Microscopic description of spherical to $\mathit{\gamma }$-soft shape transitions in Ba and Xe nuclei

The rapid transition between spherical and $\gamma$-soft shapes in Ba and Xe nuclei in the mass region $A⩾130$ is analyzed using excitation spectra and collective wave functions obtained by diagonalization of a five-dimensional Hamiltonian for quadrupole vibrational and rotational degrees of freedom, with parameters determined by constrained self-consistent relativistic mean-field calculations for triaxial shapes. The results reproduce the characteristic evolution of excitation spectra and E2 transition probabilities, and in general, a good agreement with available data is obtained. The calculated spectra display fingerprints of a second-order shape phase transition that can approximately be described by analytic solutions corresponding to the E(5) dynamical symmetry.

Figure 1

Self-consistent $\mathrm{RMF}+\mathrm{BCS}$ triaxial quadrupole binding energy maps of the even-even ${}^{136–130}\mathrm{Ba}$ isotopes in the $\beta$-$\gamma$ plane ($0⩽\gamma ⩽{60}^{°}$). All energies are normalized with respect to the binding energy of the absolute minimum; the contours join points on the surface with the same energy (in megaelectronvolts). For each nucleus in the corresponding inset, we plot the axial projection of the binding energy for oblate (negative-$\beta$) and prolate (positive-$\beta$) deformations.

Figure 2

Same as Fig. 1, but for the isotopes ${}^{134–128}\mathrm{Xe}$.

Figure 3

Self-consistent $\mathrm{RMF}+\mathrm{BCS}$ binding energy curves of the ${}^{134}\mathrm{Ba}$ nucleus, as functions of the deformation parameter $\gamma$, for four values of axial deformation, $\beta =0.05$, $0.1$, $0.15$, and $0.2$.

Figure 4

Same as Fig. 3, but for the ${}^{132}\mathrm{Xe}$ nucleus.

Figure 5

Evolution of the characteristic collective observables $R_{4/2}$ and $B$(E2; $2_1^{+}\to 0_1^{+}$) (in Weisskopf units) with mass number in Ba isotopes. Microscopic values calculated with the PC-F1 energy density functional are shown in comparison with data [24, 25].

Figure 6

Same as Fig. 5, but for the isotopes ${}^{128–134}\mathrm{Xe}$.

Figure 7

Fluctuations of the quadrupole deformations $\Delta \beta /\langle \beta \rangle$ and $\Delta \gamma /\langle \gamma \rangle$ for the sequence of ground states of Ba isotopes.

Figure 8

Same as Fig. 7, but for the isotopes of Xe.

Figure 9

The low-energy spectrum of ${}^{134}\mathrm{Ba}$ calculated with the PC-F1 relativistic density functional (middle), compared with the data (left), and the $E$(5) symmetry predictions (right) for the excitation energies and intraband and interband $B$(E2) values (in Weisskopf units). Theoretical spectra are normalized to the experimental energy of state $2_1^{+}$, and $E$(5) transition strengths are normalized to the experimental $B$(E2; $2_1^{+}\to 0_1^{+}$).

Figure 10

Probability densities, Eq. (15), in the $\beta$-$\gamma$ plane for states $2_1^{+}$, $2_2^{+}$, and $4_1^{+}$ of ${}^{134}\mathrm{Ba}$.

Figure 11

Same as Fig. 9, but for the ${}^{132}\mathrm{Xe}$ nucleus.

Figure 12

Same as Fig. 9 but for the ${}^{130}\mathrm{Xe}$ nucleus.

Figure 13

Experimental low-energy spectrum of ${}^{128}\mathrm{Xe}$ (left), compared with the level scheme and decay pattern predicted by the solution of the microscopic collective Hamiltonian with the PC-F1 relativistic density functional.