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+\ensuremath{\alpha}${\mathit{e}}^{2}$/R+O(${\mathit{R}}^{\mathrm{\ensuremath{-}}2}$), where W is the work function. Two different classical values for \ensuremath{\alpha} 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 \ensuremath{\alpha}=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 \ensuremath{\alpha}=1/2. The experimental deviation from \ensuremath{\alpha}=1/2 is a quantum effect.