Investigation of the symmetry energy of nuclear matter using isospin-dependent quantum molecular dynamics

Simulations of infinite nuclear matter at different densities,isospin asymmetries and temperatures are performed using the isospin-dependent quantum molecular dynamics(IQMD)model to study the equation of state and symmetry energy.A rigorous periodic boundary condition is used in the simulations.Symmetry energies are extracted from the binding energies under different conditions and compared to the classical molecular dynamics(CMD)model using the same method.The results show that both models can reproduce the experimental results for the symmetry energies at low densities,but IQMD is more appropriate than CMD for nuclear matter above the saturation density.This indicates that IQMD may be a reliable model for the study of the properties of infinite nuclear matter.

Constraining nuclear symmetry energy with the charge radii of mirror-pair nuclei

The nuclear charge radius plays a vital role in determining the equation of state of isospin asymmetric nuclear matter.Based on the correlation between the differences in charge radii of mirror-partner nuclei and the slope parameter(L)of symmetry energy at the nuclear saturation density,an analysis of the calibrated slope parameter L was performed in finite nuclei.In this study,relativistic and nonrelativistic energy density functionals were employed to constrain the nuclear symmetry energy through the available databases of the mirror-pair nuclei^(36)Ca–^(36)S,^(38)Ca–^(38)Ar,and ^(54)Ni–^(54)Fe.The deduced nuclear symmetry energy was located in the range 29.89–31.85 MeV,and L of the symmetry energy essentially covered the range 22.50–51.55 MeV at the saturation density.Moreover,the extracted L_(s) at the sensitivity density p_(s)=0.10 fm^(-3) was located in the interval range 30.52–39.76 MeV.

Symmetry energy of strange quark matter and tidal deformability of strange quark stars

Research performed during the past decade revealed an important role of symmetry energy in the equation of state(EOS)of strange quark matter(SQM).By introducing an isospin-dependent term into the quark mass scaling,the SQM stability window in the equivparticle model was studied.The results show that a sufficiently strong isospin dependence C_(I)can significantly widen the SQM region of absolute stability,yielding results that simultaneously satisfy the constraints of the astrophysical observations of PSR J1614-2230 with 1.928±0.017 Mand tidal deformability 70≤Λ_(1:4)≤580 measured in the event GW170817.With increasing C_(I),the difference between the u,d,and s quark fractions for the SQM inβ-equilibrium becomes inconspicuous for C>0,leading to small isospin asymmetryδ,and further resulting in similar EOS and structures of strange quark stars(SQSs).Moreover,unlike the behavior of the maximum mass of ud QSs,which varies with C_(I)depending on the sign of the parameter C,the maximum mass of the SQSs decreases monotonously with increasing CI.

First-Order Symmetry Energy Induced by Neutron-Proton Mass Difference

The 1st-order symmetry energy coefficient of nuclear matter induced merely by the neutron-proton （n p） mass difference is derived analytically, which turns out to be completely model-independent. Based on this result, （npDM） the 1st-order symmetry energy Esym,1 （A） of heavy nuclei such as 2~spb induced by the np mass difference is investigated with the help of a local density approximation combined with the Skyrme energy density functionals. Although /U（npDM） Esym,1 （A） is small compared with the second-order symmetry energy, it cannot be dropped simply for an accurate estimation of nuclear masses as it is still larger than the rms deviation given by some accurate mass formulas. It is therefore suggested that one perhaps needs to distinguish the neutron mass from the proton one in the construction of nuclear density funetionals.

Effects of the symmetry energy slope on the axial oscillations of neutron stars

The impact of symmetry energy slope L on the axial w-mode oscillations is explored, where the range of the con- strained slope L of symmetry energy at saturation density is adopted from 25 MeV to 115 MeV while keeping the equation of state （EOS） of symmetric nuclear matter fixed. Based on the range of the symmetry energy slope, a constraint on the frequency and damping time of the wi-mode of the neutron star is given. It is found that there is a perfect linear relation between the frequency and the stellar mass for a fixed slope L, and the softer symmetry energy corresponds to a higher frequency. Moreover, it is confirmed that both the frequencies and damping times have a perfect universal scaling behavior for the EOSs with different symmetry energy slopes at saturation density.

Impact of Crust Matter on Properties of Neutron Star with Supersoft Symmetry Energy

By employing three typical equations of states （EOSs） of the crust matter, the effect of the crust on the structure and properties are investigated, where the core matter is described by the MDIxl model and the non-Newtonian gravity （described by the Yukawa contribution） is considered. It is shown that the EOSs of the crust matter have a notable effect on the mass-radius relation and the moment of inertia.

