Isospin Dependence of Fragmentation

Isospin Dependence of Fragmentation

Constraining Constraining the the properties properties of of dense dense matter matter William Lynch, Michigan State University A. A. What Whatisisthe theEOS EOS 1.1.Theoretical Theoreticalapproaches approaches 2.2.Example:T=0 Example:T=0with withSkyrme Skyrme 3.3.Present Presentstatus status a)a)symmetric symmetricmatter matter b)b)asymmetric asymmetricmatter matterand andsymmetry symmetryterm. term. 4.4.Astrophysical Astrophysicalrelevance relevance B. B.Summary Summaryofoffirst firstlecture lecture C. C.What Whatobservables observablesare aresensitive sensitivetotothe theEOS EOSand andatatwhat whatdensities? densities? 1.1.Binding energies Binding energies 2.2.Radii Radiiofofneutron neutronand andproton protonmatter matterininnuclei nuclei 3.3.Giant Giantresonances resonances 4.4.Particle Particleflow flowand andparticle particleproduction: production:symmetric symmetricEOS EOS 5.5.Particle Particleflow flowand andparticle particleproduction: production:symmetry symmetryenergy energy DDSummary Summary T he EO S: F in ite te m p e ra tu re : T h e E O S d e s c rib e s h o w th e p re s s u re P d e p e n d s o n th e te m p e ra tu re T , th e d e n s ity a n d th e a s y m m e try = ( n- p)/ :

P P ( ,T , ) H e re , d e s ig n a te s th e n u m b e r d e n s ity o f n u c le o n s ( n u c l e o n s / f m 3 ) f o r n u c l e a r m a t t e r a t s a t u r a t i o n d e n s i t y . O n e c a n u s e = E /A , th e a v e ra g e e n e rg y p e r p a rtic le in th e s y s te m , to c a lc u la te P : F P P ( ,T , ) V where f ( , T , ) F/A T , 2 f ( ,T , ) (1 ) T , ( , T , ) T ( , T , ). H e re f is th e h e lm h o ltz fre e e n e rg y p e r n u c le o n a n d is th e e n tro p y p e r n u c le o n . c a n b e o b ta in e d fro m : where 0 T d V , T c V T 0 T c v dT , T i s t he V , heat capacity per nucleon . (2 ) H o m e w o rk 1 2T /(2 F( )) a s s u m in g a ll k in e tic e n e rg : O b t a i n t h e a p p r o x i m a t e e x p r e s s i o n ) fo r th e fe rm io n ic n u c le a r s y s te m a t d e n s ity , te m p e ra tu re d e p e n d e n c e in re s id e s in th e n u c le o n ie s . A s s u m e fo r s im p lic ity th a t n= p . Z e ro te m p e ra tu re If o n e is a t lo w e n o u g h te m p e ra tu re , o n e c a n ig n o re th e d e p e n d e n c e o f th e E O S o n te m p e ra tu re a n d e v a lu a te th e E O S a t T =0. F or T =0, 2 f ( ,0 , ) P P ( ,0 , )

T , 2 ( ,0 , ) (3 ) , P h a se tra n s itio n s: P h a se tra n s itio n s c a n m a n ife s t th e m se lv e s in th e E O S if th e re a re re g io n s w h e re d P /d | < 0 , m a k in g th e m a tte r m e c h a n ic a lly u n sta b le . W h e re th is o c c u rs, o n e m u st m a tc h th e c h e m ic a l p o te n tia ls fo r th e d e n s e r a n d m o re d ilu te p h a s e s b y m a k in g a M a x w e ll c o n s tru c tio n , 20 0 D e m o n str a tio n o f M a x w e ll c o n s tr u c tio n in w h ic h th e a re a , A Vdp b e tw e e n th e 15 0 B o g o ta 2 10 0 P M a x w e ll c o n s tru c tio n lin e a n d th e o rig in a l E O S is e q u a l o n th e le ft a n d th e rig h t sid e . bog2_pt 50 0 T h e stra ig h t lin e s h o w s a n E O S (in re d ) -5 0 a n d its M a x w e ll c o n stru c tio n in b lu e . (In 1 1 .5 2 m o st p la c e s , th e re d a n d b lu e c o in c id e a n d o n ly th e b lu e is v is ib le . T h e re d c u rv e c a n o n ly b e se e n in th e u n sta b le re g io n .) 2 .5 3 3 .5 4 / 0 N o te , th e sy ste m m a y fo llo w th e d a s h e d c u rv e fo r a w h ile e v e n in th e m ix e d p h a se re g io n if th e e x p a n sio n o r c o m p re ssio n is fa s t enough. O Theoretical Theoretical Approaches Approaches Variational Variationaland andBruekner Brueknermodel modelcalculations calculationswith withrealistic realistictwo-body two-bodynucleonnucleonnucleon nucleoninteractions: interactions:(see (seeAkmal Akmaletetal., al.,PRC PRC58, 58,1804 1804(1998) (1998)and andrefs refstherein.) therein.) Variational Variationalminimizes minimizes with withelaborate elaborategrounds

