Critical charges on strange quark nuggets and other extended objects

We investigate the behavior of the critical charge for spontaneous pair production, ZC, defined as the charge at which the total energy of a K-shell electron is E=-me, as a function of the radius R of the charge distribution. Our approach is to solve the Dirac equation for a potential V(r) consisting of a spherically symmetrical charge distribution of radius R and a Coulomb tail. For a spherical shell distribution of the type usually associated with color-flavor locked strange quark nuggets, we confirm the relation ZC=0.71R (fm) for sufficiently large R obtained by Madsen, who used an approach based on the Thomas-Fermi model. We also present results for a uniformly charged sphere and again find that ZC∼R for large enough R. Also discussed is the behavior of ZC when simple ad hoc modifications are made to the potential for 0≤r<R.