Atomic scattering from a sinusoidal hard wall: Comparison of approximate methods with exact quantum results

Several approximate procedures for the calculation of the intensities of atoms elastically scattered from solids are compared with nearly exact quantum calculations from a sinusoidal hard wall potential. The quasiclassical method reproduces the qualitative features of the ’’rainbow’’ envelope of the scattered intensities present in the exact results but it does not contain any of the quantum interference arising from multiple scattering within a single unit cell and thus does not reproduce the rich structure in the intensities of individual diffraction beams. CCGM theory predicts only specular scattering for the hard wall, independent of roughness. Semiclassical theory at normal incidence gives the same analytical results as the Kirchoff approximation. This result agrees to within 3% of the exact calculations for surface roughness from 0.02 to 0.1 of a lattice constant a at ka vectors greater than 22. At lower ka vectors deviations as large as 60% are observed for the 0.1 roughness while at 0.02 roughness deviations are still less than 2%. Renormalization of the results from the 0.1 surface reduces its deviations from the exact calculation to less than 10% even at low ka vectors. The Kirchoff result agrees with the semiclassical one only at normal incidence and can be used only for hard wall potenials.