Evolution of ground-state quadrupole and octupole stiffnesses in even-even barium isotopes

Quadrupole and octupole stiffnesses in the ground states of even-even ${}^{112-150}\mathrm{Ba}$ isotopes have been systematically investigated by means of potential-energy-surface calculations. The calculations are carried out in both $\left({\beta }_2,\gamma ,{\beta }_4\right)$ and (${\beta }_2,{\beta }_3,{\beta }_4,{\beta }_5$) deformation spaces with the inclusion of triaxial and reflection-asymmetric shape degrees of freedom, respectively. The present results are compared with previous calculations and available experiments. The shape instabilities are evaluated by analyzing the potential energy curves with respect to both the quadrupole and octupole deformations, which is consistent with the previous discussions predicting the $\gamma$ softness or triaxiality and octupole instability. In addition, taking the near-drip-line ${}^{114}\mathrm{Ba}$ nucleus as an example, we briefly investigate the effects of potential parameters (e.g., the strength of the spin-orbit potential $\lambda$, and the nuclear surface diffuseness $a$) on the deformation energy curve, showing almost negligible modifications of nuclear shape and stiffness but considerable changes in the depth of the minimum and the height of the fission barrier (which may be very important for the study of heavy and superheavy nuclei).

Figure 1

Calculated ground-state ${\beta }_2$ (a), $\gamma$ (b), and ${\beta }_3$ (c) deformations for even-even ${}^{112-150}\mathrm{Ba}$ nuclei, compared with the FY+FRDM calculations [39], the Grodzins estimate [63, 64], and the partial experimental values [62]. Cal.1 and Cal.2 denote the calculations are performed in $\left({\beta }_2,\gamma ,{\beta }_4\right)$ and (${\beta }_2,{\beta }_3,{\beta }_4,{\beta }_5$) deformation spaces, respectively.

Figure 2

Deformation energy curves against ${\beta }_2$ (the upper panel) and $\gamma$ (the lower panel) for even-even ${}^{112-150}\mathrm{Ba}$ nuclei, calculated in $\left({\beta }_2,\gamma ,{\beta }_4\right)$ deformation space. At each ${\beta }_2 \left(\gamma \right)$ point, the energy has been minimized with respect to the $\gamma \left({\beta }_2\right)$ and ${\beta }_4$ deformations.

Figure 3

Similar to Fig. 2 but against ${\beta }_2$ (the upper panel) and ${\beta }_3$ (the lower panel) deformations, calculated in (${\beta }_2,{\beta }_3,{\beta }_4,{\beta }_5$) deformation space.

Figure 4

Energies of the first excited $2_1^{+},4_1^{+},3_1^{-}$ and the second $2_2^{+}$ states (a), the ratio $R_{4/2}$ (b), the quantity $E_S/E_{2^{+}}$ (c) and the energy difference $\delta E\left(3\right)$ (d) in even-even Ba isotopes as a function of the neutron (mass) number. See text for more details. The available experimental data are taken from Ref. [75].

Figure 5

Calculated stiffness coefficients $C_2$ (a), $C_{\gamma }$ (b), and $C_3$ (c) towards ${\beta }_2,\gamma ,$ and ${\beta }_3$ deformations for even-even ${}^{112-150}\mathrm{Ba}$ nuclei. The square and circle symbols indicate that the calculations are performed in $\left({\beta }_2,\gamma ,{\beta }_4\right)$ and (${\beta }_2,{\beta }_3,{\beta }_4,{\beta }_5$) deformation spaces, respectively.

Figure 6

(a) Calculated neutron and proton Fermi energy levels for even-even ${}^{112-150}\mathrm{Ba}$ nuclei with the calculated equilibrium shapes as input quantities [90]. Note that the filled and open symbols indicate that the calculations are performed in $\left({\beta }_2,\gamma ,{\beta }_4\right)$ and (${\beta }_2,{\beta }_3,{\beta }_4,{\beta }_5$) deformation spaces, respectively. (b) Available two-proton and two-neutron separation energies for the Ba isotopic chain. (c) Available half-lives of known Ba isotopes [75], indicating the nuclear instabilities. Note that the half-life for ${}^{132}\mathrm{Ba}$ is greater than $10{}^{28}$ s.

Figure 7

Calculated LN pairing gaps of proton and neutron for even-even ${}^{112-150}\mathrm{Ba}$ nuclei, compared with the FY+FRDM calculations [90] and experiments. The insets indicate the contributions of the pairing gap $\mathrm{\Delta }$ and the number-fluctuation constant ${\lambda }_2$ to the LN pairing gap ${\mathrm{\Delta }}_{\mathrm{LN}}$. See text for more details.

Figure 8

Calculated proton (a) and neutron (b) single-particle levels near their Fermi surfaces for neutron-deficient ${}^{114}\mathrm{Ba}$ based on different set of WS parameters [58]. The red levels connected by dotted lines denote the Fermi levels.

Figure 9

Calculated proton single-particle levels for ${}^{114}\mathrm{Ba}$ as functions of quadrupole deformation ${\beta }_2$. Part (a) shows results for three selected surface diffuseness $a$ values (0.67, 0.70, and 0.73 fm) and all other potential parameters are identical with those of universal values [58]. Part (b) is similar to (a) but for different spin-orbit strengths $\lambda$ (34.0, 36.0, and 38.0).

Figure 10

Similar to Fig. 2 but for different parameter combinations of the surface diffuseness $a$ and spin-orbit strength $\lambda$ for the selected ${}^{114}\mathrm{Ba}$ nucleus. All other potential parameters are identical with those of universal values [58].