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

7.7ANISOTROPY CONSTANTS

Table 7.4 shows the values of the room temperature anisotropy constants of several magnetic materials. It is usually found that lower symmetry leads to larger anisotropy, so materials with hexagonal crystal structure have larger anisotropy constants than cubic materials. The extremely large anisotropy of YCo5 is typical of that of other RCo5 phases, where R is Y (yttrium) or a rare-earth element.

The values of these constants are usually known only to two significant figures (sometimes only one) and the accuracy of K2 is less than that of K1. Furthermore, the results of different investigators are sometimes in poor agreement. The lack of reproducibility may result from differences between samples, different experimental techniques, and different maximum applied fields.

Anisotropy constants almost always decrease as the temperature increases and become essentially zero before the Curie temperature is reached. There is then no preferred crystallographic direction for the magnetization of a domain. Figure 7.24 shows the behavior of iron and nickel and Fig. 7.25 that of cobalt. (The data for iron, which are quite old, are in poor agreement with the room-temperature values given in Table 7.4.) The thermal variation of K1 and K2 for cobalt is such that the easy axis is (a) parallel to the c axis up to 2458C, (b) inclined to the c axis, but having any azimuthal position about it, at an angle u which increases from 0 to 908 as the temperature increases from 245 to 3258C, and (c) in any direction in the basal plane above 3258C. Note that cobalt undergoes a sluggish phase transformation from hexagonal close-packed to face centered cubic at about 4108C, with no change in saturation magnetization.

One model for the temperature dependence of anisotropy is based on the idea that, as the temperature increases, the local magnetization direction spreads over a range of angles about the mean direction. This raises the energy of the easy direction and lowers the energy of the hard direction, and leads to the prediction that anisotropy will decrease as a power of the reduced magnetization M/M0, where M0 is the saturation magnetization at 0K. Specifically, the prediction is that

Fig. 7.25 Temperature dependence of anisotropy constants of cobalt [Y. Barnier, R. Pauthenet, and G. Rimet, Cobalt, 15 (1962) p. 1].

where n is the power of the anisotropy function. So for uniaxial anisotropy (n ¼ 2),

This model fits the data quite well for iron, but fails badly for some other materials.

The anisotropy constants of alloys vary markedly with composition. These values will be given later when particular alloy systems are discussed.

7.8POLYCRYSTALLINE MATERIALS

If the constituent crystals (grains) of a polycrystalline body are oriented randomly in space (which is not often the case), then the anisotropy of the individual grains will average out, and the body on the whole will exhibit no crystal anisotropy. If, on the other hand, the crystals have a preferred orientation, also called a crystallographic texture, then the polycrystalline aggregate itself will have an anisotropy dictated by the weighted average of the individual crystals.

The kind of texture possessed by a body depends on its shape and how it was formed. Thus a round wire, rod, or bar generally has a fiber texture: each grain has a certain

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crystallographic direction kuvwl parallel, or nearly parallel, to a single direction called the fiber axis, which, in the case of a wire, coincides with the wire axis. However, the grains can have any rotational position about this axis. Iron wire, for example, has a k110l fiber texture. Double textures are also possible. Wires of some face-centered-cubic metals have a k111l þ k100l texture; some grains have k111l directions parallel to the wire axis, others have k100l. Electrodeposited or evaporated layers, deposited on a flat surface, may also have a fiber texture, but here the fiber axis is normal to the surface of deposition.

Sheet made by rolling has a texture in which, in each grain, a certain plane fhklg tends to be parallel to the sheet surface and a certain direction kuvwl in that plane tends to be parallel to the direction in which the sheet was rolled. Such sheet textures are described by the symbolism: frolling planeg krolling directionl. Thus the “cube texture,” found in some metals and alloys, is f100g k001l; a cube plane f100g is parallel to the sheet surface, and a cube-edge direction k001l is parallel to the rolling direction.

Textures in metals are also distinguished by whether they are formed during deformation (deformation textures) or during a recrystallization heat treatment (recrystallization textures).

Crystal anisotropy is often exploited in the manufacture of magnetic materials by inducing a texture such that the easy directions of magnetization in all grains are at least approximately parallel. The polycrystalline body as a whole then has an easy direction. Some control of the degree and, to a lesser extent, of the kind of preferred orientation is possible in metals and alloys formed by the usual processes of casting and working by rolling or wire drawing. But in general it is not possible to produce a particular desired texture at will. Thus both the deformation and recrystallization textures of iron wire are k110l, and it stubbornly resists any attempts to rotate the k100l easy directions into parallelism with the wire axis. Efforts to make marked changes of any kind in the k110l texture, including recrystallization in a magnetic field, have so far been unsuccessful.

