Density Dependence Of Symmetry Energy Research Articles

Cluster production within transport theory

Clusters are quite important at various situations in heavy-ion collisions and in excited low-density nuclear matter. Antisymmetrized molecular dynamics has been improved to take into account the correlations to form light clusters such as deuterons and alpha particles. The present status of this approach for the description of multifragmentation is reported. Calculated results demonstrate strong impacts of clusters in various observables including those usually regarded as probes of the density dependence of symmetry energy.

Cluster production within antisymmetrized molecular dynamics

Clusters are quite important at various situations in heavy-ion collisions. Antisymmetrized molecular dynamics was improved to take into account the correlations to form light clusters, such as deuterons and α particles, and light nuclei composed of several clusters. The momentum fluctuations of emitted particles are also taken into account by a simple method. Formation of fragments and light clusters in a wide range of heavy-ion collisions was well described with a single set of model parameters. Fragmentation in a proton induced reaction was also well reproduced by introducing cluster correlations. Calculated results demonstrate strong impacts of clusters in various observables including those usually regarded as probes of the density dependence of symmetry energy.

Decomposition of the sensitivity of the symmetry energy observables

To exactly answer which density region that some frequently used symmetry-energy-sensitive observables probe, for the first time, we make a study of decomposition of the sensitivity of some symmetry-energy-sensitive observables. It is found that for the Au+Au reaction at incident beam energies of 200 and 400 MeV/nucleon, frequently used symmetry-energy-sensitive observables mainly probe the density-dependent symmetry energy around 1.25$\rho_{0}$ (for pionic observables) or 1.5$\rho_{0}$ (for nucleonic observables). Effects of the symmetry energy in the low-density region is in general small but observable. The fact that the symmetry-energy-sensitive observables are not sensitive to the symmetry energy in the maximal baryon-density region increases the difficulty of studying nuclear symmetry energy at super-density.

Probing the momentum dependence of the symmetry potential by the freen/pratio of pre-equilibrium emission

Based on an isospin and momentum-dependent transport model, we studied the effect of momentum-dependent symmetry potential on the free neutron to proton ratio of preequilibrium nucleon emission. It is found that for $^{132}Sn+^{124}Sn$ reaction at 400 MeV/nucleon incident beam energy, free n/p ratio of preequilibrium nucleon emission mainly probe the momentum-dependence of symmetry potential at nucleon momentum around 400 $\sim$ 600 MeV/c. Whereas for 200 MeV/nucleon incident beam energy, this observable mainly probe the momentum-dependence of symmetry potential at nucleon momentum around 200 $\sim$ 400 MeV/c. To probe the symmetry energy/potential using free n/p ratio, not all the details of the momentum-dependence of the symmetry potential are important, the values of symmetry potential at only certain momentum range are crucial for an observable. It is important to input reasonable density- and momentum-dependence of the symmetry potential according to the magnitude of incident beam energy of heavy-ion collisions. The present experimental data on the symmetry potential are not enough for probing the density-dependent symmetry energy. More experimental data (such as nucleon and nuclei's scattering experiments at different nucleonic momenta and densities) on the symmetry potential are therefore needed to pin down the density-dependent symmetry energy.

Impact of the equation-of-state–gravity degeneracy on constraining the nuclear symmetry energy from astrophysical observables

There is a degeneracy between the equation of state (EOS) of superdense neutron-rich nuclear matter and the strong-field gravity in understanding properties of neutron stars. While the EOS is still poorly known, there are also longstanding ambiguities in choosing General Relativity or alternative gravity theories in the not-so-well tested strong-field regime. Besides possible appearance of hyperons and new phases, the most uncertain part of the nucleonic EOS is currently the density dependence of nuclear symmetry energy. To provide information that may help break the EOS-gravity degeneracy, we investigate effects of symmetry energy within its uncertain range determined by terrestrial nuclear laboratory experiments on the gravitational binding energy and spacetime curvature of neutron stars within GR and the scalar-tensor (ST) theory of gravity. In particular, we focus on effects of the following parameters characterizing the EOS of neutron-rich nucleonic matter: (1) the incompressibility $K_0$ of symmetric nuclear matter, (2) the slope $L$ of symmetry energy at saturation density and (3) the high-density behavior of symmetry energy. We find that the variation of either the density slope $L$ or the high-density behavior of symmetry energy leads to large changes in both the binding energy and curvature of neutron stars while effects of varying the more constrained $K_0$ are negligibly small. The difference in predictions using the GR and the ST theory appears only for massive neutron stars, and is significantly smaller than the differences resulting from variations in the symmetry energy. We conclude that within the ST theory of gravity, the EOS-gravity degeneracy has been broken by the recent relativistic pulsar measurements, and that measurements of neutron star properties sensitive to the compactness constrain mainly the density dependence of the symmetry energy.

