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

258 MAGNETOSTRICTION AND THE EFFECTS OF STRESS

The rare-earth metals are exceptions to the above statements. Many of them are ferromagnetic, at temperatures mostly well below room temperature, and their orbital moments are not quenched, i.e., the spin–orbit coupling is strong. Moreover, the electron cloud about each nucleus is decidedly nonspherical. Therefore, when an applied field rotates the spins, the orbits rotate too and considerable distortion results. At 22K the saturation magnetostriction of dysprosium is about 4.5 1023 in the basal plane, or some 100 times that of “normal” metals and alloys. Some rare-earth compounds with transition metals Fe, Ni, and Co have Curie points above room temperature together with abnormally large values of magnetostriction.

Inasmuch as magnetostriction and crystal anisotropy are both due to spin–orbit coupling, we would expect some correlation between the two. In fact, a large value of the anisotropy constant K1 is usually accompanied by a large value of lsi. For example, hexagonal substances tend to have larger values of both jK1j and jlsij than cubics. And in binary alloys, the addition of a second element in solid solution often decreases both jK1j and jlsij. These are just general trends, however, and there are exceptions.

The physical origin of magnetostriction is treated in some detail by E. du Tre´molet de Lacheisserie [Magnetism, Volume 1, Chapter 12, Kluwer Academic (2002)].

8.4.1Form Effect

When a specimen is magnetized, a strain (Dl/l )f occurs which has a physical origin entirely different from that of magnetostriction but which can be erroneously ascribed to magnetostriction. Because this strain depends on the shape of the specimen, it is referred to as the form effect. This effect occurs because of the tendency of a magnetized body to minimize its magnetostatic energy. Suppose a specimen is magnetized to its saturation Ms in a direction along which its demagnetizing factor is Nd. Then, according to Equation 7.51, its magnetostatic energy is of the form 12NdM2s . The specimen can decrease this energy by lengthening a fractional amount (Dl/l )f in the direction of Ms, because a longer specimen has a smaller value of Nd. The magnitude of the elongation will depend on the elastic constants of the material. The strain lsi due to magnetostriction is superimposed on the formeffect strain (Dl/l )f, and the two together make up the observed strain. For an iron sphere, (Dl/l )f is about 4 1026. This value decreases rapidly as the sample is made more elongated in the direction of magnetization. So the form effect is generally small, but not negligible for short specimens.

Because it is difficult to calculate (Dl/l )f with accuracy, it is better (and usual) to avoid the form effect by choosing a specimen shape with small Nd. The magnetostatic energy will then be so small that (Dl/l )f will become negligible. For most materials, a disk sample with a diameter/thickness ratio of 10 or more will not require a correction for the form effect.

8.5EFFECT OF STRESS ON MAGNETIC PROPERTIES

Although the magnetostrictive strain is small in most magnetic materials, the existence of magnetostriction means that an applied mechanical stress can alter the domain structure and create a new source of magnetic anisotropy. These effects can have a substantial effect on the low-field magnetic properties, such as permeability and remanence. Figure 8.16 shows the marked effects of applied stress on the magnetization behavior of polycrystalline nickel. At a field H of 10 Oe (800 A/m), a compressive stress of 10,000 lb/in2 (70 MPa) almost

Fig. 8.16 Effect of applied tensile (þ) and compressive (2) stress on the magnetization curve of polycrystalline nickel; 10,000 lb/in2 69 MPa. [D. K. Bagchi, unpublished.]

doubles the permeability m, while the same amount of tensile stress reduces m to about onetenth of the zero-stress value and makes the M, H curve practically linear. Nickel is not unique in this respect. Materials are known in which the low-field permeability is changed by a factor of 100 by an applied stress of the order of 10,000 lb/in2.

The magnetostriction of nickel is negative. For a material with positive magnetostriction, such as 68 permalloy, the effect of stress is just the opposite. (The word permalloy now refers to a family of Ni–Fe alloys, sometimes containing small additions of other elements. The number before the alloy name gives the nickel content. Thus “68 permalloy” means an alloy containing 68% Ni and 32% Fe.) Applied tensile stress increases the permeability of this alloy, as Fig. 8.17 shows.

The magnetostriction of polycrystalline iron is positive at low fields, then zero, then negative at higher fields, as shown in Fig. 8.13. As a result, the magnetic behavior under stress is complicated. At low fields tension raises the B, H curve and at higher fields lowers it; the crossover of the two curves at a particular field strength, which depends on the stress and on preferred orientation, is called the Villari reversal. In the measurements shown in Fig. 8.18, tension has no appreciable effect until B exceeds about 10 kilogauss; the Villari reversal occurs at about 20 Oe. Compression has a reverse, and larger, effect, lowering B at low fields and raising it at large fields.

