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PHYSICAL REVIEW C 67, 024608 ~2003!

Influence of a momentum dependent interaction on the isospin dependence of fragmentationand dissipation in intermediate energy heavy ion collisions

Jian-Ye Liu,1,2,3 Wen-Jun Guo,2 Yong Zhong Xing,2,4 Wei Zou,1,2 and Xi-Guo Lee1,2

1Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, People’s Republic of C2Institute of Modern Physics, Chinese Academy of Sciences, P.O. Box 31, Lanzhou 730000, People’s Republic of China

3Department of Chemistry and Physics, P.O. Box 419, Arkansas State University, State University, Arkansas 72467-04194Department of Physics, Tianshui Normal College, Gansu Tianshui 741000, People’s Republic of China

~Received 2 April 2002; published 27 February 2003!

We studied the influence of a momentum dependent interaction in the context of isospin effects on frag-mentation and dissipation in intermediate energy heavy ion collisions by using an isospin dependent quantummolecular dynamics model. It is shown that nuclear stopping, the number of nucleons emitted, and the mul-tiplicity of intermediate mass fragments are larger with a momentum dependent interaction than without. Inparticular, the differences for these observables, when using an isospin dependent in-medium nucleon-nucleoncross section versus an isospin independent one, are also larger at high energies for a momentum dependentinteraction than without one. Therefore, momentum dependence enhances the sensitivities of those observablesto the isospin effect of the in-medium nucleon-nucleon cross section towards high beam energies.

DOI: 10.1103/PhysRevC.67.024608 PACS number~s!: 25.70.Pq, 02.70.Ns, 24.10.Lx

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I. INTRODUCTION

The rapid progress in producing energetic radioactbeams has offered an excellent opportunity to investigvarious isospin effects in the dynamics of nuclear reacti@1–5#. This study has made it possible to obtain crucialformation about the equation of state of isospin asymmenuclear matter and about the isospin dependent in-mednucleon-nucleon cross section. This information is importfor understanding both novel properties of neutron-proton-rich nuclei as well as explosion mechanisms of supnovae and the cooling rates of protoneutron stars. Receseveral interesting isospin effects in heavy ion collisiowere explored both experimentally@6–17# and theoretically@18–32#. For a recent review, see Ref.@1#. However, twoessential ingredients in heavy ion collision dynamics,symmetry potential of the mean field and the isospin depdent in-medium nucleon-nucleon cross section, are stillwell determined. Moreover, the nonlocal property of tnuclear interaction leads to a repulsive momentum depeninteraction in the dynamics of intermediate energy heavycollisions. Even though the roles of momentum dependinteractions in heavy ion collisions have been studiedmany years@33–38#, the underlying influence of momentumdependence on isospin effects in intermediate energy heion collisions is poorly known.

Based on an isospin dependent quantum molecularnamics model we study in this work the underlying influenof a momentum dependent interaction~MDI ! on isospin ef-fects in intermediate energy heavy ion collisions. We fipronounced isospin effects in the nuclear stoppingR, themultiplicity of intermediate mass fragmentsNimf , and thenumber of nucleons emittedNn (Np) due mainly to the isos-pin dependent in-medium nucleon-nucleon cross sectiothe high energy region@39–41#. Furthermore, we investigatthe influence of momentum dependence on isospin effecfragmentation and dissipation with~MDI ! and without~no-

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MDI ! a momentum dependent interaction in the high eneregion. It is found that a momentum dependent interactenhances the sensitivities of nuclear stopping, the multipity of intermediate mass fragments, and the number of nuons emitted to the isospin effect on in-medium nucleonucleon cross sections.

II. THEORETICAL MODEL

A quantum molecular dynamics~QMD! model containstwo dynamical ingredients, the density dependent mean fiand the in-medium nucleon-nucleon cross section. In ordedescribe the isospin dependence appropriately, the Qmodel must be modified properly. The density dependmean field must contain correct isospin terms, includingsymmetry potential and the Coulomb potential. Themedium nucleon-nucleon cross section should be differfor neutron-neutron~or proton-proton! and neutron-protoncollisions, in which Pauli blocking should be included bdistinguishing between neutrons and protons. In addition,initial condition of the ground states of two colliding nuclmust also contain isospin effects.

