Slope Of Symmetry Energy Research Articles

Nuclear matter and neutron star properties with the extended Nambu-Jona-Lasinio model * * Supported by Hunan Provincial Natural Science Foundation of China (2015JJ2005), the Science Research Foundation of Education Department of Hunan Province (15C0029), the National Natural Science Foundation of China (11103001), and the construct program of the key discipline in Hunan province

An extended Nambu-Jona-Lasinio (eNJL) model with nucleons as the degrees of freedom is used to investigate properties of nuclear matter and neutron stars (NSs), including the binding energy and symmetry energy of the nuclear matter, the core-crust transition density, and mass-radius relation of NSs. The fourth-order symmetry energy at saturation density is also investigated. When the bulk properties of nuclear matter at saturation density are used to determine the model parameters, the double solutions of parameters are obtained for a given nuclear incompressibility. It is shown that the isovector-vector interaction has a significant influence on the nuclear matter and NS properties, and the sign of isovector-vector coupling constant is critical in the determination of the trend of the symmetry energy and equation of state. The effects of the other model parameters and symmetry energy slope at saturation density are discussed.

Delineating effects of nuclear symmetry energy on the radii and tidal polarizabilities of neutron stars

What can we learn about the density dependence of nuclear symmetry energy from precise measurements of the radius (R1.4) and/or tidal polarizability (Λ1.4) of canonical neutron stars with a mass of 1.4 ? With the parameterized using three parameters L, Ksym, and Jsym which have the asymptotic meaning of being respectively the slope, curvature, and skewness of symmetry energy at saturation density, we found that, while both the R1.4 and Λ1.4 depend strongly on the slope L, the Ksym and Jsym parameters characterizing the high-density behavior of Esym(ρ) also play appreciable roles. Thus, there is not a simple relation between the Λ1.4/R1.4 and L alone. Precise measurements of just the Λ1.4 and R1.4 cannot completely determine the Esym(ρ) but limit combinations of its parameters. In particular, stringent constraints approximately independent of the Jsym on the L–Ksym correlations can be obtained. However, infinite combinations of the larger (smaller) L and smaller (larger) Ksym can lead to the same Λ1.4 and R1.4. Additional observables including those from terrestrial nuclear experiments are thus necessary to break this degeneracy in order to completely determine the density dependence of nuclear symmetry energy Esym(ρ).

Crust-core properties of neutron stars in the Nambu–Jona-Lasinio model**Supported by National Natural Science Foundation of China (11775049, 11275048) and the China Jiangsu Provincial Natural Science Foundation (BK20131286)

We adopt the Nambu–Jona-Lasinio (NJL) model to study the crust-core transition properties in neutron stars (NSs). For a given momentum cutoff and symmetry energy of saturation density in the NJL model, decreasing the slope of the symmetry energy gives rise to an increase in the crust-core transition density and transition pressure. Given the slope of the symmetry energy at saturation density, the transition density and corresponding transition pressure increase with increasing symmetry energy. The increasing trend between the fraction of the crustal moment of inertia and the slope of symmetry energy at saturation density indicates that a relatively large momentum cutoff of the NJL model is preferred. For a momentum cutoff of 500 MeV, the fraction of the crustal moment of inertia clearly increases with the slope of symmetry energy at saturation density. Thus, at the required fraction (7%) of the crustal moment of inertia, the NJL model with momentum cutoff of 500 MeV and a large slope of the symmetry energy of saturation density can give the upper limit of the mass of the Vela pulsar to be above 1.40 .

Testing the Formation Scenarios of Binary Neutron Star Systems with Measurements of the Neutron Star Moment of Inertia

Abstract Two low-mass (M < 1.4 M ⊙) neutron stars, J0737-3039B and the companion to J1756-2251, show strong evidence of being formed in an ultra-stripped supernova explosion (US-SN) with a ONeMg or Fe progenitor. Using systematically generated sets of equations of state we map out the relationship between the moment of inertia of J0737-3039A, a candidate for a moment of inertia measurement within a decade, and the binding energy of the two low-mass neutron stars. This relationship, similar to the I-Love-Q relations, is more robust than a previously explored correlation between the binding energy and the slope of the nuclear symmetry energy L. We find that, if either J0737-3039B or the J1756-2251 companion were formed in a US-SN, no more than 0.06 M ⊙ could have been lost from the progenitor core. Furthermore, a measurement of the moment of inertia of J0737-3039A to within 10% accuracy can discriminate between formation scenarios and, given current constraints on the predicted core mass loss, potentially rule them out. Advanced LIGO can potentially measure the neutron star tidal polarizability to equivalent accuracy which, using the I-Love-Q relations, would obtain similar constraints on the formation scenarios. Such information would help constrain important aspects of binary evolution used for population synthesis predictions of the rate of binary neutron star mergers and resulting electromagnetic and gravitational wave signals. Further progress needs to be made in modeling the core-collapse process that leads to low-mass neutron stars, particularly in making robust predictions for the mass loss from the progenitor core.

