Using an isospin-dependent transport model, we study the effects of nuclear
symmetry energy on two-nucleon correlation functions in heavy ion collisions
induced by neutron-rich nuclei. We find that the density dependence of the
nuclear symmetry energy affects significantly the nucleon emission times in
these collisions, leading to larger values of two-nucleon correlation functions
for a symmetry energy that has a stronger density dependence. Two-nucleon
correlation functions are thus useful tools for extracting information about
the nuclear symmetry energy from heavy ion collisions.Comment: Revised version, to appear in Phys. Rev. Let

B. A. Li

Bao-An Li

Che Ming Ko

D. H. Boal

I. Bombaci

Lie-Wen Chen

Vincenzo Greco

W. Bauer

Physical Review Letters

English

arXiv

Crossref

Effects of Symmetry Energy on Two-Nucleon Correlation Functions in Heavy-Ion Collisions Induced by Neutron-Rich Nuclei

Chen, Lie-Wen

Greco, Vincenzo

Ko, Che Ming

Li, Bao-An

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:nucl-th/0211002v2  22 Mar 2003Effects of symmetry energy on two-nucleon correlation functions in heavy-ioncollisions induced by neutron-rich nucleiLie-Wen Chen,1, ∗ Vincenzo Greco,1 Che Ming Ko,1 and Bao-An Li21Cyclotron Institute and Physics Department, Texas A&M University, College Station, Texas 77843-33662Department of Chemistry and Physics, P.O. Box 419,Arkansas State University, State University, Arkansas 72467-0419(Dated: January 23, 2018)Using an isospin-dependent transport model, we study the effects of nuclear symmetry energyon two-nucleon correlation functions in heavy ion collisions induced by neutron-rich nuclei. Wefind that the density dependence of the nuclear symmetry energy affects significantly the nucleonemission times in these collisions, leading to larger values of two-nucleon correlation functions fora symmetry energy that has a stronger density dependence. Two-nucleon correlation functions arethus useful tools for extracting information about the nuclear symmetry energy from heavy ioncollisions.PACS numbers: 25.70.-z, 25.70.Pq., 24.10.LxThe nuclear symmetry energy is a measure of the dif-ference between the binding energy of an asymmetric nu-clear matter and that of a nuclear matter with equalnumbers of protons and neutrons. Knowledge on thedensity-dependence of the nuclear symmetry energy isessential for understanding not only the structure of ra-dioactive nuclei [1, 2, 3, 4] but also many important is-sues in astrophysics [5, 6, 7, 8, 9]. For example, thenuclear symmetry energy plays an important role in un-derstanding the nucleosynthesis during the pre-supernovaevolution of massive stars, the mechanisms for super-nova explosions, the cooling rate of protoneutron starsand the associated neutrino fluxes, and the kaon con-densation as well as the hadron to quark-gluon plasmaphase transitions in neutron stars [5, 6]. However, ourknowledge of nuclear symmetry energy remains limitedin spite of extensive theoretical studies [7, 10, 11]. Onthe other hand, radioactive beams, particularly the veryenergetic ones to be available at the planned Rare Iso-tope Accelerator (RIA) and the new accelerator facilityat the German Heavy Ion Accelerator Center (GSI), pro-vide a great opportunity to study the density dependenceof the nuclear symmetry energy. Significant progress hasalready been achieved in identifying the observables thatare sensitive to the symmetry energy. It was proposedthat information about the nuclear symmetry energy canbe extracted from measurements of the neutron-skins ofradioactive nuclei via their total reaction cross sections[1] and those of stable heavy nuclei via parity-violatingelectron scatterings [3]. Promising probes to the nuclearsymmetry energy have also been found in heavy-ion col-lisions induced by neutron-rich nuclei, and these includethe pre-equilibrium neutron/proton ratio [12], the isospinfractionation [13, 14, 15, 16, 17], the isoscaling in mul-tifragmentation [18], the proton differential elliptic flow[19] and the neutron-proton differential transverse flow[20] as well as the pi− to pi+ ratio [21].Since two-particle correlation functions, through finalstate interactions and quantum statistical effects, havebeen shown to be a sensitive