Equation Of State For Symmetric Nuclear Matter Research Articles

Re-visiting the role of short-range correlations on neutron-star properties

The impact of short-range correlations (SRCs) on neutron star (NS) properties are re-examined across various classes of relativistic mean-field (RMF) models. The coupling strengths of the models are so adjusted that the low-density part of the equation of state (EoS), complemented with saturation properties such as binding energy of symmetric nuclear matter and nuclear symmetry energy responsible for the bulk properties of finite nuclei, remains practically unaffected with the inclusion of SRCs. The EoS for symmetric nuclear matter and the nuclear symmetry energy at supra-saturation densities which governs the EoS for NS matter become softer or stiffer in the presence of the SRCs, depending on the type of RMF model considered. These distinct effects of SRCs are observed to be more significant at higher densities, as expected, when behaviour of the EoS at low densities, which govern the finite nuclei physics, is not compromised. For most of the models with self-couplings of scalar and vector mesons, the EoS of symmetric nuclear matter stiffens, with a slight effect on nuclear symmetry energy with the inclusion of SRCs. Conversely, in the models incorporating cross-couplings between mesons, the addition of SRCs leads to softer symmetry energy, compensating the stiffening effect of the EoS of symmetric nuclear matter. With the inclusion of SRCs, the values of radius and tidal deformability of canonical mass star and maximum mass of NSs for realistic EoSs align well the present constraints on astrophysical observations.

Toward the System Size Dependence of Anisotropic Flow in Heavy-Ion Collisions at sNN= 2–5 GeV

The study of the high-density equation of state (EOS) and the search for a possible phase transition in dense baryonic matter is the main goal of beam energy scan programs with relativistic heavy ions at energies sNN= 2–5 GeV. The most stringent constraints currently available on the high-density EOS of symmetric nuclear matter come from the present measurements of directed (v1) and elliptic flow (v2) signals of protons in Au + Au collisions. In this energy range, the anisotropic flow is strongly affected by the presence of cold spectators due to the sizable passage time. The system size dependence of anisotropic flow may help to study the participant–spectator contribution and improve our knowledge of the EOS of symmetric nuclear matter. In this work, we discuss the layout of the upgraded BM@N experiment and the anticipated performance for differential anisotropic flow measurements of identified hadrons at Nuclotron energies: sNN= 2.3–3.5 GeV.

Bayesian reconstruction of nuclear matter parameters from the equation of state of neutron star matter

The nuclear matter parameters (NMPs), those underlie in the construction of the equation of state (EoS) of neutron star matter, are not directly accessible. The Bayesian approach is applied to reconstruct the posterior distributions of NMPs from the EoS of neutron star matter. The constraints on lower-order parameters as imposed by the finite nuclei observables are incorporated through appropriately chosen prior distributions. The calculations are performed with two sets of pseudo data on the EoS whose true models are known. The median values of second or higher order NMPs show sizeable deviations from their true values and associated uncertainties are also larger. The sources of these uncertainties are intrinsic in nature, identified as (i) the correlations among various NMPs and (ii) the variations in the EoS of symmetric nuclear matter, symmetry energy, and the neutron-proton asymmetry in such a way that the neutron star matter EoS remain almost unaffected.

Determination of the symmetry energy from the neutron star equation of state

We analyze the uncertainties introduced in the determination of the neutron star matter proton fraction, in a range of densities close to the saturation density, if the cold $\ensuremath{\beta}$-equilibrium neutron star matter equation of state (EOS) is known. In particular, we discuss the effect of neglecting the muon contribution and of considering that the energy density of nuclear matter is well described by taking only terms until second order in the proton-neutron asymmetry. It is shown that two types of uncertainties may be associated with the extraction of the symmetry energy from the $\ensuremath{\beta}$-equilibrium EOS: an overestimation if terms above the parabolic approximation on the asymmetry parameter are neglected, or an underestimation if the muon contribution is not considered. The effect of the uncertainty on the symmetric nuclear matter EOS on the determination of the proton fraction is discussed. It could be shown that the neutron star mass-radius curve is sensitive to the parabolic approximation on the asymmetry parameter.

