Application of a Variational Principle to the Calculation of Low-Energy Electron Diffraction Intensities. II. The Generalized Formalism of Three-Dimensional Problems

Abstract

A variational principle for the reflection coefficient is derived for elastic scattering from three-dimensional crystals. The crystal is assumed to possess perfect two-dimensional periodicity in the plane parallel to the crystal surface, but arbitrary variations of potential and lattice spacing in the perpendicular direction. As a result, this theory is valid for crystals with or without impurity layers. With linear trial functions, the reflection coefficient is uniquely given by the ratio of two determinants without any additional calculation of the wave field inside the crystal. Another variational principle is derived for the transmission coefficient of a plane wave transmitted through a three-dimensional crystal with a finite width. A modified Born approximation, which leads to the Born approximations, is obtained by inserting the incident and the adjoint plane waves in the variational equation without applying the variational principle. A general modified Born approximation is also obtained. For crystals having triperiodicity, the employment of Bloch waves as trial functions determines the reflection coefficient in terms of integrals over known functions and the perpendicular components of the wave numbers of the Bloch waves.