PROBABILISTIC DISTRIBUTION OF ONE-PHASE STRUCTURE SEMINVARIANTS FOR AN ISOMORPHOUS PAIR OF STRUCTURES - THEORETICAL BASIS AND INITIAL APPLICATIONS

Given a special type of triplet of reciprocal-lattice vectors in the m onoclinic and orthorhombic systems, there exist eight three-phase stru cture seminvariants (3PSSs) for a pair of isomorphous structures. The first neighborhood of each of these 3PSSs is defined by the six magnit udes and the joint probability distribution of the corresponding six s tructure factors is derived according to Hauptman's neighborhood princ iple. This distribution leads to the conditional probability distribut ion of each of the 3PSSs, assuming as known the six magnitudes in its first neighborhood. The conditional probability distributions can be d irectly used to yield the reliable estimates (0 or pi) of the one-phas e structure seminvariants (1PSSs) in the favorable case that the varia nces of the distributions happen to be small [Hauptman (1975). Acta Cr yst. A31, 680-687]. The relevant parameters in the formulas for the mo noclinic and orthorhombic systems are given in a tabular form. The app lications suggest that the method is efficient for estimating the 1PSS s with values of 0 or pi.