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

Microscopy and Microanalysis, vol. 28, pp. 454–468

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.

Authors 3

1. Jonas Bünger corresponding Aachen Applied and Computational Mathematics

   RWTH Aachen University

   Affiliation as printed

   Applied and Computational Mathematics, RWTH Aachen University, Schinkelstrasse 2, 52062Aachen (North Rhine-Westphalia), Germany

2. Silvia Richter Aachen Gemeinschaftslabor für Elektronenmikroskopie (GFE)

   RWTH Aachen University

   Affiliation as printed

   Gemeinschaftslabor für Elektronenmikroskopie, RWTH Aachen University, Ahornstrasse 55, 52074Aachen (North Rhine-Westphalia), Germany

3. Manuel Torrilhon Aachen Applied and Computational Mathematics

   RWTH Aachen University

   Affiliation as printed

   Applied and Computational Mathematics, RWTH Aachen University, Schinkelstrasse 2, 52062Aachen (North Rhine-Westphalia), Germany