Complex Diffraction Amplitude

X-ray Bragg diffraction is a powerful diagnostic tool for the non-destructive analysis of crystalline materials. The widely used method of least-squares fitting of a calculated reflectivity to the experimental diffraction profile relies on an a priori model of crystal-lattice deformation. The method works well for a relatively large class of crystalline structures and allows one to obtain information about the crystal lattice strain profiles. However, there is no evidence that the solution obtained through least-squares fitting is unique, since it is an intensity which is fitted, while the phase information (representing half of the information) is not taken into account. Since the crystal structure-factor is a complex function, the analysis of the intensity profile alone, which is a real function, is not able to give a physically sound result for a structure-factor. For the common case of 100 points in the analyzed profile the number of possible complex diffraction amplitude profiles is more than 1030. Generating and analyzing one solution per second we would spend about 3x1022 years (!!!) to obtain a result using this inversion procedure. The fact that we can obtain an enormous number of identical diffraction intensity profiles, which can be calculated for the same number of crystal structure-factor distributions, shows explicitly the hopelessness of the least-squares fitting methodology.

A method for the solution of the inverse problem, based on the calculation of the reflectivity phase profile via a logarithmic dispersion relation has been suggested for x-ray Bragg diffraction [1]. Phase retrieval techniques based on the use of a logarithmic dispersion relation are complicated by the problem of the localization of zeros of the complex diffraction amplitude. A new approach to the unambiguous solution of the inversion problem has been developed recently [2-3]. It was suggested that one must distinguish between the physical and mathematical zeros in the analytical continuation of the complex diffraction amplitude using experimental data collected for two radiation energies. Only the true zeros should be used in the complete phase profile calculation [4].

The present page presents the concept of the complex diffraction amplitude in x-ray Bragg diffraction as a unique product of its zeros. It is shown how physically reasonable assumptions about the complex diffraction amplitude allow us to formulate the inversion procedure in terms of a discrete (physical) representation. This discrete formalism, which corresponds directly to an experiment, allows one to obtain a unique solution for the structure-factor profile, which is consistent with the experimental observations. As a practical example the formalism is applied to x-ray Bragg diffraction data collected at two different radiation energies.

[1] P. V. Petrashen' and F. N. Chukhovskii, Sov. Phys. Dokl. 34, 957 (1989). [2] A. Y. Nikulin, P. Zaumseil and P. V. Petrashen, J. Appl. Phys., 80, 6683 (1996). [3] A. Y. Nikulin, P. Zaumseil and P. V. Petrashen, J. Phys. D: Appl. Phys., 30, 2373 (1997). [4] A. Y. Nikulin, Phys. Rev. B, 57, 11178 (1998).

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