The Brueckner-Hartree-Fock Equation of State for Nuclear Matter and Neutron Skin

The equation of state for nuclear matter is presented within the Brueckner Hartree-Fock （BHF） scheme, by using the realistic Argonne VI8 or Bonn B two-nucleon potentials plus their corresponding microscopic three-nucleon forces. It is then applied to calculate the properties of finite nuclei within a simple liquid-drop model, and we compare the calculated volume, surface, and Coulomb parameters with the empirical ones from the liquid drop model. Nuclear density distributions and charge radii in good agreement with the experimental data are obtained~ and we predict the neutron skin thickness of various nuclei.

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.

Apparent softening of the symmetry energy with the inclusion of non-nucleonic components in nuclear matter

Apparent softening of the symmetry energy with the inclusion of hyperon and quark degrees of freedom is demonstrated by the fact that the phase transition causes the change of the interaction and the suppression of nucleon fractions.The demonstration is fulfilled in the relativistic mean-field model.

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.

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.

How tightly is the nuclear symmetry energy constrained by a unitary Fermi gas?

We examine critically how tightly the density dependence of nuclear symmetry energy E_(sym)(q) is constrained by the universal equation of state of the unitary Fermi gas EUG(q) considering currently known uncertainties of higher order parameters describing the density dependence of the equation of state of isospin asymmetric nuclear matter. We found that E_(UG)(q) does provide a useful lower boundary for the E_(sym)(q). However, it doesnot tightly constrain the correlation between the magnitude E_(sym)(q_0) and slope L unless the curvature K_(sym)of the symmetry energy at saturation density q_0 is more precisely known. The large uncertainty in the skewness parameters affects the E_(sym)(q_0) versus L correlation by the same almost as significantly as the uncertainty in K_(sym).

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.

Probing the density dependence of the symmetry energy with central heavy ion collisions

An improved isospin dependent Boltzmann Langevin model,in which the inelastic channels and momentum dependent interactions are incorporated,is used to investigate the high-density behavior of nuclear symmetry energy.By taking several forms of nuclear symmetry energy,we calculate the time evolutions of neutron over proton ratio,π multiplicity and π-/π+ ratio,and the kinetic energy and transverse momentum spectra of π-/π+ ratio in the heavy ion collisions at 400A MeV.It is found that the neutron over proton ratio and π-/π+ ratio are very sensitive to the nuclear symmetry energy,and the π-is more sensitive to the nuclear symmetry energy than the π+.A supersoft symmetry energy results in a larger π-/π+ ratio.

Development of large acceptance multi-purpose spectrometer in Korea for symmetry energy

The Rare Isotope Accelerator complex for ONline experiments(RAON) is a new radioactive ion beam accelerator facility under construction in Korea. The large acceptance multi-purpose spectrometer(LAMPS) is one of the experimental devices for nuclear physics at RAON. It focuses on the nuclear symmetry energy at supra-saturation densities. The LAMPS Collaboration has developed and constructed various detector elements, including a time projection chamber(TPC) and a forward neutron detector array. From the positron beam test, the drift velocity of the secondary electrons in the TPC is 5:3±0:2 cm/ls with P10 gas mixture, and the position resolution for pads with dimensions of 4×15 mm^2 is in the range of$ 0.6–0.8 mm, depending on the beam position. From the neutron beam test, the energy resolution of the prototype neutron detector module is determined to be 3.4%, and theposition resolution is estimated to be better than 5.28 cm.At present, the construction of the LAMPS neutron detector system is in progress.

The symmetry energy and incompressibility constrained by the observations of glitching pulsars

We investigate the masses of glitching pulsars in order to constrain their equation of state(EOS). The observations of glitches(sudden jumps in rotational frequency) may provide information on the interior physics of neutron stars. With the assumption that glitches are triggered by superfluid neutrons, the masses of glitching neutron stars can be estimated using observations of maximum glitches.Together with the observations of thermal emission from glitching pulsars Vela and J1709–4429, the slope of symmetry energy and incompressibility of nuclear matter at saturation density can be constrained.The slope of symmetry energy L should be larger than 67 MeV while the lower limit of incompressibility for symmetric nuclear matter K_0 is 215 MeV. We also obtain a relationship between L and K_0:6.173 MeV + 0.283 K_0≤ L ≤ 7.729 MeV + 0.291 K_0. The restricted EOSs are consistent with the observations of 2-solar-mass neutron stars and gravitational waves from a binary neutron star inspiral.

