Introductory Material

Abstract

Abstract The determination of crystal and molecular structures achieved its present respectable status thanks to advances in techniques of data collection, the widespread availability of computing power and the development of methods for the interpretation of experimental data in terms of crystal symmetry and atomic and molecular models. Some of the most successful of these rely heavily on statistical methodology. Structure-factor statistics play an important role among these methods and many investigations of this subject can be found in the crystallographic literature, since its introduction by Wilson (1949). Early work on statistical properties of a single structure factor (or normalized intensity), as applied to intensity statistics, is summarized in the book of Srinivasan and Parthasarathy (1976). This work is mainly based on central limit theorem methods. A detailed treatment of approximate joint probability density functions (hereafter: pdfs) of several structure factors, relevant to the implementation of direct methods of phase determination, is given in a book by Giacovazzo (1980), which is also a good source of references to earlier work on the subject. There is no book or monograph which develops or summarizes exact expressions for the pdfs required to implement structure-factor statistics, and certainly none that addresses numerical problems in evaluating such expressions. Our main purpose is to develop pdfs of the magnitude of the structure factor. In addition, we discuss some of the approximations that result in less accurate but more conveniently computable expressions for these densities. In many instances we recall the mathematical tools that may be required by the reader for following details of the derivations. The quantities to be discussed in this monograph are of importance to the theories and practice of intensity statistics and direct methods.