Multiple-Scattering Treatment of Low-Energy Electron-Diffraction Intensities

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Abstract
A theory of intensities in the diffraction of low‐energy (≲100‐eV) electrons by crystals is formulated on the basis of Lax's multiple‐scattering equations. The formulation is self‐consistent and is thus applicable regardless of the magnitude of the atomic‐scattering factor. The qualitative conclusions of the theory are discussed in detail with reference to simplified model crystals. The theory predicts that the reflectivity curves (reflectivity associated with a given beam versus electron energy) should show two types of peaks in addition to the ordinary Bragg peaks: (a) a resonance peak in the specular reflectivity curve; (b) secondary Bragg peaks in all reflectivity curves. The resonance peak is predicted at an energy just below that of the first appearance of a nonspecular beam. It derives from multiple scattering in a single atom layer and is associated with a resonance (zero and phase discontinuity) in the effective field. The secondary Bragg peaks are predicted at energies given by conditions similar in form to the ordinary Bragg condition. They derive from multiple scattering between atom layers and are associated with the interference between four or more plane wave components of the effective field. Both types of peaks are prominent only when the atomic‐scattering cross section is large. All qualitative results are illustrated by an exact numerical application to a model crystal of isotropic (s‐wave) scatterers. The relationship with existing theories is discussed. It is shown that the conventional modified kinematical approach to low‐energy electron diffraction is a limiting case that is approached when inelastic scattering is dominant. It is also shown that when certain assumptions are introduced, the most important being that of neglecting intralayer multiple scattering and the ``two‐beam'' assumption, the theory reduces to the Darwin dynamical theory.