Representation of Crystallographic Texture using the Symmetric Bingham Distribution

Abstract

Crystalline materials often exhibit a strong anisotropy, or directionally dependent response, with respect to their properties. Most structural materials are polycrystalline, composed of many microscopic crystalline regions each with a different orientation with respect to the sample coorrdinate frame. Crystallographic texture, or the presence of a statistically preferred orientation, is a key predictor of macroscale properties and can greatly affect the material the material response. The estimation of texture from X-ray diffraction data is a classical problem in materials science and geology. Traditional methods for estimating texture have utilized a Fourier series expansion by generalized spherical harmonics. Estimation of texture from discretely sample orientations, via spherical harmonics, requires making several ad-hoc assumptions that can strongly influence the final result.The Bingham distribution, the maximum entropy probability distribution on the space of 3-D rotations (SO(3)), has been proposed as a model distribution for crystallographic texture. However the Bingham distribution does not account for the inherent crystallographic symmetry, so its application has been severely limited. In this work we present a symmetrized Bingham distribution that can be applied to materials with arbitrary symmetries, and efficient numerical scheme for the estimation of distribution parameters from discretely sampled orientation data. We compare texture estimates from discrete orientation data using the symmetrized Bingham model and the classical spherical harmonic technique.