Relativistic energy density functionals: Low-energy collective states of ${}^{240}\mathrm{Pu}$ and ${}^{166}\mathrm{Er}$

The empirical relativistic density-dependent, point-coupling energy density functional, adjusted exclusively to experimental binding energies of a large set of deformed nuclei with $A\approx 150–180$ and $A\approx 230–250$, is tested with spectroscopic data for ${}^{166}\mathrm{Er}$ and ${}^{240}\mathrm{Pu}$. Starting from constrained self-consistent triaxial relativistic Hartree-Bogoliubov calculations of binding energy maps as functions of the quadrupole deformation in the $\beta –\gamma$ plane, excitation spectra and $E2$ transition probabilities are calculated as solutions of the corresponding microscopic collective Hamiltonian in five dimensions for quadrupole vibrational and rotational degrees of freedom and compared with available data on low-energy collective states.

Figure 1

Self-consistent RHB triaxial quadrupole binding energy maps of ${}^{240}\mathrm{Pu}$ 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 MeV).

Figure 2

Deformation energy curves and the inner barrier of ${}^{240}\mathrm{Pu}$ as functions of the axial deformation $\beta$. The two curves correspond to the axially symmetric RHB calculation (solid line) and to the projection on the $\beta$ axis of the triaxial PES (dashed line), calculated with the functional DD-PC1. The experimental values for the ground-state deformation, the barrier height, and the energy of the second minimum are indicated with an arrow, a symbol with error bars, and three lines indicating the value and its errors, respectively. The data are taken from Refs. [28, 29, 30, 31].

Figure 3

(left) Low-energy spectrum of ${}^{240}\mathrm{Pu}$ calculated with the DD-PC1 relativistic density functional, compared with (right) data for the three lowest positive-parity bands at normal deformation, and the lowest $\pi =+$ superdeformed band.

Figure 4

Same as described in the caption of Fig. 1 but for the nucleus ${}^{166}\mathrm{Er}$.

Figure 5

(left) Low-energy spectrum of ${}^{166}\mathrm{Er}$ calculated with the DD-PC1 relativistic density functional compared with (right) data for the ground-state band, the $\gamma$ band, and the two-phonon $\gamma$-vibrational states $K^{\pi }=4^{+}$ and $0^{+}$.

Figure 6

Same as described in the caption of Fig. 1 but for the binding energy map of the nucleus ${}^{166}\mathrm{Er}$ calculated using the energy density functional PC-F1.

Figure 7

(a) Self-consistent RHB binding energy curves and (b) cranking mass parameters $B_{\gamma \gamma }$ of ${}^{166}\mathrm{Er}$ at the axial deformation $\beta =0.35$, calculated with the energy density functionals DD-PC1 and PC-F1, as functions of the deformation parameter $\gamma$.

Figure 8

Same as described in the caption of Fig. 5 but the collective spectrum of ${}^{166}\mathrm{Er}$ is calculated using the energy density functional PC-F1.

Figure 9

Staggering $S(J)$ defined in Eq. (29) for the $\gamma$ band of ${}^{166}\mathrm{Er}$. The results calculated with the density functionals DD-PC1 and PC-F1 are compared with data.