Quantitative assessment of symmetry in CBED patterns

Abstract

Experimental convergent-beam electron diffraction (CBED) patterns never exhibit the ideal symmetries predicted by the diffraction group tables because of residual strains in the crystal and minor aberrations in the electron optics. For comparison with dynamical calculations used to measure structure factors, it is essential that CBED patterns show a close approach to ideal symmetry. Linear algebra and group theory is used to construct a single quantitative measure of the symmetry error by treating the experimental pattern as a function vector in a higher dimensional Euclidean space. By using projection operators, expressions are derived for the orthogonal irreducible representations, where the optimum location for the point group centre maximises the vector component parallel to the identity representation. Analysis of the low order reflections in 〈1 10〉 zone axis patterns from Cu, Ni and Si showed that errors varied between 10% and 3%, with a maximum acceptable error component of about 5%. A further requirement was that errors should be uniformly distributed in sign and magnitude across the pattern.