Nuclear fourth-order symmetry energy and its effects on neutron star properties in the relativistic Hartree-Fock theory

Adopting the density dependent relativistic mean-field (RMF) and relativistic Hartree-Fock (RHF) approaches, the properties of the nuclear fourth-order symmetry energy $S_4$ are studied within the covariant density functional (CDF) theory. It is found that the fourth-order symmetry energies are suppressed in RHF at both saturation and supranuclear densities, where the extra contribution from the Fock terms is demonstrated, specifically via the isoscalar meson-nucleon coupling channels. The reservation of $S_4$ and higher-order symmetry energies in the nuclear equation of state then affects essentially the prediction of neutron star properties, which is illustrated in the quantities such as the proton fraction, the core-crust transition density as well as the fraction of crustal moment of inertia. Since the Fock terms enhance the density dependence of the thermodynamical potential, the RHF calculations predict systematically smaller values of density, proton fraction and pressure at the core-crust transition boundary of neutron stars than density dependent RMF ones. In addition, a linear anti-correlation between the core-crust transition density $\rho_t$ and the density slope of symmetry energy $L$ is found which is then utilized to constrain the core-crust transition density as $\rho_t\thicksim[0.069, 0.098]~\rm{fm}^{-3}$ with the recent empirical information on $L$. The study clarifies the important role of the fourth-order symmetry energy in determining the properties of nuclear matter at extreme isospin or density conditions.

Bootstrap Technique to Study Correlation Between Neutron Skin Thickness and the Slope of Symmetry Energy in Atomic Nuclei

We present a new statistical tool based on random sampling to assess the confidence interval of Pearson's and Spearman's correlation coefficients. These estimators are then used to quantify the statistical correlations among the neutron skin thickness of atomic nuclei and the slope of the symmetry energy in the infinite nuclear medium.

Symmetry Parameter Constraints from a Lower Bound on Neutron-matter Energy

Abstract We propose the existence of a lower bound on the energy of pure neutron matter (PNM) on the basis of unitary-gas considerations. We discuss its justification from experimental studies of cold atoms as well as from theoretical studies of neutron matter. We demonstrate that this bound results in limits to the density-dependent symmetry energy, which is the difference between the energies of symmetric nuclear matter and PNM. In particular, this bound leads to a lower limit to the volume symmetry energy parameter S 0. In addition, for assumed values of S 0 above this minimum, this bound implies both upper and lower limits to the symmetry energy slope parameter L ,which describes the lowest-order density dependence of the symmetry energy. A lower bound on neutron-matter incompressibility is also obtained. These bounds are found to be consistent with both recent calculations of the energies of PNM and constraints from nuclear experiments. Our results are significant because several equations of state that are currently used in astrophysical simulations of supernovae and neutron star mergers, as well as in nuclear physics simulations of heavy-ion collisions, have symmetry energy parameters that violate these bounds. Furthermore, below the nuclear saturation density, the bound on neutron-matter energies leads to a lower limit to the density-dependent symmetry energy, which leads to upper limits to the nuclear surface symmetry parameter and the neutron-star crust–core boundary. We also obtain a lower limit to the neutron-skin thicknesses of neutron-rich nuclei. Above the nuclear saturation density, the bound on neutron-matter energies also leads to an upper limit to the symmetry energy, with implications for neutron-star cooling via the direct Urca process.

Warm unstable asymmetric nuclear matter: Critical properties and the density dependence of the symmetry energy

The spinodal instabilities in hot asymmetric nuclear matter and some important critical parameters derived thereof are studied using six different families of relativistic mean-field (RMF) models. The slopes of the symmetry energy coefficient vary over a wide range within each family. The critical densities and proton fractions are more sensitive to the symmetry energy slope parameter at temperatures much below its critical value ($T_c\sim$14-16 MeV). The spread in the critical proton fraction at a given symmetry energy slope parameter is noticeably larger near $T_c$, indicating that the warm equation of state of asymmetric nuclear matter at sub-saturation densities is not sufficiently constrained. The distillation effects are sensitive to the density dependence of the symmetry energy at low temperatures which tend to wash out with increasing temperature.

