Three-Dimensional Modelling 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材料内部存在的不同化学元素的分布。一种成熟的表征技术是电子探针显微分析(EPMA)，其中电子束与材料相互作用，导致发射具有局部成分特征的x射线。这项技术具有独特的优势，可以在微米到纳米级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。虽然这种方法是成功的，但它是基于这样的假设，即样品在电子束的相互作用体积内是均匀的，因此要进行分析，感兴趣的结构必须大于相互作用体积。因此，为了将量化过程应用到更小的尺度上，有必要建立复杂几何结构中电子-X射线-物质相互作用的快速、准确的数学和数值模型。在这个项目中，通常的方法通过简单的分析模型或昂贵的蒙特卡罗模拟将被描述为遵循玻耳兹曼输运方程的连续的电子散射过程所取代。玻尔兹曼方程的高维可以通过矩近似和基于最大熵原理的闭合过程进行降维。简化的模型由偏微分方程给出，并将用有限体积法进行数值求解，以提供对材料内部电子能量分布的快速而准确的预测。由此产生的数值软件将根据现有的分析模型、标准化样本和蒙特卡罗模拟进行仔细验证。

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 well-established 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, this method is based on the assumption that the sample is homogeneous within the interaction volume of the electron beam, hence 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. In this project the usual approaches by simple analytical models or expensive Monte-Carlo simulations will be replaced by the description of electron scattering as a continuous process following the Boltzmann transport equation. The high dimensionality of the Boltzmann equation can be reduced by moment approximations and a closure procedure based on the maximum-entropy principle. The reduced model is given by a partial differential equation and will be solved numerically by a finite volume method in order to provide a fast and accurate prediction of the electron energy distribution inside a material. The resulting numerical software will be carefully validated against existing analytical models, standardized samples and Monte-Carlo simulations.