Nuclear Matter Saturation Density Research Articles

Study of the Symmetry Energy and the Nuclear Equation of State for 13O -13B and 13N-13C Mirror Nuclei

In parallel with the shell model using the harmonic oscillator's single-particle wave functions, the Hartree-Fock approximation was also used to calculate the neutron skin thickness, the mirror charge radii, and the differences in proton radii for 13O-13B and 13N-13C mirror nuclei. The calculations were done for both mirror nuclei in the psdpn model space. Depending on the type of potential used, the calculated values of skin thickness are affected. The symmetry energy and the symmetry energy's slope at nuclear saturation density were also determined, and the ratio of the density to the saturation density of nuclear matter and the symmetry energy has a nearly linear correlation. The mirror energy displacement was calculated, and the findings corresponded well with the available experimental data for the binding energies of the studied mirror nuclei. The measured values of the symmetry energy coefficient for the pair of mirror nuclei agreed with the computed ones, and this coefficient's value rises exponentially as the difference in charge radius increases.

Impact of symmetry energy on sound speed and spinodal decomposition in dense neutron-rich matter

Using a meta model for nuclear Equation of State (EOS) with its parameters constrained by astrophysical observations and terrestrial nuclear experiments, we examine effects of nuclear EOS especially its symmetry energy \esym term on the speed of sound squared $C^2_s(\rho)$ and the critical density $\rho_t$ where $C^2_s(\rho_t)$ vanishes (indicating the onset of spinodal decomposition) in both dense neutron-rich nucleonic matter relevant for relativistic heavy-ion collisions and the cold $n+p+e+\mu$ matter in neutron stars at $\beta$-equilibrium. Unlike in nucleonic matter with fixed values of the isospin asymmetry $\delta$, in neutron stars with a density dependent isospin profile $\delta(\rho)$ determined consistently by the $\beta$ equilibrium and charge neutrality conditions, the $C^2_s(\rho)$ almost always show a peak and then vanishes at $\rho_t$. The latter strongly depends on the high-density behavior of \esym if the skewness parameter $J_0$ characterizing the stiffness of high-density symmetric nuclear matter (SNM) EOS is not too far above its currently known most probable value of about $-190$ MeV inferred from recent Bayesian analyses of neutron star observables. Moreover, in the case of having a super-soft \esym that is decreasing with increasing density above about twice the saturation density of nuclear matter, the $\rho_t$ is significantly lower than the density where the \esym vanishes (indicating the onset of isospin-separation instability and pure neutron matter formation) in neutron star cores.

Non-radial oscillation modes in hybrid stars: consequences of a mixed phase

We study the possibility of the existence of a deconfined quark matter in the core of neutron star (NS)s and its relation to non-radial oscillation modes in NSs and hybrid star (HS)s. We use relativistic mean field (RMF) models to describe the nuclear matter at low densities and zero temperature. The Nambu-Jona-Lasinio (NJL) model is used to describe the quark matter at high densities and zero temperature. A Gibbs construct is used to describe the hadron-quark phase transition (HQPT) at large densities. Within the model, as the density increases, a mixed phase (MP) appears at density about 2.5 times the nuclear matter saturation density (ρ 0) and ends at density about 5ρ 0 beyond which the pure quark matter phase appears. It turns out that a stable HS of maximum mass, M = 2.27 M ⊙ with radius R = 14 km (for NL3 parameterisation of nuclear RMF model), can exist with the quark matter in the core in a MP only. HQPT in the core of maximum mass HS occurs at radial distance, rc = 0.27R where the equilibrium speed of sound shows a discontinuity. Existence of quark matter in the core enhances the non-radial oscillation frequencies in HSs compared to NSs of the same mass. This enhancement is significantly large for the g modes. Such an enhancement of the g modes is also seen for a density dependent Bayesian (DDB) parmeterisation of the nucleonic EOS. The non-radial oscillation frequencies depend on the vector coupling in the NJL model. The values of g and f mode frequencies decrease with increase the vector coupling in quark matter.

