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.

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.

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.

Neutron skin thickness of ^(90)Zr and symmetry energy constrained by charge exchange spin-dipole excitations

The charge exchange spin-dipole(SD) excitations of ^(90)Zr are studied using the Skyrme Hartee-Fock plus proton-neutron random phase approximation with SAMi-J interactions.The experimental value of the model-independent sum rule obtained from the SD strength distributions of ^(90)Zr(p,n) ^(90)Nb and ^(90)Zr(n,p) ^(90)Y is used to deduce the neutron skin thickness.The neutron skin thickness Δr_(np) of ^(90)Zr is extracted as 0.083±0.032 fm,which is similar to the results of other studies.Based on the correlation analysis of the neutron skin thickness Δr_(np) and the nuclear symmetry energy J as well as its slope parameter L,a constraint from the extracted Δr_(np) leads to the limitation of J to 29.2±2.6 MeV and L to 53.3±28.2 MeV.

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.

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.

Experiment of constraining symmetry energy at supra-saturation density with π^-/π^+ at HIRFL-CSR

The possibility of the experiment for constraining the symmetry energy Esym（ρ） at supra-densities via π^-/π^＋ probe on the external target experiment of phase I （ ETE（I） ） with part coverage at forward angle at HIRFL-CSR is studied for the first time by using the isospin and momentum dependent hadronic transport model IBUU04. Based on the transport simulation with Au＋Au collisions at 400 MeV/u, it is found that the differential π^-/π^＋ ratios are more sensitive to Esym（ρ） at forward angles in laboratory reference, compared with the total yield ratio widely proposed. The insufficient coverage at lower transverse momentum maintains the sensitivity of the dependence of π^-/π^＋ ratio on the Esym（ρ） at high density, indicating that the ETF （I） under construction in Lanzhou provides the possibility of performing the experiment for probing the asymmetric nuclear equation of state.

Constraint on nuclear symmetry energy imposed by f-mode oscillation of neutron stars

Due to improvements in the sensitivity of gravitational wave(GW)detectors,the detection of GWs originating from the fundamental quasi-normal mode(f-mode)of neutron stars has become possible.The future detection of GWs originating from the f-mode of neutron stars will provide a potential way to improve our understanding of the nature of nuclear matter inside neutron stars.In this work,we investigate the constraint imposed by the f-mode oscillation of neutron stars on the symmetry energy of nuclear matter using Bayesian analysis and parametric EOS.It is shown that if the frequency of the f-mode of a neutron star of known mass is observed precisely,the symmetry energy at twice the saturation density(E_(sym)(2_(ρ0)))of nuclear matter can be constrained within a relatively narrow range.For example,when all the following parameters are within the given intervals:220≤K0≤260 Me V,28≤E_(sym)(ρ0)≤36 Me V,30≤L≤90 Me V,-800≤J0≤400 Me V,-400≤K_(sym)≤100 Me V,-200≤Jsym≤800 Me V,E_(sym)(2ρ0)will be constrained to within-+48.85.56.6 Me V if the f-mode frequency of a canonical neutron star(1.4 M■)is observed to be 1.720 k Hz with a 1%relative error.Furthermore,if only f-mode frequency detection is available,i.e.there is no stellar mass measurement,a precisely detected f-mode frequency can also impose an accurate constraint on the symmetry energy.For example,given the same parameter space and the same assumed observed f-mode frequency mentioned above,and assuming that the stellar mass is in the range of 1.2–2.0 Me,E_(sym)(2ρ0)will be constrained to within 49.5■MeV.In addition,it is shown that a higher slope of 69≤L≤143 Me V will give a higher posterior distribution of E_(sym)(2ρ0),53.8■MeV.

Effect of Wigner energy on the symmetry energy coefficient in nuclei

The nuclear symmetry energy coefficient（including the coefficient asym^（4） of the I^4 term） of finite nuclei is extracted by using the differences of available experimental binding energies of isobaric nuclei.It is found that the extracted symmetry energy coefficient asym^＊（A,I） decreases with increasing isospin asymmetry I,which is mainly caused by Wigner correction,since esym^＊ is the summation of the traditional symmetry energy esym and the Wigner energy ew.We obtain the optimal values J = 30.25±0.10 MeV,ass=56.18±1.25 MeV,asym^（4） = 8.33±1.21 MeV and the Wigner parameter x= 2.38 ±0.12 through a polynomial fit to 2240 measured binding energies for nuclei with20 ≤ A ≤ 261 with an rms deviation of 23.42 keV.We also find that the volume symmetry coefficient J≌ 30 MeV is insensitive to the value x,whereas the surface symmetry coefficient ass and the coefficient asym^（4） are very sensitive to the value of x in the range 1≤x≤4.The contribution of the asym^（4） term increases rapidly with increasing isospin asymmetry I.For very neutron-rich nuclei,the contribution of the asym^（4） term will play an important role.

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.

Influence of coupling constants on nuclear symmetry energy

By studying the energy of neutron star matter, we discuss the nuclear symmetry energy at different baryon densities and different coupling constants in the relativistic mean field approximation. The results show that the symmetry energy increases with baryon density at various coupling constants and incompressibilities. Further-more, the symmetry energy at saturation density increases with increasing incompressibility at fixed d, and decreases at fixed c. Specifically, when coupling constants gv and gs are fixed, respectively, the symmetry energy has a little change with increasing incompressibility. It is demonstrated that the NN coupling constants have greater influences on the symmetry energy than the self-coupling constants.

