In two recent articles we have formulated nuclear mean-field theory predictions of existence of a new form of magic numbers, referred to as fourfold magic numbers. These predictions stipulate the presence of strong shell closures at the neutron numbers $N=136$ (actinide region) and $N=196$ (superheavy region) simultaneously at nonvanishing all four octupole deformations ${\ensuremath{\alpha}}_{3\ensuremath{\mu}=0,1,2,3}\ensuremath{\ne}0$. In contrast to the traditional notion of magic numbers, the new notion refers to simultaneous nonspherical configurations (${\ensuremath{\alpha}}_{3\ensuremath{\mu}}\ensuremath{\ne}0, {\ensuremath{\alpha}}_{2\ensuremath{\mu}}=0$). In this article we study the nuclear equilibrium deformations with ${\ensuremath{\alpha}}_{33}\ensuremath{\ne}0$ combined with nonvanishing quadrupole deformation, ${\ensuremath{\alpha}}_{20}\ensuremath{\ne}0$. One easily shows that such geometrical shapes have a threefold symmetry axis and are invariant under the symmetry operations of the ${\mathrm{D}}_{3h}$ point group. We employ a realistic phenomenological mean-field approach with the so-called universal deformed Woods-Saxon potential and its recently optimized parametrization based on actualized experimental data with the help of the inverse problem theory methods. The presence of parametric correlations among 4 of 12 parameters in total was detected and removed employing Monte Carlo approach leading to stabilization of the modeling predictions. Our calculations predict the presence of three nonoverlapping groups of nuclei with ${\mathrm{D}}_{3h}$ symmetry, referred to as islands on the nuclear ($Z,N$) plane (mass table). These islands lie in the rectangle $110\ensuremath{\le}Z\ensuremath{\le}138$ and $166\ensuremath{\le}N\ensuremath{\le}206$. The ``repetitive'' structures with the ${\mathrm{D}}_{3h}$ symmetry minima are grouped in three zones of oblate quadrupole deformation, approximately, at ${\ensuremath{\alpha}}_{20}\ensuremath{\in}[\ensuremath{-}0.10,\ensuremath{-}0.20]$ (oblate normal deformed), around ${\ensuremath{\alpha}}_{20}\ensuremath{\approx}\ensuremath{-}0.5$ (oblate superdeformed) and ${\ensuremath{\alpha}}_{20}\ensuremath{\approx}\ensuremath{-}0.85$ (oblate hyperdeformed). Importantly, the energies of those latter exotic deformation minima are predicted to be very close to the ground-state energies. We illustrate, compare, and discuss the evolution of the underlying shell structures. Nuclear surfaces parametrized as usual with the help of real deformation parameters, ${{\ensuremath{\alpha}}_{\ensuremath{\lambda}\ensuremath{\mu}}^{}={\ensuremath{\alpha}}_{\ensuremath{\lambda}\ensuremath{\mu}}^{*}}$, are invariant under ${\mathcal{O}}_{xz}$-plane reflection, the symmetry also referred to as $y$ simplex (${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}_{y}$). For the shapes with odd-multipolarity ($\ensuremath{\lambda}\ensuremath{\rightarrow}{\ensuremath{\lambda}}_{\mathrm{odd}}=3,5,7,...$) it follows that $E(\ensuremath{-}{\ensuremath{\alpha}}_{{\ensuremath{\lambda}}_{\mathrm{odd}},\ensuremath{\mu}})=E(+{\ensuremath{\alpha}}_{{\ensuremath{\lambda}}_{\mathrm{odd}},\ensuremath{\mu}})$. It turns out that the predicted equilibrium deformations generate symmetric double (or ``twin'') minima separated by potential barriers, whose heights vary with the nucleon numbers, possibly inducing the presence of parity-doublets in the spectra. To facilitate possible experimental identification of such structures, we examine the appearance of such doublets solving the collective Schr\"odinger equation. Implied suggestions are illustrated and discussed.
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Open Access
Jun 24, 2024
Jeremy Lantis + 47