Bayesian Inference of the Symmetry Energy of Superdense Neutron-rich Matter from Future Radius Measurements of Massive Neutron Stars

Abstract Using as references the posterior probability distribution functions of the equation of state (EOS) parameters inferred from the radii of canonical neutron stars (NSs) reported by the LIGO/VIRGO and NICER Collaborations based on their observations of GW170817 and PSR J0030+0451, we investigate how future radius measurements of more massive NSs will improve our current knowledge about the EOS of superdense neutron-rich nuclear matter, especially its symmetry energy term. Within the Bayesian statistical approach using an explicitly isospin-dependent parametric EOS for the core of NSs, we infer the EOS parameters of superdense neutron-rich nuclear matter from three sets of imagined mass–radius correlation data representing typical predictions by various nuclear many-body theories, that is, the radius stays the same, decreases, or increases with increasing NS mass within ±15% between 1.4 and 2.0 M ⊙. The corresponding NS average density increases quickly or slowly or slightly decreases as the NS mass increases from 1.4 to 2.0 M ⊙. While the EOSs of symmetric nuclear matter (SNM) inferred from the three data sets are approximately the same, the corresponding symmetry energies above about twice the saturation density of nuclear matter are very different, indicating that the radii of massive NSs carry important information about the high-density behavior of nuclear symmetry energy with little influence from the remaining uncertainties of the SNM EOS at suprasaturation densities.

Key factor for determining relation between radius and tidal deformability of neutron stars: Slope of symmetry energy * * This work is partly Supported by the Shandong Natural Science Foundation (JQ201701), the Natural Science Foundation of China (11622540, 11675094, 11705102, 11775133), the China Postdoctoral Science Foundation (2019M652358), the Young Scholars Program of Shandong University, Weihai (2015WHWLJH01), and the Fundamental Research

The constraints on tidal deformability of neutron stars were first extracted from GW170817 by LIGO and Virgo Collaborations. However, the relationship between the radius and tidal deformability is still under debate. Using an isospin-dependent parameterized equation of state (EOS), we study the relation between and and its dependence on parameters of symmetry energy and EOS of symmetric nuclear matter when the mass is fixed at , , and . We find that, although the changes of high order parameters of and can shift individual values of and , the relation remains approximately at the same fitted curve. The slope of the symmetry energy plays the dominant role in determining the relation. By investigating the mass dependence of the relation, we find that the well fitted relation for 1.4 is broken for massive neutron stars.

Analysis of nuclear structure in a converging power expansion scheme

In the framework of the KIDS generalized energy density functional (EDF), the nuclear equation of state (EoS) is expressed as an expansion in powers of the Fermi momentum or the cubic root of the density ($\rho^{1/3}$). Although an optimal number of converging terms was obtained in specific cases of fits to empirical data and pseudodata, the degree of convergence remains to be examined not only for homogeneous matter but also for finite nuclei. One goal of the present work is to validate the minimal and optimal number of EoS parameters required for the description of homogeneous nuclear matter over a wide range of densities relevant for astrophysical applications. The major goal is to examine the validity of the adopted expansion scheme for an accurate description of finite nuclei. To this end we vary the values of the high-order derivatives of the EoS, namely the skewness of the energy of symmetric nuclear matter and the kurtosis of the symmetry energy, at saturation and examine the relative importance of each term in $\rho^{1/3}$ expansion for homogeneous matter. For given sets of EoS parameters determined in this way, we define equivalent Skyrme-type functionals and examine the convergence in the description of finite nuclei focusing on the masses and charge radii of closed-shell nuclei. The EoS of symmetric nuclear matter is found to be efficiently parameterized with only 3 parameters and the symmetry energy (or the energy of pure neutron matter) with 4 parameters when the EoS is expanded in the power series of the Fermi momentum. Higher-order EoS parameters do not produce any improvement, in practice, in the description of nuclear ground-state energies and charge radii, which means that they cannot be constrained by bulk properties of nuclei.

