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

Abstract

Stimulated by recent experimental discoveries [{Phys. Lett. B \textbf{847}, 138310 (2023)} and {Phys. Rev. Lett. \textbf{132}, 072502 (2024)}], two-quasiparticle $K^π=8^+$ 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 $β_2$. With decreasing spin-orbit coupling interaction, the structure of the $K^π=8^+$ isomeric state will evolute from $νh_{9/2}f_{7/2}$ ($ν9/2^-[505] \otimes 7/2^-[503]$) to the mixing of $νh_{9/2}f_{7/2}$ and $νh_{9/2}^2$ ($ν9/2^-[505] \otimes 7/2^-[514]$) to $νh_{9/2}^2$, 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$_{84}$.

Figures (6)

• Figure 1: Comparison between the available experimental data and the calculated single-particle energies in $^{146}$Gd. Note that theoretical single-particle levels are solved by using the Woods-Saxon Hamiltonian with uni. and StkI model parameters, .
• Figure 2: Calculated single-particle energies near the Fermi surface for neutrons (similarly for protons) as functions of quadrupole deformation $\beta_{2}$ by using the Woods-Saxon Hamiltonian with the StkI (a) and universal (b) parameters Meng2018Yang2016Bhagwat2010Bhagwat2023 for $^{160}$Os. The energy levels with positive and negative parities are respectively denoted by red solid and blue dotted lines. At $\beta_2 = 0.0$, the spherical quantum numbers $nlj$ are adopted for the single-particle labels. In the present deformation space, the single-particle levels at the positive and negative $\beta_2$ values are calculated by setting the triaxial deformation $\gamma= 0^\circ$ and $+60^\circ$ (or $-60^\circ$), respectively. See text for more details.
• Figure 3: Discrepancies between experimental and calculated $l$-level (labeled as {nl} quantum numbers) splittings, due to the spin-orbit interaction, as a function of spin-orbit coupling strength $\lambda$ for $^{132}$Sn and $^{146}$Gd. Theoretically calculated values $\Delta$E$_{theo.}$ are obtained based on the Woods-Saxon Hamiltonian with the StkI parameter set. Note that one can see, e.g., Eq.(2.3.1) in Ref.[34] for the definition of spin-orbit potential and the $\lambda$ value is 29.494 in the StkI parameter set. More information see text.
• Figure 4: Calculated PESs in ($\beta_2$, $\gamma$) and ($\beta_2$, $\beta_4$) planes for ground state (a) and (c), and $K^{\pi}=8^{+} \{\nu7/2^{-}[503]$$\otimes$$\nu9/2^{-}[505]\}$ isomeric state (b) and (d) for $^{160}_{76}$Os$_{84}$. According to the Lund convention Andersson1976, the Cartesian coordinates $X = \beta_2$cos($\gamma +30^\circ$) and $Y = \beta_2$sin($\gamma +30^\circ$) are adopted in subplots (a) and (b), avoiding that the minimum appears at the boundary. All the energy intervals between neighboring contour lines are 50 keV. The circle dot denotes the energy minimum which is normalized to zero. More details, see the text, e.g., cf. Table \ref{['table2']}.
• Figure 5: Calculated single-particle levels (a), (d), (g), quasi-particle levels (b), (e), (h) and excitation energies (c), (f), (i) of $K^\pi = 8^+$ isomer in functions of $\beta_2$ at three selected strengths of spin-orbit coupling (namely, $\lambda$ = 24.0, 29.494 and 34.0) for $^{160}$Os. Note that the single- and quasi-particle levels with red solid and blue dotted lines correspond to positive- and negative-parity states, respectively. The negative $\beta_2$ deformation indicates the oblate shape, realizing by fixing $\gamma$ to $60^\circ$ in the calculations. The blue squared symbol at $\beta_2=0.071$ in (f) denotes the observed excitation energy of $K^\pi = 8^+$ isomeric state. See the context for more explanations.