Variational approximations to exact solutions in shell-model valence spaces: Systematic calculations in the $sd$ shell

We study the ability of variational approaches based on self-consistent mean-field and beyond-mean-field methods to reproduce exact energies and electromagnetic properties of the nuclei defined within the $sd$-shell valence space using the nontrivial USD Hamiltonian. In particular, Hartree-Fock-Bogoliubov (HFB), variation after particle-number projection (VAPNP), and projected generator coordinate methods (PGCM) are compared to exact solutions obtained by the full diagonalization of the Hamiltonian. We analyze the role played by the proton-neutron $\left(pn\right)$ mixing as well as the quadrupole and pairing degrees of freedom (including both isoscalar and isovector channels) in the description of the spectra of even-even, even-odd, and odd-odd nuclei in the whole $sd$ shell.

Figure 1

Total HFB energy (first row), isoscalar (second row), and isovector (third row) pairing amplitudes, and quadrupole moments (fourth row) obtained in 650 solutions of the unconstrained HFB equations with (a)–(d) random axial $pn$-no seeds, (e)–(h) random general $pn$-no seeds, and (i)–(l) random general $pn$-yes seeds. The nucleus considered is ${}^{24}\mathrm{Ne}$ ($Z=2$ and $N=6$ in the valence space) and was calculated using the USD interaction.

Figure 2

Same as Fig. 1 but for the VAPNP method.

Figure 3

Distribution in eigenstates of the proton and neutron number operators of the intrinsic wave functions ($\langle \mathrm{\Phi }\left|{\widehat{P}}^Z{\widehat{P}}^N\right|\mathrm{\Phi }\rangle$) obtained from minimizing HFB/VAPNP energy functionals with different seed wave functions: (a) HFB axial $pn$-no, (b) HFB general $pn$-no, (c) HFB general $pn$-yes, (d) VAPNP axial $pn$-no, (e) VAPNP general $pn$-no, and (f) VAPNP general $pn$-yes. The nucleus considered is ${}^{24}\mathrm{Ne}$ ($Z=2$ and $N=6$ in the valence space) and was calculated using the USD Hamiltonian.

Figure 4

(a) HFB and (b) VAPNP energies calculated with axial $pn$-no (red bullets), general $pn$-no (blue squares), and general $pn$-yes (black diamonds) seeds as a function of the label of the initial blocked neutron level in the $sd$ shell (from left to right: $0d_{3/2}$, $0d_{5/2}$, $1s_{1/2}$ orbits). The nucleus considered is ${}^{25}\mathrm{Ne}$ ($Z=2$ and $N=7$ in the valence space) and was calculated using the USD Hamiltonian.

Figure 5

Same as Fig. 3 but for the ${}^{25}\mathrm{Ne}$ nucleus ($Z=2$ and $N=7$ in the valence space).

Figure 6

(a) HFB and (b) VAPNP energies calculated with axial $pn$-no (red bullets), general $pn$-no (blue squares), and general $pn$-yes (black diamonds) seeds as a function of the label of the initial blocking levels (protons and neutrons) in the $sd$ shell. The nucleus considered is ${}^{24}\mathrm{Na}$ ($Z=3$ and $N=5$ in the valence space) and was calculated with the USD Hamiltonian.

Figure 7

Same as Fig. 3 but for the ${}^{24}\mathrm{Na}$ ($Z=3$ and $N=5$ in the valence space).

Figure 8

Energy difference with respect to the exact solution for unconstrained HFB (red symbols) and VAPNP (black symbols) calculations for even-even and even-odd nuclei in the $sd$ shell and the USD interaction. Two different choices of Bogoliubov transformations are used: general $pn$-no (boxes and bullets) and general $pn$-yes (circles and filled boxes).

Figure 9

Same as Fig. 8 but for odd-odd isotopes. The HFB results only contains general $pn$-no seeds.

Figure 10

Total energy surfaces as a function of the triaxial deformations $({\overline{\beta }}_2,\gamma )$ calculated for ${}^{24}\mathrm{Ne}$ within the following approaches: (a) HFB $pn$-no, (b) VAPNP $pn$-no, and (c) VAPNP $pn$-yes. Contour lines are separated by intervals of 0.25 MeV and the scale of the color code is different for each plot.

