Constraining the Symmetry Term in the Nuclear Equation of State at Sub-Saturation Densities and Finite Temperatures
P. Marini, A. Bonasera, A. McIntosh, R. Tripathi, S. Galanopoulos, K. Hagel, L. Heilborn, Z. Kohley, L. W. May, M. Mehlman, S. N. Soisson, G. A. Souliotis, D. V… 展开作者
P. Marini, A. Bonasera, A. McIntosh, R. Tripathi, S. Galanopoulos, K. Hagel, L. Heilborn, Z. Kohley, L. W. May, M. Mehlman, S. N. Soisson, G. A. Souliotis, D. V. Shetty, W. B. Smith, B. C. Stein, S. Wuenschel, S. J. Yennello
备注：
12 pages, 9 figures, submitted to Phys. Rev. C and accepted for publication
Methods of extraction of the symmetry energy (or enthalpy) coefficient to temperature ratio from isobaric and isotopic yields of fragments produced in Fermi-energy heavy-ion collisions are discussed. We show that the methods are consistent when the hot fragmenting source is well characterized and its excitation energy and isotopic composition are properly taken into account. The results are independent of the mass number of the detected fragments, which suggests that their fate is decided very early in the reaction.
Using FOPI data on elliptic flow in Au+Au collisions between 0.4 and 1.5A GeV we extract constraints for the equation of state (EOS) of compressed symmetric nuclear matter using the transport code IQMD by introducing an observable describing the evolution of the size of the elliptic flow as a function of rapidity. This observable is sensitive to the nuclear EOS and a robust tool to constrain the compressibility of nuclear matter up to 2 $ρ_0$.
Within the interacting Fermi gas model for isospin asymmetric nuclear matter, effects of the in-medium three-body interaction and the two-body short-range tensor force due to the $ρ$ meson exchange as well as the short-range nucleon correlation on the high-density behavior of the nuclear symmetry energy are demonstrated respectively in a transparent way. Possible physics origins of the extremely uncertain nuclear symmetry energy at supra-saturation densities are discussed.
The density dependence of the nuclear symmetry energy governs important aspects of very neutron rich systems such as heavy nuclei and their collisions, neutron stars and their mergers. Many analyses of experimental data have generated constraints on the symmetry energy and its first derivative at saturation density, rho_0 ~ 2.7xE14 g/cm3. We show that each analysis does not accurately constrain the symmetry energy at rho_0, but rather at a lower density, rho_s, that is most sensitively probed by that analysis. Using published constraints on the symmetry energy and its first derivative at rho_0…
Symmetry energy for asymmetric nuclear matter at subsaturation densitieswas investigated in the framework of an isospin-dependent quantum molecular dynamics model.Asingle ratio of neutrons and protons is comparedwith the experimental data of Famiano et al. [Phys. Rev. Lett. 97, 052701 (2006)]. We have also performed a comparison for the double ratio with experimental as well as different theoretical results of Boltzmann-Uehling-Uhlenbeck in 1997, Isospin-dependent Boltzmann-Uehling-Uhlenbeck in 2004, Boltzmann-Nordeim-Vlasov, and Improved QuantumMolecular Dynamics models. It is found that the …
Within the newly updated version of the Ultra-relativistic quantum molecular dynamics (UrQMD) model in which the Skyrme potential energy-density functional is introduced, the yield ratio between $^3$H and $^3$He clusters emitted from central $^{40}$Ca+$^{40}$Ca, $^{96}$Zr+$^{96}$Zr, $^{96}$Ru+$^{96}$Ru, and $^{197}$Au+$^{197}$Au collisions in the beam energy range from 0.12 to 1 GeV$/$nucleon is studied. The recent FOPI data for the $^3$H$/$$^3$He ratio are compared with UrQMD calculations using 13 Skyrme interactions (all exhibiting similar values of iso-scalar incompressibility but very diff…
