BULKNUCLEARPROPERTIESFROMDYNAMICALDESCRIPTIONOFHEAVY-IONCOLLISIONSByJunHongADISSERTATIONSubmittedtoMichiganStateUniversityinpartialfulÞllmentoftherequirementsforthedegreeofPhysics-DoctorofPhilosophy2016ABSTRACTBULKNUCLEARPROPERTIESFROMDYNAMICALDESCRIPTIONOFHEAVY-IONCOLLISIONSByJunHongMappingouttheequationofstate(EOS)ofnuclearmatterisalongstandingprobleminnuclearphysics.Bothexperimentalistsandtheoreticalphysicistssparenoeffortinimprovingunderstand-ingoftheEOS.Inthisthesis,weexamineobservablessensitivetotheEOSwithinthepBUUtransportmodelbasedontheBoltzmannequation.Bycomparingtheoreticalpredictionswithex-perimentaldata,wearriveatnewconstraintsfortheEOS.Furtherweproposenovelpromisingobservablesforanalysisoffutureexperimentaldata.OnesetofobservablesthatweexaminewithinthepBUUmodelarepionyields.First,weÞndthatnetpionyieldsincentralheavy-ioncollisions(HIC)arestronglysensitivetothemomentumdependenceoftheisoscalarnuclearmeanÞeld.WereexaminethemomentumdependencethatisassumedintheBoltzmannequationmodelforthecollisionsandoptimizethatdependencetodescribetheFOPImeasurementsofpionyieldsfromtheAu+Aucollisionsatdifferentbeamener-gies.Alassuchoptimizeddependenceyieldsasomewhatweakerbaryonicellipticßowthanseeninmeasurements.Subsequently,weusethesamepBUUmodeltogeneratepredictionsforbaryonicellipticßowobservableinHIC,whilevaryingtheincompressibilityofnuclearmatter.Inparallel,wetestthesensitivityofpionmultiplicitytothedensitydependenceofEOS,andinparticulartoincompress-ibility,andoptimizethatdependencetodescribeboththeellipticßowandpionyields.Uponarriv-ingatacceptableregionsofdensitydependenceofpressureandenergy,wecompareourconstraintsonEOSwiththoserecentlyarrivedatbythejointexperimentandtheoryeffortFOPI-IQMD.Weshouldmentionthat,forthemoreadvancedobservablesfromHIC,thereremaindiscrepanciesofupto30%,dependingonenergy,betweenthetheoryandexperiment,indicatingthelimitationsofthetransporttheory.Next,weexploretheimpactofthedensitydependenceofthesymmetryenergyonobserv-ables,motivatedbyexperimentsaimingatconstrainingthesymmetryenergy.IncontradictiontoIBUUandImIQMDmodelsintheliterature,thatclaimsensitivityofnetchargedpionyieldstothedensitydependenceofthesymmetryenergy,albeitindirectionoppositefromeachother,weÞndpracticallynosuchsensitivityinpBUU.However,weÞndaratherdramaticsensitivityofdiffer-entialhigh-energycharged-pionyieldratiotothatdensitydependence,whichcanbequalitativelyunderstood,andweproposethatthatdifferentialratiobeusedinfutureexperimentstoconstrainthesymmetryenergy.Finally,wepresentGaussianphase-spacerepresentationmethodforstudyingstronglycorre-latedsystems.Thisapproachallowstofollowtimeevolutionofquantummany-bodysystemswithlargeHilbertspacesthroughstochasticsampling,providedtheinteractionsaretwo-bodyinnature.WedemonstratetheadvantageoftheGaussianphase-spacerepresentationmethodincopingwiththenotoriousnumericalsignproblemforfermionsystems.Lastly,wediscussthedifÞcultyintryingtostabilizethesystemduringitstimeevolution,withintheGaussianphase-spacemethod.CopyrightbyJUNHONG2016ACKNOWLEDGEMENTSIthasbeenalongjourneyforme,andIowemydeepappreciationtothemanypeoplewhoprovidedcareandsupportalongtheway.Firstofall,IwouldliketothankmyadvisorPawelDanielewicz.Hisbroadknowledgeandpassionshowsmeagreatexampleofatheoreticalnuclearphysicist.Despitehisbusyschedule,hemakeseveryefforttoprovideweeklyadviseonourresearchprojects,evenitmeansskypemeetingduringalayoverintheairport.SpecialthanksareaddressedtoBettyTsangandBillLynch.Fromthemanyhelpfuldiscus-sions,Igainedbetterunderstandingintherelatedresearchprojectsandbeyond.Iamalsothankfulfortheirkindhelpinsomeofmypersonalissues.IgainedtremendousexperiencecollaboratingwithHiRAgroup.Fromsymmetryenergyclubtocloseworkwithsomeofthegraduatestudents,IlearnedmanyspeciÞcsofexperimentalmeasurementsandanalysis,aswellasmakingmanynewfriends.IenjoyedconversationswithJustinEstee,RachelHodges,SetiawanHananiel,DanielCoupland,MikeYoungs.IwouldliketoacknowledgetheNationalScienceFoundationforsupportingmyresearchprojects.IenjoyedtheniceworkingenvironmentprovidedbytheDepartmentofPhysicsandAstronomyandtheNationalSuperconductingCyclotronLaboratory.BrentBarkerhasbeenmyofÞcemate,groupmember,housemateandwillbeagoodfriendofmineforever.HishelpandencouragementliftedmeduringmydifÞculttimesovertheyears.Wesharedgoodmemoriesinourdailymeetings,wehadmanyfruitfulconversationsonlineandoff.YuanyuanZhangwasanothergroupmemberthatlaterbecamemyclosefriend.Welearnandgrowtogether,andIthankherforhergenuinefriendship.IwouldliketoexpressmygratitudetoProf.FilomenaNunes.Myone-yearresearchexperiencewithherwaswonderful.Andshehasbeenveryunderstandingandsupportiveeversince.IthankProf.ArtimesSpyrouforherkindnessinprovidingmanyniceclothesforlittleAudreyandusefultipsforme.vRebeccaShanehasbeenagoodfriendformanyyears.Havingthanksgivingdinneratherplacehasbecomeoneofmyfamilytraditionsince2011.Iowehermanythanksforallherhelp.IamgratefultohaveknownSarahComstockandZajeHarrellduringourlunchtalksatthedaycare.Thosewillbesomeoftheunforgettablemomentsinmylife.IalsothankmanyofthefriendsImadeinthebeginningofmygraduateprogram,AdamFritsch,ScottBustabad,LingyingLin,MichaelScott,RiverHuang,DingWang,thelistgoeson.Lastbutnotleast,IwouldliketothankmyfatherWenzhangHongandmymotherChunhuaWufortheirunconditionalloveandsupport.IammostthankfulforhavingmyhusbandYaxingZhanganddaughterAudreyZhangtheseyears.Thanksforeverything.viTABLEOFCONTENTSLISTOFFIGURES.......................................ixCHAPTER1INTRODUCTION...............................11.1HeavyIonCollisions.................................21.2Pions..........................................31.3Transporttheory....................................41.3.1Densityfunctionalmethod..........................41.3.2Transportmodels...............................51.3.3DetailsintheBUUmodel..........................6CHAPTER2CONSTRAINTSONTHEMOMENTUMDEPENDENCEOFNUCLEARMEANFIELD.................................102.1MomentumdependenceofnuclearmeanÞeld....................102.1.1Momentum-dependentmeanÞeldinpBUUmodel.............112.2Pionobservables...................................122.3Opticalpotentialcomparison.............................162.4Ellipticßow......................................222.5Conclusions......................................32CHAPTER3CONSTRAINTSONNUCLEARINCOMPRESSIBILITY.........333.1Introduction......................................333.2IncompressibilityandisoscalarGiantMonopoleResonance.............343.3Ellipticßow......................................343.3.1Ellipticßowandimpactparameter......................393.3.2Ellipticßowandeffectivemass........................413.3.3Ellipticßowandincompressibility......................433.4Constraintsonnuclearincompressibilityfromßowandpionobservables......453.5Conclusion......................................54CHAPTER4CONSTRAINTSONSYMMETRYENERGYATSUPRANORMALDEN-SITIES.....................................554.1Introductiontosymmetryenergy...........................554.2Motivation.......................................584.3Chargedpionratios..................................594.4Pionpotential.....................................634.5Differentialpionratios................................654.6Isospinfractionation..................................714.7Conclusions......................................76CHAPTER5GAUSSIANQMCMETHOD.........................78vii5.1Introduction......................................785.2Phasespacemethods.................................795.2.1Classicalphase-spacerepresentations....................795.2.2Quantumphase-spacerepresentations....................795.3Gaussianphase-spacerepresentation.........................805.3.1Gaussianphase-spacerepresentationsforBosons..............815.3.2Gaussianphase-spacerepresentationsforFermions.............845.4PropertiesofGaussianphase-spacemethodforFermions..............855.4.1Single-modeGaussianoperator........................855.4.2Completeness.................................865.5Freegas........................................875.6Fermi-Bosemodeling.................................885.7Conclusion......................................93CHAPTER6CONCLUSIONS................................95BIBLIOGRAPHY........................................97viiiLISTOFFIGURESFigure1.1Thebeforeandaftersketchofanintermediateenergynuclearcollision......3Figure2.1PionmultiplicityincentralAu+Aucollisions.SymbolsrepresentdataoftheFOPICollaboration[1].ThelinesrepresentpBUUcalculationswhenfol-lowingeitherthemomentum-independentMF(leftpanel)orthepastßow-optimizedmomentum-dependentMF(rightpanel).Solidlinesarepredic-tionsfor!!,anddashedlinesarepredictionsfor!+.Theexperimentalerrorbarsareaboutthesizeofsymbols......................14Figure2.2PionmultiplicityincentralAu+Aucollisions,asafunctionofbeamenergy.SymbolsrepresentdataoftheFOPIcollaboration[1],,whilelinesrepresentthepBUUcalculationswiththeN!-adjustedmomentum-dependentMF.Theexperimentalerrorbarsareaboutthesizeofsymbols...............15Figure2.3Opticalpotentialinnuclearmatteratdifferentindicateddensities,asafunc-tionofmomentum.Dashedandsolidlinesrepresent,respectively,thev2-optimizedandN!-adjustedMFs..........................17Figure2.4Opticalpotentialinnuclearmatteratdifferentindicateddensities,asafunc-tionofnucleonenergy.Dashedandsolidlinesrepresent,respectively,UV14+UVIIvariationalcalculationsandourN!-adjustedMF..................18Figure2.5Opticalpotentialinnuclearmatteratdifferentindicateddensities,asafunc-tionofnucleonenergy.Dashedandsolidlinesrepresent,respectively,AV14+UVIIvariationalcalculationsandourN!-adjustedMF..................19Figure2.6Opticalpotentialinnuclearmatteratdifferentindicateddensities,asafunc-tionofnucleonmomentum.Dashedandsolidlinesrepresent,respectively,Dirac-Brueckner-Hartree-FockcalculationsandourN!-adjustedMF.......20Figure2.7Opticalpotentialinnuclearmatteratdifferentindicateddensities,asafunc-tionofnucleonenergy.Dashedandsolidlinesrepresent,respectively,UV14+TNIvariationalcalculationsandourN!-adjustedMF..................21Figure2.8SchematicdrawingofthegeometryinaHIC.Thebeamlineisalongzaxis,xaxisisparalleltotheimpactparameterdirection.x-zplaneisdeÞnedasthereactionplane,andyaxisisperpendiculartothereactionplane........22Figure2.9Particleemissionprocessesareshownwithrespecttothereactionplane.Theblockageofspectatorparticlesleadstotheout-of-planeemissionintheearlystageofthereactions................................23ixFigure2.10Ratioofoutofreactionplanetoin-planeprotonyields,asafunctionoftrans-versemomentum.SymbolsrepresentdatafromthemeasurementsoftheKaoSCollaborationofmid-peripheralBi+Bicollisionsatthebeamenergyof400AMeV(b!8.7fm)[2].SolidlinerepresentspBUUcalculationswiththeN!