Thermodynamical instabilities under strong magnetic fields

The thermodynamical instabilities of low densities in the $np$ matter and $npe$ matter are studied within several relativistic nuclear models under some values of magnetic fields. The results are compared between each other and the effects of the symmetry energy slope at saturation density on the instability are investigated. The instability regions can exhibit bands due to the presence of Landau levels for very strong magnetic fields of the order of ${10}^{17}$ G, while for weaker magnetic fields, the bands are replaced by many diffused or scattered pieces. It also shows that the proton fraction in the inner crust of neutron stars may be complex under strong magnetic fields.

Strong correlations of neutron star radii with the slopes of nuclear matter incompressibility and symmetry energy at saturation

We examine the correlations of neutron star radii with the nuclear matter incompressibility, symmetry energy, and their slopes, which are the key parameters of the equation of state (EoS) of asymmetric nuclear matter. The neutron star radii and the EoS parameters are evaluated using a representative set of 24 Skyrme-type effective forces and 18 relativistic mean field models, and two microscopic calculations, all describing 2$M_\odot$ neutron stars. Unified EoSs for the inner-crust-core region have been built for all the phenomenological models, both relativistic and non-relativistic. Our investigation shows the existence of a strong correlation of the neutron star radii with the linear combination of the slopes of the nuclear matter incompressibility and the symmetry energy coefficients at the saturation density. Such correlations are found to be almost independent of the neutron star mass in the range $0.6\text{-}1.8M_{\odot}$. This correlation can be linked to the empirical relation existing between the star radius and the pressure at a nucleonic density between one and two times saturation density, and the dependence of the pressure on the nuclear matter incompressibility, its slope and the symmetry energy slope. The slopes of the nuclear matter incompressibility and the symmetry energy coefficients as estimated from the finite nuclei data yield the radius of a $1.4M_{\odot}$ neutron star in the range $11.09\text{-}12.86$ km.

Neutron star radii and crusts: Uncertainties and unified equations of state

The uncertainties in neutron star (NS) radii and crust properties due to our limited knowledge of the equation of state (EOS) are quantitatively analysed. We first demonstrate the importance of a unified microscopic description for the different baryonic densities of the star. If the pressure functional is obtained matching a crust and a core EOS based on models with different properties at nuclear matter saturation, the uncertainties can be as large as $\sim 30\%$ for the crust thickness and $4\%$ for the radius. Necessary conditions for causal and thermodynamically consistent matchings between the core and the crust are formulated and their consequences examined. A large set of unified EOS for purely nucleonic matter is obtained based on 24 Skyrme interactions and 9 relativistic mean-field nuclear parametrizations. In addition, for relativistic models 17 EOS including a transition to hyperonic matter at high density are presented. All these EOS have in common the property of describing a $2\;M_\odot$ star and of being causal within stable NS. A span of $\sim 3$ km and $\sim 4$ km is obtained for the radius of, respectively, $1.0\;M_\odot$ and $2.0\;M_\odot$ star. Applying a set of nine further constraints from experiment and ab-initio calculations the uncertainty is reduced to $\sim 1$ km and $2$ km, respectively. These residual uncertainties reflect lack of constraints at large densities and insufficient information on the density dependence of the EOS near the nuclear matter saturation point. The most important parameter to be constrained is shown to be the symmetry energy slope $L$ which exhibits a linear correlation with the stellar radius, particularly for masses $\sim 1.0\;M_\odot$. Potential constraints on $L$, the NS radius and the EOS from observations of thermal states of NS are also discussed. [Abriged]

Model dependence of the neutron-skin thickness on the symmetry energy

The model dependence in the correlations of the neutron-skin thickness in heavy nuclei with various symmetry energy parameters is analyzed by using several families of systematically varied microscopic mean field models. Such correlations show a varying degree of model dependence once the results for all the different families are combined. Some mean field models associated with similar values of the symmetry energy slope parameter at saturation density $L$, and pertaining to different families, yield a greater-than-expected spread in the neutron-skin thickness of the $^{208}$Pb nucleus. The effective value of the symmetry energy slope parameter $L_{\rm eff}$, determined by using the nucleon density profiles of the finite nucleus and the density derivative $S^\prime(\rho)$ of the symmetry energy starting from about saturation density up to low densities typical of the surface of nuclei, seems to account for the spread in the neutron-skin thickness for the models with similar $L$. The differences in the values of $L_{\rm eff}$ are mainly due to the small differences in the nucleon density distributions of heavy nuclei in the surface region and the behavior of the symmetry energy at subsaturation densities.

