Thermal attenuation in atom-surface scattering: The multiphonon contribution

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Abstract
A resummation procedure is presented which allows one to include the contribution of virtual multiphonon exchange in the T-matrix element for the intensity of the specular beam. This procedure uses the exact-T-matrix elements of the one and two virtual-phonon processes as the starting point, quantities which have been calculated previously by the authors. Results are given for the scattering of helium, molecular hydrogen, and neon by the flat (100) face of copper. The intensities yielded by the resummation always lie between those given by the one-phonon and the one- plus two-phonon processes. It seems that the addition of an even number of phonon events increases the intensity, whereas an odd number yields the reverse effect. The minima observed on the intensity versus crystal-temperature curves are shifted to higher temperature with the inclusion of an increasing number of phonon events. The curve shape cannot be fitted by the usual Debye-Waller relation, a result in agreement with experimental results. However, the calculated intensities are in many cases greater than the measured ones. This can be due to the deficiencies of the chosen potential interaction or to an increase of anharmonic effects in the surface plane with respect to bulk anharmonicity.