Figure 11

VAPNP total energy surfaces for ${}^{24}\mathrm{Ne}$ as a function of the quadrupole deformation, ${\overline{\beta }}_2$, and (a) isoscalar pairing, ${\delta }_{M_{J_p}=0;M_{T_p}=0}^{J_p=1;T_p=0}$, (b) isovector $pn$ pairing, ${\delta }_{M_{J_p}=0;M_{T_p}=0}^{J_p=0;T_p=1}$, (c) $nn$ pairing, ${\delta }_{M_{J_p}=0;M_{T_p}=+1}^{J_p=0;T_p=1}$, and (d) $pp$ pairing, ${\delta }_{M_{J_p}=0;M_{T_p}=-1}^{J_p=0;T_p=1}$. Contour lines are separated by intervals of 0.25 MeV and the scale of the color code is the same in all the plots.

Figure 12

PGCM and exact energies of the yrast states for ${}^{24}\mathrm{Ne}$ calculated with the USD interaction. (a) PGCM exploring triaxial $({\overline{\beta }}_2,\gamma )$ deformations, and (b) PGCM exploring deformation and pairing $({\overline{\beta }}_2,{\delta }_{M_{J_p}{M}_{T_p}}^{J_p{T}_p})$ degrees of freedom.

Figure 13

Energy difference with respect to the exact solution PGCM calculations using $({\overline{\beta }}_2,\gamma )$ as generating coordinates for even-even and even-odd nuclei in the $sd$ shell and the USD interaction. Three different sets of Bogoliubov states are used: general $pn$-no seeds (black circles) and general $pn$-yes seeds (black filled boxes) with VAPNP minimization, and general $pn$-no seeds (red bullets) with HFB minimization.

Figure 14

Same as Fig. 13 but for odd-odd isotopes. The HFB result only contains general $pn$-no seeds. The magenta crosses show the VAPNP-PGCM results including $M_{T_p}=0$ pairing degrees of freedom apart from $({\overline{\beta }}_2,\gamma )$ (see text for details).

Figure 15

$0_2^{+}$ (top panel), $2_1^{+}$ (middle panel), and $4_1^{+}$ (bottom panel) excitation energies for the even-even isotopes in the $sd$-shell nuclei calculated exactly (black diamonds) and using VAPNP-PGCM with $pn$ mixing (red dots), VAPNP-PGCM without $pn$ mixing (blue squares), and HFB-PGCM (green pentagons) techniques.

Figure 16

First (top panel), second (middle panel), and third (bottom panel) excitation energies for the even-odd isotopes in the $sd$-shell nuclei calculated exactly (black diamonds) and using VAPNP-PGCM with $pn$ mixing (red dots), VAPNP-PGCM without $pn$ mixing (blue squares), and HFB-PGCM (green pentagons) techniques.

Figure 17

Excitation energies for the first excited states in odd-odd isotopes in the $sd$-shell nuclei calculated exactly (black diamonds) and using PGCM with HFB-PGCM (green pentagons) and VAPNP-PGCM with $pn$ mixing (red dots) and without $pn$ mixing (blue squares) techniques. The angular momenta used as g.s. and first excited states are those of the exact calculation even though they could not correspond to the g.s. and first excited states obtained with the variational approaches. Empty symbols represent a wrong assignment of the exact g.s. angular momentum. The magenta crosses show the results for PGCM including $M_{T_p}=0$ pairing degrees of freedom apart from $({\overline{\beta }}_2,\gamma )$ (see text for details).

Figure 18

Electromagnetic properties calculated with HFB-PGCM (green pentagons), VAPNP-PGCM with (red bullets) and without (blue squares) $pn$-mixing, and the exact values (black diamonds) for selected states in Ne isotopes: (a) $B(E2,2_1^{+}\to 0_1^{+})$ in even-even isotopes, (b) $B(E2,J_{1\mathrm{st}\mathrm{exc}.}^{+}\to J_{\mathrm{g}.\mathrm{s}.}^{+})$ in even-odd isotopes, (c) $B(M1,3_1^{+}\to 2_1^{+})$ in even-even isotopes, and (d) magnetic dipole moment for the ground state in even-odd isotopes.