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 = ∑_HE_−H∫_Vδ_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{()exp[iφ_H(Φ)]} (where FT = Fourier transform) with all values smaller than Δ = 2.5σ_P equated to zero ( 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 phasing function follows.