Unblocking of the Gamow-Teller Strength in Stellar Electron Capture on Neutron-Rich Germanium Isotopes

We propose a new model to calculate stellar electron capture rates for neutron-rich nuclei. These nuclei are encountered in the core collapse of a massive star. Using the shell model Monte Carlo approach, we first calculate the finite temperature occupation numbers in the parent nucleus. We then use these occupation numbers as a starting point for calculations using the random phase approximation (RPA). Using the RPA approach, we calculate electron capture rates including both allowed and forbidden transitions. Such a hybrid model is particularly useful for nuclei with proton numbers $Zl40$ and neutron numbers $Ng40,$ where allowed Gamow-Teller transitions are only possible due to configuration mixing by the residual interaction and by thermal unblocking of $\mathrm{pf}$-shell single-particle states. Using the even germanium isotopes ${}^{68\ensuremath{-}76}\mathrm{Ge}$ as examples, we demonstrate that the configuration mixing is strong enough to unblock the Gamow-Teller transitions at all temperatures relevant to core-collapse supernovae.