Inelastic atom-surface scattering by phonons: A comparison of different approaches

The inelastic scattering of atoms by phonons of a semi-infinite solid involves infinitely many degrees of freedom and hence cannot be treated rigorously by any of the known numerical methods. An approximation which has been used successfully to explain experimental data is the exponentiated Born approximation (EBA). Here we compare this approximation for the scattering by an Einstein-model with two types of coherent state approximations (where the wave function is approximated by a Gaussian centered at the classical trajectory) and with the results of an exact coupled channel calculation. The EBA turns out to be close to the coherent state approximations, and all three agree rather well with the exact results except near thresholds and (in particular) resonances. The calculations are done for zero temperature T but excited initial states are considered from which nonzero T results can be obtained as Boltzmann averages.