Probing the Nuclear Symmetry Energy with Heavy-Ion Reactions Induced by Neutron-Rich Nuclei
aa r X i v : . [ nu c l - t h ] A p r Probing the Nuclear Symmetry Energy with Heavy-Ion Reactions Induced byNeutron-Rich Nuclei
Lie-Wen Chen,
1, 2, 3
Che Ming Ko, Bao-An Li, and Gao-Chan Yong
5, 6, 2 Institute of Theoretical Physics, Shanghai Jiao Tong University, Shanghai 200240, China Department of Physics, Texas A&M University-Commerce, Commerce, Texas 75429-3011, USA Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China Cyclotron Institute and Physics Department, Texas A&M University, College Station, Texas 77843-3366, USA Institute of Modern Physics, Chinese Academy of Science, Lanzhou 730000, China Graduate School, Chinese Academy of Science, Beijing 100039, P.R. China (Dated: October 27, 2018)Heavy-ion reactions induced by neutron-rich nuclei provide a unique means to investigate theequation of state of isospin-asymmetric nuclear matter, especially the density dependence of thenuclear symmetry energy. In particular, recent analyses of the isospin diffusion data in heavy-ionreactions have already put a stringent constraint on the nuclear symmetry energy around the nuclearmatter saturation density. We review this exciting result and discuss its implications on nucleareffective interactions and the neutron skin thickness of heavy nuclei. In addition, we also reviewthe theoretical progress on probing the high density behaviors of the nuclear symmetry energy inheavy-ion reactions induced by high energy radioactive beams.
PACS numbers: 25.70.-z, 21.30.Fe, 24.10.Lx
I. INTRODUCTION
During the last decade, many radioactive beam facili-ties have become available in the world. At these facili-ties, nuclear reactions involving nuclei with large neutronor proton excess can be studied, providing the opportu-nities to study the properties of nuclear matter under theextreme condition of large isospin asymmetry. This hasled to a lot of interests and activities in a new researchdirection in nuclear physics, namely the isospin physics.The ultimate goal of studying isospin physics is to extractinformation on the isospin dependence of in-medium nu-clear effective interactions as well as the equation of state(EOS) of isospin asymmetric nuclear matter, particularlyits isospin-dependent term, i.e., the density dependenceof the nuclear symmetry energy. There are already ex-tensive reviews on the isospin physics in nuclear physics,and they can be found in, e.g., Refs.[1, 2, 3, 4, 5, 6].Knowledge about the nuclear symmetry energy ex-tracted from the EOS of isospin asymmetric nuclear mat-ter is essential in understanding not only many aspectsof nuclear physics, such as heavy-ion collisions inducedby radioactive nuclei and the structure of exotic nuclei,but also a number of important issues in astrophysics,such as nucleosynthesis during pre-supernova evolutionof massive stars and the cooling of protoneutron stars.Although the nuclear symmetry energy at normal nu-clear matter density is known to be around 30 MeV fromthe empirical liquid-drop mass formula [7, 8], its valuesat other densities, especially at supra-normal densities,are poorly known [1, 2]. Predictions based on variousmany-body theories differ widely at both low and highdensities [9, 10]. Empirically, the incompressibility ofasymmetric nuclear matter is essentially undetermined[11], even though the incompressibility of symmetric nu-clear matter at its saturation density ρ ≈ .
16 fm − has been determined to be 231 ± ρ < ρ < ρ has also been constrained bymeasurements of collective flows in nucleus-nucleus colli-sions [3].Theoretical studies of the EOS of isospin asymmetricnuclear matter were started by Brueckner et al. [13] andSiemens [14] in the late 60’s. Since then, There havebeen many studies on this subject based on differentmany-body theories using various two-body and three-body forces or interaction Lagrangians. These many-body theories provide very useful tools for understand-ing the properties of hot and dense nuclear matter, andthey can be roughly classified into three categories: themicroscopic many-body approach, the effective-field the-ory approach, and the phenomenological approach. Inthe microscopic many-body approach, the nuclear many-body problem is treated microscopically using nucleon-nucleon interactions fitted to high-precision experimen-tal data and is thus free of parameters. The microscopicmany-body approach mainly includes the non-relativisticBrueckner-Hartree-Fock (BHF) approach [15, 16, 17, 18],relativistic Dirac-Brueckner-Hartree-Fock (DBHF) ap-proach [19, 20, 21, 22, 23, 24, 25], self-consistent Green’sfunction approach [10, 26, 27, 28, 29], and variationalmany-body approach [30, 31, 32, 33, 34]. In the effective-field theory approach, an effective interaction is con-structed based on the effective-field theory (EFT), lead-ing to a systematic expansion of the EOS in powersof density (the Fermi momentum). The effective-fieldtheory approach can be based on the density func-tional theory [35, 36] or on chiral perturbation theory[37, 38, 39, 40, 41, 42]. Since this approach can be linkedto low energy QCD and its symmetry breaking, it has theadvantage of small number of free parameters and a cor-respondingly higher predictive power. The phenomeno-logical approach is based on effective density-dependentnuclear forces or effective interaction Lagrangians. Inthese approaches, a number of parameters have to be ad-justed to fit the properties of many nuclei. This type ofmodels mainly includes the relativistic mean-field (RMF)theory [21, 43, 44, 45, 46, 47, 48, 49, 50], relativis-tic and non-relativistic Hartree-Fock [51, 52, 53, 54, 55,56, 57, 58, 59, 60, 61, 62] or Thomas-Fermi approxima-tions [61, 63, 64]. These phenomenological approachesallow the most precise description for the properties offinite nuclei. Both the phenomenological and EFT ap-proaches contain parameters that are fixed by nuclearproperties around the saturation density and thus usu-ally give excellent descriptions for the nuclear propertiesaround or below the saturation density. Their predictionsat the high density region are, however, probably unre-liable. In addition, due to different approximations ortechniques used in different microscopic many-body ap-proaches, their predictions on the properties of nuclearmatter could be very different even for the same barenucleon-nucleon interaction [10, 65]. In particular, pre-dictions on the properties of isospin asymmetric nuclearmatter, especially the density dependence of the nuclearsymmetry energy, are still significantly different for dif-ferent many-body theory approaches.Fortunately, heavy-ion reactions induced by radioac-tive beams provide a unique opportunity to investi-gate in terrestrial laboratories the EOS of asymmetricnuclear matter, particularly the density dependence ofthe nuclear symmetry energy. During the past decade,a large amount of theoretical and experimental effortshave been devoted to the study of the properties ofisospin asymmetric nuclear matter via heavy-ion reac-tions [1, 2, 5, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76,77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91,92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104].To extract information about the EOS of neutron-richmatter, especially the density dependence of the nuclearsymmetry energy, from heavy-ion reactions induced byradioactive beams, one needs reliable theoretical tools.Transport models that include explicit isospin-dependentdegrees of freedom are especially useful for understand-ing the role of isospin degree of freedom in the dynamicsof central nuclear reactions induced by neutron nuclei atintermediate and high energies and in extracting infor-mation about the EOS of produced neutron-rich mat-ter. During past two decades, significant progresses havebeen made in developing semi-classical transport mod-els for nuclear reactions. These semi-classical modelsmainly include the following two types: the Boltzmann-Uehling-Ulenbeck (BUU) model [105] and the quantummolecular dynamical (QMD) model [106]. While it isimportant to develop practically implementable quan-tum transport theories, applications of the semi-classicaltransport models have enabled us to learn a great dealof interesting physics from heavy-ion reactions. In par-ticular, with the development of the radioactive nuclearbeam physics, some isospin-dependent transport models [66, 67, 68, 71, 72, 93, 96, 107] have been successfullydeveloped in recent years to describe the nuclear reac-tions induced by neutron nuclei at intermediate and highenergies.In studying the properties of asymmetric nuclear mat-ter from heavy-ion reactions induced by neutron-rich nu-clei, a key task is to identify experimental observablesthat are sensitive to the density dependence of the nu-clear symmetry energy, especially at high densities. Be-cause of the fact that the symmetry potentials for neu-trons and protons have opposite signs and that they aregenerally weaker than the nuclear isoscalar potential atsame density, most observables proposed so far use differ-ences or ratios of isospin multiplets of baryons, mirror nu-clei and mesons, such as the neutron/proton ratio of nu-cleon emissions [68], neutron-proton differential flow [82],neutron-proton correlation function [85], t / He [86, 96], π − /π + [84, 94, 100, 101], Σ − / Σ + [97] and K /K + ra-tios [102], etc.. In addition, in order to reduce the sys-tematical errors, multiple probes taken from several reac-tion systems using different isotopes of the same elementhave also been proposed. These multiple probes mainlyinclude double ratio or double differential flow.Indeed, recent experimental and theoretical analysisof the isospin diffusion data from heavy-ion reactionshas led to significant progress in determining the nuclearsymmetry energy at subnormal densities [107, 108, 109].Based on the same underlying Skyrme interactions asthe ones constrained by the isospin diffusion data, theneutron-skin thickness in Pb calculated within theHartree-Fock approach is consistent with available ex-perimental data [110, 111, 112]. This symmetry energyis also consistent with that from a relativistic mean-fieldmodel using an accurately calibrated parameter set thatreproduces both the giant monopole resonance in Zrand
Pb, and the isovector giant dipole resonance of
Pb [113]. It further agrees with the symmetry en-ergy recently obtained from isoscaling analyses of iso-tope ratios in intermediate-energy heavy ion collisions[114]. These different studies have provided so far thebest phenomenological constraints on the symmetry en-ergy at sub-normal densities. Information on the symme-try energy at supra-normal densities, on the other hand,remains inclusive and more efforts are needed to investi-gate the supra-normal density behavior of the symmetryenergy. Heavy-ion collisions induced by future high en-ergy radioactive beams to be available at high energyradioactive beam facilities will provide a unique oppor-tunity for determining the symmetry energy at supra-normal densities.In the present paper, we review recent progress on thedetermination of the nuclear symmetry energy in heavy-ion reactions induced by neutron-rich nuclei. In particu-lar, we review the exciting results on density dependenceof the nuclear symmetry energy at subnormal densitiesdetermined from recent analysis of the isospin diffusiondata in heavy-ion reactions. We also discuss the im-plications derived from this new information on nucleareffective interactions and the neutron skin thickness ofheavy nuclei. In addition, we review theoretical progressin studying the behavior of nuclear symmetry energy athigh density from heavy-ion reactions induced by highenergy radioactive beams.The paper is organized as follows. In Section II, wegive a brief introduction to the nuclear symmetry en-ergy. We then describe in Section III the IBUU04 hadrontransport model for nuclear reactions induced by radioac-tive beams at intermediate energies. In Section IV, wepresent the results from the IBUU04 model analysis ofthe isospin diffusion data in heavy-ion reactions and dis-cuss the stringent constraint they have imposed on thenuclear symmetry energy around the nuclear matter sat-uration density. Based on the constrained symmetry en-ergy from the isospin diffusion data, we discuss in Sec-tion V the implications of the isospin diffusion data onnuclear effective interactions and the neutron skin thick-ness of heavy nuclei. In Section VI, we review theoreticalprogress on studying the behavior of the nuclear symme-try energy at high density in heavy-ion reactions inducedby high energy radioactive beams. Finally, a summary isgiven in Section VII.
