A new approach to crystal-structure analysis

Abstract

In order to apply vector-set methods to the solution of Patterson syntheses, it is first pointed out that the Patterson synthesis is the vector set of the electron density provided that the Patterson function is defined as an integral and not as an average of the electron-density product. With this proviso, vector-set methods can be applied to solve the Patterson synthesis for the electron density. The most appropriate vector-set method is the synthesis of the vector-set points into n identical n-gons. To do this with Patterson syntheses, one sets up a function of the Patterson values at the vertices of an image polygon, called an image-seeking function. Such a function has the property of attaining a high value when the polygon is in image position, but zero when it is not. Two such functions have been proposed, a product function and a minimum function. The properties of these functions are discussed. These functions bear an inequality relation to the electron density, and the inequality is strengthened by combining functions to produce others of greater rank. Application of this theory leads to a map of the function which is essentially an approximate map of the electron density. The theory is illustrated by application to the solution of the structure of berthierite, FeSb~S 4, from its Patterson projection. By combining maps of the minimum function for this crystal, one of rank 8 is readily attained which is sufficiently close to the electron density so that seven atoms in the asymmetrical unit are clearly distinguished. These have peaks of relative ranks 4 : 2 : 1, which is approximately the relative electron counts of Sb, Fe and S, respectively.