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$β$-decay half-lives and nuclear structure of exotic proton-rich waiting point nuclei under $rp$-process conditions

We investigate even-even nuclei in the $A\sim70$ mass region within the framework of the proton-neutron quasi-particle random phase approximation (\mbox{pn-QRPA}) and the interacting boson model-1 (\mbox{IBM-1}). Our work includes calculation of the energy spectra and the potential energy surfaces $V(β,γ)$ of Zn, Ge, Se, Kr and Sr nuclei with the same proton and neutron number, $N = Z$. The parametrization of the \mbox{IBM-1} Hamiltonian was performed for the calculation of the energy levels in the ground state bands. Geometric shape of the nuclei was predicted by plotting the potential energy surfaces $V(β,γ)$ obtained from the \mbox{IBM-1} Hamiltonian in the classical limit. The \mbox{pn-QRPA} model was later used to compute half-lives of the neutron-deficient nuclei which were found to be in very good agreement with the measured ones. The \mbox{pn-QRPA} model was also used to calculate the Gamow-Teller strength distributions and was found to be in decent agreement with the measured data. We further calculate the electron capture and positron decay rates for these $N$ = $Z$ waiting point (WP) nuclei in the stellar environment employing the \mbox{pn-QRPA} model. For the $rp$-process conditions, our total weak rates are within a factor two compared with the Skyrme HF+BCS+QRPA calculation. All calculated electron capture rates are comparable to the competing positron decay rates under $rp$-process conditions. Our study confirms the finding that electron capture rates form an integral part of the weak rates under $rp$-process conditions and should not be neglected in the nuclear network calculations.