MATHEMATICAL PROPERTIES OF DIFFRACTION POLE FIGURES, H. Schaeben, pp. 129-134

Crystallographic diffraction pole figures are defined as hyperspherical x-ray transforms of an orientation density function. The range of this transform is characterized in terms of an ultrahyperbolic differential equation which is derived without referring to spherical harmonics. The general solution of the differential equation is given both in terms of its characteristics or spherical harmonics. These results will aid in the solution of actual problems of texture goniometry, e.g. the high-resolution analysis of preferred crystallographic orientation in polycrystalline materials by diffraction; it is shown in particular that solving the inverse tomographic problem of texture goniometry and determinin g additional pole figures which have not been measured are two equivalent issues of the same problem. Eventually, the paradigm [l] that the determination of additional pole figures requires first the determination of an reasonable orientation density function is shown to be obsolete. POLE FIGURES, HYPERSPHERICAL X-RAY TRANSFORMS The analysis of preferred crystallographic orientation is referred to as texture analysis. A probability density function f(g) defined ‘on the group SO(3) of rotations or a subset G of SO(3) depending on the crystal symmetry, respectively, is referred to as orientation density function describing the distribution of crystallographic orientations by volume. While orientation density functions may be mathematically determined from individual orientation measurements directly in some applications, they are not generally directly accessible in many other applications. In these applications it is common practice to measure diffraction pole figures of a few crystallographic forms h. A difh-action pole figure is mathematically represented as the projection of an orientation density function basically provided by the integral operator