Exact random-walk models in crystallographic statistics. III. Distributions of |E| for space groups of low symmetry

Abstract

Exact univariate probability density functions (p.d.f.'s) of the magnitude of the normalized structure factor, taking into account space-group symmetry and the chemical composition of the asymmetric unit, have been investigated. The p.d.f.'s that represent distributions for centrosymmetric space groups are given by single Fourier series, while those for the non-centrosymmetric ones can be obtained from double Fourier series or, more conveniently, from single Fourier-Bessel expansions. Analytical expressions for the expansion coefficients are given for all triclinic, monoclinic and orthorhombic space groups, except Fdd2 and Fddd. These results are applied to a comparison of simulated distributions, based on a C9U asymmetric unit and the space groups P1, P1, P2 (or Pm), P2/m, P222, Pmm2 and Pmmm, with the theoretical p.d.f.'s derived here and with approximate generalized p.d.f.'s given by previously published five-term Hermite and Laguerre expansions. The performance of the Fourier and Fourier-Bessel p.d.f.'s is very good throughout the range of symmetries investigated, while that of the approximate ones is rather poor for the lowest symmetries and improves - albeit not uniformly - for the higher ones. Pertinent programming considerations, which suffice for the implementation of the new results in appropriate software, are presented.