Probability calculus for quantitative HREM. Part I: Monte-Carlo and point cloud techniques

A new approach to a central question of modern high-resolution electron microscopy (HREM) is presented: How precisely can we locate atom positions in crystal defects using computer-controlled structure retrieval algorithms? The purpose is not just to give error bars for the determined atomic column positions, but to derive estimations for the continuous probability functions. In the first part of this two-part paper, we present techniques which analyse point clouds of fluctuating fit-results for atom coordinates. The point clouds are obtained in a first approach from multiple input images differing in noise, commonly known as Monte-Carlo error estimation. Furthermore, we exploit the response obtained during a global optimisation-based refinement process for which all the trial structures are evaluated resulting in a second type of point cloud. In comparison, the Monte-Carlo-type technique turns out to be the most robust one. Using examples from current research on SrTiO3-bicrystals and Cu–Al2O3 interfaces, we study two largely different crystallographic and statistical situations.