Beta-decay rates in the vicinity of N=50 nuclei

Over the past few years, $^{78}$Ni (with 50 neutrons and 28 protons) has garnered much attention being the most neutron-rich doubly-magic nuclei discovered to date. The recent spectroscopic study of $^{78}$Ni has provided direct evidence of its doubly-magic nature with an indication of competing spherical and deformed configurations. Subsequently, numerous studies employing various methods have been focused to probe the nuclear structure properties in the vicinity of $N=50$. One such method is the $\beta$-decay study that not only offers first hand information on nuclear structure but also serves as one of the most important inputs for modeling the $r$-process nucleosynthesis. Given the scarcity of experimental data near $N=50$, such $r$-process calculations depend upon the predictions from different theoretical models whose uncertainties directly effect the final $r$-process abundances. In this work, we present the truncated shell-model (SM) calculations focused on the $\beta$-decay rates of neutron-rich nuclei in $N=50$ region. To perform these calculations, we consider the PFSDG-U interaction and perform the monopole corrections to it in order to reproduce the measured $\beta$-decay half-lives. We name the resulting interaction as PFSDG-U-Mix. The model-space used in this study is larger than the previous $\beta$-decay calculations performed in this region and is comprised of $pf$ shell for protons and $pf-sdg$ shells for neutrons. A good theoretical description of $\beta$-decay properties such as half-lives ($T_{1/2}$) and one-neutron emission probability ($P_{1n}$) critically depends on the values of one-neutron separation energy ($S_{n}$) and $\beta$-decay energy ($Q_{\beta}$) used in the calculations. Using the SM code ANTOINE, we start by performing the calculations for nuclear masses at truncation level $t=4$ for the isotopic chains of $Z=22-30$ and $N=41-52$. Our results agree well with the Atomic Mass Evaluation (AME2020). Therefore, during the calculations of $T_{1/2}$ and $P_{1n}$ we use the SM calculated $S_{n}$ and $Q_{\beta}$ values. For the same isotopic chains, we then perform the truncated SM calculations for $T_{1/2}$ and $P_{1n}$ including the Gamow-Teller (GT) and First-Forbidden (FF) transitions. In the GT decay of $N=50$ waiting point nuclei, the daughter states wavefunction demonstrate a mixing of np-nh configurations. Generally, the GT transitions were found to be feeding the low-lying states in daughter nuclei whose contribution decreased with increasing neutron number. This reduction was although countered by the increased FF transitions, their effect on the half-lives in this region was determined to be small. Overall, we find good agreement between our calculated half-lives and the experimentally measured ones, enabling us to make predictions for experimentally unknown nuclei near the neutron drip-line. The one-neutron emission probabilities calculated in this work demonstrate marginally higher values compared to the measured data.