Determination of orientations from monochromatic diffraction patterns as the constellation problem

Determination of crystal orientations from diffraction patterns is directly linked to pattern indexing. The problem of indexing can be seen as matching scattering vectors to vectors of the crystal reciprocal lattice. With known crystal structure, the simplest version of indexing is formulated as the (constellation) problem of matching vectors under rotations: given two sets X and Y of unit vectors, determine a rotation carrying the largest subset of X to a position approximating a subset of Y. It is shown that algorithms for solving the constellation problem establish a framework for several orientation determination methods. A class of these algorithms is based on accumulating contributions in the rotation space. A rotation with the largest accumulation is considered to solve the problem. The contributions can be made by n-tuples of vectors with n starting from 1. Formulas for the points of accumulation are given for arbitrary n ≥ 1. Particularly simple turns out to be the case of 2-tuples. It has a potential of being robust, and it is easy to implement. (paper)