Nuclear Symmetry Energy Research Articles

Constraining Hadron-quark Phase Transition Parameters within the Quark-mean-field Model Using Multimessenger Observations of Neutron Stars

Abstract We extend the quark mean-field (QMF) model for nuclear matter and study the possible presence of quark matter inside the cores of neutron stars. A sharp first-order hadron-quark phase transition is implemented combining the QMF for the hadronic phase with “constant-speed-of-sound” parameterization for the high-density quark phase. The interplay of the nuclear symmetry energy slope parameter, L, and the dimensionless phase transition parameters (the transition density n trans/n 0, the transition strength Δε/ε trans, and the sound speed squared in quark matter ) are then systematically explored for the hybrid star properties, especially the maximum mass M max and the radius and the tidal deformability of a typical 1.4 M ⊙ star. We show the strong correlation between the symmetry energy slope L and the typical stellar radius R 1.4, similar to that previously found for neutron stars without a phase transition. With the inclusion of phase transition, we obtain robust limits on the maximum mass (M max < 3.6 M ⊙) and the radius of 1.4 M ⊙ stars (R 1.4 ≳ 9.6 km), and we find that a phase transition that is too weak (Δε/ε trans ≲ 0.2) taking place at low densities ≲1.3–1.5 n 0 is strongly disfavored. We also demonstrate that future measurements of the radius and tidal deformability of ∼1.4 M ⊙ stars, as well as the mass measurement of very massive pulsars, can help reveal the presence and amount of quark matter in compact objects.

Probing the Neutron Skin with Ultrarelativistic Isobaric Collisions.

Particle production in ultrarelativistic heavy ion collisions depends on the details of the nucleon density distributions in the colliding nuclei. We demonstrate that the charged hadron multiplicity distributions in isobaric collisions at ultrarelativistic energies provide a novel approach to determine the poorly known neutron density distributions and thus the neutron skin thickness in finite nuclei, which can in turn put stringent constraints on the nuclear symmetry energy.

Microscopic description of structural, surface, and decay properties of Z=124, 126 superheavy nuclei

A systematic investigation has been carried out for even-even isotopic series of two superheavy nuclei with Z=124 and Z=126 in the region 160≤N≤220, and 162≤N≤218, respectively, within the framework of covariant density functional theory (CDFT). Based on the potential energy curves calculated using the explicit density-dependent meson-exchange (DD-ME) and point-coupling (DD-PC) models, many sensitive observables have been analyzed to understand the structural evolution, shape coexistence, and to predict the possible candidates for neutron shell-closure within the isotopic series of Z=124 and Z=126. The shell correction energy and nuclear symmetry energy have been evaluated within the Strutinsky shell correction approach and the Coherent Density Fluctuation Model (CDFM), respectively. All the sensitive observables i.e. two-neutron separation energy, its differential, neutron-pairing energy, single-particle energy levels, and total shell-correction energy reproduce known spherical neutron magic numbers, and present evidence on new deformed neutron magic numbers at N=168, 174 and 178. The symmetry energy shows a direct correlation with the shape of the system. The α-decay chain of four isotopes 124298,302 and 126300,304 have been studied within three different semi-empirical approaches named UNIV2, SemFIS2 and ImSahu, and compared with the available experimental data, FRDM2012 and the WS4 mass model. All the models, DD-ME2, DD-PC1, FRDM2012, and WS4 have predicted around 8α to 10α decay before spontaneous fission (SF) can start.

Quantifying uncertainties and correlations in the nuclear-matter equation of state

We perform statistically rigorous uncertainty quantification (UQ) for chiral effective field theory ($\chi$EFT) applied to infinite nuclear matter up to twice nuclear saturation density. The equation of state (EOS) is based on high-order many-body perturbation theory calculations with nucleon-nucleon and three-nucleon interactions up to fourth order in the $\chi$EFT expansion. From these calculations our newly developed Bayesian machine-learning approach extracts the size and smoothness properties of the correlated EFT truncation error. We then propose a novel extension that uses multitask machine learning to reveal correlations between the EOS at different proton fractions. The inferred in-medium $\chi$EFT breakdown scale in pure neutron matter and symmetric nuclear matter is consistent with that from free-space nucleon-nucleon scattering. These significant advances allow us to provide posterior distributions for the nuclear saturation point and propagate theoretical uncertainties to derived quantities: the pressure and incompressibility of symmetric nuclear matter, the nuclear symmetry energy, and its derivative. Our results, which are validated by statistical diagnostics, demonstrate that an understanding of truncation-error correlations between different densities and different observables is crucial for reliable UQ. The methods developed here are publicly available as annotated Jupyter notebooks.

