Three-Dimensional Modelling of X-Ray Emission in Electron Probe Microanalysis Based on Deterministic Transport Equations

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The investigation of material properties and the development of new materials heavily rely on the characterization of their microstructure. A critical aspect of the characterization is to measure the distribution of the different chemical elements present inside a material. A well-established characterization technique is the electron probe microanalysis (EPMA), in which an electron beam interacts with the material causing the emission of x-rays characteristic to the local composition. This technique has the unique advantage to provide accurate quantitative information about the composition of a sample at the micrometer to nanometer scale, while allowing the investigation of a macroscopic sampling area. Although successful, this method is based on the assumption that the sample is homogeneous within the interaction volume of the electron beam, hence the structures of interest must be bigger than the interaction volume in order to be analyzed.Therefore, to apply the quantification procedures to even smaller scales it is necessary to derive fast and accurate mathematical and numerical models of electron-x-ray-matter interactions in complex geometries. In this project the usual approaches by simple analytical models or expensive Monte-Carlo simulations will be replaced by the description of electron scattering as a continuous process following the Boltzmann transport equation. The high dimensionality of the Boltzmann equation can be reduced by moment approximations and a closure procedure based on the maximum-entropy principle. The reduced model is given by a partial differential equation and will be solved numerically by a finite volume method in order to provide a fast and accurate prediction of the electron energy distribution inside a material. The resulting numerical software will be carefully validated against existing analytical models, standardized samples and Monte-Carlo simulations.