groundsstate statewavefunction wavefunctionthat that includes includesnucleon-nucleon nucleon-nucleoncorrelations. correlations. Incorporate Incorporatethree-body three-bodyinteractions. interactions. Some Someare are"fundamental" "fundamental" Others Othersmodel modelrelativistic relativisticeffects. effects. Relativistic Relativisticmean meanfield fieldcalculations calculationsusing usingrelativistic relativisticeffective effectiveinteractions, interactions,(see (see Lalasissis Lalasissisetetal., al.,PRC PRC55, 55,540 540(1997), (1997),Peter PeterRing Ringlectures) lectures) Well Welldefined definedtransformations transformationsunder underLorentz Lorentzboosts boosts Parameterization Parameterizationcan canbe beadjusted adjustedtotoincorporate incorporatenew newdata. data. Skyrme Skyrmeparameterizations: parameterizations:(Vautherin (Vautherinand andBrink, Brink,PRC PRC5, 5,626 626(1972).) (1972).) Requires Requirestransformation transformationtotolocal localrest restframe frame Computationally Computationallystraightforward straightforward--example example Example: Example: Skyrme Skyrme interaction. interaction. Hint: use the expressions for the differential increases in potential energy per unit volume above and do a parametric integration over from zero to one. D iv id in g b y th e d e n sity , o n e o b ta in s th e a v e ra g e p o te n tia l e n e rg y p e r n u c le o n :

p 2 V a b c , where n . 2 1 1 T o th is , o n e m u s t a d d th e a v e ra g e k in e tic e n e rg y , w h ic h a t z e ro te m p e ra tu re h a s th e fo rm : 3 3 f n n f p p 3 1 5 KE 5 f ( ) f ( ) 2 5 3 w h e re f is th e F e rm i e n e rg y a t d e n s ity . The average energy for symmetric matter, with =0, and neutron matter, with =1, for this expression are shown in the left figure. In the right figure, we show results for the neutron matter energy for 19 different Skyrme interactions that were used in Hartree Fock calculations that reproduce the binding energy of Sn isotopes butwith differ their symmetry energies. Calculations simpleinskyrme. 40 Brown, Phys. Rev. Lett. 85, 5296 (2001) 50 K =200 MeV Symmetric matter nm Neutron matter (MeV) 30 20 10 0 -10 -20 0 0.05 0.1 0.15 /0 0.2 0.25 0.3 Unlike Unlikesymmetric symmetricmatter, matter,the the potential potentialenergy energyof ofneutron neutron matter matterisisrepulsive. repulsive.

O The Thedensity densitydependence dependenceof of symmetry symmetryenergy energyisislargely largely unconstrained. unconstrained. Constraints Constraints on on symmetric symmetric and and asymmetric asymmetric matter matter EOS EOS 3 100 50 0 1 2 3 / 4 5 neutron matter 100 P (MeV/fm3) 150 P (MeV/fm ) Danielewicz et al., Science 298,1592 (2002). RMF:DD RMF:NL3 Boguta Akmal experiment = (n- p)/ (n+ p) = Akmal av14uvII MS (=0,=0) GWM:neutrons Fermi Gas Exp.+Asy_soft Exp.+Asy_stiff 10 1 1 1.5 2 2.5 3 / 3.5 4 4.5

Danielewicz et al., Science 298,1592 (2002). 200 E/A (, ) = E/A (,0) + 2S() (N-Z)/A1 symmetric matter EOS 5 0 0 Constraints Constraintscome comemainly mainlyfrom from collective collectiveflow flowmeasurements. measurements. Know Knowpressure pressureisiszero zeroatat= =0.0. Results Resultsfrom fromvariational variationalcalculations calculations and andRelativistic Relativisticmean meanfield fieldtheory theorywith with density densitydependent dependentcouplings couplingslie liewithin within the theallowed allowedboundaries. boundaries. Neutron Neutronmatter matterEOS EOSalso alsoincludes includesthe the poorly poorlyconstrained constrainedpressure pressurefrom fromthe the symmetry symmetryenergy. energy. The Theuncertainty uncertaintyfrom fromthe thesymmetry symmetry energy energyisislarger largerthan thanthat thatfrom fromthe the symmetric symmetricmatter matterEOS. EOS. O

Type Type IIII supernova: supernova: (collapse (collapse of of 20 20 solar solar mass mass star) star) Supernovae Supernovaescenario: scenario:(Bethe (BetheReference) Reference) Nuclei NucleiHHeC...SiFe HHeC...SiFe Fe Festable, stable,Fe Feshell shellcools coolsand andthe thestar starcollapses collapses Matter Mattercompresses compressestoto>4 >4s sand andthen thenexpands expands Relevant Relevantdensities densitiesand andmatter matterproperties properties Compressed 0, 0.40.9 Compressedmatter matterinside insideshock shockradius radius0<<10 0<<100, 0.40.9 What Whatdensities densitiesare areachieved? achieved? What Whatisisthe thestored storedenergy energyininthe theshock? shock? What Whatisisthe theneutrino neutrinoemission emissionfrom fromthe theproto-neutron proto-neutronstar? star? Clustered Clusteredmatter matteroutside outsideshock shockradius radiusmixed mixedphase phaseof ofnucleons nucleonsand and nuclear nucleardrops drops -- nuclei: nuclei:< <0,0,0.30.5 0.30.5 How

Howmuch muchenergy energyisisdissipated dissipatedininvaporizing vaporizingthe thedrops dropsduring duringthe the explosion? explosion? What Whatisisthe thenature natureofofthe thematter matterthat thatinteracts interactsand andtraps trapsthe theneutrinos? neutrinos? What Whatare arethe theseed seednuclei nucleithat thatare arepresent presentatatthe thebeginning beginningofofr-process r-process which whichmakes makesroughly roughlyhalf halfofofthe theelements? elements? O Neutron Neutron Stars Stars Neutron NeutronStar Starstability stabilityagainst against gravitational gravitationalcollapse collapse Stellar Stellardensity densityprofile profile Internal Internalstructure: structure:occurrence occurrenceofof various variousphases. phases. Observational Observationalconsequences: consequences: Stellar Stellarmasses, masses,radii radiiand and moments momentsofofinertia. inertia. Cooling Coolingrates ratesofofproto-neutron proto-neutron stars stars Cooling Coolingrates ratesfor forX-ray

