b

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a Skyrmestability

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ew

Physics Letters B 580 (2004) 137–143

www.elsevier.com/locate/physlet

Exploring proton rich systems and Coulomb induced instabilit

S.J. Leea, A.Z. Mekjianb

a Department of Physics and Institute of Natural Sciences, Kyung Hee University, Suwon, KyungGiDo, South Koreab Department of Physics, Rutgers University, Piscataway, NJ 08854, USA

Received 11 August 2003; received in revised form 29 October 2003; accepted 18 November 2003

Editor: W. Haxton

Abstract

The thermodynamic properties of proton rich systems are explored in a mean field approach which is generated frominteraction. The addition of Coulomb interactions result in asymmetries which modify the chemical and mechanical inof the system and its equilibrium properties. These properties are studied for systems with proton fractiony on the protonricher side of the valley ofβ-stability as well as the neutron rich side. Coulomb induced instabilities lead to proton diffprocesses on the proton richer side and also large asymmetries in chemical and mechanical instabilities and coexisteConsidering the whole range of 0� y � 1, we can study how the symmetry abouty = 1/2 is broken by asymmetric interactioand we can also explicitly show that the role between proton and neutron is exchanged aroundyE which is the point wherethe liquid and gas have the same proton fraction. It is shown that there are two asymmetric coexistence surfaces in(y,P,T )

space, one fory < yE and another fory > yE and touching each other atyE . These asymmetries in instabilities show up as nbranches, one fory < yE and one fory > yE , and thus form a closed loop in pressure versusρ for both chemical instabilityand coexistence regions. The branch ofy > 1/2> yE was not previously investigated since only they < 1/2 region is usuallyconsidered. In our simplified model, mechanical instability is still symmetric around a pointyE �= 1/2 even with Coulombforces present. 2003 Elsevier B.V. All rights reserved.

m-in-eri-IA)ofesnd-tarsrestich

avyisothfordy-thatnd

nu-me,mbtures ofand

Understanding properties of nuclei with large asymetry in the proton–neutron ratio had a renewedterest for several reasons. For example, future expments to be done with rare isotope accelerators (Rcan explore properties of nuclei far from the valleyβ-stability on both the neutron and proton rich sidof it. These studies would be useful for understaing the production of the elements and neutron sin the area of nuclear astrophysics. Current intestems from the liquid–gas phase transition [1] wh

E-mail address: sjlee@physsun7.rutgers.edu (S.J. Lee).

0370-2693/$ – see front matter 2003 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2003.11.049

is experimentally studied using medium energy heion reactions. A primary goal of heavy ion reactionsto explore the phase diagram of hadronic matter bat low and high density and temperature lookingsignals of phase transitions. Another reason for stuing asymmetric two component nuclear systems isthey offer a unique example of the thermodynamic astatistical properties of binary systems. Here, theclear phase diagram is governed by nuclear volusurface, symmetry energy and asymmetric Couloeffects. The interplay of these on the phase struchas been of interest in past studies [2,3]. Exampleother two component systems are binary alloys

.

138 S.J. Lee, A.Z. Mekjian / Physics Letters B 580 (2004) 137–143

upuc-low

[2,thege

ts,

rm.

issnce

in

d to

tieseralms

henthen.off theicaltryd byeont anlsondofbeinnd

-tion,me-gyre-

isis

be

.in-nu-heof

ten-

ric

Oned-testion

or-Oneform

pro-tionredrgeTo

atly.ace

byerrom

rgyeak

in-ityforli-

he

liquid 3He. For3He the two components are spin-and spin-down fluids and important new phase strture appears within the superfluid phase itself be3 mK.