Deexcitation Energy of Superdeformed Secondary Minima as a Probe to Density Dependence of Symmetry Energy

Deexcitation energies of superdeformed secondary minima of odd-odd A u and T1 isotopes are investigated with the relativistic mean field （RMF） model where the isoscalar-isovector coupling is included to change the symmetry energy. It is verified by the theoretical analysis and numerical results that the deexcitation energies of superdeformed secondary minima relative to the ground states in these heavy nuclei are sensitive to differences in the symmetry energy. In particular, the linear correlation between the deexeitation energies of odd-odd Au and T1 isotopes and the neutron skin thickness in 208Pb is established. Moreover, explorations are extended to superdeformed candidates of other mass regions. It is found that the linear correlation can even be established between the deexcitation energies and the symmetry pressure at subsaturation density. These indicate that deexcitation energies can serve as a probe to the density dependence of the symmetry energy.

Properties of collective flow and pion production in intermediate-energy heavy-ion collisions with a relativistic quantum molecular dynamics model

The relativistic mean-field approach was implemented in the Lanzhou quantum molecular dynamics transport model(LQMD.RMF). Using the LQMD.RMF, the properties of collective flow and pion production were investigated systematically for nuclear reactions with various isospin asymmetries. The directed and elliptic flows of the LQMD.RMF are able to describe the experimental data of STAR Collaboration. The directed flow difference between free neutrons and protons was associated with the stiffness of the symmetry energy, that is, a softer symmetry energy led to a larger flow difference. For various collision energies, the ratio between the π^(-) and π^(+) yields increased with a decrease in the slope parameter of the symmetry energy. When the collision energy was 270 MeV/nucleon, the single ratio of the pion transverse momentum spectra also increased with decreasing slope parameter of the symmetry energy in both nearly symmetric and neutron-rich systems.However, it is difficult to constrain the stiffness of the symmetry energy with the double ratio because of the lack of threshold energy correction on the pion production.

Effects of the momentum dependence of nuclear symmetry potential on pion observables in Sn+Sn collisions at 270 MeV/nucleon

Within a transport model,we investigated the effects of the momentum dependence of the nuclear symmetry potential on the pion observables in central Sn+Sn collisions at 270 MeV/nucleon.To this end,the quantity U^(∞)_(sym)(ρ_(0))(i.e.,the value of the nuclear symmetry potential at the saturation densityρ_(0) and infinitely large nucleon momentum)was used to characterize the momentum dependence of the nuclear symmetry potential.With a certain L(i.e.,the slope of the nuclear symmetry energy at(ρ_(0))),the characteristic parameter U^(∞)_(sym)(ρ_(0))of the symmetry potential significantly affects the production of−and+and their pion ratios.Moreover,by comparing the charged pion yields,pion ratios,and spectral pion ratios of the theoretical simulations for the reactions ^(108) Sn+^(112) Sn and ^(132)Sn+^(124)Sn with the corresponding data in the SRIT experiments,we found that our results favor a constraint on U^(∞)_(sym)(ρ_(0))(i.e.,−160_(−9)^(+18) MeV),and L is also suggested within a range of 62.7 MeV<L<93.1 MeV.In addition,the pion observable for^(197)Au+^(197)Au collisions at 400 MeV/nucleon also supports the extracted value for U^(∞)_(sym)(ρ_(0)).

Scaling of anisotropic flows and nuclear equation of state in intermediate energy heavy ion collisions

Elliptic flow （v2） and hexadecupole flow （v4） of light clusters have been studied in detail for 25 MeV/nucleon ^86Kr ＋ ^124Sn at large impact parameters by using a quantum molecular dynamics model with different potential parameters. Four sets of parameters including soft or hard equation of state （EOS） with or without symmetry energy term are used. Both number-of-nucleon （A） scaling of the elliptic flow versus transverse momentum （Pt） and the scaling of v4/A^2 versus （pt/A）^2 have been demonstrated for the light clusters in all above calculation conditions. It is also found that the ratio of v4/v2^2 maintains a constant of 1/2 which is independent of pt for all the light fragments. Comparisons among different combinations of the EOS and the symmetry potential term show that the above scaling behaviours are sound and independent of the details of potential, while the strengths of flows are sensitive to the EOS and the symmetry potential term.

New quantification of symmetry energy from neutron skin thicknesses of^(48)Ca and^(208)Pb

Precise knowledge of the nuclear symmetry energy can be tentatively calibrated using multimessenger constraints.The neutron skin thickness of a heavy nucleus is one of the most sensitive indicators for probing the isovector components of effective interactions in asymmetric nuclear matter.Recent studies have suggested that the experimental data from the CREX and PREX2 collaborations are not mutually compatible with existing nuclear models.In this study,we review the quantification of the slope parameter of the symmetry energy L from the neutron skin thicknesses of^(48)Ca and^(208)Pb.Skyrme energy density functionals classified by various isoscalar incompressibility coefficients K were employed to evaluate the bulk properties of finite nuclei.The calculated results suggest that the slope parameter L deduced from^(208)Pb is sensitive to the compression modulus of symmetric nuclear matter,but not that from^(48)Ca.The effective parameter sets classified by K=220 MeV can provide an almost overlapping range of L from^(48)Ca and^(208)Pb.