Constraints on high density equation of state from maximum neutron star mass

The low density nuclear matter equation of state is strongly constrained by nuclear properties; however, for constraining the high density equation of state it is necessary to resort to indirect information obtained from the observation of neutron stars, compact objects that may have a central density several times nuclear matter saturation density, ${n}_{0}$. Taking a metamodeling approach to generate a huge set of equations of state that satisfy nuclear matter properties close to ${n}_{0}$ and that do not contain a first-order phase transition, the possibility of constraining the high density equation of state (EOS) was investigated. The entire information obtained from the GW170817 event for the probability distribution of $\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Lambda}}}$ was used to make a probabilistic inference of the EOS, which goes beyond the constraints imposed by nuclear matter properties. Nuclear matter properties close to saturation, below $2{n}_{0}$, do not allow us to distinguish between equations of state that predict different neutron star (NS) maximum masses. This is, however, not true if the equation of state is constrained at low densities by the tidal deformability of the NS merger associated to GW170817. Above $3{n}_{0}$, differences may be large, for both approaches, and, in particular, the pressure and speed of sound of the sets studied do not overlap, showing that the knowledge of the NS maximum mass may give important information on the high density EOS. Narrowing the maximum mass uncertainty interval will have a sizable effect on constraining the high density EOS.

$$\varUpsilon $$ and $$\eta _b$$ mass shifts in nuclear matter

By extending the previous works that two of the present authors were involved, we estimate for the first time the \(\varUpsilon \) and \(\eta _b\) as well as \(B^*\) meson mass shifts (scalar potentials) in symmetric nuclear matter. The main interest is, whether the strengths of the bottomonium-(nuclear matter) and charmonium-(nuclear matter) interactions are similar or very different, in the range of a few tens of MeV at the nuclear matter saturation density. This is because, each (\(J/\varPsi ,\varUpsilon \)) and (\(\eta _c,\eta _b\)) meson group is usually assumed to have very similar properties based on the heavy charm and bottom quark masses. The estimate for the \(\varUpsilon \) is made using an SU(5) effective Lagrangian density and the anomalous coupling one, by studying the BB, \(BB^*\), and \(B^*B^*\) meson loop contributions for the self-energy in free space and in nuclear medium. As a result, we include only the BB meson loop contribution as our prediction. As for the \(\eta _b\), to be complete, we include the \(BB^*\) and \(B^*B^*\) meson loop contributions in the self-energy for the analysis. The in-medium masses of the B and \(B^{*}\) mesons appearing in the self-energy loops are calculated by the quark–meson coupling model. Form factors are used to regularize the loop integrals with a wide range of the cutoff mass values. A detailed analysis on the BB, \(BB^{*}\), and \(B^{*}B^{*}\) meson loop contributions for the \(\varUpsilon \) mass shift is made by comparing with the respectively corresponding \(DD, DD^*\), and \(D^*D^*\) meson loop contributions for the \(J/\varPsi \) mass shift. Based on the analysis for the \(\varUpsilon \), our prediction for the \(\eta _b\) mass shift is made on the same footing as that for the \(\varUpsilon \), namely including only the lowest order \(BB^*\) meson loop. The \(\varUpsilon \) mass shift is predicted to be \(-16\) to \(-22\) MeV at the nuclear matter saturation density with the cutoff mass values in the range of 2000–6000 MeV using the \(\varUpsilon BB\) coupling constant determined by the vector meson dominance model with the experimental data, while the \(\eta _b\) mass shift is predicted to be \(-75\) to \(-82\) MeV with the SU(5) universal coupling constant determined by the \(\varUpsilon BB\) coupling constant for the same range of the cutoff mass values. Our results show an appreciable difference between the bottomonium-(nuclear matter) and charmonium-(nuclear matter) interaction strengths. We also study the \(\varUpsilon \) and \(\eta _b\) mass shifts in a heavy quark (heavy meson) symmetry limit, namely, by calculating their mass shifts using the same coupling constant value as that was used to estimate the \(J/\varPsi \) and \(\eta _c\) mass shifts. For the \(\eta _b\) mass shift an SU(5) symmetry breaking case is also studied in this limit. Our predictions for these cases at nuclear matter saturation density are, \(-6\) to \(-9\) MeV for \(\varUpsilon \), \(-31\) to \(-38\) MeV for \(\eta _b\), and \(-8\) to \(-11\) MeV for \(\eta _b\) with a broken SU(5) symmetry, where the corresponding charm sector ones are, \(-5\) to \(-21\) for \(J/\varPsi \), \(-49\) to \(-87\) for \(\eta _c\), and \(-17\) to \(-51\) for \(\eta _c\) with a broken SU(4) symmetry.

Constraining nuclear matter parameters from correlation systematics: a mean-field perspective

The nuclear matter parameters define the nuclear equation of state (EoS), they appear as coefficients of expansion around the saturation density of symmetric and asymmetric nuclear matter. We review their correlations with several properties of finite nuclei and of neutron stars within mean-field frameworks. The lower order nuclear matter parameters such as the binding energy per nucleon, incompressibility and the symmetry energy coefficients are found to be constrained in narrow limits through their strong ties with selective properties of finite nuclei. From the correlations of nuclear matter parameters with neutron star observables, we further review how precision knowledge of the radii and tidal deformability of neutron stars in the mass range $1 - 2 M_\odot$ may help cast them in narrower bounds. The higher order parameters such as the density slope and the curvature of the symmetry energy or the skewness of the symmetric nuclear matter EoS are, however, plagued with larger uncertainty. From inter-correlation of these higher order nuclear matter parameters with lower order ones, we explore how they can be brought to more harmonious bounds.