Analyzing density dependent symmetry energy and dynamics for mass-asymmetric heavy-ion reactions

We attempt to study the interplay of density-dependent symmetry energy and its impact on the dynamics evolved in intermediate energy mass-asymmetric reactions. The investigation includes the theoretical analysis of various mass-asymmetric heavy-ion reactions (keeping total mass the same) based on the microscopic isospin-dependent quantum molecular dynamical (IQMD) approach. We present the time evolution, colliding geometry (impact parameter), and excitation energy dependence (100 MeV nucleon−1–1 GeV nucleon−1) of the density and temperature reached during the collisions. The mass-asymmetry factor influences the reaction dynamics drastically at higher incident energies and central collisions. Owing to different density profiles, when subjected to different mass-asymmetry combinations, the role of density-dependent symmetry energy varies accordingly.

Effect of tensor force on the density dependence of symmetry energy within the BHF framework

The effect of tensor force on the density dependence of nuclear symmetry energy has been investigated within the framework of the Brueckner-Hartree-Fock (BHF) approach. It is shown that the tensor force manifests its effect via the tensor 3SD1 channel. The density dependence of symmetry energy Esym turns out to be determined essentially by the tensor force from the π meson and ρ meson exchanges via the 3SD1 coupled channel. Increasing the strength of the tensor component due to the ρ-meson exchange tends to enhance the repulsion of the equation of state of symmetric nuclear matter and leads to the reduction of symmetry energy. The present results confirm the dominant role played by the tensor force in determining nuclear symmetry energy and its density dependence within the microscopic BHF framework.

Equation of state of nuclear matter from empirical constraints

From empirically determined values of some of the characteristic constants associated with homogeneous nuclear matter at saturation and sub-saturation densities, within the framework of a Skyrme-inspired energy density functional, we construct an equation of state (EoS) of nuclear matter.This EoS is then used to predict values of density slope parameters of symmetry energy $L(\rho)$, isoscalar incompressibility $K(\rho)$ and a few related quantities. The close consonance of our predicted values with the currently available ones for the density dependence of symmetry energy and incompressibility gleaned from diverse approaches offers the possibility that our method may help in settling their values in tighter bounds. Extrapolation of our EoS at supranormal densities shows that it is in good harmony with the one extracted from experimental data.

Reconstructed primary fragments and symmetry energy, temperature and density of the fragmenting source in Zn64+Sn112 at 40 MeV/nucleon

Symmetry energy, temperature and density at the time of the intermediate mass fragment formation are determined in a self-consistent manner, using the experimentally reconstructed primary hot isotope yields and anti-symmetrized molecular dynamics (AMD) simulations. The yields of primary hot fragments are experimentally reconstructed for multifragmentation events in the reaction system $^{64}$Zn + $^{112}$Sn at 40 MeV/nucleon. Using the reconstructed hot isotope yields and an improved method, based on the modified Fisher model, symmetry energy values relative to the apparent temperature, $a_{sym}/T$, are extracted. The extracted values are compared with those of the AMD simulations, extracted in the same way as that for the experiment, with the Gogny interaction with three different density-dependent symmetry energy terms. $a_{sym}/T$ values change according to the density-dependent symmetry energy terms used. Using this relation, the density of the fragmenting system is extracted first. Then symmetry energy and apparent temperature are determined in a self consistent manner in the AMD model simulations. Comparing the calculated $a_{sym}/T$ values and those of the experimental values from the reconstructed yields, $\rho /\rho_{0} = 0.65 \pm 0.02 $, $a_{sym} = 23.1 \pm 0.6$ MeV and $T= 5.0 \pm 0.4$ MeV are evaluated for the fragmenting system experimentally observed in the reaction studied.

