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

Fig. 7.31 Shape anisotropy constant vs axial ratio of a prolate spheroid. Numerical values calculated for cobalt (Ms ¼ 1422 emu cm3).

prolate spheroid of polycrystalline cobalt with no preferred orientation. E. C. Stoner and E. P. Wohlfarth in [Phil. Trans. R. Soc, A-240 (1948) p. 599] give values of (Na –Nc) as a function of c/a for both prolate and oblate spheroids. Figure 7.31 shows the results. At an axial ratio of about 3.5, Ks is about 45 105 erg/cm3 or 4.5 106 J/m3, which is equal to the value of the first crystal-anisotropy constant K1 of cobalt. In other words, neglecting K2, we can say that a prolate spheroid of saturated cobalt, with axial ratio 3.5 and without any crystal anisotropy, would show the same uniaxial anisotropy as a spherical cobalt crystal with its normal crystal anisotropy.

7.11MIXED ANISOTROPIES

The calculation of the last paragraph suggests that we consider a more realistic situation, in which two anisotropies are present together. The discussion will be limited to uniaxial anisotropies. We have already touched on the problem of mixed anisotropies in Section 7.8, where the effect of double textures on the anisotropy of polycrystalline sheet was discussed. Here we are interested in the combined effect of two anisotropies of different physical origin, such as crystal and shape anisotropy, on the resultant anisotropy of a single crystal.

We might have, for example, a rod-shaped crystal of a uniaxial substance like cobalt, with its easy crystal axis at right angles to the rod axis. Will it be easier to magnetize along the rod axis, as dictated by shape anisotropy, or at right angles to the rod axis, as dictated by crystal anisotropy? Both anisotropy energies are given, except for constant terms, by expressions of the form: energy ¼ (constant) . sin2 (angle between M and easy axis).

The problem is generalized in Fig. 7.32. where AA represents one easy axis and BB the other. The separate anisotropy energies, distinguished by subscripts, are

EA ¼ KA sin2u,

EB ¼ KB sin2(908 u):

238 MAGNETIC ANISOTROPY

Fig. 7.32 Mixed anisotropy.

The total energy is

E ¼ KA sin2u þ KB cos2u

(7:64)

¼ KB þ (KA KB) sin2u:

If the two anisotropies are of equal strength (KA ¼ KB), then E is independent of angle and there is no anisotropy. (Thus two equal uniaxial anisotropies at right angles are not equivalent to biaxial anisotropy.) If they are not equal, we want the value of u for which E is a minimum:

The solutions are u ¼ 0 and u ¼ 908. To find whether these are minima or maxima, we take the second derivative,

which must be positive for a minimum. Therefore, u ¼ 0 is a minimum-energy position if KA . KB, and u ¼ 908 if KA , KB. The direction of easiest magnetization is not, as might be expected, along some axis lying between AA and BB. The easy direction is along AA if the A anisotropy is stronger, and along BB if B is stronger. The basic reason for this behavior is that a uniaxial anisotropy exerts no torque on M when M is at 908 to the uniaxial axis.

If the two easy axes of Fig. 7.31 are at some angle a to each other, rather than at right angles, the reader can show that they are together equivalent to a new uniaxial axis CC, which either (a) lies midway between AA and BB, if KA ¼ KB, and has a strength of KC ¼ KA ¼ KB, or (b) lies closer to AA, if KA . KB, and has a strength KC . KA.

PROBLEMS

7.1Prove the statements in Section 7.1 regarding the equilibrium values of the angle u for various relative values of K1 and K2 in a hexagonal crystal.

PROBLEMS 239

7.2A disk sample 0.80 cm in diameter and 0.050 cm thick has a weak uniaxial anisotropy with K1 ¼ 2500 erg/cm3. What is the maximum torque acting on the sample when it is rotated in a strong magnetic field about an axis perpendicular to the disk surface?

7.3a. Find the relations between H and M for magnetization in the ,100. and ,110. directions of a cubic crystal like nickel, with K1 , 0. Assume K2 ¼ 0.

b.Compute and plot the magnetization curves of a nickel crystal in the ,100. , ,110. , and ,111. directions. Take K1 ¼ 25.0 103, K2 ¼ 0, M ¼

846 103 A/m.

c.What are the fields required to saturate in the ,110. and ,100. directions?

7.4Calculate the anisotropy energy stored in a cubic crystal with ,100. easy directions, magnetized in a ,110. direction, by finding the area between the M, H curve and the M-axis.

