Beyond the relativistic mean-field approximation: Configuration mixing of angular-momentum-projected wave functions

We report the first study of restoration of rotational symmetry and fluctuations of the quadrupole deformation in the framework of relativistic mean-field models. A model is developed that uses the generator coordinate method to perform configuration mixing calculations of angular-momentum-projected wave functions, calculated in a relativistic point-coupling model. The geometry is restricted to axially symmetric shapes, and the intrinsic wave functions are generated from the solutions of the constrained relativistic mean-field + BCS equations in an axially deformed oscillator basis. A number of illustrative calculations are performed for the nuclei ${}^{194}\mathrm{Hg}$ and ${}^{32}\mathrm{Mg}$, in comparison with results obtained in nonrelativistic models based on Skyrme and Gogny effective interactions.

Figure 1

The binding energy curves for ${}^{194}\mathrm{Hg}$, as functions of the mass quadrupole moment, calculated by expanding the Dirac spinors in $10,12,14,16$, and 18 oscillator shells.

Figure 2

The eigenvalues of the norm overlap kernel, calculated using four different values for the mesh spacing: $\Delta q=2,4,6$, and 8 b (left panel). The energies of the 12 lowest GCM states plotted as functions of the dimension of the GCM basis (right panel).

Figure 3

The energies of the 15 lowest GCM states in ${}^{194}\mathrm{Hg}$, plotted as functions of the average quadrupole moment, together with the corresponding mean-field binding energy curves. The four panels correspond to calculations in oscillator bases with $10,12,14$, and 16 major oscillator shells.

Figure 4

GCM collective wave functions $g_k(q)$ [Eq. (57)] for the lowest 14 states in ${}^{194}\mathrm{Hg}$. The vertical dashed line denotes the position of the barrier separating the main and the superdeformed potential wells.

Figure 5

The mean-field binding energy curves for ${}^{194}\mathrm{Hg}$, together with the energies and average quadrupole moments of the 15 lowest GCM states. Calculations have been performed using four values of the mesh spacing: $\Delta q=2,4,6,$ and 8 b. Zero energy is placed at the position of the minimum of the binding energy curve.

Figure 6

The binding energy curves for ${}^{32}\mathrm{Mg}$, calculated from the constrained solutions of the self-consistent relativistic mean-field equations in axially deformed oscillator bases with 8, 10, and 12 major shells.

Figure 7

The neutron (left panel) and proton (right panel) single-particle levels for ${}^{32}\mathrm{Mg}$, as functions of the mass quadrupole moment. The thick dashed curve denotes the position of the Fermi energy.

Figure 8

The expectation value $\langle {\stackrel{\wedge }{J}}_y^2\rangle$ for ${}^{32}\mathrm{Mg}$, as a function of the mass quadrupole moment.

Figure 9

A comparison between the exact values of the function $n(q;\beta )$ [Eq. (71) (dots)] and the approximate expression [Eq. (72) (curves)] for several values of the mass quadrupole moment.

Figure 10

Projected norm overlap kernel ${\mathcal{N}}^J(q,q)$ as a function of the mass quadrupole moment for ${}^{32}\mathrm{Mg}$.

Figure 11

Angular-momentum-projected ($J^{\pi }=0^{+},2^{+},4^{+},6^{+}$, and $8^{+}$) potential energy curves for ${}^{32}\mathrm{Mg}$, as functions of the mass quadrupole moment. The mean-field energies are also included (thick dotted curves). The three panels correspond to solutions in axially deformed oscillator bases with 8, 10, and 12 major shells.

Figure 12

Rotational energy correction as a function of the mass quadrupole moment for ${}^{32}\mathrm{Mg}$.

Figure 13

The energies and the average quadrupole moments of the two lowest GCM states for each angular momentum in ${}^{32}\mathrm{Mg}$, together with the corresponding projected energy curves.

Figure 14

Squares of the collective wave functions $|g_k^J(q)|{}^2$ of the two lowest GCM states for each value of the angular momentum in ${}^{32}\mathrm{Mg}$, together with the corresponding projected energy curves.