An Adjoint Method for High-Resolution EPMA Based on the Spherical Harmonics (PN) Model of Electron Transport

Electron Probe Microanalysis (EPMA) is a nondestructive technique to determine the chemical composition of material samples in the micro- to nanometer range. Based on intensity measurements of x-radiation, the reconstruction of the material composition and structure poses an inverse problem. The reconstruction methods currently applied are based on models that assume a homogeneous or a layered structure of the studied material. To increase the spatial resolution of reconstruction in EPMA the combination of a more sophisticated reconstruction method, which is based on a model that allows complex material structure, together with multiple measurements with varying beam configurations is required. We present a deterministic k-ratio model that is based on the PN model, an approximation of the radiative transfer equation for electron transport. Our goal is to approximate a maximum likelihood solution of the inverse problem using gradient-based optimization. We detail the application of the model in the context of algorithmic differentiation, in particular by deriving its continuous adjoint formulation. Algorithmic differentiation provides the flexibility to adapt the reconstruction method to various material parametrizations and thus to regularize and take into account prior knowledge. Through examples, we verify our implementation and demonstrate the flexibility of the reconstruction/differentiation method.