$eta$ and $eta$-delayed neutron decay of the $N=82$ nucleus $^{131}_{~49}$In$_{82}$

The half-lives of three $beta$ decaying states of $^{131}_{~49}$In$_{82}$ have been measured with the GRIFFIN $gamma$-ray spectrometer at the TRIUMF-ISAC facility to be $T_{1/2}(1/2^-)=328(15)$~ms, $T_{1/2}(9/2^+)=265(8)$~ms, and $T_{1/2}(21/2^+)=323(55)$~ms, respectively. The first observation of $gamma$-rays following the $beta n$ decay of $^{131}$In into $^{130}$Sn is reported. The $beta$-delayed neutron emission probability is determined to be $P_{1n} = 12(7)%$ for the $21/2^+$ state and $2.3(3)%$ from the combined $1/2^-$ and $9/2^+$ states of $^{131}_{~49}$In$_{82}$ observed in this experiment. A significant expansion of the decay scheme of $^{131}$In, including 17 new excited states and 34 new $gamma$-ray transitions in $^{131}_{~50}$Sn$_{81}$ is also reported. This leads to large changes in the deduced $beta$ branching ratios to some of the low-lying states of $^{131}$Sn compared to previous work with implications for the strength of the first-forbidden $beta$ transitions in the vicinity of doubly-magic $^{132}_{~50}$Sn$_{82}$.