Equation of state of dense matter in the multimessenger era

While the equation of state (EOS) of symmetric nuclear matter (SNM) at suprasaturation densities has been relatively well constrained from heavy-ion collisions, the EOS of high-density neutron-rich matter is still largely uncertain due to the poorly known high-density behavior of the symmetry energy. Using the constraints on the EOS of SNM at suprasaturation densities from heavy-ion collisions together with the data of finite nuclei and the existence of $2M_\odot$ neutron stars from electromagnetic (EM) observations, we show that the high-density symmetry energy cannot be too soft, which leads to lower bounds on dimensionless tidal deformability of $\Lambda_{1.4} \ge 193$ and radius of $R_{1.4} \ge 11.1$ km for $1.4M_\odot$ neutron star. Furthermore, we find that the recent constraint of $\Lambda_{1.4} \le 580$ from the gravitational wave signal GW170817 detected from the binary neutron star merger by the LIGO and Virgo Collaborations rules out too stiff high-density symmetry energy, leading to an upper limit of $R_{1.4} \le 13.3$ km. All these terrestrial nuclear experiments and astrophysical observations based on strong, EM and gravitational measurements together put stringent constraints on the high-density symmetry energy and the EOS of SNM, pure neutron matter and neutron star matter.

Combined Constraints on the Equation of State of Dense Neutron-rich Matter from Terrestrial Nuclear Experiments and Observations of Neutron Stars

Abstract Within the parameter space of the equation of state (EOS) of dense neutron-rich matter limited by existing constraints mainly from terrestrial nuclear experiments, we investigate how the neutron star maximum mass M max > 2.01 ± 0.04 M ⊙, radius 10.62 km < R 1.4 < 12.83 km and tidal deformability Λ1.4 ≤ 800 of canonical neutron stars together constrain the EOS of dense neutron-rich nucleonic matter. While the 3D parameter space of K sym (curvature of nuclear symmetry energy), J sym, and J 0 (skewness of the symmetry energy and EOS of symmetric nuclear matter, respectively) is narrowed down significantly by the observational constraints, more data are needed to pin down the individual values of K sym, J sym, and J 0. The J 0 largely controls the maximum mass of neutron stars. While the EOS with J 0 = 0 is sufficiently stiff to support neutron stars as massive as 2.37 M ⊙, supporting the hypothetical ones as massive as 2.74 M ⊙ (composite mass of GW170817) requires J 0 to be larger than its currently known maximum value of about 400 MeV and beyond the causality limit. The upper limit on the tidal deformability of Λ1.4 = 800 from the recent observation of GW170817 is found to provide upper limits on some EOS parameters consistent with but far less restrictive than the existing constraints of other observables studied.

Symmetry energy of cold nucleonic matter within a relativistic mean field model encapsulating effects of high-momentum nucleons induced by short-range correlations

