Probabilistic theory of the three-phase structure seminvariant in the space group P1.

By embedding the three-phase structure seminvariant T and its symmetry-related variants in suitable quintets Q, one obtains the extensions Q of the seminvariant T. Because of the space group-dependent relationships among the phases, T is simply related to its extensions Q. In this way the probabilistic theory of the seminvariant T is reduced to that of the quintets Q, which is well developed. In particular, the neighborhoods of T are defined in terms of the neighborhoods of the Qs. The conditional probability distribution of the structure seminvariant T, given the seven magnitudes in its first neighborhood, is described for the space group P1. The distribution yields a reliable estimate (0 or pi) for T in the favorable case that the variance of the distribution happens to be small.