In this Letter we extended our previous study3] of phase structure of nuclear systems belowliquid–gas critical point to include the entire ranof the proton fractiony = Z/(N + Z) from 0 to1. Our previous study focused on 0� y � 1/2.For systems with no asymmetric Coulomb effeca symmetry exists about they = 1/2 point. Thissymmetry is broken by the asymmetric Coulomb teDue to this symmetry, usually only they < 1/2 sidewas considered and thus such studies would mvarious symmetry breaking features and the existeof another pair of coexistence points at largey. Somepreliminary initial new findings were presentedRefs. [2,3] in a restricted region ofy � 1/2 whichare further elaborated on here and also extendethe whole range of 0� y � 1. In particular we willfocus on the proton rich side of the valley ofβ-stabilityand study the chemical and mechanical instabiliand equilibrium properties of these systems. Sevprevious mean field studies of two component systewere carried out without the Coulomb force. Tpioneering work of Müller and Serot [4] focused othe importance of the nuclear symmetry energy onequilibrium structure of liquid–gas phase transitioBaran et al. [5] investigated instability propertiessuch systems and pointed some new properties oliquid–gas phase transition associated with cheminstability and isospin distillation based on symmeenergy considerations. Calculations were extendeColonna et al. [6] to finite nuclei including surfacand Coulomb effects. Their calculation focusedthe instability modes and their results showed thaoctupolar mode dominates. Pawlowski [7] has arecently considered the importance of Coulomb asurface effects on the equilibrium phase structurethe liquid–gas phase transition and found them toimportant. An extensive summary of isospin effectsheavy-ion collisions at medium energy can be fouin the book of Li and Schröder [8].

In our study we will use a simplified Skyrme interaction to generate the nuclear long range attracshort range repulsion and also the nuclear symtry energy. We include a simplified Coulomb enerwhich generates a proton–neutron asymmetry. The

sulting interaction energy density is

(1)

U(ρ)= −a0ρ2 + a3ρ

3 + aS(2y − 1)2ρ2 +Cy2ρ2.

The a0 term is the long range attraction whichquadratic in the densityρ. Shorter range repulsionthea3 term and we take its density dependence toρ3. The nuclear symmetry energy is theaS(2y − 1)2

term withaS = 23(

12 + x0)a0 and has aρ2 dependence

Since the Coulomb energy density is infinite forfinite charged system, we need to consider finiteclear system to consider a Coulomb interaction. Tsimplest finite nuclear system is a uniform sphereradiusR and its sharp surface introduces a surfacesion which may be approximated asCsρ2/3 with Cs =4πr2

0σ/V1/3 as used in Refs. [2,3]. The asymmet

Coulomb term isCy2ρ2. TheC, which in principledepends on nuclear sizeR asC = (4π/5)e2R2 for auniformly charged sphere, is taken to be constant.can correct this with a Wigner–Seitz term by incluing a uniform background charge that approximathe presence of other charges. This approximachanges the radial dependence ofC, but it is not clearto us that the Wigner–Seitz procedure for nuclei is crect since there is no negative background charge.can also use as an ansatz a somewhat differentwhereC is taken to be independent ofR. This leads toa Coulomb force that depends on the square of theton density. It should be noted that an exact calculaof the true Coulomb force for the situation considerequires detailed knowledge of the position and chadistribution of one cluster with every other cluster.our knowledge no one has solved this problem.

For real nuclei the surface energy comes fromdiffused surface which is formed self consistenPawlowski [7] has considered a more detailed surfenergy where the diffused surface is expressedWigner function with the finite thickness. Our simplexpression replaces the surface energy coming fthe diffused surface by a surface tensionCsρ2/3 fora sharp surface. This simple form of surface eneis independent of isospin and thus it does not brisospin symmetry.

Our choice for the Coulomb energy and nuclearteraction energy greatly simplifies the full complexof the analysis and results in analytical solutionsinstability boundaries. By using a somewhat simpfied interaction, we hope to qualitatively illustrate t

S.J. Lee, A.Z. Mekjian / Physics Letters B 580 (2004) 137–143 139

tran-oththatinted

ndto

it

ndal

gnthe

andois a

n,

e

theat a

nd

stint

ate

lndbe

ular

icaltials

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tricthe

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main physical phenomena associated with phasesitions in two component systems which include bsymmetry and asymmetric terms. We also believethe main results will not change qualitatively, norany major quantitative way when more complicadependences are included.

A kinetic energy density is added toU(ρ). Sincewe will present results for moderate density atemperatures of the order of 10 MeV, we will workfirst-order in the degeneracy corrections. In this limthe pressure is

P = ρT −(a0 − T

2√

2

λ3

4γ

)ρ2 + 2a3ρ

3

+(

2

3

(1

2+ x0

)a0 + T

2√

2

λ3

4γ

)(2y − 1)2ρ2

(2)+Cy2ρ2.

