Prolate-shape dominance and dual-shell mechanism

The dominance of prolate over oblate deformations in the ground states of deformed even-even nuclei has long been known empirically. Although various theoretical approaches have been applied, the underlying mechanism is not yet considered as being adequately understood. One of those theoretical approaches has found an interesting interference effect between the spin-orbit coupling and the surface diffuseness in the single-particle potential. Recently the proxy-SU(3) model claimed that the prolate dominance would emerge naturally from the SU(3) symmetry assuming the highest-weight irreducible representations as ground-state configurations. This article presents a possible interpretation of the interference effect in the former approach in the context of the latter approach, more specifically invoking the dual-shell mechanism that includes both the three-dimensional harmonic-oscillator shells and the spin-orbit-like shells.

Figure 1

An example of triangular array for the 3D HO shell model. Here $N_{\mathrm{sh}}=5$ is assumed. The single-particle levels within a single shell are labeled by three quantum numbers ($n_z$, $n_x$, $n_y$) that are the oscillator numbers in the three Cartesian axes. These numbers are specified only for the top two rows in the figure. The actions of U(3) raising operators are drawn with solid arrows. The dotted arrows show the filling order to obtain the hw irreps.

Figure 2

The intruders for $N_{\mathrm{sh}}=5$ displayed on a triangular array similar to Fig. 1. The arrows put on both ends of each row denote spin directions. The single-particle levels are now labeled by the Nilsson asymptotic quantum numbers $K[N_{\mathrm{sh}},n_z,\mathrm{\Lambda },\mathrm{\Sigma }]$. These numbers are specified only for the leftmost and rightmost of the top three rows in the figure. The leftmost with the downward arrow and the rightmost with the upward arrow of each row are the intruders that would invade the shell below because of the spin-orbit interaction.

Figure 3

The Elliott labels for the SU(3) hw irreps as a function of the nucleon number $N$. The ordinate is $\lambda +\mu$. The solid line and the dashed line are for the 3D HO shells and the SO-like shells, respectively.

Figure 4

The proportion of prolate nuclei as a function of the parameter $\alpha$.

Figure 5

The distribution of ${\tilde{\beta }}_2$ over the plane spanned by the neutron number $N$ and the proton number $Z$. The graphs for $\alpha =0$, $\alpha =0.45$, and $\alpha =1$ are panels (a), (b), and (c), respectively.