Dynamical corrections to the image potential

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Abstract
The interaction potential V(v,z0) for a classical charged particle approaching a metal surface with velocity v is discussed within the jellium approximation. A general expression for the potential is derived in terms of the retarded density response function of the bounded electron gas including effects due to acceleration of the charge. For normal incidence and the classical infinite barrier model of the surface, the asymptotic form of the potential has been determined analytically and the lowest-order finite velocity correction to the classical image potential shown to be proportional to z0-2 ln z0, and not to z0-2 as has been suggested. Close to the surface, the potential has been determined numerically, and shows a pronounced velocity dependence. An approximate expression for V(vz,z0) is then derived for a more general model of the surface which takes into account the diffuseness of the electron density. For v<>vF, finite velocity effects dominate when the charge is still some distance from the surface.