Recent Progress and New Challenges in Isospin Physics with Heavy-Ion Reactions

Constraining the neutron–proton effective mass splitting using empirical constraints on the density dependence of nuclear symmetry energy around normal density

Density-dependent relativistic Hartree-Fock approach

Analytical expressions for the saturation density of asymmetric nuclear matter as well as its binding energy and incompressibility at saturation density are given up to fourth order in the isospin asymmetry $\ensuremath{\delta}=({\ensuremath{\rho}}_{n}\ensuremath{-}{\ensuremath{\rho}}_{p})/\ensuremath{\rho}$ using 11 characteristic parameters defined by the density derivatives of the binding energy per nucleon of symmetric nuclear matter, the symmetry energy ${E}_{\mathrm{sym}} (\ensuremath{\rho})$, and the fourth-order symmetry energy ${E}_{\mathrm{sym},4}(\ensuremath{\rho})$ at normal nuclear density ${\ensuremath{\rho}}_{0}$. Using an isospin- and momentum-dependent modified Gogny interaction (MDI) and the Skyrme-Hartree-Fock (SHF) approach with 63 popular Skyrme interactions, we have systematically studied the isospin dependence of the saturation properties of asymmetric nuclear matter, particularly the incompressibility ${K}_{\mathrm{sat}}(\ensuremath{\delta})={K}_{0}+{K}_{\mathrm{sat},2}{\ensuremath{\delta}}^{2}+{K}_{\mathrm{sat},4}{\ensuremath{\delta}}^{4}+O({\ensuremath{\delta}}^{6})$ at saturation density. Our results show that the magnitude of the higher order ${K}_{\mathrm{sat},4}$ parameter is generally small compared to that of the ${K}_{\mathrm{sat},2}$ parameter. The latter essentially characterizes the isospin dependence of the incompressibility at saturation density and can be expressed as ${K}_{\mathrm{sat},2}={K}_{\mathrm{sym}}\ensuremath{-}6L\ensuremath{-}\frac{{J}_{0}}{{K}_{0}}L$, where $L$ and ${K}_{\mathrm{sym}}$ represent, respectively, the slope and curvature parameters of the symmetry energy at ${\ensuremath{\rho}}_{0}$ and ${J}_{0}$ is the third-order derivative parameter of symmetric nuclear matter at ${\ensuremath{\rho}}_{0}$. Furthermore, we have constructed a phenomenological modified Skyrme-like (MSL) model that can reasonably describe the general properties of symmetric nuclear matter and the symmetry energy predicted by both the MDI model and the SHF approach. The results indicate that the higher order ${J}_{0}$ contribution to ${K}_{\mathrm{sat},2}$ generally cannot be neglected. In addition, it is found that there exists a nicely linear correlation between ${K}_{\mathrm{sym}}$ and $L$ as well as between ${J}_{0}/{K}_{0}$ and ${K}_{0}$. These correlations together with the empirical constraints on ${K}_{0},L,{E}_{\mathrm{sym}} ({\ensuremath{\rho}}_{0})$, and the nucleon effective mass lead to an estimate of ${K}_{\mathrm{sat},2}=\ensuremath{-}370\ifmmode\pm\else\textpm\fi{}120$ MeV.

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Higher-order effects on the incompressibility of isospin asymmetric nuclear matter

Nucleon Effective Masses in Neutron-Rich Matter

The isospin splitting of the nucleon mean field is derived from the Brueckner theory extended to asymmetric nuclear matter. The Argonne ${V}_{18}$ has been adopted as bare interaction in combination with a microscopic three-body force. The isospin splitting of the effective mass is determined from the Brueckner-Hartree-Fock self-energy: It is linear according to the Lane ansatz, such that ${m}_{n}^{*}g{m}_{p}^{*}$ for neutron-rich matter. The symmetry potential is also determined, and a comparison is made with the predictions of the Dirac-Brueckner approach and the phenomenological interactions. The theoretical predictions are also compared with the empirical parametrizations of neutron and proton optical model potentials based on the experimental nucleon-nucleus scattering and the phenomenological ones adopted in transport model simulations of heavy-ion collisions. The direct contribution of the rearrangement term due to three-body forces to the single-particle potential and symmetry potential is discussed.

L. G. Cao

Papers

Isospin splitting of the nucleon mean field