Probing the two-quasiparticle $K^\pi=8^+$ isomeric structure and enhanced stability in the proton drip-line nuclei

Stimulated by recent experimental discoveries [{Phys. Lett. B \textbf{847}, 138310 (2023)} and {Phys. Rev. Lett. \textbf{132}, 072502 (2024)}], two-quasiparticle $K^\pi=8^+$ isomeric structure (related to the neutron $h_{9/2}$ and $f_{7/2}$ orbitals) in $^{160}_{76}$Os$_{84}$ that lies at the two-proton drip line has been studied by means of the configuration-constrained potential-energy-surface calculations. Calculated results indicate that, for such an isomer, the excitation energy can be well reproduced and its oblate shape can be enhanced by the polarization effects of the two high-$K$ orbits. Comparing with experimental data, two sets of the widely used Woods-Saxon parameters, especially, the spin-orbit coupling one, are evaluated and argued. It is found that, considering the uncertainty of the spin-orbit coupling strength, the energy crossing or inversion of the $h_{9/2}$ and $f_{7/2}$ neutrons can occur, which may lead to three kinds of different evolution-trends of two-quasiparticle excitation energies with the changing quadrupole deformation $\beta_2$. With decreasing spin-orbit coupling interaction, the structure of the $K^\pi=8^+$ isomeric state will evolute from $\nu h_{9/2}f_{7/2}$ ($\nu 9/2^-[505] \otimes 7/2^-[503]$) to the mixing of $\nu h_{9/2}f_{7/2}$ and $\nu h_{9/2}^2$ ($\nu 9/2^-[505] \otimes 7/2^-[514]$) to $\nu h_{9/2}^2$, indicating that its structural probes is still of interest and an arbitrary assignment may be risky. The related theoretical calculations and experimental evidences e.g., the transition properties, are desirable. In addition, similar to that in superheavy nuclei, it is suggested that the stability inversion between high-$K$ isomeric states and ground states might occur in this proton drip-line mass region, e.g., in the hitherto unknown nucleus $^{162}_{78}$Pt$_{84}$.