Computer-aided derivation of theoretical joint probability distributions of normalized structure factors

Abstract

A general theoretical and practical procedure is presented for deriving joint probability distributions of any number of structure factors in any space group. The distributions include all higher-order terms up to a present order of N and thus may be used at any approximation. The procedure combines and extends the two different methods introduced by Naya, Nitta & Oda [Acta Cryst. (1964). 17, 421-433; Acta Cryst. (1965). 19, 734-747] for deriving joint probability distributions of phase-restricted and not-phase-restricted normalized structure factors respectively. The general algorithm for deriving joint probability distributions of structure factors has been implemented in a computer program, thus resulting in the possibility of computer-aided derivations of probabilistic relations for any set of structure factors. Optionally, the program transforms the resulting series-expansion form of the joint probability distribution into an exponential expression. In low-order approximation these exponential expressions usually turn out to be identical to expressions known from the literature.