Beyond the relativistic mean-field approximation. II. Configuration mixing of mean-field wave functions projected on angular momentum and particle number

The framework of relativistic self-consistent mean-field models 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 and particle-number projected relativistic wave functions. The geometry is restricted to axially symmetric shapes, and the intrinsic wave functions are generated from the solutions of the relativistic mean-field+Lipkin-Nogami BCS equations, with a constraint on the mass quadrupole moment. The model employs a relativistic point-coupling (contact) nucleon-nucleon effective interaction in the particle-hole channel, and a density-independent δ-interaction in the pairing channel. Illustrative calculations are performed for ${}^{24}\mathrm{Mg}$, ${}^{32}\mathrm{S}$, and ${}^{36}\mathrm{Ar}$, and compared with results obtained employing the model developed in the first part of this work, i.e., without particle-number projection, as well as with the corresponding nonrelativistic models based on Skyrme and Gogny effective interactions.

Figure 1

Comparison of the average pairing gaps calculated with a simple BCS, and the Lipkin-Nogami plus BCS approximations. For the $Z=50$ and $Z=28$ isotopes the neutron pairing gaps are shown, whereas for the $N=50$ and $N=82$ isotones the proton average pairing gaps are compared.

Figure 2

The BCS pairing energy for protons and neutrons (upper panel), and the mean-field plus BCS binding energy curve (lower panel) of ${}^{24}\mathrm{Mg}$, as functions of the mass quadrupole moment.

Figure 3

The ground-state rotational band in ${}^{24}\mathrm{Mg}$. The $B(E2)$ values are in units of $e^2\hspace{1em}{\mathrm{fm}}^4$. The GCM spectra calculated with (PN&AMP), and without (AMP) particle number projection, are compared with the experimental ground-state band [34]

Figure 4

The amplitudes of the PN&AMP GCM collective wave functions $\left|g_k^J\right|{}^2$ of the ground-state band in ${}^{24}\mathrm{Mg}$.

Figure 5

The BCS pairing energy for protons and neutrons (upper panel), and the mean-field plus BCS binding energy curve (lower panel) of ${}^{32}\mathrm{S}$, as functions of the mass quadrupole moment.

Figure 6

The energies and the average quadrupole moments of the GCM states in ${}^{32}\mathrm{S}$, plotted together with the corresponding angular momentum projected energy curves. GCM calculations with (left panel), and without (right panel) particle number projection are compared. The mean-field binding energy curves are also included in the figure (dotted curves). In both panels zero energy corresponds to the minimum of the $J=0$ projected energy curve.

Figure 7

Transition energies $\Delta E(J)=E(J)-E(J-2)$ (left panel), and the $B(E2,J\to J$-2) values (right panel), for the SD configuration in ${}^{32}\mathrm{S}$, as functions of the angular momentum.

Figure 8

The energies and the average quadrupole moments of the GCM states in ${}^{36}\mathrm{Ar}$, plotted together with the corresponding angular momentum projected energy curves. GCM calculations with (left panel), and without (right panel) particle number projection are compared. The mean-field binding energy curves are also included in the figure (dotted curves). In both panels zero energy corresponds to the minimum of the $J=0$ projected energy curve.

Figure 9

The energy differences $\Delta E(J)=E(J)-E(J-2)$ (left panel), and the $B(E2,J\to J-2)$ values (right panel), for the SD band in ${}^{36}\mathrm{Ar}$, as functions of the angular momentum. Results of GCM calculations with and without particle number projection are compared with the available data [49, 50].

Figure 10

The amplitudes of the PN&AMP GCM collective wave functions $\left|g_k^J\right|{}^2$ for the ND ground-state band ($0_1^{+},2_1^{+},4_1^{+}$ and $6_2^{+}$), and the SD band ($0_3^{+},2_2^{+},4_2^{+}$ and $6_1^{+}$ and $8_1^{+}$) in ${}^{36}\mathrm{Ar}$, as functions of the mass quadrupole moment.