Physical and mathematical foundations of multiwave electron diffraction

Abstract

The dynamical problem of multiwave electron diffraction in perfect and imperfect crystals is treated on the basis of the Bloch wave formalism. In the case of a perfect crystal the classification of the electron Bloch states in terms of the transverse energy is introduced. A perturbation theory for the analytical construction of electron Bloch functions is suggested, using the symmetry of the dynamical matrix. The contribution of bound, valence and free Bloch states to the reflections and to high-resolution electron-microscopy (HREM) images are analysed. Using the column approximation, the equations describing the propagation and the excitation of the electron Bloch waves due to defects in a crystal lattice are derived from the Schrödinger equation. A solution of these equations is obtained in the first Born approximation. The theoretical foundation of the Cowley-Moodie multislice method for computer simulations of HREM images is presented. As an illustration of the Cowley-Moodie method, HREM images of Au in the (110) orientation and Y3Al5O12 (111) are calculated for different defocusing parameters Δf. In the case of Au the fine structure of Au lattice images, when the Au atom strings are surrounded by 'halos', is clearly revealed for certain values of Δf The Y-Al garnet images are shown to have either a sixfold or a threefold axis of symmetry, depending whether or not the dimensionless thickness in terms of the lattice spacing along (111) is an integer number.