Perturbation theory in high-energy transmission electron diffraction

A perturbation theory for many-beam high-energy transmission electron diffraction in noncentrosymmetric crystals is described for both the nondegenerate and degenerate cases. This perturbation theory differs from the conventional quantum-mechanical perturbation theory by perturbing the electron wavevectors instead of the total electron energy, which is constant for elastically scattered electrons. The relations between the perturbation theory and some other approximations commonly used in electron diffraction are discussed. It is shown that the few-beam approximation and the Kambe approximation are both applications of degenerate perturbation theory. Finally, as an example, this degenerate perturbation theory is applied to obtain an analytical solution to a four-beam case with two systematic (0 and g) and two nonsystematic (h and l) beams. This four-beam solution shows that the intensity of a four-beam interaction depends on all the four three-phase invariants involved, and also shows that the effects of the g beam on the three-beam interaction of 0, h and l are localized to the region near the Bragg condition of g. This may serve as a guide for future experiments using three-beam interactions for the measurement of structure-factor phases of an unknown structure.