Sensitivity of the fusion cross section to the density dependence of the symmetry energy

It is the aim of this paper to discuss the impact of nuclear fusion on the EOS. This is a timely subject given the expected availability of increasingly exotic beams at rare isotope facilities\,\cite{balantekin2014}. In practice, we focus on $^{48}$Ca+$^{48}$Ca fusion. We employ three different approaches to calculate fusion cross-sections for a set of energy density functionals with systematically varying nuclear matter properties. Fusion calculations are performed using frozen densities, using a dynamic microscopic method based on density-constrained time-dependent Hartree-Fock (DC-TDHF) approach, as well as direct TDHF study of above barrier cross-sections. For these studies, we employ a family of Skyrme parametrizations with systematically varied nuclear matter properties. We find a slight preference for forces which deliver a slope of symmetry energy of $L\approx 50$\,MeV that corresponds to a neutron-skin thickness of $^{48}$Ca of $R_\mathrm{skin}\!=\!(0.180\!-\!0.210)$\,fm.

Correlation of the neutron star crust-core properties with the slope of the symmetry energy and the lead skin thickness

The correlations of the crust-core transition density and pressure in neutron stars with the slope of the symmetry energy and the neutron skin thickness are investigated, using different families of relativistic mean field parametrizations with constant couplings and non-linear terms mixing the $\sigma$, $\omega$ and $\rho$-meson fields. It is shown that the modification of the density dependence of the symmetry energy, involving the $\sigma$ or the $\omega$ meson, gives rise to different behaviors: the effect of the $\omega$-meson may also be reproduced within non-relativistic phenomenological models, while the effect of the $\sigma$-meson is essentially relativistic. Depending on the parametrization with $\sigma-\rho$ or $\omega-\rho$ mixing terms, different values of the slope of the symmetry energy at saturation must be considered in order to obtain a neutron matter equation of state compatible with results from chiral effective field theory. This difference leads to different pressures at the crust-core transition density. A linear correlation between the transition density and the symmetry energy slope or the neutron skin thickness of the $^{208}$Pb nucleus is obtained, only when the $\omega$-meson is used to describe the density dependence of the symmetry energy. A comparison is made between the crust-core transition properties of neutron stars obtained by three different methods, the Relativistic Random Phase Approximation (RRPA), the Vlasov equation and Thermodynamical method. It is shown that the RRPA and the Vlasov methods predict similar transition densities for $pne$ $\beta$-equilibrium stellar matter.

Influence of Skyrme-type interaction on HICs observables

A new version of the improved quantum molecular dynamics model has been developed by including Skyrme type momentum dependent interaction. 12 Skyrme like parameter sets {K0, S0, L, m ∗, m ∗} are adopted in the transport model code to calculate the isospin diffusion, single and double ratios of transverse emitted nucleons, neutron proton isoscaling ratios. The calculations and correlation analysis evidence that isospin diffusion observable at lower beam energy is sensitive to the slope of symmetry energy and m ∗. The high energy neutrons and protons yield ratios from reactions at different incident energies provide a sensitive observable to study the nucleon effective mass splitting, at higher beam energy. The properties of isospin asymmetric nuclear equation of state is of fundamental importance in our understanding of nature’s asymmetric objects including neutron stars as well as heavy nuclei with very different number of neutrons and protons. Currently, there exist, lot of effective interaction parameter sets [1, 2], for fitting the ground-state properties of stable nuclei, symmetric and asymmetric nuclear matter. It also leads very different neutron Equation of state which may have different nuclear matter (NM) parameters, such as incompressibility K0, isoscalar effective mass m ∗ ,s ymmetry energy coef fi cientS0, slope of symmetry energy L and isovector effec

Covariance analysis of symmetry energy observables from heavy ion collision

Using covariance analysis, we quantify the correlations between the interaction parameters in a transport model and the observables commonly used to extract information of the Equation of State of Asymmetric Nuclear Matter in experiments. By simulating 124Sn + 124Sn, 124Sn + 112Sn and 112Sn + 112Sn reactions at beam energies of 50 and 120 MeV per nucleon, we have identified that the nucleon effective mass splitting is most strongly correlated to the neutrons and protons yield ratios with high kinetic energy from central collisions especially at high incident energy. The best observable to determine the slope of the symmetry energy, L, at saturation density is the isospin diffusion observable even though the correlation is not very strong (∼0.7). Similar magnitude of correlation but opposite in sign exists for isospin diffusion and nucleon isoscalar effective mass. At 120 MeV/u, the effective mass splitting and the isoscalar effective mass also have opposite correlation for the double n/p and isoscaling p/p yield ratios. By combining data and simulations at different beam energies, it should be possible to place constraints on the slope of symmetry energy (L) and effective mass splitting with reasonable uncertainties.