-adjustedmomentum-dependentMFanddashedlinerepresentscal-culationswithv2-optimizedmomentum-dependentMF.Theindicatedtheo-reticalerrorsarestatistical,associatedwiththeMonte-Carlosamplinginthetransportcalculations................................25Figure2.11EllipticßowofparticlemassA=1particles,asafunctionoftransversemo-mentum.SymbolsrepresentdatafromthemeasurementsoftheFOPICol-laborationofmid-peripheralAu+Aucollisionsatthebeamenergyof400AMeV(b"2.0#5.3fm).TheshadedregionrepresentspBUUcalculationswiththeN!-adjustedmomentum-dependentMF.Theindicatedtheoreticalerrorsarestatistical,associatedwiththeMonte-Carlosamplinginthetrans-portcalculations..................................26Figure2.12Ellipticßowofproton,asafunctionoftransversemomentum.SymbolsrepresentdatafromthemeasurementsoftheFOPICollaborationofmid-peripheralAu+Aucollisionsatthebeamenergyof600AMeV(b"2.0#5.3fm).TheshadedregionrepresentspBUUcalculationswiththeN!-adjustedmomentum-dependentMF.Theindicatedtheoreticalerrorsarestatistical,as-sociatedwiththeMonte-Carlosamplinginthetransportcalculations.......28Figure2.13Transverserapiditydistributionofprotons.Rapidityvaluesyxarescaledwiththeprojectilerapidityinthecenter-of-massframe:yx0=yx/yp.Thetransverserapiditydistributionwithrespecttoyxm0isobtainedwithamidra-piditycutof|yz0|<0.5.TrianglesrepresentdatafromthemeasurementsoftheFOPICollaborationofcentralAu+Aucollisionsatthebeamenergyof400AMeV(b=1fm).ThesquaresrepresentpBUUcalculationswiththeN!-adjustedmomentum-dependentMF......................29Figure2.14Transverserapiditydistributionoftritons.Rapidityvaluesyxarescaledwiththeprojectilerapidityinthecenter-of-massframe:yx0=yx/yp.Thetrans-verserapiditydistributionwithrespecttoyxm0isobtainedwithamidrapid-itycutof|yz0|<0.5.TrianglesrepresentdatafromthemeasurementsoftheFOPICollaborationofcentralAu+Aucollisionsatthebeamenergyof400AMeV(b=1fm).ThesquaresrepresentpBUUcalculationswiththeN!-adjustedmomentum-dependentMF........................30xFigure2.15TransverserapiditydistributionofHelium3.Rapidityvaluesyxarescaledwiththeprojectilerapidityinthecenter-of-massframe:yx0=yx/yp.Thetransverserapiditydistributionwithrespecttoyxm0isobtainedwithamidra-piditycutof|yz0|<0.5.TrianglesrepresentdatafromthemeasurementsoftheFOPICollaborationofcentralAu+Aucollisionsatthebeamenergyof400AMeV(b=1fm).ThesquaresrepresentpBUUcalculationswiththeN!-adjustedmomentum-dependentMF......................31Figure3.1EllipticßowofA=1(proton)particles,asafunctionofscaledtransversemo-mentum.CloseddotsrepresentdatafromthemeasurementsoftheFOPICol-laborationofmid-peripheralAu+Aucollisionsatthebeamenergyof400AMeV(b!2.0"5.3fm).TheopensquaresrepresentpBUUcalculationswiththesoftequationofstate(K=210MeV),whichwasadjustedtoKaoSßowdata.ThetrianglesrepresentpBUUcalculationswithstiffequationofstate(K=380MeV)andnomomentumdependenceinMF...............36Figure3.2EllipticßowofA=2(deuteron)particles,asafunctionofscaledtransversemomentum.CloseddotsrepresentdatafromthemeasurementsoftheFOPICollaborationofmid-peripheralAu+Aucollisionsatthebeamenergyof400AMeV(b!2.0"5.3fm).TheopensquaresrepresentpBUUcalcula-tionswiththesoftequationofstate(K=210MeV),whichwasadjustedtoKaoSßowdata.ThetrianglesrepresentpBUUcalculationswithstiffequa-tionofstate(K=380MeV)andnomomentumdependenceinMF.........37Figure3.3EllipticßowofA=3(tritonandHelium3)particles,asafunctionofscaledtransversemomentum.CloseddotsrepresentdatafromthemeasurementsoftheFOPICollaborationofmid-peripheralAu+Aucollisionsatthebeamenergyof400AMeV(b!2.0"5.3fm).TheopensquaresrepresentpBUUcalculationswiththesoftequationofstate(K=210MeV),whichwasad-justedtoKaoSßowdata.ThetrianglesrepresentpBUUcalculationswithstiffequationofstate(K=380MeV)andnomomentumdependenceinMF....38Figure3.4Protonellipticßowasafunctionofimpactparameter,forAu+Aucollisionsat1.2AGeV.DotsarecalculationpointsforsoftmomentumdependentMFwithm*/m=0.782andK=270MeV,thelineisforguidancepurpose.Inab-senceofdeformationforcollidingnuclei,duetosymmetry,theellipticßowneedstovanishatb=0.Asgeometryofthereactionbecomesmoreasymmet-ricwithincreasingimpactparameter,sodoesthemagnitudeofellipticßow.Athigherimpactparameters,theßowsaturatesforamomentum-dependentMF.40xiFigure3.5Ratioofout-of-the-reaction-planetoin-planeprotonyields,asafunctionoftransversemomentum,inmid-peripheralBi+Bicollisionsatthebeamen-ergyof400AMeVandimpactparameterb!7.6fm.SymbolsrepresentdatafromthemeasurementsoftheKaoSCollaboration[2].Solid,dot-dashedanddashedlinesrepresentpBUUcalculationswithMFscharacterizedbyK=270MeVandeffectivemassm*/m=0.8,0.7and0.6,respectively.......42Figure3.6Ratioofout-of-the-reaction-planetoin-planeprotonyields,asafunctionoftransversemomentum,inmid-peripheralBi+Bicollisionsatthebeamenergyof400AMeVandimpactparameterb!7.6fm.SymbolsrepresentdatafromthemeasurementsoftheKaoSCollaboration[2].Dot-dashed,dashed,solidanddottedlinesrepresentpBUUcalculationswithincompressibilityK=300,270,240and210MeV,respectively,witheffectivemassm*=0.7m........44Figure3.7Ratioofoutofreactionplanetoin-planeprotonyields,asafunctionoftransversemomentum.SymbolsrepresentdatafromthemeasurementsoftheKaoSCollaborationofmid-peripheralBi+Bicollisionsatthebeamen-ergiesof400AMeVand700MeV(b=7.6fm)[2].Shadedregionsrepre-sentpBUUcalculationswithanoptimalEOSforincompressibilitybetweenK=240-300MeV..................................47Figure3.8PionmultiplicityincentralAu+Aucollisionsvsbeamenergy.Symbolsrep-resentdataoftheFOPICollaboration[1].TheshadedregionrepresentpBUUcalculationswiththerangeofnuclearequationofstateÞttedtoellipticßowandpionyieldssimultaneously..........................48Figure3.9!