II. THE NUCLEAR SYMMETRY ENERGY
In the parabolic approximation that has been verifiedby all many-body theories to date, the EOS of isospinasymmetric nuclear matter can be written as E ( ρ, δ ) = E ( ρ, δ = 0) + E sym ( ρ ) δ + O ( δ ) , (1)where ρ = ρ n + ρ p is the baryon density with ρ n and ρ p denoting the neutron and proton densities, respectively; δ = ( ρ n − ρ p ) / ( ρ p + ρ n ) is the isospin asymmetry; E ( ρ, δ =0) is the energy per nucleon in symmetric nuclear matter,and E sym ( ρ ) = 12 ∂ E ( ρ, δ ) ∂δ | δ =0 (2)is the nuclear symmetry energy. In Eq. (1), there are noodd-order δ terms due to the exchange symmetry of theproton and neutron in the nuclear matter (the chargesymmetry of nuclear forces). Higher-order terms in δ are negligible. For example, the magnitude of the δ term at ρ has been estimated to be less than 1 MeV[14, 15, 31]. As a good approximation, the density de-pendent symmetry energy E sym ( ρ ) can thus be extractedfrom E sym ( ρ ) ≈ E ( ρ, δ = 1) − E ( ρ, δ = 0) which impliesthat the symmetry energy E sym ( ρ ) is the energy cost toconvert all protons in a symmetric nuclear matter to neu-trons at the fixed density ρ .Around the nuclear matter saturation density ρ , thenuclear symmetry energy E sym ( ρ ) can be further ex-panded to second-order as E sym ( ρ ) = E sym ( ρ ) + L (cid:18) ρ − ρ ρ (cid:19) + K sym (cid:18) ρ − ρ ρ (cid:19) , (3) where L and K sym are the slope and curvature parame-ters of the nuclear symmetry energy at ρ , i.e., L = 3 ρ ∂E sym ( ρ ) ∂ρ | ρ = ρ , (4) K sym = 9 ρ ∂ E sym ( ρ ) ∂ ρ | ρ = ρ . (5)The L and K sym characterize the density dependenceof the nuclear symmetry energy around normal nuclearmatter density, and thus provide important informationon properties of the nuclear symmetry energy at bothhigh and low densities.At the nuclear matter saturation density ρ and around δ = 0, the isobaric incompressibility of asymmetric nu-clear matter can be further expressed as [56, 115] K ( δ ) = K + K asy δ (6)where K is the incompressibility of symmetric nuclearmatter at the nuclear matter saturation density ρ andthe isospin-dependent part K asy ≈ K sym − L [79] char-acterizes the density dependence of the nuclear symme-try energy. In principle, the information on K asy canbe extracted experimentally by measuring the GMR ofneutron-rich nuclei and a constraint of − ± 3, which, according to microscopic many-body calculations, reduces the nuclear matter incom-pressibility by about 30% compared to that for symmet-ric nuclear matter. Moreover, the magnitude of protonconcentration at β equilibrium in a neutron star is almostentirely determined by the symmetry energy. The pro-ton fraction affects not only the stiffness of the EOS butalso the cooling mechanisms of neutron stars [118, 119]and the possibility of kaon condensation ( e − → K − ν e )in dense stellar matter [120]. If the proton concentrationis larger than a critical value of about 15%, the directURCA process ( n → p + e − + ¯ ν e , p + e − → n + ν e )becomes possible, and would then enhance the emissionof neutrinos, making it a more important process in thecooling of a neutron star [118].Unfortunately, the density dependence of E sym ( ρ ), es-pecially its high density behavior, is poorly known and E sy m () ( M e V ) (fm -3 ) FIG. 1: (Color online) Density dependence of the symmetryenergy from the continuous choice Brueckner-Hartree-Fockwith Reid93 potential (circles), self-consistent Green’s func-tion theory with Reid93 potential (full line), variational cal-culation with Argonne Av14 potential (dashed line), Dirac-Brueckner-Hartree-Fock calculation (triangles), relativisticmean-field model (squares), and effective field theory (dash-dotted line). Data are taken from [10]. is regarded as the most uncertain among all propertiesof an isospin asymmetric nuclear matter. Even aroundthe saturation density, values of the parameters L , K sym ,and K asy are still very uncertain with different theoret-ical models giving very different predictions. This canbe seen in Fig. 1 where we show the density dependenceof the nuclear symmetry energy from some of the mostwidely used microscopic many-body theories [10]. Onesees that the theoretical predictions diverge widely atboth low and high densities. In fact, even the sign of thesymmetry energy above 3 ρ is uncertain [9]. The theoret-ical uncertainties are largely due to a lack of knowledgeabout the isospin dependence of nuclear effective inter-actions and the limitations in the techniques in solvingthe nuclear many-body problem.As mentioned in the Introduction, heavy-ion reactions,especially those induced by radioactive beams, providea unique opportunity to pin down the density depen-dence of nuclear symmetry energy in terrestrial labora-tories. Indeed, significant progress has been made re-cently both experimentally and theoretically in determin-ing the symmetry energy at subnormal densities. At sub-normal densities, a density-dependent symmetry energyof E sym ( ρ ) ≈ . ρ/ρ ) . has been found to best repro-duce both the isospin diffusion [107, 108, 109, 110, 112]and isoscaling [114] data in heavy-ion collisions as well as the presently acceptable neutron-skin thickness in Pb[110, 111, 112]. Together, these results represent the bestphenomenological constraints available so far on the sym-metry energy at sub-normal densities. Although the highdensity behavior of the symmetry energy remain largelyundetermined, future high energy radioactive beams tobe available at high energy radioactive beam facilitieswill allow us to determine the symmetry energy at supra-normal densities. III. IBUU TRANSPORT MODEL FORNUCLEAR REACTIONS INDUCED BYRADIOACTIVE BEAMS Transport models are useful theoretical tools not onlyfor studying the reaction mechanisms but also for ex-tracting information on the properties of produced hotdense matter in heavy ion collisions. For nuclear reac-tions induced by radioactive beams, comparing experi-mental data with transport model calculations allows usto extract the information about the EOS of neutron-richmatter. The IBUU model, which has been very useful inunderstanding a number of new phenomena associatedwith the isospin degree of freedom in heavy-ion reactions,is an isospin- and momentum-dependent transport modelthat is based on the Boltzmann-Uhling-Uhlenbeck equa-tion and is applicable for heavy-ion reactions induced byboth stable and radioactive beams [90].In the IBUU model, besides nucleons, ∆ and N ∗ res-onances as well as pions and their isospin-dependent dy-namics are included. The initial neutron and protondensity distributions of projectile and target nuclei areobtained from the Relativistic Mean-field model or theSkyrme-Hartree-Fock model. It has the option of usingeither the experimental free-space nucleon-nucleon (NN)scattering cross sections or the in-medium NN cross sec-tions. For NN inelastic collisions, the experimental free-space cross sections are used as their in-medium crosssections are still very much controversial. The totaland differential cross sections for all other particles aretaken either from experimental data or obtained by us-ing the detailed-balance formula. Time dependence ofthe isospin-dependent phase-space distribution functionsof involved particles are solved numerically using the test-particle method. In treating NN scattering, the isospindependent Pauli blocking factors for fermions is also in-cluded.In the following, we outline the two major ingredi-ents, i.e., the single-nucleon potential and the NN crosssections, of the version IBUU04 of the isospin- andmomentum-dependent IBUU transport model for nuclearreactions induced by radioactive beams [90]. Other de-tails, such as the initialization of the phase space distri-butions of colliding nuclei, the Pauli blocking, etc. canbe found in Refs. [1, 2, 68, 90, 107]. A. Single-nucleon potential One of the most important inputs to all transport mod-els is the single-nucleon potential. Both the isovector(symmetry potential) and isoscalar parts of this potentialshould be momentum-dependent due to the non-localityof strong interactions and the Pauli exchange effects inmany-fermion systems. In the IBUU04, we use a single-nucleon potential derived from the Hartree-Fock approx-imation based on a modified Gogny effective interaction(MDI) [121], i.e., U ( ρ, δ, ~p, τ, x ) = A u ( x ) ρ τ ′ ρ + A l ( x ) ρ τ ρ + B ( ρρ ) σ (1 − xδ ) − τ x Bσ + 1 ρ σ − ρ σ δρ τ ′ + 2 C τ,τ ρ Z d p ′ f τ ( ~r, ~p ′ )1 + ( ~p − ~p ′ ) / Λ + 2 C τ,τ ′ ρ Z d p ′ f τ ′ ( ~r, ~p ′ )1 + ( ~p − ~p ′ ) / Λ . (7)In the above τ = 1 / − / 2) for neutrons (protons) and τ = τ ′ ; σ = 4 / f τ ( ~r, ~p ) is the phase space distribu-tion function at coordinate ~r and momentum ~p . The pa-rameters A u ( x ) , A l ( x ) , B, C τ,τ , C τ,τ ′ and Λ were obtainedby fitting the momentum dependence of U ( ρ, δ, ~p, τ, x )to that predicted by the Gogny Hartree-Fock and/orthe Brueckner-Hartree-Fock calculations, the saturationproperties of symmetric nuclear matter, and the symme-try energy of 31.6 MeV at normal nuclear matter density ρ = 0 . 16 fm − [121]. The incompressibility K of sym-metric nuclear matter at ρ is set to be 211 MeV. Theparameters A u ( x ) and A l ( x ), given by A u ( x ) = − . − x Bσ + 1 , A l ( x ) = − . 57 + x Bσ + 1 , (8)depend on the parameter x that can be adjusted tomimic the predicted E sym ( ρ ) from microscopic and/orphenomenological many-body theories. The last twoterms in Eq. (7) contain the momentum-dependenceof the single-particle potential. The momentum depen-dence of the symmetry potential stems from the differentinteraction strength parameters C τ,τ ′ and C τ,τ for a nu-cleon of isospin τ interacting, respectively, with unlikeand like nucleons in the background fields. More specif-ically, we use C τ,τ ′ = − . C τ,τ = − . U neutron + U proton ) / x = − − x .The interaction part of nuclear symmetry energy can TABLE I: The parameters F (MeV), G , K sym (MeV), L (MeV), and K asy (MeV) for different values of x . Taken fromRef. [109]. x F G K sym L K asy . 232 1 . − . . . 981 1 . − . . − . 673 1 . 569 94 . . − − . 395 1 . 416 276 . . be parameterized by E potsym ( ρ ) = F ( x ) ρ/ρ + (18 . − F ( x ))( ρ/ρ ) G ( x ) (9)with F ( x ) and G ( x ) given in Table I for x = 1, 0, − − 2. Also shown in Table I are other properties ofthe symmetry energy, including its slope parameter L andcurvature parameter K sym at ρ , as well as the isospin-dependent part K asy of the isobaric incompressibility ofasymmetric nuclear matter. It is seen that the stiffness ofthe symmetry energy increases with decreasing x values. E sy m () ( M e V ) / FIG. 2: (Color online) Symmetry energy as a function of den-sity for the MDI interaction with x = 1 , , − − 2. Takenfrom Ref. [109]. What is particularly interesting and important fornuclear reactions induced by neutron-rich nuclei is theisovector (symmetry) potential. The strength of this po-tential can be estimated very accurately from ( U neutron − U proton ) / δ [90]. In Fig. 3, the strength of the sym-metry potential for four x parameters is displayed as afunction of momentum and density. Here we have onlyplotted the symmetry potential at sub-saturation densi-ties most relevant to heavy-ion reactions studies at in-termediate energies. The momentum dependence of thesymmetry potential is seen to be the same for all valuesof x as the parameter x appears by construction onlyin the density-dependent part of the single-nucleon po-tential given by Eq.