Significant progress has been made recently in constraining the isospin-dependent parameters characterizing the SRC (short-range correlation)-modified single-nucleon momentum distribution in neutron-rich nucleonic matter using both experimental data and microscopic model calculations. Using the constrained single-nucleon momentum distribution in a nonlinear relativistic mean field (RMF) model, we study the equation of state (EOS) of asymmetric nucleonic matter (ANM), especially the density dependence of nuclear symmetry energy $E_{\rm{sym}}(\rho)$. Firstly, as a test of the model, the average nucleon kinetic energy extracted recently from electron-nucleus scattering experiments using a neutron-proton dominance model is well reproduced by the RMF model incorporating effects of the SRC-induced high momentum nucleons, while it is significantly under predicted by the RMF model using a step function for the single-nucleon momentum distribution as in free Fermi gas (FFG) models. Secondly, the kinetic symmetry energy of quasi-nucleons is found to be $E^{\rm{kin}}_{\rm{sym}}(\rho_0)=-16.94\pm13.66\,\rm{MeV}$ which is dramatically different from the prediction of $E^{\rm{kin}}_{\rm{sym}}(\rho_0)\approx 12.5$ MeV by FFG models at nuclear matter saturation density $\rho_0=0.16\,\rm{fm}^{-3}$. Thirdly, comparing the RMF calculations with and without the high momentum nucleons using two sets of model parameters both reproducing identically all empirically constraints on the EOS of symmetric nuclear matter (SNM) and the symmetry energy of ANM at $\rho_0$, the SRC-modified single-nucleon momentum distribution is found to make the $E_{\rm{sym}}(\rho)$ more concave around $\rho_0$ by softening it significantly at both sub-saturation and supra-saturation densities, leading to an isospin-dependent incompressibility of ANM in better agreement with existing experimental data.

EOSDB: The database for nuclear equations of state

Abstract The nuclear equation of state (EoS) plays an important role in understanding the formation of compact objects such as neutron stars and black holes. The true nature of the EoS has been a matter of debate at any density range not only in nuclear physics but also in astronomy and astrophysics. We have constructed a database of EoSs by compiling data from the literature. Our database contains the basic properties of the nuclear EoS of symmetric nuclear matter and of pure neutron matter. It also includes detailed information about the theoretical models, for example the adopted methods and assumptions in individual models. The novelty of the database is to consider new experimental probes such as symmetry energy, its slope with respect to the baryon density, and incompressibility, which enables users to check their model dependence. We demonstrate the performance of the EOSDB through examinations of model dependence among different nuclear EoSs. It is revealed that some theoretical EoSs, commonly used in astrophysics, do not satisfactorily agree with the experimental constraints.

Coupled-cluster calculations of nucleonic matter

Background: The equation of state (EoS) of nucleonic matter is central for the understanding of bulk nuclear properties, the physics of neutron star crusts, and the energy release in supernova explosions. Purpose: This work presents coupled-cluster calculations of infinite nucleonic matter using modern interactions from chiral effective field theory (EFT). It assesses the role of correlations beyond particle-particle and hole-hole ladders, and the role of three-nucleon-forces (3NFs) in nuclear matter calculations with chiral interactions. Methods: This work employs the optimized nucleon-nucleon NN potential NNLOopt at next-to-next-to leading-order, and presents coupled-cluster computations of the EoS for symmetric nuclear matter and neutron matter. The coupled-cluster method employs up to selected triples clusters and the single-particle space consists of a momentum-space lattice. We compare our results with benchmark calculations and control finite-size effects and shell oscillations via twist-averaged boundary conditions. Results: We provide several benchmarks to validate the formalism and show that our results exhibit a good convergence toward the thermodynamic limit. While neutron matter with interactions from chiral EFT is perturbative, symmetric nuclear matter requires nonperturbative approaches. Correlations beyond the standard particle-particle ladder approximation yield non-negligible contributions. The saturation point of symmetric nuclear matter is sensitive to the employed 3NFs and the employed regularization scheme. 3NFs with nonlocal cutoffs exhibit a considerably improved convergence than their local cousins. We are unable to find values for the parameters of the short-range part of the local 3NF that simultaneously yield acceptable values for the saturation point in symmetric nuclear matter and the binding energies of light nuclei.

