The rapidity dependence of the single- and double- neutron to proton ratios
of nucleon emission from isospin-asymmetric but mass-symmetric reactions Zr+Ru
and Ru+Zr at energy range $100 \sim 800$ A MeV and impact parameter range
$0\sim 8$ fm is investigated. The reaction system with isospin-asymmetry and
mass-symmetry has the advantage of simultaneously showing up the dependence on
the symmetry energy and the degree of the isospin equilibrium. We find that the
beam energy- and the impact parameter dependence of the slope parameter of the
double neutron to proton ratio ($F_D$) as function of rapidity are quite
sensitive to the density dependence of symmetry energy, especially at energies
$E_b\sim 400$ A MeV and reduced impact parameters around 0.5. Here the symmetry
energy effect on the $F_D$ is enhanced, as compared to the single neutron to
proton ratio. The degree of the equilibrium with respect to isospin (isospin
mixing) in terms of the $F_D$ is addressed and its dependence on the symmetry
energy is also discussed.Comment: 10 pages, 2 figure

Li, Qingfeng

Li, Zhuxia

Stoecker, Horst

English

arXiv

The rapidity dependence of the single- and double- neutron to proton ratios of nucleon emission from isospin-asymmetric but mass-symmetric reactions Zr+Ru and Ru+Zr at energy range 100 ~ 800 A MeV and impact parameter range 0 ~ 8 fm is investigated. The reaction system with isospin-asymmetry and mass-symmetry has the advantage of simultaneously showing up the dependence on the symmetry energy and the degree of the isospin equilibrium. We find that the beam energy- and the impact parameter dependence of the slope parameter of the double neutron to proton ratio (F_D) as function of rapidity are quite sensitive to the density dependence of symmetry energy, especially at energies E_b ~ 400 A MeV and reduced impact parameters around 0.5. Here the symmetry energy effect on the F_D is enhanced, as compared to the single neutron to proton ratio. The degree of the equilibrium with respect to isospin (isospin mixing) in terms of the F_D is addressed and its dependence on the symmetry energy is also discussed

