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Gamma-Ray Infrastructure For Fundamental Investigations of Nuclei (GRIFFIN) is a new high-efficiency $\gamma$-ray spectrometer designed for use in decay spectroscopy experiments with low-energy radioactive ion beams provided by TRIUMF's Isotope Separator and Accelerator (ISAC-I) facility. The high-efficiency GRIFFIN array comprises 16 Compton-suppressed large-volume HPGe clovers, and is designed to be used with a suite of ancillary detectors, providing a powerful and versatile tool for studying exotic nuclei. The structures of $N = 82$ nuclei below doubly-magic $^{132}$Sn are crucial for calculations of the astrophysical r-process as these isotopes form 'waiting-points' that play an important role in the formation and shape of the second $r$-process abundance peak. Many of the most neutron-rich $N = 82$ nuclei are, however, out of reach to the current generation of radioactive beam facilities and their properties must be predicted. In the past, shell-model calculations for the half-lives of these nuclei have been performed by adjusting the quenching of the Gamow-Teller (GT) operator in order to reproduce the half-life of $^{130}$Cd. However, the calculated half-lives of the nuclei below $^{130}$Cd are known to be systematically too large. A recent measurement of the $^{130}$Cd half-life with EURICA indicated a shorter half-life of $^{130}$Cd, which would lead to a re-scaling of the GT quenching by a constant factor for all nuclei in the region and potentially resolves the discrepancy. However, the reduced quenching of the GT operator implied by these results creates a new discrepancy in the calculated half-life of the $N=82$ isotope $^{131}$In. Half-life measurements of nuclei in this region are complicated due to the presence of $\beta$-decaying isomers with comparable half-lives and large $\beta n$ branches, making $\gamma$-ray photo-peak gating with a high-resolution, high-efficiency, $\gamma$-ray spectrometer an ideal method to measure each of the isomeric half-lives. In this work, measurements of the half-lives of the $^{128-130}$Cd isotopes were performed. The half-life of the $N=82$ $r$-process waiting point nucleus $^{130}$Cd was measured to be 126(4)~ms confirming the EURICA measurements and in strong disagreement with the earlier measurements of 162(7)~ms and 195(35)~ms. A detailed data set for the $\beta$ and $\beta n$ decay of $^{131}$In was also analyzed. The $\beta$ decay from the single proton hole nucleus $^{131}$In to the single neutron hole nucleus $^{131}$Sn is of current interest with respect to advancing our understanding of nuclear forces and shell evolution in this region, including modelling the astrophysical $r$-process. In this work, three different $\beta$ decaying states in $^{131}$In were studied in detail, providing new information about the shell structure of $^{131}$Sn. The half-lives of these three $\beta$ decaying states of $^{131}$In measured in this work agree with the previous measurements. The half-life of each $\beta$ decaying state of $^{131}$In is significantly longer than the half-life predicted from shell-model calculations. |
