Coulomb charging effects for finite channel number

We consider quantum fluctuations of the charge on a small metallic grain ca used by virtual electron tunneling to a nearby electrode. The average elect ron number and the effective charging energy are determined by means of per turbation theory in the tunneling Hamiltonian. In particular, we discuss th e dependence of charging effects on the number N of tunneling channels. Ear lier results for N much greater than I are found to be approached rather ra pidly with increasing N.