probe to the space-timedistributions of emitted particles in heavy-ion collisions[22], it is of interest to investigate if they can also be usedto study the density dependence of the nuclear symmetryenergy. In this Letter, we show that the emission times ofneutrons and protons are indeed sensitive to the nuclearsymmetry energy. A stronger density dependence in thenuclear symmetry energy leads to an earlier and morecorrelated emission of pre-equilibrium neutrons and pro-tons. Consequently, strengths of the correlation functionsfor nucleon pairs with high total momenta, especiallyfor neutron-proton pairs with low relative momenta, arelarger for a stiffer symmetry energy. Measurements oftwo-nucleon correlation functions in heavy ion collisionsthus provides another possible tool for extracting usefulinformation about the nuclear symmetry energy.Our study is based on an isospin-dependentBoltzmann-Uehling-Uhlenbeck (IBUU) transport model(e.g., Refs. [12, 20, 21, 23, 24]). In this model, the initialpositions of protons and neutrons in the colliding nucleiare determined according to their density distributionspredicted by the relativistic mean-field (RMF) theory.Their initial momenta are then taken to have a uniformdistribution inside the neutron or proton Fermi spherewith its Fermi momentum determined from the localdensity using the Thomas-Fermi approximation. Forthe isoscalar potential, we use as default the Skyrmepotential with an incompressibility K0 = 380 MeV. Thispotential has been shown to approximately reproducethe transverse flow data in heavy-ion collisions, althoughthe latter are best described by a momentum-dependentsoft potential with K0 = 210 MeV [25, 26]. The isospineffects are included through the isospin-dependent totaland differential nucleon-nucleon cross sections, thedifferent Pauli blockings for protons and neutrons, thesymmetry potential, and the Coulomb potential forprotons. For nucleon-nucleon cross sections, we use as2default the experimental values in free space, in whichthe neutron-proton cross section is about a factor of 3larger than the neutron-neutron or proton-proton crosssection. For a review of the IBUU model, we refer thereader to Ref. [27].0.0 0.5 1.0 1.5-20-1001020protonsneutrons(b)  V sym (MeV)/ 00.00.51.01.5=0.2 =0.5 =2 (a)  Psym (MeV/fm3 ) FIG. 1: The symmetry pressure (upper panel) and potential(lower panel) in nuclear matter with an isospin asymmetryδ = 0.2 for the soft (solid curves) and stiff (dashed curves)symmetry energies.The energy per nucleon in an asymmetric nuclear mat-ter is usually expressed asE(ρ, δ) = E(ρ, δ = 0) + Esym(ρ)δ2 +O(δ4), (1)where ρ = ρn + ρp is the baryon density; δ = (ρn −ρp)/(ρp+ρn) is the relative neutron excess; and E(ρ, δ =0) is the energy per particle in a symmetric nuclear mat-ter, while Esym(ρ) is the nuclear symmetry energy. Forthe latter, we adopt the parameterization used in Ref.[28] for studying the properties of neutron stars, i.e.,Esym(ρ) = Esym(ρ0) · uγ , (2)where u ≡ ρ/ρ0 is the reduced density and Esym(ρ0) =35 MeV is the bulk symmetry energy at normal nuclearmatter density ρ0 = 0.16 fm−3. In the following, weconsider the two cases of γ = 0.5 (soft) and 2 (stiff) toexplore the large range of Esym(ρ) predicted by differenttheoretical models [7]. The pressure due to the symmetryenergy, given by Psym = (ρ2∂Esym/∂ρ)δ2, in asymmetricnuclear matter with an isospin asymmetry δ = 0.2 areshown in the upper panel of Fig. 1 for the two symmetryenergies. It is seen that the stiff symmetry energy leadsto a larger pressure in asymmetric nuclear matter thanthat from the soft symmetry energy at the same density.The symmetry potential acting on a nucleon derivedfrom the above nuclear symmetry energy is [19]Vsym = ±2[Esym(ρ0)uγ − 12.7u2/3]δ+[Esym(ρ0)(γ − 1)uγ + 4.2u23 ]δ2, (3)where “+” and “-” are for neutrons and protons, respec-tively. In the lower panel of Fig. 1, the symmetry poten-tials for protons and neutrons in an isospin asymmetricnuclear matter with δ = 0.2 are shown. The symmetrypotential is seen to be attractive for protons and repul-sive for neutrons and increases with density, except forthe stiff symmetry potential at densities below 0.5ρ0.100 150 200 250 300 350 400 4502040608010012052Ca+48CaE=80 AMeV, b=0 fm p,  