Considering the above ingredients, we have made imptant modifications to QMD to obtain an isospin dependquantum molecular dynamics~IQMD! approach@3,35#. TheIQMD code consists of two parts, an initialization step athe calculation of the reaction dynamics itself. For the itialization, the code IQMD is used to determine the groustate properties of the colliding nuclei, such as binding engies and root-mean-square~rms! radii, so that they agreewith experimental data. The initial density distributionsthe colliding nuclei in IQMD are obtained from the SkyrmHartree-Fock model with parameter set SKM* @42#. The in-teraction potential is

U~r!5USky1Uc1Usym1UYuk1UMDI1UPauli, ~1!

whereUc is the Coulomb potential. The density dependeSkyrme potentialUSky, the Yukawa potentialUYuk, the mo-

©2003 The American Physical Society08-1

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LIU et al. PHYSICAL REVIEW C 67, 024608 ~2003!

TABLE I. The parameters of the interaction potential.

a b g t3 m t4 t5 Vp p0 q0

~MeV! ~MeV! ~MeV! ~fm! ~MeV! (MeV22) ~MeV! (MeV/c) ~fm!

MDI 2390.1 320.3 1.14 7.5 0.8 1.57 531024 30 400 5.64no-MDI 2356 303 1.1667 7.5 0.8 0.0 0.0 30 400 5.6

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mentum dependent interactionUMDI, and the Pauli potentiaUPauli @43,44# are given by the following equations:

USky5aS r

r0D1bS r

r0D g

, ~2!

UYuk5t3 expS urW12rW2um D Y urW12rW2u

m, ~3!

UMDI5t4 ln2@ t5~pW 12pW 2!211#r

r0, ~4!

and

UPauli5VpH S \

p0q0D 3

expS 2~rW i2rW j !

2

2q02

2~pW i2pW j !

2

2p02 D J dpi pj

,

~5!

with

dpi pj5H 1 for neutron-neutron or proton-proton

0 for neutron-proton.

We used the following four different forms of the symmetpotential@2,3#:

U1sym5cudtz , ~6!

U2sym5cu2dtz1

1

2cu2d2, ~7!

U3sym56cu1/2d2

1

4cu1/2d2, ~8!

U0sym50, ~9!

where

tz5H 1 for neutron

21 for proton,

andc is the strength of the symmetry potential chosen to32 MeV. In this work,u5r/r0 is the reduced density and5rn2rp /rn1rp5rn2rp /r is the relative neutron excess.r, r

0, rn , andrp are the total, normal, neutron, an

proton densities, respectively. To investigate the influencemomentum dependence on the fragmentation and dissipawe calculated both observables, with~MDI ! and without~no-MDI ! a momentum dependent interaction. The parameterthe interaction potentials are given in Table I.

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It is worth mentioning that recent studies of collectivflow in heavy ion collisions at intermediate energy havedicated a reduction of in-medium nucleon-nucleon crosections. An empirical, in-medium density dependenucleon-nucleon cross section@27# has been suggested afollows:

sNNmed5S 11a

r

r0DsNN

free, ~10!

wheresNNfree is the experimental nucleon-nucleon cross s

tion @45#. The above expression witha'20.2 has beenfound to reproduce well the flow data. The free neutroproton cross section is about a factor of 3 larger than theproton-proton or the free neutron-neutron cross sectionlow 400 MeV, which contributes the main isospin effect fronucleon-nucleon collisions in heavy ion collisions. In fathe ratio of neutron-proton cross section to proton-proton~orneutron-neutron! cross section in the medium,snp /spp , de-pends sensitively on the evolution of the nuclear density dtribution and the beam energy. We used Eq.~10! to take intoaccount medium effects, in which the neutron-proton crsection is always larger than the neutron-neutron or protproton cross section at the beam energies considered inpaper. We constructed the clusters by means of a modcoalescence model@35#, in which coalescence occurs whethe relative particle momentum is smaller thanp05300 MeV/c and the relative distance is smaller thanR053.5 fm. The restructured aggregation model@46# has beenapplied to avoid nonphysical clusters. The main physicsgredients and their numerical realization in the IQMD modcan be found in Refs.@3,29,33–35#.

