Simplified analysis of helium scattering: supernumerary rainbows from a soft, corrugated potential

Abstract

This paper presents a simple, but quantitative approach to the analysis of helium diffraction from surfaces. Instead of calculating the intensities of helium diffraction beams, we analyze the locations of the supernumerary rainbows, or oscillations in the intensity envelope of the diffraction beams. Within a semiclassical framework, we interpret the supernumerary rainbows in terms of the phase integrals associated with the different classical scattering trajectories. Using a perturbation theory to calculate the classical trajectories, we obtain a simple, closed formula which predicts the locations of the supernumerary rainbows as a function of the scattering angle. This formula includes a correction for the softness of the scattering potential. Since the soft-potential effect depends strongly on the angle between the incoming and outgoing trajectories, our scattering formula can be used to extract both the corrugation and the decay length of the scattering potential from a series of diffraction patterns taken at various angles of incidence. The results of our perturbative semiclassical analysis are in good agreement with those of exact quantum-mechanical calculations.