Sensitivity of the neutron starr-mode instability window to the density dependence of the nuclear symmetry energy

Using a simple model of a neutron star with a perfectly rigid crust constructed with a set of crust and core equations of state that span the range of nuclear experimental uncertainty in the symmetry energy, we calculate the instability window for the onset of the Chandrasekhar-Friedmann-Schutz instability in $r$-mode oscillations for canonical neutron stars ($1.4{M}_{\ensuremath{\bigodot}}$) and massive neutron stars ($2.0{M}_{\ensuremath{\bigodot}}$). In these models the crust-core transition density, and thus crustal thickness, is calculated consistently with the core equation of state (EOS). The EOSs are calculated using a simple model for the energy density of nuclear matter and probe the dependence on the symmetry energy by varying the slope of the symmetry energy at saturation density $L$ from 25 MeV (soft symmetry energy and EOS) to 115 MeV (stiff symmetry energy and EOS) while keeping the EOS of symmetric nuclear matter fixed. For the canonical neutron star, the lower bound of the $r$-mode instability window is reduced in frequency by $\ensuremath{\approx}\phantom{\rule{-0.16em}{0ex}}\phantom{\rule{-0.16em}{0ex}}150$ Hz from the softest to the stiffest symmetry energy used, independent of mass and temperature. The instability window also drops by $\ensuremath{\approx}100$ Hz, independent of EOS when the mass is raised from $1.4{M}_{\ensuremath{\bigodot}}$ to $2.0{M}_{\ensuremath{\bigodot}}$. Where temperature estimates are available, the observed neutron stars in low-mass x-ray binaries (LMXBs) have frequencies below the instability window for the $1.4{M}_{\ensuremath{\bigodot}}$ models, while some LMXBs fall within the instability window for $2.0{M}_{\ensuremath{\bigodot}}$ stars if the symmetry energy is relatively stiff, indicating that a softer symmetry energy is more consistent with observations within this model. Thus we conclude that smaller values of $L$ help stabilize neutron stars against runaway $r$-mode oscillations. The critical temperature, below which no star can reach the instability window without exceeding its Kepler frequency, varies by nearly an order of magnitude from soft to stiff symmetry energies. When the crust thickness and core EOS are treated consistently, a thicker crust corresponds to a lower critical temperature, the opposite result to previous studies in which the transition density was independent of the core EOS.

Higher order bulk characteristic parameters of asymmetric nuclear matter

The bulk parameters characterizing the energy of symmetric nuclear matter and the symmetry energy defined at normal nuclear density ρ 0 provide important information on the equation of state (EOS) of isospin asymmetric nuclear matter. While significant progress has been made in determining some lower order bulk characteristic parameters, such as the energy E 0(ρ 0) and incompressibility K 0 of symmetric nuclear matter as well as the symmetry energy E sym(ρ 0) and its slope parameter L, yet the higher order bulk characteristic parameters are still poorly known. Here, we analyze the correlations between the lower and higher order bulk characteristic parameters within the framework of Skyrme Hartree-Fock energy density functional and then estimate the values of some higher order bulk characteristic parameters. In particular, we obtain J 0 = (−355±95) MeV and I 0 = (1473±680) MeV for the thirdorder and fourth-order derivative parameters of symmetric nuclear matter at ρ 0 and K sym = (−100 ± 165) MeV, J sym = (224 ± 385) MeV, I sym = (−1309 ± 2025) MeV for the curvature parameter, third-order and fourth-order derivative parameters of the symmetry energy at ρ 0, using the empirical constraints on E 0(ρ 0), K 0, E sym(ρ 0), L, and the isoscalar and isovector nucleon effective masses. Furthermore, our results indicate that the three parameters E 0(ρ 0), K 0, and J 0 can reasonably characterize the EOS of symmetric nuclear matter up to 2ρ 0 while the symmetry energy up to 2ρ 0 can be well described by E sym(ρ 0), L, and K sym.

