The Probability Distribution of Structure Factors with Non-Integral Indices. I. TheP1 Case

Abstract

The probability distribution of structure factors with non-integral indices is derived. The distributions are first studied in the one-dimensional case, to understand their main features, then the three-dimensional case is treated. Only the P1 group is taken into consideration. For integral values of the indices, the distributions coincide with those provided by Wilson statistics but may strongly differ from them when the indices are (or are close to) half-integrals and are sufficiently small. In these cases, the moduli and phases of the reflections may be accurately estimated in the absence of any structural information. Conditional distributions are also derived which are able to estimate moduli or phases by exploiting the prior information on the specific crystal structure.