Phase determination from new joint probability distribution; space groupP1

A procedure for calculating phases in space group P1 is described.The new formulas constitute an improvement over earlier ones in that they make more effective use of the available data.In addition, two fundamental identities are presented which play an important role in the procedure.The derivation of the new formulas is based on the joint probability distributions of several normalized structure factors, obtained by fixing the crystal structure and allowing the indices to range uniformly but not independently over the vectors in reciprocal space.From these, the expected values of the products of powers of several structure factors are found.The expected values, together with a subsequent integrating process, lead to explicit formulas for the magnitudes of certain structure invariants in terms of the observed intensities alone.By means of the first fundamental identity, the initial values for the magnitudes of the structure invariants are improved, thus facilitating the problem of selecting one of the two enantiomorphous structures permitted by the given set of observed intensities.Once an enantiomorph has been chosen the second identity leads to refined values (both signs and magnitudes) of the invariants.Initial values of the individual phases are then readily obtainable, and a final least-squares adjustment procedure yields improved values for the phases.