(7). Systematic analysis of a large MDI with x=-2 ( U n - U p ) / ( M e V ) / k ( f m - ) ( U n - U p ) / ( M e V ) / k ( f m - ) MDI with x=-1 MDI with x=0 ( U n - U p ) / ( M e V ) / k ( f m - ) MDI with x=1 ( U n - U p ) / ( M e V ) / k ( f m - ) FIG. 3: (Color online) Symmetry potential as a function of momentum and density for MDI interactions with x = 1 , , − − 2. Taken from Ref. [107]. number of nucleon-nucleus scattering experiments and(p,n) charge exchange reactions at beam energies belowabout 100 MeV has shown that the data can be well de-scribed by the parametrization U Lane = a − bE kin with a ≃ − 34 MeV and b ≃ . − . a and b are large, the symmetry potential at ρ , i.e., the Lanepotential, clearly decreases approximately linearly withincreasing beam energy E kin . This provides a stringentconstraint on the symmetry potential. The potential inEq. (7) at ρ satisfies this requirement very well as seenin Fig. 3. This can be more clearly seen from the solidline in Fig. 4 which gives the kinetic energy dependenceof ( U neutron − U proton ) / δ at normal nuclear saturationdensity given by the MDI interaction with x = 0. Alsoshown in this figure are the predicted kinetic energy de-pendence of the symmetry potential from the MDI in-teraction with x = 0 for densities away from normal nu-clear density, which are presently not known empirically.Experimental determination of both the density and mo-mentum dependence of the symmetry potential is thus ofgreat interest, and heavy-ion reactions with radioactivebeams provides a unique tool to explore this informationin terrestrial laboratories. Although the effective mass of a nucleon in nuclearmatter depends the density of nuclear matter as well asthe momentum of the nucleon [23, 128, 129], the momen-tum dependence of the symmetry potential further leadsto different effective masses for neutrons and protons inisospin asymmetric nuclear matter, i.e., m ∗ τ m τ = (cid:26) m τ p dU τ dp (cid:27) . (10)When the effective mass is evaluated at the Fermi mo-mentum p τ = p F ( τ ), Eq. (10) yields the Landau masswhich is related to the f Landau parameter of a Fermiliquid [23, 128, 129]. A detailed discussion about differentkinds of effective masses can be found in Ref. [128]. Withthe potential in Eq. (7), the nucleon effective massesare independent of the x parameter as the momentum-dependent part of the nuclear potential is independent ofthe parameter x .Shown in Fig. 5 are the effective masses of neutronsand protons at their respective Fermi surfaces as func-tions of density (upper window) and isospin asymme-try (lower window) obtained from the MDI interaction.It is seen that the effective mass of neutrons is largerthan that of protons and the splitting between them in- - . * x + . - . * x + - . * x + . - . * x + E kin (MeV) MDI interaction with x = 0 ( U n - U p ) / ( M e V ) FIG. 4: (Color online) Kinetic energy dependence of( U neutron − U proton ) / δ at different densities using the MDIinteraction with x = 0. The shaded region indicates the ex-perimental constraint. creases with both the density and isospin asymmetry ofthe medium. Although the momentum dependence of thesymmetry potential and the associated splitting betweenthe neutron and proton effective masses are still highlycontroversial among different approaches and/or usingdifferent nuclear effective interactions [93, 130, 131], theresults presented here are consistent with the predic-tions from all non-relativistic microscopic models, see,e.g., [17, 123, 132], and the non-relativistic limit ofmicroscopic relativistic many-body theories, see, e.g.,[23, 24, 25]. Recent transport model studies have in-dicated that the neutron/proton ratio at high transversemomenta and/or rapidities is a potentially useful probeof the splitting between the neutron and proton effectivemasses in neutron-rich matter [5, 90].The effect due to the momentum dependence of nu-clear mean-field potential can be studied by comparingits predictions with those obtained using the momentum-independent nucleon potential U ( ρ, δ, τ ) ≡ U ( ρ ) + U MDI( x )sym ( ρ, δ, τ ) with the isoscalar part U ( ρ ) taken fromthe original momentum-independent soft nuclear poten-tial with K = 200 MeV (SBKD) introduced by Bertsch,Kruse and Das Gupta [133], i.e., U SBKD ( ρ ) = − ρ/ρ + 303( ρ/ρ ) / . (11)For the momentum-independent symmetry po-tential U MDI( x )sym ( ρ, δ, τ ), it can be obtained from U MDI( x )sym ( ρ, δ, τ ) = ∂W sym /∂ρ τ using the isospin-dependent part of the potential energy density W sym = E pot sym ( ρ ) · ρ · δ where E pot sym ( ρ ) is given byEq. (9) from the MDI interaction. Therefore, themomentum-independent SBKD potential that has K = 200 MeV [133] and exactly the same E sym ( ρ ) as ρ / ρ m * τ / m τ δ m * τ / m τ npnp solid: δ =0.2solid: ρ = ρ dash: δ =0.4dash: ρ =2 ρ FIG. 5: (Color online) Neutron and proton effective massesin asymmetric matter as a function of density (upper win-dow) and isospin asymmetry (lower window). Taken fromRef. [107]. the MDI interaction is U SBKD ( ρ, δ, τ ) = − ρ/ρ + 303 ( ρ/ρ ) / + 4 τ E potsym ( ρ ) + (18 . − F ( x )) × ( G ( x ) − ρ/ρ ) G ( x ) δ . (12) B. In-medium nucleon-nucleon cross sections Another important quantity in the IBUU model is thein-medium NN cross sections. While much attentionhas been given to finding experimental observables thatcan constrain the symmetry energy, little effort has beenmade so far to study the NN cross sections in isospinasymmetric nuclear matter. Most of existing works onin-medium NN cross sections have concentrated on theirdensity dependence in isospin symmetric nuclear matter,see, e.g., [134, 135, 136, 137, 138, 139, 140, 141, 142, 143].One simple model for NN in-medium cross sections isthe effective mass scaling model [134, 135, 138]. In thismodel, while both the incoming current in the initialstate and the level density of the final state in an NNscattering depend on the effective masses of colliding nu-cleons, the scattering matrix elements are assumed tobe the same in free-space and in the medium. As a re-sult, the ratio of the NN cross section in nuclear medium σ medium NN to that in free-space value σ free NN is simply givenby R medium ≡ σ medium NN /σ free NN = ( µ ∗ NN /µ NN ) , (13)where µ NN and µ ∗ NN are the reduced masses of scat-tering nucleon pairs in free-space and in the medium,respectively. Since the nucleon mass becomes smallerin nuclear medium, the in-medium NN cross section issmaller than its value in free space. The relation givenin Eq. (13) was recently found to be consistent with cal-culations based on the DBHF theory [144] for nucleonpairs with relative momenta less than about 240 MeV/cand in nuclear matter with densities less than about 2 ρ .This finding thus lends a strong support to the effectivemass scaling model of in-medium NN cross sections inthis limited density and momentum ranges. We havethus extended this model to determine the in-mediumNN cross sections in the asymmetric nuclear matter pro-duced in nuclear reactions induced by radioactive beams.For nucleon-nucleon scatterings at higher energies, inelas-tic reaction channels become important. Although therewere some studies on in-medium effects in these channels[145, 146, 147], the experimental free-space cross sectionsare used in the IBUU model as the model is mainly forheavy ion collisions at intermediate energies where NNinelastic scatterings are less important than elastic scat-terings.While the effective masses and the in-medium NN crosssections have to be calculated dynamically in the evolvingenvironment created during heavy-ion reactions, it is in-structive to examine the in-medium NN cross sections inisospin asymmetric nuclear matter at zero temperature.In this situation, the integrals in Eq. (7) can be analyti-cally carried out. Specifically, it is given by [121, 148], Z d p ′ f τ ( ~r, ~p ′ )1 + ( ~p − ~p ′ ) / Λ = 2 h π Λ " p f ( τ ) + Λ − p p Λ ln ( p + p f ( τ )) + Λ ( p − p f ( τ )) + Λ + 2 p f ( τ )Λ − − p + p f ( τ )Λ − − p − p f ( τ )Λ (cid:21) . (14)The reduction factor R medium for in-medium NN crosssections can thus also be obtained analytically, albeitlengthy.As an illustration, we show in Fig. 6 the reduction fac-tor R medium for a simplified case of two colliding nucle-ons having the same momentum p . The R medium factoris examined as a function of density (top panel), isospinasymmetry (middle panel) and the momentum (bottompanel). It is interesting to note that not only in-mediumNN cross sections are reduced compared with their free-space values but also the nn and pp cross sections be-come different while their free-space cross sections are thesame. Moreover, the difference between the nn and pp F0 δ σ m ed i u m / σ f r ee ρ/ρ np nnpp npnnppnp n n p p δ =0.2, p=p F0 ρ=ρ , p=p F0 δ =0.2, ρ=ρ FIG. 6: (Color online) The reduction factor of the in-mediumnucleon-nucleon cross sections compared to their free-spacevalues as a function of density (top panel), isospin asymmetry(middle panel) and momentum (bottom panel). Taken fromRef. [107]. scattering cross sections grows in more asymmetric mat-ter. The larger in-medium cross sections for nn than forpp are completely due to the positive neutron-proton ef-fective mass splitting with the effective interaction used.This feature may serve as a probe of the neutron-protoneffective mass splitting in neutron-rich matter. We alsonote that in-medium NN cross sections are also indepen-dent of the parameter x as they are solely determined bythe momentum dependence of the nuclear potential usedin the model. IV. CONSTRAINING THE SYMMETRYENERGY AT SUB-NORMAL DENSITIES USINGISOSPIN DIFFUSION DATA The IBUU model outlined above have been used tostudy many observables in heavy-ion reactions inducedby both stable and radioactive beams. In this section, weillustrate one of its applications in extracting the densitydependence of nuclear symmetry energy from the isospindiffusion data in heavy-ion collisions.Isospin diffusion in heavy-ion collisions has been shownto depend sensitively on the density dependence ofnuclear symmetry energy [91, 92, 149]. Within amomentum-independent transport model, in which thenuclear potential depends only on local nuclear density,the isospin diffusion data from recent experiments at theNSCL/MSU (National Superconducting Cyclotron Lab-oratory at Michigan State University) was found to fa-vor a quadratic density dependence for the interactionpart of nuclear symmetry energy [108]. This conclu-sion has stimulated much interest because of its impli-cations to nuclear many-body theories and nuclear as-trophysics. However, the nuclear potential acting on anucleon is known to depend also on its momentum. Fornuclear isoscalar potential, its momentum dependence iswell-known and is important in extracting informationabout the equation of state of symmetric nuclear matter[3, 138, 150, 151, 152, 153, 154, 155, 156]. The momen-tum dependence of the isovector (symmetry) potential[9, 121, 126, 157] has also been shown to be importantfor understanding a number of isospin related phenomenain heavy-ion reactions [89, 90, 93]. It is therefore neces-sary to include momentum dependence in both isoscalarand isovector potentials for studying the effect of nuclearsymmetry energy on isospin diffusion.Isospin diffusion in heavy ion collisions can in principlebe studied by examining the average isospin asymmetryof the projectile-like residue in the final state. Since re-actions at intermediate energies are complicated by pree-quilibrium particle emission and production of neutron-rich fragments at mid-rapidity, differences of isospin dif-fusions in mixed and symmetric systems are usually usedto minimize these effects [108]. To study isospin diffusionin Sn + Sn reactions at E = 50 MeV/nucleon andan impact parameter of b = 6 fm, we thus also considerthe reaction systems Sn + S and Sn + Sn andat same energy and impact parameter as in Ref. [108].The degree of isospin diffusion in the reaction Sn + Sn is then measured by [158] R i = 2 X Sn+ Sn − X Sn+ Sn − X Sn+ Sn X Sn+ Sn − X Sn+ Sn (15)where X is the average isospin asymmetry h δ i of the Sn-like residue defined as the composition of nucle-ons with local densities higher than ρ / 20 and velocitieslarger than 1 / ρ / R i ranges between 0 . 05 and 1from complete mixing to full transparency. A. Effects of momentum-dependent interactions onisospin diffusion Effects of momentum-dependent interactions on dy-namics of heavy-ion collisions can be seen from the timeevolution of the density distribution [89]. In Fig. 7,we show the density contour ρ ( x, , z ) in the reactionplane at different times for the reaction Sn + Sn at E/A = 50 MeV and b = 6 fm calculated with x = − K and the same density dependence in the sym-metry energy. The experimental free space N - N crosssections are used in these calculations. It is seen thatboth interactions give similar time evolution of the col-lisions dynamics, namely, projectile-like and target-likeresidues can be clearly separated after about 100 fm/c.Detailed examinations indicate, however, that the reac-tion system expands more quickly and there are alsomore emitted nucleons in the case of the momentum-dependent MDI interaction than that of the momentum-independent SBKD interaction.In Fig. 8, we show the measured R i together with thepredictions from the IBUU04 for the time evolution ofboth R i and the average central densities calculated with x = − . ρ to 0 . ρ . However, the valueof R i still changes slightly with time until after about120 fm/c when projectile-like and target-like residues arewell separated as shown in Fig. 7. This is partly due tothe fact that the isovector potential remains appreciableat low density as shown in Fig. 9, where the symmetrypotential ( U n − U p ) / δ is shown as a function of nucleonmomentum (panel (a)) or density (panel (b)) for the MDIinteraction and as a function of density for the SBKD in-teraction (panel (c)). Also, evaluating isospin diffusion R i based on three reaction systems, which have differenttime evolutions for the projectile residue as a result ofdifferent total energies and numbers of nucleons, furthercontributes to the change of R i at low density. For thetwo interactions consider here, the main difference be-tween the values for R i appears in the expansion phasewhen densities in the participant region are well below ρ . The experimental data from MSU are seen to be re-produced nicely by the MDI interaction with x = − x = − R i as the strength of themomentum-independent symmetry potential is stronger(see Fig. 9), which has been shown to enhance the isospindiffusion [92, 108, 149].Effects of the symmetry energy on isospin diffusionwere also studied by varying the parameter x [109].Shown in Fig. 10 is the final saturated value for 1 − R i ,which measures the degree of isospin diffusion, as a func-tion of K asy for both MDI and SBKD interactions. It isobtained by averaging the value of 1 − R i after 120 fm/cwith error bars corresponding to its dispersion, whosemagnitude is similar to the error band shown in Ref.