Atom-surface scattering under classical conditions

agreement with the experiment and with previous theoretical calculations for the temperature dependence of the peak widths is obtained. The calculations suggest that further comparisons with the scattered lobes will produce important information about the projectile interaction and about vibrational correlations at the surface. @S0163-1829~96!06124-3# The surface scattering of atomic projectiles under quantum mechanical conditions, and particularly the scattering of He, is an experimental tool that has produced extensive information on surface dynamics and on the atom-surface interaction. 1,2 Far fewer experiments have been carried out under the classical conditions for vibrational energy transfer, which consist of large projectile masses, high incident energies, and large surface temperatures. 3‐6 Under such conditions the scattering problem becomes very complex and involves the exchange of many phonon quanta. Additionally, at high energies thresholds for the appearance of new quantum processes are crossed and such events as atomic electronic excitations of either the projectile or the crystal cores become possible. An understanding of energy and momentum exchange in the classical regime is very important for establishing the interaction potentials and the crystal dynamics at these energies, and is also necessary for an understanding of macroscopic physical processes at gas-surface interfaces such as energy accommodation, sticking, and drag and lift forces. Recently, a very interesting classical regime experiment has been reported for high-resolution, energy-resolved scattering intensities of Na 1 ions with energies of several hundred eV from a Cu~001! surface. 7 Of the several features observed in the spectra, one was identified as a peak due to single-scattering events, and the widths of this peak as a function of temperature and polar scattering angle were explained quite well with recently developed semiclassical scattering theories taken in the classical limit. 8,9 Under the conditions of this experiment, approximately half or more of the incident Na 1 translational kinetic energy was deposited in the crystal surface by the projectiles in the backscattered intensity. Theoretical comparisons were also made with the semiclassical trajectory approximation ~TA! and this approximation was found to fail completely for these high translational energy-loss experiments. The purpose of this paper is to compare these recent measurements with the complete classical theory for the scattering intensity, which is obtained from the quantum multiphonon calculations in the correspondence limit. In addition to reproducing the global width features of the measured intensities, this theory makes predictions of the angular and energy-dependent shapes of the scattered intensity lobes, which indicate that further comparison with experiment can give important information on surface interactions and vibrational correlations. Most quantum mechanical treatments of the inelastic exchange of vibrational energy in atom-surface collisions begin by describing the system in terms of a Hamiltonian of the form H5H 0 1H 0 1V, ~1!