The a0 = −(3/8)t0 and a3 = (1/16)t3 with t0 =−1089 MeV fm3 andt3 = 17 480.4 MeV fm6 as shownin Ref. [2]. Theγ = 2 is the spin degeneracy factor aλ = h/(2πmT )1/2. The proton and neutron chemicpotentials are

µq = T ln[(λ3/γ

)(1± (2y − 1)

)ρ/2

]− 2a0ρ + 3a3ρ

2

± (4/3)(1/2+ x0)a0(2y − 1)ρ + (1± 1)Cyρ

(3)+ (T /2√2)

(λ3/γ

)[1± (2y − 1)

]ρ/2,

with the upper sign for protons and the lower sifor neutrons. These three equations determineequilibrium phase structure and the mechanicalchemical instability regions of our simplified twcomponent nuclear system. The phase structurethree-dimensional surface iny, T andP . At y = 1/2,for symmetric systems with no Coulomb interactioone has the familiarP , T curve of Maxwell pressureversusT for equilibrium state which ends at thcritical point. For fixedT , the equilibrium surfaceintersects the fixedT plane to form loops inP versusy. These loops represent the proton fraction inliquid and gas phases which are different exceptpoint calledyE which is not 1/2 for the case withCoulomb interaction. These equilibrium loops eat the critical points(yc(T ),Pc(T )) at fixed T or(Pc(y), Tc(y)) at fixed y having the same densitieboth in liquid and gas phases. The point with lowesyvalue on this loop is the maximally asymmetric po

at a fixedT . Another maximal asymmetric point isy > yE with largesty value. The equilibrium surfacis generated from the chemical potentialsµq(y,T ,P )by requiring at fixedP andT the neutron chemicapotential in liquid and gas phase to be equal aproton chemical potential in these two phases toequal at the same two values ofy. Geometrically,this corresponds to the construction of a rectangbox in a plot ofµ versusy at fixedP andT wherethe horizontal lines are the same neutron chempotentials and the same proton chemical potenand the vertical lines are the two values ofy whereeach chemical potential in liquid and gas phasesequal. The gas phase will be more asymmetrichave lowery for y < yE and largery for y > yE thanthe liquid phase. The liquid phase is more symmebecause of the symmetry energy term and hasvalue of y closer toyE . Figs. 1 and 2 show variouproperties associated with the equilibrium surfaand behavior of the chemical potentials.

The point where the proton fraction is equalboth gas and liquid phase can be obtained fromcondition∂P/∂y = 0 and this point is given by

(4)yE = 1

2

(1+ C/4[2

3

(12 + x0

)a0 + T

2√

2λ3

4γ

])−1

.

An important feature of the Coulomb force isshift yE away from its value of 1/2 whenC = 0to a value closer to the valley ofβ-stability foundfrom d(E/A)/dρ = P = 0 at zeroT . This shift isimportant for understanding the phenomena of isosfractionization which favors more symmetric liquidand less symmetric gases.

Fig. 1 shows the effect of various terms on theexistence curve obtained from the intersection ofcoexistence surface with a fixedT = 10 MeV plane.Here we have used the same parameters as usRefs. [2,3] except for using a more realistic smasymmetry energy(x0 = −1/6). The curve has twosymmetric loops which are connected with minimuP aty = 1/2 when Coulomb forces are turned off athe loops are larger for a large symmetry energy (dted line with x0 = 1/2 and a corresponding nucleinteraction symmetry energy of 39 MeV(N −Z)2/A)compared to smaller symmetry energy (dashedwith x0 = −1/6 and nuclear interaction symmetenergy of 13 MeV(N − Z)2/A). It should also be

140 S.J. Lee, A.Z. Mekjian / Physics Letters B 580 (2004) 137–143

tcerms.no

ombetrytencemefrom

eVesesesen-

eineortermline.ce

es-

en-

beed.aceot-

Thehe

ando-ets

ric-ithut

cer-ilenttwogasout-

tedero-

reent.for

for

sta-es,as-

Fig. 1. PressureP versus proton fractiony for coexistence loop aT = 10 MeV. The solid line is for the case with Coulomb and surfaeffects. The dash-dotted line has surface, but no Coulomb teCoulomb effects are shown in the dash-dot-dot-dotted line withsurface terms. The dashed line is for the case without Couland surface terms while the dotted line is for a larger symmenergy. The regions of liquid and gas phases and the coexisare indicated explicitly for the dotted curve. (We believe that sozig zag behavior appearing on curves at small pressure comesa numerical error even though we were not able to remove it.)