Constraining the nuclear symmetry energy and properties of the neutron star from GW170817 by Bayesian analysis

Based on the distribution of tidal deformabilities and component masses of binary neutron star merger GW170817, the parametric equation of state (EOS) is employed to probe the nuclear symmetry energy and the properties of the neutron star. To obtain a proper distribution of the parameters of the EOS that is consistent with the observation, Bayesian analysis is used and the constraints of causality and maximum mass are considered. From this analysis, it is found that the symmetry energy and pressure at twice the saturation density of nuclear matter can be constrained within $$E_{sym}(2{\rho _{0}}) = 34.5^{+20.5}_{-2.3}$$ MeV and $$P (2{\rho _{0}}) = 3.81^{+1.18}_{-2.32}\times 10^{34}$$ dyn cm $$^{-2}$$ at 90% credible level, respectively. Moreover, the constraints on the radii and dimensionless tidal deformabilities of canonical neutron stars are also demonstrated through this analysis, and the corresponding constraints are 10.80 km $$\le R_{1.4} \le $$ 13.20 km and $$133 \le \Lambda _{1.4} \le 686$$ at 90% credible level, with the most probable value of $$\bar{R}_{1.4}$$ = 12.60 km and $$\bar{\Lambda }_{1.4}$$ = 500, respectively. With respect to the prior, our result (posterior result) prefers a softer EOS, corresponding to a lower expected value of symmetry energy, a smaller stellar radius, and a smaller tidal deformability.

Nuclear Symmetry Energies at Supra-saturation Densities Extracted from Observations of Neutron Stars and Gravitational Waves

The high-density behavior of nuclear symmetry energy is the most uncertain part of the Equation of State (EOS) of dense neutron-rich nucleonic matter. It has significant ramifications in understanding properties of nuclear reactions induced by rare isotopes, neutron stars and gravitational waves from various sources. To infer the underlying EOS of dense neutron-rich nuclear matter from observables of neutron stars is the longstanding neutron star inverse-structure problem. To make progress in solving this problem, we advanced two inversion approaches: (1) the direct inversion in the three-dimensional high-density EOS parameter space and (2) Bayesian inference of six EOS parameters statistically. Neutron star observational data obtained so far, especially their radii from analyzing gravitational waves (e.g., via the tidal deformability) from GW170817 and X-rays from quiescent low-mass X-ray binaries can already help constrain significantly the nuclear EOS up to about twice the saturation density of nuclear matter. In particular, the nuclear symmetry energy at twice the saturation density of nuclear matter E sym(2ρ 0) is found to be MeV at 68% confidence level.

Insights into nuclear saturation density from parity-violating electron scattering

The saturation density of nuclear matter $\rho_0$ is a fundamental nuclear physics property that is difficult to predict from fundamental principles. The saturation density is closely related to the interior density of a heavy nucleus, such as $^{208}$Pb. We use parity violating electron scattering to determine the average interior weak charge and baryon densities in $^{208}$Pb. This requires not only measuring the weak radius $R_{\rm wk}$ but also determining the surface thickness of the weak charge density $a$. We obtain $\rho_0=0.150\pm0.010$ fm$^{-3}$, where the 7\% error has contributions form the PREX error on the weak radius, an assumed 10\% uncertainty in the surface thickness $a$, and from the extrapolation to infinite nuclear matter. These errors can be improved with the upcoming PREX II results and with a new parity violating electron scattering experiment, at a somewhat higher momentum transfer, to determine $a$.

Analytic insights on the information content of new observables

Uncertainty quantification has emerged as a rapidly growing field in nuclear science. Theoretical predictions of physical observables often involve extrapolations to regions that are poorly constrained by laboratory experiments and astrophysical observations. Without properly quantified theoretical errors, such model predictions are of limited value. Also, one often deals with theoretical constructs that involve fundamental quantities that are not accessible to experiment or observation. Particularly relevant in this context is the pressure of pure neutron matter. In this contribution we develop an analytic framework to answer the question of "How can new data reduce uncertainties of current theoretical models?" [P.-G. Reinhard and W. Nazarewicz, Phys. Rev. C81, 051303(R) (2010)]. Simple and insightful expressions are obtained to quantify the impact of one or two new observables on theoretical uncertainties in two critical quantities: the slope of the symmetry energy at saturation density and the pressure of pure neutron matter at twice nuclear matter saturation density.

