometimes called a “Perminvar loop.” This alloy usually contains 25% Co, 45% Ni, and 30% Fe, but the composition may vary over wide limits. It has the remarkable and useful property of constant permeability (hence its name) and almost zero hysteresis loss at low field strengths, as shown in Fig. 10.8a. As the field is cycled between increasingly larger limits, the B, H line progressively opens up into a loop but with essentially zero remanence and coercive force, as in Fig. 10.8c. The “Perminvar loop” is shown in Fig. 10.8d. Its fundamental characteristic is near-zero remanence. The loop for a sample driven nearly to

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Fig. 10.7 (a) Potential energy Ep as a function of the displacement x of 908 and 1808 walls from a stabilized position. (b) Force F required to move the wall (schematic).

saturation is shown in Fig. 10.8e; the remanence is no longer near zero, but is still abnormally low. If the alloy is annealed at 6008C, a temperature too high for directional order to be established, and then quickly cooled, minor hysteresis loops of normal shape are observed at all field strengths.

The magnetic behavior shown in Fig. 10.8 can be qualitatively understood in terms of the potential wells of Fig. 10.7. There is the added complication that in a real material 908 and 1808 walls form a complex network, so that one kind of wall cannot move independently of the other. But at sufficiently low fields, neither kind of wall moves out of its potential well, and the reversible behavior of Fig. 10.8a results. Higher fields cause larger wall displacements, some of which are irreversible. However, when the field is reduced to zero, most of the walls return to their original positions, leading to the abnormally low remanence characteristic of the Perminvar loop in Fig. 10.8d. It will be noted in Fig. 10.8d that the permeability is small up to a field of about 3 Oe and then more or less abruptly becomes larger. This is the critical field (often called the stabilization field) required to move 1808 walls out of their potential wells; after they are out, subsequent motion becomes much easier and the permeability increases. Finally, if the material is driven to near saturation, as in Fig. 10.8e, the original domain structure is destroyed. Then when the field is reduced to zero, a new domain structure forms with many walls in positions unrelated to the existing potential wells. The hysteresis loop then approaches a normal shape, but with an abnormally low remanence because some domain walls (presumably mostly 908 walls) have returned to their original positions.

Fig. 10.8 Hysteresis loops of Perminvar annealed at 4258C. [R. M. Bozorth, Ferromagnetism, reprinted by IEEE Press (1993).]

10.3MAGNETIC ANNEALING (INTERSTITIAL SOLID SOLUTIONS)

The only common examples of interstitial solid solutions in magnetic materials are carbon and nitrogen in body-centered-cubic a iron or iron alloys. The C and N atoms are so small that they can fit into the spaces between the iron atoms, either at the centers of the edges or at the centers of the faces of the unit cell (Fig. 10.9), without producing too much distortion. We can distinguish x, y, and z sites for the interstitial atoms; an x site, for example, is one on a [100] edge. A face-centered site such as f is entirely equivalent to an edge site. The one shown is on a [100] line joining the body-centered iron atom to a similar atom in the next cell (not shown); this f site is therefore equivalent to an x site.

The solubility of C or N in iron is so low that not more than one of these available sites is occupied in any one unit cell. The equilibrium solubility of C, for example, ranges from nearly zero at room temperature to a maximum of about 0.025 wt% at 7238C. The latter figure corresponds to only one C atom in 500 unit cells, about the number of cells contained in a cube of eight cells on each edge. The solubility of N is larger, but still small.

For experimental purposes it may be necessary to control the amount of C and N in solution. This is done by heat treatment in an appropriate gaseous atmosphere. Both C and N can be removed from iron by annealing for several hours at 8008C in hydrogen containing a small amount of water vapor. These interstitials can then be added in

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Fig. 10.9 Positions of carbon or nitrogen atoms in the unit cell of iron.

controlled amounts by heating the iron at 7258C in a mixture of hydrogen and a carbonbearing gas like carbon monoxide or methane (to add C) or in a mixture of hydrogen and ammonia (to add N). The amount introduced at high temperature can be retained in solution at room temperature by quenching.

