Analytical models of single phonon atom–surface diffractive scattering

Simple analytical (semiclassical) models of diffractive single phonon atom–surface scattering are presented. Within the approach applied, the inelastic scattering amplitudes are expressed through the elastic ones which leads to the natural extension of the well-known Bessel approximation to inelastic (single phonon) scattering. The expression for the probability of diffractive scattering derived in this work gives the correct description of inelastic scattering via umklapp processes accompanied by considerable parallel momentum transfer. This model does not suffer from non-physical discontinuities at the edges of the surface Brillouin zone as confirmed by calculations of the single phonon diffractive scattering intensities for the He–Cu(110) system. Results of the comparison of the standard distorted-wave Born approximation with the semiclassical approach (generalized eikonal method) used in this work are also presented. It is shown that application of the semiclassical approach to the scattering problem requires the momentum formulation of quantum mechanics to describe correctly the contribution of the vicinity of the classical turning point to the semiclassical matrix element.