SEMICLASSICAL THEORY OF ATOM CRYSTAL SCATTERING DIFFRACTION AND INTERACTION WITH PHONONS

This paper deals with the problem of phonon-inelastic atom-surface sca ttering on the base of the path integral expression for the S-matrix i n the momentum representation. The Faddeev-Popov method is used to fix classical paths with respect to the symmetry of the system. The semic lassical evaluation of the path integral over projectile variables wit h the help of the generalized eikonal method, which allows us to take into consideration the recoil effect, results in an integral represent ation for the scattering probability of Van Hove type. This representa tion, taking into account a finite collision time and improving the se miclassical perturbation theory and the sudden approximation, is emplo yed to describe phonon-inelastic scattering from an ideal crystal surf ace in the thermal and diffractive regimes. Two limiting cases of sing le-phonon and multi-phonon scattering are considered. The probabilitie s of single-phonon scattering in the diffractive regime are expressed through the Bessel functions as well as the diffractive intensities. P ossible resonances during the inelastic scattering from a corrugated s urface, which may simplify the expression for the relevant probability , are discussed. We also suggest a new approximation to the dynamic st ructure factor (i.e. energy and tangential momentum transfer probabili ty) for multi-phonon scattering. It has the Gaussian form and, being a generalization of the well-known Brako-Newns formula, accounts for th e average transfer of the tangential momentum, satisfies the principle of detailed balance, and is valid for arbitrary collision times and s ur-face temperatures. An approximate expression for the average moment um transfer is proposed.