Isospin-dependent properties of asymmetric nuclear matter in relativistic mean field models

Using various relativistic mean-field models, including nonlinear ones with meson field self-interactions, models with density-dependent meson-nucleon couplings, and point-coupling models without meson fields, we have studied the isospin-dependent bulk and single-particle properties of asymmetric nuclear matter. In particular, we have determined the density dependence of nuclear symmetry energy from these different relativistic mean-field models and compared the results with the constraints recently extracted from analyses of experimental data on isospin diffusion and isotopic scaling in intermediate energy heavy-ion collisions as well as from measured isotopic dependence of the giant monopole resonances in even-$A$ Sn isotopes. Among the 23 parameter sets in the relativistic mean-field model that are commonly used for nuclear structure studies, only a few are found to give symmetry energies that are consistent with the empirical constraints. We have also studied the nuclear symmetry potential and the isospin splitting of the nucleon effective mass in isospin asymmetric nuclear matter. We find that both the momentum dependence of the nuclear symmetry potential at fixed baryon density and the isospin splitting of the nucleon effective mass in neutron-rich nuclear matter depend not only on the nuclear interactions but also on the definition of the nucleon optical potential.

Figure 1

Density dependence of the nuclear symmetry energy $E_{\mathrm{sym}}(\rho )$ for the parameter sets NL1, NL2, NL3, NL-SH, TM1, PK1, FSU-Gold, HA, NL$\rho$, and NL$\rho \delta$ in the nonlinear RMF model (a); TW99, DD-ME1, DD-ME2, PKDD, DD, DD-F, and DDRH-corr in the density-dependent RMF model (b); and PC-F1, PC-F2, PC-F3, PC-F4, PC-LA, and FKVW in the point-coupling RMF model (c). For comparison, results from the MDI interaction with $x=-1$ and 0 are also shown.

Figure 2

Symmetry energy $E_{\mathrm{sym}}(\rho )$ (a) and the scaled symmetry energy $E_{\mathrm{sym}}(\rho )/E_{\mathrm{sym}}({\rho }_0)$ (b) as functions of the scaled baryon density $\rho /{\rho }_0$ for the 23 parameter sets in the nonlinear, density-dependent, and point-coupling RMF models. Results of the MDI interaction with $x=-1$ and 0 are also included for comparison. The inset in panel (b) shows the symmetry energy $E_{\mathrm{sym}}(\rho )$ as a function of the baryon density $\rho$ without scaling.

Figure 3

Values of $L$ and $K_{\mathrm{asy}}$ for the 23 parameter sets in the nonlinear, density-dependent, and point-coupling RMF models. The constraints from the isospin diffusion data (shaded band), the isoscaling data (solid circles), and the isotopic dependence of the GMR in even-$A$ Sn isotopes (dashed rectangle) are also included.

Figure 4

Correlation between $L$ and $K_{\text{asy}}$ for the 23 parameter sets in the nonlinear, density-dependent, and point-coupling RMF models. The constraints from the isospin diffusion data (shaded band), the isoscaling data (stars), and the isotopic dependence of the GMR in even-$A$ Sn isotopes (dashed rectangle with $L$ constrained by the isospin diffusion data) are also included.

Figure 5

Energy dependence of the three different nucleon optical potentials, i.e., $U_{\text{SEP}}$ [Eq. (10)], $U_{\text{OPT}}$ [Eq. (12)], and $U_{\text{SOD}}$ [Eq. (15)] (left panels) as well as their corresponding symmetry potentials $U_{\mathrm{sym}}^{\text{SEP}}$, $U_{\text{sym}}^{\text{OPT}}$, and $U_{\mathrm{sym}}^{\text{SOD}}$ as functions of momentum (right panels), at a fixed baryon density ${\rho }_B=0.16$ fm${}^{-3}$ for parameter sets NL1, NL2, NL3, NL-SH, TM1, PK1, FSU-Gold, HA, NL$\rho$, and NL$\rho \delta$ in the nonlinear RMF model. For comparison, the energy dependence of the real part of the optical potential in symmetric nuclear matter at saturation density extracted from two different fits of the proton-nucleus scattering data in the Dirac phenomenology are also included (left panels).

Figure 6

Same as Fig. 5, but for TW99, DD-ME1, DD-ME2, PKDD, DD, DD-F, and DDRH-corr in the density-dependent RMF models.

Figure 7

Same as Fig. 5, but for PC-F1, PC-F2, PC-F3, PC-F4, PC-LA, and FKVW in the point-coupling RMF models.

Figure 8

Density dependence of different nucleon effective masses, i.e., $M_{\mathrm{Dirac}}^{*}/M$, $M_{\mathrm{Landau}}^{*}/M$, $M_{\mathrm{Lorentz}}^{*}/M$, $M_{\mathrm{OPT}}^{*}/M$, and $M_{\mathrm{SOD}}^{*}/M$ in symmetric nuclear matter as well as their corresponding isospin splittings in neutron-rich nuclear matter with isospin asymmetry $\alpha =0.5$ for the parameter sets NL1, NL2, NL3, NL-SH, TM1, PK1, FSU-Gold, HA, NL$\rho$, and NL$\rho \delta$ in the nonlinear RMF model.

Figure 9

Same as Fig. 8, but for TW99, DD-ME1, DD-ME2, PKDD, DD, DD-F, and DDRH-corr in the density-dependent RMF model.

Figure 10

Same as Fig. 8, but for PC-F1, PC-F2, PC-F3, PC-F4, PC-LA, and FKVW in the point-coupling RMF model.

Figure 11

Neutron and proton scalar densities as functions of baryon density in nuclear matter with isospin asymmetry $\alpha =0$ and 0.5 for the parameter sets NL1, NL2, NL3, NL-SH, TM1, PK1, FSU-Gold, HA, NL$\rho$, and NL$\rho \delta$ of the nonlinear RMF model (a); TW99, DD-ME1, DD-ME2, PKDD, DD, DD-F, and DDRH-corr of the density-dependent RMF model (b); PC-F1, PC-F2, PC-F3, PC-F4, PC-LA, and FKVW of the point-coupling RMF model (c).