The theory of representations: its application to the estimation of the two-phase variants and seminvariants

Abstract

The theory of representations has been used to obtain probabilistic estimates of the two-phase variants and seminvariants for the non-centrosymmetric space groups up to orthorhombic. Some practical applications show that the phase estimations can be useful in direct procedures, mostly for those crystal structures in which the enantiomorph definition is difficult.Notation h~,h 2 = reciprocal vectors N = number of atoms in the unit cell t = number of atoms in the asymmetric unit E h = normalized structure factor 1 = identity 3 × 3 matrix e=E 2-1 R h = magnitude of the normalized structure factor R v = pth rotation matrix of the point group T v = translation vector associated with the pth rotation matrix in the space group.