Direct phasing from Patterson syntheses by δ recycling

The direct methods origin-free modulus sum function [Rius (1993). Acta Cryst. A49, 406-409] includes in its definition the structure factor G(Φ) of the squared crystal structure expressed in terms of Φ, the set of φ phases of the normalized structure factors E's of the crystal structure of unit-cell volume V. Here the simpler sum function variant S'(P) = ∑(H)E(-H)∫(V)δ(P,Δ)(Φ)exp(i2πHr)dV extended over all H reflections is introduced which involves no G's and in which the δ(P,Δ) function corresponds to δ(P) = FT(-1){(E(2)(H) - <E(2)>)exp[iφ(H)(Φ)]} (where FT = Fourier transform) with all values smaller than Δ = 2.5σ(P) equated to zero (σ(2)(P) is the variance of δ(P) calculable from the experimental intensities). The new phase estimates are obtained by Fourier transforming δ(P,Δ). This iterative phasing method (δ recycling) only requires calculation of Fourier transforms at two stages. Since δ(M) ≃ δ(P)/2, similar arguments are valid for δ(M) = FT(-1)[(E(H) - <E>)exp(iφ(H))] from which the corresponding S'(M) phasing function follows.