Application of Hilbert transforms to the phase problem in crystallography

Abstract

It is possible to derive equations connecting the real and imaginary parts of the Fourier coefficients of a periodic crystal structure, analogous to the well-known Hilbert transform equations. These equations have been tested in various hypothetical examples and appear to have the ability to refine the phases to nearly the correct values from completely wrong values. However, they require the knowledge of derivatives of the structure factors at the lattice points. The application of the new relations to centrosymmetric structures seems to be feasible under certain circumstances, which are relevant to protein crystallography.