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

Fig. 8.5 Magnetostriction as a function of magnetization in single crystal iron rods. Each rod had a different crystallographic axis. [W. L. Webster, Proc. R. Soc., A109 (1925) p. 570.]

process the dimensions of the crystal do not change. Further increase of field causes Ms vectors to rotate toward [111], and this rotation causes a contraction along [111].

When H is parallel to [110] in iron, the crystal first expands in that direction and then contracts. These changes can be understood by reference to Fig. 7.4, if one imagines the

¯

addition of [001] and [001] domains to the initial state depicted there. In response to the applied field, 908 and 1808 wall motion will take place until the crystal contains only two sets of domains, those corresponding to the two easy axes nearest the applied field

¯

(Fig. 7.4c). During this process, [001] and [001] domains have disappeared. Inasmuch as these domains are spontaneously contracted in a direction parallel to the field direction [110], their removal causes an expansion in the [110] direction, as observed. With further increase in field, the magnetization in the remaining [100] and [010] domains rotates into the [110] direction, causing an additional strain of 34l111 along [110]. Because l111 is negative, this strain is a contraction, and it is large enough to make the crystal shorter at saturation than it was initially.

A nickel crystal contracts in all three principal directions when magnetized, as shown in Fig. 8.6. From the observed contraction in the [111] direction and the fact that the easy directions in nickel are k111l, it follows that the unit cell of ferromagnetic nickel is slightly distorted from cubic to rhombohedral, with one cell diagonal, the one parallel to the local direction of magnetization, slightly shorter than the other three. So when the Ms vector in a domain is rotated away from a k111l easy axis, that axis becomes longer. One can then understand, by arguments similar to those given above for iron, why l100 and l111 are both negative in nickel.

Note that, in Fig. 8.5, magnetostriction is plotted against magnetization, while in Fig. 8.6 magnetostriction is plotted against magnetic field H. These qualitative descriptions of the variation of l with H (or M ) below saturation can be made quantitative without much difficulty, although certain rather arbitrary assumptions have to be made about the sequence of 1808 and 908 wall motion.

250 MAGNETOSTRICTION AND THE EFFECTS OF STRESS

Fig. 8.6 Magnetostriction as a function of field and crystal direction for nickel single crystal. Samples were planetary (oblate) spheroids. Solid lines, {011} disk; dashed lines, [001] disk. [Y. Masiyama, Sci. Rep. Tohoku Univ., 17 (1928) p. 947.]

If the magnetostriction of a particular material is isotropic, we can put l100 ¼ l111 ¼ lsi. Then Equation 8.3 becomes, with the introduction of a new symbol,

where lu is the saturation magnetostriction at an angle u to the direction of magnetization, measured from the ideal demagnetized state. (If u is the angle between two directions defined by cosines a1, a2, a3 and b1, b2, b3, then cosu ¼ a1b1 þa2b2 þa3b3.) Because of isotropy, no reference to the crystal axes appears in Equation 8.10, and the magnetostrictive effect can be illustrated quite simply by Fig. 8.7, which shows a demagnetized sphere distorted into an ellipsoid of revolution when saturated, for a positive value of lsi. Figure 8.6 shows that the magnetostrictive behavior of nickel is approximately isotropic, and Equation 8.10 is often applied to nickel.

Table 8.1 lists lsi values for some cubic metals and ferrites. (The variation with composition of lsi for alloys will be described later.) In general, the magnetostriction of the ferrites is of about the same order of magnitude as that of the metals, with the notable exception of cobalt ferrite. Here the spontaneous distortion of the crystal unit cell, from cubic to tetragonal, is so large that it can be detected by X-ray diffraction. This ferrite also has an unusually large value of crystal anisotropy (Table 7.4).

252 MAGNETOSTRICTION AND THE EFFECTS OF STRESS

Fig. 8.8 Hexagonal and orthogonal axes in a hexagonal crystal.

cell is outlined. The c and z axes are normal to the plane of the drawing. Equation 8.11 is valid only for crystals in which the c-axis is the easy direction.

When the magnetostriction is measured in the same direction as the magnetization, then b1, b2, b3 ¼ a1, a2, a3, and Equation 8.11 reduces to a two-constant expression

because a21 þ a22 þ a23 ¼ 1. Inasmuch as only a3 appears in Equation 8.12, the value of lsi in, for example, the basal plane, is the same in any direction. Equations 8.11 and 8.12 therefore express cylindrical, rather than hexagonal, symmetry. Hexagonal symmetry appears only in the next approximation. R. M. Bozorth [Ferromagnetism, Van Nostrand (1951); reprinted by IEEE (1993)] finds that the behavior of cobalt is adequately described by the following constants:

Magnetostriction as a function of field strength is shown in Fig. 8.9. As expected, lsi parallel to the c-axis is zero, because only 1808 wall motion is involved. The contraction observed at 608 to the c-axis is much larger than in the basal plane (u ¼ 908) and is, in fact, the maximum contraction for any value of u.

