The kink effect of the nuclear charge radii in some isotopic chains and the nucleon-nucleon tensor force within nonlinear relativistic models in the Hartree-Fock approximation.

. Relativistic nonlinear models in the Hartree-Fock approach, with σ , ω , ρ and π mesons, are used to explore the influence of the nucleon-nucleon tensor force ( tf) on the behaviour of the nuclear charge radii of the Pb isotopic chain. It is found that a part of its effect on the charge radii is channelled through its opposite effect to the spin-orbit interaction. The kink effect seems to be produced by the combination of the geometrical features of the 1 i 11/2 neutron orbital and its binding energy.


Introduction
The marked change in trend of the evolution of the charge radii of some isotopic families of nuclei versus the mass number A is known as kink effect (KE).This is a consequence of the shell structure of nuclei.The fact that the density-dependent Hartree-Fock model with standard Skyrme functionals [1] or Gogny forces [2] were not able to reproduce this effect for the Pb isotopic chain, for example, whereas the relativistic models in the simple mean-field approach did reasonably well [3,4], has increased the interest in understanding the mechanism responsible for the KE [5][6][7][8].However, it seems that the full theoretical understanding has not been reached yet.
The aim of this work is to consider a nuclear relativistic Lagrangian and use the Hartree-Fock approximation, without pairing correlations, to explore the influence of the NN tensor force (tf) on the behaviour of the nuclear charge radii of the Pb isotopic chain.

Effect of the tensor force
Fig. 1 shows the neutron single-particle energies of 208 Pb.It can be seen that the tf reduce the spin-orbit splittings.Fig. 2  Figs. 3 and 4 show the charge radii of the proton nodeless orbitals of the Pb isotopic chain.The effect of the tf on 〈  〉 for the two orbitals of a spin-orbit doublet is opposite to each other and decreases as l increases.
The effect of the tf is observable and it seems (only) qualitatively equivalent to that of reducing the spin-orbit interaction.

Conclusions
The effect of the tensor force (tf) on the nuclear charge radii of the nodeless proton orbitals of the Pb isotopic chain is opposite to that of the spin-orbit interaction and it decreases as l increases.For orbitals with nodes, the effect is more involved and less important.The global effect of the tf on the nuclei charge radii is very small.The same as Fig. 3, but for the g-conf.

Figure 3 :
Figure 3: The sp r.m.s.charge radii of the proton nodeless orbitals of the Pb isotopic chain for the HF0 (η=0) and HF1 (η=1) models and i-conf.

Figure 2 :
Figure 2: Charge radius isotope shift Δ 〈 rc 2 〉 for the Pb isotopic chain with respect to that of the 208 Pb for models HF0 (η=0) and HF1 (η =1) and for the neutron i-and g-confs.

Figure 4 :
Figure 4:The same as Fig.3, but for the g-conf.