  Phonon lineshapes in atom-surface scattering are obtained from a simple
stochastic model based on the so-called Caldeira-Leggett Hamiltonian. In this
single-bath model, the excited phonon resulting from a creation or annihilation
event is coupled to a thermal bath consisting of an infinite number of harmonic
oscillators, namely the bath phonons. The diagonalization of the corresponding
Hamiltonian leads to a renormalization of the phonon frequencies in terms of
the phonon friction or damping coefficient. Moreover, when there are adsorbates
on the surface, this single-bath model can be extended to a two-bath model
accounting for the effect induced by the adsorbates on the phonon lineshapes as
well as their corresponding lineshapes.