noted that a kinetic symmetry energy term of 12 M(N−Z)2/A also arises from Fermi energy differencbetween protons and neutrons. A comparison of thtwo curves show that the symmetry energy has estially no effect on the minimum point aty = 1/2 (sym-metry energy is zero at thisy) while the maximumpoints (which are not aty = 1/2) has larger pressurfor a larger symmetry energy. The dash-dotted lwith minimum aty = 1/2 includes a surface energysurface tension term besides a symmetry energywhose value is the same as that for the dashedBy comparing these two loops, the effect of a surfaterm is to shift the coexistence surface to lower prsure. The loops are still symmetric aroundy = 1/2and slightly compressed by the droplet’s surface t

sion. The shifting by the surface term can alsoseen from other curves with Coulomb terms includThe solid curve contains the symmetry energy, surfenergy, and Coulomb energy while the dash-dot-ddotted curve has symmetry and Coulomb effects.main effects of the Coulomb term are: (1) to shift tintersection point of the two loops toyE = 0.41 forthe parameters considered and to higher pressure(2) to make the loops very asymmetric with the cexistence loop fory < yE becoming smaller and thcoexistence loop fory > yE becoming larger. Mosof the loop fory > yE was not studied and thus wanot closed in previous studies [2,3] due to the resttion of y � 1/2. The existence of coexistence pair wmore proton in gas than in liquid was first pointed oin Ref. [2] but in a small window of pressure sinwe restrictedy � 1/2. For each case, two loops intesect atyE which occurs at the lowest pressure whthe maximum pressure for each loop has a differeyvalue. The liquid and gas coexist inside of theseloops since the boundaries are the pure liquid andphases. The liquid and gas phases are separatedside of these loops; liquid phase fory value betweentwo maxima and gas phase fory value outside the twomaxima. These are indicated explicitly for the dotcurve. A system withy < yE inside the coexistencregion is composed of a proton richer liquid and a pton deficit gas compared to the originaly, while a sys-tem with y > yE in the coexistence region has moprotons in the gas phase than in the liquid componThe Coulomb term shrinks the coexistence regiony < yE while it enlarges the coexistence regiony > yE (Fig. 1).

The boundaries for mechanical and chemical inbility are easy to generate. Of these two instabilitithe region of mechanical instability is somewhat eier to obtain and follows from(dP/dρ)y,T = 0 whichleads to a quadratic equation,aρ2 + 2bρ + c = 0. Us-ing Eq. (2), this condition gives

ρ± = −b± √b2 − aca

,

a = 6a3,

b= −(a0 − d

4

)+

(2

3

(1

2+ x0

)a0 + d

4

)(2y − 1)2

+Cy2,

(5)c= T ,

S.J. Lee, A.Z. Mekjian / Physics Letters B 580 (2004) 137–143 141

ratems

veopde

. 2.b--

-ntom-nge

Theines-

ta-

ion,so-

micales

hasillndt

in

s ofl

rate

r

where d = T

2√

2λ3

γ. The presence ofd represents a

first-order quantal correction to the non-degenelimit. The Coulomb term and symmetry energy terappear inb. The critical point (yc(T ),Pc(T )) or(Pc(y), Tc(y)) occurs whenb2 = ac with the criticaldensityρc = −b/a.

We note that the coefficientb can be rewritten ina form involving b = d1(y − yE)

2 + d2 with d1 =(8

3(12 + x0)a0 + d) + C andd2 = C

4 /(1 + C/[83(

12 +

x0)a0 + d])− (a0 − d4 ). Thusb is symmetric around

yE and sincey does not appear ina or c, the solutionρ± as a function ofy is also symmetric aroundyE .Therefore, the mechanical instability curve ofP ver-susρ will have a symmetry that results in one curinstead of a loop. Also the mechanical instability loin P versusy andy versusρ are symmetric arounyE . This fact is not true for chemical instability and thcoexistence curve. These facts will be shown in Fig

