Pseudospin-orbit splitting and its consequences for the central depression in nuclear density

The occurrence of the bubble-like structure has been studied, in the light of pseudospin degeneracy, within the relativistic Hartree-Fock-Bogoliubov (RHFB) theory. It is concluded that the charge/neutron bubble-like structure is predicted to occur in the mirror system of {${}^{34}\mathrm{Si},{}^{34}\mathrm{Ca}$} commonly by the selected Lagrangians, due to the persistence of $Z\left(N\right)=14$ subshell gaps above which the $\pi \left(\nu \right)2s_{1/2}$ states are not occupied. However, for the popular candidate ${}^{46}\mathrm{Ar}$, the RHFB Lagrangian PKA1 does not support the occurrence of the bubble-like structure in the charge (proton) density profiles, due to the almost degenerate pseudospin doublet $\{\pi 2s_{1/2},\pi 1d_{3/2}\}$ and coherent pairing effects. The formation of a semibubble in heavy nuclei is less possible as a result of small pseudospin-orbit (PSO) splitting, while it tends to appear at $Z=120$ superheavy systems which coincides with large PSO splitting of the doublet $\{\pi 3p_{3/2},\pi 2f_{5/2}\}$ and couples with significant shell effects. Pairing correlations, which can work against bubble formation, significantly affect the PSO splitting. Furthermore, we found that the influence on semibubble formation due to different types of pairing interactions is negligible. The quenching of the spin-orbit splitting in the $p$ orbit has been also stressed, and it may be considered the hallmark for semibubble nuclei.

Figure 1

Canonical proton single-particle energies along the $N=20$ isotonic chain calculated by RHFB with PKA1 and RHB with DD-ME2. The lengths of thick bars correspond with the occupation probabilities of the proton orbits and the filled stars denote the experimental data that are taken from Ref. [85].

Figure 2

Charge distributions of $N=20$ isotones calculated by RHFB with PKA1 and PKO3 and by RHB with DD-ME2. The experimental data for ${}^{36}\mathrm{S}$ are taken from Ref. [92].

Figure 3

Same as Fig. 1, but for $N=28$ isotones.

Figure 4

Charge distributions of $N=28$ isotones calculated by RHFB with PKA1 and PKO3 and by RHB with DD-ME2. The experimental data for ${}^{48}\mathrm{Ca}$ are taken from Ref. [97].

Figure 5

Charge distributions of $N=126$ isotones, calculated by RHFB with PKA1 and PKO3 and by RHB with DD-ME2. The experimental data for ${}^{208}\mathrm{Pb}$ are taken from Ref. [97].

Figure 6

Charge distributions of Hg isotopes, calculated by RHFB with PKA1 and PKO3 and by RHB with DD-ME2. The experimental data for ${}^{204}\mathrm{Hg}$ are taken from Ref. [101].

Figure 7

Canonical proton single-particle energies along the Hg isotopic chain calculated by RHFB with PKA1 and by RHB with DD-ME2. The lengths of thick bars correspond to the occupation probabilities of the proton orbits. The size of pseudospin-orbit splitting is indicated with the value in units of MeV.

Figure 8

Neutron distributions of $N=14$ isotones ${}^{22}\mathrm{O},{}^{34}\mathrm{Ca}$ and $N=16$ isotones ${}^{24}\mathrm{O},{}^{36}\mathrm{Ca}$, calculated by RHFB with PKA1 and PKO3 and by RHB with DD-ME2.

Figure 9

Proton single-particle states at spherical shape in the nuclei ${}^{292}120,{}^{304}120$, and ${}^{318}120$, calculated by RHFB with PKA1 and RHB with DD-ME2. The lengths of thick bars correspond to the occupation probabilities of the orbits.

Figure 10

Neutron and proton distributions of $Z=120$ isotopes calculated by RHFB with PKA1 and PKO3 and by RHB with DD-ME2.

Figure 11

The contour of neutron and proton densities for the semibubble candidates ${}^{34}\mathrm{Si},{}^{34}\mathrm{Ca},{}^{190}\mathrm{Gd}$, and ${}^{292}120$, calculated by RHFB with PKA1.

Figure 12

Charge distributions of ${}^{46}\mathrm{Ar}$ calculated by RH(F)B with different pairing interactions.

Figure 13

The $\pi 1\tilde{p}$ PSO and $\pi 2p$ SO splittings in ${}^{46}\mathrm{Ar}$ as a function of the neutron pairing gap, calculated from PKA1 with Gogny D1S (a,c) and DDDI (b,d) pairing forces. The lengths of thick bars correspond to the occupation probabilities of the orbits.

Figure 14

The SO splittings of $\nu 2p$, calculated by different Lagrangians. The experimental value (star) of ${}^{40}\mathrm{Ca}$ [85] is also displayed as a reference.