Sum Rule for Crystalline Metal Surfaces

We have derived a sum rule for the electronic scattering phase shifts in a metal, produced by a planar boundary surface. The sum rule, which states that a weighted Fermi-surface average of the asymptotic phase of the electronic wave function in the presence of a metal surface is equal to $\frac{-\pi }{4}$, is derived for the case where explicit account is taken of the energy bands and crystalline structure of the solid. This extends the sum rule originally stated by Sugiyama for a jellium model of a metal. Modifications of the sum rule in the presence of an applied electric field and for complex band structure are also considered.