A model calculation of the vibrational line shape of adsorbates at metal surfaces damped by the generation of electron-hole pairs is presented. Two vibrational excitations are considered, one fundamental and one overtone. To calculate the total absorption of a radiation field by the system, we use the formalism developed in the preceding paper to include contributions, ${\alpha }_{\perp }$ and ${\alpha }_{?}$, from the respective E-field components perpendicular and parallel to the surface. The generalized asymmetric Fano line-shape formula is shown to apply to both the fundamental and the overtone. For each case, the isotope effect on the line-shape parameters is obtained both in the limits of weak and strong breakdown of adiabaticity, where the weak- or strong-breakdown limit is modeled by assuming a flat spectral density near the Fermi level, or a sharp structure in that region on top of the flat part. It is found that a strong breakdown of adiabaticity weakens the isotope dependence of the asymmetry factor and the linewidth, while the isotope effect for the excitation strength remains the same as in the weak-breakdown limit. The magnitude of the excitation strength is modified as the strong-breakdown limit is reached. These conclusions remain valid whether the vibration contributes to ${\alpha }_{\perp }$ or to ${\alpha }_{?}$. When applied to the wag overtone of hydrogen on W(100), reasonably good agreement is found between the theory in the strong-breakdown limit and experiment assuming that the overtone absorption is given by ${\alpha }_{\perp }$, the perpendicular response. We also discuss other possibilities for the measured absorption, including the contribution from the overtone by ${\alpha }_{?}$, the tangential response, and from other excitations nearby in frequency.

• Received 7 October 1988

DOI:https://doi.org/10.1103/PhysRevB.39.10028