Quantitative analysis of tensor effects in the relativistic Hartree-Fock theory

Tensor force is identified in each meson-nucleon coupling in the relativistic Hartree-Fock theory. It is found that all the meson-nucleon couplings, except the $\sigma$-scalar one, give rise to the tensor force. The effects of tensor force on various nuclear properties can now be investigated quantitatively, which allows fair and direct comparisons with the corresponding results in the nonrelativistic framework. The tensor effects on nuclear binding energies and the evolutions of the $Z,N=8,20$, and 28 magic gaps are studied. The tensor contributions to the binding energies are shown to be tiny in general. The $Z,N=8$ and 20 gaps are sensitive to the tensor force, but the $Z,N=28$ gaps are not.

Figure 1

Dirac mass $M^{*}$ at the Fermi momentum as a function of matter density $\rho$ calculated by the RHF theory with the PKA1 effective interaction. The $M^{*}$ of the symmetric nuclear matter are shown with the solid line, while the $M_n^{*}$ and $M_p^{*}$ of the asymmetric matter with $\beta =0.2$ are shown with the dashed and short-dashed lines, respectively.

Figure 2

Proton gap $Z=8$ in the O isotopes and the contributions from the tensor force in each meson-nucleon coupling as a function of neutron number $N$, calculated by the RHF theory with the PKA1 effective interaction. The results with and without the tensor contributions are shown with filled and open circles, respectively. The total tensor contributions are shown with filled squares, while the contributions from each coupling are denoted with different symbols. See the text for the details of the empirical values.

Figure 3

Same as described in the caption of Fig. 2, but for the neutron gap $N=8$ in the $N=8$ isotones as a function of proton number $Z$.

Figure 4

Same as described in the caption of Fig. 2, but for the proton gap $Z=20$ in the Ca isotopes.

Figure 5

Same as described in the caption of Fig. 3, but for the neutron gap $N=20$ in the $N=20$ isotones.

Figure 6

Same as described in the caption of Fig. 2, but for the proton gap $Z=28$ in the Ni isotopes.

Figure 7

Same as described in the caption of Fig. 3, but for the neutron gap $N=28$ in the $N=28$ isotones.

Figure 8

Ratio between the starred momentum and its undressed counterpart ${\mathit{p}}^{*}/\mathit{p}$ as a function of $p/p_F$ for the (a) symmetric ($\beta =0$) and (b) asymmetric ($\beta =0.2$) nuclear matter with $\rho =0.1{\rho }_{\mathrm{sat}.},0.5{\rho }_{\mathrm{sat}.}$, and ${\rho }_{\mathrm{sat}.}$ calculated by RHF with PKA1.

Figure 9

Ratio between the Dirac mass and nucleon mass $M^{*}/M$ as a function of $p/p_F$ for the (a) symmetric ($\beta =0$) and (b) asymmetric ($\beta =0.2$) nuclear matter with $\rho =0.1{\rho }_{\mathrm{sat}.},0.5{\rho }_{\mathrm{sat}.}$, and ${\rho }_{\mathrm{sat}.}$ calculated by RHF with PKA1.