Spin-orbit splittings of neutron states in $N=20$ isotones from covariant density functionals and their extensions

Spin-orbit splitting is an essential ingredient for our understanding of the shell structure in nuclei. One of the most important advantages of relativistic mean-field (RMF) models in nuclear physics is the fact that the large spin-orbit (SO) potential emerges automatically from the inclusion of Lorentz-scalar and -vector potentials in the Dirac equation. It is therefore of great importance to compare the results of such models with experimental data. We investigate the size of $2p$ and $1f$ splittings for the isotone chain ${}^{40}\mathrm{Ca}$, ${}^{38}\mathrm{Ar}$, ${}^{36}\mathrm{S}$, and ${}^{34}\mathrm{Si}$ in the framework of various relativistic and nonrelativistic density functionals. They are compared with the results of nonrelativistic models and with recent experimental data.

Figure 1

Proton densities of the nuclei ${}^{40}\mathrm{Ca}$, ${}^{38}\mathrm{Ar}$, ${}^{36}\mathrm{S}$, and ${}^{34}\mathrm{Si}$ for the functional DD-ME2.

Figure 2

Radial profiles of the $2p_{1/2}$ (full) and the $1f_{5/2}$ (dashed) neutron state for ${}^{40}\mathrm{Ca}$ and ${}^{34}\mathrm{Si}$.

Figure 3

Evolution of spin-orbit splittings for the neutron levels $p$ (left panel) and $f$ (right panel) with respect to the mass number $A$, without pairing.

Figure 4

Same as Fig. 3 but with TMR pairing.

Figure 5

Radial dependence of the total density ${\rho }_{\mathrm{tot}}$ and the proton density ${\rho }_p$ for NL3 with and without pairing correlations for the nuclei ${}^{38}\mathrm{Ar}$, ${}^{36}\mathrm{S}$, and ${}^{34}\mathrm{Si}$.

Figure 6

Neutron $2p_{1/2}\text{−}2p_{3/2}$ splitting relative reduction with respect to occupation change in the proton state $2s_{1/2}$ going from ${}^{36}\mathrm{S}$ to ${}^{34}\mathrm{Si}$. More details are given in the text.

Figure 7

The evolution of the energy splittings for the NL3RHF0.5 functional, where boxes (a)–(d) correspond to ${}^{40}\mathrm{Ca}$, ${}^{38}\mathrm{Ar}$, ${}^{36}\mathrm{S}$, and ${}^{34}\mathrm{Si}$, respectively. The red dashed lines represent the experimental values of the centroids for ${}^{40}\mathrm{Ca}$ [22].

Figure 8

Change of single-particle energies of the $1f_{5/2}$ and $1f_{7/2}$ and of the $2p_{1/2}$ and $2p_3/2$ neutron states as we move down the chain of isotones $N=20$. The red lines correspond to the tensor results and the blue lines to the standard NL3 with frozen $\mathrm{\Delta }$.

Figure 9

Evolution of the $p$ and $f$ SO splittings for NL3.

Figure 10

Distribution of the major fragments of the single-particle strengths of (a) ${}^{37}\mathrm{S}$ and (d) ${}^{35}\mathrm{Si}$ as given in Ref. [19] and the same distribution calculated with PVC for (b), (e) the force ${\mathrm{NL}3}^{*}$. Panels (c) and (f) show results obtained without particle-vibration coupling using the same density functional.