Stöcker, Horst

Hochschulschriftenserver - Universität Frankfurt am Main

arXiv:nucl-th/0603050 v1   21 Mar 2006Probing the symmetry energy and the degree of isospinequilibriumQingfeng Li 1∗ , Zhuxia Li 2, and Horst St¨ ocker 1,31) Frankfurt Institute for Advanced Studies (FIAS),Johann Wolfgang Goethe-Universit¨ at,Max-von-Laue-Str. 1,D-60438 Frankfurt am Main, Germany2) China Institute of Atomic Energy,P.O. Box 275 (18),Beijing 102413, P.R. China3) Institut f¨ ur Theoretische Physik,Johann Wolfgang Goethe-Universit¨ at,Max-von-Laue-Str. 1,D-60438 Frankfurt am Main, Germany∗ Fellow of the Alexander von Humboldt Foundation.1AbstractThe rapidity dependence of the single- and double- neutron to proton ratios of nucleon emissionfrom isospin-asymmetric but mass-symmetric reactions Zr+Ru and Ru+Zr at energy range 100 ∼800 A MeV and impact parameter range 0 ∼ 8 fm is investigated. The reaction system with isospin-asymmetry and mass-symmetry has the advantage of simultaneously showing up the dependence onthe symmetry energy and the degree of the isospin equilibrium. We ﬁnd that the beam energy- andthe impact parameter dependence of the slope parameter of the double neutron to proton ratio (FD)as function of rapidity are quite sensitive to the density dependence of symmetry energy, especiallyat energies Eb ∼ 400 A MeV and reduced impact parameters around 0.5. Here the symmetryenergy eﬀect on the FD is enhanced, as compared to the single neutron to proton ratio. The degreeof the equilibrium with respect to isospin (isospin mixing) in terms of the FD is addressed and itsdependence on the symmetry energy is also discussed.PACS numbers: 24.10.Lx, 25.75.Dw, 25.75.-q2High energy collision of heavy ions provides a unique tool to study the unbroken prop-erties of dense nuclear matter at ﬁnite temperature, outside of the reach of conventionednuclear physics [1, 2]. The FOPI Collaboration has recently performed the mixing experi-ment by using four mass 96+96 systems 9644Ru+9644Ru, 9640Zr+9640Zr, 9644Ru+9640Zr, and 9640Zr+9644Ruto investigate the degrees of the isospin-equilibrium in the intermediate energy heavy ioncollisions (HICs) [3, 4]. It was found that complete isospin equilibration is not reached bycomparing the emitted proton number counted in rapidity (see [3, 4, 5, 6, 7]). The advantageof taking mixed reactions is that the isospin asymmetry shows up more obviously becausethe eﬀects of the iso-scalar part of the equation of state (EoS) largely cancel. This advan-tage could also be used for probing the density dependence of the symmetry energy. So farthe density dependence of the nuclear symmetry energy, Esym(u), with u = ρ/ρ0 being thereduced nuclear density, is still largely uncertain, although a lot of theoretical and experi-mental eﬀorts have been undertaken in the past decade [8, 9, 10]. Recently, the symmetryenergy has been extracted to Esym(u) ≃ 31.6u1.05 at subnormal densities by comparing withexperimental data [11] when a free nucleon-nucleon (NN) elastic cross section has been used[12], while Esym(u) ≃ 31.6u0.69 after considering partly medium modiﬁcations of the nucleontransport [13, 14] were subtracted . The symmetry energy can be investigated by the doubleneutron to proton and π−/π+ ratios. The neutron-rich 124,132Sn +124 Sn system and moreisospin-symmetric 112Sn +112 Sn system serve as robust probes of the nuclear symmetry en-ergy [15, 16]. A very attractive feature of taking double ratios is the fact that the systematicerrors can be strongly reduced, while the sensitivity of the symmetry energy is close to thesingle neutron to proton and π−/π+ ratios.In the present work, we take two sets of isospin-asymmetric but mass-symmetric reactions,Zr+Ru and Ru+Zr, to study the symmetry energy eﬀect on the rapidity dependence of both,single as well as double unbound neutron to proton ratios. We take advantage of the mixingreactions to study the inﬂuence of the isospin dependent part of the EoS on the equilibrationprocess. We show that the slope of the double neutron to proton ratio as distributed in therapidity space is — for mixing reactions — closely related to degree of isospin equilibration.The UrQMD transport model has been updated for simulating intermediate energy HICs[2, 17, 18, 19, 20, 21]. It is also used