Beyond-mean-field study of the possible “bubble” structure of ${}^{34}$Si

Recent self-consistent mean-field calculations predict a substantial depletion of the proton density in the interior of ${}^{34}\text{Si}$. In the present study, we investigate how correlations beyond the mean field modify this finding. The framework of the calculation is a particle-number and angular-momentum projected generator coordinate method based on Hartree-Fock-Bogoliubov+Lipkin-Nogami states with axial quadrupole deformation. The parametrization SLy4 of the Skyrme energy density functional is used together with a density-dependent pairing energy functional. For the first time, the generator coordinate method is applied to the calculation of charge densities. The impact of pairing correlations, symmetry restorations and shape mixing on the density profile is analyzed step by step. All these effects significantly alter the radial density profile, and tend to bring it closer to a Fermi-type density distribution.

Figure 1

Energy curves for the particle-number-projected HFB states ($N\&Z$) and particle-number and angular-momentum projected states ($J=0$, 2, 4, 6 curves) for ${}^{34}\text{Si}$ as a function of the intrinsic quadrupole deformation of the mean-field states they are projected from. The solid square dots correspond to the lowest GCM solutions, which are plotted at their average deformation ${\sum }_{q}q{\left|g_{\mu }^J\left(q\right)\right|}^2$ (see text).

Figure 2

Nilsson diagram of the eigenvalues of the single-particle Hamiltonian for neutrons (a) and protons (b) as obtained with the Skyrme interaction SLy4 for ${}^{34}\text{Si}$ as a function of the quadrupole deformation. Solid (dotted) lines represent levels of positive (negative) parity, and black, red, green and blue color represents levels with expectation values of $\langle j_z\rangle =1/2$, $3/2$, $5/2$, and $7/2$. The thick long-dashed line represents the Fermi energy. Single-particle levels are labeled for the spherical configuration only.

Figure 3

Comparison between the experimental (left) and calculated (right) excitation energies $E_x$ and $B\left(E2\right)$ values (in units of $e^2$ fm${}^4$) for the low-lying states of ${}^{34}\text{Si}$. Experimental data are taken from Ref. [61].

Figure 4

Collective wave functions $g_{\mu }^J\left(q\right)$ [cf. Eq. (6)] of the two lowest GCM solutions for $J=0$.

Figure 5

Neutron, proton, and total radial densities at $x=y=0$ for ${}^{34}\text{Si}$ for the spherical HF state (a), the spherical HFB+LN state (b), and its projection on good particle numbers (c), as well as for the GCM $0_1^{+}$ ground state (d). Neutron and proton densities have been rescaled with the factors $A/N$ and $A/Z$, respectively.

Figure 6

Contour plots of the neutron (upper panels) and proton (lower panels) densities in the $y=0$ plane for the particle number projected HFB+LN state with ${\beta }_2=0.26$ (left column), its projection on both particle numbers and total angular momentum $J=0$ (middle column) and for the $0^{+}$ GCM ground state (right column).

Figure 7

Comparison of point-proton densities (a) with the folded charge densities (b) for ${}^{34}\text{Si}$ for the same states as in Fig. 5.