How Well Do We Know the Neutron-Matter Equation of State at the Densities Inside Neutron Stars? A Bayesian Approach with Correlated Uncertainties.

We introduce a new framework for quantifying correlated uncertainties of the infinite-matter equation of state derived from chiral effective field theory (χEFT). Bayesian machine learning via Gaussian processes with physics-based hyperparameters allows us to efficiently quantify and propagate theoretical uncertainties of the equation of state, such as χEFT truncation errors, to derived quantities. We apply this framework to state-of-the-art many-body perturbation theory calculations with nucleon-nucleon and three-nucleon interactions up to fourth order in the χEFT expansion. This produces the first statistically robust uncertainty estimates for key quantities of neutron stars. We give results up to twice nuclear saturation density for the energy per particle, pressure, and speed of sound of neutron matter, as well as for the nuclear symmetry energy and its derivative. At nuclear saturation density, the predicted symmetry energy and its slope are consistent with experimental constraints.

Unveiling the correlations of tidal deformability with the nuclear symmetry energy parameters

The chi-squared based covariance approach allows one to estimate the correlations among desired observables related to nuclear matter directly from a set of fit data without taking recourse to the distributions of the nuclear matter parameters (NMPs). Such an approach is applied to study the correlations of tidal deformability of neutron star with the slope and the curvature parameters of nuclear symmetry energy governed by an extensive set of fit data on the finite nuclei together with the maximum mass of the neutron star. The knowledge of the distributions of NMPs consistent with the fit data is implicitly inbuilt in the Hessian matrix which is central to this covariance approach. Comparing our results with those obtained with the explicit use of the distributions of NMPs, we show that the appropriate correlations among NMPs as induced by the fit data are instrumental in strengthening the correlations of the tidal deformability with the symmetry energy parameters, without it, the said correlations tend to disappear. The interplay between isoscalar and isovector NMPs is also emphasized.

Proton and neutron skins and symmetry energy of mirror nuclei

The neutron skin of nuclei is an important fundamental property, but its accurate measurement faces many challenges. Inspired by charge symmetry of nuclear forces, the neutron skin of a neutron-rich nucleus is related to the difference between the charge radii of the corresponding mirror nuclei. We investigate this relation within the framework of the Hartree-Fock-Bogoliubov method with Skyrme interactions. Predictions for proton skins are also made for several mirror pairs in the middle mass range. For the first time the correlation between the thickness of the neutron skin and the characteristics related with the density dependence of the nuclear symmetry energy is investigated simultaneously for nuclei and their corresponding mirror partners. As an example, the Ni isotopic chain with mass number A=48−60 is considered. These quantities are calculated within the coherent density fluctuation model using Brueckner and Skyrme energy-density functionals for isospin asymmetric nuclear matter with two Skyrme-type effective interactions, SkM* and SLy4. Results are also presented for the symmetry energy as a function of A for a family of mirror pairs from selected chains of nuclei with Z=20, N=14, and N=50. The evolution curves show a similar behavior crossing at the N=Z nucleus in each chain and a smooth growing deviation when N≠Z starts. Comparison of our results for the radii and skins with those from the calculations based on high-precision chiral forces is made.

Spin-polarized β -stable neutron star matter: The nuclear symmetry energy and GW170817 constraint

Magnetic field of rotating pulsar might be so strong that the equation of state (EOS) of neutron star (NS) matter is significantly affected by the spin polarization of baryons. In the present work, the EOS of the spin-polarized nuclear matter is investigated in the nonrelativistic Hartree-Fock formalism, using a realistic density dependent nucleon-nucleon interaction with its spin and spin-isospin dependence accurately adjusted to the Brueckner-Hartree-Fock results for the spin-polarized nuclear matter. The nuclear symmetry energy and proton fraction are found to increase significantly with the increasing spin polarization of baryons, leading to a larger probability of the direct Urca process in the cooling of magnetar. The EOS of the $\beta$-stable np$e\mu$ matter obtained at different spin polarizations of baryons is used as the input for the Tolman-Oppenheimer-Volkov equations to determine the hydrostatic configuration of NS. Based on the GW170817 constraint on the radius $R_{1.4}$ of NS with $M\approx 1.4_\odot$, our mean-field results show that up to $60~\%$ of baryons in the NS merger might be spin-polarized. This result supports the magnetar origin of the blue kilonova ejecta of GW170817 suggested by Metzger et al., and the spin polarization of baryons needs, therefore, to be properly treated in the many-body calculation of the EOS of NS matter before comparing the calculated NS mass and radius with those constrained by the multi-messenger GW170817 observation.