Alternative Notation An alternative notation for magnetostriction in single crystals [E. R. and H. B. Callen, Phys. Rev., 129 (1963) p. 578; A139 (1965) p. 455] is sometimes used, especially in theoretical treatments of the subject.

General Magnetostriction constants usually decrease in absolute magnitude as the temperature increases, and approach zero at the Curie point. However, there are exceptions. Figure 8.10 shows the behavior of iron; it is clear that l100 and l111 have very different temperature dependences.

Before leaving the topic of single crystals it is important to realize that any demagnetized crystal that contains 908 walls is never completely stress free at room temperature. The various domains simply do not fit together exactly. Figure 8.11a depicts a single crystal of iron, for example, at a temperature above the Curie point; the dashed lines indicate where domain walls will form below Tc. As the crystal cools below Tc, it spontaneously

Substituting Equation 8.6 for lsi and integrating, we obtain

On the other hand, H. E. Callen and N. Goldberg [J. Appl. Phys., 36 (1965) p. 976] claim that an assumption intermediate between those of uniform stress and uniform strain leads to an equation which better represents the data available for a number of polycrystalline metals, alloys, and ferrites. This assumption leads to the following equation

where c ¼ 2c44/(c11 –c12), and c44, c11, and c12 are single-crystal elastic constants. A crystal is elastically isotropic if c ¼ 1; in that case, Equations 8.14 and 8.15 are the same.

When the single-crystal data for iron are substituted into these relations, ¯si is found to be l

24 1026 from Equation 8.14 and 29 1026 from Equation 8.15. The usually accepted experimental value of lp is about 27 1026. However, it should be noted that the magnetostriction of polycrystalline materials is often measured on rod specimens, and iron rods almost invariably have a more or less pronounced k110l fiber texture, as mentioned in Section 7.8. Inasmuch as the value of l110 is 210 1026, the presence of a k110l component would tend to make the value of lp measured on a rod specimen more negative than the value to be expected for a polycrystal with randomly oriented grains. This suggests that Equation 8.14 is more accurate for iron.

The usually accepted experimental value of lp for nickel, 234 1026, compares reasonably well with 233 1026 given by Equation 8.14, and with 230 1026 given by Equation 8.15. However, the values of lp reported by individual investigators cover a surprisingly wide range, from 225 1026 to 247 1026. Part of this spread may be due to differences in preferred grain orientation. While magnetostriction in nickel is commonly regarded as approximately isotropic, the magnitude of l100 is in fact almost double that of l111. Like other face-centered-cubic metals, nickel in rod form can be expected to have a double k100l þ k111l fiber texture, as mentioned in Section 7.8. Variations in the relative amounts of these two components could cause large changes in lp. So could insufficient annealing of the specimens. The magnetic measurements might then have been made on specimens containing residual stress left over from the cold-worked state. This stress could introduce large errors, because the magnetic properties of nickel are very stress sensitive, as we shall see in Sections 8.5 and 8.6.

If we wish to know the magnetostriction at an angle u to the magnetization, we can use

where lp has been substituted for lsi. Because this equation was derived for an isotropic specimen, its application to a polycrystal requires that the specimen have no preferred orientation or that it be composed of grains which are themselves magnetically isotropic.

Figure 8.13 shows typical l, H curves for polycrystalline iron, cobalt and nickel. The shapes of such curves, as reported by different investigators, can vary widely, usually because of differences in preferred orientation. Unfortunately, however, the measurement of magnetostriction is hardly ever accompanied by a determination of the kind and degree of preferred orientation.

256 MAGNETOSTRICTION AND THE EFFECTS OF STRESS

Fig. 8.13 Magnetostriction of polycrystalline iron, cobalt, and nickel. [E. W. Lee, Rep. Progr. Phys., 18 (1955) p. 184.]

Magnetostriction is usually measured, and the results presented, as strain vs field. Since magnetostriction results directly from changes in magnetization, it would be more appropriate from a fundamental viewpoint to plot magnetostrictive strain vs magnetization, although this makes the experimental work more complicated. Figure 8.14 shows plots of strain vs

Fig. 8.14 Magnetostriction vs field and vs magnetization for polycrystalline FeCo þ 2% V. [B. E. Lorenz and C. D. Graham, IEEE Trans. Mag., 42 (2006) p. 3886.]