TEXTURE AND MAGNETIC PROPERTIES

. Expressions are derived for the dependence of the magnetic properties of cubic materials on texture. The theoretical development generally parallels that of Bunge, but employs Roe’s formalism. The expressions, together with limited experimental data, enable one to express magnetic property variations in the plane of the sheet, to obtain property values for comparable texture-free specimens, and to separate magnetic properties into texture-dependent and texture-independent components.


INTRODUCTION
An equation giving the magnetization energy of cubic polycrystals as a function of direction in the plane of a sheet from the magneto-crystalline anisotropy coefficients and the coefficients of a spherical surface harmonic expansion of a pole figure was first derived by Bunge. 1 Development of methods by Bunge 2 and Roe , in which generalized spherical harmonic expansions of the crystallite orientation distribution (COD) were obtained from spherical surface harmonic expansions of the pole figure made it possible to express the orientation dependence of the magnetization energy of a polycrystal in terms of the coefficients of the series development of the COD. This was done for orthorhombic (sheet) sample symmetry and cubic crystal symmetry by Bunge. In the present work we present a theoretical development which generally parallels that of Bunge, S but employs the formalism of Roe. '4 A principal advantage of these techniques results from their ability to separate texture-dependent and texture-independent contributions to such properties as magnetic permeability and core loss, as noted by Hutchinson Equations (10) are of the form and have a solution of the form Once B0, B, B6 have been determined according to Equation

Alternative Treatment
An alternative representation has frequently been used to that of Equation (I), where 4 () 2m2 + m2n 2 + n2 2, 6 () 2m2n 2, 4, m, n are direction cosines with respect tO the X, Y, Z cubic crystal axes, and higher order terms ar+e formed by roducts of= ( g. This representation has a disadvantage with respect to that of Equation (i), in that the ii(h) form a series of orthoonal functions [which simplifies integration of the product F(h) P() leading to Equation (7) where T4 () T () are defined by Equations (7) and (8) Equation ( The materials used for experimental work were specimens of nominal 3% silicon-iron which are commercially designated as "nonoriented" electrical steels. Specimens A and C were cold-rolled to final thickness in a single-stage 81% cold reduction. Specimens B and D were cold-rolled to final thickness in two stages with cold reductions of 63% and 54%, separated by an intermediate anneal. After cold rolling all specimens were annealed at 815C to effect primary recrystallization and then annealed at I040C to allow some primary grain growth. After processing specimens A and B had grain sizes of about ASTM 1 while specimens C and D had grain sizes of about ASTM 5.
Determinations of {ii0}, {200}, and {112} pole figures were made with an Enraf-Nonius automated X-ray diffraction unit on composite samples 9 from fully-processed specimens. The pole figure data were used to calculate the COD to 16th order by a method similar to one previously described. I Measurements of the magnetic permeability at 796 A/M and the core loss at 1.5 T were made at 60 Hz on fully-processed samples from specimens A and B by conventional Epstein tests. The Epstein strips were given a stress-relief anneal at 815C for 1 hr. prior to magnetic testing. Magnetic torque curves were determined for specimens C and D at Westinghouse Research each specimen and also show the COD values for some ideal orientations of special interest. The strongest single orientations present in specimens A and C, processed using a single stage of cold reduction, were centered near ideal {iii}<112> orientations.
Specimen B, processed with two stages of cold reduction, had a strong maximum near the ideal {ii0}<001> orientation. Specimen D, also processed using two stages of cold reduction, had a weaker near {ii0}<001> component than specimen B, but this component was significantly stronger for specimen D than for either of the single stage specimens. As mentioned earlier, specimens A and B underwent significantly more grain growth during the final anneal than specimens C and D and the amount of grain growth may have a strong influence on the final textures.
Hutchinson and Swift also found a fourth-order fit of the form of Equation (29) to be satisfactory to describe the variation of core loss with test direction in a low-silicon electrical steel.
Permeability and core loss data are plotted vs. (e) in Figure 2 where the data for specimens A and B have been pooled.
The permeability at 796 A/M is known to be strongly dependent on the average orientation in electrical steels, I and the data for both specimens fit well to a common leastsquares line. This implies that any difference in permeability between samples of the two specimens cut in the same test direction is almost entirely due to texture differences. The pooled core loss data at 1.5 T are also described fairly well by a common least-squares line, which implies that texture differences are the most significant cause for core loss differences between the two specimens in any test direction. However, the data for two-stage specimen A appear to be slightly higher at any value of (e) than those for singlestage specimen B. Other metallurgical factors sch as grain fourth-order texture parameter. 12 size differences are known to affect core loss values, so there may be texture-independent factors which contribute to the apparent offset between the core loss values for the two specimens.
The fourth-order calculated values of permeability and core loss vs. test angle in the plane of the sheet for both specimens A and B are compared to the measured values in The agreement is generally good. The two-stage specimen B has better values for both properties when tested parallel to the rolling direction, but has inferior properties at test angles near 50 The much higher amount of material near the ideal {ii0}<001> orientation for specimen B compared to specimen A (see Figure i) is probably a major contributing factor to these differences.
Since the magnetic properties studied can be described used to display the variation of permeability and core loss in the plane of the sheet if two or more test values are available. An example of such a plot is given in Figure 4 for specimen B. For materials with relatively weak multicomponent textures such as "nonoriented" electrical steels, such plots would seem to be of moEe immediate practical use for studying texture-property relationships than either conventional pole figures or COD sections. It may also be noted that when a fourth-order expansion is adequate to describe magnetic property variations, a property value for a "random" sample (no preferred orientation) The random property values of permeability and core loss for specimen B are indicated in Figure 4. The random values are texture-independent properties for a given specimen and differences between the random values for two or more specimens represent property differences which are produced by factors other than texture. Hutchinson and Swift 6 have pointed out that the constant term A0 in Equation (29) represents the property value for a hypothetical specimen with the same microstructure but no misorientation.
Differences among the A0 terms for different specimens could also be used to measure texture-independent property differences.
The measured magnetic torque curves for single-stage specimen C and two-stage specimen D are given in Figure 5 along with torque values calculated from COD data using the fourth order form of Equation (28). A value of 380,000 ergs/cm was used for the first anisotropy constant (K i) both for calibrating the measured curves and calculating the fourth-order values. I Both specimens have low maximum torque values compared to highly-oriented silicon-iron specimens but two-stage specimen D has a significantly higher maximum torque than single-stage sepcimen C. The agreement between measured and calculated torque values is generally good.

CONCLUSIONS i)
Expressions were derived for the dependence of magnetic properties on texture. The theoretical development generally parallels that of Bunge, but employs Roe's formalism.
2) Core loss, magnetic permeability and torque values for specimens of comparatively weakly-oriented electrical steels could be fitted to simple fourth-order equations. 3) The method, together with limited experimental data, enables one to express magnetic property variations in the plane of the sheet, to obtain property values for comparable specimens with no preferred orientation and to separate the magnetic properties into texture-dependent and textureindependent components.