X-ray bursters. bursters. Neutron Star Structure: Pethick and Ravenhall, Ann. Rev. Nucl. Part. Sci. 45, 429 (1995) O Some Some examples examples Neutron star radii: 0.5 0 S pot const. F (u ); u / 0 These Theseequations equationsofofstate statediffer differonly only inintheir density dependent symmetry their density dependent symmetry terms. terms. Clear Clearsensitivity sensitivitytotothe thedensity density dependence of the symmetry dependence of the symmetryterms terms 10 TS (deg) x oo 1.5 Lattimer , Ap. J., 550, 426 (2001). 2 Cooling of proto-neutron stars: 7 Lattimer et al., Ap. J. 425 (1994) 802. Standard cooling Direct URCA Isothermal t W 10 6 10 5 1 10 4 100 1000 10 Age (y) 5 10

6 10 Neutrino Neutrinosignal signalfrom fromcollapse. collapse. O Feasibility of URCA processes for Feasibility of URCA processes for proto-neutron proto-neutronstar starcooling coolingififffp p>>0.1. 0.1. This occurs if S() is strongly density This occurs if S() is strongly density dependent. dependent. p+e nnp+e p+e-n+ n+ p+e+-+ Summary Summary of of last last lecture lecture The TheEOS EOSdescribes describesthe themacroscopic macroscopicresponse responseofofnuclear nuclearmatter matterand andfinite finitenuclei. nuclei. 2 (,0,) (n+ p) = (N-Z)/A 2S(); ; ==( (,0,)==(,0,0) (,0,0)++S() (n-n-p)/ p)/ (n+ p) = (N-Z)/A ItItcan canbe becalculated calculatedby byvarious varioustechniques. techniques.Skyrme Skyrmeparameterizations parameterizationsare areaa relatively relativelyeasy easyand andflexible flexibleway waytotodo doso. so.. . The Thehigh highdensity densitybehavior behaviorand andthe

thebehavior behavioratatlarge largeisospin isospinasymmetries asymmetriesofofthe the EOS EOSare arenot notwell wellconstrained. constrained. The Thebehavior behavioratatlarge largeisospin isospinasymmetries asymmetriesisisdescribed describedby bythe thesymmetry symmetryenergy. energy. The Thesymmetry symmetryenergy energyhas hasaaprofound profoundinfluence influenceon onneutron neutronstar starproperties: properties: stellar stellarradii, radii,maximum maximummasses, masses,cooling coolingof ofproto-neutron proto-neutronstars, stars,phases phasesininthe the stellar stellarinterior, interior,etc. etc. O Binding Binding energies energies as as probes probes of of the the EOS EOS 2/3 1/3 -1/2 BBA,Z ==aav[1-b --aas[1-b 2/3- ac Z/A 1/3+ A,ZA -1/2+ CdZ/A, 1((N-Z)/A)]A 2((N-Z)/A)]A [1-b ((N-Z)/A)]A [1-b ((N-Z)/A)]A a Z/A + A + CdZ/A, 1 2 A,Z v s c A,Z Fits Fitsofofthe

theliquid liquiddrop dropbinding bindingenergy energyformula formulaexperimental experimentalmasses massescan canprovide provide values valuesfor foraav,v,aas,s,aac,c,bb1,1,bb2,2,CCd dand andAA,Z,Z. . 2 E / A ( , 0 , ) ( , 0 , 0 ) S ; n p / Relationship to EOS Relationship to EOS aav ==( avb1=S(s) (s,0,0); v s,0,0); avb1=S(s) aas and asb2 provide information about the density dependence of (s,0,0) s and asb2 provide information about the density dependence of (s,0,0) and andS( S(s)s)atatsubsaturation subsaturationdensities densities1/2 1/2s s. .(See (SeeDanielewicz, Danielewicz,Nucl. Nucl. Phys. Phys.AA727 727(2003) (2003)233.) 233.) The Thevarious variousparameters parametersare arecorrelated. correlated.Coulomb Coulomband andsymmetry symmetryenergy energyterms terms are strongly correlated. Shell effects make masses differ

from LDM. are strongly correlated. Shell effects make masses differ from LDM. Measurement Measurementtechniques: techniques: Penning Penningtraps: traps:=qB/m =qB/m Time Timeofofflight: flight:TOF=distance/v TOF=distance/v B=mv/q B=mv/q 2 Transfer c-mD)c 2 Transferreactions: reactions:A(b,c)D A(b,c)D Q=(m Q=(mAA+m +mb-m b-mc-mD)c Mass Masscompilations compilationsexist: exist:e.g. e.g.Audi Audietetal,.,NPA al,.,NPA595, 595,(1995) (1995)409. 409. O Neutron Neutron and and proton proton matter matter radii radii 0.1 Pb n (r) (fm ) 0.08 -3 (fm-3) AAsimple simpleapproximation approximationtotothe thedensity density profile profileisisaaFermi Fermifunction function(r)= (r)=0/(1+exp(r0/(1+exp(rR)/a). R)/a). For Forstable stablenuclei, nuclei,RRp phas hasbeen beenmeasured measuredby by electron electronscattering scatteringtotoabout about0.02 0.02fm fm accuracy. accuracy. (see (seeG. G.Fricke Frickeetetal.,

