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) -) expiφ (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) -) exp (iφ (H) ) from which the corresponding S' (M) phasing function follows.