Phonons at metal surfaces

A hydrodynamic model of collective motion at a metal surface is presented which treats ion and electron degrees of freedom on the same basis. In order to examine modes at a surface, the bulk equations of motion must be supplemented by additional boundary conditions. It is argued that at a free surface, continuity of surface traction is more appropriate than continuity of normal current density. The acoustic-phonon modes for several model systems are determined. Both the surface-mode dispersion and the bulk-and surface-mode external coupling to a charged probe show considerable sensitivity to the choice of boundary conditions. The implications of these results for electron-loss spectroscopy are discussed.