Microscopic description of octupole shape-phase transitions in light actinide and rare-earth nuclei

A systematic analysis of low-lying quadrupole and octupole collective states is presented based on the microscopic energy density functional framework. By mapping the deformation constrained self-consistent axially symmetric mean-field energy surfaces onto the equivalent Hamiltonian of the $sdf$ interacting boson model (IBM), that is, onto the energy expectation value in the boson condensate state, the Hamiltonian parameters are determined. The study is based on the global relativistic energy density functional DD-PC1. The resulting IBM Hamiltonian is used to calculate excitation spectra and transition rates for the positive- and negative-parity collective states in four isotopic chains characteristic for two regions of octupole deformation and collectivity: Th, Ra, Sm, and Ba. Consistent with the empirical trend, the microscopic calculation based on the systematics of ${\beta }_2$-${\beta }_3$ energy maps, the resulting low-lying negative-parity bands and transition rates show evidence of a shape transition between stable octupole deformation and octupole vibrations characteristic for ${\beta }_3$-soft potentials.

Figure 1

Axially symmetric energy surfaces of the isotopes ${}^{222–232}$Th in the $({\beta }_2,{\beta }_3)$ plane, calculated using the self-consistent RHB model with the microscopic functional DD-PC1. The contours join points on the surface with the same energy (in MeV), and the color scale varies in steps of 0.2 MeV. The energy difference between neighboring contours is 1 MeV. Note that energy surfaces are symmetric with respect to the ${\beta }_3=0$ axis. Open circles denote the absolute energy minima.

Figure 2

Same as the caption to Fig. 1, but for ${}^{218–228}$Ra.

Figure 3

Same as the caption to Fig. 1, but for ${}^{146–156}$Sm.

Figure 4

Same as the caption to Fig. 1, but for ${}^{140–150}$Ba.

Figure 5

Same as the caption to Fig. 1, but for the mapped IBM energy surfaces of ${}^{222–232}$Th.

Figure 6

Same as the caption to Fig. 1, but for the mapped IBM energy surfaces of ${}^{218–228}$Ra.

Figure 7

Same as the caption to Fig. 1, but for the mapped IBM energy surfaces of ${}^{146–156}$Sm.

Figure 8

Same as the caption to Fig. 1, but for the mapped IBM energy surfaces of ${}^{140–150}$Ba.

Figure 9

Mean value of the octupole deformation parameter $\langle {\beta }_3\rangle$ for ${}^{220–232}$Th, ${}^{218–228}$Ra, ${}^{146–156}$Sm, and ${}^{140–150}$Ba, computed with the IBM intrinsic state (a), and the corresponding variance $\Delta {\beta }_3=\sqrt{\langle {\beta }_3^2\rangle -{\langle {\beta }_3\rangle }^2}$ (b), as functions of the boson number $N_B$.

Figure 10

Total boson number $N_B$ dependence of the parameters for the $sdf$ IBM Hamiltonian Eq. (2): ${\epsilon }_d$ (a), ${\epsilon }_f$ (b), ${\chi }_d$ (b), ${\chi }_f$ (c), ${\chi }_f$ (d), and ${\chi }_3$ (e), and the coefficient of the shape variables $C_2$ (f) and $C_3$ (g), determined by mapping the microscopic RHB energy surfaces onto the corresponding expectation values of the IBM Hamiltonian. Solid, dotted, dashed, and dot-dashed lines connect the parameters of the Th, Ra, Sm, and Ba isotopic chains, respectively.

Figure 11

Excitation energies of low-lying yrast positive-parity collective states of ${}^{220–232}$Th, ${}^{218–228}$Ra, ${}^{146–156}$Sm, and ${}^{140–150}$Ba, as functions of the mass number. In each panel lines and symbols denote the theoretical and experimental [64] values, respectively. Figure legends are found in (a).

Figure 12

Same as in the caption to Fig. 11 but for the negative-parity states.

Figure 13

Theoretical and experimental [64] energy ratios $E\left(J^{\pi }\right)/E\left(2_1^{+}\right)$ for states of the positive-parity ground-state band ($J$ even) and the lowest negative-parity band ($J$ odd), as functions of the angular momentum $J$, for ${}^{220–232}$Th (a), (b), ${}^{218–228}$Ra (c), (d), ${}^{146–156}$Sm (e), (f), and ${}^{140–150}$Ba (g), (h).

Figure 14

The energy displacement $\delta E\left(J\right)$ [defined in Eq. (14)] normalized with respect to the excitation energy of $2_1^{+}$, as a function of the mass number. The limit of stable octupole deformation $\delta E\left(J\right)=0$ is indicated by the dotted horizontal line. Experimental data, represented symbols in each panel, are taken from Ref. [64].

Figure 15

Isotopic dependence of the $B$($E3;3_1^{-}\to 0_1^{+}$) [(a) to (d)] and $B$($E1;1_1^{-}\to 0_1^{+}$) [(e) to (h)] values for ${}^{220–232}$Th, ${}^{218–228}$Ra, ${}^{146–156}$Sm, and ${}^{140–150}$Ba. The vertical axes for (e)–(h) are in logarithmic scale. Solid lines connect calculated values, symbols denote data taken from Refs. [4, 5, 6, 64, 67, 68, 69].

Figure 16

Partial level scheme of ${}^{226}$Th. The theoretical low-lying spectra, in-band $B$($E2$) values (in Weisskopf units, dotted arrows), and interband $B$($E1$) values (in units of $e^2$b$\times {10}^{-5}$, dashed arrows), are compared to available data [4, 5, 64].

Figure 17

Partial level scheme of ${}^{230}$Th. The theoretical low-lying spectra and in-band $B$($E2$) values (in Weisskopf units, dotted arrows) are compared to available data [64].

Figure 18

Comparison between the experimental [6] and calculated quadrupole and octupole intrinsic moments of ${}^{224}$Ra, as functions of the angular momentum $J^{\pi }$ ($\pi =+1$ and $-1$ for $J$ even and odd, respectively).