Anisotropic relativistic cross sections for inelastic electron scattering, and the magic angle

Abstract

The knowledge of the inelastic double differential scattering cross section (DDSCS) is of utmost importance for microanalysis in electron energy loss spectrometry (EELS) in the electron microscope and for the investigation of electron energy loss near edge structure (ELNES). Calculations are usually based on a nonrelativistic first-order Born approximation. Relativistic corrections were thought to be important only for primary beam energies much higher than $300\phantom{\rule{0.3em}{0ex}}\mathrm{keV}$. A semirelativistic correction taking into account the relativistic mass change of the probe electron was considered sufficient. We review the relativistic expression of the inelastic double differential scattering cross section (DDSCS) originally given by M\o{}ller. We show that the virtual photon exchange proposed by Fano need not be considered separately. The interference terms between longitudinal and transverse exitations cannot be neglected in anisotropic systems. This explains the surprisingly small magic angle.