A Model for Characteristic X-Ray Emission in Electron Probe Microanalysis Based on the (Filtered) Spherical Harmonic () Method for Electron Transport

Abstract

Classical$k$-ratio models, for example, ZAF and$\phi ( \rho z)$, used in electron probe microanalysis (EPMA) assume a homogeneous or multilayered material structure, which essentially limits the spatial resolution of EPMA to the size of the interaction volume where characteristic X-rays are produced. We present a new model for characteristic X-ray emission that avoids assumptions on the material structure to not restrict the resolution of EPMAa priori. Our model bases on the spherical harmonic ($P_{\rm N}$) approximation of the Boltzmann equation for electron transport in continuous slowing down approximation.$P_{\rm N}$models have a simple structure, are hierarchical in accuracy and well-suited for efficient adjoint-based gradient computation, which makes our model a promising alternative to classical models in terms of improving the resolution of EPMA in the future. We present results of various test cases including a comparison of the$P_{\rm N}$model to a minimum entropy moment model as well as Monte-Carlo (MC) trajectory sampling, a comparison of$P_{\rm N}$-based$k$-ratios to$k$-ratios obtained with MC, a comparison with experimental data of electron backscattering yields as well as a comparison of$P_{\rm N}$and MC based on characteristic X-ray generation in a three-dimensional material probe with fine structures.