Configuration mixing of angular-momentum-projected triaxial relativistic mean-field wave functions

The framework of relativistic energy-density functionals is extended to include correlations related to the restoration of broken symmetries and to fluctuations of collective variables. The generator coordinate method is used to perform configuration mixing of angular-momentum-projected wave functions, generated by constrained self-consistent relativistic mean-field calculations for triaxial shapes. The effects of triaxial deformation and of $K$ mixing is illustrated in a study of spectroscopic properties of low-spin states in ${}^{24}\mathrm{Mg}$.

Figure 1

Distributions of the $K$ quantum number (K) and $q$ values (iq) in the full $K\otimes q$ space. The dimension of the direct product of the $K$ subspace and the $q$ subspace is $D=(J+2)n_q/2$ for even $J$ or $D=(J-1)n_q/2$ for odd $J$. $n_q$ is the number of points on the mesh in $q$ space, and $J$ is the total angular momentum.

Figure 2

The distribution of mesh points in the $\beta ,\gamma$ plane for the sets AI, AII, AIII, AI$+$TI, AI$+$TII, AI$+$TIII, AIII$+$TI, AIII$+$TII, and AIII$+$TIII.

Figure 3

(Upper panel) Energies and the average axial deformations for the two lowest GCM states with angular momentum $0^{+},2^{+},4^{+},6^{+}$ in ${}^{24}\mathrm{Mg}$, together with the mean-field (dotted) and the corresponding angular-momentum-projected energy curves. (Lower panel) Squares of collective wave functions $|g_{\alpha }^J(q)|{}^2$ with $q_{22}=0$ for the corresponding lowest GCM states in ${}^{24}\mathrm{Mg}$. These results are obtained in the axial 1DAMP$+$GCM calculation. Positive (negative) values of the axial deformation $\beta$ correspond to prolate (oblate) configurations.

Figure 4

Lowest energy levels of angular momentum $J^{\pi }=0^{+}$, $2^{+}$, $4^{+}$, $6^{+}$ in ${}^{24}\mathrm{Mg}$, and reduced $E2$ transition probabilities in ${\mathrm{e}}^2$fm${}^4$, calculated with the 3DAMP$+$GCM and 1DAMP$+$GCM models. See text for details.

Figure 5

Self-consistent RMF$+$BCS energy surface (left panel) of ${}^{24}\mathrm{Mg}$ in the $\beta$-$\gamma$ plane ($0⩽\gamma ⩽{60}^0$), and angular-momentum-projected energy surface with $J^{\pi }=0^{+}$ (right panel). The contours join points on the surface with the same energy. The difference between neighboring contours is 1.0 MeV.

Figure 6

Average neutron and proton pairing gaps of ${}^{24}\mathrm{Mg}$ in the $\beta$-$\gamma$ plane ($0⩽\gamma ⩽{60}^0$). The contours join points on the surface with the same pairing gap. The difference between neighboring contours is 0.2 MeV.

Figure 7

The low-spin level scheme of ${}^{24}\mathrm{Mg}$ calculated using the 3DAMP$+$GCM model and 1DAMP$+$GCM model with the PC-F1 relativistic density functional, in comparison with data [46, 47, 48]. The $B(E2)$ values are given in units of $e^2$ fm${}^4$.

Figure 8

Contour plots of the probability distributions $\left|g_{\alpha }^J\right|{}^2$ for the ground state $0_1^{+}$ and the first excited state $2_1^{+}$ (both the $K=0$ and $K=2$ components) in ${}^{24}\mathrm{Mg}$.

Figure 9

Same as in Fig. 8 but for the excited states $3_1^{+}$ and $2_2^{+}$ in ${}^{24}\mathrm{Mg}$.

Figure 10

Same as in Fig. 8 but for the excited states $4_1^{+}$ and $4_2^{+}$ in ${}^{24}\mathrm{Mg}$.

Figure 11

Average particle numbers for the 3DAMP$+$GCM states belonging to the three bands of ${}^{24}\mathrm{Mg}$ in Fig. 7.