Coulomb charging effects for finite channel number

We consider quantum fluctuations of the charge on a small metallic grain caused by virtual electron tunneling to a nearby electrode. The average electron number and the effective charging energy are determined by means of perturbation theory in the tunneling Hamiltonian. In particular, we discuss the dependence of charging effects on the number N of tunneling channels. Earlier results for N ≫ 1 are found to be approached rather rapidly with increasing N.