III. RESULTS AND DISCUSSIONS

The isospin effect of the in-medium nucleon-nuclecross section on the observables is defined by the differebetween the observable for an isospin dependent nuclnucleon cross sections iso and that for an isospin independenucleon-nucleon cross sectionsno-iso in the medium. Heres iso means snp>snn5spp and sno-iso means snp5snn5spp , wheresnp , snn , and spp are the neutron-protonneutron-neutron, and proton-proton in-medium cross stions, respectively.

To study the influence played by the MDI on isospeffects of the fragmentation and dissipation in intermedienergy heavy ion collisions we investigated the time evotions of the following variations on the nuclear stoppingR,the multiplicity of intermediate mass fragmentNimf , andthe number of nucleon emissionsNn(Np) due to isospin

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INFLUENCE OF A MOMENTUM DEPENDENT . . . PHYSICAL REVIEW C 67, 024608 ~2003!

FIG. 1. Time evolution of the variation of nuclear stoppingDR due to the isospin effect from the in-medium nucleon-nucleon crsection for head-on collisions of76Kr176Kr ~top panel! and 76Zn176Zn ~bottom panel! at beam energies of 50, 150, and 300 MeV/nuclewith ~MDI, solid lines! and without~no-MDI, dotted lines! a momentum dependent interaction.

tiot ing

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effects from the in-medium nucleon-nucleon cross secwith ~MDI ! and without~no-MDI! a momentum dependeninteraction. They are indicated byDR(Ds,U2

sym),DNimf(Ds,U1

sym), DNn(Ds,U1sym), and DNp(Ds,U1

sym),respectively. It is more quantitative to define

DRMDI ~no-MDI!~Ds,U2sym!

[RMDI ~no-MDI!~s iso,U2sym!

2RMDI ~no-MDI!~sno-iso,U2sym!, ~11!

DNimfMDI ~no-MDI!~Ds,U1

sym!

[NimfMDI ~no-MDI!~s iso,U1

sym!

2NimfMDI ~no-MDI!~sno-iso,U1

sym! , ~12!

DNnMDI ~no-MDI!~Ds,U1

sym!

[NnMDI ~no-MDI!~s iso,U1

sym!

2NnMDI ~no-MDI!~sno-iso,U1

sym!, ~13!

DNpMDI ~no-MDI!~Ds,U1

sym!

[NpMDI ~no-MDI!~s iso,U1

sym!

2NpMDI ~no-MDI!~sno-iso,U1

sym!. ~14!

02460

nU2sym in Eq. ~11! andU1

sym in Eqs.~12!, ~13!, and~14! are thesymmetry potentials used for calculating the correspondvariations.

To describe the nuclear stopping in heavy ion collisiowe used the transverse-parallel ratio of the momentumfined byR5(2/p)(( i

AuP'( i )u)/(( iAuPi( i )u), where the total

massA is the sum of the projectile massAp and the targetmassAt . The transverse and the parallel components ofmomentum of thei th nucleon areP'( i )5APx( i )

21Py( i )2

andP//( i )5Pz( i ), respectively. The multiplicity of the intermediate mass fragments is defined as the number of fments with charge numbers from 4 to 18.

Figure 1 shows the time evolution of the variatioDRMDI(Ds,U2

sym) ~solid lines! and DRno-MDI(Ds,U2sym)

~dotted lines! for systems76Kr176Kr ~top panels! and 76Zn176Zn ~bottom panels! at E550 MeV/nucleon~left panels!,150 MeV/nucleon~middle panels!, and 300 MeV/nucleon~right panels! andb50.0 fm. From Fig. 1 it is clear that alof the solid lines are always higher than the dotted lines,the MDI enhances the sensitivity of the nuclear stoppingR tothe isospin effect of the in-medium nucleon-nucleon crosection.

