Calculation of potential cutoff for one-phonon atom-surface scattering

A cluster calculation for Cu(111), Ag(111), and Au(111) shows that the lateral Fourier transform of the charge deformation due to a displacement of a single surface atom is a Gaussian $\exp (-\frac{Q^2}{2{Q_c}^2})$. This is proportional to the scattering amplitude for one-phonon exchange atom-surface scattering and produces a cutoff in the cross section. The values of $Q_c$ are evaluated for the above-mentioned surfaces, explaining very well the experimental observations. We also find a simple formula that should be useful to experimentalists in calculating the cutoff.