On the direct-methods theory of single isomorphous replacement

The joint probability distribution of six structure factors for a pair of isomorphous structures [Hauptman (1982). Acta Cryst. A38, 289-294] is used for probability calculations in which doublet invariant phase information (i.e. phase-difference information) is employed as conditional information together with intensity data. This information is obtained from the structure-factor magnitudes of the structure formed by the replacement atoms (assumed to be known) and the two isomorphous structures. First, conditional probability distributions of the triplet invariants of the native structure are derived. An alternative to the approach of Fortier, Moore & Fraser [Acta Cryst. (1985), A41,571-577] is presented, based on a new enantiomorph-sensitive distribution. It is argued that application of enantiomorph-sensitive distributions obtained by restriction of phase invariants can be widened by using various enantiomorph-defining invariants. Second, the ambiguity in single isomorphous replacement is resolved by calculating the probability of the two possible solutions as was proposed by Fan Hai-fu, Han Fu-son, Qian Jin-zi & Yao Jia-xing [Acta Cryst. (1984), A40, 489-495], but using a different probabilistic basis. It turns out that the formulae of the latter authors are a special case of formulae derived in the present paper.