Contribution of electric quadrupole transitions to scattering cross section and collected current in a transmission electron microscope

Abstract

In this paper, the validity of the dipole approximation in the calculation of the double-differential scattering cross section (DDSCS) is investigated. A new expression of the DDSCS is obtained by adding to the classical term associated with the electric dipole transitions two terms taking the electric quadrupole transitions and the interference between dipole and quadrupole transition channels into account. In this study, we restrict to cases where the probed atom site is tetrahedral or octahedral. Moreover, the calculation of the average collected current (ACC) falling into the transition electron microscope collector is carried out by assuming that the incident electron beam is parallel. Then, DDSCS and ACC reduce to a linear combination of five intrinsic components, which only depend on the energy loss. The dominant intrinsic component is associated with the electric dipole transitions, three other intrinsic components are associated with the electric quadrupole transitions, and the last one, which cancels when the probed atom site is an inversion center, is associated with the interferences between dipole and quadrupole transition channels. These intrinsic components are rather hard to calculate, but their weights can be readily evaluated. In the case of the ACC, they essentially depend on the incident beam orientation and the collector aperture. The amplitudes of these weights allow us to evaluate the actual importance of the electric quadrupole transitions. At large collector aperture, the weights of the three quadrupole intrinsic components strongly increase and, thus, the contribution of the electric quadrupole transitions can not be neglected. At smaller collection angles, the importance of the electric quadrupole transitions can be estimated in each particular situation by comparing the amplitudes of the dipole and quadrupole intrinsic components.