[108] for the theoretical results from the momentum-independent BUU model. For the SBKD interactionwithout momentum dependence, the isospin diffusion de-creases monotonically (i.e., increasing value for R i ) with0 -20-1001020-20 -10 0 10 MDI with x=-1- - /8- /20 t=0 fm/c -20 -10 0 10t=40 fm/c -20 -10 0 10t=70 fm/c -20 -10 0 10t=100 fm/c -20 -10 0 10 20 -20-1001020-20 -10 0 10 MDI(-1) - - /8- /20 t=0 fm/c -20 -10 0 10t=40 fm/c -20 -10 0 10t=70 fm/c -20 -10 0 10t=100 fm/c -20 -10 0 10 20 SBKD + U sym t=130 fm/c x ( f m ) z (fm) Sn + SnE/A=50 MeV and b=6 fm t=130 fm/c x ( f m ) FIG. 7: (Color online) Density contour ρ ( x, , z ) in the reaction plane at different times for the reaction Sn + Sn at E/A = 50 MeV and b = 6 fm by using momentum-dependent interaction MDI with x = − U MDI( − ( ρ, δ, τ ) (lower panels). Thick solidlines represent ρ /20 while dashed lines represent ρ /8. x=-1 MDI SBKD / (MDI) / (SBKD) t (fm/c) Sn + Sn MSU data R i FIG. 8: (Color online) Degree of isospin diffusion as a functionof time with the MDI and SBKD interactions. Correspondingtime evolutions of central density are also shown. Taken fromRef. [109]. increasing strength of K asy as the corresponding isovec-tor potential is mostly positive and decreases with in-creasing stiffness of E sym ( ρ ) in the whole range of consid-ered x parameter. The isospin diffusion is reduced whenthe momentum-dependent interaction MDI is used as themomentum dependence weakens the strength of symme-try potential except for x = − 2. As seen in Fig. 9, thesymmetry potential in the MDI interaction has the small-est strength for x = − k ≈ . − and ρ/ρ ≈ . 5, and increases again with furtherhardening of the symmetry energy, e.g., x = − 2, when itbecomes largely negative at all momenta and densities.The MDI interaction with x = − x=1 x=0 x=-1 x=-2 ( U n - U p ) / ( M e V ) k (fm -1 ) -1 / / FIG. 9: (Color online) Symmetry potential as a function ofnucleon momentum (a) or density (b) with the MDI interac-tion and SBKD interaction (c). Taken from Ref. [109]. degree of isospin diffusion among the interactions consid-ered here and reproduces the MSU data as already shownin Fig. 8.The symmetry energy in the MDI interaction with x = − E sym ( ρ ) ≈ . ρ/ρ ) . . Itleads to a value of K asy ≈ − 550 MeV for the isospindependent part of the isobaric incompressibility of asym-metric nuclear matter, which should be compared to thepublished constraint of − ± < K asy < ± -700 -600 -500 -400 -3000.40.60.81.0 x=-2( =1.60)MSU data x=-1( =1.05) x=0( =0.69) x=1 E/A=50 MeV and b=6 fm MDI interaction SBKD interaction Sn + Sn - R i K asy (MeV) FIG. 10: (Color online) Degree of isospin diffusion as a func-tion of K asy with the MDI and SBKD interactions. γ is theparameter for fitting the corresponding symmetry energy with E sym ( ρ ) = 31 . ρ/ρ ) γ . Taken from Ref. [109]. B. Effects of in-medium NN cross sections onisospin diffusion The isospin degree of freedom plays an important rolein heavy-ion collisions through both the nuclear EOS andthe nucleon-nucleon (NN) scatterings [1, 2]. In partic-ular, the transport of isospin asymmetry between twocolliding nuclei is expected to depend on not only thesymmetry potential through the density dependence ofthe symmetry energy E sym ( ρ ) but also the in-mediumNN cross sections. The two are related, respectively, tothe long-range and the short-range parts of the isospin-dependent in-medium nuclear effective interactions. Forinstance, the drifting contribution to the isospin trans-port in a nearly equilibrium system is proportional to theproduct of the mean relaxation time τ np and the isospinasymmetric force F np [91]. While F np is directly relatedto the gradient of the symmetry potential, τ np is inverselyproportional to the neutron-proton (np) scattering crosssection σ np [91]. Furthermore, the collisional contribu-tion to the isospin transport in non-equilibrium systemis generally expected to be proportional to the np scat-tering cross section. In the above study on isospin dif-fusion, the free-space NN cross sections were used. Inthis section, we review the effects of in-medium NN crosssections on the isospin diffusion in heavy ion collisions.Shown in Fig. 11 are the time evolution of R i re-calculated using the in-medium NN cross sections andthe MDI interaction with four x parameters. It is seenthat the net isospin transport and the influence of the x parameter show up mainly in the expansion phase ofthe reaction after about 40 fm/c and become relativelystable after about 80 fm/c. In the late stage of the reac-tion, the values of R i from x = − MDI + med x=-2 x=-1 x= 0 x= 1t (fm/c) Sn + SnMSU data R i FIG. 11: (Color online) Time evolution of the isospin diffusion R i using for MDI interactions with different x parameters andthe in-medium nucleon-nucleon cross sections. Taken fromRef. [107]. the experimental data from MSU, which is shown by theshaded band. -700 -600 -500 -400 -3000.20.40.60.81.0 x=-2( =1.60) MSU data x=-1( =1.05) x=0( =0.69) x=1 E/A=50 MeV and b=6 fm MDI + exp MDI + med Sn + Sn - R i K asy (MeV) FIG. 12: (Color online) Degree of isospin diffusion as a func-tion of K asy ( ρ ) with the free (filled squares) and in-medium(open squares) nucleon-nucleon cross sections. Taken fromRef. [107]. For a more meaningful comparison with the experi-mental data, the time average of R i between t = 120fm/c and 150 fm/c has been used as in Fig. 8. Shownin Fig. 12 is a comparison of the averaged strength ofisospin transport 1 − R i obtained with the free and in-medium NN cross sections, respectively, as a function ofthe asymmetric part of the isobaric incompressibility ofnuclear matter at ρ , i.e., K asy ( ρ ). The error bars weredrawn to indicate fluctuations and were obtained fromthe dispersion in the time evolution of R i [109]. First, itis interesting to note that with the in-medium NN crosssections the strength of isospin transport 1 − R i decreasesmonotonically with decreasing value of x . With the free-2space NN cross sections, there appears, however, to be aminimum at around x = − 1. Moreover, this minimum isthe point closest to the experimental data. This allowedus to extract the value of K asy ( ρ ) = − ± 100 MeV.With the in-medium NN cross sections we can now fur-ther narrow down the K asy ( ρ ) to be about − ± Sn to Sn isotopes by Fujiwara et al at Osaka [159].Also shown in the figure are the γ values used in fittingthe symmetry energy with E sym ( ρ ) = 31 . ρ/ρ ) γ . Theresults with the in-medium NN cross sections constrainthe γ parameter to be between 0 . 69 and 1 . 05. The lowervalue is close to what is extracted from studying giantresonances [160, 161]. The value of γ = 1 . 05 extractedearlier using the free-space NN cross sections sets an up-per limit.The difference in 1 − R i obtained with the free-spaceand the in-medium NN cross sections is small for x = 1and x = 0, but becomes large for x = − x = − K asy ( ρ ) or x parameter can be under-stood from consideration of the contributions from thesymmetry potential and the np scatterings. As we havementioned above, both contributions to the isospin trans-port depend on the np scattering cross section σ np . Whilethe collisional contribution is proportional to the np scat-tering cross section σ np , the mean-field contribution isproportional to the product of the isospin asymmetricforce F np and the inverse of σ np . The overall effect of thein-medium NN cross sections on isospin transport is a re-sult of a complicated combination of both the mean fieldand the NN scatterings. Generally speaking, the symme-try potential effects on the isospin transport will becomeweaker when the NN cross sections are larger while thesymmetry potential effects will show up more clearly ifsmaller NN cross sections are used. This feature can beseen from Fig. 9. For x = 1 and x = 0, the symmetry po-tential and its gradient with respect to density, as shownin Fig. 9, are large at low densities where the majority ofnet isospin transport occurs. The F np factor makes thecontribution due to the mean field to dominate over thatdue to the collisions. Therefore, the reduced in-medium σ np leads to about the same or a slightly higher isospintransport. As the x parameter decreases to x = − x = − 2, however, the symmetry potential decreases andits density gradient can be even negative at low densi-ties. In these cases, either the collisional contributiondominates or the mean-field contribution becomes neg-ative. Therefore, the reduced in-medium np scatteringcross section σ np leads to a weaker isospin transport com-pared with the case with the free-space NN cross sections.Based on an isospin- and momentum-dependentIBUU04 transport model with free-space experimentalNN cross sections, comparing the theoretical results withthe experimental data has allowed us to extract a nuclearsymmetry energy of E sym ( ρ ) ≈ . ρ/ρ ) . . Includingalso medium-dependent NN cross sections, which are im- x= -2 MSU data x= -1 x= 0x= 1 E/A=50 MeV and b=6 fm IBUU04 with MDI + med Sn + Sn - R i L (MeV) L=88+25 MeV- FIG. 13: (Color online) Degree of the isospin diffusion 1 − R i as a function of L using the MDI interaction with x = − − 1, 0, and 1. The shaded band indicates the data fromNSCL/MSU [108]. The solid square with error bar represents L = 88 ± 25 MeV. Taken from Ref. [112]. portant for isospin-dependent observables [107, 162], theisospin diffusion data leads to an even softer nuclear sym-metry energy of E sym ( ρ ) ≈ . ρ/ρ ) γ with γ ≈ . L of the nuclear symmetryenergy gives an important constraint on the density de-pendence of the nuclear symmetry energy, in particular,it has been shown that the slope parameter L is relatedto the neutron skin thickness of heavy nuclei, it is in-teresting to see how the isospin diffusion data constrainthe value of L . In Fig. 13, we show the results from theIBUU04 transport model with in-medium NN cross sec-tions, that are consistent with the mean-field potentialobtained with the MDI interactions used in the model,for the degree of the isospin diffusion 1 − R i as a functionof L . The shaded band in Fig. 13 indicates the data fromNSCL/MSU [108]. It is seen that the strength of isospindiffusion 1 − R i decreases monotonically with decreasingvalue of x or increasing value of L . This is expected asthe parameter L reflects the difference in the pressureson neutrons and protons. From comparison of the theo-retical results with the data, we can clearly exclude theMDI interaction with x = 1 and x = − − R i compared tothat of data. The range of x or L values that give valuesof 1 − R i falling within the band of experimental valuescould in principle be determined in our model by detailedcalculations. Instead, we determine this schematically byusing the results from the four x values. For the centroidvalue of L , it is obtained from the interception of the lineconnecting the theoretical results at x = − − R i data in Fig. 13, i.e., L = 88MeV. The upper limit of L = 113 MeV is estimated fromthe interception of the line connecting the upper errorbars of the theoretical results at x = − − − R i . Similarly, thelower limit of L = 65 MeV is estimated from the inter-ception of the line connecting the lower error bars of thetheoretical results at x = 0 and − − R i . This leads to an extractedvalue of L = 88 ± 25 MeV as shown by the solid squarewith error bar in Fig. 13. V. CONSTRAINING THE SKYRMEEFFECTIVE INTERACTIONS AND THENEUTRON SKIN THICKNESS OF NUCLEIUSING ISOSPIN DIFFUSION DATA FROMHEAVY ION COLLISIONS Information on the density dependence of the nuclearsymmetry energy can in principle also be obtained fromthe thickness of the neutron skin in heavy nuclei as thelatter is strongly correlated with the slope parameter L of the nuclear symmetry energy at saturation density[10, 163, 164, 165, 166, 167]. Because of the large un-certainties in measured neutron skin thickness of heavynuclei, this has not been possible. Instead, studies havebeen carried out to use the extracted nuclear symmetryenergy from the isospin diffusion data to constrain theneutron skin thickness of heavy nuclei [107, 110]. Us-ing the Hartree-Fock approximation with parameters fit-ted to the phenomenological EOS that was used in theIBUU04 transport model to describe the isospin diffusiondata from the NSCL/MSU, it was found that a neutronskin thickness of less than 0 . 