Allowed $\beta^-$ decay of bare atoms with A$\sim$ 60-80 in stellar environments
We have calculated $\beta^-$ decay rates to the continuum and bound states of some fully ionized atoms in the stellar s-process environment having free electron density and temperature in the range $n_e = 10^{26} $ cm$^{-3} - 10^{27} $ cm$^{-3}$ and $T = 10^8$ K - $5 \times 10^{8}$ K, respectively. The presence of bare atoms in these particular situations has been confirmed by solving Saha ionization equation taking into account the ionization potential depression (IPD). At these temperatures, low lying excited energy levels of parent nuclei may have thermal equilibrium population and those excited levels may also decay via $\beta^-$ emission. The Nuclear Matrix Element (NME) of all the transitions of the set of 15 nuclei is calculated using nuclear shell-model. These NME are then used to calculate the comparative half-life ($ft_{1/2}$) of the transitions. Calculated terrestrial half-lives of the $\beta^-$ decays are in good agreement with the experimental results in most of the cases. Decay to bound and continuum states of bare atoms from ground/isomeric levels and excited nuclear levels have been calculated separately. The ratio of bound state to continuum state decay rates as a function of IPD modified $Q$-value reveals that bound state $\beta^-$ decay rate may compete and even dominate for $Q$-value $<$ 100 keV. The importance of the bound state $\beta^-$ decay in stellar situations has been shown explicitly. We have calculated total $\beta^-$ decay rates (bound state plus continuum state) taking into account IPD corrected neutral atom $Q$-value as a function of density and temperature. We have also presented results for the stellar $\beta^-$ half-lives and compared the ratio of neutral atom to bare atom half-lives for different density and temperature combinations. These results may be useful for s-process nucleosynthesis calculations.