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Isomerism in the "south-east" of $^{132}$Sn and a predicted neutron-decaying isomer in $^{129}$Pd

Excited states in neutron-rich nuclei located south-east of $^{132}$Sn are investigated by shell-model calculations. A new shell-model Hamiltonian is constructed for the present study. The proton-proton and neutron-neutron interactions of the Hamiltonian are obtained through the existing CD-Bonn $G$ matrix results, while the proton-neutron interaction across two major shells is derived from the monopole based universal interaction plus the M3Y spin-orbit force. The present Hamiltonian can reproduce well the experimental data available in this region, including one-neutron separation energies, level energies and the experimental $B(E2)$ values of isomers in $^{134,136,138}$Sn, $^{130}$Cd, and $^{128}$Pd. New isomers are predicted in this region, $e.g.$ in $^{135}$Sn, $^{131}$Cd, $^{129}$Pd, $^{132,134}$In and $^{130}$Ag, in which almost no excited states are known experimentally yet. In the odd-odd $^{132,134}$In and $^{130}$Ag, the predicted very long $E2$ life-times of the low-lying $5^{-}$ states are discussed, demanding more information on the related proton-neutron interaction. The low-lying states of $^{132}$In are discussed in connection with the recently observed $γ$ rays. The predicted $19/2^{-}$ isomer in $^{129}$Pd could decay by both electromagnetic transitions and neutron emission with comparable partial life-times, making it a good candidate for neutron radioactivity, a decay mode which is yet to be discovered.