Nuclear matter fourth-order symmetry energy in nonrelativistic mean-field models

Abstract

Background: Nuclear matter fourth-order symmetry energy Esym,4(ρ) may significantly influence the properties of neutron stars such as the core-crust transition density and pressure as well as the proton fraction at high densities. The magnitude of Esym,4(ρ) is, however, largely uncertain. Purpose: Based on systematic analyses of several popular non-relativistic energy density functionals with mean-field approximation, we estimate the value of the Esym,4(ρ) at nuclear normal density ρ0 and its density dependence, and explore the correlation between Esym,4(ρ0) and other macroscopic quantities of nuclear matter properties. Method: We use the empirical values of some nuclear macroscopic quantities to construct model parameter sets by Monte Carlo method for four different energy density functionals with mean-field approximation, namely, the conventional Skyrme-Hartree-Fock (SHF) model, the extended SkyrmeHartree-Fock (eSHF) model, the Gogny-Hartree-Fock (GHF) model, and the momentum-dependent interaction (MDI) model. With the constructed samples of parameter sets, we can estimate the density dependence of Esym,4(ρ) and analyze the correlation of Esym,4(ρ0) with other macroscopic quantities.Results: The value of Esym,4(ρ0) is estimated to be 1.02 ± 0.49 MeV for the SHF model, 1.02 ± 0.50 MeV for the eSHF model, 0.70 ± 0.60 MeV for the GHF model, and 0.74 ± 0.63 MeV for the MDI model. Moreover, our results indicate that the density dependence of Esym,4(ρ) is model dependent, especially at higher densities. Furthermore, we find that the Esym,4(ρ0) has strong positive (negative) correlation with isoscalar (isovector) nucleon effective mass m * s,0 (m * v,0 ) at ρ0. In particular, for the SHF and eSHF models, the Esym,4(ρ) is completely determined by the isoscalar and isovector nucleon effective masses m * s (ρ) and m * v (ρ), and the analytical expression is given. Conclusions: In the mean-field models, the magnitude of Esym,4(ρ0) is generally less than 2 MeV, and its density dependence depends on models, especially at higher densities. Esym,4(ρ0) is strongly correlated with m * s,0 and m * v,0 .

Anomalous quartic term in the expansion of the symmetry energy

“…In the relativistic mean field (RMF) framework, the quartic term has been analyzed in [7] and [8] and appears to be small, typically, one order of magnitude smaller than S 2 . Other theoretical predictions for S 4 based on the Hartree-Fock theory [9][10][11][12] lead to similar conclusions.…”

“…Summing up, for S 2 one needs to know ∂δ ∂x and S 4 requires ∂δ ∂x , ∂ 3 δ ∂x 3 , ∂ 2 σ ∂x 2 , which can be obtained by the differentiation of the equations of motion, Eqs. (11) and (12). In the following, we assume k p = k n = k and m * p = m * n = m * for symmetric nuclear matter.…”

Accurate nuclear symmetry energy at finite temperature within a BHF approach

“…Our results are J 4 (0) = 0.41, 0.93, 1.17, 1.17 MeV for the V18, BOB, SFHo, Shen EOS, respectively. Within energy density functionals with mean-field approximation, for example Skyrme-Hartree-Fock and Gogny-Hartree-Fock models, the values of J 4 reported in the literature are around 1.0 MeV [59], and around 0.66 MeV within RMF models [55], while values extracted from Quantum Molecular Dynamics models could be larger depending on the specific interaction [57]. From the view point of finite nuclei, J 4 can be related to the second-order symmetry energy a sym,4 (A) in a semi-empirical mass formula, in which the latter can be inferred from the double difference of "experimental" symmetry energies by analyzing the binding energies of a large number of measured nuclei [69,70].…”

“…which is usually a good approximation at zero temperature [17,18,53], and also used at finite temperature [19]. It has, however, been pointed out [20][21][22][56][57][58][59][60][61][62][63] that at least the kinetic part of the free energy density [first term in Eq. ( 1)] violates the parabolic law, in particular at high temperature.…”

Basic quantities of the Equation of State in isospin asymmetric nuclear matter

“…It should be emphasized that, at both sub-saturation and supra-saturation densities, the quadratic symmetry energy is not well constrained, especially at supra-saturation densities [30][31][32][33]. The quartic symmetry energy E sym,4 (ρ 0 ) is predicted to be less than 1 MeV [34][35][36]. In contrast to the quadratic one, few studies have been conducted on the quartic density slope L 4 (ρ 0 ) and the corresponding incompressibility coefficient K 4 (ρ 0 ) [37].…”