ON-LINE TEXTURE ANALYSIS FOR MAGNETIC PROPERTY CONTROL

Magnetic properties of hard and soft magnetic materials are strongly anisotropic, i.e. they depend on the crystal direction in which they are being considered. Technological materials are usually polycrystalline. Hence, their properties are orientation mean values of the properties of the crystallites with the texture of the material as the weight function. Inspection and control of magnetic properties of materials thus requires inspection and control of the materials texture which can be carded out off-line by taking samples from the finished material and investigating them in the laboratory. If, however, the texture of the material is to be controlled during the production process then a fast non-destructive on-line texture analyser is required the output signal of which can be used to control the production process.

shows the principles of a fixed angle on-line texture analyzer, developed and applied for continuous r,-determination in the cold rolling mill quality control of the Hoesch Stahl AG in Dortmund F.R.G. (see e.g. Kopineck, 1986;Kopineck and Otten, 1987;Kopineck and Bunge, 1989). The same principle can also be used for the determination of the direction dependence of many other anisotropic physical properties. This paper gives the mathematical basis for the determination of the texture coefficients needed for on-line determination of magnetic properties. (For the relationship between these coefficients and the property values themselves see e.g. Bunge (this volume)).
The texture of a polycrystalline material is the orientation distribution of its crystallites Thereby g is the orientation of the crystal coordinate system KB of an individual crystallite (e.g. the cubic axes) with respect to a sample coordinate system e.g. rolling, transverse, and normal direction of a sheet. The orientation g can be expressed by the Euler angles tpl, , tp2 and f(g) describes the volume fraction of crystals which have an orientation g within the angular limits dg. The texture function is needed in order to calculate texture mean values of direction depend physical properties. If E(h) is the value of any physical property (.g. the magnetization energy) in the crystal direction h then the mean value E(y) of this property in the sample direction y is given by where A(h, y) is the volume fraction of crystals, the crystal direction h of which is parallel to the sample direction y (sec e.g. Bungc, 1982; Bungc, this volume).
This functon is an integral over the texture function f(g) The texture function f(g) can be expressed in terms of a series expansion L M(A) N(A) f (g)= l + . , E C " '(g) (4) ,=Ao /=1 v=l of generalized spherical harmonics T(g). The texture is then described by its coefficients C . For a complete description of the texture, the series must be extended up to relatively high L-values, e.g. up to L 22. In the case of cubic crystals and textures in sheet materials (orthorhombic sample symmetry) the total ON-LINE TEXTURE ANALYSIS 263 number of coefficients C is then 185. This total number of coefficients is, however, not needed if only mean values of physical properties according to Eq.
(2) are being considered. The direction dependence E(h) of most of all physical properties is a low-order function of h (e.g. fourth or sixth order in the case of magnetic properties, see e.g. Bunge, 1982). Then only very few of the coefficients C of Eq. (4) enter the mean value expression Eq. (2) (e.g. up to L = 4 or L = 6 in the case of magnetic properties). With cubic crystal symmetry and orthorhombic sample symmetry, these are only three, or seven coefficients out of a total of 185. It is then possible to obtain these few texture coefficients by a fixed angle method which can be applied non-destructively and "on-line" in a sheet production line. In contrast hereto, a complete texture anlysis requires a sample to be cut from the sheet and to measure its texture "off-line" on a texture goniometer.
The principles of the fixed-angle texture analyzer of Figure 1 are shown in   (15) , where to(y) is the "transparency" function of the pole figure "window" defined by the apertures of the ith detector Figure 4. Then in Eq. (5) lj and I, are to be replaced by the values according to Eq. (15) and/(y) in Eqs. (6,8,9,14) is to be replaced by the averaged value according to Eq. (16). In Eq. (5) the calibration of the measurements with a random sample was assumed. It is, however, difficult to prepare a sample which is--on the one hand, exactly random within the required accuracymand which is on the other hand Figure 4 The divergencies of incident and reflected beam define a pole figure "window". Table 1 Texture coefficients C" up to L --6 of an annealed low-carbon steel compared with approximations of these coefficients calculated with various numbers of pole figure points and in two different approximations L = 6 and L = 4 (one pole figure means 343 points). correction factors near to one. When the coefficients _a (,1) have been calculated then Eq. (13) provides an approximation zC to the correct coefficients C which can be proven if the theoretical values C are known. This test can be achieved purely "mathematical" without taking recourse to real measurements. For this purpose the A: in Eq. (13) can be replaced with A:-values calculated by Eq. 6 with a sufficiently high degree L. The A: are then ideal "experimental" values belonging to an exactly known texture with the coefficients C .T hese A:-values have furthermore the advantage that they are free of experimental errors. Hence, they are very well suited to test the proposed mathematical procedure. Tests of this type were carried out for several assumed experimental conditions some examples of which are given in Table 1. It is seen that, with a reasonable choice of the experimental conditions, the degree of approximation is quite satisfactory.