Slope Parameter Of Symmetry Energy Research Articles

Stripping Model for Short GRBs: The Impact of Nuclear Data

We investigate the impact of forthcoming nuclear data on the predictions of the neutron star (NS) stripping model for short gamma-ray bursts. The main area to which we pay attention is the NS crust. We show that the uncertain properties of the NS equation of state can significantly influence the stripping time tstr, the main dynamical parameter of the model. Based on the known time delay (tstr≈1.7 s) between the peak of the gravitational wave signal GW170817 and the detection of gamma photons from GRB170817A, we obtain new restrictions on the nuclear matter parameters, in particular, the symmetry energy slope parameter: L<114.5MeV. In addition, we study the process of nucleosynthesis in the outer and inner crusts of a low-mass NS. We show that the nucleosynthesis is strongly influenced by both the forthcoming nuclear data and the equation of state of the NS matter.

Constraints on Neutron Skin Thickness and Nuclear Deformations Using Relativistic Heavy-ion Collisions from STAR

In these proceedings, we present the measurements of neutron skin thickness and nuclear deformation using isobar $^{96}_{44}$Ru+$^{96}_{44}$Ru and $^{96}_{40}$Zr+$^{96}_{40}$Zr collisions at $\sqrt{s_{_{\rm NN}}}=200$ GeV by the STAR detector. The significant deviations from unity of the isobar ratios of elliptic flow $v_{2}$, triangular flow $v_{3}$, mean $p_{\rm T}$ fluctuations $\langle\delta p_{\rm T}^{2}\rangle/\langle p_{\rm T}\rangle^{2}$, and asymmetric cumulant ${\rm ac}_{2}\{3\}$ indicate large differences in their quadrupole and octuple deformations. The significant deviations of the isobar ratios of produced hadron multiplicity $N_{\rm ch}$, mean transverse momentum $\langle p_{\rm T}\rangle$, and net charge number $\Delta Q$ indicate a halo-type neutron skin for the Zr nucleus, much thicker than for the Ru nucleus, consistent with nuclear structure calculations. We discuss how we extract the neutron skin thickness, the symmetry energy slope parameter, and deformation parameters from data.

Isospin symmetry breaking in the charge radius difference of mirror nuclei

Isospin symmetry breaking (ISB) effects in the charge radius difference $\mathrm{\ensuremath{\Delta}}{R}_{\mathrm{ch}}$ of mirror nuclei are studied using the test example of $^{48}\mathrm{Ca}$ and $^{48}\mathrm{Ni}$. This choice allows for a transparent study of ISB contributions since pairing and deformation effects, commonly required for the study of mirror nuclei, can be neglected in this specific pair. The connection of $\mathrm{\ensuremath{\Delta}}{R}_{\mathrm{ch}}$ with the nuclear equation of state and the effect of ISB on such a relation are discussed according to an energy density functional approach. We find that nuclear ISB effects may shift the estimated value for the symmetry energy slope parameter $L$ by about 6 to 14 MeV while Coulomb corrections can be neglected. ISB effects on the ground-state energy and charge radii in mirror nuclei have been recently predicted by ab initio calculations to be relatively small, pointing to a negligible effect for the extraction of information on the nuclear EoS. These contrasting results call for a dedicated theoretical effort to solve this overarching problem that impacts not only the neutron-skin thickness or the difference in mass and charge radii of mirror nuclei but also other observables such as the isobaric analog state energy.

Decoding the nuclear symmetry energy event-by-event in heavy-ion collisions with machine learning

Inferences of the nuclear symmetry energy from heavy-ion collisions are currently based on the comparison of measured observables and transport model simulations. Only the expectation values of observables over all considered events are used in these approaches, however, observables can be obtained event-by-event both in experiments and transport model simulations. By using the light gradient boosting machine (LightGBM), a modern machine-learning algorithm, we present a framework for inferring the density-dependent nuclear symmetry energy from observables in heavy-ion collisions on the event-by-event analysis. The ultrarelativistic quantum molecular dynamics (UrQMD) model simulations are used as training data. The symmetry energy slope parameter extracted with LightGBM event-by-event from test data also by UrQMD has an average spread of approximately 30 MeV from the truth, and is found to be robust against variations in model parameters. In addition, LightGBM can identify features that have the greatest effect on the physics of interest, thereby offering valuable insights. Our study suggests that the present framework can be a powerful tool and may offer a new paradigm to study the underlying physics in heavy-ion collisions.

