A study of generalized intensity statistics: extension of the theory and practical examples

Generalized probability density functions, cumulative distribution functions and moments of the normalized structure amplitude |E|, depending on space-group symmetry of the crystal and on the composition of the asymmetric unit, were extended to include the tenth moment of |E| and five-term expansions. The formalism was also simplified and is presented in a concise and unified form. The equations linking the formalism to practical problems, the composition and space-group terms, are discussed from a practical point of view and a convenient implementation of the above statistics in a computer program is indicated. The generalized cumulative distributions of |E| and of the normalized intensity z = |E|² are compared with corresponding distributions based on five published structures, each containing one outstandingly heavy atom (Pt, Rh and Br) and about twenty light ones in the asymmetric unit, excluding hydrogens. These examples indicate that the above formalism is a valuable tool for resolving space-group ambiguities which cannot be treated by conventional methods because of effects of atomic heterogeneity. N(z) distributions for a structure belonging to the space group Fddd show that the theoretical expressions correctly predict the existence of different intensity distributions in reflection subsets with hkl all even and hkl all odd for this space group.