Phase shifts, image planes, and surface states at metal surfaces

Crystal- and image-potential-induced surface states of Cu, Ag, and Ni are studied with use of the phase-analysis model, with an accurate expression for the classical image-barrier phase shift. The energies of the first three surface states of the (111) and (001) faces of these metals are obtained as a function of the image plane distance to the surface, $z_{\mathrm{im}}$. The experimental results are reproduced if a small effective $z_{\mathrm{im}}$ is used. The effects of changing the plateau of the image potential are also analyzed.