Microscopic analysis of the octupole phase transition in Th isotopes

A shape phase transition between stable octupole deformation and octupole vibrations in Th nuclei is analyzed in a microscopic framework based on nuclear density functional theory. The relativistic density dependent point coupling (DD-PC1) functional is used to calculate axially symmetric quadrupole-octupole constrained energy surfaces. Observables related to order parameters are computed using an interacting-boson Hamiltonian, with parameters determined by mapping the microscopic energy surfaces to the expectation value of the Hamiltonian in the boson condensate. The systematics of constrained energy surfaces and low-energy excitation spectra point to the occurrence of a phase transition between octupole-deformed shapes and shapes characterized by octupole-soft potentials.

Figure 1

Microscopic DD-PC1 self-consistent relativistic Hartree-Bogoliubov (RHB) axially symmetric energy surfaces of the nuclei ${}^{222-232}$Th in the $({\beta }_2,{\beta }_3)$ plane, symmetric with respect to ${\beta }_3=0$ axis. The contours join points on the surface with the same energy, and the separation between neighboring contours is 1 MeV. In each panel, the minimum is indicated by a open diamond.

Figure 2

Variation of the derived parameters ${\epsilon }_d$, ${\epsilon }_f$, ${\chi }_d$, ${\chi }_f$, and ${\chi }_3$ as functions of the mass number.

Figure 3

Isotopic dependence of the excitation energies of levels of the positive-parity ground-state band ($K^{\pi }=0_1^{+}$) (a), and the lowest negative-parity band ($K^{\pi }=0_1^{-}$) (b), for ${}^{220-232}$Th. In each panel lines and symbols denote the theoretical and the experimental [37] values, respectively.

Figure 4

Theoretical (a) and experimental [37] (b) energy ratios $E\left(J\right)/E\left(2_1^{+}\right)$ of the yrast states of ${}^{220-232}$Th, including both positive ($J$ even) and negative ($J$ odd) parity, as functions of the angular momentum $J$.

Figure 5

Experimental [37, 38] and calculated yrast states of positive and negative parity in ${}^{230}$Th. The in-band $B$(E2) values (dotted) and the $B(E3;3_1^{-}\to 0_1^{+})$ (solid) (both in Weisskopf units), and the branching ratio $B(E1;1_1^{-}\to 2_1^{+})/B(E1;1_1^{-}\to 0_1^{+})$ (dashed-dotted) are also shown.