Probabilistic Theory of the Structure Seminvariants

Abstract

The major goal of this chapter is to exhibit the structure seminvariant as the central concept, not only in the development of the theoretical basis of direct methods, but also as a practical tool that is useful in phase determination as well. To this end conditional probability distributions of selected seminvariants are described, given, in the first instance, an appropriate set of structure-factor magnitudes | E | and, in the second instance, the values of one or more structure seminvariants as well as magnitudes | E |. Each of these distributions leads to an estimate for the structure seminvariant. This estimate is particularly good when the variance of the distribution is small. Since the structure seminvariants lead in turn directly and unambiguously to the values of the individual phases ɸ, the structure seminvariants serve to link the known magnitudes | E | with the desired phases ɸ. Thus the derivation of appropriate conditional probability distributions of the structure seminvariants constitutes in effect a program for phase determination.