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A Model for Characteristic X-Ray Emission in Electron Probe Microanalysis Based on the (Filtered) Spherical Harmonic ($P_{\rm N}$) Method for Electron Transport

Published online by Cambridge University Press:  11 April 2022

Jonas Bünger ,
Silvia Richter  and
Manuel Torrilhon
Jonas Bünger*
Affiliation:
Applied and Computational Mathematics, RWTH Aachen University, Schinkelstrasse 2, 52062 Aachen (North Rhine-Westphalia), Germany
Silvia Richter
Affiliation:
Gemeinschaftslabor für Elektronenmikroskopie, RWTH Aachen University, Ahornstrasse 55, 52074 Aachen (North Rhine-Westphalia), Germany
Manuel Torrilhon
Affiliation:
Applied and Computational Mathematics, RWTH Aachen University, Schinkelstrasse 2, 52062 Aachen (North Rhine-Westphalia), Germany
*
*Corresponding author: Jonas Bünger, E-mail: buenger@mathcces.rwth-aachen.de
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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 EPMA a 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.

Keywords

characteristic X-ray emissiondeterministic electron transportEPMAk-ratiomoment methodspherical harmonic PN approximation
Type
Software and Instrumentation
Information
Microscopy and Microanalysis , Volume 28 , Issue 2 , April 2022 , pp. 454 - 468
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of the Microscopy Society of America

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