Quadrupole-octupole coupling and the onset of octupole deformation in actinides

The evolution of quadrupole and octupole collectivity and their coupling is investigated in a series of even-even isotopes of the actinides Ra, Th, U, Pu, Cm, and Cf with neutron number in the interval $130⩽N⩽150$. The Hartree-Fock-Bogoliubov approximation, based on the parametrization D1M of the Gogny energy density functional, is employed to generate potential energy surfaces depending upon the axially symmetric quadrupole and octupole shape degrees of freedom. The mean-field energy surface is then mapped onto the expectation value of the $sdf$ interacting-boson-model Hamiltonian in the boson condensate state as to determine the strength parameters of the boson Hamiltonian. Spectroscopic properties related to the octupole degree of freedom are produced by diagonalizing the mapped Hamiltonian. Calculated low-energy negative-parity spectra, $B(E3;3_1^{-}\to 0_1^{+})$ reduced transition rates, and effective octupole deformation suggest that the transition from nearly spherical to stable octupole-deformed, and to octupole vibrational states occurs systematically in the actinide region.

Figure 1

SCMF-PESs computed with the Gogny-D1M EDF for the nuclei ${}^{224\text{–}234}\mathrm{Pu}$ (left), and the corresponding mapped $sdf$-IBM PESs (right). The color code indicates the total HFB and IBM energies in MeV units, plotted up to 10 MeV with respect to the global minimum. The energy difference between neighboring contours is 1 MeV. For each nucleus, the global minimum is indicated by an open circle.

Figure 2

Same as for Fig. 1, but for ${}^{248\text{–}238}\mathrm{Cf}$.

Figure 3

The (a) ${\beta }_{2,\min }$ and (b) ${\beta }_{3,\min }$ values, corresponding to the ground-state minimum, are plotted as a function of neutron number. (c) The deformation energy $\mathrm{\Delta }{E}_{\mathrm{def}}$, defined as the energy difference between the global minimum and the spherical configuration, is plotted as a function of neutron number. (d) The octupole deformation energy $\mathrm{\Delta }{E}_{\mathrm{oct}}$, defined as the energy difference between the global minimum and the quadrupole deformed minimum along the ${\beta }_3=0$ axis, for the mapped IBM-PESs is plotted. (e)–(h) The corresponding quantities for the SCMF-PESs are also plotted on the right-hand side. The results for the Ra and Th nuclei are taken from Ref. [45]. See the main text for definitions of the above quantities.

Figure 4

The strength parameters (a) ${\epsilon }_d$, (b) ${\epsilon }_f$, (c) ${\kappa }_2$, (d) ${\chi }_f$, (e) ${\chi }_d$, (f) ${\kappa }_3$, (g) $\rho$, and (h) ${\chi }_3$ of the $sdf$-IBM Hamiltonian, Eq. (1), are plotted as a function of neutron number for the four isotopic chains considered. The coefficients (i) $C_2$ and (j) $C_3$ connecting the IBM and microscopic quadrupole and octupole deformation parameters are plotted. Finally, the boson effective charges for the (k) quadrupole $e_2$ and (l) octupole $e_3$ transitions are plotted as a function of the neutron number. The parameters for Ra and Th isotopes can be found in Ref. [45].

Figure 5

Low-energy excitation spectra of positive-parity even-spin yrast states of ${}^{222\text{–}242}\mathrm{U}$, ${}^{224\text{–}244}\mathrm{Pu}$, ${}^{226\text{–}246}\mathrm{Cm}$, and ${}^{228\text{–}248}\mathrm{Cf}$ computed by the diagonalization of the mapped $sdf$-IBM Hamiltonian, Eq. (1). Experimental data are taken from Ref. [63]. Results for Ra and Th isotopes can be found in Ref. [45].

Figure 6

Same as the caption of Fig. 5, but for the odd-spin negative-parity yrast states.

Figure 7

Expectation values of the $f$-boson number operator $\langle \widehat{n_f}\rangle$ in the IBM wave functions of the states $0_1^{+}$, $2_1^{+}$, $0_2^{+}$, $2_2^{+}$, $1_1^{-}$, and $3_1^{-}$ of the Ra, Th, U, Pu, Cm, and Cf nuclei, plotted as functions of the neutron number. The values for the Ra and Th isotopes are taken from Ref. [45].

Figure 8

The energy displacement $\delta E\left(J^{-}\right)$ [defined in Eq. (8)], normalized with respect to the excitation energy of the $2_1^{+}$ state, is shown as a function of the neutron number. The theoretical values are connected by lines. The corresponding experimental [63] values for the $J^{-}=1^{-}$, $3^{-}$, $5^{-}$, $7^{-}$, and $9^{-}$ yrast states are represented by the solid circles, squares, diamonds, triangles, and stars, respectively. Results for the Ra and Th nuclei are from Ref. [45]. The limit of stable octupole deformation $\delta E\left(J^{-}\right)=0$ is indicated in each panel by a broken horizontal line.

Figure 9

Evolution of the (a) theoretical and (b) experimental excitation energy of the $3_1^{-}$ state, and the (c) theoretical and (d) experimental $B(E3;3_1^{-}\to 0_1^{+}$) transition strength in Weisskopf units for the Ra, Th, U, Pu, Cm, and Cf isotopes as functions of the neutron number. Theoretical values for the Ra and Th nuclei are taken from Ref. [45]. The experimental data are from Refs. [63, 64].

Figure 10

Evolution of the (a) theoretical and (b) experimental $B(E2;2_1^{+}\to 0_1^{+}$) and (c) theoretical and (d) experimental $B(E1;1_1^{-}\to 0_1^{+}$) transition strength in Weisskopf units for the Ra, Th, U, Pu, Cm, and Cf isotopes as functions of the neutron number. Theoretical values for the Ra and Th nuclei are taken from Ref. [45]. The experimental data are from Refs. [63].

Figure 11

Effective (a) quadrupole $\langle {\beta }_2\rangle$ and (b) octupole $\langle {\beta }_3\rangle$ deformation parameters, (c), (d) variance $\delta {\beta }_{\lambda }$, and (e), (f) the fluctuations $\delta {\beta }_{\lambda }/\langle {\beta }_{\lambda }\rangle$ for ${}^{218\text{–}238}\mathrm{Ra}$, ${}^{220\text{–}240}\mathrm{Th}$, ${}^{222\text{–}242}\mathrm{U}$, ${}^{224\text{–}244}\mathrm{Pu}$, ${}^{226\text{–}246}\mathrm{Cm}$, and ${}^{228\text{–}248}\mathrm{Cf}$, as functions of the neutron number. Values for the Ra and Th nuclei are calculated based on the results from Ref. [45]. See the main text for details.

Figure 12

Comparison of theoretical and experimental low-energy level schemes of ${}^{240}\mathrm{Pu}$.