+multiplicityincentralAu+Aucollisionsvsbeamenergy.Symbolsrep-resentdataoftheFOPICollaboration[1].SolidanddashedlinesrepresentcalculationscarriedoutwithincompressibilityK=210and270MeV,respec-tively,andtheeffectivemassm*/m=0.75forboth.................49Figure3.10!"multiplicityincentralAu+Aucollisions.SymbolsrepresentdataoftheFOPICollaboration[1].SolidanddashedlinesrepresentcalculationswithincompressibilityK=210and270MeV,respectively,bothwiththeeffectivemassm*/m=0.75..................................50Figure3.11Ratioofoutofreactionplanetoin-planeprotonyields,asafunctionoftrans-versemomentum.SymbolsrepresentdatafromthemeasurementsoftheKaoSCollaborationofmid-peripheralBi+Bicollisionsatthebeamenergyof400AMeV(b=8.7fm)[2].DashedandsolidlinesrepresentpBUUcalcu-lationswithincompressibilityK=210and270MeV,respectively,bothwiththeeffectivemassm*/m=0.75...........................51xiiFigure3.12Energypernucleonforsymmetricnuclearmatterasafunctionofscaleddensity.Thesolidlinesrepresentconstraints,upperandlower,ontheenergyarrivedatbytheFOPI-IQMDPartnership.Theshadedregionrepresentsourconclusions,withthepionyieldsandellipticßowtestingthesupranormalregion.Theverticaldashedlinesshowtheroughdensityregionthatgetsprobedbytheobservablesinthecalculations...................52Figure3.13Nuclearequationofstateplottedaspressureintermsofscaleddensity.Pat-ternedshadedareasrepresentconstraintsdeducebycomparingtransportthe-orytodataonkaonmultiplicityandondirectedandellipticßow.Thesolidlineontheleftrepresentstheequationofstatepreferredbyanalysisoftherecentgiantmonopoleresonance(GMR)experiment.DotsrepresentsthenoninteractingFermigasandtrianglesrepresenttherelativisticmeanÞeld(RMF)modelNL3.Theshadedregions,pinkandblue,representrespec-tivepressureconstraintsthatcanbededucedwhenrelyingoneitheranEOSwithK=240MeVorK=300MeVwithinthepBUUtransportmodel.Theverticaldashedlinesshowtheroughdensityregionthatgetsprobedbytheobservablesinthecalculations...........................53Figure4.1Themultifacetedinßuenceofthenuclearsymmetryenergy.[3].........57Figure4.2Densitydependenceofnuclearsymmetryenergyfor21setsofSkyrmeinter-actionparameters.Symbolsrepresentmomentum-dependentinteractionsinIBUU04[4].....................................58Figure4.3(Coloronline)PionratiosincentralAu+Aucollisions,asafunctionofbeamenergy.DataoftheFOPICollaborationarerepresentedbyÞlledtriangles.TheleftpanelcomparespredictionsfromIBUUandImIQMDmodelstothedata.TheIBUUcalculationsemployingstiffandsoftsymmetryenergiesarerepresentedtherebyÞlleddiamondsandÞlledcircles,respectively.TheImIQMDemployingstiffandsoftsymmetryenergiesare,ontheotherhand,representedtherebyÞlledsquaresandcrosses,respectively.TherightpanelcomparespredictionsfrompBUUmodeltothedata.Calculationsemployingv2-optimizedMFandN!-adjustedMFarerepresentedbyÞlledcirclesandÞlledsquares,respectively.Inourcalculationshere,thepotentialpartofthesymmetryenergyislinearindensity........................61Figure4.4(Coloronline)RatioofnetchargedpionyieldsincentralAu+Aucollisionsat400AMeVand200AMeV,asafunctionofthestiffnessofsymmetryenergy",frompBUUcalculationsusingN!-adjustedMF.Thedashedregionrepresentsthe400AMeVFOPImeasurement.ThetheoreticalerrorsareduetostatisticalsamplinginthepBUUcalculations..................62xiiiFigure4.5(Coloronline)S-wavecontributionto!!197Auopticalpotential.SolidlinerepresentstheworkofToki[5].Short-dash,dottedanddash-dottedlinesrepresentpionpotentialsfrompBUUparameterizationfor"=0.5,1.0,2.0,respectively,intheinteractionpartofthesymmetryenergy.LongdashedlinerepresentsthelackofcorrespondingpotentialsintheIBUUandImIQMDmodels.......................................64Figure4.6(Coloronline)ChargedpionratioincentralAu+Aucollisionsat200AMeV,asafunctionofkineticenergyinthecenterofmassframe,fordifferentvaluesofthestiffness"ofthesymmetryenergy,from0.5to2.0.Thehorizontallinerepresentstheratioofnetchargedpionyields...................66Figure4.7(Coloronline)Chargedpionratioincentral124Sn+132Sncollisionsat300AMeV,asafunctionofkineticenergyinthecenterofmassframe,fordifferentvaluesofthestiffness"ofthesymmetryenergy,from0.5to2.0.........67Figure4.8(Coloronline)Averagecenter-of-masskineticenergyof!+and!!incen-tralAu+Aucollisionsat200AMeV,plottedagainststiffness"ofthesym-metryenergy....................................68Figure4.9(Coloronline)Differencebetweenaveragec.m.kineticenergyof!