An approach to the numerical solution of the problem of electron scattering in thin crystals, based on the Galerkin method

Abstract

The problem of electron scattering in a thin crystal (sample), arising in high-resolution transmission electron microscopy (HRTEM), is considered. In contrast to the theory of electron diffraction, the sample is supposed to be of arbitrary structure. The crystal potential is supposed to be, in general, complex; this allows inelastic scattering to be taken into account phenomenologically. The high-energy approximation to the three-dimensional time-independent Schrödinger equation, represented by the two-dimensional time-dependent Schrödinger equation, is used. To solve the problem numerically, the latter is reduced, by the semidiscrete Galerkin method, to an initial-value problem for a system of ordinary differential equations. Orthonormed families of functions, taking into account the features of the problem and having good approximation properties, are introduced for this. In contrast to the system of ordinary differential equation (ODE) for the complex amplitudes of diffracted beams, describing electron diffraction in the dynamical theory, the system derived in this paper corresponds to the discretization of the problem in the coordinate space. This allows the problem of electron scattering in a thin crystal (sample) having arbitrary structure to be solved.