Curvature-slope correlation of nuclear symmetry energy and its imprints on the crust-core transition, radius, and tidal deformability of canonical neutron stars

Background: The nuclear symmetry energy $E_{sym}(\rho)$ encodes information about the energy necessary to make nuclear systems more neutron-rich. While its slope parameter L at the saturation density $\rho_0$ of nuclear matter has been relatively well constrained by recent astrophysical observations and terrestrial nuclear experiments, its curvature $K_{\rm{sym}}$ characterizing the $E_{sym}(\rho)$ around $2\rho_0$ remains largely unconstrained. Over 520 calculations for $E_{sym}(\rho)$ using various nuclear theories and interactions in the literature have predicted several significantly different $K_{\rm{sym}}-L$ correlations. Purpose: If a unique $K_{\rm{sym}}-L$ correlation of $E_{sym}(\rho)$ can be firmly established, it will enable us to progressively better constrain the high-density behavior of $E_{sym}(\rho)$ using the available constraints on its slope parameter L. We investigate if and by how much the different $K_{\rm{sym}}-L$ correlations may affect neutron star observables. Method: A meta-model of nuclear Equation of States (EOSs) with three representative $K_{\rm{sym}}-L$ correlation functions is used to generate multiple EOSs for neutron stars. We then examine effects of the $K_{\rm{sym}}-L$ correlation on the crust-core transition density and pressure as well as the radius and tidal deformation of canonical neutron stars. Results:The $K_{\rm{sym}}-L$ correlation affects significantly both the crust-core transition density and pressure. It also has strong imprints on the radius and tidal deformability of canonical neutron stars especially at small L values. The available data from LIGO/VIRGO and NICER set some useful limits for the slope L but can not distinguish the three representative $K_{\rm{sym}}-L$ correlations considered.

Symmetry energy effect on emissions of light particles in coincidence with fast fission

The emission of light particles in coincidence with fast fission in near peripheral reactions of 40Ar+197Au at 30 MeV/u beam energy has been studied by using the Improved Quantum Molecular Dynamics Model (ImQMD). It is demonstrated that in the symmetrical fast fission of the heavy target-like fragment (TLF), the emission of the coalescence-invariant neutrons (CIN) is enhanced particularly at large angles in comparison to that in the events without fission. Due to the enhanced CIN emission and the accumulation of the isospin effect in the long-time process, the yield ratio of the CIN to the coalescence-invariant protons (CIP) in the fission events exhibits significant dependence on the symmetry energy Esym(ρ) varying with density. Considering that the large angle neutrons are experimentally advantageous to measure for the lower flying velocity and lower multiplicity, we emphasize that the yield ratio RCINP at large angles in the fast fission events can be used as a novel probe of the nuclear symmetry energy at sub-saturation densities.

Bayesian inference of nuclear symmetry energy from measured and imagined neutron skin thickness in Sn116,118,120,122,124,130,132, Pb208 , and Ca48

The neutron skin thickness $\Delta r_{np}$ in heavy nuclei has been known as one of the most sensitive terrestrial probes of the nuclear symmetry energy around $\frac{2}{3}$ of the saturation density $\rho_0$ of nuclear matter. Existing neutron skin data mostly from hadronic observables suffer from large uncertainties and their extraction from experiments are often strongly model dependent. While waiting eagerly for the promised model-independent and high-precision neutron skin data for $^{208}$Pb and $^{48}$Ca from the parity-violating electron scattering experiments (PREX-II and CREX at JLab as well as MREX at MESA), within the Bayesian statistical framework using the Skyrme-Hartree-Fock model we infer the posterior probability distribution functions (PDFs) of the slope parameter $L$ of the nuclear symmetry energy at $\rho_0$ from imagined $\Delta r_{np}(^{208}$Pb$)=0.15$, 0.20, and 0.30 fm as well as $\Delta r_{np}(^{48}$Ca$)=0.12$, 0.15, and 0.25 fm, with different $1\sigma$ error bars. The results are compared with the PDFs of $L$ inferred using the same approach from the available $\Delta r_{np}$ data for Sn isotopes from hadronic probes. They are also compared with results from a recent Bayesian analysis of the radius and tidal deformability data of canonical neutron stars from GW170817 and NICER. The neutron skin data for Sn isotopes gives $L=45.5^{+26.5}_{-21.6}$ MeV surrounding its mean value or $L=53.4^{+18.6}_{-29.5}$ MeV surrounding its maximum {\it a posteriori} value, respectively, with the latter smaller than but consistent with the $L=66_{-20}^{+12}$ MeV from the neutron star data within their 68\% confidence intervals. In order to provide additionally useful information on $L$ extracted from the $\Delta r_{np}$ of Sn isotopes, the experimental error bar of $\Delta r_{np}(^{208}$Pb) should be at least smaller than 0.06 fm aimed by some current experiments.

