Physics

Symmetry (geometry)

Mathematics

Algorithm

Orientation (vector space)

Mathematical analysis

Tangent

Unit sphere

Geometry

Hypersphere

Computation

Quantum mechanics

Configuration space

Journal of Applied Crystallography

A method is presented for generating nearly uniform distributions of three-dimensional orientations in the presence of symmetry. The method is based on the Thomson problem, which consists in finding the configuration of minimal energy of N electrons located on a unit sphere – a configuration of high spatial uniformity. Orientations are represented as unit quaternions, which lie on a unit hypersphere in four-dimensional space. Expressions of the electrostatic potential energy and Coulomb's forces are derived by working in the tangent space of orientation space. Using the forces, orientations are evolved in a conventional gradient-descent optimization until equilibrium. The method is highly versatile as it can generate uniform distributions for any number of orientations and any symmetry, and even allows one to prescribe some orientations. For large numbers of orientations, the forces can be computed using only the close neighbourhoods of orientations. Even uniform distributions of as many as 10 6 orientations, such as those required for dictionary-based indexing of diffraction patterns, can be generated in reasonable computation times. The presented algorithms are implemented and distributed in the free (open-source) software package Neper .

Nearly uniform sampling of crystal orientations