15 fm [107, 110] for Pbwas incompatible with the isospin diffusion data.In the following, we study more systematically the cor-relation between the density dependence of the nuclearsymmetry energy and the thickness of the neutron skin ina number of nuclei within the framework of the SkyrmeHartree-Fock model. Using the extracted values of L from the isospin diffusion data in heavy-ion collisions, weobtain stringent constraints on the neutron skin thick-ness of the nuclei Pb, Sn, and Sn. The extractedvalue of L also limits the allowed parameter sets for theSkyrme interaction. A. Constraining the Skyrme effective interactionsusing isospin diffusion data In the standard Skyrme Hartree-Fock model, the in-teraction is taken to have a zero-range, density- andmomentum-dependent form [61, 168, 169, 170, 171], i.e., V ( R , r ) = t (1 + x P σ ) δ ( r )+ 16 t (1 + x P σ ) ρ σ ( R ) δ ( r )+ 12 t (1 + x P σ )( K ′ δ ( r ) + δ ( r ) K )+ t (1 + x P σ ) K ′ · δ ( r ) K + iW K ′ · δ ( r )[( σ + σ ) × K ] , (16) E sy m () ( M e V ) (fm -3 ) Skyrme-Hartree-Fock with 21 parameter sets MDI interaction with x= 0 MDI interaction with x= -1 FIG. 14: (Color online) Density dependence of the nuclearsymmetry energy E sym ( ρ ) for 21 sets of Skyrme interactionparameters. The results from the MDI interaction with x = − with r = r − r and R = ( r + r ) / 2. In the above,the relative momentum operators K = ( ∇ − ∇ ) / i and K ′ = − ( ∇ −∇ ) / i act on the wave function on the rightand left, respectively. The quantities P σ and σ i denote,respectively, the spin exchange operator and Pauli spinmatrices. The σ , t − t , x − x , and W are Skyrmeinteraction parameters that are chosen to fit the bind-ing energies and charge radii of a large number of nucleiin the periodic table. For infinite nuclear matter, thesymmetry energy from the Skyrme interaction can beexpressed as [170, 171] E sym ( ρ ) = 13 ¯ h m (cid:18) π (cid:19) / ρ / − t (2 x + 1) ρ − t (2 x + 1) ρ σ +1 + 124 (cid:18) π (cid:19) / [ − t x + (4 + 5 x ) t ] ρ / . (17)The coefficient of the δ term in Eq. (1) can also be ob-tained analytically and has been shown to be very smallover a large range of nuclear density ( ≤ − ) andisospin asymmetry. The parabolic law of Eq. (1) withoutthe δ and higher-order terms in δ is thus justified [170].Fig. 14 displays the density dependence of E sym ( ρ )for 21 sets of Skyrme interaction parameters, i.e., SKM , SKM ∗ , RAT P , SI , SII , SIII , SIV , SV , SV I , E , E σ , G σ , R σ , Z , Z σ , Z ∗ σ , T , T SkX , SkXce , and SkXm .The values of the parameters in these Skyrme interac-tions can be found in Refs. [61, 168, 169]. For compari-son, we also show in Fig. 14 results from the phenomeno-logical MDI interactions with x = − E sym ( ρ ) are all in the range of 26-35 MeV, the val-ues of L and K sym are in the range of − − L = 88 ± 25 MeV gives a ratherstringent constraint on the density dependence of the nu-clear symmetry energy and thus puts strong constraintson the nuclear effective interactions as well. For theSkyrme effective interactions shown in Fig. 14, for in-stance, all of those lie beyond x = 0 and x = − L . Actually, we note that only 4 sets ofSkyrme interactions, i.e., SIV, SV, G σ , and R σ , in the 21sets of Skyrme interactions considered here have nuclearsymmetry energies that are consistent with the extracted L value. B. Constraining the neutron skin thickness ofnuclei using isospin diffusion data The neutron skin thickness S of a nucleus is defined asthe difference between the root-mean-square radii p h r n i of neutrons and p h r p i of protons, i.e., S = p h r n i − q(cid:10) r p (cid:11) . (18)It has been known that S is sensitive to the density de-pendence of the nuclear symmetry energy, particularlythe slope parameter L at the normal nuclear matter den-sity [10, 163, 164, 165, 166, 167]. Using above 21 setsof Skyrme interaction parameters, we have evaluated theneutron skin thickness of several nuclei. In Figs. 15(a),(b) and (c), we show, respectively, the correlations be-tween the neutron skin thickness of Pb with L , K sym ,and E sym ( ρ ). It is seen from Fig. 15(a) that there existsan approximate linear correlation between S and L . Thecorrelations of S with K sym and E sym ( ρ ) are less strongand even exhibit some irregular behavior. The solid linein Fig. 15(a) is a linear fit to the correlation between S and L and is given by the following expression: S ( Pb) = (0 . ± . . ± . × − ) × L, (19)or L = ( − . ± . . ± . × S ( Pb) , (20)where the units of L and S are MeV and fm, respectively.Therefore, if the value for either S ( Pb) or L is known,the value for the other can be determined. (a) Pb L (MeV) S ( f m ) -600 -400 -200 0 (b) K sym (MeV) 26 28 30 32 34 36 (c) E sym ( ) (MeV) FIG. 15: (Color online) Neutron skin thickness S of Pbas a function of (a) L , (b) K sym , and (c) E sym ( ρ ) for 21sets of Skyrme interaction parameters. The line in panel (a)represents a linear fit. Taken from Ref. [112]. It is of interest to see if there are also correlations be-tween the neutron skin thickness of other neutron-richnuclei and the nuclear symmetry energy. Fig. 16 showsthe same correlations as in Fig. 15 but for the neutron-rich nuclei Sn, Sn, and Ca. For the heavy Snand Sn, we obtain a similar conclusion as for Pb,namely, S exhibits an approximate linear correlation with L but weaker correlations with K sym and E sym ( ρ ). Forthe lighter Ca, on the other hand, all the correlationsbecome weaker than those of heavier nuclei. Therefore,the neutron skin thickness of heavy nuclei is better cor-related with the density dependence of the nuclear sym-metry energy. As in Eqs. (19) and (20), a linear fit tothe correlation between S and L can also be obtained for Sn and Sn, and the corresponding expressions are S ( Sn) = (0 . ± . . ± . × − ) × L, (21) L = ( − . ± . . ± . × S ( Sn) , (22)and S ( Sn) = (0 . ± . . ± . × − ) × L. (23) L = ( − . ± . . ± . × S ( Sn) , (24)Similar linear relations between S and L are also ex-pected for other heavy nuclei. This is not surprising asdetailed discussions in Refs. [10, 163, 164, 165, 166, 167]have shown that the thickness of the neutron skin inheavy nuclei is determined by the pressure difference be-tween neutrons and protons, which is proportional to theparameter L .To give a quantitative estimate of above discussed cor-relations, we define the following linear correlation coef-ficient C l : C l = p − q/t, (25)5 (a) Sn Sn Ca L (MeV) S ( f m ) -600 -400 -200 0 (b) K sym (MeV) 26 28 30 32 34 36 (c) E sym ( ) (MeV) FIG. 16: (Color online) Same as Fig. 2 but for nuclei Sn(Solid squares), Sn (Open squares) and Ca (Triangles).Taken from Ref. [112].TABLE II: Linear correlation coefficients C l of S with L , K sym and E sym ( ρ ) for Pb, Sn, Sn, and Ca from 21 setsof Skyrme interaction parameters. Taken from Ref. [112]. C l (%) Pb Sn Sn Ca S - L . 25 98 . 76 98 . 75 93 . S - K sym . 26 92 . 06 92 . 22 86 . S - E sym . 89 85 . 74 85 . 77 81 . where q = n X i =1 [ y i − ( A + Bx i )] , (26) t = n X i =1 ( y i − y ) , y = n X i =1 y i /n. (27)In the above, A and B are the linear regression coeffi-cients, ( x i , y i ) are the sample points, and n is the num-ber of sample points. The linear correlation coefficient C l measures the degree of linear correlation, and C l = 1corresponds to an ideal linear correlation. Table II givesthe linear correlation coefficient C l for the correlation of S with L , K sym and E sym ( ρ ) for Pb, Sn, Sn,and Ca shown in Figs. 15 and 16 for different Skyrmeinteractions. It is seen that these correlations becomeweaker with decreasing nucleus mass, and a strong lin-ear correlation only exists between the S and L for theheavier nuclei Pb, Sn, and Sn. Therefore, theneutron skin thickness of these nuclei can be extractedonce the slope parameter L of the nuclear symmetry en-ergy at saturation density is known.The extracted L value from isospin diffusion data al-lows us to determine from Eqs. (19), (21), and (23),respectively, a neutron skin thickness of 0 . ± . 04 fmfor Pb, 0 . ± . 04 fm for Sn, and 0 . ± . 04 fmfor Sn. Experimentally, great efforts were devoted tomeasure the thickness of the neutron skin in heavy nu-clei [172, 173], and a recent review can be found in Ref.[174]. The data for the neutron skin thickness of Pbindicate a large uncertainty, i.e., 0 . . 28 fm. Our re-sults for the neutron skin thickness of Pb are thus consistent with present data but give a much strongerconstraint. A large uncertainty is also found experimen-tally in the neutron skin thickness of Sn, i.e., its valuevaries from 0 . . Pb at the JeffersonLaboratory is expected to give another independent andmore accurate measurement of its neutron skin thickness(within 0 . 05 fm), thus providing improved constraints onthe density dependence of the nuclear symmetry energy[175, 176].Recently, an accurately calibrated relativisticparametrization based on the relativistic mean-fieldtheory has been introduced to study the neutron skinthickness of finite nuclei [113]. This parametrization candescribe simultaneously the ground state properties offinite nuclei and their monopole and dipole resonances.Using this parametrization, the authors predicted aneutron skin thickness of 0 . 21 fm in Pb, 0 . 27 fm in Sn, and 0 . 19 fm in Sn [113, 160]. These predictionsare in surprisingly good agreement with our resultsconstrained by the isospin diffusion data in heavy-ioncollisions.In addition, the neutron skin thickness of the nucleus Zr has recently been determined to be 0 . ± . 04 fmfrom the model-independent spin-dipole sum rule valuemeasured from the charge-exchange spin-dipole excita-tions [177]. This value is reproduced by the symmetryenergy with L = 88 ± 25 MeV extracted from the isospindiffusion data in heavy-ion collisions, which predicts aneutron skin thickness of 0 . ± . 04 fm for Zr. VI. PROBING THE HIGH DENSITYBEHAVIOR OF THE NUCLEAR SYMMETRYENERGY IN HEAVY-ION REACTIONSINDUCED BY HIGH ENERGY RADIOACTIVEBEAMS Although significant progress has been made in thedetermination of the density dependence of the nuclearsymmetry energy at sub-normal densities, the high den-sity behavior of the the nuclear symmetry energy is stillpoorly known. Fortunately, heavy-ion reactions, espe-cially those induced by high energy radioactive beams tobe available at high energy radioactive beam facilities,provide a unique opportunity to pin down the high den-sity behavior of the symmetry energy. In this section,we illustrate via transport model simulations several ex-perimental observables which are sensitive to the highdensity behavior of the symmetry energy. A. Isospin asymmetry of dense matter formed inhigh energy heavy-ion reactions To see the maximum baryon density and isospin asym-metry that can be achieved in central heavy ion colli-6 FIG. 17: (Color online) Central baryon density (upper win-dow) and isospin asymmetry (lower window) of high densityregion for the reaction of Sn + Sn at a beam energy of400 MeV/nucleon and an impact parameter of 1 fm. Takenfrom Ref. [94]. sions induced by high energy radioactive beams in fu-ture radioactive beam facilities, we show in Fig. 17 thecentral baryon density (upper window) and the average( n/p ) ρ ≥ ρ ratio (lower window) of all regions with baryondensities higher than ρ in the reaction of Sn+ Snat a beam energy of 400 MeV/nucleon and an impactparameter of 1 fm. It is seen that the maximum baryondensity is about 2 times normal nuclear matter density.Moreover, the compression is rather insensitive to thesymmetry energy because the latter is relatively smallcompared to the EOS of symmetric matter around thisdensity. The high density phase lasts for about 15 fm/cfrom 5 to 20 fm/c for this reaction. The isospin asym-metry of the high density region is, however, sensitive tothe symmetry energy. The soft (e.g., x = 1) symmetryenergy leads to a significantly higher value of ( n/p ) ρ ≥ ρ than the stiff one (e.g., x = − δ in the high density region for asofter symmetry energy functional E sym ( ρ ) as a resultof the E sym ( ρ ) δ term in the EOS of asymmetric nuclearmatter. Since the symmetry energy changes from be-ing soft to stiff when the parameter x varies from 1 to − 2, the value of ( n/p ) ρ ≥ ρ becomes lower in the supra-normal density region as the parameter x changes from1 to − 2. Because of neutron-skins of the colliding nu-clei, especially that of the projectile Sn, the initialvalue of the quantity ( n/p ) ρ ≥ ρ , which is about 1.4, isless than the average n/p ratio of 1.56 of the reactionsystem. Also, in neutron-rich nuclei, the n/p ratio onthe low-density surface is much higher than that in their interior. The dense matter region in heavy ion collisionscan thus become either neutron-richer or neutron-poorerwith respect to the initial state depending on the sym-metry energy functional E sym ( ρ ) used. B. Isospin fractionation and n-p differential flow The degree of isospin equilibration or translucencyin heavy ion collisions can be measured by the rapid-ity distribution of nucleon isospin asymmetry δ free ≡ ( N n − N p ) / ( N n + N p ), where N n and N p are multi-plicities of free neutrons and protons, respectively [90].Although it might be difficult to measure directly δ free because it requires the detection of neutrons, similar in-formation