Simple model for phonon spectroscopy using fast electrons

We propose a simple approach to simulating the phonon sector in electron energy-loss spectroscopy (EELS), as implemented in scanning transmission electron microscopy. Simplification of the problem is obtained by working with the phonon density of states (PDOS), a function of phonon energy, which is an integral over the details of the dispersion relations due to the correlated motions of the atoms. For a given PDOS, we derive a spectral distribution function, to distribute the total inelastic scattering, as calculated within the quantum excitation of phonons model, into an energy-loss/gain spectrum. The spectral distribution is obtained assuming a linear relationship between inelastic phonon scattering and atomic mean-squared displacements, a good approximation for phonon EELS with a detector covering only moderate scattering angles. We provide examples of the usefulness of the proposed approach in the modeling and interpretation of experimental phonon EELS data.