Puzzle on isomeric configurations in and around N=126 closed shell

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The puzzle of finding consistent nuclear configurations to explain both the decay probabilities and moments of the ${9/2}^{\ensuremath{-}}, {8}^{+}$, and ${21/2}^{\ensuremath{-}}$ isomers in and around the $N=126$ closed shell has been approached in the generalized seniority scheme. Though ${h}_{9/2}$ is the dominant orbital near Fermi energy, the role of configuration mixing from the surrounding ${f}_{7/2}$ and ${i}_{13/2}$ orbitals is found to be very important for the consistent explanation of all the isomeric properties such as the $B(E2)$ rates, $Q\phantom{\rule{0.28em}{0ex}}\mathrm{moments},$ and $g\phantom{\rule{0.28em}{0ex}}\mathrm{factors}$. The structural behavior of the closed shell $N=126$ isotonic isomers turns out to be very similar to that of the $N=124$ and $N=128$ isotonic isomers, which have two neutron holes and two neutron particles, respectively. This is due to the pairing symmetries of nuclear many-body Hamiltonian. As confirmation, the microscopic shell model occupancies are also calculated for these isomers in the $N=126$ chain which support the generalized seniority results. Additional arguments using the systematics of odd-proton ${9/2}^{\ensuremath{-}}$ states in Tl ($Z=81$), Bi ($Z=83$), At ($Z=85$), and Fr ($Z=87$) isotopes are also presented.