Bubble nuclei within the self-consistent Hartree-Fock mean field plus pairing approach

The depletion of the nuclear density at its center, called the nuclear bubble, is studied within the Skyrme Hartree-Fock mean field consistently incorporating the superfluid pairing. The latter is obtained within the finite-temperature Bardeen-Cooper-Schrieffer theory and within the approach using the exact pairing. The numerical calculations are carried out for ${}^{22}\mathrm{O}$ and ${}^{34}\mathrm{Si}$ nuclei, whose bubble structures, caused by a very low occupancy of the $2s_{1/2}$ level, were previously predicted at $T=0$. Among 24 Skyrme interactions under consideration, the MSk3 is the only one which reproduces the experimentally measured occupancy of the $2s_{1/2}$ proton level as well as the binding energy, and consequently produces the most pronounced bubble structure in ${}^{34}\mathrm{Si}$. As compared to the approaches employing the same BSk14 interaction, our approach with exact pairing predicts a pairing effect which is stronger in ${}^{22}\mathrm{O}$ and weaker in ${}^{34}\mathrm{Si}$. The increase in temperature depletes the bubble structure and completely washes it out when the temperature reaches a critical value, at which the factor measuring the depletion of the nucleon density vanishes.

Figure 1

Binding energy per nucleon BE/A and two proton (neutron) separation energy $S_{2p}$ ($S_{2n}$) obtained from different methods using the MSk3 interaction at $T=$ 0. The results of HFB with BSk14 are extracted from the mass excesses taken from RIPL3 [29]. The dashed lines represent the experimental data.

Figure 2

Proton density ${\rho }_Z\left(r\right)$ for ${}^{34}\mathrm{Si}$ obtained within the FTHF, FTBCS, FTEP, and HFB at $T=$ 0.

Figure 3

Neutron density ${\rho }_N\left(r\right)$ for ${}^{22}\mathrm{O}$. Notations are the same as in Fig. 2.

Figure 4

Proton pairing gaps (a) and root-mean-square (rms) radii for protons (b) and neutrons (c) in ${}^{34}\mathrm{Si}$ obtained within the FTHF, FTBCS, and FTEP at $T\ne 0$.

Figure 5

Single-particle energies for protons (a) and neutrons (b) in ${}^{34}\mathrm{Si}$ obtained within the FTEP (solid lines) and FTBCS (dashed lines) as functions of temperature. The dash-dotted line in (a) is the exact proton chemical potential, obtained as ${\lambda }^{\left(N\right)}=[\mathcal{F}(N+2,T)+\mathcal{F}(N-2,T)]/4$, where $\mathcal{F}(N\pm 2,T)$ are the exact free energies of the systems with $N\pm 2$ protons, whereas the dotted line in (a) is the proton chemical potential obtained within the FTBCS. The dotted and dash-dotted lines in (b) are the chemical potential obtained within the IPM calculation (without pairing) using the neutron single-particle spectra obtained within the FTBCS and FTEP, respectively.

Figure 6

Occupation numbers of proton single-particle levels $1d_{5/2}$ (a), $2s_{1/2}$ (b), and $1d_{3/2}$ (c) in ${}^{34}\mathrm{Si}$ obtained within FTHF, FTBCS, and FTEP.

Figure 7

Proton density ${\rho }_Z\left(r\right)$ in ${}^{34}\mathrm{Si}$ obtained within the FTHF, FTBCS, and FTEP at several temperatures.

Figure 8

Neutron density ${\rho }_N\left(r\right)$ in ${}^{34}\mathrm{Si}$ obtained within the FTHF, FTBCS, and FTEP at several temperatures.

Figure 9

Depletion factor $F$ for the proton density ${\rho }_Z\left(r\right)$ obtained within the FTHF, FTBCS, and FTEP for ${}^{34}\mathrm{Si}$.

Figure 10

Wave function of the proton $2s_{1/2}$ level in ${}^{34}\mathrm{Si}$ obtained within the FTEP at several temperatures.

Figure 11

Neutron pairing gaps (a) and root-mean-square (rms) radii for protons (b) and neutrons (c) in ${}^{22}\mathrm{O}$ obtained within the FTHF, FTBCS, and FTEP.

Figure 12

Occupation numbers of proton single-particle levels $1d_{5/2}$ (a), $2s_{1/2}$ (b), and $1d_{3/2}$ (c) in ${}^{22}\mathrm{O}$ obtained within FTHF, FTBCS, and FTEP.

Figure 13

Proton [(a1)–(a3)] and neutron [(b1)–(b3)] densities in ${}^{22}\mathrm{O}$ obtained within the FTHF, FTBCS, and FTEP at several temperatures.

Figure 14

(a): Depletion factor $F$ for the neutron densities ${\rho }_N$ in ${}^{22}\mathrm{O}$ obtained within the FTHF, FTBCS, and FTEP. (b) and (c) Same as in Fig. 13 but at two critical temperatures $T_F$ correspond to the FTEP calculations which are carried out for the MSk3 and BSk14 interaction.