The experimental results of Figs 8.15–8.18 show that there is a close connection between the magnetostriction l of a material and its magnetic behavior under stress. As a result, the effect of stress on magnetization is sometimes called the inverse magnetostrictive effect, but more commonly is referred to simply as a magnetomechanical effect. It is entirely distinct from the magnetomechanical factor of Section 3.7. These results could have been anticipated by a general argument based on Le Chatelier’s principle. If, for

Fig. 8.19 Effect of tension on the magnetization of a material with positive magnetostriction (schematic).

origin. (The symbol s is used here and in the following paragraphs for applied mechanical stress, not, as previously, for magnetization per unit mass.)

So far we have tacitly assumed that H, M (or B), and s are all parallel, but, in general, M and s may not be parallel. We know from Equation 8.3 that the amount of magnetostrictive strain exhibited by a crystal in a particular direction depends on the direction of the magnetization. If we impose an additional strain by applying a stress, we expect that the direction of the magnetization will change. We therefore need a general relation between the direction of Ms within a domain and the direction and magnitude of s. But we know that, in the absence of stress, the direction of Ms is controlled by crystal anisotropy, as characterized by the first anisotropy constant K1. Therefore, when a stress is acting, the direction of Ms is controlled by both s and K1. These two quantities are therefore involved in the expression for that part of the energy which depends on the direction of Ms, which is, for a cubic crystal,

where a1, a2, a3, are the direction cosines of Ms, as before, and g1, g2, g3 are the direction cosines of the stress s. The units of E are erg/cm3 if s is expressed in dyne/cm2 (1 dyne/

cm2 ¼ 1.02 1028 kg/mm2 ¼ 1.45 1025 lb/in2) or joule/m3 if s is expressed in pascals ¼ N/m2.

Note that here and subsequently the applied stress s is an elastic stress; that is, a stress less than the elastic limit or at least less than the yield stress. If a higher stress is applied, the material will undergo a permanent change in dimensions, and various magnetic properties will be affected.

The first term of Equation 8.17 is the crystal anisotropy energy, taken from Equation 7.1 in its abbreviated form. The next two terms, which involve the magnetostrictive strains and the stress, comprise what is usually called the magnetoelastic energy Eme. The equilibrium direction of Ms is that which makes E a minimum, and this direction is seen to be a complicated function of K1, l100, l111, and s, for any given stress direction g1, g2, g3. But we can note, qualitatively, that the direction of Ms will be determined largely by crystal

This form of the equation states that the magnetoelastic energy is zero when Ms and s are parallel and that it increases to a maximum of 3/2lsis when they are at right angles, provided that lsis is positive. If this quantity is negative, the minimum of energy occurs when Ms and s are at right angles.

We are now in a position to understand why it is so easy to magnetize a material with positive l, such as 68 permalloy, when it is stressed in tension (Fig. 8.17). If the crystal anisotropy is weak, as it is in this alloy, the direction of Ms in the absence of a field will be controlled largely by stress, and, in a polycrystalline specimen with no preferred orientation, Equations 8.18 or 8.19 will apply. Let Fig. 8.21a represent a small portion of the specimen, comprising four domains. The application of a small tensile stress to the demagnetized specimen, as in Fig. 8.21b, will cause domain walls to move in such a way as to decrease the volume of domains magnetized at right angles to the stress axis, because such domains have a high magnetoelastic energy. These domains are completely eliminated by some higher value of the stress, as in Fig. 8.21c, and Eme is now a minimum. The domain structure is now identical with that of a uniaxial crystal, shown in Fig. 7.6. Only a small applied field is now required to saturate the specimen, because the transition from Fig. 8.21c to d can be accomplished solely by the relatively easy process of 1808 wall motion. If the crystal anisotropy is zero, and if no impediments to wall motion (to be discussed in Chapter 9) exist, then an infinitesimal stress and an infinitesimal field should suffice for saturation. Of course these conditions are never met, and we find in practice that a nonzero stress and a nonzero field are required. For 68 permalloy, Fig. 8.17 shows that þ2840 lb/in2 (19.6 MPa) and 0.5 Oe (40 A/m) are enough to raise the magnetization almost to saturation.

Fig. 8.21 Magnetization of a material with positive magnetostriction under tensile stress (schematic).

264 MAGNETOSTRICTION AND THE EFFECTS OF STRESS

In this example, lsis is a positive quantity. The mechanism of Fig. 8.21 will therefore also apply to nickel under compression, because lsis is again positive, and we see in Fig. 8.16 that compression of nickel does indeed make magnetization easier.

Two points emerge from this examination of magnetization under stress:

1.In the demagnetized state, stress alone can cause domain wall motion. This motion will be such as to retain zero net magnetization for the whole specimen. This condition is not difficult to meet, however, because there is an infinite number of domain arrangements which make M equal to zero.