+and!!incentralAu+Aucollisionsat200AMeV,plottedagainststiffness"ofthesymmetryenergy................................70Figure4.10(Coloronline)Ratioofneutron-to-protonnumbersatsupranormalnetdensi-ties,#>#0,incentralAu+Aucollisionsat200AMeV,asafunctionoftime.Atearlytimes,thenumbersintheratioaremarginal,andtheratio,thus,notverymeaningful...................................71Figure4.11Ratioofneutron-to-protonnumbers,incentralAu+Aucollisionsatthebeamenergyof200AMeVandimpactparameterb=1fm,asafunctionofkineticenergyincenterofmassframe..........................72Figure4.12Ratioofneutron-to-protonnumbers,incentral132Sn+124Sncollisionsatthebeamenergyof200AMeVandimpactparameterb=1fm,asafunctionofkineticenergyincenterofmassframe.......................73Figure4.13Ratioofneutron-to-protonnumbers,incentral132Sn+124Sncollisionsatthebeamenergyof300AMeVandimpactparameterb=1fm,asafunctionofkineticenergyincenterofmassframe.......................74Figure4.14Ratioofneutron-to-protonnumbers,incentral132Sn+124Sncollisionsatthebeamenergyof300AMeVandimpactparameterb=3fm,asafunctionofkineticenergyincenterofmassframe.......................75xivFigure5.1Averagenumberofbosonmoleculesasafunctionoftime,inasystemofsingle-modebosonmoleculesdissociatingintotwotwo-modefermions,fromasimulationwithintheGaussianoperatorrepresentation.Dashedlinerep-resentanalyticsolutionusingmeanÞeldapproximationinEq.5.49.Thesystemstartswith10moleculesandnofermions.InthespeciÞccase,thenumberbosonicmoleculesiscalculatedfromtheaverage4000trajectories.Thesamplingerrorisundercontrolforashorttimeandthengrowsdramatically.91Figure5.2Averagenumberofbosonmoleculesasafunctionoftime,inasystemofsingle-modebosonmoleculesdissociatingintotwotwo-modefermions,fromasimulationwithintheGaussianoperatorrepresentation.Dashedlinerep-resentanalyticsolutionusingmeanÞeldapproximationinEq.5.49.Thesystemstartswith20moleculesandnofermions.InthespeciÞccase,thenumberbosonicmoleculesiscalculatedfromtheaverage4000trajectories.Thesamplingerrorisundercontrolforashorttimeandthengrowsdramatically.92xvCHAPTER1INTRODUCTIONExplorationofbulknuclearpropertiesunderawiderangeofdensityandtemperatureisoneofthecentralgoalsofnuclearphysics.Nuclearequationsofstate(EOS)relatedifferentthermodynamiccharacteristicsofnuclearmatter,suchasenergyorpressurewithdensityandtemperature.NuclearmatteritselfstandsforaninÞniteuniformnucleonsystematsomeÞxedratioofneutrontoprotondensity,withCoulombinteractionsswitchedoff.TheEOSrelationsarerelevantformanyphysicalprocesses,e.g.excitationofgiantcollectiveresonances[6],thedynamicsinheavyioncollision(HIC)[7],thepropertiesofneutronstars[3],etc.InHIC,awiderangeofdensityandtemperatureisachievedinthecourseofsystemevolution,providingstudygroundstounderstandtheEOS.Inastrophysicalscenarios,pressureisoneimportantmacroscopicquantitythatlinkstothedatafromHIC.Forasystematnetdensity!,proton-neutronasymmetry"=(!n!!p)/(!n+!p),andtemperatureT=0,thepressurePisrelatedtoenergypernucleonEA(!,")withP(!,")=!2#EA(!,")#!.(1.1)Inmanysituationsinnuclearphysics,includingvariousmicroscopiccalculations,itisEA(!,")thatisarriveddirectly,andthentherelationaboveisusedtogetthepressure.Energypernucleoninnuclearmattercanbeexpandedinpowersoftheneutron-protonasym-metry"ofthesystem:EA(!,")=EA(!,0)+S(!)"2+O("4).(1.2)Onlyevenpowerssurviveintheaboveexpressionresultsfromthefactthat,nuclearinteractionissymmetricforprotonandneutron.TheÞrsttermE/A(!,0)representstheEOSforsymmetricnu-clearmatter(SNM),whichhasbeensigniÞcantlyconstrained.Atzero-temperatureE/A=(!,0)minimizesat-16MeVpernucleon,atnormaldensityof!0=0.16fm!3.Thenuclearincompress-1ibilityKisthescaledcurvatureofenergyatnormaldensity,followingistherelationbetweenincompressibilityandenergy:K=9!2"2EA"!2.(1.3)Nuclearincompressibilityhasbeendeterminedinnonrelativisticcalculationstobe240MeV±20MeV,bystudyingexcitationstotheGiantMonopoleResonance(GMR).However,relativisticcalculationsclaimabithighervalues,withK!250-270MeV[8].Nuclearincompressibilitycon-clusionsimpactresearchoncollectivemotioninHICandresearchonsupernovaeexplosionsandconversely.Inthisthesis,wereassessassumptionsonincompressibilityemployedintransportforHIC,beinginspiredbytherecentFOPI-IQMDanalysis[9].Inthesecondtermintheexpansionofenergyforasymmetricmatter,thecoefÞcientS(!)iscalledthesymmetryenergy.Differenteffortshavebeenundertaken,withmoderatesuccessatbest,toconstrainthedensitydependenceofsymmetryenergyat!<!0,suchasusingexperimentaldataonisospindiffusion,Pygmydipoleresonances,giantdipoleresonances,etc[10,11,12].For!>!0,ourknowledgeaboutthedensitydependenceofSdeÞnitelyremainspoor[13,4].Therefore,EOSforasymmetricnuclearmatterstillhaslargeuncertainties.1.1HeavyIonCollisionsIntermediateenergy(100AMeV-2AGeV)heavyioncollisionshavebeenapowerfultoolforextractinginformationonbulkpropertiesofnuclearmatter.Foremost,theyremainanimportanttestinggroundinnuclearphysicsforstudyingtheEOSathightemperatureandhighdensity.Heavyionnominallyreferstonucleiheavierthan4He,butsigniÞcantlyheaviernucleiaremoresuitableforEOSstudies.Figure1.1showstheschematicsketchofbeforeandafteranintermediate-energynuclearcollision.Theasymptoticdistancebetweenthecentersoftwonucleiabouttocollide,indirectiontransversetothemotion,isdeÞnedastheimpactparameterb.2Figure1.1Thebeforeandaftersketchofanintermediateenergynuclearcollision.Duringacollision,theregionofoverlapbetweenthenucleimayreachdensityashighastwotothreetimesthenormaldensity,dependingontheincidentenergyandimpactparameter.Theparticipantswithinthatregiongothroughcomplicatedinteractionsthatgiverisetoanexcited,nearlyequilibratedsystem.Newparticlesarecreatedifrelativeinter-particleenergiesareabovethethreshold;whenthenewlycreatedparticlestravelthroughthemedium,theyfurtherexperiencere-scatteringandmaygetreabsorbed.Astimeprogresses,elementaryparticlesandheaviernuclearfragmentsordertheirmotiontowardsaHubble-typeexpansion,withalocalcoolingdown,untiltheyÞnallyßyoutofthereactionregion.1.2PionsInsubatomicworld,particlesinteractwitheachotherbyexchangingaforcecarrier.Pionsactastheforcecarrierbetweennucleons(protonsandneutrons).Theattractiveresidualstrongforceholdsthenucleustogether.In1930,themassofpionswerepredictedbyHidekiYukawa,based3ontheuncertaintyprinciple,frommeasuringtherangeofthestrongforcebetweennucleons.Healsopredictedthatpionshavethreechargestates:positive,negativeandneutral.Chargedpionswerediscoveredin1947inthecosmicrayinteractionsandtheneutronpionwasnÕtdiscovereduntilacceleratorexperimentsin1950.Pionsarethelightestmesonsandarecomposedofupanddownquarks.Themassofchargedpionsis139.6MeV/c2,andofneutronpionis135.0MeV/c2.InintermediateenergyHIC,theyaretheÞrstmesonstobecreatedasenergyisraised.ChargedpionratiohasbeenidentiÞedasasensitiveobservabletosymmetryenergy,anditisfurtherstudiedinthisthesis.1.3Transporttheory1.3.1DensityfunctionalmethodTounderstandthemeasurementsinHIC,theoreticalmodelsareneededtofollowreactionprocessfromcontacttoproductdetection,andtoprovideguidancetotheunderlyingphysics.ThenumberofnucleonsinintermediateenergyHICrangefromtenstohundreds;thedegreesoffreedominvolvedaretoocomplicatedtobetreatedinafullquantummechanicalmanner.Transportmodelhasbeensuccessfulincharacterizingthenon-equilibriumdynamicsinanuclearreaction[14,15,16,17,18,19,20,21,22,23].Grossfeaturesofmanybodyquantumsystemsarenotlikelytodependondetailspertainingsimultaneouslytoallparticlesinasystem.Eventheinteractionofanyindividualparticlewiththerestisgenerallynotlikelytodependonsuchdetails.Withthatinmind,onecantrytoapproximatethecomplicatedmany-bodydynamicswithasimpliÞedonewherethedetailsofthemany-bodydynamicsareaveragedout.Oneprimaryexample,wherethisisemployed,istheHartree-Fockmethod[24,25].Initsbasicformofthemethod,thetwo-bodyinteractionsareaveragedoutoverpositionsofallparticles.Ifmore-particleeffectsmatterwithinasystem,adensityfunctionalisconstructeddependentonsingle-particleorbitals.ThecharacteristicsofaspeciÞcgroundstatearefoundthroughaminimizationofthatfunctional.Inthetime-dependentcase,equationsofmotion4arederivedfromavariationalprinciplewiththedensityfunctional.Innucleartransport,theelementarydynamicquantityistheparticlephase-spacedistribution.Withthis,thedensityfunctionaltheorybecomestheLandautheorythathistoricallyhasbeenputforwardinthecontextofFermiliquids.WithinthattheorysimpliÞcationsinthedirectionoftheHartree-Fockpracticearepossible,yieldinginparticulartheSkyrme-Hartree-Focklimitwherethefunctionalispresumedtodependonphase-spacedistributionsonlythroughsimplespatialdensitiesconstructedfromthephase-spacedistributions.Opticalpotentialsinthesingle-particledescriptionintermsofawavefunctionoraphase-spacedensityhavesimilarphysicscontentandthesedescriptionsformallymergeinthenuclearmatterlimit.AfunctionaldiscussedabovecanbeusedtodescribetheenergyanddensityproÞleforthegroundstateand,inamoregeneralsituation,theequationofstateforanexcitedsystem.However,thereisanotheraverageaspectofinteractions,inthatshort-rangeencountersbetweenparticles[26]canabruptlychangemomentaoftheparticlesandevenparticleexistence.Thisobviouslyofutmostimportanceinadynamicsituation,evenwhenitisofmarginalimpactonstationaryquantities.Inadescriptionintermsofsingle-particlewave-functionorbitals,thoseencountersareaccountedforintermsofimaginarypartofopticalpotential.IntheLandautheory,theshort-rangeencountersgetdescribedintermsofratesforprocessesoccurringovershorttimesanddistances.BothapproachesgetgeneralizedwithinnonequilibriumGreenÕsfunctiontheorywheresingle-particleequationscontaintermsthatdescribepropagationaswellassinkandsourceterms.1.3.2TransportmodelsEarlyon,cascademodelwasdevelopedtotreatnucleoncollisionsinafullymicroscopicmanner,itisabletodescribeinclusiveenergyspectrumsinintermediateenergyHIC[27].Thelimitationtocascademodelisthatitignoresthemean-Þeldeffectswhichareimportantfordescribingcollectivemotionsfornuclearmatter.Thisinspiresthedevelopmentofmoresophisticatedsemi-classicaltransportmodelslatertotakeintoaccountthenuclearmeanÞeldinthetheories.Currently,therearetwomaintypesoftransportmodelsusedinsimulatingHICdynamics.5TheBoltzmann-Uehling-Uhlenbeck(BUU)transportmodelutilizesBoltzmannequationstosimulateHICdynamics,itcontainingbothmeanÞeldandhardnucleon-nucleoncollisions.TheBoltzmannequationcanbederivedfromtheKadanoff-Baymequation[28]-itsamplesthequasi-particledistributionanddescribesthetimeevolutionoftheensemble.IthasbeenverysuccessfulinunderstandingthephysicsdrivingtheHIC.Effortsweremadetoimprovethemodelovertheyears.Bertschetal.[22]introducedquasi-particleinteractionsintheBoltzmannequations.DanielewiczandBertsch[23]introducedthree-bodyinteractionsforparticleformations.Andtheenergyfunc-tionalmethodforthemeanÞeldwasintroducedbyDanielewicz[29].Anotherimportanttypeoftransportmodeliscalledquantummoleculardynamic(QMD),whereindividualparticlesarerepresentedbyaGaussianwavepacketwithÞxedwidth,thetotalwavefunctionofthesystemisaproductofalltheGaussianwavefunctions.Theevolutionofthecoordinatesandmomentaforthewavepacketsaresolvedclassically,followingtheHamiltonianequationsofmotion.BothBUUandQMDmodelsaresemi-classical,andassignmomentumandpositiontoindividualparticles.Thenucleon-nucleoninteractionsareaccountedforinaneffectivemeanÞeldandinresidualinteraction.ParticlestravelthroughthemeanÞeld,andtheresidualinteractionsleadtocollisions.Phenomenologicalexpressionsforeffectivenuclearinteractions,yieldingtheEOS,areemployedinthemodels.Inthisthesis,weusetheBUUtransportmodeldevelopedbyDanielewiczetal.