n ( =0.5) p,  n ( =2)  Average emission time (fm/c)p (MeV/c)FIG. 2: The average emission times of protons or neutronsas functions of their momenta for soft (solid curves) and stiff(dashed curves) symmetry energies.As an example, we study in this letter central collisionsof 52Ca + 48Ca at E=80MeV/nucleon. This reaction sys-tem with an isospin asymmetry δ = 0.2 can be studiedat RIA. Nucleons are considered as being emitted whentheir local densities are less than ρ0/8 and subsequentinteractions with the mean field do not cause their re-capture into regions of higher density. Other emissioncriteria, such as taking the nucleon emission time as itslast collision time in the IBUU model, do not changeour conclusions. Shown in Fig. 2 are the average emis-sion times of protons and neutrons as functions of theirmomenta. It is seen that the average emission time ofnucleons with a given momentum is earlier for the stiffsymmetry energy than for the soft one, as nucleon emis-sions are mainly governed by the pressure of the excitedmatter created during the collisions [19, 29], which islarger for the stiffer symmetry energy than for the softone. Moreover, there is a significant delay in the emis-sion of protons when the symmetry energy is soft. Thisis due to the fact that the symmetry potential is gen-erally repulsive for neutrons and attractive for protons,and their magnitudes at low densities, where most nucle-3ons are emitted, are larger for the soft symmetry energythan for the stiff symmetry potential as seen in Fig. 1.The relative emission times of neutrons and protons inthe case of the stiff symmetry energy depends on thedensity at which they are emitted, as the stiff symmetrypotential changes sign when the nuclear density is below0.5ρ0. It is interesting to note that high momentum nu-cleons are emitted during the early pre-equilibrium stageof the collision when the density is relatively high, whilelow momentum ones are mainly emitted when the systemis close to equilibrium and the density is low.It is well known that the space-time distributions ofemitted particles can be extracted from the two-particlecorrelation functions; see, e.g., Refs. [22, 30, 31], forreviews. The two-proton correlation function has beenstudied most extensively. Recently, experimental dataon two-neutron and neutron-proton correlation functionshave also become available, and this has made it possibleto deduce the relative emission times of neutrons andprotons [32]. Theoretically, the two-nucleon correlationfunction can be evaluated in the standard Koonin-Prattformalism [33, 34] by convoluting the emission functiong(p, x), i.e., the probability for emitting a particle withmomentum p from space-time point x = (r, t), with therelative wave function of the two particles, i.e.,C(P,q) =∫d4x1d4x2g(P/2, x1)g(P/2, x2) |φ(q, r)|2∫d4x1g(P/2, x1)∫d4x2g(P/2, x2).(4)In the above, P(= p1 + p2) and q(=12(p1 − p2)) are, re-spectively, the total and relative momenta of the par-ticle pair; and φ(q, r) is their relative wave functionwith r being their relative position, i.e., r = (r2−r1)−12(v1 + v2)(t2 − t1).Using the program Correlation After Burner [35],which takes into account final-state nucleon-nucleon in-teractions, we have evaluated the two-nucleon correla-tions from the emission function given by the IBUUmodel. Shown in Fig. 3 are the two-nucleon corre-lation functions gated on the total momentum (P ) ofnucleon pairs from central collisions of 52Ca + 48Ca atE=80 MeV/nucleon. The left and right panels are forP < 300 MeV/c and P > 500 MeV/c, respectively.For both neutron-neutron and neutron-proton correla-tion functions, they peak at q ≈ 0 MeV/c. The proton-proton correlation function is, on the other hand, peakedat about q = 20 MeV/c due to the strong final-states-wave attraction but is suppressed at q = 0 as a resultof Coulomb repulsion and wave-function antisymmetriza-tion between the two protons. Since the emission timesof low momentum nucleons are not affected much by thedifferent symmetry energies used in the IBUU model asshown in Fig. 2, the two-nucleon correlation functionsare thus similar for the stiff and soft symmetry energies.On the other hand, the emission times of high momentumnucleons, which are dominated by those with