  The diffraction of fast atoms at crystal surfaces is ideal for a detailed
investigation of the surface electronic density. However, instead of sharp
diffraction spots, most experiments show elongated streaks characteristic of
inelastic diffraction. This paper describes these inelastic profiles in terms
of individual inelastic collisions with surface atoms taking place along the
projectile trajectory and leading to vibrational excitation of the local Debye
oscillator. A quasi-elastic regime where only one inelastic event contributes
is identified as well as a mixed quantum-classical regime were several
inelastic collision are involved. These regimes describe a smooth evolution of
the scattering profiles from sharp spots to elongated streaks merging
progressively into the classical diffusion regime.