Neutron skin and its effects in heavy-ion collisions

Neutron skin is an exotic phenomenon that occurs in unstable nuclei.In this study,the various efects of the neutron skin on nuclear reactions and their relationship with the properties of nuclear structures are reviewed.Based on numerous studies using theoretical models,strong correlations have been found between the neutron skin thickness and the neutron removal cross section,neutron/proton yield ratio,t∕3He yield ratio,neutron-proton momentum diference,isoscaling parameter,photon production,reaction cross sections for neutron-induced reactions,charge-changing cross-sectional diferences of mirror nuclei,astrophysical S-factor,and other quantities in nuclear reactions induced by neutron-rich nuclei.Moreover,the relationships between the neutron skin thickness and certain properties of the nuclear structure,such asα-cluster formation,αdecay,nuclear surface,nuclear temperature,and proton radii diference of mirror nuclei,have also been investigated.Furthermore,it has also been shown that the neutron skin plays a crucial role in relativistic heavy-ion collisions.Experimentally,an unstable nucleus with a neutron skin can be generated by radioactive nuclear beam facilities,and the thickness of the neutron skin can be extracted by measuring the sensitive probes,which further helps impose stringent constraints on the equation of state of asymmetric nuclear matter and the properties of neutron stars.

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.

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.

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.

The spin-isospin decomposition of the nuclear symmetry energy from low to high density

The energy per particle BA in nuclear matter is calculated up to high baryon density in the whole isospin asymmetry range from symmetric matter to pure neutron matter.The results,obtained in the framework of the Brueckner-Hartree-Fock approximation with two-and three-body forces,confirm the well-known parabolic dependence on the asymmetry parameterβ=(N?Z)/A(β^2 law)that is valid in a wide density range.To investigate the extent to which this behavior can be traced back to the properties of the underlying interaction,aside from the mean field approximation,the spin-isospin decomposition of BA is performed.Theoretical indications suggest that theβ^2 law could be violated at higher densities as a consequence of the three-body forces.This raises the problem that the symmetry energy,calculated according to theβ^2 law as a difference between BA in pure neutron matter and symmetric nuclear matter,cannot be applied to neutron stars.One should return to the proper definition of the nuclear symmetry energy as a response of the nuclear system to small isospin imbalance from the Z=N nuclei and pure neutron matter.

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.

Bulk Properties of Symmetric Nuclear and Pure Neutron Matter

We study the equation of state (EOS) of symmetric nuclear and neutron matter within the framework of the Brueckner-Hartree-Fock (BHF) approach which is extended by including a density-dependent contact interaction to achieve the empirical saturation property of symmetric nuclear matter. This method is shown to affect significantly the nuclear matter EOS and the density dependence of nuclear symmetry energy at high densities above the normal nuclear matter density, and it is necessary for reproducing the empirical saturation property of symmetric nuclear matter in a nonrelativistic microscopic framework. Realistic nucleon-nucleon interactions which reproduce the nucleon-nucleon phase shifts are used in the present calculations.

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.

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.

Higher order bulk characteristic parameters of asymmetric nuclear matter

The bulk parameters characterizing the energy of symmetric nuclear matter and the symmetry energy defined at normal nuclear density ρ0 provide important information on the equation of state (EOS) of isospin asymmetric nuclear matter. While significant progress has been made in determining some lower order bulk characteristic parameters, such as the energy E0(ρ0) and incompress ibility K0 of symmetric nuclear matter as well as the symmetry energy Esym(ρ0) and its slope parameter L, yet the higher order bulk characteristic parameters are still poorly known. Here, we analyze the correlations between the lower and higher order bulk char acteristic parameters within the framework of Skyrme Hartree-Fock energy density functional and then estimate the values of some higher order bulk characteristic parameters. In particular, we obtain J0 = (-355±95) MeV and I0 = (1473±680) MeV for the third order and fourth-order derivative parameters of symmetric nuclear matter at ρ0 and Ksym = (-100 ± 165) MeV, Jsym = (224 ± 385) MeV, Isym = (-1309 ± 2025) MeV for the curvature parameter, third-order and fourth-order derivative parameters of the symmetry energy at ρ0, using the empirical constraints on E0(ρ0), K0, Esym(ρ0), L, and the isoscalar and isovector nucleon effective masses. Furthermore, our results indicate that the three parameters E0(ρ0), K0, and J0 can reasonably characterize the EOS of symmetric nuclear matter up to 2ρ0 while the symmetry energy up to 2ρ0 can be well described by Esym(ρ0), L, and Ksym.