for the calculations presented in this work. A softequation of state (S-EoS) is adopted [20] (except otherwise stated). We consider here twosets of symmetry energy parameterizations: Esym(u) = 34uγ with γ = 0.5 (soft) and 1.53(hard). A non-relativistic medium modiﬁcation of the nucleon-nucleon elastic cross sectionis also considered here (please see Ref. [21] for details). The method to construct thefreeze-out clusters is the same as presented in our previous work [19, 21].We calculate the rapidity (Y(0)cm = ycm/ybeam in the center-of-mass system) dependence ofunbound neutrons and protons in Zr+Ru and Ru+Zr reactions at the beam energies Eb =100, 200, 400, 600, and 800 A MeV and at the reduced impact parameters b/b0 = 0, 0.25,0.5, and 0.75, where b0 = Rproj +Rtarg is the maximum impact parameter leading to nuclearreactions by adopting ’sharp-cutoﬀ’ approximation. The calculated rapidity dependence ofthe neutron to proton ”n/p” ratios for Zr+Ru and Ru+Zr reactions are shown in Fig. 1 (a)-(d) for Eb = 100 A MeV, b/b0 = 0.5 in plot (a); Eb = 400 A MeV, b/b0 = 0.5 in (b); Eb = 400A MeV, b/b0 = 0 in (c); and Eb = 800 A MeV, b/b0 = 0 in (d). If equilibrium with respectto the isospin degree of freedom was reached, the rapidity distributions of the neutron toproton ratios for the two diﬀerent mixing reactions should coincide with each other. Figs.1 (a) and (b) show clearly that for b/b0 = 0.5 equilibrium (with respect to isospin degree offreedom) is not reached, because the centroid of the rapidity dependence of the neutron toproton ratio is not located at mid-rapidity [5]. For Zr+Ru reactions, the neutron to protonratio distribution peaks at the positive rapidity, while for the inverse Ru+Zr reactions, thedistribution peaks at the negative rapidity. The rapidity dependence of the neutron to protonratio is symmetric with respect to Y(0)cm = 0 for central reactions at high beam energy (Fig.1 (d)), where the equilibration is nearly reached. It is interesting to see that the statisticsin the UrQMD calculations is perfectly guaranteed since the calculated results for Zr+Ruand Ru+Zr reactions are symmetrically distributed very well with respect to Y(0)cm = 0, asone expects. This is also a fundamental but important checkup in the experimental side. Asseen for reactions at Eb = 400 A MeV and b/b0 = 0 in Fig. 1 (c), the rapidity distributionof the neutron to proton ratio almost coincides for reactions Zr+Ru and Ru+Zr when thesoft-symmetry energy is chosen, but it deviates strongly when the stiﬀ-symmetry energy ischosen. This shows the closely relationship between the eﬀect of the symmetry energy andthe equilibration in HICs at intermediate energies.Figs. 1 (e) and (f) present the corresponding rapidity distributions of double neutronto proton ratio ”(n/p)ZrRu/(n/p)RuZr” with the same conditions as Figs. 1 (a) and (d),respectively. From Fig. 1 (e) it is seen that the double ratio increases almost linearly fromtarget (Y(0)cm = −1) to projectile (Y(0)cm = 1) rapidity region for the case of Eb=100 A MeV and4b/b0 = 0.5. The two curves corresponding to soft and stiﬀ symmetry energies cross at thepoint of rapidity equal to zero, with positive but diﬀerent slopes. And obviously, the slopefor γ = 0.5 case is larger than that for γ = 1.5 case. It is clear that for isospin asymmetricand mass symmetric reactions Zr+Ru and Ru+Zr, if the equilibrium with respect to isospindegree of freedom is totally reached, the double ratio should be unity at all rapidities.The deviation of (n/p)Zr+Ru/(n/p)Ru+Zr from unit thus implies the deviation from isospinequilibration. Fig. 1 (e) shows clearly that the isospin equilibration is not reached and thechoice of symmetry energy inﬂuences the isospin equilibrium process for this case. While forthe reactions at Eb = 800 A MeV and b/b0 = 0 presented in Fig. 1 (f), the slope is ∼ 0 andbecomes horizontal in large rapidity region, which means that the isospin equilibration canbe roughly reached. Thus the double neutron to proton ratio as function of rapidity takingfrom the mixing reaction system can provide both the information of density dependence ofthe symmetry energy and the degree of isospin equilibrium reached in the reaction.Further, we study the beam energy and the centrality dependence of the slope