A generalized polynomial for the extraction of the periodic vector set from the Patterson function

Abstract

Crystal-structure analysis via the Patterson function may be considered as consisting of two distinct steps. In the first step, the weighted periodic vector set is determined by establishing the location of each peak in the Patterson function. In the second step, the weighted periodic vector set is analysed to determine the crystal structure. The second step apparently offers little difficulty, since existing procedures for the analysis of periodic vector sets appear to be capable of dealing with complex structures, provided of course that the vector set is accurately determined. Unfortunately, a general and powerful method for the location of peaks in the Patterson function has not yet been developed and therefore it is the first step in the solution process which now prevents the formulation of a general method of structure analysis via the Patterson function. Such a method would be extremely useful, since the Patterson function is not restricted to centrosymmetric structures. In the present paper a way of representing the Patterson function as a linear generalized polynomial in a system of independent interatomic functions is developed. The coefficients of this polynomial determine the weighted periodic vector set. This approach, therefore, reduces the problem of extracting the periodic vector set from the Patterson function to a relatively simple problem in linear approximation, namely the determination of the coefficients of a generalized polynomial.