Surface imaging of inelastic Friedel oscillations

Impurities that are present on the surface of a metal often have internal degrees of freedom. Inelastic scattering due to impurities can be revealed by observing local features seen in the tunneling current with scanning tunneling microscope (STM). We consider localized vibrational modes coupled to the electronic structure of a surface. We argue that vibrational modes of impurities produce Fermi momentum $k_F$ oscillations in second derivative of current with respect to voltage $\partial^2I(\br, V)/\partial V^2$. These oscillations are similar to the well known Friedel oscillations of screening charge on the surface. We propose to measure inelastic scattering generated by the presence of the vibrational modes with STM by imaging the $\partial^2I/\partial V^2$ oscillations on the metal surface.