Phase extension in crystallography using the iterative Fienup -- Gerchberg -- Saxton algorithm and Hilbert transforms.

The article proposes a procedure for phase extension in crystallography using the iterative Fienup-Gerchberg -Saxton algorithm and Hilbert transforms. Based on an incomplete set of phases, the magnitudes of many non-Bragg reflections with fractional indices can be calculated with high accuracy using equations derived from the Shannon sampling theorem and the Hilbert transform. Constraints play a very important role in the iterative algorithm. Constraints in reciprocal and real space are the key points that drive the iterations toward a unique solution. In this way, it is shown that the iterative algorithm conventionally used for phasing diffuse scattering from non-periodic objects can also be applied to problems in conventional crystallography to find the phase of high-order beams from a known set of low-order phases.