# Three-dimensional modelling of x-ray emission in electron probe microanalysis based on deterministic transport equations

- Drei-dimensionale Modellierung von Röntgenstrahlemissionen in der Elektronenstrahlmikroanalyse basierend auf deterministischen Transportgleichungen

Bünger, Jonas; Torrilhon, Manuel (Thesis advisor); Mayer, Joachim (Thesis advisor)

*Aachen : RWTH Aachen University (2021)*

Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2021

Abstract

Electron probe microanalysis (EPMA) is a well-established technique to image the mass concentration of the chemical elements inside a material probe on the micro- to nanometer scale. The material image follows from solving the inverse problem of finding the mass concentration fields that best reproduce characteristic X-Rays intensities measured in an electron microscope for a given synthetic model of characteristic X-Ray emission under excitation by a focused electron beam. Within the last decades significant improvements in the spatial resolution of EPMA were obtained by instrumental enhancements, while the models of characteristic X-Ray emission andinversion procedures essentially remained unchanged. Since classical analytical models of characteristic X-Ray emission, like ZAF and φ(ρz), assume either a homogeneous or a multi-layered material structure, the spatial resolution of EPMA is essentially limited by the size of the volume where characteristic X-Rays are produced in interactions between beam electrons and X-Rays with material atoms. In this thesis we develop a deterministic model of characteristic X-Ray emission for EPMA that bases on deterministic modelling of electron transport using moment models of the Boltzmann equation of electron transport in continuous slowing down (CSD) approximation. Modelling the transport of beam electrons allows us to omit drastic a-priory assumptions on the material structure, which constitutes the first step towards high-resolution EPMA through reconstruction of material structures on scales smaller than the interaction volume. We give an introduction to classical EPMA, motivate our approach to high-resolution EPMA as well as give an overview of the physical processes involved in the emission of characteristic X-Rays before formulating our model of characteristic X-Ray emission as a function of the fluence of beam electrons. We then motivate the method of moments as a technique to derive models of electron transport that balance accuracy and computational cost of numerical solutions - an essential ingredient for the practicability of high-resolution EPMA. In this thesis we consider moment models for electron transport from two families: First and second order minimum entropy (MN) moment models, M1 and M2, and the classic spherical harmonic (PN) approximation up to moderately high moment order N as well as PN with filtering (FPN). Besides motivating and deriving the MN and PN moment equations of electron transport in CSD approximation, two important parts of this thesis are the explicit approximation of the M2 closure to avoid computationally expensive numerical minimization schemes, and the formulation of compatible and energy stable strong-form formulations of boundary conditions for PN equations. Numerical schemes tailored to M1, M2 and (F)PN equations are given in the appendix. We conduct several numerical experiments in which we verify the accuracy of the considered moment models of electron transport by comparison with Monte-Carlo simulation of electron trajectories as well as experimental data, compare our moment-based model of characteristic X-Ray generation to classical φ(ρz) models and demonstrate the efficiency of moment models in terms of accurate and noise-free computation of characteristic X-Ray generation. From a first systematic comparison of the different moment models of electron transport considered here, we found the P N approximation with appropriate filtering is the most attractive moment model for our purpose: The simple structure of P N equations provides flexibility in accuracy by adapting the moment order N at acceptable increase of computational cost and with appropriate filtering the fluence of beam electrons is captured very accurately even in challenging situations like a very narrow electron beam or when changes in material properties are very sharp. On the example of our (F)PN-based k-ratio model we furthermore give an outlook on the development of efficient iterative inversion procedures using gradient-based minimization techniques together with the adjoint-state method for efficient gradient computation. To demonstrate the iterative reconstruction of fine material structures we also present results of a first numerical high resolution EPMA test case.

Institutions

- Department of Mathematics [110000]
- Chair of Applied and Computational Mathematics [115010]

### Identifier

- DOI: 10.18154/RWTH-2021-05180
- RWTH PUBLICATIONS: RWTH-2021-05180