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

This paper presents a detailed assessment of the ability of the 240 Skyrme interaction parameter sets in the literature to satisfy a series of criteria derived from macroscopic properties of nuclear matter in the vicinity of nuclear saturation density at zero temperature and their density dependence, derived by the liquid-drop model, in experiments with giant resonances and heavy-ion collisions. The objective is to identify those parametrizations which best satisfy the current understanding of the physics of nuclear matter over a wide range of applications. Out of the 240 models, only 16 are shown to satisfy all these constraints. Additional, more microscopic, constraints on the density dependence of the neutron and proton effective mass β-equilibrium matter, Landau parameters of symmetric and pure neutron nuclear matter, and observational data on high- and low-mass cold neutron stars further reduce this number to 5, a very small group of recommended Skyrme parametrizations to be used in future applications of the Skyrme interaction of nuclear-matter-related observables. Full information on partial fulfillment of individual constraints by all Skyrme models considered is given. The results are discussed in terms of the physical interpretation of the Skyrme interaction and the validity of its use in mean-field models. Future work on application of the Skyrme forces, selected on the basis of variables of nuclear matter, in the Hartree-Fock calculation of properties of finite nuclei, is outlined.

Skyrme interaction and nuclear matter constraints

We provide convenient parametrizations of the high-density nuclear equation of state obtained within the Brueckner-Hartree-Fock approach using different modern nucleon-nucleon potentials together with compatible microscopic nuclear three-body forces. The corresponding neutron star mass-radius relations are also presented.

Neutron star structure with modern nucleonic three-body forces

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

Nuclear equation of state from ground and collective excited state properties of nuclei

We determine the saturation properties of nuclear matter within the Brueckner-Hartree-Fock approach based on a large set of modern nucleon-nucleon potentials and confirm the validity of the Coester band. The improvement of the saturation point when including nuclear three-body forces is pointed out and comparison with the Dirac-Brueckner-Hartree-Fock results is made.

U. Lombardo

Papers

Nuclear matter saturation point and symmetry energy with modern nucleon-nucleon potentials