Subscale Inversion of X-Ray Emission in Electron Probe Microanalysis Based on Deterministic Transport Equations

材料性能的研究和新材料的开发在很大程度上依赖于其微观结构的表征。表征的一个关键方面是测量存在于材料内部的不同化学元素的分布。一种成熟的表征技术是电子探针微分析（EPMA），其中电子束与材料相互作用，导致局部成分的x射线特征发射。该技术具有独特的优势，可以在微米到纳米尺度上提供关于样品组成的准确定量信息，同时允许对宏观采样区域进行调查。虽然成功，但EPMA中使用的重建技术是基于假设样品在电子束的相互作用体积内是均匀的，因此，通常感兴趣的结构必须大于相互作用体积才能进行分析。因此，为了将量化过程应用于更小的尺度，有必要在复杂几何形状和相互作用体积以下的非均匀材料中推导出快速准确的电子-x射线-物质相互作用的数学和数值模型。在本项目的第一部分中，通常采用简单解析模型或蒙特卡罗模拟的方法已被玻尔兹曼输运方程及其矩方程近似的电子散射描述所取代。该简化模型是由一个确定性的偏微分方程系统给出的，可以在无噪声的情况下高效地求解。新提出的项目包括建模和实验部分。确定性模型将被用于求解重构逆问题，并采用高效的基于伴随的优化方法。重建需要足够的数据采集，这将在一系列使用人工和现实世界样本的对照实验中发展。此外，基于蒙特卡罗模拟的重建将在需要时与确定性模型相结合，以提高物理精度。

The investigation of material properties and the development of new materials heavily rely on the characterization of their microstructure. A critical aspect of the characterization is to measure the distribution of the different chemical elements present inside a material. A wellestablished characterization technique is the electron probe microanalysis (EPMA), in which an electron beam interacts with the material causing the emission of x-rays characteristic to the local composition. This technique has the unique advantage to provide accurate quantitative information about the composition of a sample at the micrometer to nanometer scale, while allowing the investigation of a macroscopic sampling area. Although successful, the reconstruction technique used in EPMA is based on the assumption that the sample is homogeneous within the interaction volume of the electron beam, hence, typically the structures of interest must be bigger than the interaction volume in order to be analyzed. Therefore, to apply the quantification procedures to even smaller scales it is necessary to derive fast and accurate mathematical and numerical models of electron-x-ray-matter interactions in complex geometries and inhomogeneous material below the interaction volume. In the first part of this project the usual approaches by simple analytical models or Monte-Carlo simulations have been replaced by the description of electron scattering following the Boltzmann transport equation and its approximation by moment equations. The reduced model is given by a deterministic system of partial differential equation and can be solved efficiently without noise. The newly proposed project contains a modelling and an experimental part. The deterministic model will be used to solve the inverse problem of the reconstruction using efficient adjoint-based optimization methods. The reconstruction requires sufficient data aquisition and this will be developed in a series of controlled experiments using artificial and real-world samples. Additionally, a reconstruction based on Monte-Carlo simulations will be coupled to the deterministic model to increase physical accuracy when needed.