On the average Coulomb potential (Σ₀) and constraints on the electron density in crystals

The rôle of the average Coulomb potential (mean inner potential), Φ₀, in electron diffraction and methods for its calculation are reviewed. "From an examination of model and real crystals, it is shown that a prescription of Becker & Coppens [Acta Cryst. (1990), A46, 254–258] for calculating the average potential from X-ray data may lead to incorrect results because of an artificial surface-dipole term that is independent of crystal size. It is suggested that measurements of Φ₀ for crystals with adsorbed monolayers could be used to measure the density and sign of a real surface dipole moment. Published data for MgO, silicon and aluminium show that Φ₀ is very sensitive to bonding effects when atoms combine to form crystals. The often-reported expansion of the silicon valence shell deduced from pseudo-atom refinements of X-ray diffraction data is shown to be an artifact of the refinement method, as recent accurate measurements of Φ₀ for silicon instead require a contraction of the outer part of the valence shell. It is concluded that X-ray diffraction data alone do not allow a determination of Φ₀. However, Φ₀ provides information about valence-electron distributions that is not available in practice from X-ray data and places powerful constraints on pseudo-atom refinements.