2.Stress alone can create an easy axis of magnetization. Therefore, when stress is present, stress anisotropy must be considered, along with any other anisotropies that may exist. It is a uniaxial anisotropy, and the relation which governs it, namely Equation 8.19, is of exactly the same form as Equation 7.4 for uniaxial crystal anisotropy or Equation 7.53 for shape anisotropy. We therefore write for the stress anisotropy energy, which is a magnetoelastic energy,

where the stress anisotropy constant Ks is given by 32lsis, from Equation 8.19. The axis of stress is an easy axis if lsis is positive. If this quantity is negative, the stress axis is a hard axis and the plane normal to the stress axis is an easy plane of magnetization. The three anisotropies we have met so far are summarized in Table 8.2, in terms of a general uniaxial anisotropy constant Ku.

When lsis is negative, as it is for nickel under tension, the stress axis becomes a hard axis, because the field now has to supply energy equal to the magnetoelastic energy in order to rotate the Ms vector of each domain by 908 into the field direction (Fig. 8.22c). When this rotation is complete, the domain wall simply disappears, and the saturated state of Fig. 8.22d results. The magnetization curve is then expected to be a straight line, just like the M, H curve of a uniaxial crystal such as cobalt when H is perpendicular to the easy axis. This latter example was discussed in Section 7.6, and the relation between M and H was given by Equation 7.29, in terms of a single anisotropy constant. Translating this equation into terms of stress anisotropy, we have

The lower curve of Fig. 8.16 shows that the M, H behavior of nickel under tension is indeed linear, as required by Equation 8.24, at least up to a field of 60 Oe. However, over the whole

TABLE 8.2 Summary of Some Uniaxial Anisotropies

Fig. 8.22 Magnetization of a material with positive magnetostriction under compressive stress (schematic).

range of about 500 Oe required to saturate, the M, H relation is decidedly nonlinear, as shown by the insert to Fig. 8.16.

This disagreement suggests that we examine the validity of Equation 8.24. It is derived on the basis that only stress anisotropy is present, i.e., that any other anisotropy which may exist, such as crystal anisotropy, is negligible in comparison. Now crystal anisotropy, for example, is a constant of the material, but stress anisotropy, for a given material, depends on the stress. We might therefore ask: At what stress does the stress anisotropy become equal to the crystal anisotropy? This stress can be found approximately by equating the stress anisotropy energy 3/2lsis to the crystal anisotropy energy K1. For nickel, this critical stress will be (in cgs units)

(The sign of lsi is irrelevant in this calculation.) Therefore, at a stress of 10,000 lb/in2, at which the data of Fig. 8.16 were obtained, stress anisotropy is actually weaker than crystal anisotropy. The stress would have to be many times 15,000 lb/in2 before Equation 8.24 would strictly apply, all the way to saturation.

The calculation just made is only approximate, because it ignores the complex details of domain rotation. In a single domain of nickel, for example, there are four easy axes determined by the crystal anisotropy and one easy axis determined by the stress. As the Ms vector

266 MAGNETOSTRICTION AND THE EFFECTS OF STRESS

of this domain rotates from its initial position, in response to an increasing applied field, its rotation is sometimes helped and sometime hindered by the crystal anisotropy, depending on the relative orientation at that time of Ms and the easy crystal axes. The calculation also assumes isotropic magnetostriction.

Of the three cases illustrated in Figs 8.16 to 8.18 (nickel, 68 permalloy, and iron), the effect of stress on magnetization is smallest for iron. This is because iron has a relatively large value of K1 and a small value of lsi. This combination of properties means that crystal anisotropy is more important than stress anisotropy; the critical stress at which the two become equal is found from Equation 8.25 to have the very high value of 660,000 lb/in2 ¼ 4500 MPa. This stress is some ten times the breaking stress of iron and is therefore unattainable.

This section has been devoted mainly to the effect of stress on domain rotation. It can also affect domain wall motion, and this topic will be discussed in Section 9.11.

8.6EFFECT OF STRESS ON MAGNETOSTRICTION

Stress not only alters the character of a magnetization curve, but it can produce large changes in the observed magnetostriction ls. This effect is nicely illustrated by the results shown in Fig. 8.23. Nickel rods were subjected to axial tension and compression and the saturation magnetostriction lsi was measured in the axial direction. Compression reduced the magnitude of lsi and, at a stress of about 12 kg/mm2 (17,000 lb/in2 or 115 MPa), the magnetostriction disappeared. Tension, on the other hand, increased the magnitude of lsi and, at a tensile stress of about 12 kg/mm2, ls had become 3/2 of the zero-stress value, which was found to be 240 1026.

This behavior can be understood in terms of the preferred domain orientation set up in the demagnetized state by the applied stress before the magnetostriction measurement is begun. Thus, for nickel under compression, lsis is positive, and sufficient compression will produce the domain arrangement shown in Fig. 8.21c. This specimen can be

Fig. 8.23 Saturation magnetization of nickel under tension and compression. [H. Kirchner, Ann. Phys., 27 (1936) p. 49.]