Abstract
The spatial resolution of electron probe microanalysis (EPMA), a non-destructive method to determine the chemical composition of materials, is currently restricted to a pixel size larger than the volume of interaction between beam electrons and the material, as a result of limitations on the underlying k-ratio model. Using more sophisticated models to predict k-ratios while solving the inverse problem of reconstruction offers a possibility to increase the spatial resolution. Here, a k-ratio model based on the deterministic M1-model in Boltzmann Continuous Slowing-Down approximation (BCSD) will be utilized to present a reconstruction method for EPMA which is implemented as a PDE-constrained optimization problem. Iterative gradient-based optimization techniques are used in combination with the adjoint state method to calculate the gradient in order to solve the optimization problem efficiently. The accuracy of the spatial resolution still depends on the number and quality of the measured data, but in contrast to conventional reconstruction methods, an overlapping of the interaction volumes of different measurements is permissible without ambiguous solutions. The combination of k-ratios measured with various electron beam configurations is necessary for a high resolution. Attempts to reconstruct materials with synthetic data show challenges that occur with small reconstruction pixels, but also indicate the potential to improve the spatial resolution in EPMA using the presented method.
Highlights
Increasing the Spatial Resolution in electron probe microanalysis (EPMA)Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
The spatial resolution of electron probe microanalysis (EPMA), a non-destructive method to determine the chemical composition of materials, is currently restricted to a pixel size larger than the volume of interaction between beam electrons and the material, as a result of limitations on the underlying k-ratio model
The results presented in this paper illustrate the potential of using high-resolution deterministic models for the implementation of a reconstruction method that improves the spatial resolution of EPMA
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The ZAF model uses correction factors for the atomic number Z (c), the absorption A(c), and the fluorescence F (c) and is implemented as a fixed-point iteration to reconstruct homogeneous mass fractions from k-ratios obtained from a single beam configuration. This idea is extended to layered samples using a correction model based on the depth distribution of X-ray generation φ(ρz). Monte Carlo models can be used to implement a k-ratio model that allows inhomogeneous material structures by first estimating the electron transport on the basis of Monte Carlo sampling and approximating the measured X-ray intensity. //github.com/tam724/m1epma (accessed on 9 July 2021) [21]
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