Uniqueness of the complex diffraction amplitude in x-ray Bragg diffraction

The concept of the complex diffraction amplitude for x-ray Bragg diffraction is discussed in terms of a unique product of its zeros. This formalism allows the inverse scattering problem in x-ray Bragg diffraction to be solved unambiguously. The phase-retrieval technique, via a logarithmic dispersion relation, has associated with it the problem of localization of zeros of the complex diffraction amplitude. The mathematical approach predicts an infinite number of zeros of the complex diffraction amplitude. However, a physical (discrete) representation of the inversion technique limits the number of zeros that should be considered and allows one to obtain a unique solution for the structure-factor profile. Practical examples of the analytical continuation of the complex diffraction amplitude are presented. Distinctions between the artificial, mathematical, and the true, physical, features of the analytical continuation are elucidated. © 1998 The American Physical Society.