al.,At. At.Data DataNucl. Nucl.Data Data Tables Tables60, 60,177 177(1995).) (1995).) 208 0.06 -3 0.04 (fm ) n -3 (fm ) 0.02 0 p -2 0 2 4 6 R (fm) 8 10 12 r (fm) Neutron Neutronradii radiican canbe bemeasured measuredby byhadronic hadronicscattering, scattering,which whichisismore moremodel modeldependent dependent and less accurate (R 0.2 fm) because the interaction is mainly on the surface. and less accurate (Rn n 0.2 fm) because the interaction is mainly on the surface. aa0.5 0.50.6 0.6fm fmfor forstable stablespherical sphericalnuclei, nuclei,but butnear nearthe theneutron

neutrondripline, dripline,aan ncan canbe be much muchlarger. larger. 11 208 Strong Stronginteraction interactionradius radiusfor for 11Li Liisisabout aboutthe thesame sameasasthat thatfor for 208Pb. Pb. Comparison Comparison of of RRnnand and RRpp Brown, Phys. Rev. Lett. 85, 5296 (2001) 0.25 2 1/2 0.2 p - 0.15 n 2 1/2 The Theasymmetry asymmetryininthe thenuclear nuclearsurface surfacecan can be larger when S() is strongly density be larger when S() is strongly density dependent dependentbecause becauseS() S()vanishes.more vanishes.more rapidly at low density when rapidly at low density whenS() S()isisstiff. stiff. Stiff Stiffsymmetry symmetryenergy energy larger larger neutron neutronskins. skins.(See (SeeDanielewicz Danielewicz lecture.) lecture.) 208 Measurements Measurementsofof 208Pb Pbusing usingparity

parity violating electron scattering are violating electron scattering areexpected expected 2 1/2totoprovide strong constraints on 1/2provide strong constraints on 2 1/2 and on S() for < . 1/2 and on S() for < s Uncertainties are of order 0.06 Uncertainties are of order 0.06fm. fm.(see (see Horowitz et al., 63, 025501(2001).) Horowitz et al., 63, 025501(2001).) The Theupper upperfigure figureshows showshow howthe thepredicted predicted 2dS()/d neutron skins depend on P = 2 neutron skins depend on Psym sym= dS()/d 2 1/2- 1/2 for Na Analyses p 2>1/2 for Na 2>1/2- p isotopes have placed some isotopes have placed someconstraints constraintson on S() S()for for< - n

p softer stiffer 0.1 0.05 0 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 3 P (MeV/fm ) at =0.1 fm sym O Radii Radii of of Na Na isotopes isotopes Suzuki, et al., PRL 75, 3241 (1995) 1/2 ~ 0.1 fm N transmitted N incident exp int c x The Therelationship relationshipbetween betweencrosscrosssection sectionand andNa Nainteraction interactionradius radius 2 ( R R ) is: int c , int Na , int is: Getting Gettingthe theactual actualneutron neutron radius radiusisismodel modeldependent. dependent. O Proton Protonradii radiiare aredetermined determinedby by measuring atomic transitions measuring atomic transitionsininNa, Na, which whichhas hasaa3s 3sg.s. g.s.orbit. orbit. Neutron Neutronradii radiiincrease increasefaster fasterthan

than 1/3 R=r , reflecting the thickness of 1/3 R=r0A 0A , reflecting the thickness of neutron neutronskin, skin,e.g. e.g.RMF RMFcalculation. calculation. Giant Giant resonances resonances 0 (MeV) Imagine Imagineaamacroscopic, macroscopic,i.e. i.e.classical classicalexcitation excitationof ofthe thematter matter ininthe nucleus. the nucleus. e.g. e.g.Isoscaler IsoscalerGiant GiantMonopole Monopole(GMR) (GMR)resonance resonance GMR GMRprovides providesinformation informationabout aboutthe the2curvature curvatureofof(,0,0) (,0,0) about 1 2 aboutminimum. minimum. ( ,0,0) 16MeV s 2 2 90 90 Inelastic Inelastic particle particlescattering scatteringe.g. e.g. 90Zr(, Zr(,))90Zr* Zr*can can excite excitethe theGMR. GMR.(see (seeYoungblood Youngbloodetetal., al.,PRL PRL92, 92,691 691 (1999).) (1999).) Peak Peakisisstrongest strongestatat00

-5 -10 -15 -20 0 0.1 0.2 / 0 0.3 Giant Giant resonances resonances 22 HW HW3:3:Assume Assumethat thatwe wecan canapproximate approximateaanucleus nucleusasashaving havingaasharp sharpsurface surfaceatatradius radius RRand andignore ignorethe thesurface, surface,Coulomb Coulomband andsymmetry symmetryenergy energycontributions contributionstotothe thenuclear nuclear energy. energy. R0 3 InInthe theadiabatic adiabaticapproximation approximationshow showthat that PE A s ,0,0 R 2 Show Showthat that KE 1 / 2 3 / 5 MR Show Showthat that EGMR K nm 9 s2 2 ; where K nm 2 2 mr s InInpractice practicethere thereare aresurface, surface,Coulomb Coulomband andsymmetry symmetryenergy energycorrections correctionstotothe theGMR GMR energy. energy.(see