momenta12345678P<300 MeV/c(b) pp    P>500 MeV/c(d) nn(c) np    0.00.51.01.552Ca+48CaE=80 AMeV, b=0 fm(a) nnC(q)q (MeV/c)    (e) pp    10 20 30 40 501234 =0.5 =2   10 20 30 40 50(f) np    FIG. 3: Two-nucleon correlation functions gated on the totalmomentum of nucleon pairs. Left panels are for P < 300MeV/c while right panels are for P > 500 MeV/c.near 250 MeV/c, differ appreciably for the two symme-try energies considered here. The correlation functionsof these nucleon pairs thus show a strong dependence onsymmetry energy. Gating on nucleon pairs with high to-tal momentum thus allows one to select those nucleonsthat have stronger correlations as a result of small spa-tial separations at emissions. From Fig. 3, it is seenthat the correlation functions for neutron-neutron andneutron-proton pairs with high total momentum but lowrelative momentum of q = 5 MeV/c are, respectively,about 20% and 30% higher for the stiff symmetry energythan for the soft symmetry energy, while the correlationfunction for proton-proton pairs with high total momen-tum and relative momentum of q = 20 MeV/c is about20% higher for the stiff symmetry energy than for the softsymmetry energy. The neutron-proton correlation func-tion thus has the largest sensitivity to nuclear symmetryenergy. As shown in Fig. 2 and discussed earlier, the rel-ative emission times of neutrons and protons is sensitiveto nuclear symmetry energy. Since the pre-equilibriumneutrons and protons are emitted almost simultaneouslyin the case of the stiff symmetry energy, they are stronglycorrelated, leading thus to a larger value for the neutron-proton correlation function. On the other hand, protonsand neutrons are less correlated in the case of the softsymmetry energy as proton emissions are delayed rela-tive to neutron emissions. As a result, the soft symmetryenergy gives a smaller value for the neutron-proton cor-4relation function. Furthermore, with the stiff symmetryenergy, neutrons as well as protons have smaller spatialseparations at emissions as they are emitted earlier dur-ing the collision. Therefore, neutrons and also protonsare more correlated among themselves for the stiff sym-metry energy than for the soft one. Our results thusclearly demonstrate that correlation functions of nucleonpairs with high total momentum can indeed reveal sen-sitively the effect of the nuclear symmetry energy on thespace-time distributions of nucleons at emissions.We have also studied the dependence of two-nucleoncorrelation functions on the incompressibility K0 andnucleon-nucleon cross sections. We find that changingthe value of K0 from 380 to 210 MeV or using the in-medium nucleon-nucleon cross sections predicted by theDirac-Brueckner approach based on the Bonn A poten-tial [36], which has a smaller magnitude and weakerisospin dependence than the free ones but a strong den-sity dependence, changes both the neutron-proton andproton-proton correlation functions by about 5%, simi-lar to those found in Ref. [22]. The weak dependenceof the two-nucleon correlation functions on the stiffnessof isoscalar energy is due to the reduced difference inthe pressure of the excited matter as a result of differentmaximum densities reached in the collision, with the stiffone giving a lower density than the soft one. Details onthese results as well as the dependence of the symmetryenergy effects studied here on the impact parameter, in-cident energy, and masses of the colliding system will bereported elsewhere [37].In conclusion, we have studied the two-nucleon corre-lation functions in intermediate-energy heavy-ion colli-sions induced by neutron-rich nuclei in the framework ofan isospin-dependent transport model. We find that thenuclear symmetry energy affects significantly the emis-sion times of neutrons and protons. A stiffer symme-try energy leads to an earlier and nearly simultaneousemission of pre-equilibrium neutrons and protons. 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Effect of symmetry energy on two-nucleon correlation functions in
  heavy-ion collisions induced by neutron-rich nuclei

Effect of symmetry energy on two-nucleon correlation functions in heavy-ion collisions induced by neutron-rich nuclei

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