Relativistic effects in three-nucleon forces for nuclear matter and finite nuclei

In order to simulate the relativistic effects of the Dirac-Brueckner-Hartree-Fock approach for finite nuclei, the part of the Urbana three-nucleon (3N) force is considered, which represents the enhancement of the small components of the Dirac spinors for the nucleons in the nuclear medium. This 3N force is included in a Brueckner-Hartree-Fock calculation with rearrangement terms using a realistic model for the NN interaction. The strength of the 3N force is adjusted to reproduce the empirical saturation point of nuclear matter and then used in corresponding studies of the closed shell nuclei ${}^{16}\mathrm{O}$ and ${}^{40}\mathrm{Ca}$. Special attention is paid to a consistent treatment of the spectrum of particle states in the NN propagator of the Bethe-Goldstone equation.

Figure 1

Results for symmetric nuclear matter calculations using the $pn$ interaction of the CD-Bonn potential. The left panel presents results for the single-particle potential $U\left(k\right)$ assuming a Fermi momentum $k_F$ of $1.36 {\mathrm{fm}}^{-1}$, which corresponds to the empirical saturation densities. Results are displayed for the BHF approximation using the angle -average in the Bethe-Goldstone equation (BHF AA) and BHF, RBHF [see Eq. (16)] and DHF [see Eq. (20)] calculations solving the Bethe-Goldstone equation without this approximation. The right panel shows the corresponding results for the energy per nucleon calculated at various Fermi momenta.

Figure 2

Results for symmetric nuclear matter calculations using the CD-Bonn potential with (red curves) and without inclusion (black curves) of the 3N potential. The left panel presents results for the single-particle potential $U\left(k\right)$ assuming a Fermi momentum $k_F$ of $1.36 {\mathrm{fm}}^{-1}$ for the RBHF approximation. The right panel shows results for the energy per nucleon calculated at various Fermi momenta for the of BHF, RBHF, and DHF approximations.

Figure 3

Results for the energy per nucleon and the radius of the charge distribution $\left(R_c\right)$ for ${}^{16}\mathrm{O}$ (left panel) and ${}^{40}\mathrm{Ca}$ are presented in an energy versus $1/R_c$ plot to enable comparison with the corresponding figures for nuclear matter (Figs. 1 and 2). Results referring to BHF calculations with different values of $C$ in Eq. (10) defining the particle spectrum in the Bethe-Goldstone equation are connected by a red solid line; those of RBHF calculations are connected by a blue dashed line. Results of calculations with just the 2N interaction, with inclusion of the 3N term, and including the 3N term via a global density approximation are visualized by a dot, a box, and a cross, respectively. The experimental result is given in terms of a green diamond symbol.