7.5Derive Equation 7.50.

7.6An oblate or planetary spheroid and a prolate spheroid have the same ratio of major axis to minor axis. Which has the greater shape anisotropy?

7.7A cobalt single crystal is made into an oblate or planetary ellipsoid with ratio of major to minor axis ¼ 2. The ,0001. easy axis of the crystal is normal to the plane of the ellipsoid. Take K1 ¼ 4.5 106 erg/cm3, K2 ¼ 0, Ms ¼ 1422 emu/cm3. In which direction is it easiest to magnetize the sample to saturation?

7.8At what value of the dimensional ratio will the sample of Problem 7.7 be equally easy to saturate parallel and perpendicular to the surface?

242 MAGNETOSTRICTION AND THE EFFECTS OF STRESS

20 1026 per K, so a strain of 1025 results from a temperature change of just 0.5K. In weakly magnetic substances the effect is even smaller, by about two orders of magnitude, and can be observed only in very strong fields. We will not be concerned with magnetostriction in such materials.

Although the direct magnetostrictive effect is small, and not usually important in itself, there exists an inverse effect (Section 8.5) which causes such properties as permeability and the size and shape of the hysteresis loop to be strongly dependent on stress in many materials. Magnetostriction therefore has many practical consequences, and a great deal of research has accordingly been devoted to it.

The value of the saturation longitudinal magnetostriction ls can be positive, negative, or, in some alloys at some temperature, zero. The value of l depends on the extent of magnetization and hence on the applied field, and Fig. 8.1 shows how l typically varies with H for a substance with positive magnetostriction. As mentioned in the preceding chapter, the process of magnetization occurs by two mechanisms, domain-wall motion and domain rotation. Most of the magnetostrictive change in length usually occurs during domain rotation.

Between the demagnetized state and saturation, the volume of a specimen remains very nearly constant. This means that there will be a transverse magnetostriction lt very nearly equal to one-half the longitudinal magnetostriction and opposite in sign, or

When technical saturation is reached at any given temperature, in the sense that the specimen has been converted into a single domain magnetized in the direction of the field, further increase in field causes a small further strain (Section 4.2). This causes a slow change in l with H called forced magnetostriction, and the logarithmic scale of H in Fig. 8.1 roughly indicates the fields required for this effect to become appreciable. It is caused by the increase in the degree of spin order which very high fields can produce (the paraprocess).

The longitudinal, forced-magnetostriction strain l shown in Fig. 8.1 is a consequence of a small volume change, of the order of DV/V ¼ 10210 per oersted, occurring at fields

Fig. 8.1 Dependence of magnetostriction on magnetic field (schematic). Note that the field scale is logarithmic.

beyond saturation and called volume magnetostriction. It causes an equal expansion or contraction in all directions. Forced magnetostriction is a very small effect and has no bearing on the behavior of practical magnetic materials in ordinary fields.

The measurement of longitudinal magnetostriction is straightforward but not trivial, especially over a range of temperatures. While early investigators used mechanical and optical levers to magnify the magnetostrictive strain to an observable magnitude, today this measurement on bulk samples is commonly made with an electrical-resistance strain gage cemented to the specimen. The gage is made from an alloy wire or foil grid, embedded in a thin paper or polymer sheet, which is cemented to the sample. When the sample changes shape, so does the grid, and the change in shape causes a change in the electrical resistance of the gage. With ordinary gages, the fractional change in resistance is about twice the elastic strain. This is typically a small resistance change, but one fairly easily measured with a bridge circuit, either ac or dc.

Commercial strain gages commonly are magnetoresistive (show a change in resistance when subjected to a magnetic field). This effect can be compensated by including in the bridge circuit used for detection a dummy gage that is subject to the same magnetic field as the active gage, but that is not attached to the sample and so undergoes no strain. The sensitivity, or gage factor, of a strain gage may also be temperature dependent. Semiconductor strain gages are much more sensitive than ordinary alloy gages, but have larger magnetoresistance and larger temperature dependence.

Capacitance or inductance or optical measuring systems of various designs may also be used. These have the advantage that nothing has to be cemented to the specimen, and by appropriate design they can be used over a wide temperature range. Careful control of the sample temperature is always necessary, since (as noted above) a one-degree temperature change can easily produce a strain larger than the magnetostrictive strain to be measured. Other, more subtle sources of error, due to uncertainties about the nature of the demagnetized state, are considered later.

