A 'three-beam' analysis of resonance scattering in reflection high-energy electron diffraction

Enhanced reflection of fast electrons from a crystal surface and a decrease in the depth of penetration of the primary beam occurs when diffraction conditions are such as to set up a wave travelling just beneath the crystal surface. This is the surface resonance condition for reflection high-energy electron diffraction (RHEED). Quantitative prediction of these effects can be achieved by assuming that only the primary and two diffracted beams are significant. Expressions for the coefficient of reflection and the depth of penetration in terms of a few Fourier coefficients of an effective potential are derived. These coefficients depend sensitively on incident-beam direction and are significantly different from the values for the bulk crystal. In particular, the mean potential experienced by the electrons in the resonance state is increased. It can be estimated using Bethe's perturbation approach. Predictions of the position, height and width of the peak in reflectivity resulting from resonance scattering from the (111) surface of platinum are in reasonable agreement with the values obtained from many-beam computations. The three-beam approach gives insight into resonance scattering using the standard formalism of diffraction theory.