Nuclear matter properties of finite nuclei using relativistic mean field formalism

In this theoretical study, we establish a correlation between the neutron skin thickness and the nuclear symmetry energy for the even–even isotones for magic neutron N = 20, 40, 82, and 126 within self-consistent relativistic mean-field formalism for non-linear NL3⁎ and density-dependent DD-ME2 parameter sets. The local density approximation is used to formulate the symmetry energy, and its co-efficient, namely, neutron pressure of finite nuclei over the isotonic chains. We find a few moderate signatures of pick and/or depth over the isotonic chains at and/or near the proton magic for symmetry energy and neutron pressure, which is a manifestation of the persistence of shell/sub-shell closure. We determine the symmetry energy for the isotonic chain of expected neutron shell closures N = 172 and also find similar behavior as in the case of known magic neutrons. Furthermore, we show the symmetry energy as a function of neutron-proton asymmetry, which results in similar behavior as persisted in the mass-dependence curve. In addition to these, a comparative study is performed using Coherent-Density-Fluctuation-Model to examine the surface effect over the isotonic chain in terms of nuclear matter quantities at local density. The obtained results are of considerable importance since due to shell closure over the isotonic chain, will act as awaiting point in nucleosynthesis of r-process and experimental investigations towards the drip-line region of the nuclear chart.

GW190814's Secondary Component with Mass 2.50–2.67 M⊙ as a Superfast Pulsar

Abstract We use Stergioulas’s RNS code for investigating fast pulsars with equations of state (EOSs) on the causality surface (where the speed of sound is equal to that of light) of the high-density EOS parameter space satisfying all known constraints from both nuclear physics and astrophysics. We show that one possible explanation for GW190814's secondary component, which has mass 2.50–2.67 M ⊙, is that it is a superfast pulsar spinning faster than 971 Hz, about 42% below its Kepler frequency. If confirmed, it would be the fastest pulsar with the highest mass yet observed. There is a large and physically allowed EOS parameter space below the causality surface where pulsars heavier than 2.50 M ⊙ are supported if they can rotate even faster with critical frequencies that depend strongly on the high-density behavior of nuclear symmetry energy.

Constraining isovector nuclear interactions with giant resonances within a Bayesian approach

We put a stringent constraint on the isovector nuclear interactions in the Skyrme-Hartree-Fock model from the centroid energy E−1 of the isovector giant dipole resonance in 208Pb as well as its electric polarizability αD. Using the Bayesian analysis method, E−1 and αD are found to be mostly determined by the nuclear symmetry energy Esym at about ρ⋆=0.05 fm−3 and the isovector nucleon effective mass mv⋆ at the saturation density. At 90% confidence level, we obtain Esym(ρ⋆)=16.4−0.9+1.0 MeV and mv⋆/m=0.79−0.06+0.06.

Nuclear symmetry energy and neutron skin thickness of 208Pb using a finite range effective interaction

We use a finite range simple effective interaction to construct nuclear equations of state for the study of the density dependence of the nuclear symmetry energy. The EoSs provide good descriptions of the nuclear symmetry energy at a subsaturation density ρc = 0.11 fm−3 and at a density around two times the saturation density ρ0. We obtain a correlation between the neutron skin thickness in 208Pb and the density slope parameter at the subsaturation density. A linear relation is obtained between the neutron skin thickness and the parameter , where Es(ρ0) and L(ρc) are respectively the nuclear symmetry energy at saturation density and the density slope parameter at the subsaturation density.

