A six-dimensional approach to microtexture analysis

A mathematical framework is presented for microtexture derivation and analysis. A six-dimensional space is introduced, defined as the product of direct space (x, y, z) and orientation space, parametrized by the Rodrigues vector (r ₁, r ₂, r ₃). The framework is adapted to diffraction experiments, where orientation information from many grains or deformation microstructures is recorded in parallel, with the prime aim of allowing fast measurements and thus the ability to study microstructure dynamics. Assuming the use of a monochromatic beam the geometry of the projection surfaces is deduced. The derivation of the microtexture is formulated as an inverse problem similar to reconstruction problems in absorption or emission tomography. The applicability of the two major types of reconstruction algorithm, namely transform methods and algebraic methods, is discussed. An explicit solution in terms of the iterative algebraic reconstruction technique algorithm is given. The choice of basis function is discussed. Furthermore, the framework is applied to a number of important simplifying cases. These include grain-by-grain reconstruction, classical pole figure inversion and the generation of three-dimensional maps of non-deformed grains.