Symmetry energy from neutron-rich fragments in heavy-ion collisions,and its dependence on incident energy,and impact parameters

The yields of fragments produced in the ^60Ni＋ ^12C reactions at 80 A and 140 A MeV, and with maximum impact parameters of 1.5, 2 and 7.3 fm at 80 A MeV are calculated by the statistical abrasion-ablation model. The yields of fragments are analyzed by the isobaric yield ratio （IYR） method to extract the coefficient of symmetry energy to temperature （asym/T ）. The incident energy is found to influence asym/T very little. It’s found that asym/T of fragments with the same neutron-excess I = N-Z increases when A increases, while asym/T of isobars decreases when A increases. The asym/T of prefragments is rather smaller than that of the final fragments, and the asym/T of fragments in small impact parameters is smaller than that of the larger impact parameters, which both indicate that asym/T decreases when the temperature increases. The choice of the reference IYRs is found to have influence on the extracted a sym /T of fragments, especially on the results of the more neutron-rich fragments. The surface-symmetry energy coefficient （bs/T ） and the volume-symmetry energy coefficient （bv/T） are also extracted, and the bs/bv is found to coincide with the theoretical results.

Effects of the density dependence of the symmetry energy on neutron stars

In this paper,we include the density dependence behavior of the symmetry energy in the improved quark mass density dependent （IQMDD） model.Under the mean field approximation,this model is applied to investigate neutron star matter and neutron stars successfully.Effects of the density dependence of the symmetry energy on neutron stars are described.

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 p meson exchanges via the 3SD1 coupled channel. Increasing the strength of the tensor component due to the p-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.

Constraints on the nuclear symmetry energy and its density slope from theαdecay process

We study the impact of the nuclear symmetry energy and its density dependence on the α-decay process.Within the framework of the preformed cluster model and the energy density formalism, we use different parameterizations of the Skyrme energy density functionals that yield different equations of state（EOS）. Each EOS is characterized by a particular symmetryenergy coefficient（asym） and a corresponding density-slope parameter L. The stepwise trends of the neutron（proton） skin thickness of the involved nuclei with both asym and L do not clarify the oscillating behaviors of the α-decay half-life Tα with these quantities. We find that the change of the skin thickness after α-decay satisfactorily explains these behaviors. The presented results provide constraints on asym centered around an optimum value asym = 32 MeV, and on L between 41 and 57 MeV. These values of asym and L, which indicate larger reduction of the proton-skin thickness and less increase in the neutron-skin thickness after an α-decay,yield a minimum calculated half-life with the same extracted value of the α-preformation factor inside the parent nucleus.

Key factor for determining relation between radius and tidal deformability of neutron stars: Slope of symmetry energy

The constraints on tidal deformability Λ of neutron stars were first extracted from GW170817 by LIGO and Virgo Collaborations.However,the relationship between the radius R and tidal deformability Λ is still under debate.Using an isospin-dependent parameterized equation of state(EOS),we study the relation between R andΛand its dependence on parameters of symmetry energy Esym and EOS of symmetric nuclear matter E0 when the mass is fixed at 1.4 M⊙,1.0 M⊙,and 1.8 M⊙.We find that,although the changes of high order parameters of Esym and E 0 can shift individual values of R1.4 and Λ1.4,the R1.4~Λ1.4 relation remains approximately at the same fitted curve.The slope L of the symmetry energy plays the dominant role in determining the R1.4~Λ1.4 relation.By investigating the mass dependence of the R~Λ relation,we find that the well fitted R~Λ relation for 1.4 M⊙ is broken for massive neutron stars.

Ground-state properties of light kaonic nuclei signaling symmetry energy at high densities

A sensitive correlation between the ground-state properties of light kaonic nuclei and the symmetry energy at high densities is constructed under the framework of relativistic mean-field theory. Taking oxygen isotopes as an example, we see that a high-density core is produced in kaonic oxygen nuclei, due to the strongly attractive antikaonnucleon interaction. It is found that the 1 S_（1/2） state energy in the high-density core of kaonic nuclei can directly probe the variation of the symmetry energy at supranormal nuclear density, and a sensitive correlation between the neutron skin thickness and the symmetry energy at supranormal density is established directly. Meanwhile, the sensitivity of the neutron skin thickness to the low-density slope of the symmetry energy is greatly increased in the corresponding kaonic nuclei. These sensitive relationships are established upon the fact that the isovector potential in the central region of kaonic nuclei becomes very sensitive to the variation of the symmetry energy. These findings might provide another perspective to constrain high-density symmetry energy, and await experimental verification in the future.

Symmetry energy and experimentally observed cold fragments in intermediate heavy-ion collisions

An attempt is made to study the symmetry energy at the time of primary fragment formation from the experimentally observed cold fragments for a neutron-rich system of ^64Ni ＋ ^9Be at 140 MeV/nucleon,utilizing the recent finding that the excitation energy becomes lower for more neutron-rich isotopes with a given Z value.The extracted α（sym）/T values from the cold fragments,based on the Modified Fisher Model（MFM）,are compared to those from the primary fragments of the antisymmetrized molecular dynamics（AMD） simulation and become consistent with the simulation when the I = N —Z value becomes larger,indicating that the excitation energy of these neutron-rich isotopes is indeed lower.