Effects of space-group symmetry and atomic heterogeneity on intensity statistics

Probability functions, moments and cumulative distributions of the normalized intensity z = |E|², depending on the space-group symmetry of the crystal and its chemical composition, have been investigated. The probability functions were reduced to a simple, unified representation, by reconsidering some mathematical properties of the corresponding asymptotic expansions. A subsequent unified derivation of the first four even moments of |E|, in terms of symmetry and composition, leads to (i) simple and readily computable expressions for (|E|⁴), (|E|⁶) and (|E|⁸) and (ii) a significant simplification of the expansion coefficients which appear in the above asymptotic expansions. The convergence of these expansions is discussed and illustrated by a numerical example. It is shown that the Edgeworth arrangement of these asymptotic expansions is superior to the frequently given Gram-Charlier one. The dependence of the fourth moment of |E| on atomic heterogeneity and the generalized cumulative distribution functions N(|E|) are illustrated for all the symmorphic space groups. The results of this study are directly applicable to practical intensity statistics for structures containing all the atoms in general positions.