Beyond the relativistic mean-field approximation. III. Collective Hamiltonian in five dimensions

The framework of relativistic energy density functionals is extended to include correlations related to the restoration of broken symmetries and fluctuations of collective variables. A new implementation is developed for the solution of the eigenvalue problem of a five-dimensional collective Hamiltonian for quadrupole vibrational and rotational degrees of freedom, with parameters determined by constrained self-consistent relativistic mean-field calculations for triaxial shapes. The model is tested in a series of illustrative calculations of potential energy surfaces and the resulting collective excitation spectra and transition probabilities of the chain of even-even gadolinium isotopes.

Figure 1

Self-consistent RMF$+$BCS binding energy curves of the ${}^{152-160}\mathrm{Gd}$ isotopes, as functions of the axial deformation parameter $\beta$. Negative values of $\beta$ correspond to the $(\beta >0,\gamma ={180}^{°})$ axis on the $\beta \text{−}\gamma$ plane.

Figure 2

Self-consistent RMF$+$BCS triaxial quadrupole binding energy map of ${}^{152}\mathrm{Gd}$ in the $\beta \text{−}\gamma$ plane ($0⩽\gamma ⩽{60}^{°}$). All energies are normalized with respect to the binding energy of the absolute minimum (large red dot). The contours join points on the surface with the same energy (in MeV).

Figure 3

Same as Fig. 2, but for the nucleus ${}^{154}\mathrm{Gd}$.

Figure 4

Same as Fig. 2, but for the nucleus ${}^{156}\mathrm{Gd}$.

Figure 5

Same as Fig. 2, but for the nucleus ${}^{158}\mathrm{Gd}$.

Figure 6

Same as Fig. 2, but for the nucleus ${}^{160}\mathrm{Gd}$.

Figure 7

Map of the Inglis-Belyaev inertia parameter $B_x$ of ${}^{154}\mathrm{Gd}$ in the $\beta \text{−}\gamma$ plane ($0⩽\gamma ⩽{60}^{°}$).

Figure 8

Cranking mass parameter $B_{\beta \beta }$ of ${}^{154}\mathrm{Gd}$ in the $\beta \text{−}\gamma$ plane ($0⩽\gamma ⩽{60}^{°}$).

Figure 9

Proton pairing energy of ${}^{154}\mathrm{Gd}$ in the $\beta \text{−}\gamma$ plane ($0⩽\gamma ⩽{60}^{°}$). The contours join points on the surface with the same energy (in MeV).

Figure 10

Neutron pairing energy of ${}^{154}\mathrm{Gd}$ in the $\beta \text{−}\gamma$ plane ($0⩽\gamma ⩽{60}^{°}$). The contours join points on the surface with the same energy (in MeV).

Figure 11

Rotational zero-point energy of ${}^{154}\mathrm{Gd}$ in the $\beta \text{−}\gamma$ plane ($0⩽\gamma ⩽{60}^{°}$).

Figure 12

Potential $V_{\mathrm{coll}}$ [Eq. (52)] of ${}^{154}\mathrm{Gd}$ in the $\beta \text{−}\gamma$ plane ($0⩽\gamma ⩽{60}^{°}$). The contours join points on the surface with the same energy (in MeV).

Figure 13

Level scheme of ${}^{154}\mathrm{Gd}$ calculated with the PC-F1 relativistic density functional, in comparison with experimental data [45]. The relative excitation energies are scaled by the common factor $\approx 0.69$, adjusted to the experimental energy of the state $2_1^{+}$. The $B(E2)$ values are given in Weisskopf units.

Figure 14

Relative ground-state band excitation energies in ${}^{152-160}\mathrm{Gd}$. For each nucleus, the theoretical energies are scaled by a common factor, adjusted to the experimental energy of the $2_1^{+}$ state.

Figure 15

Comparison between the theoretical and experimental $\gamma$-band excitation energies for ${}^{152-160}\mathrm{Gd}$. The scaling factors are the same as for the ground-state bands in Fig. 14.

Figure 16

Same as in Fig. 15, but for the $\beta$ bands in ${}^{152-160}\mathrm{Gd}$. The scaling factors are the same as for the ground-state bands in Fig. 14.

Figure 17

Distribution of the $K$ components (projection of the angular momentum on the body-fixed symmetry axis) in the collective wave functions for the states $2_2^{+}$ and $2_3^{+}$.

Figure 18

Same as Fig. 17, but for the states $4_2^{+}$ and $4_3^{+}$.

Figure 19

Same as Fig. 17, but for the states $6_2^{+}$ and $6_3^{+}$.

Figure 20

Excitation energies of the second and third states $2^{+}$, $4^{+}$, and $6^{+}$ in the Gd isotopic chain, as functions of the neutron number.

Figure 21

Staggering $S(J)$ [Eq. (53)] in the $\gamma$ bands of ${}^{152-160}\mathrm{Gd}$. Theoretical predictions are compared with experimental values.

Figure 22

$B(E2)$ values (in Weisskopf units) for the ground-state band transitions $J_1^{+}\to (J-2){}_1^{+}$ in ${}^{152-160}\mathrm{Gd}$. Theoretical values calculated with the PC-F1 relativistic density functional are compared with data.

Figure 23

Self-consistent RMF$+$BCS binding energy curves of ${}^{154}\mathrm{Gd}$, as functions of the axial deformation parameter $\beta$. Negative values of $\beta$ correspond to the $(\beta >0,\gamma ={180}^{°})$ axis on the $\beta \text{−}\gamma$ plane. The four energy curves correspond to calculations in the three-dimensional harmonic oscillator basis with 10, 12, 14, and 16 major oscillator shells.