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.

Measurement and microscopic description of odd\u2013even staggering of charge radii of exotic copper isotopes

Nuclear charge radii globally scale with atomic mass number A as A1∕3, and isotopes with an odd number of neutrons are usually slightly smaller in size than their even-neutron neighbours. This odd–even staggering, ubiquitous throughout the nuclear landscape1, varies with the number of protons and neutrons, and poses a substantial challenge for nuclear theory2–4. Here, we report measurements of the charge radii of short-lived copper isotopes up to the very exotic 78Cu (with proton number Z = 29 and neutron number N = 49), produced at only 20 ions s–1, using the collinear resonance ionization spectroscopy method at the Isotope Mass Separator On-Line Device facility (ISOLDE) at CERN. We observe an unexpected reduction in the odd–even staggering for isotopes approaching the N = 50 shell gap. To describe the data, we applied models based on nuclear density functional theory5,6 and A-body valence-space in-medium similarity renormalization group theory7,8. Through these comparisons, we demonstrate a relation between the global behaviour of charge radii and the saturation density of nuclear matter, and show that the local charge radii variations, which reflect the many-body polarization effects, naturally emerge from A-body calculations fitted to properties of A ≤ 4 nuclei.

Benchmark calculations of pure neutron matter with realistic nucleon-nucleon interactions

We report benchmark calculations of the energy per particle of pure neutron matter as a function of the baryon density using three independent many-body methods: Brueckner-Bethe-Goldstone, Fermi hypernetted chain/single-operator chain, and auxiliary-field diffusion Monte Carlo. Significant technical improvements are implemented in the latter two methods. The calculations are made for two distinct families of realistic coordinate-space nucleon-nucleon potentials fit to scattering data, including the standard Argonne $v_{18}$ interaction and two of its simplified versions, and four of the new Norfolk $\Delta$-full chiral effective field theory potentials. The results up to twice nuclear matter saturation density show some divergence among the methods, but improved agreement compared to earlier work. We find that the potentials fit to higher-energy nucleon-nucleon scattering data exhibit a much smaller spread of energies.

Strength of tensor forces from neutron drops in ab initio relativistic Brueckner-Hartree-Fock theory

CITATION: Wang, S., et al. 2019. Strength of tensor forces from neutron drops in ab initio relativistic Brueckner-Hartree-Fock theory. Physical Review C, 100(6):064319, doi:10.1103/PhysRevC.100.064319.

Effects of a dense medium on parameters of doubly heavy baryons

The spectroscopic properties of the doubly heavy spin-$1/2$ baryons $\Xi_{QQ'}$, $\Xi'_{QQ'}$, $\Omega_{QQ'}$ and $\Omega'_{QQ'}$, with heavy quarks $Q$ and $Q'$ being $b$ or/and $c$, are investigated in cold nuclear matter. In particular, the behavior of the mass of these particles with respect to the density of the medium in the range $\rho\in [0,1.4] ~\rho_{sat}$, with $\rho_{sat}=0.11^3 ~GeV^3$ being the saturation density of nuclear matter, is investigated. From the shifts in the mass and vector self energy of the states under consideration, it is obtained that $\Xi_{QQ'}$ and $\Xi'_{QQ'}$ baryons with two heavy quarks and one $u$ or $d$ quark are affected by the medium, considerably. It is also seen that the $\Omega_{QQ'}$ and $\Omega'_{QQ'}$ states, containing two heavy quarks and one $s$ quark do not see the dense medium, at all. The value of mass for the $\Xi_{cc}$ state obtained at $\rho\rightarrow 0$ limit is nicely consistent with the experimental data. Our results on parameters of other members can be useful in the search for these states. The obtained results may also shed light on the future in-medium experiments aiming to search for the behavior of the doubly heavy baryons under extreme conditions.

Implications of the Mass M⊙ of PSR J0740+6620 on the Equation of State of Super-dense Neutron-rich Nuclear Matter

Abstract We study the implications of the very recently reported mass M ⊙ of PSR J0740+6620 on the equation of state (EOS) of super-dense neutron-rich nuclear matter with respect to existing constraints on the EOS based on the mass M = 2.01 ± 0.04 M ⊙ of PSR J0348+0432, the maximum tidal deformability of GW170817, and earlier results of various terrestrial nuclear laboratory experiments. The lower limit of the skewness J 0 measuring the stiffness of super-dense isospin-symmetric nuclear matter is raised from about −220 MeV to −150 MeV, significantly reducing its current uncertainty range. The lower bound of the high-density symmetry energy also increases appreciably leading to a rise of the minimum proton fraction in neutron stars at β-equilibrium from about 0% to 5% around three times the saturation density of nuclear matter. The difficulties for some of the most widely used and previously well tested model EOSs to simultaneously predict both a maximum mass higher than 2.17 M ⊙ and a pressure consistent with that extracted from GW170817 present some interesting new challenges for nuclear theories.