Density dependence of the nuclear symmetry energy and neutron skin thickness in the KIDS framework

The KIDS framework for the nuclear equation of state (EoS) and energy density functional (EDF) offers the possibility to explore systematically the effect of EoS parameters on predictions for a variety of observables. The EoS parameters can be varied independently of each other and independently of assumptions regarding the in-medium nucleon effective mass. Here I present a pilot study of the neutron skin thickness (NST) in nuclei of current interest. The results indicate that variations of the symmetry energy slope parameter L by roughly 10 MeV and variations of the droplet-model counterpart of the curvature parameter Kτ by roughly 20 MeV affect predictions by comparable amounts. However, structural details may also have sizable effects on predictions, notably in the cases of 68Ni and 208Pb. This work is part of a systematic investigation of the NST within the KIDS framework and of a broader effort to constrain the density dependence of the nuclear symmetry energy.

Density dependence of symmetry energy and neutron skin thickness revisited using relativistic mean field models with nonlinear couplings

The correlation between neutron skin-thickness of a nucleus with neutron excess and density slope parameter of symmetry energy is assessed as a function of density using relativistic mean-field models containing nonlinear couplings among different mesons. Models with larger skin were found to probe the density slope parameter even at suprasaturation densities, whereas models with smaller skin were observed to be sensitive only at specific subsaturation density, connected to the average density of nuclei. Possible reasons behind this density dependence are explored systematically. These results might be model specific, which need to be reassessed in other types of interactions existing in the literature. Nevertheless, extrapolating the predictions at high densities from models, which are optimized by data at saturation or subsaturation densities, needs to be handled with care.

Rapidity distributions of Z = 1 isotopes and the nuclear symmetry energy from Sn+Sn collisions with radioactive beams at 270 MeV/nucleon

The rapidity distributions of hydrogen isotopes emitted from central collisions of neutron-rich 132Sn+124Sn and neutron-deficient 108Sn+112Sn systems at 270 MeV/nucleon were investigated at RIKEN-RIBF. The data are compared with antisymmetrized molecular dynamics (AMD) calculations and the rapidity distributions can be reproduced after adjusting the in-medium nucleon-nucleon cross sections. The double ratios between the two reaction systems taken for the relative yields of deuteron to proton (d/p) and triton to proton (t/p) are further examined in the midrapidity domain, where the adjustments in the AMD calculations do not affect much on them. The d/p and t/p double ratios at midrapidity agree well with the ratio of the system neutron numbers and its squared value, respectively, and the rapidity dependence of these double ratios is consistent with a picture of partial mixing of colliding nuclei. By comparing with the AMD model which shows a strong symmetry energy dependence of the t/p double ratio, the experimental result in the midrapidity domain favors the calculation with a symmetry-energy slope parameter around L=46 MeV rather than L=108 MeV.

Electric dipole polarizability in neutron-rich Sn isotopes as a probe of nuclear isovector properties

The determination of nuclear symmetry energy, and in particular, its density dependence, is a long-standing problem for nuclear physics community. Previous studies have found that the product of electric dipole polarizability $\alpha_D$ and symmetry energy at saturation density $J$ has a strong linear correlation with $L$, the slope parameter of symmetry energy. However, current uncertainty of $J$ hinders the precise constraint on $L$. We investigate the correlations between electric dipole polarizability $\alpha_D$ (or times symmetry energy at saturation density $J$) in Sn isotopes and the slope parameter of symmetry energy $L$ using the quasiparticle random-phase approximation based on Skyrme Hartree-Fock-Bogoliubov. A strong and model-independent linear correlation between $\alpha_D$ and $L$ is found in neutron-rich Sn isotopes where pygmy dipole resonance (PDR) gives a considerable contribution to $\alpha_D$, attributed to the pairing correlations playing important roles through PDR. This newly discovered linear correlation would help one to constrain $L$ and neutron-skin thickness $\Delta R_\textnormal{np}$ stiffly if $\alpha_D$ is measured with high resolution in neutron-rich nuclei. Besides, a linear correlation between $\alpha_D J$ in a nucleus around $\beta$-stability line and $\alpha_D$ in a neutron-rich nucleus can be used to assess $\alpha_D$ in neutron-rich nuclei.

