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The evolution of the energy difference between the neutron states $1i_{13/2}$and $1h_{9/2}$ in the $N=82$ isotones and that between the proton states$1h_{11/2}$ and $1g_{7/2}$ in the $Z = 50$ isotopes are investigated within theframework of the relativistic Hartree-Fock theory, using the density-dependenteffective interactions PKA1 and PKO$i$ ($i = 1 $, $ 2 $, $ 3 $). By identifyingthe contributions of the tensor force, which is naturally induced via the Fockterms, we find that the tensor force plays crucial roles in the evolution ofthe shell structure. The strength of the tensor force is also explored. It isfound that moderately increasing the coupling strength of pion-nucleoncoupling, i.e., $f_{\pi}$, will significantly improve the description of theshell-structure evolution. In particular, reducing the density dependence of$f_{\pi}$ is shown to be more preferable, in comparison to enlarging $f_{\pi}$with a factor. This is in consistence with the idea of "tensor renormalizationpersistency" and provides valuable guidance for the development of nuclearenergy density functional in the relativistic framework.

Tensor-force effects on shell-structure evolution in $N = 82$ isotones and $Z = 50$ isotopes in the relativistic Hartree-Fock theory