On the other hand, control of easy-axis orientation is relatively easy when the manufacturing operation is one of pressing and sintering a powder, either metallic or nonmetallic, if the individual particles are single crystals. It is simply a matter of applying a strong magnetic field in the required direction during the pressing operation. When the powder particles are still a loose assemblage, the field lines them up with their easy axes parallel to one another and to the field; the compacting die then locks in this preferred orientation as the powder is compressed. Note that alignment will not occur if the individual particles are themselves random polycrystals.

The aligning effect of a magnetic field can also be used in a method for determining the easy axis of a material, when a single-crystal specimen is not available. A powder is prepared by grinding or filing, annealed to remove the effects of deformation, and mixed with a solution of a binder so that it forms a powder suspension. A few drops are then placed on a glass slide, or other flat plate, and allowed to dry in the presence of a magnetic field applied normal to the slide surface. The dried powder specimen on the slide then has a marked preferred orientation, with easy axes kuvwl normal to the slide surface. It is subsequently examined in an X-ray diffractometer in the usual way, i.e., with the incident and diffracted beams making equal angles with the slide surface. Under these circumstances, only those grains which have their fhklg planes parallel to the slide surface can contribute to a particular hkl reflection. For the field-oriented specimen, certain X-ray reflections will be abnormally strong, namely, those from planes at right angles to the kuvwl easy axis. The direction kuvwl can then be determined from a knowledge of the crystal structure of the

shaded triangle OCD, according to Equation 7.40. This energy is that associated with the demagnetizing field of the specimen, and is variously called the magnetostatic energy, the self-energy, or the energy of a magnet in its own field. From Equation 7.43, this energy is

where dv is an element of volume and the integration extends over all space. The distribution of Hd in space is seldom known accurately and, even when it is, the evaluation of this integral would be difficult. It is easier to compute the area of the triangle OCD in Fig. 7.28:

where M is the level of magnetization at point C. This energy can be written in vector form as

because Hd is antiparallel to M.

On the other hand, the potential energy per unit volume of a magnet in an applied field Ha is, from Equation 1.6,

The expressions for the energy of a magnet in its own field and in an applied field are therefore similar in form, except for the factor 12. Equation 7.58 can be written in terms of Nd, by the substitution Hd ¼ NdM:

The reasoning leading to Equation 7.59 is more physically meaningful when one realizes that the point representing the state of the specimen, or part of it, in Fig. 7.29 can be made to move back and forth along the line OC. If C represents the room-temperature state, then the point moves from C to O when the specimen is heated from room temperature to the Curie point. If one now imagines a small volume, smaller that the normal volume of a domain, at a temperature above Tc, then, on cooling through Tc, this small volume becomes spontaneously magnetized to a level Ms which increases as the temperature decreases. Simultaneously, a demagnetizing field Hd is set up. The point representing this small volume therefore moves from O toward C. At room temperature the magnetization of the whole specimen, which is the sum of the Ms vectors in all the domains, is at point C.

To return to shape anisotropy, we now consider a specimen in the shape of a prolate spheroid (rod) with semi-major axis c and semi-minor axes a of equal length (Fig. 7.30).

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Fig. 7.30 Prolate ellipsoid.

Let it be magnetized to a value M at an angle u to c. Then, taking components of M parallel and perpendicular to c, we have

where Nc and Na are demagnetizing coefficients along c and a, respectively, as given by Equations (2.20) and (2.21). Substituting cos2u ¼ 1 2 sin2u, we find

This expression for the magnetostatic energy has an angle-dependent term of exactly the same form as uniaxial crystal anisotropy energy (Equation 7.4). The long axis of the specimen plays the same role as the easy axis of the crystal, and the shape-anisotropy constant Ks is given by

Magnetization is easy along the c-axis and equally hard along any axis normal to c. If c shrinks until it equals a, the specimen becomes spherical, Na ¼ Nc, Ks ¼ 0, and shape anisotropy disappears. The long axis of the spheroid is labeled 2c in Fig. 7.30 to conform with the earlier definition of ellipsoids. It is not to be confused with the c axis of a hexagonal crystal.

Magnetization of a planetary or oblate spheroid (disk) is difficult along the short a-axis and equally easy along any axis normal to a, i.e., in the plane of the disk, which is why specimens for crystal-anisotropy measurements are made in the form of disks.

As Equation 7.62 shows, the “strength” of shape anisotropy depends both on the axial ratio c/a of the specimen, which determines (Na –Nc), and on the magnitude of the magnetization M. To illustrate the sort of numbers involved, we put M ¼ 1422 emu/cm3 or 1.422 106 A m2/m3, which is the room-temperature saturation magnetization of uniaxial cobalt, and calculate the value of the shape-anisotropy constant Ks as a function of c/a for a