can be extracted from ratios of light clusters,such as, t/ He, as demonstrated recently within a coa-lescence model [86, 89]. Shown in Fig. 18 are rapiditydistributions of δ free with (upper window) and without(lower window) the Coulomb potential. It is interest-ing to see that the δ free at midrapidity is particularlysensitive to the symmetry energy. As the parameter x increases from − δ free at midrapidity decreasesby about a factor of 2. Moreover, the forward-backwardasymmetric rapidity distributions of δ free with all four x parameters indicates the apparent nuclear translucencyduring the reaction [95]. FIG. 18: (Color online) Isospin asymmetry of free nucleonswith and without the Coulomb force for different symmetryenergies. Taken from Ref. [95]. Another observable that is sensitive to the high densitybehavior of the symmetry energy is the neutron-proton7differential flow first introduced in Ref. [82] F xn − p ( y ) ≡ N ( y ) X i =1 ( p xi w i ) /N ( y ) , (28)where w i = 1( − 1) for neutrons (protons) and N ( y ) isthe total number of free nucleons at rapidity y . Sincethe differential flow depends on the symmetry potentialthrough the latter’s effects on the isospin fractionationand the collective flow, it has the advantage of maximiz-ing the effects of the symmetry potential while minimiz-ing those of the isoscalar potential. Shown in Fig. 19 isthe n-p differential flow for the reaction of Sn+ Snat a beam energy of 400 MeV/nucleon and an impactparameter of 5 fm. Effects of the symmetry energy areclearly revealed by changing the parameter x . FIG. 19: (Color online) Neutron-proton differential flow fordifferent symmetry energies. Taken from Ref. [103]. C. Pion yields and π − /π + ratio At beam energy of about 400 MeV/nucleon, pion pro-duction becomes non-negligible and may also carry inter-esting information about the EOS of dense neutron-richmatter [94, 178]. In Fig. 20, we show the π − and π + yields as functions of the x parameter in the MDI inter-action. It is seen that when the x parameter is changedfrom -2 to 1, the π − multiplicity increases by about 20%,although the π + multiplicity remains about the same.The π − multiplicity is thus more sensitively to the sym-metry energy than that of π + . Also, the multiplicity of π − is about 2 to 3 times that of π + . This is because π − mesons are mostly produced from neutron-neutron colli-sions, which are more frequent in collisions of neutron-rich nuclei. Since the high density region is more neutron FIG. 20: (Color online) The π − and π + yields as functions ofthe x parameter. Taken from Ref. [94]. rich for the softer symmetry energy as a result of isospinfractionation [94], the π − multiplicity is thus more sensi-tive to the isospin asymmetry of the reaction system andthe symmetry energy. However, it is well known that thepion yield is also sensitive to the symmetric part of thenuclear EOS, and it is thus hard to extract reliable in-formation about the symmetry energy from the π − yieldalone. The π − /π + ratio is, on the other hand, a bet-ter probe as this ratio is sensitive only to the differencein the chemical potentials for neutrons and protons [179].As well demonstrated in Fig. 21, the π − /π + ratio is quitesensitive to the symmetry energy, especially at low trans-verse momenta, and can be used to probe the high densitybehavior of nuclear symmetry energy E sym ( ρ ). FIG. 21: (Color online) The π − /π + ratio as a function oftransverse momentum. Taken from Ref. [94]. D. Double n/p and π − /π + ratio Because of the facts that the symmetry potentials haveopposite signs for neutrons and protons and are alsogenerally smaller compared to the isoscalar potential atthe same density, most observables proposed so far forstudying the density dependence of the nuclear symme-try energy employ differences or ratios of isospin multi-plets of baryons, mirror nuclei and mesons, such as, theneutron/proton ratio of emitted nucleons [68], neutron-proton differential flow [82], neutron-proton correlationfunction [85], t / He [86, 96], π − /π + [84, 94, 100, 101],Σ − / Σ + [97] and K /K + ratios [102], etc. Among theseobservables, the ratio of emitted neutrons to protonshas probably the highest sensitivity to the symmetry en-ergy as the symmetry potential acts directly on nucleonsand emitted nucleons are also rather abundant in typi-cal heavy-ion reactions. However, it is very challengingto measure some of these observables, especially thoseinvolving neutrons. The measurement of neutrons, par-ticularly the low energy ones, always suffers from lowdetection efficiencies even for the most advanced neu-tron detectors. Therefore, observables involving neutronsnormally have large systematic errors. Moreover, foressentially all of these observables, the Coulomb forceon charged particles plays an important role and some-times competes strongly with the symmetry potential.One has to disentangle carefully effects of the symme-try potential from those due to the Coulomb potential.It is thus very desirable to find experimental observableswhich can reduce the influence of both the Coulomb forceand the systematic errors associated with neutrons. Apossible candidate for such an observable is the dou-ble ratios of emitted neutrons and protons taken fromtwo reaction systems using four isotopes of the same el-ement, namely, the neutron/proton ratio in the neutron-rich system over that in the more symmetric system, asrecently proposed by Lynch et al. [180]. They have actu-ally demonstrated the feasibility of measuring the doubleneutron/proton ratios in central reactions of Sn+ Snand Sn+ Sn at a beam energy of 50 MeV/nucleonat the National Superconducting Cyclotron Laboratory[180].Both the double neutron/proton ratio and the dou-ble π − /π + ratio in Sn+ Sn and Sn+ Sn reac-tions at 400 MeV/nucleon have been studied in the IBUUmodel in order to demonstrate the effect of symmetry en-ergy at high density. It was found that these ratios haveabout the same sensitivity to the density dependence ofsymmetry energy as the corresponding single ratio in therespective neutron-rich system involved. Given the ad-vantages of measuring the double ratios over the singleones, the study of double ratios will be more useful forfurther constraining the symmetry energy of neutron-richmatter. Furthermore, the systematic errors associatedwith transport model calculations are mostly related tothe uncertainties in the in-medium NN cross sections,techniques of treating collisions, sizes of the lattices in calculating the phase space distributions, techniques inhandling the Pauli blocking, etc. Since the double ratio isa relative observable from two similar reaction systems,these systematic errors are expected to be reduced. med R n / p [ ( S n + S n ) / ( S n + S n ) ] (E kin ) c.m. (MeV)(b) b=5 fm FIG. 22: (Color online) The double neutron/proton ratioof free nucleons in the reaction of Sn + Sn and Sn+ Sn at 400 MeV/nucleon and an impact parameter of 1fm (left window) and 5 fm (right window), respectively. Takenfrom Ref. [183]. In Fig. 22, we show the double neutron/proton ratiosfrom the reactions of Sn+ Sn and Sn+ Sn ata beam energy of 400 MeV/nucleon and an impact pa-rameter of 1 fm (left window) and 5 fm (right window)predicted by the IBUU model using the MDI interactionwith x = 0 and x = − 1, which are consistent with thesymmetry energy used for sub-saturation densities. Atboth impact parameters, effects of the symmetry energyare about 5% − 10% changing from the case with x = 0 to x = − 1. One notices here that the low energy nucleonshave the largest sensitivity to the variation of the sym-metry energy for such high energy heavy-ion collisions.In fact, the neutron/proton ratio of midrapidity nucleonswhich have gone through the high density phase of thereaction are known to be most sensitive to the symmetryenergy [95]. Compared to the results at the beam energyof 50 MeV/nucleon [95], it is interesting to see a clearturnover in the dependence of the double neutron/protonratio on the x parameter, namely the double ratio is lowerat 50 MeV/nucleon but higher at 400 MeV/nucleon with x = − x = 0. The maximum densityreached at the beam energy of 50 and 400 MeV/nucleonis about 1 . ρ and 2 ρ [95], respectively. The turnoverclearly indicates that the double neutron/proton ratioreflects closely the density dependence of the symmetryenergy as shown in Fig. 2. This observation also indicatesthat systematic studies of the double neutron/proton ra-tio over a broad beam energy range will be important formapping out the density dependence of the symmetryenergy.As shown before from both the total yields and the mo-9 15 30 45 60 75 9012345 Sn+ Sn, x= 0 Sn+ Sn, x= -1 Sn+ Sn, x= 0 Sn+ Sn, x= -1 Sn+ Sn and x= 0 without Coulomb E/A=400 MeVb=1 fm (E kin ) c.m. (MeV) FIG. 23: (Color online) Kinetic energy distribution of thesingle π − /π + ratio for Sn+ Sn and Sn+ Sn ata beam energy of 400 MeV/nucleon and an impact param-eter of b = 1 fm with the stiff ( x = − 1) and soft ( x = 0)symmetry energies. The dash-dotted line is the single π − /π + ratio obtained by turning off the Coulomb potentials in the Sn+ Sn reaction. Taken from Ref. [184]. mentum spectra, the π − /π + ratio is a promising probeof the symmetry energy at high densities [84, 94, 97, 100,101]. In Fig. 23, we show again the kinetic energy dis-tribution of the single π − /π + ratio for the reactions of Sn+ Sn and Sn+ Sn at a beam energy of 400MeV/nucleon and an impact parameter of b = 1 fm withthe stiff ( x = − 1) or soft ( x = 0) symmetry energy, ob-tained with 12000 IBUU events for each reaction. It isseen that the overall magnitude of π − /π + ratio is largerfor the neutron-rich system Sn+ Sn than for theneutron-deficient system Sn+ Sn as expected. Also,the soft symmetry energy ( x = 0) leads to a larger single π − /π + ratio than the stiff one ( x = − π − /π + ratio is more sensitive tothe symmetry energy in the reaction Sn+ Sn than inthe reaction Sn+ Sn as a result of the larger isospinasymmetry in the more neutron-rich system.Fig. 23 shows that the single π − /π + ratio exhibitsa peak at a pion kinetic energy of about 45 MeV inall cases considered here. The origin of this peak canbe understood from the single π − /π + ratios in bothreactions from turning off the Coulomb potentials forall charged particles. As an example, shown in Fig.23 with the dash-dotted line is the single π − /π + ra-tio obtained by turning off the Coulomb potentials inthe Sn+ Sn reaction. It is seen that the single π − /π + ratio now becomes approximately a constant of about 2 . 4. This value agrees with the predicted valueof (5 N + N Z ) / (5 Z + N Z ) ≈ ( N/Z ) ≈ . 43, where N and Z are the total neutron and proton numbers inthe participant region, from the ∆ resonance model [181]for central Sn+ Sn collisions. This is not surpris-ing as at 400 MeV/nucleon, pions are almost exclusivelyproduced via the ∆ resonances [182]. Comparison of cal-culated results with and without the Coulomb potentialsthus indicates clearly that the peak observed in the sin-gle π − /π + ratio is due to the Coulomb effects. Althoughthe Coulomb potential distorts the spectra of pions, theeffect of symmetry potential is still seen in the result-ing π − /π + , particularly near its peak value where pionshave relatively low kinetic energies. These pions are pro-duced in the high density nucleonic matter (about 2 ρ )through the ∆ resonances and experience many rescat-terings with nucleons at both high and low densities aswell as the Coulomb potential from protons at differentdensities. Since the soft ( x = 0) and stiff ( x = − 1) sym-metry energies have slight difference at low densities andthe large difference appears at high densities (about 2 ρ )as shown in Fig. 2, the observed symmetry energy effectson the energy dependence of the π − /π + ratio thus mainlyreflect (though not completely) information on the highdensity behavior of the symmetry energy [101]. 