In paramagnetic a-iron, the x, y, and z sites are crystallographically equivalent and equally populated by any C or N in solution. (We will consider only C from now on. Both interstitials have the same effect.) Below the Curie point, however, in the presence or absence of an applied field, some sites will be preferred over others, depending on the direction of the local magnetization. This preference was first suggested by Ne´el [J. Phys. Radium, 12 (1951) p. 339; 13 (1952) p. 249], who was following up an earlier idea due to J. L. Snoek [Physica, 8 (1941) p. 711]. Just what sites are preferred was demonstrated by G. DeVries et al. [Physica, 25 (1959) p. 1131] by means of an ingenious experiment which is described here in a slightly simplified form. They took a single-crystal disk of iron cut parallel to (001) and containing 0.008 wt% C (80 ppm C) in solution, and cemented a strain gage on it to measure strain in the [010] direction, as indicated in Fig. 10.10a. The strain gage was connected to a recorder which plotted strain vs time. The disk was placed in a saturating field parallel to [100] and maintained at a temperature of 2238C. After the C atoms had reached equilibrium positions with H parallel to [100], the disk was quickly rotated 908 at time t1 to make [010], and the strain gage, parallel to the field. What happened is shown in Fig. 10.10b. Because the magnetostriction l100 is positive in iron, the gage was contracted before the rotation. When rotated, the gage showed an instantaneous elongation because of the positive magnetostrictive strain along its length, followed immediately by a slow contraction, amounting to about 4% of l100. Now the C atoms in iron are somewhat too big for the holes they occupy, so they cause a slight expansion of the lattice. The observed contraction shows that C atoms originally in y sites began to diffuse away from these sites as soon as the Y-axis became parallel to the magnetization. Evidently, before the rotation when H was parallel to X, there were more C atoms in y and z sites than in x sites. The C atoms therefore preferentially occupy sites in a plane normal to Ms.

This result was unexpected. Because each domain in iron is spontaneously elongated parallel to Ms, and contracted at right angles, there should be more room for a C atom in an x site than in a y or z site in a domain that is magnetized in the [100] or X direction. The converse is true. This result can be understood in a general way by assuming that the electron cloud around an iron atom is elongated parallel to the spontaneous

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alloys to decide between these possibilities, analogous to the experiment of DeVries et al., has not yet been done.

We see then that the interstitial sites in iron are preferentially occupied by C atoms in a manner governed by the local magnetization, namely, in a plane normal to the Ms direction. This direction becomes an easy axis, and the resulting anisotropy is described by Equation 10.1. The value of Ku is quite small, being 166 ergs/cm3 [1660 J/m3] for C at 2348C and 71 ergs/cm3 [710 J/m3] for N at 2478C [J. C. Slonczewski, Magnetism, Volume I, p. 205, G. T. Rado and H. Suhl, eds., Academic Press (1963)]. The cause of this anisotropy, like that due to directional order of solute–atom pairs, is assumed to be entirely magnetic, i.e., if the local magnetization Ms is in, say, the X direction, then the energy of a C atom in an x site is larger than it is in a y or z site.

Although the induced anisotropy in an Fe–Ni alloy and the induced anisotropy in iron containing C are formally the same (both are due to an anisotropic distribution of solute atoms), there is a large difference in the kinetics of the two, because interstitial atoms diffuse much faster than substitutional atoms. For example, the frequency at which C atoms jump from one interstitial site to a neighboring one in iron at room temperature is about once a second. This means that it is impossible to set up a preferred distribution of C atoms by magnetic annealing at an elevated temperature and preserve it by fast cooling to room temperature with or without a field, because soon after the annealing field is removed, diffusion destroys the anisotropy. It also means that iron containing C will self-magnetically anneal at room temperature and that domain-wall stabilization will occur by virtue of the potential wells of Fig. 10.7. Although the potential wells due to directional order of solute–atom pairs and those due to substitutional atoms are of slightly different shape, domain wall stabilization due to either cause can be qualitatively discussed in terms of Fig. 10.7.

All quantitative measurements mentioned in this section were made at subzero temperatures, in order to slow down the diffusion of carbon to a point where the measurement can be made in a convenient time. Even then the measurement is dynamic, rather than static, in character. For example, in the experiment of Fig. 10.10, the disturbed distribution of C atoms immediately began to change to a new distribution and the change was complete in about 20 minutes at 2238C.

Domain-wall stabilization does nevertheless lead to observable magnetic effects at room temperature, but the experiments required are of a different nature than those we have so far encountered. The effects are called “time effects,” and we shall postpone their discussion to Chapter 12.

10.4STRESS ANNEALING

When a uniaxial stress is applied to a solid solution, magnetic or not, interstitial or substitutional, the distribution of solute atoms will become anisotropic if the temperature is high enough for rapid diffusion. In an interstitial solution, for example, x sites will be preferred if a tensile stress is applied along the X axis, simply because of the elongation along that axis. In a substitutional solution, the axes of like-atom pairs may be oriented parallel or perpendicular to the axis of tension, depending on the alloy. In either case, the resulting anisotropic distribution of solute can be frozen in by cooling to a temperature where diffusion is negligible.

If the material is magnetic, the effect becomes more complicated. Because of the magnetoelastic interaction, the stress s will change the domain structure by reorienting