The boundary of chemical instability can be otained from( dµq

dy)P,T = 0. These conditions for pro

ton and neutron give the same relation sincey dµp +(1 − y) dµn = (1/ρ) dP . For two component system with a fixed total fraction, if one componechanges from one phase to other then the other cponent must compensate this concentration chasince they are not independent of each other.chemical instability condition can be rewrittenterms of derivatives of chemical potential and prsure with respect to the variablesρ andy which ap-pear in Eqs. (2) and (3). Namely, the chemical insbility boundary can be obtained from(dP/dρ)y,T =(dP/dy)ρ,T (dµq/dρ)y,T /(dµq/dy)ρ,T . Then usingthe equation of state and chemical potential equatthe chemical instability boundary is determined bylutions to a cubic equation

c3ρ3 + c2ρ2 + c1ρ + c0 = 0,

c3 =[32

(1

2+ x0

)a0 + 12C + 12d

]a3y(1− y),

c2 = 6a3T −[

32

3

(1

2+ x0

)a2

0 + 8

3(1− x0)a0C − d2

+ 32

3(1− x0)a0d − 2dC

]y(1− y),

c1 =[−4

3(1− x0)a0 + 2Cy + d

]T ,

(6)c0 = T 2.

Fig. 2. Figures show the coexistence curves (dashed line), cheinstability curves (thick solid line) and mechanical instability curv(thin solid line). Also shown are the∂µq/∂ρ = 0 curves for proton(dotted line) and for neutron (dash-dotted line) atT = 10 MeV. TheCoulomb and surface terms are included here.

We note that the term asymmetric iny ↔ (1 − y)is the coefficientc1, while c2 and c3 are invariantunder this interchange. The solution to the cubicthree roots. For a physical situation, two roots wbe positive and physical and the third negative aunphysical. Sincec0 andc3 are positive, the producof the 3 roots is negative. For eachy, the solutionto the cubic generates a chemical instability curveρ andT and a three-dimensional surface inρ, y, T .Each point on this surface has corresponding valueP andµq . Cutting the(ρ, y,T ) surface of chemicainstability with a fixedT plane givesρ, y curve forthe chemical instability and on this curveP andµqare specified. Then using this fact, we can geneµq versusy, P versusρ, and P versusy curvesfor the chemical instability shown in Fig. 2. Simila

142 S.J. Lee, A.Z. Mekjian / Physics Letters B 580 (2004) 137–143

n be

trve.ve,ryhe

thesrve

ne

calnt

d

lchase,

icesareheve

tged

ealsourectsws.

easeop,

nhe

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ionefea-e ah

ame

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l-ens.

es

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mechanical instability and coexistence surfaces cagenerated. These results are also shown in Fig. 2.

Fig. 2 contains plots ofy versusρ, the loops ofµpandµn versusy, curves ofP versusρ and curves ofP versusy. Interesting points occur atdρ/dy = 0 onthese curves which correspond to the cusp points ayEin P versusρ which have two different branches fothe coexistence curve and chemical instability curThe mechanical instability region has only one curwhich starts and ends atyE , because of the symmetof it aroundyE . The upper curve corresponds to tproton richer side with respect toyE and the lowerbranch is for the neutron richer side ofyE . These twocurves form a closed loop in a constantT plane. Theupper branch did not appear in Ref. [3] due torestriction of y � 1/2. When the Coulomb force iturned off these two branches in the coexistence cuand chemical instability curve would collapse into ocurve also due to the symmetry aroundy = 1/2. Themaximal asymmetry points correspond tody/dρ = 0,one at lowy and one at highy. In they versusρ curve,the mechanical instability loop is inside the chemiinstability loop except at two points where it is tangeto it. The tangent is atyE . The chemical instabilityloop in y versusρ is inside the coexistence loop antangent to it at two points, one aty < yE and one aty > yE . These values ofy correspond to the criticayc(T ), points with maximum pressure of each branhaving the same densities both in liquid and gas phat a fixed temperature, 10 MeV for Fig. 2. ForPversusy the mechanical instability loop is symmetraboutyE , while the chemical and coexistence curvare not when Coulomb forces are present, butsymmetric when Coulomb forces are turned off. TCoulomb instability leads to proton diffusion aboyE and neutron diffusion belowyE . The chemicalpotentialµq and the∂µq/∂ρ = 0 curves show thathe role between proton and neutron is exchanaroundyE . In Ref. [3], the∂µp/∂ρ = 0 curve had twodisconnected portions due to the cut aty = 1/2 andwas quite different from the∂µn/dρ = 0 curve.

