Pair production in non-uniform electric fields

Treating the production of electron and positron pairs in vacuum by a strong electric field as a quantum tunneling process, we derive in semiclassical approximation the pair production rate for nonuniform fields E(z) pointing the z-direction. In addition, we discuss tunneling processes in which an empty atomic bound state is spontaneously filled with a negative-energy electron creating a positron. The general expression is applied to a confined field, a semi-confined field, and a linearly increasing field. The boundary effects of the confined fields on pair-production rates are explicitly evaluated. Finally, we calculate the rate at which the atomic level of a bare nucleus of finite size and large Z are filled by electrons from the vacuum under positrons emission.