Inversion in X-ray diffraction: Zeros of the complex diffraction amplitude

The formalism for the solution of the inverse scattering problem in X-ray diffraction is considered. The method relies on the retrieval of the phase of a scattered X-ray wave from its intensity profile via a logarithmic dispersion relation. This phase-retrieval technique is always associated with the problem of localisation of zeros of the complex diffraction amplitude. A physically substantiated procedure to distinguish true and virtual zeros is discussed. To determine the locations of the true zeros it is necessary to search for the matching zeros in the analytical continuation of the diffraction amplitude performed using experimental data collected at two different radiation energies. This allows the phase of the diffracted X-ray wave to be retrieved unambiguously. A discrete representation of the numerical implementation of the inversion technique limits the spatial resolution which can be achieved in real experiments, yet allows one to obtain a unique solution for the structure-factor profile. A practical example of the applicability of the approach is presented. (C) 1998 Elsevier Science B.V. All rights reserved.