Image Formation in Electron Microscopes at Low Energies

In the past, the spatial resolution in electron microscopy was mainly limited by the unavoidable aberrations of round electron lenses. Nowadays, the successful realization of aberration correction allows achieving atomic resolution for electron energies as low as 20 keV. In this energy range, existing models for the image formation process must be augmented by previously neglected effects, since the commonly used paraxial approximation breaks down. Starting from quantum mechanical principles, this work will show that electron microscopic image contrast from elastic scattering can be calculated by solving a series of eigenvalue problems. Thereby, new theoretical insights into the scattering process are gained. Furthermore, the derived analytical matrix equations turn out to be suited for numerical calculations as well. Through the inclusion of backscattering effects, a unified description of the image formation process in different types of electron microscopes is obtained. Thus, for the first time, the elastic scattering channel in a scanning electron microscope (SEM) can be simulated based on the wave-mechanical description of the electron. Moreover, this work will show that the direct quadrature of the Schrödinger equation can be used to simulate electron micrographs. With the help of these new scattering solutions, the validity of the paraxial approximation is investigated in detail