Quantum and Coulomb repulsion effects on the bubble structures in ${}^{204,206}\mathrm{Hg}$

The decreasing proton and charge densities from around 5.0 fm towards the center of ${}^{204,206}\mathrm{Hg}$ are investigated by a covariant density functional theory at the beyond mean-field level. The charge-density difference between ${}^{208}\mathrm{Pb}$ and ${}^{204}\mathrm{Hg}$ is improved significantly and a central depression is still visible in the ground-state density of ${}^{204,206}\mathrm{Hg}$ when the dynamic correlations associated with symmetry restoration and shape mixing are taken into account. For the $0_2^{+}$ and $2_1^{+}$ excited states of ${}^{204,206}\mathrm{Hg}$, their densities remain decreasing from 5.0 fm to around 2.0 fm, but become flat in the interior region. The results show that the bubble structure in ${}^{204,206}\mathrm{Hg}$ within 2.0 fm is mainly attributed to the quantum effect, while that beyond 2.0 fm is formed by the Coulomb repulsion.

Figure 1

Comparison between the radial distribution of proton densities (a) without (w/o) and (b) with (w/) Coulomb potential, calculated from the RMF calculations for the spherical states of ${}^{208}\mathrm{Pb}$, ${}^{206}\mathrm{Hg}$, and ${}^{204}\mathrm{Hg}$ using the PC-PK1 force. The results without (w/o) pairing are also given for ${}^{206}\mathrm{Hg}$ and ${}^{204}\mathrm{Hg}$. Pairing collapse takes place in the spherical state of ${}^{208}\mathrm{Pb}$ labeled by “w/o pairing”.

Figure 2

The proton density distributions (in units of ${\mathrm{fm}}^{-3}$) of the mean-field spherical state in $y\text{−}z$ plane at $x=0$ fm by the PC-PK1 force for ${}^{208}\mathrm{Pb}$ (a), ${}^{206}\mathrm{Hg}$ (b), and ${}^{204}\mathrm{Hg}$ (c), respectively.

Figure 3

The (a) proton and (b) charge density distributions by the PC-PK1 force for ${}^{208}\mathrm{Pb}$ and ${}^{204}\mathrm{Hg}$, as well as the comparison between (c) proton and (d) charge densities for ${}^{206}\mathrm{Hg}$ by both the PC-F1 and PC-PK1. The labels “$N\&Z$(Sph. w/o)”, “$N\&Z$(Sph. w/)”, “$N\&Z,J=0$(Min.)”, and “$N\&Z,J=0$(GCM)” represent the results based on four different configurations. The densities for ${}^{208}\mathrm{Pb}$ in (a) and (b), and the density by the PC-F1 for ${}^{206}\mathrm{Hg}$ in (c) and (d) have been shifted up by 0.015 ${\mathrm{fm}}^{-3}$. The shadow area in (b) denotes the experimental uncertainty. The experimental data are taken from Refs. [4, 6]. See text for more details.

Figure 4

The radial distribution of the difference $\mathrm{\Delta }\rho \left(r\right)$ in the densities between ${}^{208}\mathrm{Pb}$ and ${}^{204}\mathrm{Hg}$. The experimental data are taken from Refs. [4, 6, 37]. The shadow area denotes the experimental uncertainty.

Figure 5

Longitudinal form factor $|F_0{\left(q\right)|}^2$ as a function of the momentum transfer $q$ (${\mathrm{fm}}^{-1}$) for the electron elastic scattering of the ground state for (a) ${}^{208}\mathrm{Pb}$, (b) ${}^{206}\mathrm{Hg}$, and (c) ${}^{204}\mathrm{Hg}$, respectively. The experimental data are taken from Ref. [37].

Figure 6

(a) Proton and (b) neutron single-particle energies corresponding to the spherical configuration of ${}^{208}\mathrm{Pb}$, ${}^{206}\mathrm{Hg}$, and ${}^{204}\mathrm{Hg}$ from the mean-field calculations with the PC-PK1 force. The results without (w/o) pairing correlations, denoted with ${}^{206}\mathrm{Hg}^{*}$ and ${}^{204}\mathrm{Hg}^{*}$, are also given for comparison. The size of spin-orbit splitting is indicated with the value in units of MeV.

Figure 7

Splitting of proton spin-orbit doublets with different principle quantum number as a function of orbital angular momentum. The results corresponding to the spherical configuration of ${}^{208}\mathrm{Pb}$, ${}^{206}\mathrm{Hg}$, and ${}^{204}\mathrm{Hg}$ are from the RMF calculations using the PC-PK1 force.

Figure 8

Comparison between (a) proton and (b) charge density distributions for the $0_1^{+},0_2^{+}$, and $2_1^{+}$ states in ${}^{204,206}\mathrm{Hg}$. For ${}^{204}\mathrm{Hg}$, the density distributions have been shifted up by 0.015 ${\mathrm{fm}}^{-3}$.