Description of nuclear systems with a self-consistent configuration-mixing approach. II. Application to structure and reactions in even-even $sd$-shell nuclei

Background: The variational multiparticle-multihole configuration mixing approach to nuclei has been proposed about a decade ago. While the first applications followed rapidly, the implementation of the full formalism of this method has only been recently completed and applied in C. Robin, N. Pillet, D. Peña Arteaga, and J.-F. Berger, [Phys. Rev. C 93, 024302 (2016)] to ${}^{12}\mathrm{C}$ as a test-case.

Purpose: The main objective of the present paper is to carry on the study that was initiated in that reference, in order to put the variational multiparticle-multihole configuration mixing method to more stringent tests. To that aim we perform a systematic study of even-even $sd$-shell nuclei.

Method: The wave function of these nuclei is taken as a configuration mixing built on orbitals of the $sd$-shell, and both the mixing coefficients of the nuclear state and the single-particle wave functions are determined consistently from the same variational principle. As in the previous works, the calculations are done using the D1S Gogny force.

Results: Various ground-state properties are analyzed. In particular, the correlation content and composition of the wave function as well as the single-particle orbitals and energies are examined. Binding energies and charge radii are also calculated and compared to experiment. The description of the first excited state is also examined and the corresponding transition densities are used as input for the calculation of reaction processes such as inelastic electron and proton scattering. Special attention is paid to the effect of the optimization of the single-particle states consistently with the correlations of the system.

Conclusions: The variational multiparticle-multihole configuration mixing approach is systematically applied to the description of even-even $sd$-shell nuclei. Globally, the results are satisfying and encouraging. In particular, charge radii and excitation energies are nicely reproduced. However, the chosen valence-space truncation scheme precludes achieving maximum collectivity in the studied nuclei. Further refinement of the method and a better-suited interaction are necessary to remedy this situation.

Figure 1

HFB potential energy surface (PES) of the neon isotopes. The red curve includes zero-point energy corrections.

Figure 2

HFB PES of ${}^{24}\mathrm{Mg}$ (top left), ${}^{28}\mathrm{Si}$ (top right), and ${}^{32}\mathrm{S}$ (bottom).

Figure 3

Proton correlations ${\sigma }^{\pi }$ (left), neutron correlations ${\sigma }^{\nu }$ (center), and proton-neutron correlations ${\sigma }^{\pi \nu }$ (right), for ${}^{28}\mathrm{Ne}$, ${}^{24}\mathrm{Ne}$, and ${}^{20}\mathrm{Ne}$. They are calculated when full consistency between orbitals and correlations is reached (level 3 of the method).

Figure 4

Proton correlations ${\sigma }^{\pi }$ (left), neutron correlations ${\sigma }^{\nu }$ (center), and proton-neutron correlations ${\sigma }^{\pi \nu }$ (right), for ${}^{24}\mathrm{Mg}$, ${}^{28}\mathrm{Si}$, and ${}^{32}\mathrm{S}$. They are calculated when full consistency between orbitals and correlations is reached (level 3 of the method).

Figure 5

Differences $\mathrm{\Delta }\varepsilon ={\varepsilon }_{\text{HF}}-\varepsilon$ between Hartree-Fock and self-consistent single-particle energies, expressed in MeV. The Fermi level is marked by a dashed line.

Figure 6

Squared modulus of the radial part of the single-particle orbitals. The proton (neutron) self-consistent (SC) orbitals are shown with full red (black) lines while the pure Hartree-Fock (HF) orbitals are shown with dashed lines.

Figure 7

Difference between theoretical and experimental binding energies (in MeV) at the different levels of implementation of the MPMH method. Experimental data are taken from Ref. [40].

Figure 8

Comparison between experimental and MPMH charge radii (in fm). Experimental values are taken from Ref. [41].

Figure 9

Radial proton density ${\rho }_{\pi }\left(r\right)$ (left) and neutron density ${\rho }_{\nu }\left(r\right)$ (right) for ${}^{20}\mathrm{Ne}$, ${}^{28}\mathrm{Ne}$, ${}^{24}\mathrm{Mg}$, ${}^{26}\mathrm{Mg}$, ${}^{28}\mathrm{Si}$, and ${}^{32}\mathrm{S}$, and contribution of each orbital, in ${\mathrm{fm}}^{-3}$.

Figure 10

Theoretical excitation energies of the $2_1^{+}$ states compared to experiment. Experimental data are taken from Ref. [42]. Results are expressed in MeV.

Figure 11

Low-energy spectra of neon and magnesium isotopes. For each nucleus the theoretical spectrum is on the right while the corresponding experimental states are shown on the left. $0^{+},2^{+},4^{+}$, and $6^{+}$ states are shown with black, blue, red, and green lines, respectively. The experimental plain lines show states with spin and parity that have been surely assigned, while the dashed lines denote states for which the spin and/or parity have been assigned “based on weak arguments” [42].

Figure 12

Same as Fig. 11 for the silicon, sulfur, and argon chains.

Figure 13

Charge transition densities and corresponding form factors for ${}^{28}\mathrm{Si}$, ${}^{32}\mathrm{S}$, and ${}^{20}\mathrm{Ne}$. The experimental data are taken from Refs. [46, 47, 48, 49].

Figure 14

Transition densities and corresponding form factors for ${}^{20}\mathrm{Ne}$ obtained with two valence spaces: $sd$ shell and $sd+fp$ shells.

Figure 15

Comparison of theoretical and experimental quadrupole transition probabilities $B(E2;2_1^{+}\to 0_1^{+})$ expressed in ${\mathrm{e}}^2{\mathrm{fm}}^4$. The experimental data (in black) are taken from Ref. [42]. Results without and with full self-consistency are shown in red and blue, respectively. Results obtained with introducing rearrangement terms in the first variational equation are shown in green. Theoretical results for ${}^{28}\mathrm{Si}$, ${}^{32}\mathrm{S}$, and ${}^{20}\mathrm{Ne}$ obtained using the $sd$-shell as a valence space are shown on the left of the figure. For comparison, the results obtained for ${}^{20}\mathrm{Ne}$ with an $sd+fp$ valence space, without and with full self-consistency, are displayed on the right.

Figure 16

Inelastic proton scattering cross section for the transition $0_1^{+}\to 2_1^{+}$ in ${}^{28}\mathrm{Si}$.