Revisiting the nuclear island of negative hexadecapole deformations in A$\approx$180 mass region: focusing on moments of inertia and quadrupole-hexadecapole coupling

TL;DR

The paper investigates an island of negative hexadecapole deformations in the A≈180 mass region and its impact on rotation by combining a macroscopic-microscopic approach with Woods-Saxon mean fields and cranked Hartree-Fock-Bogoliubov calculations at fixed shapes. Shape dynamics are explored in a deformation space spanned by β2, γ, and β4, with Strutinsky shell corrections and Lipkin-Nogami pairing to refine energies and pairing contributions, while rotational properties are assessed via the cranking framework and moment-of-inertia calculations. The key findings show that axial hexadecapole deformation (β4) lowers the total energy by about 1–3 MeV and yields nonzero β4 in equilibrium, with β4 strongly influencing single-particle spectra and shell gaps through Δl=4 couplings; the moment of inertia trends from HFBC closely follow those from rigid-body estimates, highlighting a useful link between complex microscopic structure and simpler geometric models. These results enhance understanding of static and dynamic hexadecapole effects on nuclear rotational properties and offer practical guidance for interpreting experiments and informing reaction and dinuclear-system calculations.

Abstract

For even-even nuclei $^{180-184}$Yb, $^{182-186}$Hf and $^{184-188}$W located on an island of hexadecapole-deformation archipelago, the structure properties, especially under rotation, are reinvestigated by using the Hartree-Fock-Bogliubov-Cranking (HFBC) calculation with a fixed shape (e.g., the ground-state equilibrium shape). The equilibrium deformations, extracted from the potential energy surface, are calculated based on the phenomenological Woods-Saxon mean-field Hamiltonian within the framework of macroscopic-microscopic (MM) model. The impact of different deformation degrees of freedom on, e.g., single-particle levels, total energy, and moment of inertia, is revealed, especially concentrating on the hexadecapole-deformation effects and the quadrupole-hexadecapole coupling. Considering the axially hexadecapole deformation, the present calculations can well reproduce available experimental data, including the quadrupole deformations and moments of inertia. Interestingly, it is found that the impact of different deformation degrees of freedom on moment of inertia exhibits a similar trend in the HFBC and rigid-body calculations though the latter ignores the pairing effects. Before starting or constructing a complex theory-model, to some extent, such a similarity can provide an alternative way of understanding the effect of, e.g., exotic deformations, on moment of inertia by the calculation of a simple rigid-body approximation. The present findings could offer insights into the static and dynamic effects of hexadecapole deformations, contributing valuable information for the corresponding research in nuclear structure and reaction.

Figures (12)

• Figure 1: Illustrations of nuclear surfaces, defined by Eq. (\ref{['eqn.07']}), for single deformation parameter $\alpha_{20}= +0.3$ (a), $\alpha_{20}= -0.3$ (a$^\prime$), $\alpha_{22} = +0.3$ (b), and $\alpha_{22}= -0.3$ (b$^\prime$).
• Figure 3: Projections of total energy on the ($\beta_2,\gamma$) (a), ($\beta_4,\gamma$) (b) and ($\beta_2,\beta_4$) (c) planes with contour-line separations of 0.5 MeV, minimized respectively at each deformation point over the remaining deformation, $\beta_4$, $\beta_2$ and $\gamma$, for the central nucleus $^{184}$Hf. Note that, for each subfigure, the energy normalization is specified by setting the minimum to zero at the equilibrium shape. See the text for more details.
• Figure 4: Similar to the preceding illustration in Fig. \ref{['Fig03']}, but projected on the $(\alpha_{20},\alpha_{4\mu=0,2,4})$ planes for $^{184}$Hf.
• Figure 5: Calculated proton (a, b) and neutron (c, d) single-particle energies as functions of the quadrupole deformation $\beta_2$ (a, c) and hexadecapole deformation $\beta_4$ (b, d) for the central nucleus $^{184}_{72}$Hf$_{112}$, focusing on the window of interest near the Fermi surface. Red solid (blue dotted) lines refer to positive and negative parity. In (a) and (c), the single-particle orbitals at $\beta_2 = 0.0$ are labelled by the spherical quantum numbers $nlj$ and the calculations extend to the equilibrium deformation ($\beta_2 = 0.237$), for further details, e.g. see Table \ref{['tab2']}. In (b) and (d), the deformation $\beta_2$ is always set to the equilibrium value.
• Figure 6: Calculated total energy curves as function of the quadrupole deformation $\beta_2$ for nine selected even-even nuclei $^{180-184}$Yb, $^{182-186}$Hf and $^{184-188}$W. Note that, at each deformation point $\beta_2$, the energy is minimized over the triaxial deformation $\gamma$ and the hexadecapole deformation $\beta_4$ if $\beta_4$ is included (e.g., the green line). For more details, see the text.