Improvement of the tangent formula by constraints based on additional information

Abstract

The first part of this communication describes a simple procedure by which the non-centrosymmetric form of the tangent formula is adapted to incorporate the `centrosymmetry constraint' for centrosymmetric structures, thus allowing refinement of phases uniformly distributed from 0 to 2π to the expected values 0 or 2π. The convergence of the resulting formula is illustrated with two structures. In the second part, a modified tangent formula including the constraint based on the zero points of the Patterson function is derived. To do this, both the Cochran integral ∫v ρ3 dV and the sum over all zero points of the Patterson function of ρ2 are expressed in terms of the phases of the strong E's. The modified tangent formula is then obtained assuming that the difference between the two corresponds to a large positive maximum for the correct phases. Finally, it is shown how the information supplied by the weak E's and by the zero points can be treated in an unified way, so that a combined tangent formula can be derived.