# On improving the trajectory approximation in atom-surface scattering:I. Formalism

- Authors
- Journal
- Surface Science 0039-6028
- Publisher
- Elsevier
- Publication Date
- Volume
- 154
- Identifiers
- DOI: 10.1016/0039-6028(85)90056-1

## Abstract

Abstract We consider a one-dimensional model in which an atomic particle is linearly coupled to surface phonons. An exact expression of Feynman-Hibbs type, involving a double functional integral over two paths for the particle, is given for the probability ρ( E) that the atom lose energy E in collision with the surface. The first functional integral is evaluated in an approximation which is shown to involve dropping the de Broglie wavelength of the particle, resulting in the Quasiclassical Approximation. This expresses Π(τ), the Fourier Transform of ρ( E), as a weighted path integral over R( t) of the function Π TA(τ, [ R]), calculated in trajectory approximation for a given path R( t) of the particle. Expanding about the stationary trajectory, a series of corrections to the trajectory approximation are generated. The principal ones are the introduction of an optical potential coming from the response of the surface to the particle and a recoil term. The optical potential is elastic when calculating the Debye-Waller factor but otherwise inelastic in a manner consistent with the calculated energy transfer. The recoil term leads to energy gain for a slow particle on a hot surface. By an approximate scaling argument for rare gas atoms, we show that the trajectory approximation with optical potential correction is the leading contribution for large particle mass, i.e. for the heavy rare gases.

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