FIG. 1: The upper, middle and lower panels show, respectively, the electron density (full line), electric potential and electric field, calculated for the given proton density profile (dashed line in the upper panel) as given by Eq.(1) with ρ$_{p}$$_{0}$ = 0 . 01 fm − 3 and a =5 fm. a potential would certainly lead to a spontaneous electron-positron pair production. In this process the negative-energy electrons from the Dirac sea (below the mass gap) would penetrate through the barrier and occupy a state in the upper continuum (above the mass gap) leaving a hole (positron) behind. But all electron states up to ϵ = µ are occupied (Fermi sea) to balance the bulk positive charge of protons. The situation is changed at finite temperatures when the particle-hole excitations are produced in the Fermi sea. Then, the electrons from the Dirac sea can occupy the vacancies in the Fermi sea. The process of thermal emission of electron-positron pairs from the sharp surface was considered earlier in ref. [20]. Here we want to point out other interesting processes, namely, the spontaneous 5