The investigation of superstructures by means of partial Patterson functions

Abstract

The difference in electron density between a superstructure and the structure averaged over all subcells of the superstructure is a continuously variable function having positive and negative values. The partial Patterson functions based upon supercell reflections only are the Patterson functions of such a difference electron density. Therefore the geometry of the partial Patterson functions can be investigated by the aid of the vector set of a set of positive and negative unit points. The characteristics of such a vector set have been investigated in terms of the theory of image sets (B uerger , 1961) for a set of n pairs of positive and negative unit points which well represents the distributions of the positive and negative peaks in the difference electron densities of distortion structures. The vector set is characterized by the occurrence of the sets of quadruple images, each consisting of two positive and two negative point images. In the vector space, there are n ( n – 1) such quadruplets and 2 n negative point images around the origin, in addition to 2 n positive images at the origin. The shape defined by a set of quadruple images varies in accordance with the relative orientation of two pairs that define the quadruplet. There are two alternate solutions to the same vector set. For a vector set arising from a centrosymmetric set of pairs, the two solutions are related by an anti-inversion operation. For the noncentrosymmetric case, one solution is negative of the other. The result can readily be applied to superstructures generally.