Discrete-continuum hybrid model for gas-surface collisions. II. Closed form perturbation solutions for single collisions

A perturbation scheme for the scattering of particle beams from single crystal surfaces is presented. The procedure is based on the assumption that displacements of the lattice points are small compared to the lattice spacing and that surface corrugation is weak. The phonon interaction is accounted for by representing the crystal as an elastic continuum following the hybrid model introduced in the preceding paper. The projectile-surface interaction is described in terms of compliance coefficients which are effective spring and damping constants of the solid, This makes the handling of the solid extremely practical. The perturbation theory and the linear solid model permit the study of collision-induced waves and the thermal waves separately. The total energy exchange is obtained as the sum of the energy exchanges between the projectile and the two types of waves. The perturbation solution yields closed form expressions for the energy exchange with both the exponential repulsive and Morse potentials. These results compare favorably with the exact (i.e., numerical) calculations of the proceeding paper when multiple collisions, resonances and capture are absent. The availability of explicit expressions for the energy transfer permits the calculation of the necessary thermal averages in closed form as well. Quantitative results indicate in addition that for low energies the Morse potential results are markedly different from those with the purely repulsive cases that is commonly used for its simplicity. At high energies, the results of the two potentials converge, as expected.