Correlation between the charge radii difference in mirror partner nuclei and thesymmetry energy slope

A correlation between the charge radii difference of mirror partner nucleiΔR_(ch) and the slope parameter L of symmetry energy has been built to ascertain the equation of state of isospin asymmetric nuclear matter.In this work,the influences of pairing correlations and isoscalar compression modulus on theΔRch are systematically investigated based on the Skyrme energy density functional theory.The calculated results suggest that the linear correlation betweenΔR_(ch) and L is decreased by the surface pairing correlations.The slope parameter deduced from the difference of charge radii of mirror-pair nuclei ^(32)Ar-^(32)Si,^(36)Ca-^(36)S,^(38)Ca-^(38)Ar,and ^(54)Ni-^(54)Fe falls into the range of L=42.57−50.64 MeV,that is,the rather soft equation of state of asymmetric nuclear matter.Besides,the range of the slope parameter can also be influenced by the effective forces classified by various isoscalar incompressibility coefficients.

Basic quantities of the equation of state in isospin asymmetric nuclear matter

Based on the Hugenholtz-Van Hove theorem six basic quantities of the EoS in isospin asymmetric nuclear matter are expressed in terms of the nucleon kinetic energy t(k),the isospin symmetric and asymmetric parts of the single-nucleon potentials U_(0)(ρ,k)and U_(sym,i)(ρ,k).The six basic quantities include the quadratic symmetry energy E_(sym,2)(ρ),the quartic symmetry energy E_(sym,4)(ρ),their corresponding density slopes L_(2)(ρ)and L_(4)(ρ),and the incompressibility coefficients K_(2)(ρ)and K_(4)(ρ).By using four types of well-known effective nucleon-nucleon interaction models,namely the BGBD,MDI,Skyrme,and Gogny forces,the density-and isospin-dependent properties of these basic quantities are systematically calculated and their values at the saturation density q_(0)are explicitly given.The contributions to these quantities from t(k)U_(0)(ρ,k),and U_(sym,i)(ρ,k)are also analyzed at the norma nuclear density q_(0).It is clearly shown that the first-order asymmetric term U_(sym,1)(ρ,k)(also known as the symmetry potential in the Lane potential)plays a vital role in determining the density dependence of the quadratic symmetry energy E_(sym,2)(ρ).It is also shown that the contributions from the high-order asymmetric parts of the single-nucleon potentials(U_(sym,i)(ρ,k)with i>1)cannot be neglected in the calculations of the other five basic quantities Moreover,by analyzing the properties of asymmetric nuclear matter at the exact saturation densityρ_(sat)(δ),the corresponding quadratic incompressibility coefficient is found to have a simple empirical relation K_(sat,2)=K_(2)(ρ_(0))-4.14L_(2)(ρ_(0))

The effect of symmetry potential on the balance energy of light particles emitted from mass symmetric heavy-ion collisions with isotopes,isobars and isotones

Using the Ultrarelativistic Quantum Molecular Dynamics(UrQMD) model,the balance energies of free neutrons,free protons and Z=1 particles(including free protons,deuterons and tritons) from mass symmetric heavy-ion collisions with isotopes,isobars and isotones are studied.The influence of nuclear symmetry potential energy on the balance energy is emphasized.It is found that the balance energy of free neutrons is sensitive to the nuclear symmetry energy,while that of free protons is not.Particularly,the initial neutron/proton ratio dependence of the balance energy of free neutrons from Sn isotopes can be taken as a useful probe to constrain the stiffness of the nuclear symmetry energy.

Influence of the symmetry energy on the balance energy of the directed flow

The influence of the density-dependent symmetry energy on the balance energy （Ebal） of directed flow from heavy ion collisions （HICs） at incident energies covered by INDRA and MSU experiments is studied, using the updated version of the ultra- relativistic quantum molecular dynamics （UrQMD） model, especially adapted to low-energy heavy ion collisions （HICs）. Four mass-symmetric reactions with total mass numbers between 192 and 394 are chosen for investigating the influence of the symmetry energy on the system-mass dependence of Eual. The results show that the uncertainty in the density dependence of the symmetry potential causes changes of Ebal of the order of several MeV, depending on the type of particle considered. The Ebal of neutrons from HICs is particularly sensitive to the density dependence of the symmetry potential energy, while the system-mass dependence of EbaI of Z = 1 particles is not.