Effect of short-range correlation on the density dependence of nuclear symmetry energy at supra-saturation densities

Short-range correlation (SRC) induces a significant high-momentum component in the nucleon momentum distribution n(k) in both finite nuclei and nuclear matter. Recent experiments suggest that the amount of correlation in pure neutron matter (PNM) may be much less than that in symmetric nuclear matter (SNM). In this paper we investigate on a qualitative level how this affects the density dependence of the symmetry energy, especially at supra-saturation densities, in the framework of the well-established MDI energy density functional. In the MDI formalism, SRC affects both the kinetic energy and the potential energy through its effect on n(k). For the n(k) of SNM, we use a formula proposed based on an interesting relation between n(k) at high momentum k (i.e. k > k F ) and the average nucleon density of finite nuclei. For the n(k) of PNM, we use the prediction of a free Fermi gas model. The result is compared to the symmetry energy calculated assuming both the SNM and PNM to be Fermi-gas like. Our study shows that, to have a realistic estimation of the relative contributions to the symmetry energy from the kinetic part and the two-body and three-body potential part, one has to take the SRC effect into account. Furthermore, it is found that the density dependence of the symmetry energy at supra-saturation densities is sensitive to the competition between the SRC effect and the three-body force effect.

Impact of density-dependent symmetry energy and Coulomb interactions on the evolution of intermediate mass fragments

Within the framework of isospin-dependent quantum molecular dynamics (IQMD) model, we demonstrate the evolution of intermediate mass fragments in heavy-ion collisions. In this paper, we study the time evolution, impact parameter, and excitation energy dependence of IMF production for the different forms of density-dependent symmetry energy. The IMF production and charge distribution show a minor but considerable sensitivity towards various forms of density-dependent symmetry energy. The Coulomb interactions affect the IMF production significantly at peripheral collisions. The IMF production increases with the stiffness of symmetry energy.

Analyzing the role of isospin content and colliding geometry on the transition energy for symmetric colliding nuclei

We explore the effect of isospin content (N/Z) and colliding geometry on the transition energy throughout the mass range between 72 and 394 for isotopes, isobars and isotones. The analysis has been performed within the framework of the isospin dependent quantum molecular dynamics model for symmetric colliding nuclei. Our findings indicate that the impact parameter dependence of transition energy follows a power law behavior. Moreover, the percentage change in the transition energy with and without symmetry energy (or with an isospin independent nucleon–nucleon cross-section) is higher for the lighter colliding nuclei. Further, the N/Z dependence of the transition energy for lighter nuclei can act as a good probe to constrain the density dependence of symmetry energy at a supra-saturation density. Our findings are in good agreement with the experimental measurements of the Identification de Noyaux et Detection avec Resolutions Accrues, Michigan State University and Gesellschaft für Schwerionenforschung collaborations.

Density dependence of the symmetry energy from neutron skin thickness in finite nuclei

The density dependence of the symmetry energy around saturation density, characterized by the slope parameter L , is studied using information provided by the neutron skin thickness in finite nuclei. An estimate for L is obtained from experimental data on neutron skins extracted from antiprotonic atoms. We also discuss the ability of parity-violating elastic electron scattering to obtain information on the neutron skin thickness in 208Pb and to constrain the density dependence of the nuclear symmetry energy. The size and shape of the neutron density distribution of 208Pb predicted by mean-field models is briefly addressed. We conclude with a comparative overview of the L values predicted by several existing determinations.

Three-body force effect on nuclear symmetry energy and single-particle properties of asymmetric nuclear matter

We present an upgraded review of our microscopic investigation on the single-particle properties and the EOS of isospin asymmetric nuclear matter within the framework of the Brueckner theory extended to include a microscopic three-body force. We pay special attention to the discussion of the three-body force effect and the comparison of our results with the predictions by other ab initio approaches. Three-body force is shown to be necessary for reproducing the empirical saturation properties of symmetric nuclear matter within nonrelativistic microscopic frameworks, and also for extending the hole-line expansion to a wide density range. The three-body force effect on nuclear symmetry energy is repulsive, and it leads to a significant stiffening of the density dependence of symmetry energy at supra-saturation densities. Within the Brueckner approach, the three-body force affects the nucleon s.p. potentials primarily via its rearrangement contribution which is strongly repulsive and momentum-dependent at high densities and high momenta. Both the rearrangement contribution induced by the three-body force and the effect of ground-state correlations are crucial for predicting reliably the single-particle properties within the Brueckner framework.