Using a transport model coupled with a phase-space coalescence after-burner we study the triton-$^3$He ratio, relative and differential transverse flows in semi-central $^{132}Sn+^{124}Sn$ reactions at a beam energy of 400 MeV/A. The neutron-proton ratio, relative and differential flows are also discussed as a reference. We find that similar to the neutron-proton pairs the triton-$^3$He pairs also carry interesting information about the density dependence of the nuclear symmetry energy. Moreover, the nuclear symmetry energy affects more strongly the t-$^3$He relative and differential flows tha…
The elliptic-flow ratio of neutrons with respect to protons in reactions of neutron rich heavy-ions systems at intermediate energies has been proposed as an observable sensitive to the strength of the symmetry term in the nuclear Equation Of State (EOS) at supra-saturation densities. The recent results obtained from the existing FOPI/LAND data for $^{197}$Au+$^{197}$Au collisions at 400 MeV/nucleon in comparison with the UrQMD model allowed a first estimate of the symmetry term of the EOS but suffer from a considerable statistical uncertainty. In order to obtain an improved data set for Au+Au …
An extension of the generalized relativistic mean-field (gRMF) model with density dependent couplings is introduced in order to describe thermodynamical properties and the composition of dense nuclear matter for astrophysical applications. Bound states of light nuclei and two-nucleon scattering correlations are considered as explicit degrees of freedom in the thermodynamical potential. They are represented by quasiparticles with medium-dependent properties. The model describes the correct low-density limit given by the virial equation of state (VEoS) and reproduces RMF results around nuclear s…
Nuclear structure observables usually most effectively probe the properties of nuclear matter at subsaturation densities rather than at saturation density. We demonstrate that the electric dipole polarizibility $α_ {\text{D}}$ in $^{208}$Pb is sensitive to both the magnitude $E_{\text{sym}}(ρ_{\text{c}})$ and density slope $L(ρ_{\text{c}})$ of the symmetry energy at a subsaturation cross density $ρ_{\text{c}} = 0.11$ fm$^{-3}$. Using the experimental data of $α_ {\text{D}}$ in $^{208}$Pb from RCNP and the recent accurate constraint of $E_{\text{sym}}(ρ_{\text{c}})$ from the binding energy diff…
Nuclear symmetry energy $E_{\rm{sym}}(ρ)$ at density $ρ$ is normally expanded or simply parameterized as a function of $χ=(ρ-ρ_0)/3ρ_0$ in the form of $E_{\rm{sym}}(ρ)\approx S+Lχ+2^{-1}K_{\rm{sym}}χ^2+6^{-1}J_{\rm{sym}}χ^3+\cdots$ using its magnitude $S$, slope $L $, curvature $K_{\rm{sym}}$ and skewness $J_{\rm{sym}}$ at the saturation density $ρ_0$ of nuclear matter. Much progress has been made in recent years in constraining especially the $S$ and $L$ parameters using various terrestrial experiments and astrophysical observations. However, such kind of expansions/parameterizations do not c…
The charged pion multiplicity ratio in intermediate energy heavy-ion collisions, a probe of the density dependence of symmetry energy above the saturation point, has been proven in a previous study to be extremely sensitive to the strength of the isovector $Δ$(1232) potential in nuclear matter. As there is no knowledge, either from theory or experiment, about the magnitude of this quantity, the extraction of constraints on the slope of the symmetry energy at saturation by using exclusively the mentioned observable is hindered at present. It is shown that, by including the ratio of average $p_T…
