The influence of rational dependence on the probability distribution of structure factors

An index subgroup of strong main reflexions and cosets of weak reflexions are typical features of crystal structures with systematic rational dependence of the atom coordinates exhibiting a pseudotranslational symmetry. The mean squares of these normalized structure-factor sets which deviate significantly from unity are interpreted in terms of correlation coefficients of the atom coordinates. An asymptotic form of the von Mises distribution of a structure factor phase is derived which allows for rational dependence and makes explicit use of the ||² values of the different structure-factor sets. The formula provides a basis for the use of phase relationships of the type 'weak-strong-weak' proposed in the recent literature. The limits of the method are estimated. In particular, symmetry and homometry problems in superstructures are more complex than in usual cases and their careful consideration is essential for the success of procedures intending an automatic solution.