Spectroscopy of quadrupole and octupole states in rare-earth nuclei from a Gogny force

Collective quadrupole and octupole states are described in a series of Sm and Gd isotopes within the framework of the interacting boson model (IBM), whose Hamiltonian parameters are deduced from mean-field calculations with the Gogny energy density functional. The link between both frameworks is the (${\beta }_2{\beta }_3$) potential energy surface computed within the Hartree-Fock-Bogoliubov framework in the case of the Gogny force. The diagonalization of the IBM Hamiltonian provides excitation energies and transition strengths of an assorted set of states including both positive- and negative-parity states. The resultant spectroscopic properties are compared with the available experimental data and also with the results of the configuration mixing calculations with the Gogny force within the generator coordinate method (GCM). The structure of excited $0^{+}$ states and its connection with double-octupole phonons is also addressed. The model is shown to describe the empirical trend of the low-energy quadrupole and octupole collective structure fairly well and turns out to be consistent with GCM results obtained with the Gogny force.

Figure 1

Axially symmetric (${\beta }_2, {\beta }_3$) PESs for the nuclei ${}^{146–156}\mathrm{Sm}$ and ${}^{148–158}\mathrm{Gd}$ calculated within the constrained Gogny-HFB approach based on the D1M parametrization. The contour lines join points with the same energy (in MeV) and the color scale varies in steps of 100 keV. The energy difference between neighboring contours is 0.5 MeV. These (${\beta }_2,{\beta }_3$) energy surfaces are symmetric with respect to the ${\beta }_3=0$ axis. Thus, they are only plotted for ${\beta }_3\ge 0$. For each nucleus the absolute minimum is identified by an open circle.

Figure 2

The same as Fig. 1 but for the mapped IBM PESs.

Figure 3

The parameters of the $sdf$-IBM Hamiltonian $\widehat{H}$ in Eq. (1), as well as the proportionality coefficients $C_2$ and $C_3$, are plotted as functions of the neutron number for the considered nuclei. The parameters ${\chi }_{dd}, {\chi }_{ff}, {\chi }_{df}, C_2$, and $C_3$ are are dimensionless.

Figure 4

The energy spectra of the lowest-lying even-spin positive-parity states up to $J^{\pi }={10}^{+}$ for the considered Sm and Gd isotopes. All the experimental data are taken from the NNDC compilation [77].

Figure 5

The same as in Fig. 4, but for the lowest-lying odd-spin negative-parity states up to $J^{\pi }=9^{-}$.

Figure 6

The excitation energies of the $1_1^{-}$ states predicted with the mapped IBM Hamiltonian are compared with the ones obtained in the framework of two-dimensional GCM calculations [19] for Sm (a) and Gd (b) nuclei. In both methods, the Gogny-D1M parametrization has been used. The experimental energy levels are also included in the figure.

Figure 7

Signature splitting $S\left(J\right)$ of ${}^{150,152,156}\text{Sm}$ nuclei as a function of spin $J$. For more details, see the main text.

Figure 8

Theoretical and experimental transition probabilities $B(E3;3_1^{-}\to 0_1^{+})$ and $B(E1;1_1^{-}\to 0_1^{+})$ for ${}^{146–156}\mathrm{Sm}$ and ${}^{148–158}\mathrm{Gd}$. The experimental data are taken from Refs. [77, 78, 79, 80].

Figure 9

Comparison of the low-energy spectrum predicted within the IBM framework for the nucleus ${}^{150}\mathrm{Sm}$ with the available experimental excitation energies [77].

Figure 10

Same as in Fig. 9 but for the nucleus ${}^{158}\mathrm{Gd}$.

Figure 11

Energy distribution of the theoretical lowest 15 $0^{+}$ states and expectation value of the $f$-boson number operator $\langle {\widehat{n}}_f\rangle$ for the considered Sm and Gd isotopes. Note that, concerning ${}^{146}\mathrm{Sm}$ (a) and ${}^{148}\mathrm{Gd}$ (g), the $0^{+}$ states with an energy higher than 8 MeV are not shown.

Figure 12

The correlation energies obtained from the IBM and the two-dimensional (2D) GCM for ${}^{146–156}\mathrm{Sm}$ and ${}^{148–158}\mathrm{Gd}$ isotopes.