Theory of Metal Surfaces: Induced Surface Charge and Image Potential

This paper contributes to the theory of the electron density distribution induced at a metal surface by a small static external charge distribution. As a first application, profiles of the charge induced by a uniform external electric field are obtained for metals of different bulk electron densities. A quantity of particular interest is the position of the center of mass, ${x}_{0}$, of these profiles, for which we present numerical values. (The $x$ axis is taken along the surface normal.) Next, the case of a small point charge $q$ with $x$ coordinate ${x}_{1}$ well outside the surface is treated. It is shown that the image potential experienced by such a charge has the form $\ensuremath{-}\frac{{q}^{2}}{[4({x}_{1}\ensuremath{-}{x}_{0})]}$, where ${x}_{0}$ is the above-mentioned quantity. We locate ${x}_{0}$, the effective position of the metal surface, relative to the last lattice plane of the crystal. We discuss the implications of these results for alkali adsorption on metal substrates, the capacitances of small-gap condensers, and field-emission experiments.