(seeHarakeh Harakehand andvan vander derWoude, Woude,Giant GiantResonances Resonances Oxford OxfordScience...) Science...) Leptodermous Leptodermousexpansion: expansion: O Giant Giant Resonances Resonances 33 P N Isovector IsovectorGiant GiantDipole DipoleResonance: Resonance:neutrons neutronsand andprotons protonsoscillate oscillateagainst againsteach each other. other.The Therestoring restoringforce forceisisthe thesurface surfaceenergy energyofofthe thenucleus. nucleus. Danielewicz depends on the surface symmetry energy but Danielewiczhas hasshown shownthat thatEEGDR GDR depends on the surface symmetry energy but not noton onthe thevolume volumesymmetry symmetryenergy. energy.(Danielewicz, (Danielewicz,NP NPAA727 727(2003) (2003)233.) 233.) O Probes Probes of of the the symmetric symmetric matter matter EOS EOS Nuclear Nuclearcollisions collisionsare arethe theonly onlyway waytotomake makevariations variationsininnuclear nucleardensity densityunder under experimentally experimentallycontrolled controlledconditions

conditionsand andobtain obtaininformation informationabout aboutthe theEOS. EOS. Theoretical Theoreticaltool: tool:transport transporttheory: theory: Example ExampleBoltzmann-Uehling-Uhlenbeck Boltzmann-Uehling-Uhlenbeckeq. eq.(Bertsch (BertschPhys. Phys.Rep. Rep.160, 160,189 189 (1988).): (1988).): Describes Describesthe thetime timeevolution evolutionofofthe theWigner Wignertransform transformofofthe theone-body one-body density analogue densitymatrix: matrix:(quantum (quantum analoguetotoclassical classicalphase phasespace space distribution) distribution) ! ! 3rd3p at r and p ) . f ( r , p , t ) classically, f= ( the number of nucleons/d 3 3 classically, f= ( the number of nucleons/d rd p at ). Semiclassical: Semiclassical:time timedependent dependentThomas-Fermi Thomas-Fermitheory theory Each Eachnucleon nucleonisisrepresented representedby by~1000 ~1000test testparticles particlesthat thatpropogate propogateclassically classically under the influence of

the mean field U and subject to collisions due under the influence of the mean field U and subject to collisions duetotothe the residual interaction. The mean field is self consistent, at each time step, one: residual interaction. The mean field is self consistent, at each time step, one: propogates propogatesnucleons, nucleons,etc. etc.subject subjecttotothe themean meanfield fieldand andcollisions, collisions,and and recalculates recalculatesthe themean meanfield fieldpotential potentialaccording accordingtotothe thenew newpositions. positions. Constraining Constrainingthe theEOS EOSatathigh highdensities densitiesby bylaboratory laboratory collisions collisions Au+Au Au+Aucollisions collisionsE/A E/A==11GeV) GeV) pressure contours density contours Two Twoobservable observableconsequences consequencesof ofthe thehigh highpressures pressuresthat thatare areformed: formed: Nucleons Nucleonsdeflected deflectedsideways sidewaysininthe thereaction reactionplane. plane. Nucleons

Nucleonsare aresqueezed squeezedout outabove aboveand andbelow belowthe thereaction reactionplane. plane.. . Procedure Procedure to to study study high high pressures pressures Measure Measurecollisions collisions Simulate Simulatecollisions collisionswith withBUU BUUororother othertransport transporttheory theory Identify Identifyobservables observablesthat thatare aresensitive sensitivetotoEOS EOS(see (seeDanielewicz Danielewiczetetal., al., Science 298,1592 (2002). for flow observables) Science 298,1592 (2002). for flow observables) Directed Directedtransverse transverseflow flow(in-plane) (in-plane) Elliptical Ellipticalflow flowout outofofplane, plane,e.g. e.g.squeeze-out squeeze-out Kaon Kaonproduction. production.(Schmah, (Schmah,PRC PRCCC71, 71,064907 064907(2005)) (2005)) Analyze Analyzedata dataand andmodel modelcalculations calculationstotomeasured measuredand andcalculated calculatedobservable observable assuming assumingsome somespecific specificforms formsofofthe themean meanfield fieldpotentials potentialsfor forneutrons neutronsand and protons.

protons.At Atsome someenergies, energies,produced producedparticles, particles,like likepions, pions,etc. etc.must mustbe be calculated as well. calculated as well. Find Findthe themean meanfield(s) field(s)that thatdescribes describesthe thedata. data.IfIfmore morethan thanone onemean meanfield field describes the data, resolve the ambiguity with additional data. describes the data, resolve the ambiguity with additional data. Constrain Constrainthe theeffective effectivemasses massesand andin-medium in-mediumcross crosssections sectionsby byadditional additional data. data. Use Usethe themean meanfield fieldpotentials potentialstotoapply applythe theEOS EOSinformation informationtotoother othercontexts contexts like neutron stars, etc. like neutron stars, etc. Directed Directed transverse transverse flow flow Partlan, PRL 75, 2100 (1995). target px y Au+Au collisions EOS TPC data Ebeam/A projectile Event Eventhas haselliptical ellipticalshape

shapeinin momentum momentumspace. space. The Thelong longaxis axislies liesininthe thereaction reaction plane, plane,perpendicular perpendiculartotothe thetotal total angular momentum. angular momentum. Analysis Analysisprocedure: procedure: Find Findthe thereaction reactionplane plane Determine in this plane Determine

x(y)> in this plane v 1 E pz c || y ln note: 2 E pz c c note: non - relativistically y/ybeam (in C.M) The Thedata datadisplay displaythe thes sshape shape characteristic of directed transverse characteristic of directed transverse flow. flow. The TheTPC TPChas hasin-efficiencies in-efficienciesatat y/y < -0.2. y/ybeam beam< -0.2. d p x / A / dy is Slope Slope is determined at 0.2

(2002). px / A y O The Thecurves curveslabeled labeledby byKKnmnmrepresent represent calculations calculationswith withparameterized parameterized Skyrme mean fields Skyrme mean fields They Theyare areadjusted adjustedtotofind findthe the pressure that replicates the pressure that replicates the observed observedtransverse transverseflow. flow. The Theboundaries boundariesrepresent representthe therange range ofofpressures pressuresobtained obtainedfor forthe themean mean fields that reproduce the data. fields that reproduce the data. They Theyalso alsoreflect reflectthe theuncertainties uncertainties from fromthe theeffective effectivemasses massesininininmedium cross sections. medium cross sections. Probes Probes of of the the symmetry symmetry energy energy 2 (,0,) (n+ p) = (N-Z)/A 2S(); ; ==( (,0,)==(,0,0) (,0,0)++S() (n-n-p)/ p)/ (n+ p) = (N-Z)/A F2 F1

