Shape transitions in exotic Si and S isotopes and tensor-force-driven Jahn-Teller effect

We show how shape transitions in the neutron-rich exotic Si and S isotopes occur in terms of shell-model calculations with a newly constructed Hamiltonian based on $V_{\mathrm{MU}}$ interaction. We first compare the calculated spectroscopic-strength distributions for the proton $0d_{5/2,3/2}$ and $1s_{1/2}$ orbitals with results extracted from a ${}^{48}\mathrm{Ca}(e,e^{\prime }p)$ experiment to show the importance of the tensor-force component of the Hamiltonian. Detailed calculations for the excitation energies, $B\left(E2\right)$, and two-neutron separation energies for the Si and S isotopes show excellent agreement with experimental data. The potential-energy surface exhibits rapid shape transitions along the isotopic chains towards $N=28$ that are different for Si and S. We explain the results in terms of an intuitive picture by involving a Jahn-Teller-type effect that is sensitive to the tensor-force-driven shell evolution. The closed subshell nucleus ${}^{42}\mathrm{Si}$ is a particularly good example of how the tensor-force-driven Jahn-Teller mechanism leads to a strong oblate rather than a spherical shape.

Figure 1

Spectroscopic factors of proton-hole states measured by ${}^{48}\mathrm{Ca}(e,e^{\prime }p)$ [9] (upper) and its theoretical calculation (lower left). The cross-shell tensor force is removed in the lower right panel. The black, blue, and red bars correspond to $1s_{1/2},0d_{3/2}$, and $0d_{5/2}$ states, respectively.

Figure 2

Intuitive illustration of the structure of the intrinsic state at $Z=14$. Single-particle states of magnetic quantum numbers, denoted as $m$, are shown. $q$ implies the intrinsic quadrupole moment (fm${}^2$) obtained with harmonic oscillator $\hslash \omega =11.8\mathrm{MeV}$.

Figure 3

(a) and (b) $2_{1,2}^{+}$ (blue lines and red circles) and $4_1^{+}$ (green lines and red triangles) energy levels and (c) and (d) $B(E2;0_1^{+}\to 2_1^{+})$ values of Si and S isotopes for $N=22–28$. Symbols are experimental data [13, 14, 15, 16, 17, 18, 19]. Solid (dashed) lines are calculations with (without) the cross-shell tensor force.

Figure 4

Potential-energy surfaces of Si isotopes for $\gamma =0\sim {60}^{\circ }$ from $N=22$ to 28 calculated with (left) and without (right) the cross-shell tensor force. The energy minima are indicated by red circles.

Figure 5

Potential-energy surfaces of S isotopes from $N=22$ to 28. See the caption of Fig. 4.

Figure 6

Two-neutron separation energies of Si and S isotopes from $N=22$ to 28. Solid (dashed) lines are calculations with (without) the cross-shell tensor force. Points are experimental data [40, 41].