Effect of inner crust EoS on neutron star properties

The neutron star maximum mass and the radius are investigated within the framework of the relativistic mean-field (RMF) model. The variation in the radius at the canonical mass, R1.4, using different inner crust equation of state (EoS) with different symmetry energy slope parameter is studied. It is found that although the NS maximum mass and the corresponding radius do not vary much with different inner crust EoSs, the radius and the tidal deformability at 1.4M⊙ vary with the different choice of crust EoS and variation of about 1-2 km is seen in the radius at the canonical mass. For non-unified EoSs, the crust with a low symmetry energy slope parameter produces a low NS radius at the canonical mass. The properties of maximally rotating neutron stars are also studied. The variation in the radius of rotating star at the canonical mass 1.4M⊙ is also seen with the slope parameter. Similar to the static neutron star, the radius at 1.4M⊙ of rotating neutron star is affected by slope parameter of the inner crust. Other important quantities like moment of inertia, frequency, rotational kinetic energy to gravitational energy ratio are also calculated. The variation in these quantities with the crust slope parameter is found to be more proportional to the mass and the radius of NS.

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.

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.

Fully self-consistent study of charge-exchange resonances and the impact on the symmetry energy parameters

We have examined within a fully self-consistent theoretical framework the energy difference between the anti-analog giant dipole resonance (AGDR) and the isobaric analog state (IAS), ${E}_{\mathrm{AGDR}}\ensuremath{-}{E}_{\mathrm{IAS}}$, as an indicator of the neutron skin and of the density behavior of the symmetry energy. We have improved two specific points in our HF+RPA calculations: (1) the exchange term of the two-body Coulomb interaction is treated exactly without Slater approximation; and (2) the two-parameters spin-orbit interaction is treated in a consistent way within the energy density functional theory. The estimated values for the neutron skin in $^{208}\mathrm{Pb}$ and the slope parameter of symmetry energy are compared with previous analysis available in the literature.

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.

Properties of nuclear matter from macroscopic–microscopic mass formulas

Based on the standard Skyrme energy density functionals together with the extended Thomas–Fermi approach, the properties of symmetric and asymmetric nuclear matter represented in two macroscopic–microscopic mass formulas: Lublin–Strasbourg nuclear drop energy (LSD) formula and Weizsäcker–Skyrme (WS*) formula, are extracted through matching the energy per particle of finite nuclei. For LSD and WS*, the obtained incompressibility coefficients of symmetric nuclear matter are K∞=230±11 MeV and 235±11 MeV, respectively. The slope parameter of symmetry energy at saturation density is L=41.6±7.6 MeV for LSD and 51.5±9.6 MeV for WS*, respectively, which is compatible with the liquid-drop analysis of Lattimer and Lim [4]. The density dependence of the mean-field isoscalar and isovector effective mass, and the neutron–proton effective masses splitting for neutron matter are simultaneously investigated. The results are generally consistent with those from the Skyrme Hartree–Fock–Bogoliubov calculations and nucleon optical potentials, and the standard deviations are large and increase rapidly with density. A better constraint for the effective mass is helpful to reduce uncertainties of the depth of the mean-field potential.

Core–crust transition properties of neutron stars within systematically varied extended relativistic mean-field model

The model dependence and the symmetry energy dependence of the core–crust transition properties for the neutron stars (NS) are studied using three different families of systematically varied extended relativistic mean field model. Several forces within each of the families are so considered that they yield wide variations in the values of the nuclear symmetry energy a sym and its slope parameter L at the saturation density. The core–crust transition density is calculated using a method based on random-phase-approximation. The core–crust transition density is strongly correlated, in a model independent manner, with the symmetry energy slope parameter evaluated at the saturation density. The pressure at the transition point does not show any meaningful correlations with the symmetry energy parameters at the saturation density. At best, pressure at the transition point is correlated with the symmetry energy parameters and their linear combination evaluated at the some sub-saturation density. Yet, such correlations might not be model independent. The correlations of core–crust transition properties with the symmetry energy parameter are also studied by varying the symmetry energy within a single model. The pressure at the transition point is correlated once again with the symmetry energy parameter at the sub-saturation density.