We also show the time evolution of the variationsDNimfdue to the isospin dependence of the in-medium nuclenucleon cross section with~MDI ! and without ~no-MDI!momentum dependent interaction in Fig. 2 for the syste58Fe158Fe ~left panel! at E5100 MeV/nucleon andb54.0 fm as well as for 112Sn140Ca ~right panel! at E

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with

LIU et al. PHYSICAL REVIEW C 67, 024608 ~2003!

FIG. 2. The time evolution of the variation ofDNimf due to the isospin effect from the in-medium nucleon-nucleon cross section~MDI, solid lines! and without~no-MDI, dotted lines! a momentum dependent interaction for the colliding systems58Fe158Fe ~left panel!at E5100 MeV/nucleon,b54.0 fm, and112Sn140Ca ~right panel! at E580 MeV/nucleon,b50.0 fm.

g

s

s

pinion

. 3

ne

580 MeV/nucleon andb50.0 fm. All DNimfMDI(Ds,U1

sym)~solid lines! are higher than the correspondinDNimf

no-MDI(Ds,U1sym) ~dotted lines! for the two systems. This

means that the momentum dependence also enhancesensitivity of Nimf to the isospin effect from the in-mediumnucleon-nucleon cross section.

In Fig. 3 we show the time evolution of the variation

02460

the

DNn of the number of emitted neutrons due to the isoseffect from the in-medium nucleon-nucleon cross sectwith ~MDI, solid lines! and without~no-MDI, dotted lines! amomentum dependent interaction. The left part of Figshows the results for the system76Kr140Ca at E5150 MeV/nucleon andb54 fm. The right part shows thecorresponding results for proton emission. From Fig. 3 o

n

FIG. 3. The time evolution of the variation of the number of emitted neutronsDNn

MDI ~no-MDI!(Ds, U1sym) due to the isospin effect from

the in-medium nucleon-nucleon cross section with~MDI, solid line! and without~no-MDI, dotted line! a momentum dependent interactio~left panel!. The right panel is the same as the left panel but for emitted protons atE5150 MeV/nucleon andb54 fm for the system76Kr140Ca.

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INFLUENCE OF A MOMENTUM DEPENDENT . . . PHYSICAL REVIEW C 67, 024608 ~2003!

can see that MDI enhances the sensitivities ofNp andNn tothe isospin effect from the in-medium nucleon-nucleon crsection similar toNimf andR.

To summarize, it is clear that a momentum dependinteraction enhances the sensitivities ofR, Nimf , Np , andNnto the isospin effect from the in-mediumNN cross section. Amomentum dependent interaction deflects the nucleons intransverse direction during the initial phase of the reactwhen two nuclei with large relative momentum penetreach other. This produces a larger momentum dampingthus leads to more dissipation and fragmentation duringreactions, than when the interaction does not depend onmomentum. Therefore, with a momentum dependent intetion the observablesR, Nimf , andNp(Nn) are more sensitiveto the isospin effect of the in-medium nucleon-nucleon crsection in high energy heavy ion collisions.

IV. SUMMARY AND CONCLUSIONS

We have studied the influence of a momentum dependinteraction on the isospin effects of fragmentation and dipation from mainly the in-medium nucleon-nucleon crosection by using the IQMD model. The results show that

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nuclear stopping, the multiplicity of intermediate mass framents, and the number of emitted nucleons with the momtum dependent interaction are always larger than those wout the momentum dependent interaction in the eneregion studied here. In particular, the variations due toisospin effect from the in-medium nucleon-nucleon crosection with the momentum dependent interaction are grethan those without the momentum dependent interactTherefore, the momentum dependent interaction enhanthe sensitivities of these observables to the isospin effrom the in-medium nucleon-nucleon cross sections in heion collisions towards high beam energies.

ACKNOWLEDGMENTS

We thank Professor Bao-An Li for helpful discussionThis work was supported by the Major State Basic ReseaDevelopment Program in China Under Contract NG2000077400, ‘‘One Hundred Person Project’’ of the Cnese Academy of Sciences, the National Natural ScienFoundation of China under Grant Nos. 10175080, 10004010175082, and The CAS Knowledge Innovation Project NKJCX2-SW-N02.

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