Tchebycheff extraction of the periodic vector set from the Patterson function

Abstract

The image-seeking method of Buerger and the procedure of Tokonami and Hosoya appear to be capable of dealing with complex crystal structures via the Patterson function, provided that the periodic 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 it is the lack of such a method which now prevents the formulation of a general Patterson method of structure analysis. This paper presents further results in the author's attempts to formulate a general method of vector-set extraction by representing the Patterson function as a linear generalized polynomial in a system of independent interatomic functions. This approach has the advantage in that the essentially non-linear problem of vector-set extraction is reduced to an apparently simple linear problem, namely that of determining the coefficients of the approximating polynomial. In the present paper, the Tchebycheff approximation norm is employed with coefficient determination by linear-programming procedures. Since linear-programming methods are flexible and extremely powerful, this Tchebycheff vector-set-extraction procedure is much more promising than the author's earlier published methods, which were based on interpolatory approximations.