Thin film samples present special challenges in the measurement of magnetostriction, since the films are almost always bonded to a nonmagnetic substrate. If the substrate is thin enough, a change in dimension of the film may produce a measurable curvature in the substrate, from which the magnetostrictive strain can be deduced. Another approach is to apply a known stress to the sample and measure the resulting change in magnetic anisotropy. This method makes use of the concept of the stress anisotropy, as described later in this chapter.

8.2MAGNETOSTRICTION OF SINGLE CRYSTALS

When an iron single crystal is magnetized to saturation in a [100] direction, the length of the crystal in the [100] direction is found to increase. From this we infer that the unit cell of ferromagnetic iron is not exactly cubic, but slightly tetragonal. (A tetragonal cell has three axes at right angles; two are equal to each other, and the third is longer or shorter than the other two.) This conclusion follows from what we already know about the changes occurring in an iron crystal during magnetization in a [100] direction. These changes consist entirely of domain wall motion and were described with reference to Fig. 7.3. If the saturated crystal is longer in the direction of its magnetization than the demagnetized crystal, then the single domain which comprises the saturated crystal

244 MAGNETOSTRICTION AND THE EFFECTS OF STRESS

Fig. 8.2 Magnetostriction of an iron crystal in the [100] direction.

must be made up of unit cells which are slightly elongated in the direction of the magnetization vector.

The same is true of each separate domain in the demagnetized state. Figure 8.2a depicts

¯ ¯

this state in terms of four sets of domains, [100], [100], [010], and [010]. Unit cells are shown by dashed lines; their tetragonality is enormously exaggerated and so is their size relative to the domain size. But the main point to notice is that these cells are all longer in the direction of the local Ms vector than they are in directions at right angles to this

¯

vector. Thus, when a region originally occupied by, say, a [010] domain is replaced by a [100] domain, by the mechanism of wall motion, that region must expand in the [100] direction and contract in directions at right angles. The length of the whole crystal therefore changes from l to l þ Dl, where Dl/l ¼ ls ¼ the saturation magnetostriction in the [100] direction.

The unit cell of iron is exactly cubic only when the iron is above the Curie temperature, i.e., only when it is paramagnetic, and subject to no applied field. As soon as it cools below Tc, spontaneous magnetization occurs and each domain becomes spontaneously strained, so that it is then made up of unit cells which are slightly tetragonal. The degree of tetragonality in iron and most other materials is less than can be detected by X-ray diffraction, so iron is regarded as cubic in the crystallographic literature.

There are therefore two basic kinds of magnetostriction: (1) spontaneous magnetostriction, which occurs in each domain when a specimen is cooled below the Curie point; and

(2) forced magnetostriction, which occurs when a saturated specimen is exposed to fields large enough to increase the magnetization of the domain above its spontaneous value. Both kinds are due to an increase in the degree of spin order. The spontaneous magnetostriction is difficult to observe directly, but it is evidenced by a local maximum at Tc in the variation of the thermal expansion coefficient with temperature. (This is the reason why Invar, an Fe–Ni alloy containing 36% Ni, has such a low expansion coefficient near room temperature. Its Curie point is near room temperature, and its spontaneous magnetostriction varies with temperature in such a way that it almost compensates the normal thermal expansion.) The “ordinary,” field-induced magnetostriction which concerns us, and in which l changes from 0 to ls (Fig. 8.1), is caused by the conversion of a demagnetized

specimen, made up of domains spontaneously strained in various directions, into a saturated, single-domain specimen spontaneously strained in one direction. Figure 8.2 shows one special case of such a conversion, in which the only mechanism of magnetization change is domain wall motion.

Domain walls are described by the angle between the Ms vectors in the two domains on either side of the wall. Two types exist: 1808 walls and non-1808 walls. The uniaxial crystal of Fig. 7.6 has only 1808 walls, while the iron crystal of Fig. 8.2a has only 908 walls. Nickel, which has k111l easy directions, can have 1808, 1108, or 718 walls. Non-1808 walls are often called 908 walls for brevity, whether the actual angle is 908, 1108, or 718. The domain structure of real single crystals is normally such that both 1808 and non-1808 walls exist.