parametersof the double neutron to proton ratio over the rapidity, namely the FD, which is obtained bylinearly ﬁtting the rapidity distribution of the double ratio over |Y(0)cm | ≤ 1. Fig. 2 (a) showsthe excitation function of FD for impact parameters b/b0 = 0 and 0.5 and for symmetryenergies γ = 0.5 and 1.5, respectively. The reactions with larger impact parameters showlarger FD values, which implies that less equilibration is reached. The FD values ﬁrst rise andthen drop slightly with increasing beam energy. It shows a broad maximum at Eb ∼ 200−400A MeV for intermediate impact parameters, while central collisions peak at 200 A MeV. Theinﬂuence of the symmetry energy on the FD is stronger for b/b0 = 0.5 than for b/b0 = 0. Theisoscalar part of the EoS aﬀects the equilibrium process of the colliding system as well [5].Fig. 2 (a) shows FD at Eb = 400 A MeV and b/b0 = 0.5 using a soft EoS with momentumdependence (SM-EoS, open dots). It is seen that the diﬀerence between the S-EoS and theSM-EoS is very small. This indicates that the eﬀect of the isoscalar part of the EoS on FDis largely cancelled, while the eﬀect of the isovector part on FD is nicely preserved.Fig. 2 (a) also shows that — for central collisions— FD as calculated with a stiﬀ symmetryenergy is larger than with a soft one. For b/b0 = 0.5, the relative magnitude of FD ascalculated with the stiﬀ symmetry energy becomes smaller than that with the soft symmetryenergy. The centrality dependence (with b/b0 = 0, 0.25, 0.5, and 0.75) of FD at beam energy400 A MeV is presented in Fig. 2 (b) . This shows that both curves of FD, as calculated with51.01.11.21.31.41.51.61.71.8-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.60.70.80.91.01.11.21.3-1.0 -0.5 0.0 0.5 1.0 1.51.01.11.21.31.41.51.61.7(a) n/p n/pEb=0.1A GeV; b/b0=0.5(d) Zr+Ru,  =0.5 Zr+Ru,  =1.5 Ru+Zr,  =0.5 Ru+Zr,  =1.5Eb=0.8A GeV; b/b0=0(e)Eb=0.1A GeV; b/b0=0.5Y(0)cm(n/p)ZrRu/(n/p)RuZr  =0.5  =1.5(f)Eb=0.8A GeV; b/b0=0  =0.5  =1.5(c)Eb=0.4A GeV; b/b0=0(b)Eb=0.4A GeV; b/b0=0.5 FIG. 1: (Color online) (a)-(d): Rapidity dependence of neutron to proton (n/p) ratios with sym-metry energies of γ = 0.5 and 1.5. The Zr+Ru and Ru+Zr reactions with Eb = 100 A MeV,b/b0 = 0.5 in plot (a), Eb = 400 A MeV, b/b0 = 0.5 in (b), Eb = 400 A MeV, b/b0 = 0 in (c), andEb = 800 A MeV, b/b0 = 0 in (d) are chosen. (e)-(f): the corresponding double neutron to protonratios ((n/p)ZrRu/(n/p)RuZr) for (a) and (d) cases (see text).6soft and stiﬀ symmetry energies, increase monotoneously with impact parameter. The FDvalue obtained with the stiﬀ symmetry energy is larger than that with the soft symmetryenergy, when b/b0 . 0.2. And they reverse when b/b0 & 0.2. The reversal of the relativemagnitudes of FD with stiﬀ or soft symmetry energy at small and large impact parametersis due to the gradual change of the mechanism of particle emission [22]. Central collisionsare more violent, hence the compression is stronger than for the collisions with large impactparameters. Correspondingly, the degree of isospin equilibration is both higher and reachedmore quickly. Nucleons emitted from central collisions stem mainly from the mid-rapidityregion, where matter is much more compressed than at the projectile/target region. Duringthe compression phase, the stiﬀ symmetry potential provides stronger repulsion (attraction)than the soft one for neutrons (protons) in the neutron-rich matter. Therefore, the centerof collisions is more strongly compressed. However, higher pressure caused by the stiﬀsymmetry potential leads to faster emission of particles. Hence, fewer two-body collisionsoccur and the equilibration decreases. For peripheral reaction, the average compression isrelatively weak, Nucleon emission takes place mainly at projectile/target rapidity, wherethe nuclear density is low. The low-density nuclear surface thus plays an important role fornucleon emission. At low densities, the stiﬀ symmetry potential provides weaker repulsion(attraction) on neutrons (protons) than the soft one in the neutron-rich nuclear medium.Higher densities result in the projectile/target region, leading to an enhanced number oftwo-body collisions. The smaller FD value is evident with the stiﬀ symmetry energy.The eﬀect of the symmetry energy on FD becomes the most pronounced at Eb ∼ 400 AMeV and at large impact parameters (Fig. 2). Compare the isospin eﬀects in Fig. 1 (b) andFig. 