On the probabilistic theory of isomophous data sets: general joint distributions for the SIR, SAS and partial/complete structure cases

By characterizing isomorphism in reciprocal space [i.e. diffraction data sets are isomorphous if they have the same geometry (the same reciprocal-lattice unit cell) and the same symmetry] it is shown that the diffraction data of a native protein and of its heavy-atom derivatives, the calculated data of a partial structure and the observed data of its associated complete structure, and the Friedel-pair data of an anomalously scattering crystal structure all belong to the more general class of isomorphous data sets. Their joint probability distributions for two- and three- phase structure invariants are shown to be isomorphous: they have the same functional form and differ only in individual atomic scattering factors. General joint probability distributions, which can be used for any isomorphous data pairs, are presented.