Curvature of the total electron density at critical coupling: Attractive impurity in an electron gas

Abstract

Structural properties of the spherically averaged total charge density ρλ (r) of a degenerate free electron gas distorted by the action of an attractive, screened potential λV(r) are studied. The total charge density is described as the sum over the occupied bound and scattering states of independent electrons moving in an effective central field. Each of the individual (bound, scattering) terms in the density presents discontinuities at threshold values of λ for which new bound states appear. Nevertheless, it is mathematically proven that these discontinuities cancel each other and that the density ρλ (0) at the origin, as well as its two radial derivatives ρ′λ(0) and ρ″λ(0), are differentiate functions of λ for any λ ≥ 0. The smooth behaviour implies that the curvature ρ″λ(0) at the origin does not provide a criterion for the bound-to-free transition in metallic environments. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.