On homometric structures

Abstract

Structures which have the same squared transform but which are neither congruent nor enantiomorphic are defined as homometric structures. Homometric structures can be expressed systematically by means of the convolution operation; their analysis starts out from the so-called Q-function. They may be conveniently classified as follows: I. Pseudohomometric structures. (a) Homometric structures which under suitable affine transformations degenerate to enantiomorphic or congruent structures. (b) Structures which are homometric only for infinitely large crystals. II. Homomorphs. Homometric structures which remain homometric under any arbitrary affine transformation. A general expression of finite homometric structures which covers all the known examples is given by an integral equation of the folding type; its degenerate cases are discussed.