A phenomenological equation of state for isospin asymmetric nuclear matter

A phenomenological momentum-independent (MID) model is constructed to describe the equation of state (EOS) for isospin asymmetric nuclear matter, especially the density dependence of the nuclear symmetry energy E sym(ρ). This model can reasonably describe the general properties of the EOS for symmetric nuclear matter and the symmetry energy predicted by both the sophisticated isospin and momentum dependent MDI model and the Skyrme-Hartree-Fock approach. We find that there exists a nicely linear correlation between K sym and L as well as between J 0/K 0 and K 0, where L and K sym represent, respectively, the slope and curvature parameters of the symmetry energy at the normal nuclear density ρ 0, while K 0 and J 0 are, respectively, the incompressibility and the third-order derivative parameter of symmetric nuclear matter at ρ 0. These correlations together with the empirical constraints on K 0, L and E sym(ρ 0) lead to an estimation of −477 MeV ≤ K sat,2 ≤ −241 MeV for the second-order isospin asymmetry expansion coefficient for the incompressibility of asymmetric nuclear matter at the saturation point.

Nuclear Constraints on the Moments of Inertia of Neutron Stars

The properties and structure of neutron stars are determined by the equation of state (EOS) of neutron-rich stellar matter. While the collective flow and particle production in relativistic heavy-ion collisions have tightly constrained the EOS of symmetric nuclear matter up to about 5 times the normal nuclear matter density, more recent experimental data on isospin diffusion and isoscaling in heavy-ion collisions at intermediate energies have constrained considerably the density dependence of the nuclear symmetry energy at subsaturation densities. Although there are still many uncertainties and challenges to pin down completely the EOS of neutron-rich nuclear matter, heavy-ion reaction experiments in terrestrial laboratories have limited the EOS of neutron-rich nuclear matter to a range much narrower than that spanned by the various EOSs currently used in astrophysical studies in the literature. These nuclear physics constraints could thus provide more reliable information about the properties of neutron stars. Within well-established formalisms using the nuclear-constrained EOSs, we study the moments of inertia of neutron stars. We place special emphasis on component A of the extremely relativistic double neutron star system PSR J0737–3039. Its moment of inertia is found to be between 1.30 × 1045 and 1.63 × 1045 g cm2. Moreover, the transition density at the crust-core boundary is shown to lie in the narrow range ρt = 0.091-0.093 fm−3.

Nuclear equation of state at high baryonic density and compact star constraints

A mean field calculation is carried out to obtain the equation of state (EoS) of nuclear matter from a density-dependent M3Y interaction (DDM3Y). The energy per nucleon is minimized to obtain ground state of the symmetric nuclear matter (SNM). The constants of density dependence of the effective interaction are obtained by reproducing the saturation energy per nucleon and the saturation density of SNM. The energy variation of the exchange potential is treated properly in the negative energy domain of nuclear matter. The EoS of SNM, thus obtained, is not only free from the superluminosity problem but also provides excellent estimate of nuclear incompressibility. The EoS of asymmetric nuclear matter is calculated by adding to the isoscalar part, the isovector component of M3Y interaction. The SNM and pure neutron matter EoS are used to calculate the nuclear symmetry energy which is found to be consistent with that extracted from the isospin diffusion in heavy-ion collisions at intermediate energies. The β equilibrium proton fraction calculated from the symmetry energy and related theoretical findings are consistent with the constraints derived from the observations on compact stars.

Equation of state of supernova matter

The equation of state (EOS) of dense supernova matter composed of protons, neutrons, and relativistic electrons is calculated within finite temperature Brueckner Goldstone approach with effective two-body Sussex interaction and compared with asymmetric nuclear matter. Thermal effect on the EOS is studied at temperature 5, 7, and 10 MeV. The EOS of supernova matter is found to be stiffer than the corresponding asymmetric nuclear matter equation of state as expected. Single particle properties like distribution functions and chemical potentials of proton, neutron, and electron are discussed for various values of electron fractions, densities, and temperatures. Distribution function is found to depend on Fermi kinetic energy of respective particles as well as on thermal energy. Chemical potential depends on number density of particles and temperature. It is also seen that the EOS for symmetric nuclear matter at low temperature has same order of magnitude with the recently extracted experimental values within the density range 0.35-0.56 fm{sup -3}.