  We present a comparative assessment of the features of inelastic atom-surface
scattering spectra that are produced by several different forms of linear and
nonlinear phonon coupling to the projectile atom. Starting from a simple
theoretical model of atom-surface scattering and employing several recently
developed exact numerical and approximate analytical methods we calculate and
compare the scattering probabilities ensuing from each form of interaction and
from each calculational scheme. This enables us to demonstrate that in the
regime of thermal energy atom scattering from surfaces the dominant
contributions to the zero-, one- and multi-phonon excitation probabilities
obeying unitarity arise from linear coupling treated to all orders in the
interaction.