Similar calculations as shown in Fig. 2 for thcases without Coulomb and/or surface terms canbe done even though we did not included the figin this Letter to keep the figure simple. The effeof each term when treated separately are as folloThe surface energy brings the pressure down, incrthe chemical potential of the upper part of each lo

and slightly enlarges the loops inρ–y plot. TheCoulomb term breaks they ↔ (1 − y) symmetry andshrink theρ–y plot to the lower density and protoconcentration while increasing the lower part of tchemical potential and pressure curves.

In summary, in this Letter we investigated proerties of the coexistence region, and mechanicalchemical instability regions when asymmetric Colomb forces are also included besides volume, surfand symmetry energy terms in a mean field desction of hot nuclear matter for the whole range of pton fraction 0� y � 1 (only the range of 0� y � 1/2was considered previously for the coexistence reg[2,3] and for the instability region [3]). Including thCoulomb interactions leads to a changes in manytures that are physically important. These includshift in yE (the value of the proton fraction at whicthe liquid and gas phases would coexist at the sproton fraction) fromy = 1/2 to a value ofy closerto the valley ofβ-stability. The shifting ofyE from1/2 to a lower value by the addition of the Coulomforce is an important effect for isospin fractionizatior distillation [9]. An investigation of the chemical instability properties shows that proton diffusion woccur above the value ofyE while neutron diffusionoccurs below it. By extending our calculation to vaues ofy > 1/2> yE we were also able to show somnew features of the coexistence and instability regioFirst, the constantT cuts of the corresponding surfacnow form closed loops while the constantT cuts of∂µq/∂ρ = 0 surfaces form continuous curves. Preously, the proton curve∂µp/∂ρ = 0 had two discon-nected pieces due to the restrictiony � 1/2 [3]. Themain behavior of the temperature dependence of thsurfaces remain the same as in Ref. [3] with the rof proton and neutron exchanged in going fromy < yE region to they > yE region. The mechanical instability region inP versusρ or P versusy issymmetric around the shiftedyE when Coulomb in-teractions are included in our model. This symmedoes not occur for the coexistence curve and chemequilibrium curves. Rather, these have new brancwhich appear inP versusρ and the two branches cusat the twoρ ’s which havey = yE . The upper branchcorresponds to the proton rich side ofyE , or y > yE ,while the lower branch corresponds to the neutronside ofyE , or y < yE . The coexistence curve now htwo asymmetric loops inP versusy that meet at the

S.J. Lee, A.Z. Mekjian / Physics Letters B 580 (2004) 137–143 143

e

ton-nsm-temef-suruid.

nto.

ro-da-

le,er

v.

li-unt-

shiftedyE . The upper branch of the instability curvwith y > 1/2 and coexistence loops aty > yE weremissing in previous studies. For systems withy > yE ,the gas phase can become proton richer from prodiffusion induced by a Coulomb instability of the liquid phase. Fory < yE , the gas phase will be neutroricher from neutron diffusion. This Letter also showhow the Coulomb interaction breaks some of the symetries associated with isospin in a hot nuclear sysand its associated liquid–gas phase transition. Thefects of a surface energy terms are discussed. Theface tension associated with the presence of a liqdrop shifts the coexistence curve to lower pressure

Acknowledgements

This work was supported in part by the DOE GraNo. DE-FG02-96ER-40987 and in part by Grant N

-

R05-2001-000-00097-0 from the Basic Research Pgram of the Korea Science and Engineering Fountion.

References

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[2] S.J. Lee, A.Z. Mekjian, Phys. Rev. C 63 (2001) 044605.[3] S.J. Lee, A.Z. Mekjian, Phys. Rev. C 68 (2003) 014608.[4] H. Müller, B.D. Serot, Phys. Rev. C 52 (1995) 2072.[5] V. Baran, M. Colonna, M. Di Toro, V. Greco, Phys. Re

Lett. 86 (2001) 4492.[6] M. Colonna, et al., Phys. Rev. Lett. 88 (2002) 122701.[7] P. Pawlowski, Phys. Rev. C 65 (2002) 044615.[8] B.-A. Li, W.U. Schröder, Isospin Physics in Heavy-Ion Col

sions at Intermediate Energies, Nova Science Publishers, Hington, NY, 2001.

[9] H.S. Xu, et al., Phys. Rev. Lett. 85 (2000) 716.