Journal of Applied Crystallography

Volume 26, Part 1 (February 1993)

research papers

J. Appl. Cryst. (1993). 26, 112-121 [ doi:10.1107/S0021889892009270 ]

Towards statistics of crystal orientations in quantitative texture anaylsis

H. Schaeben

Abstract: By the application of Rodrigues parameters, crystal orientations are represented as points on the unit semi-hypersphere S⁴₊ [superset of] $\bb R$ ⁴ or equivalently on the projective hyperplane H³ $\bb R$ ⁴. For the statistical analysis of orientation data, probability models on S⁴₊ [equivalent to] H³ are required. Among different hyperspherical analogs of the normal distribution in Euclidean space, corresponding to various of its characterizations, the Bingham model distribution is characterized as the hyperspherical analog for statistical purposes. Maximum-likelihood estimates of its parameter matrices and single-sample significance tests of uniformity and rotational symmetry are presented. In terms of probability theory and stochastic processes, the hyperspherical Brownian-diffusion distribution is the analog of the normal distribution in Euclidean space. An orientation distribution derived by Savelova [Ind. Lab. (USSR), (1984), 50, 468-474] and erroneously related to a central-limit-type theorem for rotations by Matthies, Muller & Vinel [Textures Microstruct. (1988), 10, 77-96], with the Brownian-diffusion distribution on S⁴₊ [equivalent to] H³, recalls the fact that no simple analog of the central limit theorem in Euclidean space exists for arbitrary spaces and manifolds, e.g. for S^P, S^P₊, SO(p), p [greater-than or equal to] 2. Therefore, interpretations of preferred orientations in terms of mechanisms of texture development related to an imaginary simple analog of the central limit theorem would generally be misleading. Individual crystal-orientation measurements may initially require different nonparametric methods of analysis. As a counterpart of pole-to-orientation density inversion of diffraction data, hyperspherical kernel orientation density estimation is suggested for individual orientation measurements. With the application of non-negative kernels, the estimated orientation density is always non-negative. The critical smoothing parameter of this method is determined on information-theoretical grounds. For patterns of preferred orientation too complex to be sufficiently approximated by the parametric Bingham model distribution, a clustering of individual orientation measurements into disjunct classes is suggested that relates similarity of orientations to multimodality of the estimated orientation density; each class of orientations may then be individually analyzed by parametric methods.