Unblocking of stellar electron captures for neutron-rich $N=50$ nuclei at finite temperature

We have calculated electron capture rates for neutron-rich $N=50$ nuclei ($^{78}$Ni, $^{82}$Ge, $^{86}$Kr, $^{88}$Sr) within the Thermal QRPA approach at temperatures $T=0$, corresponding to capture on the ground-state, and at $T=10$~GK (0.86~MeV), which is a typical temperature at which the $N=50$ nuclei are abundant during a supernova collapse. In agreement with recent experiments, we find no Gamow-Teller (GT$_+$) strength at low excitation energies, $E<7$ MeV, caused by Pauli blocking induced by the $N=50$ shell gap. At the astrophysically relevant temperatures this Pauli blocking of the GT$_+$ strength is overcome by thermal excitations across the $Z=40$ proton and $N=50$ neutron shell gaps, leading to a sizable GT contribution to the electron capture. At the high densities, at which the $N=50$ nuclei are important for stellar electron capture, forbidden transitions contribute noticeably to the capture rate. Our results indicate that the neutron-rich $N=50$ nuclei do not serve as an obstacle of electron capture during the supernova collapse.