=0.3 stiff 50 V (MeV) (MeV) U asy asy Common Commonfeatures featuresofofsome someofofthese thesestudies studies Vary Varyisospin isospinofofdetected detectedparticle particle Sign SignininUUasyasyisisopposite oppositefor fornnvs. vs.p.p. Shape Shapeisisinfluenced influencedby bystiffness. stiffness. Vary Varyisospin isospinasymmetry asymmetryofofreaction. reaction. Low Lowdensities densities(< (<0): 0): Isospin Isospindiffusion diffusion Neutron/proton Neutron/protonspectra spectraand andflows flows Neutron, Neutron,proton protonradii, radii,E1 E1collective collectivemodes. modes. High Highdensities densities(2 (20)0): : Neutron/proton Neutron/protonspectra spectraand andflows flows + +vs. vs.-production production Li et al., PRL 78 (1997) 1644 100 F3 soft Neutron Proton 0 -50 F1=2u2/(1+u)

F2=u -100 F3=u 0 0.5 1 u = /o 1.5 2 Constraining Constrainingthe thedensity densitydependence dependenceof ofthe thesymmetry symmetryenergy energy Observable: Observable:Isospin Isospindiffusion diffusionin inperipheral peripheralcollisions collisions DD the isospin diffusion coef. the isospin diffusion coef. Two Twoeffect effectcontribute contributetotodiffusion diffusion Random Randomwalk walk Potential Potential(EOS) (EOS)driven drivenflows flows DD governs the relative flow of governs the relative flow of neutrons neutronsand andprotons protons DD decreases with np decreases with np DD increases increaseswith withSSint( ()) softer stiffer int RRisisthe theratio ratiobetween betweenthe thediffusion diffusion coefficient coefficientwith withaasymmetry symmetrypotential potential and andwithout withoutaasymmetry symmetrypotential. potential. Shi et al, C 68, 064604 (2003) InInaareference

referenceframe framewhere wherethe thematter matter / n p isisstationary: ! stationary: n vn p v p D Probe: Probe: Isospin Isospin diffusion diffusion in in peripheral peripheral collisions collisions Vary Varyisospin isospindriving drivingforces forcesby bychanging changing the isospin of projectile and target. the isospin of projectile and target. Probe Probethe theasymmetry asymmetry=(N-Z)/(N+Z) =(N-Z)/(N+Z)ofof the theprojectile projectilespectator spectatorafter afterthe thecollision. collision. The Theasymmetry asymmetryofofthe thespectator spectatorcan can change changedue duetotodiffusion, diffusion,but butititalso alsocan can changed due to pre-equilibrium emission. changed due to pre-equilibrium emission. The Theuse useofofthe theisospin isospintransport transportratio ratio RRi() isolates the diffusion effects: i() isolates the diffusion effects:

124 112 mixed mixed 124Sn+ Sn+112Sn Sn 124 124 n-rich n-rich 124Sn+ Sn+124Sn Sn 112 112 p-rich p-rich 112Sn+ Sn+112Sn Sn 1.0 No isospin diffusion P 0.0 ( both _ neut. rich both _ prot. rich ) / 2 R i () 2 both _ neut. rich both _ prot. rich 124 112 Useful Usefullimits limitsfor forRRi ifor for 124Sn+ Sn+112Sn Sn collisions: collisions: RRi =1: no nodiffusion diffusion i =1: RRi 0: Isospin Isospinequilibrium equilibrium i 0: neutron-rich Complete mixing N -1.0 proton-rich Ri Sensitivity Sensitivity to to symmetry symmetry energy energy ( Neutron rich Pr oton rich ) / 2 Ri ( ) 2 Neutron rich Pr oton rich The Theasymmetry asymmetryofof the thespectators spectatorscan can change changedue duetoto diffusion, diffusion,but butitit also

alsocan canchanged changed due to predue to preequilibrium equilibrium emission. emission. The Theuse useofofthe the isospin transport isospin transport ratio isolates ratioRRi() i() isolates the thediffusion diffusion effects: effects: Lijun Shi, thesis Stronger density dependence Weaker density dependence Tsang et al., PRL92(2004) Probing Probing the the asymmetry asymmetry of of the the Spectators Spectators The Themain maineffect effectofofchanging changingthe the asymmetry asymmetryofofthe theprojectile projectile spectator remnant is spectator remnant istotoshift shiftthe the isotopic isotopicdistributions distributionsofofthe the products productsofofits itsdecay decay Liu et al., (2006) The Thethe theshift shiftcan canbe becompactly compactly described describedby bythe theisoscaling isoscalingparameters parameters and andobtained obtainedby bytaking takingratios ratiosofof the theisotopic

isotopicdistributions: distributions: Y2 N, Z C exp(N Z) Y1 N, Z Tsang et. al.,PRL 92, 062701 (2004) Determining Determining R Ri() i() ( Neutron rich Pr oton rich ) / 2 Ri ( ) 2 Neutron rich Pr oton rich Y2 N, Z C exp(N Z) Y1 N, Z Statistical Statisticaltheory theoryprovides: provides: 1 / T , n T 2Csym 1 / T p T 2Csym and n p T 4Csym / T, where : 2 1 Consider Considerthe theisoscaling isoscalingratio ratioRRi(X), i(X), where X = or where X = or X (X Neutron rich X Pr oton rich ) / 2 R i (X) 2 X Neutron rich X Pr oton rich When WhenXXdepends dependslinearly linearlyon on2:2: X a 2 b By Bydirect directsubstitution: substitution: R i X R i 2 true production truefor forknown known production models models linear lineardependences dependencesconfirmed confirmed by bydata. data. Probing Probing the the asymmetry asymmetry of of the the Spectators Spectators

