Size-dependent ionization energy of a metallic cluster: Resolution of the classical image-potential paradox

The ionization energy for a metallic cluster of radius R is I(R)=W+α${\mathit{e}}^2$/R+O(${\mathit{R}}^{\mathrm{-}2}$), where W is the work function. Two different classical values for α have been derived: 1/2 from the spherical-capacitor approach, and 3/8 from the image-potential approach. We present a ‘‘classical jellium’’ or Wigner-crystal model in which α=1/2 and W=9/10${\mathit{e}}^2$/${\mathit{r}}_{\mathit{s}}$. In the image-potential approach to the ‘‘perfect-conductor’’ model, we discover an additional work 1/8${\mathit{e}}^2$/R, resolving the contradiction in favor of α=1/2. The experimental deviation from α=1/2 is a quantum effect.