Density-dependent symmetry energy at subsaturation densities from nuclear mass differences

We extract the mass-dependent symmetry energy coefficients ${a}_{\mathrm{sym}}(A)$ with the nuclear mass differences reducing the uncertainties as far as possible. The estimated ${a}_{\mathrm{sym}}(A)$ of ${}^{208}\mathrm{Pb}$ is $22.4\ifmmode\pm\else\textpm\fi{}0.3$ 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 ${\ensuremath{\rho}}_{0}$ is $L=50.0\ifmmode\pm\else\textpm\fi{}15.5$ MeV. Furthermore, it is found that at the density of $\ensuremath{\rho}=0.69{\ensuremath{\rho}}_{0}=0.11$ fm${}^{\ensuremath{-}3}$, the symmetry energy $S(\ensuremath{\rho}=0.11\phantom{\rule{0.28em}{0ex}}{\mathrm{fm}}^{\ensuremath{-}3})=25.98\ifmmode\pm\else\textpm\fi{}0.01$ MeV and the corresponding slope parameter is $L=49.6\ifmmode\pm\else\textpm\fi{}6.2$ MeV, which are consistent with other independent analyses.

Sensitive dependence of isotope and isobar distributions of limiting temperatures on the symmetry energy

The mass, isotope, and isobar distributions of limiting temperatures for finite nuclei are investigated by using a thermodynamics approach together with the Skyrme energy density functional. The relationship between the width of the isotope (isobar) distribution of limiting temperatures and the stiffness of the density dependence of the symmetry energy clearly is observed. The nuclear symmetry energy with smaller slope parameter ${L}_{\mathrm{sym}}$ causes a wider isotope (isobar) distribution of limiting temperatures. The widths of the isotope (isobar) distributions of limiting temperatures could be useful observables for exploring the information of the density dependence of the nuclear symmetry energy at finite temperatures.

Assessment of multifragmentation under the effect of symmetry energy

We study the effect of symmetry energy on the fragment production for the reactions 2010Ne10+2010Ne10 and 19779Au118 + 19779Au118 for the incident energy range of 50–1000 MeV/nucleon using isospin-dependent quantum molecular dynamics (IQMD) model. Our study shows that the density-dependent symmetry energy plays a significant role in multiplicity of fragments produced at low energy. Moreover, sensitivity of density-dependent symmetry energy decreases with increase in neutron content of colliding system. We also compare the symmetry energy suggested by different groups. A comparative study of experimental results with theoretical calculations of IQMD shows that the density-dependent symmetry energy is a good probe to explain the multiplicity of the fragments at low energy.

EFFECT OF ISOSPIN DEGREE OF FREEDOM ON TRANSVERSE MOMENTUM SPECTRA

We study the effect of isospin degree of freedom, incident energy as well as system mass on the behavior of transverse momentum spectra, dN/ptdpt, of neutrons and protons. We find that most of the nucleons suffer soft collisions. The effect of isospin degree of freedom on transverse spectra diminishes with the increase in the incident energy. In Fermi energy region, transverse momentum spectra of both protons and neutrons show sensitivity toward the density dependence of symmetry energy.

Nuclear Symmetry Energy from a Relativistic Mean Field Theory

We discuss the density dependence and coupling-constant dependence of the nuclear symmetry energy in the relativistic mean field approximation of the σ − ω − ρ model. An extremely stiff density dependence of symmetry energy is obtained. At the same time, we find that the coupling constants gσ and gω have greater effect on the nuclear symmetry energy than the self-coupling constants c and d. In particular, gω and gσ determine the slope L and curvature Ksym of nuclear symmetry energy at saturation density, respectively.

Possible constraints on the density dependence of the nuclear symmetry energy from quasi-periodic oscillations in soft gamma repeaters

We systematically examine the fundamental frequencies of shear torsional oscillations in neutron star crusts in a manner that is dependent on the parameter $L$ characterizing the poorly known density dependence of the symmetry energy. The identification of the lowest quasiperiodic oscillation (QPO) among the observed QPOs from giant flares in soft-gamma repeaters as the $/ell=2$ fundamental torsional oscillations enables us to constrain the parameter $L$ as $L/ge 47.4$ MeV, which is the most conservative restriction on $L$ obtained in the present work that assumes that the mass and radius of the flaring neutron stars range 1.4-1.8 $M_{/odot}$ and 10-14 km. Next, we identify one by one a set of the low-lying frequencies observed in giant flares as the fundamental torsional oscillations. The values of $L$ that can reproduce all the observed frequencies in terms of the torsional oscillations coupled with a part of dripped neutrons via entrainment effects are then constrained as 101.1 MeV $/le L /le$ 131.0 MeV. Alternatively, if only the second lowest frequency observed in SGR 1806$-$20 has a different origin, one obtains relatively low $L$ values ranging 58.0 MeV $/le L /le$ 85.3 MeV, which seem more consistent with other empirical constraints despite large uncertainties.