On direct-methods phase information from differences between isomorphous structure factors

An efficient procedure is presented for the derivation of joint probability distributions of isomorphous data sets. The new technique is based on the use of the differences of isomorphous structure factors as random variables. It will be shown that the usual probabilistic techniques, applied to these random variables, finally result in the joint probability distribution of three single differences of isomorphous structure factors comprising three doublet and eight triplet phase sums. An advantage of the new technique is that the inherent correlation between the isomorphous data sets is removed if a probabilistic procedure is set up for the small difference itself. In this way, an enormous mathematical simplification is obtained while the final results are much better than those obtainable by previous probabilistic expressions. The final triplet distribution seems to be of sufficient quality to be used in a normal direct-methods procedure. In contrast to usual approaches, the heavy-atom substructure need not be solved first. The probabilistic expression will be explained in detail for one and three single differences. Applications for the cases of single anomalous scattering, two different wavelengths and single isomorphous replacement (excluding anomalous-scattering effects) for both real and randomly generated data show the strength of the method.