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 $^{208}$Pb as well as its electric polarizability $α_D$. Using the Bayesian analysis method, $E_{-1}$ and $α_D$ are found to be mostly determined by the nuclear symmetry energy $E_{sym}$ at about $ρ^\star=0.05$ fm$^{-3}$ and the isovector nucleon effective mass $m_v^\star$ at the saturation density. At $90\%$ confidence level, we obtain $E_{sym}(ρ^\star) = 16.4 ^{+1.0}_{-0.9}$ MeV and $m_v^\star/m = 0.79 ^{+0.06}_{-0.0…
Cluster formation is a fundamental aspect of the equation of state (EOS) of warm and dense nuclear matter such as can be found in supernovae (SNe). Similar matter can be studied in heavy-ion collisions (HIC). We use the experimental data of Qin et al. [Phys. Rev. Lett. 108, 172701 (2012)] to test calculations of cluster formation and the role of in-medium modifications of cluster properties in SN EOSs. For the comparison between theory and experiment we use chemical equilibrium constants as the main observables. This reduces some of the systematic uncertainties and allows deviations from ideal…
We show that the neutron skin thickness $Δr_{np}$ of heavy nuclei is uniquely fixed by the symmetry energy density slope $L(ρ)$ at a subsaturation cross density $ρ_c \approx 0.11$ fm$^{-3}$ rather than at saturation density $ρ_0$, while the binding energy difference $ΔE$ between a heavy isotope pair is essentially determined by the magnitude of the symmetry energy $E_{\text{sym}}(ρ)$ at the same $ρ_c$. Furthermore, we find a value of $L({ρ_c})$ leads to a negative $E_{\text{sym}}({ρ_{0}})$-$L({ρ_{0}})$ correlation while a value of $E_{\text{sym}}({ρ_{c})}$ leads to a positive one. Using data o…
We show in the Skyrme-Hartree-Fock approach that unambiguous correlations exist between observables of finite nuclei and nuclear matter properties. Using this correlation analysis to existing data on the neutron skin thickness of Sn isotopes, we find important constraints on the value E_{sym}(rho_0) and density slope L of the nuclear symmetry energy at saturation density. Combining these constraints with those from recent analyses of isospin diffusion and double neutron/proton ratio in heavy ion collisions leads to a value of L=58\pm 18 MeV approximately independent of E_{sym}(ρ_0).
Probes for the small-x parton densities and predicted effects of gluon saturation are discussed. At very low x and intermediate Q, only results on hadronic observables at the LHC are available, which do not provide unambiguous information. It is shown that the measurement of direct photons at forward rapidity at the LHC is particularly promising to provide a unique signal. We further discuss the possibilities to perform such measurements with a detector upgrade in the ALICE experiment and present the R&D activities ongoing.
The elliptic flow in collisions of neutron-rich heavy-ion systems at intermediate energies emerges as an observable sensitive to the strength of the symmetry energy at supra-saturation densities. First results obtained by comparing ratios or differences of neutron and hydrogen flows with predictions of transport models favor a moderately soft to linear density dependence, consistent with ab-initio nuclear-matter theories. Comprehensive data sets of high accuracy can be expected to improve our knowledge of the equation of state of asymmetric nuclear matter.