The Themain maineffect effectofofchanging changingthe the asymmetry asymmetryofofthe theprojectile projectile spectator remnant is spectator remnant istotoshift shiftthe the isotopic isotopicdistributions distributionsofofthe the products productsofofits itsdecay decay Liu et al., (2006) The Thethe theshift shiftcan canbe becompactly compactly described describedby bythe theisoscaling isoscalingparameters parameters and andobtained obtainedby bytaking takingratios ratiosofof the theisotopic isotopicdistributions: distributions: Y2 N, Z C exp(N Z) Y1 N, Z Tsang et. al.,PRL 92, 062701 (2004) 1.0 no diffusion 0.33 Ri() -0.33 -1.0 Constraints Constraints from fromIsospin Isospin Diffusion Diffusion Data Data M.B. Tsang et. al., PRL 92, 062701 (2004) L.W. Chen, C.M. Ko, and B.A. Li, PRL 94, 032701 (2005) 124 Sn+112Sn data C B A C.J. Horowitz and J. Piekarewicz, PRL 86, 5647 (2001) B.A. Li and A.W. Steiner, nucl-th/0511064 Approximate representation of the various asymmetry terms used in BUU calcuations:

Esym() ~ 32(/0) [(n - p) /(n +p)]2 for cases A, B, C) O Interpretation Interpretationrequires requiresassumptions assumptionsabout about isospin isospindependence dependenceof ofin-medium in-mediumcross cross sections sectionsand andeffective effectivemasses masses Final Final Summary Summary The TheEOS EOSdescribes describesthe themacroscopic macroscopicresponse responseof ofnuclear nuclearmatter matterand andfinite finitenuclei. nuclei. 2 (,0,) (n+ p) = (N-Z)/A 2S(); ; ==( (,0,)==(,0,0) (,0,0)++S() (n-n-p)/ p)/ (n+ p) = (N-Z)/A ItItcan canbe becalculated calculatedby byvarious varioustechniques. techniques.Skyrme Skyrmeparameterizations parameterizationsare are relatively easy. relatively easy. The Thehigh highdensity densitybehavior behaviorand andthe thebehavior behavioratatlarge largeisospin isospinasymmetries asymmetriesof ofthe the EOS EOSare arenot notwell wellconstrained. constrained. The Thebehavior behavioratatlarge largeisospin isospinasymmetries asymmetriesisisdescribed describedby bythe thesymmetry symmetryenergy.

energy. ItItinfluences influencesmany manynuclear nuclearphysics physicsquantities: quantities:binding bindingenergies, energies,neutron neutronskin skin thicknesses, thicknesses,isovector isovectorgiant giantresonances, resonances,isospin isospindiffusion, diffusion,etc. etc.Measurements Measurementsof of these thesequantities quantitiescan canconstrain constrainthe thesymmetry symmetryenergy. energy. The Thesymmetry symmetryenergy energyhas hasaaprofound profoundinfluence influenceon onneutron neutronstar starproperties: properties: stellar stellarradii, radii,maximum maximummasses, masses,cooling coolingofofproto-neutron proto-neutronstars, stars,phases phasesininthe the stellar stellarinterior, interior,etc. etc. Constraints Constraintson onthe thesymmetry symmetryenergy energyand andon onthe theEOS EOSwill willbe beimproved improvedby byplanned planned experiments. Some of the best ideas have not yet been discovered. experiments. Some of the best ideas have not yet been discovered. Influence Influenceof ofproduction productionmechanism mechanismon onisoscaling

isoscalingparameters parameters Primary: Before decay of excited fragments, Final: after decay of excited fragments Dynamical Dynamicaltheories: theories: Final Finalisoscaling isoscalingparameters parametersare are often smaller than those of often smaller than those of primary primarydistribution distribution Both Bothdepend dependlinearly linearlyon on R()=R() R()=R() Doesn't Doesn'tmatter matterwhich whichone oneisis correct. correct. Statistical Statisticaltheory: theory: Final Finalisoscaling isoscalingparameters parametersare are often oftensimilar similartotothose thoseofofthe the primary primarydistribution distribution Both Bothdepend dependlinearly linearlyon on R()=R() R()=R() 0.3 AMD - Gogny: Ca+Ca collisions 1 0.4 Statistical Multifragmentation Model Sn+Sn collisions primary secondary 0.8 final 0.6 0.2 0.4 0.1 Primary_stiff

Final_stiff 0 -0.1 0.1 0.12 0.14 0.16 0.18 0.2 0.2 0 R 0 0.05 0.1 0.15 0.2 Test Test of of linearity linearity using using central central collisions collisions Data Dataanalyzed analyzedininwell-mixed well-mixedregion region atat70 70cmcm110. 110. Linearity Linearityisisdemonstrated demonstratedfor for, , 7 7 and 7Li)/Y(Be))- 7Be))- andln(Y( ln(Y(Li)/Y( Liu et al., (2006) Liu et al., (2006) R Asymmetry Asymmetry term term studies studies at at 2 200 Densities ofof 2 be atatE/A400 MeV. 0 can Densities ofof 2 be atatE/A400