15 30 45 60 751.01.21.41.61.8 x= 0 x= -1E/A=400 MeV, b=1 fm R [ ( S n + S n ) / ( S n + S n ) ] (E kin ) c.m. (MeV) FIG. 24: (Color online) Kinetic energy dependence of thedouble π − /π + ratio of Sn+ Sn over Sn+ Sn at abeam energy of 400 MeV/nucleon and an impact parameter b = 1 fm with the stiff ( x = − 1) and soft ( x = 0) symmetryenergies. Taken from Ref. [184]. In order to reduce the systematic errors related to thesymmetry energy effect on the π − /π + ratio, it is moreuseful to study the double π − /π + ratio in the reactionsof Sn+ Sn and Sn+ Sn as for the double pro-ton/neutron ratio. Fig. 24 shows the double π − /π + ratiofor these two reactions. It is seen that the kinetic energydependence of the double π − /π + ratio is rather differ-0ent for the stiff ( x = − 1) and soft ( x = 0) symmetryenergies. While the double π − /π + ratio is quite flat for x = 0, it displays a concave structure for x = − π − /π + ratios inthe two reactions shown in Fig. 23. It is reassuring to seethat around the Coulomb peak the double π − /π + ratiois still sensitive to the symmetry energy. Compared withthe single π − /π + ratio, the kinetic energy dependence ofthe double π − /π + ratio becomes, however, weaker. Thisis because effects of the Coulomb potentials are reducedin the double π − /π + ratio. We note that the double π − /π + ratio displays an opposite symmetry energy de-pendence compared with the double n/p ratio for freenucleons shown in Fig. 22. This is understandable sincethe soft symmetry energy leads to a more neutron-richdense matter in heavy-ion collisions induced by neutron-rich nuclei and thus a smaller n/p ratio for free nucleonsdue to the charge conservation. On the other hand, more π − ’s would be produced due to more neutron-neutron in-elastic scatterings in the more neutron-rich matter. E. Double neutron-proton differential transverseflow The neutron-proton differential transverse flow definedin Eq. (28) can be further expressed as [82, 84, 94] F xn − p ( y ) ≡ N ( y ) N ( y ) X i =1 p xi ( y ) w i = N n ( y ) N ( y ) h p xn ( y ) i − N p ( y ) N ( y ) h p xp ( y ) i (29)where N ( y ), N n ( y ) and N p ( y ) are the numbers of freenucleons, neutrons, and protons, respectively, at rapidity y ; p xi ( y ) is the transverse momentum of a free nucleonat rapidity y ; w i = 1 ( − 1) for neutrons (protons); and h p xn ( y ) i and h p xp ( y ) i are, respectively, the average trans-verse momenta of neutrons and protons at rapidity y . Eq.(29) shows that the neutron-proton differential transverseflow depends not only on the proton and neutron trans-verse momenta but also on their relative multiplicities,i.e., the isospin fractionation. This can be more clearlyseen from the following two special cases. If neutronsand protons have the same average transverse momen-tum in the reaction plane but different multiplicities ineach rapidity bin, i.e., h p xn ( y ) i = h p xp ( y ) i = h p x ( y ) i , and N n ( y ) = N p ( y ), then Eq. (29) is reduced to F xn − p ( y ) = N n ( y ) − N p ( y ) N ( y ) h p x ( y ) i = δ ( y ) · h p x ( y ) i , (30)reflecting the effects of isospin fractionation. On theother hand, if neutrons and protons have the same mul-tiplicity but different average transverse momenta, i.e., N n ( y ) = N p ( y ) but h p xn ( y ) i 6 = h p xp ( y ) i , then Eq. (29) is reduced to F xn − p ( y ) = 12 ( h p xn ( y ) i − h p xp ( y ) i ) (31)and reflects directly the difference between the neutronand proton transverse flows. Since a stiffer symmetrypotential is expected to lead to a higher isospin fraction-ation and also a larger transverse flow for neutrons thanfor protons in heavy-ion collisions at higher energies, theneutron-proton differential flow is thus a measure of thesetwo combined effects of the symmetry potentials on neu-trons and protons [183, 184]. -0.6 -0.3 0.0 0.3 0.6-80816 -0.6 -0.3 0.0 0.3 0.6 (c) F x n - p ( M e V / c ) (y/y beam ) c.m. -15015 (b) F x n - F x p ( M e V / c ) (a) x= 0 x= -1 Sn+ Sn, E/A=400 MeV, b=5 fm f r ee (f) (y/y beam ) c.m. (e) (d) E/A=800 MeV FIG. 25: (Color online) Rapidity distributions of the isospinasymmetry of free nucleons (upper panels), the difference ofthe average nucleon transverse flows (middle panels) and theneutron-proton differential transverse flow (lower panels) from Sn+ Sn reaction at the incident beam energies of 400,800 MeV/nucleon and b = 5 fm with two symmetry energiesof x = 0 and x = − 1. Taken from Ref. [188]. Shown in Fig. 25 are the rapidity distributions of theisospin asymmetry of free nucleons (upper panels), thedifference of the average nucleon transverse flows (mid-dle panels) and the neutron-proton differential transverseflow (lower panels) from the Sn+ Sn reaction at in-cident beam energies of 400 and 800 MeV/nucleon andan impact parameter of b = 5 fm with the two sym-metry energies of x = 0 and x = − 1. It is seen thatthe stiffer symmetry energy ( x = − 1) leads to a largerisospin asymmetry of free nucleons (stronger isospin frac-tionation) (upper panels) than the softer symmetry en-ergy ( x = 0). As a result, the neutron-proton differentialtransverse flow from the stiff symmetry energy ( x = − x = 0) compared to the difference be-tween the average neutron and proton transverse flowsobtained from the two symmetry energies (middle pan-els). Also, the negative (positive) values of the neutronand proton average transverse flow F xn − F xp at forward1(backward) rapidities as a result of the Coulomb poten-tial effect on protons is reversed for the neutron-protondifferential flow F xn − p ( y ) after taking into account the ef-fect due to isospin fractionation.From panels (c) and (f) of Fig. 25, it is seen that theslope of the neutron-proton differential transverse flowaround the mid-rapidity obtained from the same sym-metry energy is larger for the higher incident beam en-ergy. This is mainly because a denser nuclear matteris formed at higher incident beam energy, leading to astronger symmetry potential and thus higher transversemomenta for neutrons compared to protons. Althoughthe net effect of the symmetry potential on the neutron-proton differential transverse flow at 800 MeV/nucleon isnot much larger than that at 400 MeV/nucleon, its mag-nitude is much larger and is thus easier to be measuredexperimentally. -0.6 -0.3 0.0 0.3 0.6-12-6061218 x= 0 x= -1E/A=400 MeV Sn+ Snb=5 fm F x n - p ( M e V / c ) -0.6 -0.3 0.0 0.3 0.6 E/A=800 MeV (y/y beam ) c.m. FIG. 26: (Color online) Same as panels (c) and (f) of Fig. 25but for the reaction system of Sn+ Sn. Taken from Ref.[188]. The systematic errors in the neutron-proton differen-tial flow can be reduced by studying its values in twosimilar reaction systems. Besides the Sn+ Sn re-action involving neutron-rich nuclei, one can consideranother less neutron-rich reaction system Sn+ Sn.Fig. 26 shows the rapidity distribution of the neutron-proton differential transverse flow in the semi-central re-action of Sn+ Sn at the same incident beam en-ergies of 400 and 800 MeV/nucleon. In comparisonwith the Sn+ Sn reaction, the slope of the neutron-proton differential transverse flow around mid-rapidityand effects of the symmetry energy are much smallerdue to the smaller isospin asymmetry in the reaction of Sn+ Sn.The double neutron-proton differential flow is de-fined as the difference of the neutron-proton differentialflows in the two reaction systems of Sn+ Sn and Sn+ Sn. Fig. 27 shows the rapidity distribution ofthe double neutron-proton differential transverse flow in -0.6 -0.3 0.0 0.3 0.6-80816 x= 0 x= -1E/A=800 MeV D F x n - p ( M e V / c ) (y/y beam ) c.m. -80816 Sn+ Sn vs Sn+ Sn E/A=400 MeV FIG. 27: (Color online) Rapidity distribution of the doubleneutron-proton differential transverse flow in semi-central re-actions of Sn+Sn isotopes at incident beam energies of 400and 800 MeV/nucleon with two symmetry energies of x = 0and x = − 1. Taken from Ref. [188]. the semi-central reactions of Sn+Sn isotopes. At bothincident beam energies of 400 and 800 MeV/nucleon, thedouble neutron-proton differential transverse flow aroundmid-rapidity is essentially zero for the soft symmetry en-ergy of x = 0. It displays, however, a finite slope withrespect to the rapidity for the stiffer symmetry energyof x = − 1. Moreover, the double neutron-proton dif-ferential transverse flow at the higher incident energyexhibits a stronger symmetry energy effect as expected.Since the double neutron-proton differential transverseflow retains about the same symmetry energy effect asthe Sn+ Sn reaction, it is less sensitive to the sys-tematic uncertainties in experiments [158].Also, the Coulomb effect, which competes stronglywith the symmetry potentials, is less important in thedouble neutron-proton differential transverse flow thanin the neutron-proton differential transverse flow. Thiscan be seen in Fig. 28 which shows the neutron-protondifferential transverse flow (upper two panels) and thedouble neutron-proton differential transverse flow (low-est panel) in the semi-central reactions of Sn+Sn isotopesat the incident beam energy of 400 MeV/nucleon with the2 -0.6 -0.3 0.0 0.3 0.6-80816 (c) Sn+ Sn vs Sn+ Sn (y/y beam ) c.m. -80816 (b) without Coulomb with Coulomb Sn+ Sn D F x n - p ( M e V / c ) F x n - p ( M e V / c ) -80816 E/A=400 MeV, b=5 fm, x= 0 Sn+ Sn (a) FIG. 28: (Color online) Coulomb effects on the neutron-proton differential transverse flow (upper two panels) andthe double neutron-proton differential transverse flow (low-est panel) in the semi-central reactions of Sn+Sn isotopes atthe incident beam energy of 400 MeV/nucleon with the sym-metry energy of x = 0. Taken from Ref. [188]. symmetry energy of x = 0 for the two cases of with andwithout the Coulomb potential. From the upper two pan-els of Fig. 28, one sees that the Coulomb effect reducesthe strength of the neutron-proton differential transverseflow as it makes more protons unbound and to have largetransverse momenta in the reaction-plane. The Coulombeffect is, however, largely reduced in the double neutron-proton differential transverse flow shown in the bottompanel of Fig. 28. F. The n/p ratio of squeezed-out nucleons It is well known that in noncentral heavy-ion collisionsnucleons in the participant region is squeezed out of thereaction plane as a result of the large density gradientin this direction and the absence of spectator nucleonsto block their emissions. These nucleons can thus carrydirect information about the high density phase of thereaction and have been widely used in probing the EOS ofdense matter, particularly that of the symmetric nuclear matter, see, e.g., Refs. [3, 105, 106, 185, 186, 187] for areview. protonsneutrons x= 0 x= -1 Sn+ Sn, E/A=400 MeV, b=5 fm and |(y/y beam ) c.m. |<0.5 d N / da r b i t r a r y un i t ) (degree) FIG. 29: (Color online) Azimuthal distribution of midrapiditynucleons emitted in the reaction of Sn+ Sn at an incidentbeam energy of 400 MeV/nucleon and an impact parameterof b = 5 fm. Taken from Ref. [104]. The possibility of using the squeezed out nucleons tostudy the high density behavior of the nuclear symme-try energy has been studied recently using the IBUU04model [104]. Shown in Fig. 29 are the azimuthal dis-tributions of free nucleons in the midrapidity region( | ( y/y beam ) c.m. | < . 5) of the reaction Sn+ Sn atan incident beam energy of 400 MeV/nucleon and animpact parameter of b = 5 fm, predicted by the IBUUmodel using the MDI interactions with x = 0 and − x = − 1) than for a soft ( x = 0) nuclear symmetry en-ergy in the neutron-rich matter, in addition to the strongnuclear isoscalar potential. For protons, their azimuthaldistributions is less sensitive to the stiffness of the nu-clear symmetry energy. This is due to the additionalrepulsive Coulomb potential which works against the at-tractive symmetry potential experienced by protons inthe neutron-rich matter. Although it is not easy to mea-sure neutrons in experiments, both the transverse flowand squeeze-out of neutrons together with other chargedparticles were measured at both the BEVALAC [189]and SIS/GSI [190, 191, 192]. These experiments andthe associated theoretical calculations, see, e.g., Refs.[193, 194], have, however, all focused on extracting in-formation about the EOS of symmetric nuclear matterwithout paying much attention to the effects due to the3nuclear symmetry energy. (b) , ( n / p ) f r ee p t (GeV/c)1.21.51.82.1 (a) Without azimuthal angle cuts x= 0 x= -1 Sn+ Sn, E/A=400 MeV, b=5 fm and |(y/y beam ) c.m. |<0.5 FIG. 30: (Color online) Transverse momentum distributionof the ratio of midrapidity neutrons to protons emitted in thereaction of Sn+ Sn at the incident beam energy of 400MeV/nucleon and impact parameter of b = 5 fm, with (lowerwindow) and without (upper window) an azimuthal angle cutof 80 ◦ < φ < ◦ and 260 ◦ < φ < ◦ , which corresponds tosqueezed-out nucleons emitted in the direction perpendicularto the reaction plane. Taken from Ref. [104]. To reduce the effect due to uncertainties associatedwith the EOS of symmetric nuclear matter, it is usefulto consider the ratio of squeezed-out neutrons to protons,particularly its transverse momentum dependence. Asshown in Refs. [1, 68], the n/p ratio is determined mostlyby the density dependence of the symmetry energy andalmost not affected by the EOS of symmetric nuclearmatter. In Fig. 30, we show the transverse momentumdependence of the neutron/proton (n/p) ratio of midra-pidity nucleons emitted in the reaction of Sn+ Snat the incident beam energy of 400 MeV/nucleon andimpact parameter of b = 5 fm. For squeezed-out nucle-ons emitted in the direction perpendicular to the reactionplane, which are obtained by introducing an azimuthalangle cut of 80 ◦ < φ < ◦ and 260 ◦ < φ < ◦ , thesymmetry energy effect on the n/p ratio increases withthe increasing transverse momentum p t as shown in thelower window. The effect can be as large as 40% at atransverse momentum of 1 GeV/c. Since high p t par- ticles most likely come from the high density region inthe early stage during heavy-ion collisions, they are thusmore sensitive to the high density behavior of the sym-metry energy. Without the cut on the azimuthal angle,the