Quasi-SU(3) coupling of ($1h_{11/2},2f_{7/2}$) across the $N=82$ shell gap: Enhanced $E2$ collectivity and shape evolution in Nd isotopes

The essence of the quasi-SU(3) coupling scheme suggested by Zuker et al. is that the high-$j$ intruder orbit alone is insufficient to induce required large collectivity, and it is necessary for the high-$j$ orbit to correlate through the quadrupole interaction with another higher-lying orbit with $\mathrm{\Delta }j=2$. To extend this idea to the medium-heavy mass region, we investigate the systematics of energy levels and $B\left(E2\right)$ values for Sn, Te, Xe, Ba, Ce, Nd, and Sm isotopes by applying the recently proposed realistic PMMU shell model. The calculations are performed by using the projected Hartree-Fock-Bogolyubov plus generator coordinate method in the model space of ($1g_{9/2},1g_{7/2},2d_{5/2},2d_{3/2},3s_{1/2},1h_{11/2}$). The calculations describe well the experimental data over a wide range of nuclei. However, it is found that, in some nuclei close to the neutron midshell, ${}^{124,126}\mathrm{Ce}$, ${}^{130,132}\mathrm{Nd}$, and ${}^{134}\mathrm{Sm}$, the calculated $B(E2;2_1^{+}\to 0_1^{+})$ values underestimate the data considerably. This problem can be resolved by inclusion of the $2f_{7/2}$ orbit into the model space, which, with $1h_{11/2}$, forms a quasi-SU(3) coupling scheme. It is shown that the $QQ$ interaction between the SU(3)-partner $1h_{11/2}\text{−}2f_{7/2}$ is a driving force for enhanced $E2$ collectivity in the above nuclei and is responsible for the shape evolution in Nd isotopes. In addition, the participation of the $2f_{7/2}$ orbit in the model space favors prolate shape in the ground state of $N\le 76$ isotopes, thus subverting the oblate result from the same calculation but without $2f_{7/2}$.

Figure 1

Comparison between HFB $+$ gcm and exact shell-model calculation using the PMMU model for ${}^{56}\mathrm{Ni}$.

Figure 2

Energy levels and $B\left(E2\right)$ values for Sn, Te, Xe, Ba, Ce, Nd, and Sm isotopes. (a) The upper graph shows $B\left(E2\right)$ values for Sn isotopes. Experimental data are indicated by different symbols: ORNL [7, 8, 13, 18], MSU [10, 15], RIKEN [19], ISOLDE [9, 11], GSI [12, 16, 17, 20], GSIANU [14], IUAC [22]. The middle graph shows $E2$ matrix elements for protons and neutrons. The lower graph displays energy levels for the $2_1^{+}$ and $4_1^{+}$ states. Panels (b) and (c) denote energy levels for $2_1^{+}$ and $4_1^{+}$ states and $B(E2;2_1^{+}\to 0_1^{+})$ values, respectively, for Te, Xe, Ba, Ce, Nd, and Sm isotopes. Experimental data are taken from Refs. [40, 41].

Figure 3

(a) Comparison between calculated energy levels in the $gdshf$ model space and experiments for Nd isotopes. (b) Comparison between calculated $B(E2;2_1^{+}\to 0_1^{+})$ values and experiments for Nd isotopes. The $gdsh$ and $gdshf$ calculations with the effective charges $e_p=1.4e$ and $e_n=0.6e$ are denoted by red and broken curves, respectively. The blue curve indicates the $gdshf$ calculation with the effective charges $e_p=1.3e$ and $e_n=0.5e$. (c) Calculated spectroscopic quadrupole moments. Results of the $gdsh$ ($gdshf$) calculations are indicated by red (blue) curves. (d) Extra occupation numbers $n_{h11/2+f7/2}$ for protons and neutrons. See text for discussion. Panels (e)–(g) display the calculated PESs for ${}^{130}\mathrm{Nd}$, ${}^{138}\mathrm{Nd}$, and ${}^{142}\mathrm{Nd}$, respectively. In the figures, experimental data are taken from Ref. [40].

Figure 4

Effects of the $QQ$ component between $1h_{11/2}$ and $2f_{7/2}$ orbits on (a) $B\left(E2\right)$ values, and (b) extra occupancies $n_{h11/2+f7/2}$ in ${}^{130}\mathrm{Nd}$ (see text for details).

Figure 5

Excitation energies as functions of $I(I+1)$ for ${}^{130}\mathrm{Nd}$, ${}^{138}\mathrm{Nd}$, and ${}^{142}\mathrm{Nd}$. Data [40] are shown in solid and calculation in open symbols. For ${}^{142}\mathrm{Nd}$, the calculated points with the $gdsh$ ($gdshf$) space are connected by dashed (solid) lines. The inset compares the calculated energy spectrum with experiment for ${}^{138}\mathrm{Nd}$.