Using a set of model equations of state satisfying the latest constraints from both terrestrial nuclear experiments and astrophysical observations as well as state-of-the-art nuclear many-body calculations of the pure neutron matter equation of state, the tidal polarizability of canonical neutron stars in coalescing binaries is found to be a very sensitive probe of the high-density behavior of nuclear symmetry energy which is among the most uncertain properties of dense neutron-rich nucleonic matter. Moreover, it changes less than $\pm 10%$ by varying various properties of symmetric nuclear ma…
A Gaussian Process framework for constraining the nuclear equation of state from microscopic calculations with correlated uncertainties
Ohio University（俄亥俄大学）；Argonne National Laboratory（阿贡国家实验室）；University of Washington（华盛顿大学）；University of California, Berkeley（加州大学伯克利分校）；Michigan State University（密歇根州立大学）
We show that the gravitational binding energy of a neutron star of a given mass is correlated with the slope of the nuclear symmetry energy at 1-2 times nuclear saturation density for equations of state without significant softening (i.e., those that predict maximum masses $M_{\rm max} > 1.44M_{\odot}$ in line with the largest accurately measured neutron star mass). Applying recent laboratory constraints on the slope of the symmetry energy to this correlation we extract a constraint on the baryon mass of the lower mass member of the double pulsar binary system, PSR J0737-3039B. We compare with…
Efficiency corrected single ratios of neutron and proton spectra in central $^{112}$Sn+$^{112}$Sn and $^{124}$Sn+$^{124}$Sn collisions at 120 MeV/u are combined with double ratios to provide constraints on the density and momentum dependencies of the isovector mean-field potential. Bayesian analyses of these data reveal that the isoscalar and isovector nucleon effective masses, $m_s^* - m_v^*$ are strongly correlated. The linear correlation observed in $m_s^* - m_v^*$ yields a nearly independent constraint on the effective mass splitting $Δm_{np}^*= (m_n^*-m_p^*)/m_N = -0.05_{-0.09}^{+0.09}δ$.…
We present a systematic survey the range of predictions of the neutron star inner crust composition, crust-core transition densities and pressures, and density range of the nuclear `pasta' phases at the bottom of the crust provided by the compressible liquid drop model in the light of current experimental and theoretical constraints on model parameters. Using a Skyrme-like model for nuclear matter, we construct baseline sequences of crust models by consistently varying the density dependence of the bulk symmetry energy at nuclear saturation density, $L$, under two conditions: (i) that the magn…
The nuclear symmetry energy determines the birth of neutron stars and the elementary nucleosynthesis in the universe. To constrain the nuclear symmetry energy, besides the studies in astrophysics, many terrestrial nuclear experiments are being carried out using a wide variety of advanced new facilities. In this paper, by cross-checking two unrelated experimental measurements using the newly updated transport model, we obtained a slightly soft symmetry energy at supra-saturation nuclear densities. While extrapolating to low nuclear densities, this result is compatible with the recent finding by…
The saturation density of nuclear matter $ρ_0$ 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 $ρ_0=0.150\pm0.010$ fm$^{-3}$, where the 7\% error has contributions form …
According to the Hugenholtz-Van Hove theorem, the nuclear symmetry energy $S(ρ)$ and its slope $L(ρ)$ at arbitrary densities can be decomposed in terms of the density and momentum dependence of the single-nucleon potentials in isospin-asymmetric nuclear matter which are potentially accessible to experiment. We quantify the correlations between several well-known isovector observables and $L(ρ)$ to locate the density range in which each isovector observable is most sensitive to the density dependence of the $S(ρ)$. We then study the correlation coefficients between those isovector observables a…
Constraining the nuclear symmetry energy with electric dipole polarizability and neutron skin characteristics in \texorpdfstring{$^{208}\mathrm{Pb}$}{208Pb} within antisymmetrized molecular dynamics
China Institute of Atomic Energy（中国原子能科学研究院）；Department of Physics, Graduate School of Science, Tohoku University（东北大学理学研究科物理学系）；Guangxi Key Laboratory of Nuclear Physics and Technology, Guangxi Normal University（广西师范大学核物理与技术重点实验室）；Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所南方核科学理论中心）