Effects of density-dependent scenarios of in-medium nucleon-nucleon interactions in heavy-ion collisions

Using a more reasonable separate density-dependent scenario instead of the total density-dependent scenario for in-medium $nn$, $pp$ and $np$ interactions, we examine effects of differences of in-medium nucleon-nucleon interactions in two density-dependent scenarios on isospin-sensitive observables in central $^{197}$Au+$^{197}$Au collisions at 400 MeV/nucleon. Moreover, to more physically detect the differences between the nucleon-nucleon interactions in two density-dependent scenarios, we also map the nucleon-nucleon interaction in the separate density-dependent scenario into that in the total density-dependent scenario through fitting the identical constraints for symmetric nuclear matter as well as the identical slope parameter of nuclear symmetry energy at the saturation density. It is shown that two density-dependent scenarios also lead to essentially different symmetry potentials especially at high densities although they can lead to the identical equation of state for the symmetry nuclear matter as well as the identical symmetry energy for the isospin asymmetric nuclear matter. Consequently, these isospin-sensitive observables are also appreciably affected by the different density-dependent scenarios of in-medium nucleon-nucleon interactions. Therefore, according to these findings, it is suggested that effects of the separate density-dependent scenario of in-medium nucleon-nucleon interactions should be taken into account when probing the high-density symmetry energy using these isospin-sensitive observables in heavy-ion collisions.

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.

Nuclear pasta in hot and dense matter and its influence on the equation of state for astrophysical simulations

We explore the properties of nuclear pasta appearing in supernova matter, i.e., matter at finite temperature with a fixed proton fraction. The pasta phases with a series of geometric shapes are studied using the compressible liquid-drop (CLD) model, where nuclear matter separates into a dense liquid phase of nucleons and a dilute gas phase of nucleons and $\alpha$ particles. The equilibrium conditions for two coexisting phases are derived by minimization of the total free energy including the surface and Coulomb contributions, which are clearly different from the Gibbs conditions for phase equilibrium due to the finite-size effects. Compared to the results considering only spherical nuclei, the inclusion of pasta phases can delay the transition to uniform matter and enlarge the region of nonuniform matter in the phase diagram. The thermodynamic quantities obtained in the present calculation with the CLD model are consistent with those in the realistic equation of state table for astrophysical simulations using the Thomas--Fermi approximation. It is found that the density ranges of various pasta shapes depend on both the temperature $T$ and the proton fraction $Y_p$. Furthermore, the nuclear symmetry energy and its density dependence may play crucial roles in determining the properties of pasta phases. Our results suggest that the pasta phase diagram is most sensitively dependent on the symmetry energy slope $L$ especially in the low-$Y_p$ and high-$T$ region.

Quarkyonic matter equation of state in beta-equilibrium

Quark matter may appear due to a hadronic-quark transition in the core of a hybrid star. Quarkyonic matter is an approach in which both quarks and nucleons appear as quasiparticles in a crossover transition, and provides an explicit realization of early ideas concerning quark matter (e.g., the MIT bag model). This description has recently been employed by McLerran and Reddy to model chargeless (pure neutron) matter with an approach that has the virtue that the speed of sound rises quickly at a neutron-quark transition so as to satisfy observational constraints on the neutron star maximum mass ($\ensuremath{\gtrsim}2\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$) and the radius of a $1.4\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ star (${R}_{1.4}\ensuremath{\lesssim}13.5\text{ }\text{ }\mathrm{km}$). Traditional models involving first-order transitions result in softer pressure-energy density relations that have difficulty satisfying these constraints except with very narrow choices of parameters. We propose a variation of quarkyonic matter involving protons and leptons whose energy can be explicitly minimized to achieve both chemical and beta equilibrium, which cannot be done in the chargeless formulation. Quarkyonic stellar models are able to satisfy observed mass and radius constraints with a wide range of model parameters, avoiding the obligatory fine-tuning of conventional hybrid star models, including requiring the transition density to be very close to the nuclear saturation density. Our formulation fits experimental and theoretical properties of the nuclear symmetry energy and pure neutron matter, and contains as few as three free parameters. This makes it an ideal tool for the study of high-density matter that is an efficient alternative to piecewise polytrope or spectral decomposition methods.

Effect of Λv coupling on liquid gas phase transition in warm asymmetric nuclear matter

We present a comprehensive analysis on several features of liquid gas (LG) phase transition in warm asymmetric nuclear matter using relativistic mean field model with isoscalar-isovector Λv coupling. At subsaturation as well as supranormal densities, the Λv coupling systematically controls the density dependence of nuclear symmetry energy in the relativistic mean field model. We found that the boundary of LG phase-coexistence region increases with the higher values of Λv coupling. Similarly the maximal isospin asymmetry, critical pressure and critical temperature increases with the increasing values of Λv, which shows that the liquid gas phase transition is quite sensitive to the density dependence of symmetry energy in hot asymmetric nuclear matter.