A Bayesian Approach to Surface X-ray Diffraction

Abstract

We report on the development of an iterative method to directly invert surface x-ray diffraction (SXRD) data and thereby provide a map of electron density in the near-surface region of a solid. We have termed this method PARADIGM, which stands for Phase and Amplitude Recovery And Diffraction Image Generation Method. Significant advances in the PARADIGM theory were made during the grant period, and experimental milestones have also been achieved. The two components of the research program worked in concert, each spurring progress in the other. The method works by iteratively recovering the phases of surface scattering factors. Initially, random phases are assigned to the structure factors. After subtracting off the known bulk component, a Fourier transform converts these factors into an estimate of the real-space electron density map. This map is subjected to a support constraint, which holds that the electron density may only be non-zero near the solid surface. The modified electron density is then subjected to an inverse Fourier transform, and the bulk contributions are added back in. This renders an improved estimate of the phases of the surface structure factors. A constraint in reciprocal space is then applied, namely, the amplitudes of the scattering factors are set equal to the experimentally observed ones. This cycle is repeated, transforming between real and reciprocal space and applying constraints in each, until convergence is reached. The result renders a good initial model of the unknown surface structure. Such a direct method is important because conventional structural refinement methods rely on having a guess of the starting structure that sufficiently good that it may be refined into a model with the correct atomic positions. If the starting model has, for example, the wrong number or identity of atoms in the surface unit cell, it can never refine to the correct model. Even in cases where the starting model contains the correct number and identity of atoms, it is relatively easy for refinement routines to get trapped in false minima; finding a global minimum of a multi-parameter phase space is a notoriously difficult problem. The utility of the present method, then, stems from its ability to, independently of preconceived notions, identify robust starting models for testing by conventional refinement techniques. The method has been shown to work well on three independent experimental data sets. First, the efficacy of the method was demonstrated on a known reconstruction, the well-known missing-row Au(110)-(2x1) surface. The method recovered all known structural features of this reconstruction. Next, the method was applied to two heretofore unknown reconstructions of the Au(110) surface that are induced by Sb adsorption, the c(2x2) and the (rt(3)xrt(3))R54.7 reconstructions. In each case, the direct method provided an excellent starting model for later refinement by conventional means.