  Prompted by recent experimental developments, a theory of surface scattering
of fast atoms at grazing incidence is developed. The theory gives rise to a
quantum mechanical limit for ordered surfaces that describes coherent
diffraction peaks whose thermal attenuation is governed by a Debye-Waller
factor, however, this Debye-Waller factor has values much larger than would be
calculated using simple models. A classical limit for incoherent scattering is
obtained for high energies and temperatures. Between these limiting classical
and quantum cases is another regime in which diffraction features appear that
are broadened by the motion in the fast direction of the scattered beam but
whose intensity is not governed by a Debye-Waller factor. All of these limits
appear to be accessible within the range of currently available experimental
conditions.