The magnetostrictive effect of the motion of the two kinds of walls is quite different. Because the spontaneous strain is independent of the sense of the magnetization, the dimensions of a domain do not change when the direction of its spontaneous magnetization is reversed. Since passage of a 1808 wall through a certain region reverses the magnetization of that region, we conclude that 1808 wall motion does not produce any magnetostrictive change in dimensions. Thus, when the uniaxial crystal of Fig. 7.6 is saturated in the axial direction by an applied field, only 1808 wall motion is involved, and the length of the crystal does not change in the process. On the other hand, magnetization of the iron crystal of Fig. 8.2 is accomplished by 908 wall motion, and a change in the length of the crystal does occur.

Rotation of the Ms vector of a domain always produces a dimensional change, because the spontaneous magnetostriction depends on the direction of the Ms vector relative to the crystal axes. Thus, in the general case of a crystal being magnetized in a noneasy direction, the magnetization process will involve 1808 and 908 wall motion and domain rotation. The last two of these three processes will be accompanied by magnetostriction.

We now need expressions for the strain which a crystal undergoes in a certain direction when it is magnetized either in the same direction or in some arbitrary direction.

8.2.1Cubic Crystals

The saturation magnetostriction lsi undergone by a cubic crystal in a direction defined by the direction cosines b1, b2, b3 relative to the crystal axes, when it changes from the demagnetized state to saturation in a direction defined by the direction cosines a1, a2, a3, is given by

where l100 and l111 are the saturation magnetostrictions when the crystal is magnetized, and the strain is measured, in the directions k100l and k111l, respectively. This equation is valid for crystals having either k100l or k111l as easy directions. Most commonly, we want to know the strain in the same direction as the magnetization; then b1, b2, b3 ¼ a1, a2, a3, and Equation 8.3 becomes

246 MAGNETOSTRICTION AND THE EFFECTS OF STRESS

This can be further reduced, by means of the relation

Equation 8.3 is called the “two-constant” equation for magnetostriction. Like the rather similar equation for crystal anisotropy energy (Equation 7.1), it can be expanded to higher powers of the direction cosines. The next approximation involves five constants; its use is rarely justified by the accuracy of available magnetostriction data.

Equation 8.3 and others derived from it require a word of caution. They give the fieldinduced strain when the crystal is brought from the demagnetized to the saturated state. The saturated state is, by definition, one in which the whole specimen consists of a single domain with its Ms vector parallel to the applied field. The demagnetized state, on the other hand, is not well-defined. All that is required is that all the domain magnetizations, each properly weighted by its volume, add vectorially to zero. The difficulty is that there are an infinite number of domain arrangements and relative volumes that can result in zero net magnetization of the whole specimen. Equation 8.3 is based on a paticular definition of the demagnetized state, namely, one in which all possible types of domains have equal volumes. For example, in a cubic crystal like iron, with k100l easy directions, this state is one in which the total volume of the crystal is divided equally among six

¯ ¯ ¯

kinds of domains: [100], [100], [010], [010], [001], and [001]. If this “ideal” demagnetized state is not achieved, Equation 8.3 is invalid, and the measured magnetostriction will be larger or smaller than the calculated value. Differences in the magnetostriction values observed by different investigators for the same material are usually due to differences in the demagnetized states of their specimens.

The special symbol lsi is used in Equation 8.3 and similar ones, where the subscript “i” refers to the ideal demagnetized state. This symbol is not widely used in the scientific literature. It is introduced here in an attempt to inject greater clarity into the discussion of magnetostrictive strains. The single symbol ls is ambiguous, because it is used in the literature to refer both to ideal and nonideal demagnetized states. We therefore have two kinds of saturation magnetostriction:

1.lsi, measured from the ideal demagnetized state. This value is a constant of the material. Because it is defined, through Equation 8.3, in terms of l100 and l111, the two latter are also lsi values, measured in these particular crystal directions.

2.ls, measured on a particular specimen having a particular, nonideal demagnetized state. This value is a property only of that particular specimen. The quantity ls is highly structure sensitive, in that it depends on the mechanical, thermal, and magnetic

history of the specimen. If the demagnetized state is nonideal, i.e., if all possible domains are not present in equal volumes, it is said to be a state of preferred domain orientation. Thus a preferred domain orientation can exist in the demagnetized state of a single crystal, or in the individual grains of a polycrystal, in addition to the preferred grain orientation that may exist in a polycrystal.

The domain arrangement shown for the demagnetized crystal in Fig. 8.2a is an example of preferred domain orientation, because it contains only four kinds of domains. If the

¯

missing domains [001] and [001] were present, the demagnetized crystal would be