Nuclear Matter Bulk Parameter Scales and Correlations

We study the arising of correlations among some isovector bulk parameters in nonrelativistic and relativistic hadronic mean-field models. For the former, we investigate correlations in the nonrelativistic (NR) limit of relativistic point-coupling models. We provide analytical correlations, for the NR limit model, between the symmetry energy and its derivatives, namely, the symmetry energy slope, curvature, skewness and fourth order derivative, discussing the conditions in which they are linear ones. We also show that some correlations presented in the NR limit model are reproduced for relativistic models presenting cubic and quartic self-interactions in its scalar field. As a direct application of such linear correlations, we remark its association with possible crossing points in the density dependence of the linearly correlated bulk parameter.

Thermal properties of asymmetric nuclear matter with an improved isospin- and momentum-dependent interaction

Thermal properties of asymmetric nuclear matter, including the temperature dependence of the symmetry energy, single-particle properties, and differential isospin fractionation, are investigated with different neutron-proton effective mass splittings using an improved isospin- and momentum-dependent interaction. In this improved interaction, the momentum-dependence of the isoscalar single-particle potential at saturation density is well fitted to that extracted from optical model analyses of proton-nucleus scattering data up to nucleon kinetic energy of 1 GeV, and the isovector properties, i.e., the slope of the nuclear symmetry energy, the momentum-dependence of the symmetry potential, and the symmetry energy at saturation density can be flexibly adjusted via three parameters $x$, $y$, and $z$, respectively. Our results indicate that the nucleon phase-space distribution in equilibrium, the temperature dependence of the symmetry energy, and the differential isospin fractionation can be significantly affected by the isospin splitting of nucleon effective mass.

Impact of the symmetry energy on nuclear pasta phases and crust-core transition in neutron stars

We study the impact of the symmetry energy on properties of nuclear pasta phases and crust-core transition in neutron stars. We perform a self-consistent Thomas-Fermi calculation employing the relativistic mean-field model. The properties of pasta phases presented in the inner crust of neutron stars are investigated and the crust-core transition is examined. It is found that the slope of the symmetry energy plays an important role in determining the pasta phase structure and the crust-core transition. The correlation between the symmetry energy slope and the crust-core transition density obtained in the Thomas-Fermi approximation is consistent with that predicted by the liquid-drop model.

Modifications of the pion-production threshold in the nuclear medium in heavy ion collisions and the nuclear symmetry energy

Using the relativistic Vlasov-Uehling-Uhlenbeck (RVUU) equation based on mean fields from the nonlinear relativistic $\text{NL}\ensuremath{\rho}$ and $\text{NL}\ensuremath{\rho}\ensuremath{\delta}$ models, which have same nuclear equation of state and symmetry energy but different symmetry energy slope parameters, we study the effect of medium modification of the pion-production threshold on the total pion yield and the ${\ensuremath{\pi}}^{\ensuremath{-}}/{\ensuremath{\pi}}^{+}$ ratio in Au+Au collisions. We find that the in-medium threshold effect enhances both the total pion yield and the ${\ensuremath{\pi}}^{\ensuremath{-}}/{\ensuremath{\pi}}^{+}$ ratio, compared to those without this effect. Furthermore, including the medium modification of the pion-production threshold leads to a larger ${\ensuremath{\pi}}^{\ensuremath{-}}/{\ensuremath{\pi}}^{+}$ ratio for the $\text{NL}\ensuremath{\rho}\ensuremath{\delta}$ model with a larger symmetry energy parameter than the $\text{NL}\ensuremath{\rho}$ model with a smaller symmetry energy parameter, opposite to that found without the in-medium threshold effect. To reproduce the total pion yield measured by the FOPI Collaboration, we introduce a density-dependent cross section for $\ensuremath{\Delta}$ baryon production from nucleon-nucleon collisions, which suppresses the total pion yield but hardly changes the ${\ensuremath{\pi}}^{\ensuremath{-}}/{\ensuremath{\pi}}^{+}$ ratio. Because of the small difference in the stiffness of their symmetry energies, the ${\ensuremath{\pi}}^{\ensuremath{-}}/{\ensuremath{\pi}}^{+}$ ratios obtained from both the $\text{NL}\ensuremath{\rho}$ and $\text{NL}\ensuremath{\rho}\ensuremath{\delta}$ models are consistent with the FOPI data within the experimental errors.