The calculation of Binding Energy and Incompressibility, Pressure, and Velocity of Sound of Infinite Nuclear matter Using New One Boson Interaction.

The use of effective nucleon – nucleon (N N) interactions for the determination of nuclear matter properties such as, binding energy per nucleon, incompressibility,K of infinite nuclear matter, pressure 0 and velocity of sound of nuclear matter has been a subject of great interest to nuclear physicists for many decades. The effective interaction usually involved in these calculations has been the Michigan three Yukawa (M3Y) effective interactions whose origin is from G- matrix approach. In this research work however, we have used a newly developed interaction known as new one boson (NOB) effective interaction to carry out similar calculations. This new interaction is based on the Lowest Order Constrained Variational (LOCV) technique. The interaction reproduces the saturation energy of spin and isospin infinite nuclear matter of approximately -16MeV at the normal nuclear matter saturation density consistent with the best available density-dependent interaction derived from the G-matrix approach. The results of the incompressibility obtained using the NOB interaction ranges from 304 to 309 MeV. These values are in good agreement with the values of incompressibility obtained for similar calculations using the M3Y – Reid effective interaction, in which values for K range from 304 to 310 MeV. The results of 0 pressure and velocity of sound of infinite nuclear matter obtained in the present calculations are also in excellent agreement with results of other workers. The results of our present calculations indicate that, the NOB interaction has passed the basic test for an effective interaction. The NOB may therefore be applied to other nuclear matter and optical model calculations to ascertain its reliability.

Universal correlations in the nuclear symmetry energy, slope parameter, and curvature

From general Fermi liquid theory arguments, we derive correlations among the symmetry energy (J), its slope parameter (L), and curvature (Ksym) at nuclear matter saturation density. We argue that certain properties of these correlations do not depend on details of the nuclear forces used in the calculation. We derive as well a global parametrization of the density dependence of the symmetry energy that we show is more reliable, especially at low densities, than the usual Taylor series expansion around saturation density. We then benchmark these predictions against explicit results from chiral effective field theory.

Competition between delta isobars and hyperons and properties of compact stars

The Δ-isobar degrees of freedom are included in the covariant density functional (CDF) theory to study the equation of state (EoS) and composition of dense matter in compact stars. In addition to Δ's we include the full octet of baryons, which allows us to study the interplay between the onset of delta isobars and hyperonic degrees of freedom. Using both the Hartree and Hartree–Fock approximation we find that Δ's appear already at densities slightly above the saturation density of nuclear matter for a wide range of the meson–Δ coupling constants. This delays the appearance of hyperons and significantly affects the gross properties of compact stars. Specifically, Δ's soften the EoS at low densities but stiffen it at high densities. This softening reduces the radius of a canonical 1.4M⊙ star by up to 2 km for a reasonably attractive Δ potential in matter, while the stiffening results in larger maximum masses of compact stars. We conclude that the hypernuclear CDF parametrizations that satisfy the 2M⊙ maximum mass constraint remain valid when Δ isobars are included, with the important consequence that the resulting stellar radii are shifted toward lower values, which is in agreement with the analysis of neutron star radii.

Tensor Fermi liquid parameters in nuclear matter from chiral effective field theory

We compute from chiral two- and three-body forces the complete quasiparticle interaction in symmetric nuclear matter up to twice nuclear matter saturation density. Second-order perturbative contributions that account for Pauli-blocking and medium polarization are included, allowing for an exploration of the full set of central and noncentral operator structures permitted by symmetries and the long-wavelength limit. At the Hartree-Fock level, the next-to-next-to-leading order three-nucleon force contributes to all noncentral interactions, and their strengths grow approximately linearly with the nucleon density up that of saturated nuclear matter. Three-body forces are shown to enhance the already strong proton-neutron effective tensor interaction, while the corresponding like-particle tensor force remains small. We also find a large isovector cross-vector interaction but small center-of-mass tensor interactions in the isoscalar and isovector channels. The convergence of the expansion of the noncentral quasiparticle interaction in Landau parameters and Legendre polynomials is studied in detail.