n/p ratio of free nucleons in the midrapidity region ismuch less sensitive to the symmetry energy in the wholerange of transverse momentum. is as shown in the upperwindow. It is worth mentioning that the n/p ratio of freenucleons perpendicular to the beam direction in the CMSframe in Sn+ Sn reactions at 50 MeV/nucleon wasrecently measured at the NSCL/MSU [195]. This mea-surement was useful for studying the density dependenceof the symmetry energy at sub-normal densities. To in-vestigate the symmetry energy at supra-normal densi-ties, similar measurements that allow the constructionof the reaction plane using a TPC (Time-Projection-Chamber) and simultaneous detection of neutrons to-gether with charged particles at much higher energies arebeing planned [196]. The results reviewed here providestrong scientific motivations and support for such exper-imental efforts. Compared to other potential probes, then/p ratio of squeezed-out nucleons is complementary butcarries more direct information about the symmetry en-ergy at high densities. The sensitivity to the high densitybehavior of the nuclear symmetry energy observed in then/p ratio of squeeze-out nucleons is probably the high-est found so far among all observables studied within thesame transport model. G. K /K + and Σ − / Σ + ratios Since the proposal of Aichelin and Ko that the kaonyield in heavy ion collisions at energies that are belowthe threshold for kaon production in a nucleon-nucleoncollision in free space may be a sensitive probe of the EOSof nuclear matter at high densities [197], a lot of workshave been done both theoretically and experimentally onthis problem [186, 198, 199, 200, 201, 202]. Since thekaon is an iso-doublet meson with the quark content of ds for K and us for K + , the K /K + ratio provides apotentially good probe of the nuclear symmetry energyas the n/p and π − /π + ratios, especially its high densitybehavior as kaons are produced mainly from the highdensity region during the early stage of the reaction andsuffer negligible absorption effects.Using the UrQMD model (version 1.3), Li et al. haveinvestigated the symmetry energy effects on the K /K + ratio by studying K and K + production from the central Sn+ Sn collisions at a beam energy 1 . A GeV withtwo different forms of the symmetry energy, namely, theF15 and Fa3, and the results are shown in Fig. 31 [203].Since the beam energy is close to the kaon productionthreshold, which is about 1 . 58 GeV for nucleon-nucleoncollisions in free space, the K /K + ratio displays onlya small symmetry energy effect. With decreasing beamenergy, the symmetry energy effect becomes larger. Forexample, the kaon yields from the reaction Pb+ Pb4 10 20 30 401.01.11.21.31.40.020.040.060.080.100.12 K / K + t [fm/c] F15 Fa3 Sn+ Snb=0~2 fm K K + < N > F15 Fa3 FIG. 31: (Color online) Top: Time evolution of the K andK + abundances for central Sn + Sn collisions at a beamenergy 1 . A GeV and with the symmetry potentials F15and Fa3. Bottom: the corresponding time evolution of the K /K + ratios. Taken from Ref. [203]. at E b = 0 . A GeV and b = 7 ∼ K /K + ratio for the stiff F15 is about 1 . 25, whereas it is about1 . b = 0 fm impact pa-rameter) Au+Au collisions [102]. Their results, shownin Fig. 32, indicate that at beam energies below andaround the kinematical threshold of kaon production, the K /K + inclusive yield ratio is more sensitive to the sym-metry energy than the π − /π + , and subthreshold kaonproduction thus could provide a promising tool to ex-tract information on the density dependence of the nu-clear symmetry energy.Experimentally, the FOPI collaboration has reportedrecently the results on K + and K meson production in Ru + Ru and Zr + Zr collisions at a beam kineticenergy of 1 . A GeV, measured with the FOPI detec-tor at GSI-Darmstadt [204]. The measured double ratio( K + /K ) Ru /( K + /K ) Zr is compared in Fig. 33 to the π - / π + E beam (AGeV) K / K + DDFNLNL ρ NL ρδ NLDD ρ FIG. 32: (Color online) π − /π + (upper) and K + /K (lower)ratios as a function of the incident energy for central ( b = 0 fmimpact parameter) Au+Au collisions with the RBUU model .In addition, for E beam = 1 AGeV , NLρ results with a densitydependent ρ -coupling (triangles) are also presented. The open symbols at 1 . AGeV show the corresponding results for a Sn + Sn collision, more neutron rich. Note the differentscale for the π − /π + ratios. Taken from Ref. [102]. predictions of a thermal model and the RBUU transportmodel using two different collision scenarios and underdifferent assumptions on the stiffness of the symmetryenergy. From Fig. 33, one can see a good agreement withthe thermal model prediction and the assumption of asoft symmetry energy for infinite nuclear matter whilemore realistic transport simulations of the collisions showa similar agreement with the data but exhibit a reducedsensitivity to the symmetry term. We note that in thepresent RBUU calculations, the isospin dependence ofthe K + - and K -nucleon potentials in the asymmetricnuclear medium has been neglected. Recently, Mishra etal. studied the isospin dependent kaon and antikaon op-tical potentials in dense hadronic matter using a chiralSU(3) model and their results indicate that the densitydependence of the isospin asymmetry is appreciable forthe kaon and antikaon optical potentials. On the otherhand, subthreshold kaon production in heavy-ion colli-sions depends on some detailed implementations of thetransport model [201, 202]. Therefore, extracting usefulinformation on the high density behavior of the nuclearsymmetry energy from subthreshold kaon production inheavy-ion collisions induced by neutron-rich nuclei needsfurther studies from both experimental and theoreticalsides.Besides the K /K + ratio, the Σ − / Σ + ratio has alsobeen proposed as a probe of the high density behaviorof the nuclear symmetry energy based on the UrQMDmodel (version 1.3) calculations [97]. Shown in Fig. 34 isthe time evolution of the π − /π + ratios (left-hand side)and the Σ − / Σ + ratios (right-hand side) calculated with a5 ρ NL ρδ THERM.DATA 30 75 100 Data Vs Models ( K + / K ) R u / ( K + / K ) Z r E sym (MeV, ρ B =2.5 ρ ) FIG. 33: (Color online) Experimental ratio( K + /K ) Ru /( K + /K ) Zr (star) and theoretical predic-tions of the thermal model (cross) and the transport modelwith 3 different assumptions on the symmetry energy: NL(circles), NL ρ (squares) and NL ρδ (triangles). The INM andHIC calculations are represented by open and full symbols,respectively (see text for more details). The statistic andsystematic errors are represented by vertical bars andbrackets, respectively. Taken from Ref. [204]. stiff symmetry energy F γ =11 and a soft symmetry energy F a =32 for the reaction Sn+ Sn at E beam = 1 . A ,2 . A , 3 . A GeV and b = 2 fm, and Sn+ Sn at E b = 3 . A GeV and b = 2 fm. It is seen that theΣ − / Σ + ratio is sensitive to the density dependence ofthe symmetry energy for neutron-rich Sn+ Sn col-lisions, but insensitive to that for the nearly symmetric Sn+ Sn collisions. For Sn+ Sn at E b = 1 . A GeV, the Σ − / Σ + ratio calculated with the stiff symmetryenergy ( F γ =11 ) is higher than the one with the soft sym-metry energy ( F a =32 ). As the beam energy increases, theΣ − / Σ + ratio falls and the difference between the Σ − / Σ + ratios calculated with F γ =11 and F a =32 reduces strongly.As the beam energy increases further, at E b = 3 . A GeVthe Σ − / Σ + ratio falls further but the difference betweenthe Σ − / Σ + ratios calculated with F γ =11 and F a =32 ap-pears again, the Σ − / Σ + ratio with soft symmetry energynow becoming higher than that with the stiff one. Forpions, the results indicate that the ratio π − /π + at highenergies (as in the case with E b = 3 . A GeV) becomesinsensitive to the symmetry energy. The difference be-tween the Σ − / Σ + ratio and the π − /π + ratio can be un-derstood from the fact that, like nucleons, Σ ± hyperonsare under the influence of the mean field produced by thesurrounding nucleons, as soon as they are produced. Thesymmetry potential of hyperons thus play an importantdynamic role and results in a strong effect on the ratio 12 18 24 306 12 18 24 301.01.11.21.31.41.51.61.71.81.92.0 Sn+ Sn, b=2 fm t (fm/c) X - / X + Sn+ Sn,E b =3.5A GeV F =1 F a=3 E b (A GeV) F =1 F a=3 FIG. 34: The ratios π − /π + (left) and Σ − / Σ + (right) for thecollisions Sn + Sn ( E b = 1 . A , 2 . A , and 3 . A GeV; b = 2 fm) and Sn + Sn ( E b = 3 . A GeV; b = 2 fm),calculated with the different symmetry potentials F γ =11 and F a =32 . Taken from Ref. [97]. of the negatively to positively charged Σ hyperons. VII. SUMMARY AND OUTLOOK Heavy-ion collisions induced by neutron-rich nucleiprovides a unique opportunity for investigating the prop-erties of the isospin asymmetric nuclear matter, espe-cially the density dependence of the nuclear symmetryenergy. To extract useful information from these colli-sions, transport models have been found to be extremelyuseful. Applications of these models have helped usunderstand not only the isospin dependence of the in-medium nuclear effective interactions but also that of thethermal, mechanical and transport properties of asym-metric nuclear matter. These information, particularlythe density dependence of the nuclear symmetry energy,are very important for both nuclear physics and astro-physics. Significant progress has been made in recentyears in determining the density dependence of the nu-clear symmetry energy. Based on transport model cal-culations, a number of sensitive probes of the symmetryenergy have been identified. In particular, the momen-tum dependence in both the isoscalar and isovector partsof the nuclear potential was found to play an importantrole in extracting accurately the density dependence ofthe symmetry energy. From comparison of results fromthe transport model with recent experimental data onisospin diffusion from NSCL/MSU, a symmetry energy of E sym ( ρ ) ≈ . ρ/ρ ) γ with γ = 0 . − . 05 at subnormaldensities, which corresponds to the isospin and momen-tum dependent MDI interaction with x = 0 and − 1, hasbeen extracted. This conclusion is consistent with thoseextracted from studying other observables such as the6isoscaling data and the neutron-skin thickness in Pb. NL3 ChPT DBHF DD- DD-TW var AV + V+3-BF BHF(AV +3BF) FSU-gold x= -1 x= 0 E sy m ( M e V ) / FIG. 35: (Color online) Density dependence of the nuclearsymmetry energy using the MDI interaction with x = 0 and x = − Although considerable progress has been made in de-termining the density dependence of the nuclear sym-metry energy at sub-normal densities, probing the highdensity behavior of the nuclear symmetry energy remainsa major challenge. Our current knowledge on the densitydependence of the nuclear symmetry energy can be seenfrom Fig. 35, where predictions from several typical the-oretical model [10, 18, 205] are compared with the phe-nomenological constraints that have been obtained frommodel analysis of experimental data. The constraints la-beled x = 0 and x = − Sn + Sn at E beam = 50 AMeV within an isospin and momentum de-pendent transport model [91, 107, 108, 109]. For this par-ticular reaction, the maximum density reached is about1 . ρ . Moreover, it was shown that the neutron-skinthickness in Pb calculated within the Hartree-Fock ap-proach using the same underlying Skyrme interactions asthe ones labeled x = 0 and x = − Zr and Pb as well as the isovector gi-ant dipole resonance of Pb [113]. We note that theconstraint obtained from the isoscaling analysis is alsoconsistent with the FSU-Gold and the x = 0 case [114].At present, these results represent the best phenomeno-logical constraints on the nuclear symmetry energy atsub-normal densities.Although all predicted nuclear symmetry energiesshown in Fig. 35 are close to the existing constraintsat low densities, they diverge widely at supra-normaldensities, including those from the MDI interaction with x = − x = 0 as well as the FSU-Gold.Since there are currently no experimental constraints onthe high density behavior of the nuclear symmetry en-ergy, more work is thus needed. In particular, experimen-tal data including neutrons from reactions with neutron-rich beams in a broad energy range will be useful forstudying the behavior of the symmetry energy at highdensities. We have reviewed recent theoretical progressin identifying the observables in heavy-ion collisions in-duced by high energy radioactive nuclei that are sensitiveto the high density behavior of the nuclear symmetry en-ergy. A plethora of potentially sensitive probes have beenfound, and they include the π − /π + ratio, isospin frac-tionation, n-p differential flow, double n/p and π − /π + ratio, double n-p differential transverse flow as well as the K /K + and Σ − / Σ + ratios. Studying these observable infuture experiments at high energy radioactive beam facil-ities is expected to lead to significant constraints on thebehavior of the symmetry energy at supra-normal den-sities. 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