Direct methods and structures showing superstructure effects.II. A probabilistic theory of triplet invariants

Abstract

A probabilistic theory of triplet invariants is provided which may be used when pseudotranslations occur in the crystal structure. The final formula for estimating a triplet phase invariant in centrosymmetric space groups is of Cochran-Woolfson type, in non-centrosymmetric space groups of von Mises type with maximum at 2π single phases are determined via a special tangent formula. Thus the usual algorithms for phase expansion and refinement can be employed with few modifications. Parameters occur in the von Mises and tangent formulae which are markedly different from the usual ones. In particular, the reliability of each triplet depends not only on |Eh|, |Ek|, |Eh - k|, but also on the actual h, k, h - k indices and on the nature of the pseudotranslations. An automatic phasing procedure and some applications are also described for the solution of superstructures and other structures showing pseudotranslations.