Polarization dependence in ELNES: Influence of probe convergence, collector aperture and electron beam incidence angle

The differential scattering cross section in electron energy loss near edge spectroscopy (ELNES) generally depends on the orientation of the QQ wave vector transferred from the incident electron to an atomic core electron. In the case where the excited atom belongs to a threefold, fourfold or sixfold main rotation axis, the dipole cross section depends on the angle of QQ with respect to this axis. In this paper, we restrict to this situation called dichroism. Furthermore, if we take into account the relativistic effects due to the high incident electron velocity, this dipole cross section also depends on the angle of QQ with respect to the electron beam axis. It is due to these dependences that the shape of measured electron energy loss spectra varies with the electron beam incidence, the collector aperture, the incident beam convergence and the incident electron energy. The existence of a particular beam incidence angle for which the scattering cross section becomes independent of collection and beam convergence semi-angles is clearly underscored. Conversely, it is shown that EELS spectra do not depend on the beam incidence angle for a set of particular values of collection and convergence semi-angles. Particularly, in the case of a parallel incident beam, there is a collection semi-angle (often called magic angle) for which the cross section becomes independent of the beam orientation. This magic angle depends on the incident beam kinetic energy. If the incident electron velocity V is small compared with the light velocity c  , this magic angle is about 3.975θE3.975θE (θEθE is the scattering angle). It decreases to 0 when V approaches c. These results are illustrated in the case of the K boron edge in the boron nitride.