MeV. 0 can Densities 2 beachieved achieved E/A400 0 can Densities 2 beachieved achieved E/A400 MeV. MeV. 0 can Provides Provides information about direct URCA cooling ininproto-neutron stars, information about direct URCA cooling ininproto-neutron stars, Provides information about direct URCA cooling proto-neutron stars, Provides information about direct URCA cooling proto-neutron stars,stability stability and phase transitions of dense neutron star interior. stability phase of neutron and phaseand transitions of dense neutron star interior. stability and phasetransitions transitions ofdense dense neutronstar starinterior. interior. S() S()influences influencesdiffusion diffusionofofneutrons neutronsfrom fromdense denseoverlap overlapregion regionatatb=0. b=0. Diffusion Diffusionisisreduced, reduced,neutron-rich

neutron-richdense denseregion regionisisformed formedfor forsoft softS(). S(). Yong et al., Phys. Rev. C 73, 034603 (2006) R First First observable: observable: pion pion production production Yong et al., Phys. Rev. C 73, 034603 (2006) The Theenhanced enhancedneutron neutronabundance abundance atathigh highdensity densityfor forthe thesoft soft asymmetry asymmetryterm term(x=0) (x=0)leads leadstotoaa stronger strongeremission emissionofofnegative negative pions pionsfor forthe thesoft softasymmetry asymmetryterm term (x=0) (x=0)than thanfor forthe thestiff stiffone one(x=-1). (x=-1). InIndelta deltaresonance resonancemodel, model, + 2 Y( n,/ p) 2 -)/Y()( +)( Y()/Y( n,/p) + + - -/ -)/Y() +) /+means meansY( Y()/Y( Coulomb Coulombinteraction interactionhas hasaastrong strong effect effecton onthe thepion pionspectra: spectra: + Coulomb Coulombrepels

repels +and and attracts attracts .-. soft stiff The Thedensity densitydependence dependenceofofthe the asymmetry asymmetryterm termchanges changesratio ratioby byabout about 15% 15%for forneutron neutronrich richsystem. system. How Howdoes doesone onereduce reducesensitivity sensitivitytoto systematic systematicerrors? errors? Double Double ratio: ratio: pion pion production production Double Doubleratio ratioinvolves involvescomparison comparison 132 124 between betweenneutron neutronrich rich 132Sn+ Sn+124Sn Sn 112 112 and andneutron neutrondeficient deficient 112Sn+ Sn+112Sn Sn reactions. reactions. R / 132 Y Y / Y / Y Sn 124 Sn / 132 124

112 112 112 Sn 112 Sn 132 124 112 112 Yong et al., Phys. Rev. C 73, 034603 (2006) Double Doubleratio ratiomaximizes maximizessensitivity sensitivity totoasymmetry asymmetryterm. term. Largely Largelyremoves removessensitivity sensitivitytoto + difference differencebetween between-and and+ acceptances. acceptances. soft stiff R Independent Independent observable: observable: n/p n/p spectra spectra Rn/p 132 Sn 124 Sn / 112 Sn 112 Sn stiff Y n 132124 / Y p 132124 Y n 112 112 / Y p 112 112 Removes Removessensitivity sensitivitytotocalibration calibration and andefficiency efficiencyproblems problems soft Li et al., arXiv:nucl-th/0510016 (2005)

Neutrons Neutronsare arerepelled repelledand andprotons protons are areattracted attractedby bythe theasymmetry asymmetry term term(in (inneutron neutronrich richmatter). matter). The TheCoulomb Coulombinteraction interactionhas has somewhat somewhatthe theopposite oppositeeffect. effect. Sensitivity Sensitivitycan canbe bemaximized maximizedby by constructing constructingaadouble doubleratio: ratio: Alternate Alternate observable: observable: n-p n-p differential differential transverse transverseflow flow Transverse Transversedirected directedflow flowisisusually usually obtained obtainedby byplotting plottingthe themean mean x vs. the transverse momentum

vs. the transverse momentum

rapidity rapidityy.y. The Theneutron-proton neutron-protondifferential differentialflow flow isisdefined definedhere heretotobe: be: 1 N( y) x Fn p w i p ix N y i w i 1 (-1); neut. (prot.)

Sensitivity Sensitivitytotoacceptance acceptanceeffects effects might mightbe beminimized minimizedby byconstructing constructing the thedifference: difference: D nx p Fnx p 132 Sn 124 Sn Fnx p 112 Sn 112 Sn Li et al., arXiv:nucl-th/0504069 (2005) F Constraints Constraints on on momentum momentumdependence dependenceof of mean mean fields fields and and in-medium in-medium cross cross sections sections Li et al., Phys. Rev. C 69, 011603(R) (2004) =1 np =(N-Z)/(N+Z) 0.5 pp = <> =1 free cross sections =2 = = <> np 0.4 pp =2 free cross sections 0.3 0.2 Ca+100Zn E/A=200 MeV 40 0.1 0 -1 -0.5 (y/y ) beam cm We Weneed needcalculations calculationsofofthe the corresponding double ratios. corresponding double ratios. Not Notclear clearthat

thatwe wehave haveaagood goodway way totodistinguish distinguishmomentum momentumand and density dependencies. density dependencies. = 0 0.5 (normal kinematics) Important Importanttotocontrol controlthe thenumber numberofofn-p n-p collisions, p-p and n-n collisions collisions, p-p and n-n collisions 37 112 37 124 compare compare 37Ca+ Ca+112Sn Sntoto 37Ca+ Ca+124Sn Sn 52 112 52 124 compare compare 52Ca+ Ca+112Sn Sntoto 52Ca+ Ca+124Sn. Sn. 1 Li et al., Phys. Rev. C 71, 054603 (2005) 0.6 R

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