Quantum Theory of Inelastic Atom-Surface Scattering

Abstract

A quantum-mechanical treatment is developed for the inelastic scattering of atoms by thermal fluctuations near a surface using Green's functions and diagrammatic methods. A basic set of equations is derived for the Green's function and $T$ matrix by treating the thermal fluctuations, which give inelastic scattering, stochastically. This enables us to express scattering cross sections, adsorption or capture rates of atoms by the surface field, etc., in terms of a fluctuation correlation function which characterizes the surface excitations. The approach presented treats the scattering surface in a continuum approximation making it especially applicable to liquids and to solids when surface-diffraction effects are neglected. The derivation leads to a Dyson equation for the Green's function (the propagator in the presence of the fluctuating surface field) which is formally similar to the equation for the dressed-electron Green's function in the electron-phonon problem. Some essential differences exist, however, and these are discussed. The similarity between these problems is exploited to some extent to generate approximate solutions of the scattering problem to all orders in perturbation theory. The simplest case, corresponding to summing a class of diagrams with no vertex corrections, is examined. The use of separable matrix elements makes the approximation scheme particularly transparent. Expressions for the scattering and capture (adsorption) cross section are given and a resonancelike form of the cross section is shown. Results from lowest-order perturbation theory are presented for comparative purposes and their connection to previous work is described. An analogy with neutron scattering theory as formulated by Van Hove is also examined.