Reciprocity in electron diffraction and microscopy

The Reciprocity Theorem of scattering theory is shown to hold generally for electrons scattered elastically in an absorbing medium, and also for inelastically scattered electrons, to a certain approximation. Two scattering symmetry conditions are defined by applying the theorem to crystals having mirror symmetry parallel to the crystal surface, and to centrosymmetric crystals. These are general forms of conditions demonstrated by Fukuhara (J. Phys. Soc. Japan (1966), 21, 2645). A number of symmetry effects observable in electron microscope images of crystals result from one or other of these conditions. The variation of intensity of diffracted beams as a function of angular deviation from the Bragg condition is considered in detail. The symmetry of bright-field images of defects lying at equal distances from the crystal surfaces was first explained by Howie & Whelan (Proc. Roy. Soc. (1961), A 263, 217); such symmetry properties can be conveniently classified and explained by the reciprocity theorem. A method of obtaining high-resolution dark-field images, due to Cowley, is based on the use of the reciprocity theorem. Examples of its application for both Bragg and diffusely scattered electrons are given.