Improving the direct-methods sign-unconstrained S-FFT algorithm. XV

In order to extend the application field of the direct-methods S-FFT phase-refinement algorithm to density functions with positive and negative peaks, the equal-sign constraint was removed from its definition by combining ρ² with an appropriate density function mask [Rius & Frontera (2008). Acta Cryst. A64, 670–674]. This generalized algorithm (S₂-FFT) was shown to be highly effective for crystal structures with at least one moderate scatterer in the unit cell but less effective when applied to structures with only light scatterers. To increase the success rate in this second case, the mask has been improved and the convergence rate of S₂-FFT has been investigated. Finally, a closely related but simpler phase-refinement function (S_m) combining ρ (instead of ρ²) with a new mask is introduced. For simple cases at least this can also treat density peaks in the absence of the equal-sign constraint.