When fast electrons are used to study matter at subnanometer length scales, it is often necessary to model the inelastic cross section and the absorptive effect of phonon excitation on elastically scattered electrons. The inelastic cross section for the excitation of a phonon in a crystal by a fast electron is well modeled by using an effective absorptive potential. In this paper, the absorption potential for phonon excitation by fast electrons is rigorously derived from many-body quantum mechanics taking into account correlated atomic motion. This potential is calculated for a silicon crystal at room temperature from the force constants and dispersion curves for the crystal. It is shown that the total absorption for a crystal at room temperature predicted by a phonon model with correlated atomic motion agrees with the Einstein-model potential, based on independent atomic motions. This suggests that ignoring correlated atomic motion is not likely to contribute to the well-known quantitative discrepancy in contrast between simulated and experimental transmission electron microscopy images (the so-called “Stobbs factor”). The quantum-mechanical formulation allows us to further investigate the form of the inelastically scattered waves and the nonlocality of the absorption potential in directions both perpendicular and parallel to the direction of propagation, providing deeper insight into underlying physics of phonon excitation by fast electrons.

• Received 31 March 2009

DOI:https://doi.org/10.1103/PhysRevB.80.024308