Hamiltonian theory for vibrational line shapes of atoms adsorbed on surfaces

The vibrational motions of atomic adsorbates on surfaces can be probed by helium atom scattering. The experimental observable is the dynamic structure factor, which shows an inelastic peak around the vibrational frequency of the isolated adsorbates known as the frustrated translational or T-mode peak. In this paper we develop a theory for the line shape of this peak, as well as for its temperature-dependent shift and broadening, based on a Hamiltonian equivalent of the generalized Langevin equation. The theory can be used to infer physical parameters of the adatom-surface interaction, such as the friction coefficient, the barrier height to diffusion, and the anharmonicity parameter. Numerical simulations are used to ascertain the range of validity of the theory, which is also generalized to describe multidimensional systems and to include quantum corrections. We compare the theoretical predictions for the shift and broadening with experimental results for the Na/Cu(001) system, showing quantitative agreement within experimental resolution.