Equation of state and thickness of the inner crust of neutron stars

The cell structure of $\ensuremath{\beta}$-stable clusters in the inner crust of cold and warm neutron stars is studied within the Thomas--Fermi approach by using relativistic mean-field nuclear models. The relative size of the inner crust and the pasta phase of neutron stars is calculated, and the effect of the symmetry energy slope parameter $L$ on the profile of the neutron star crust is discussed. It is shown that, while the size of the total crust is mainly determined by the incompressibility modulus, the relative size of the inner crust depends on $L$. It is found that the inner crust represents a larger fraction of the total crust for smaller values of $L$. Finally, it is shown that, at finite temperature the pasta phase in $\ensuremath{\beta}$-equilibrium matter essentially melts above $5\phantom{\rule{4.pt}{0ex}}\text{to}\phantom{\rule{4.pt}{0ex}}6$ MeV, and that the onset density of the rod-like and slab-like structures does not depend on the temperature.

Magnetar giant flare oscillations and the nuclear symmetry energy

If the observed quasi-periodic oscillations in magnetar flares are partially confined to the crust, then the oscillation frequencies are unique probes of the nuclear physics of the neutron star crust. We study crustal oscillations in magnetars including corrections for a finite Alfven velocity. Our crust model uses a new nuclear mass formula that predicts nuclear masses with an accuracy very close to that of the finite range droplet model. This mass model for equilibrium nuclei also includes shell corrections and an updated neutron-drip line. We perturb our crust model to predict axial crust modes and assign them to observed giant flare quasi-periodic oscillation frequencies from the soft gamma-ray repeater SGR 1806-20. We find magnetar crusts that match observations for various magnetic field strengths, entrainment of the free neutron gas in the inner crust, and crust-core transition densities. We find that observations can be reconciled with smaller values of the symmetry energy slope parameter, L, if there is a significant amount of entrainment of the neutrons by the superfluid or if the crust-core transition density is large. We also find neutron star masses and radii which are in agreement with expectations from what is known about low-density matter from nuclear experiment. Matching observations with a field-free model we obtain the approximate values of M =1.35 Msun and R = 11.9 km. Matching observations using a model with the surface dipole field of SGR 1806-20 (B=2.4x10^15 G) we obtain the approximate values of M = 1.25 Msun and R = 12.4 km.

Electric dipole polarizability in208Pb: Insights from the droplet model

We study the electric dipole polarizability $\alpha_D$ in ${}^{208}$Pb based on the predictions of a large and representative set of relativistic and non-relativistic nuclear mean field models. We adopt the droplet model as a guide to better understand the correlations between $\alpha_D$ and other isovector observables. Insights from the droplet model suggest that the product of $\alpha_D$ and the nuclear symmetry energy at saturation density $J$ is much better correlated with the neutron skin thickness $\Delta r_{np}$ of ${}^{208}$Pb than the polarizability alone. Correlations of $\alpha_D J$ with $\Delta r_{np}$ and with the symmetry energy slope parameter $L$ suggest that $\alpha_D J$ is a strong isovector indicator. Hence, we explore the possibility of constraining the isovector sector of thenuclear energy density functional by comparing our theoretical predictions against measurements of both $\alpha_D$ and the parity-violating asymmetry in ${}^{208}$Pb. We find that the recent experimental determination of $\alpha_D$ in ${}^{208}$Pb in combination with the range for the symmetry energy at saturation density $J=[31\pm (2)_{\rm est.}]$\,MeV suggests $\Delta r_{np}({}^{208}{\rm Pb}) = 0.165 \pm (0.009)_{\rm exp.} \pm (0.013)_{\rm theo.} \pm (0.021)_{\rm est.} {\rm fm}$ and $L= 43 \pm(6)_{\rm exp.} \pm (8)_{\rm theo.}\pm(12)_{\rm est.}$ MeV.

Constraining the density dependence of the symmetry energy from nuclear masses

Empirically determined values of the nuclear volume and surface symmetry energy coefficients from nuclear masses are expressed in terms of density distributions of nucleons in heavy nuclei in the local density approximation. This is then used to extract the value of the symmetry energy slope parameter $L$. The density distributions in both spherical and well deformed nuclei calculated within microscopic framework with different energy density functionals give $L=59.0\ifmmode\pm\else\textpm\fi{}13.0$ MeV. Application of the method also helps in a precision determination of the neutron skin thickness of nuclei that are difficult to measure accurately.