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}e²/R+O(R^-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=9/10e²/rₛ. In the image-potential approach to the ``perfect-conductor'' model, we discover an additional work 1/8e²/R, resolving the contradiction in favor of {α}=1/2. The experimental deviation from {α}=1/2 is a quantum effect.