Bloch waves and higher order Laue zone effects in high energy electron diffraction

Abstract

The theory of the scattering of fast electrons by a thin crystalline slab is formulated in terms of the Bloch waves of an infinite perfect crystal. In the symmetric Laue case, effects due to the variation of the crystal potentialU(r)along the zone axis parallel to the surface normal, are investigated by expanding these Bloch waves in terms of the Bloch functions of a two dimensional potential obtained by averagingU(r)along the zone axis. A high energy and forward scattering approximation is introduced which allows the scattering to be treated as an initial value problem. Perturbation expansions are used to analyse the changes in the dispersion surface and the Bloch waves when the variation of the potential along the zone axis is included. It is found that the most important perturbations are due to interactions associated with reciprocal lattice points in the Laue zones. These lead to hybridization of the Bloch functions of the two dimensional projected potential. A [111] zone axis of silicon at 293 K is studied as an example. In this case, the first order Laue zone leads to the strongest effects which can appear as fine bright lines in reflections in this zone, and also as fine lines in the strong reflexions in the zero Laue zone.The latter are usually dark, but can sometimes be bright. It is shown how it is often possible to separate the effects into the geometry of the intersection of free wave dispersion spheres centred on points in the non-zero Laue zones with the dispersion surface of the projected potential, and the strength of the matrix elements of the deviation of the potentialU(r)from the projected potential.