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 the
rmal 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.