Symmetry energy at subsaturation densities and the neutron skin thickness of ^(208)Pb

The mass-dependent symmetry energy coefficients asym（A） has been extracted by analysing the heavy nuclear mass differences reducing the uncertainties as far as possible in our previous work. Taking advantage of the obtained symmetry energy coefficient asym（A） and the density profiles obtained by switching off the Coulomb interaction in ^208Pb, we calculated the slope parameter L0.11 of the symmetry energy at the density of 0.11 fm^-3. The calculated L0.11 ranges from 40.5 MeV to 60.3 MeV. The slope parameter L0.11 of the symmetry energy at the density of 0.11 fm^-3 is also calculated directly with Skyrme interactions for nuclear matter and is found to have a fine linear relation with the neutron skin thickness of ^208spb, which is the difference of the neutron and proton rms radii of the nucleus. With the linear relation the neutron skin thickness ARnp of ^208spb is predicted to be 0.15-0.21 fm.

Constraints on Non-Radial Oscillation Modes by Symmetry Energy Slope

The effect of the nuclear symmetry energy slope on the non-radial oscillation f-modes in neutron stars is calculated and discussed. Based on a conservative range of the symmetry energy slope constrained by the experiment and theoretical analysis, a constraint on the frequency and damping time of the gravitational radiation from the f-mode in neutron star is obtained. It is also shown that a higher symmetry energy slope corresponds with a smaller frequency and a longer damping time. Meanwhile, a new set of parameters is given to present the universal properties of the scaled frequency and damping time.

Exploring nuclear symmetry energy with isospin dependence in neutron skin thickness of nuclei

We propose an alternative way to constrain the density dependence of the symmetry energy from the neutron skin thickness of nuclei which shows a linear relation to both the isospin asymmetry and the nuclear charge with a form of Z2/3. The relation of the neutron skin thickness to the nuclear charge and isospin asymmetry is systematically studied with the data from antiprotonic atom measurement, and with the extended Thomas-Fermi approach incorporating the Skyrme energy density functional. An obviously linear relationship between the slope parameter L of the nuclear symmetry energy and the isospin asymmetry dependent parameter of the neutron skin thickness can be found, by adopting 70 Skyrme interactions in the calculations. Combining the available experimental data, the constraint of -20 MeV 〈～ L 〈～ 82 MeV on the slope parameter of the symmetry energy is obtained. The Skyrme interactions satisfying the constraint are selected.

The nuclear symmetry energy from relativistic Brueckner-Hartree-Fock model

The microscopic mechanisms of the symmetry energy in nuclear matter are investigated in the framework of the relativistic Brueckner-Hartree-Fock(RBHF)model with a high-precision realistic nuclear potential,pvCDBonn A.The kinetic energy and potential contributions to symmetry energy are decomposed.They are explicitly expressed by the nucleon self-energies,which are obtained through projecting the G-matrices from the RBHF model into the terms of Lorentz covariants.The nuclear medium effects on the nucleon self-energy and nucleon-nucleon interaction in symmetry energy are discussed by comparing the results from the RBHF model and those from Hartree-Fock and relativistic Hartree-Fock models.It is found that the nucleon self-energy including the nuclear medium effect on the single-nucleon wave function provides a largely positive contribution to the symmetry energy,while the nuclear medium effect on the nucleon-nucleon interaction,i.e.,the effective G-matrices provides a negative contribution.The tensor force plays an essential role in the symmetry energy around the density.The scalar and vector covariant amplitudes of nucleon-nucleon interaction dominate the potential component of the symmetry energy.Furthermore,the isoscalar and isovector terms in the optical potential are extracted from the RBHF model.The isoscalar part is consistent with the results from the analysis of global optical potential,while the isovector one has obvious differences at higher incident energy due to the relativistic effect.

Isospin splitting of nucleon effective mass and symmetry energy in isotopic nuclear reactions

Within an isospin and momentum dependent transport model, the dynamics of isospin particles（nucleons and light clusters） in Fermi-energy heavy-ion collisions are investigated for constraining the isospin splitting of nucleon effective mass and the symmetry energy at subsaturation densities. The impacts of the isoscalar and isovector parts of the momentum dependent interaction on the emissions of isospin particles are explored, i.e., the mass splittings of m_n^＊=m_p^＊ and m_n^＊〉 m_p^＊（m_n^＊〈 m_p^＊）. The single and double neutron to proton ratios of free nucleons and light particles are thoroughly investigated in the isotopic nuclear reactions of ^112Sn＋^112Sn and ^124Sn＋^124Sn at incident energies of 50 and 120 MeV/nucleon, respectively. It is found that both the effective mass splitting and symmetry energy impact the kinetic energy spectra of the single ratios, in particular at the high energy tail（larger than 20 Me V）. The isospin splitting of nucleon effective mass slightly impacts the double ratio spectra at the energy of 50 MeV/nucleon. A soft symmetry energy with stiffness coefficient of γ_s =0.5 is constrained from the experimental data with the Fermi-energy heavy-ion collisions.