2 for reactions at Eb =400 A MeV and b/b0 = 0.5: The double neutron to proton ratioexhibits an enhanced sensitivity (∼ 30%) to the density dependence of the symmetry energyas compared to the single neutron to proton ratio (∼ 10%). Thus, the double neutron toproton ratio (taken from mass symmetric, but isospin asymmetric mixing systems at energiesaround 400 A MeV and at semi-peripheral impact parameters) is a prominent probe of thedensity dependence of the symmetry energy.In summary, we have investigated the rapidity dependence of the single- and doubleneutron to proton ratios as taken from isospin-asymmetric but mass-symmetric systemsZr+Ru and Ru+Zr. The beam energy- and the centrality dependence of the slope parametersof the double neutron to proton ratio (FD) as function of rapidity are presented. We ﬁnd70.00 0.25 0.50 0.75 1.000.00.10.20.30.40.5(b)FDb/b0  =0.5  =1.5Eb=400A MeV0 100 200 300 400 500 600 700 800 9000.000.050.100.150.200.25(a) FDEb (A MeV)                    b/b0 0.5       0.5 1.5       0.5 0.5       0 1.5       0FIG. 2: (Color online) (a): Beam energy dependence of the slope parameter FD with symmetryenergies with γ = 0.5 and 1.5, for impact parameters b/b0 = 0 and 0.5. The FD values at Eb = 400A MeV and at b/b0 = 0.5 with SM-EoS are shown with open dots. (see text) (b): The centralitydependence of FD at Eb = 400 A MeV.that both, the rapidity distribution of the single neutron to proton ratio and the FD aresensitive to the density dependence of symmetry energy. The study of the beam energy-and the centrality dependence of the FD values shows that the (semi-) peripheral HICs atenergies around Eb ≃ 400 A MeV are most suited for probing the density dependence of8the symmetry energy. Here the symmetry energy eﬀect is enhanced most strongly as isobservable in the rapidity dependence of the double neutron to proton ratio, as comparedto the single ratio.AcknowledgmentsQ. Li thanks the Alexander von Humboldt-Stiftung for a fellowship. This work is partlysupported by the National Natural Science Foundation of China under Grant No. 10235030and the Major State Basic Research Development Program of China under Contract No.G20000774, as well as by GSI, BMBF, DFG, and Volkswagenstiftung. We gratefully ac-knowledge support by the Frankfurt Center for Scientiﬁc Computing (CSC).[1] H. St¨ ocker and W. Greiner, Phys. Rept. 137 (1986) 277.[2] S. A. Bass et al., Prog. Part. Nucl. Phys. 41, 225 (1998)[3] F. Rami et al. [FOPI Collaboration], Phys. Rev. Lett. 84, 1120 (2000)[4] B. Hong et al., [FOPI Collaboration], Phys. Rev. C 66, 034901 (2002).[5] A. Hombach, W. Cassing and U. Mosel, Eur. Phys. J. A 5, 77 (1999)[6] Q. Li and Z. Li, Phys. Rev. C 64, 064612 (2001)[7] T. Gaitanos, M. Colonna, M. Di Toro and H. H. Wolter, Phys. Lett. B 595, 209 (2004)[8] B. A. Li, C. M. Ko and W. Bauer, Int. J. Mod. Phys. E 7, 147 (1998)[9] Isospin Physics in Heavy Ion Collisions at Intermediate Energies, edited by Bao-An Li andW. Udo Schroeder, NOVA Science Publishers, Inc., New York, 2001[10] V. Baran, M. Colonna, V. Greco, M. DiToro, Phys. Rep. 410, 235 (2005)[11] M. B. Tsang, et al., Phys. Rev. Lett. 92, 062701 (2004)[12] L. W. Chen, C. M. Ko and B. A. Li, Phys. Rev. Lett. 94, 032701 (2005)[13] B. A. Li and L. W. Chen, Phys. Rev. C 72, 064611 (2005)[14] B. A. Li, L. W. Chen, C. M. Ko and A. W. Steiner, arXiv:nucl-th/0601028.[15] B. A. Li, L. W. Chen, G. C. Yong and W. Zuo, Phys. Lett. B 634, 378 (2006)[16] G. C. Yong, B. A. Li, L. W. Chen and W. Zuo, Phys. Rev. C 73, 034603 (2006)[17] H. Weber, E. L. Bratkovskaya and H. St¨ ocker, Phys. Lett. B 545, 285 (2002)9[18] Q. Li, Z. Li, S. Soﬀ, R. K. Gupta, M. Bleicher and H. St¨ ocker, J. Phys. G: Nucl. Part. Phys.31, 1359 (2005)[19] Q. Li, Z. Li, S. Soﬀ, M. Bleicher and H. St¨ ocker, Phys. Rev. C 72, 034613 (2005)[20] Q. Li, Z. Li, S. Soﬀ, M. Bleicher and H. St¨ ocker, J. Phys. G: Nucl. Part. Phys. 32, 151 (2006)[21] Q. Li, Z. Li, S. Soﬀ, M. Bleicher and H. St¨ ocker, J. Phys. G 32, 407 (2006)[22] Y. Zhang and Z. Li, Phys. Rev. C 71, 024604 (2005).10

Probing the symmetry energy and the degree of isospin equilibrium

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