The LIGO-Virgo collaboration ground-breaking detection of the binary neutron-star merger event, GW170817, has intensified efforts towards the understanding of the equation of state (EoS) of nuclear matter. In this letter, we compare directly the density-pressure constraint on the EoS obtained from a recent analysis of the neutron-star merger event to density-pressure constraints obtained from nuclear physics experiments. To relate constraints from nuclear physics to the radii and the tidal deformabilities of neutron stars, we use a large collection of Skyrme density functionals that describe p…
We extract the mass-dependent symmetry energy coefficients $a_{\text{sym}}({A})$ with the nuclear mass differences reducing the uncertainties as far as possible. The estimated $a_{\text{sym}}({A})$ of $^{208}\text{Pb}$ is $22.4\pm 0.3 $ MeV, which is further used to analyze the density-dependent nuclear matter symmetry energy at subsaturation densities. The slope parameter of the symmetry energy at the saturation density $ρ_{0}$ is $L=50.0\pm15.5$ MeV. Furthermore, it is found that, at the density of $ρ=0.69ρ_{0}=0.11$fm$^{-3}$, the symmetry energy $S(ρ=0.11\text{fm}^{-3})=25.98\pm0.01$ MeV an…
We report on an analysis of the impact of available experimental data on hard processes in proton-lead collisions during Run I at the Large Hadron Collider on nuclear modifications of parton distribution functions. Our analysis is restricted to the EPS09 and DSSZ global fits. The measurements that we consider comprise production of massive gauge bosons, jets, charged hadrons and pions. This is the first time a study of nuclear PDFs includes this number of different observables. The goal of the paper is twofold: i) checking the description of the data by nPDFs, as well as the relevance of these…
New observational data of neutron stars since GW170817 have helped improve our knowledge about nuclear symmetry energy especially at high densities. We have learned particularly: (1) The slope parameter $L$ of nuclear symmetry energy at saturation density $ρ_0$ of nuclear matter from 24 new analyses is about $L\approx 57.7\pm 19$ MeV at 68\% confidence level consistent with its fiducial value, (2) The curvature $K_{\rm{sym}}$ from 16 new analyses is about $K_{\rm{sym}}\approx -107\pm 88$ MeV, (3) The magnitude of nuclear symmetry energy at $2ρ_0$, i.e. $E_{\rm{sym}}(2ρ_0)\approx 51\pm 13$ MeV …
We summarize the current status on constraining the density dependence of the symmetry energy from terrestrial laboratory measurements and astrophysical observations. While the value $E_{sym}({ρ_{0}})$ and density slope $L$ of the symmetry energy at saturation density $ρ_{0}$ can vary largely depending on the data or methods, all the existing constraints are essentially consistent with $E_{sym}({ρ_{0}}) = 31 \pm 2$ MeV and $L = 50 \pm 20$ MeV. The determination of the supra-saturation density behavior of the symmetry energy remains a big challenge.
A thorough understanding of properties of neutron stars requires both a reliable knowledge of the equation of state (EOS) of super-dense nuclear matter and the strong-field gravity theories simultaneously. To provide information that may help break this EOS-gravity degeneracy, we investigate effects of nuclear symmetry energy on the gravitational binding energy of neutron stars within GR and the scalar-tensor subset of alternative gravity models. We focus on effects of the slope $L$ of nuclear symmetry energy at saturation density and the high-density behavior of nuclear symmetry energy. We fi…
We introduce a new framework for quantifying correlated uncertainties of the infinite-matter equation of state derived from chiral effective field theory ($χ$EFT). 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 $χ$EFT 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 $χ$EFT expansion. This produce…
Within the nonlinear relativistic mean field (NL-RMF) model, we show that both the pressure of symmetric nuclear matter at supra-saturation densities and the maximum mass of neutron stars are sensitive to the skewness coefficient $J_0$ of symmetric nuclear matter. Using experimental constraints on the pressure of symmetric nuclear matter at supra-saturation densities from flow data in heavy ion collisions and the astrophysical observation of a large mass neutron star PSR J0348+0432, with the former favoring a smaller $J_0$ while the latter a larger $J_0$, we extract a constraint of $-494 \math…
Quantum chromodynamics (QCD) is the theory of strong interactions of quarks and gluons collectively called partons, the basic constituents of all nuclear matter. Its non-abelian character manifests in nature in the form of two remarkable properties: color confinement and asymptotic freedom. At high energies, perturbation theory can result in the growth and dominance of very gluon densities at small-x. If left uncontrolled, this growth can result in gluons eternally growing violating a number of mathematical bounds. The resolution to this problem lies by balancing gluon emissions by recombinati…