[23](oftencalledpBUU)asourtheoreticaltooltostudytheEOS.WewillelaborateonthemethodologyoftheBUUtransportmodelinthefollowingsubsection.1.3.3DetailsintheBUUmodelTheBoltzmannequationwasoriginallyderivedbyL.Bolzmannin1872foragasofclassicalparticleswithbinarycollisions[30],laterUehlingandUhlenbeckextendedtheequationtoquantumgas[31].TheBUUmodeladoptedthequasiparticleapproximationfromLandautheory,wherequasiparticlesaretheexcitationsofthestronglyinteractingsystem.ThesystemcanbespeciÞedbyobtainingtheoccupationofquasiparticlestates,i.e.thephase-spacedistributionfunctions.The6modelhasbeendevelopedbymanytheoristswithdifferentvariations[32,33,34,35,36,37,38,39,40].Inwhatfollows,wegivetheformulasthatunderlinedynamicsinthepBUUmodel.Intheenergyrangeofinterest,thespeciesaccountedforarenucleons,pions,!,N*resonances,andlight(A!3)clusters.MoredetailsontheproductionofparticlescanbefoundinRef.[41].TheBoltzmannequationforstableparticleshasthefollowingparticularform:!fX!t+!"X!!p!fX!!r"!"X!!r!fX!!p=K<X(1#fX)"K>XfX.(1.4)TheindexXaboveisfordifferentspeciesofparticles,and"Xisthesingleparticleenergy.Thesingleparticleenergyandmomentumformacovariantvectorpµ=(",!p).ThefactorsK<andK>,arethefeedingandremovalrates,respectively,forspeciÞcmomentumstates,andtheupperandlowersignsintheirexpressionsareforfermionsandbosons.ThepropagationofparticlesthroughthenuclearmeanÞeldisaccountedforonthelefthandsideoftheequation.Ontherighthandsideoftheequation,theelasticandinelasticinteractionsareincluded,whereparticlesgetdeßectedorabsorbed,newparticlesareformed,etc.Notethatinthecollisionterms,in-mediumcross-sectionsforthenucleonsareused.AndPauliblockingeffectsaretakenintoaccountbyexaminingthephasespaceoccupationoftheÞnalstates.Therateforremovalin2-bodyscatteringintheequationiswrittenas:K>X(!p1)=gx#1!d!p2(2$)3#2!d!p$1(2$)3#$1!d!p$2(2$)3#$2|M2X%2X$|2&(2$)3%(!p1+!p2"!p$1"!p$2)&2$%("1+"2""$1""$2)f2(1"f$1)(1"f$2)=gX#1!d!p2(2$)3#2!d"'$p'$24$2#'$1#'$2v'$12|M2X%2X$|2&f2(1"f$1)(1"f$2)=gx!d!p2(2$)3!d"'$v12d&d"'$f2(1#f1$)(1#f2$),(1.5)wheregXisthespindegeneracy,andthestarsrefertoquantitiesinthecenterofmassframe.Thecovariantvelocityisuµ=(#,#!v),andfactorsare#=1/"1"v2.v12istherelativevelocity7betweentheincomingparticles,anditisobtainedthroughthefollowingexpression:!1!2v12=[![(Páu2)u1!(Páu1)U2]2P2]1/2,(1.6)wherePisthe4-momentumofthe2-bodysystem.Thecrosssectionsintheformulasared"d!"#=p"#24#2!"1!"2v"12!"#1!"#2v"#12|M2x$2x#|2.(1.7)|M|2inthecrosssectionformularepresentsmatrixelementforscatteringamplitudebetweeninitialandÞnalstates.InpBUUmodel,pionsareproducedthroughthedecayof"orN*resonances.Thetransportequationsfortheresonanceshaveamoreelaborateform:$fXAX$t+$%X$!p$fXAX$!r!$%X$!r$fXAX$!p=K<X(1%fX)AX!K>XfXAX,(1.8)whereAXdescribesthemassdistributionoftheresonances,withawidthof#X:AX=#X(m!mX)2+1/4#X2.(1.9)Thederivativesonthelefthandsideoftheequationaretakenatconstantm!mX.InpBUUmodel,adensityfunctionalforthenetenergy(Hamiltonian)ofthesystemiscon-structedtoachieveenergyandmomentumconservations.Thesingle-particleenergyisrelatedtothenetenergyEofthespin-symmetricsystemwith:%X(!p,!r,t)=(2#)3gX&E&fX(!p,!r,t).(1.10)TheresultingsingleparticleenergyfromEq.(1.10)isusedastheinputonthelefthandsideofEq.(1.4)and(1.8).Thefourcomponentsofthenetenergyare:avolume,surface,isospin-dependentcomponentandaCoulombcontribution:E=!ed!r+Es+ET+Ecoul.(1.11)8ThesurfacetermisEs=a12!0!d!r(!!)2.(1.12)TheisospincontributionisET=aT2!0!d!r(!T)2,(1.13)wheretheisospindensityiscalculatedthroughsummingoverthethirdisospincomponentofallparticles:!T="Xt3X!X(1.14)AndtheCoulombtermisEcoul=18"#0!d!r!d!r!!ch(!r)!ch(!r!)|!r"!r!|.(1.15)Tosolvethenon-linearintegral-differentialBoltzmannequations,testparticletechniqueisoftenusedtosimulatethesolution.Thenon-equilibriumtimeevolutionofthesystemissimulatedthroughaMonte-Carloprocedure.EOSatzerotemperatureandthegroundstateofthesystemareextrapolatedfromÞnitetemperaturebehaviorandthenon-equilibriumstatesofthesystemthroughenergydensityfunctional.Inthenextchapter,wewilldiscussinmoredetailstheexpressionforthebulkenergydensity.9CHAPTER2CONSTRAINTSONTHEMOMENTUMDEPENDENCEOFNUCLEARMEANFIELD2.1MomentumdependenceofnuclearmeanÞeldIntransporttheory,bothmomentumindependentandmomentumdependentmeanÞeldhavebeenusedtodescribenucleon-nucleusinteractions.WiththemomentumindependentmeanÞeld,trans-portmodelswereabletodescribesidewardßowinHIC.However,asimpledensitydependenceinthemeanÞeldisnotsufÞcienttoexplainthemomentumdependenceoftheellipticßowobserv-able.CorrespondinglyamomentumdependentmeanÞeldwaslaterimplementedintothetransportmodels[42].Ontheotherhand,amomentumdependenceinthenuclearmeanÞeldhasalsobeenobservedinnucleon-nucleusscatteringexperiments[43].Thatis,anucleonexperiencesdifferentinteractionstrengthwhenapproachingthenucleusatdifferentmomenta.Whentherelativemo-mentumiszero,thenucleonfeelsanattractivepotentialwiththemagnitudeofabout50MeV.Thisattractivepotentialresultsfromthesumofanattractivescalarpotentialandarepulsivevectorpo-tential.ThemeanÞeldbecomeslessattractivewhennucleonapproachesathighermomenta,andÞnallyrepulsiveatk>3-4fm!1.Themomentumdependenceoriginatesfromthenon-localityofindividualnucleon-nucleoninteractions[44,45],fromtheexchangetermintheopticalpotential,fromintrinsicenergydepen-denceinthenucleon-nucleoninteractions(ortimenon-locality),etc.ItimpactsthedynamicsofnucleonsinHICandultimatelythefreenucleonemissionandfragmentproduction.Wecanin-vestigatethemomentumdependencebystudyingparticledifferentialyields,collectivemotionofnucleonsandstoppingobservables.102.1.1Momentum-dependentmeanÞeldinpBUUmodelInpBUUmodel,withamomentumindependentmeanÞeld,thesingle-particleenergiesarepa-rameterizedthroughfollowingexpression:!X=!p2+m2X(")+AXU1+T3XUT+ZX!,(2.1)wheremX(")=mX+AXU("),AXisthebaryonnumber,t3XisthethirdcomponentofisospinandZXisthechargenumber,forparticlespeciesX.WetakeU(#)=!a#+b#$1+(#/2.5)$!1,(2.2)where#="/"0,anda,b,$areparametersthatwillbedeterminedbyÞndingtherightminimumofEOSinnuclearmatteratnormaldensity,andalsobyrequiringtheincompressibilitytobecertainvalue.WehaveU1=-a1"2("/"0),UT=aT"T/"0,and!istheCoulombpotential."Tisthedensityofthethirdcomponentofisospin.ThecoefÞcientsa1andaTarethestrengthofthegradientandisospininteractions.NotethatinpBUU,thebulkofmeanÞeldU(")onlyactsonbaryons.PionsareinfrequentintheintermediateenergyHIC,thereforeareassumedtoonlybesubjectedtoisospindependentpartofthemeanÞeldinteraction.InthecaseofmomentumdependentmeanÞeld,thesingleparticleenergyisparameterizedinadifferentform:!X=mX+"p0dp"v#X+AX#"$"p10dp"%v%"+U(")%+AXU1+T3XUT+ZX!.(2.3)Here,themomentumdependencehasbeenimplementedforsymmetricnuclearmatter,throughtheparametrizationoflocalparticlevelocity,inthefollowingform[29]:v#X(p,#)=p&p2+m2X/(1+CmNmXAX#(1+&p2/m2X))2.(2.4)ThetwofreeparametersCand&arealsotobeÞxedincalculatingtheminimumoftheEOS.Stableparticlescontributetothetotaldensitythroughthefollowingexpression:"X=gX"d!P(2')3mX0e(!p)fX(!p),(2.5)11andthedensityassociatedwithresonancesare!X=gX!d!P(2")3!dE2"mX0mfX(!p,E)AX.(2.6)Whenparticletravelsthroughthemediumandisaffectedbyamomentum-dependentmeanÞeld,itappearstohaveadifferentmassthanwhenmovinginfreespace.ThisÕapparentmassÕ,relatingmomentumandvelocity,iscalledtheeffectivemass,deÞnedby:m!=p/v,(2.7)wherep=|!p|,andv=|!v|.TheeffectivemassisaconvenientwaytorepresentthemomentumdependenceofthemeanÞeldand,incomparedifferentMFs,itiscommontospecifytheeffectivemassatFermimomentumincoldnormalmatter.WiththeaboveparametrizationofthemomentumdependentmeanÞeld,pBUUhasbeensuc-cessfulindescribingvariousexperimentaldata[23,46].HoweverthemodelhasnotbeentestedagainstmeasurementsofpionmultiplicityatincidentenergiesnearNNpionproductionthreshold(e.g.400AMeV).Inthefollowing,weexaminethemomentumdependenceonthepionproductionandpionspectraincentralHIC.2.2PionobservablesWithinpBUUmodel,pionsareproducedthroughthedecayof!orN*resonancesinintermediateenergyHIC(100AMeV-2AGeV).Thechargeoftheproducedpionsfollowsfromrepresenting!isospinasasuperpositionofnucleonandpionisospinstates.n+n"n+!/N!,(2.8)!/N!"n+".(2.9)Toremindreaders,weshowhereagainthetransportequationsfor!orN*resonances:#fXAX#t+#$X#!p#fXAX#!r##$X#!r#fXAX#!p=K<X(1$fX)AX#K>XfXAX,(2.10)12wheresubscriptXrepresentdifferentparticlespecies,andAXdescribesthemassdistributionoftheresonances,withawidthof!X:AX=!X(m!mX)2+1/4!X2.(2.11)Intheabove,mXisthevacuummassforresonanceparticles,m"=1232MeV/c2,mN"=1440MeV/c2.Theproductionandabsorptionofpionsaredescribedthroughdecayoftheresonancesandasequenceofinverseprocesses.Fig.2.1showsnetpionmultiplicityobtainedwhenusingthemomentum-independentandmomentum-dependentmeanÞeldinpBUU,adjustedpreviouslytodifferentnuclearcharacteristicsanddata[29].SpeciÞcallyFig.2.1(a)showscalculationsdonewithmomentum-independentmeanÞeldandFig.2.1(b)withthepreviousßow-optimizedmomentum-dependentmeanÞeld.ThedatarepresentedintheÞgurearefromtheFOPImeasurementsofAu+Aucentralcollisions(impactparameterb<2fm)at400AMeV,800AMeVand1.5AGeV[1].Ascanbeseen,pBUUwithmomentum-independentmeanÞeldoverestimates,byafactoroftwo,themeasuredmultiplicitiesatallenergies.Withmomentum-dependentmeanÞeld,thecalculationsareconsistentwithdataatthetwohigherenergies,butat400AMeV,thepredictedyieldsareonlyabouthalfofthosemeasured.TheresultsofthecalculationssuggestthatsomeweakeningofthemomentumdependenceisrequiredinordertoarriveatanagreementbetweenthepBUUresultsandFOPIdataatthelowestofthebeamenergies.Otherthanmomentumdependence,weexploredpotentialimpactofin-mediumchangesinthe!and"productionrates[23,29]consistentwithdetailedbalance,butwefoundtheimpactofsuchchanges,withinplausiblerange,tobenegligibleontheÞnalyields.Furtheron,theparametrizationusedontherightpanelofFig.2.1willbereferredtoasv2-optimizedMF.13Figure2.1PionmultiplicityincentralAu+Aucollisions.SymbolsrepresentdataoftheFOPICollaboration[1].ThelinesrepresentpBUUcalculationswhenfollowingeitherthemomentum-independentMF(leftpanel)orthepastßow-optimizedmomentum-dependentMF(rightpanel).Solidlinesarepredictionsfor!!,anddashedlinesarepredictionsfor!+.Theexperimentalerrorbarsareaboutthesizeofsymbols.previousparameteriza-tion(v2-optimizedMF)newparameterization(N!-adjustedMF)C0.6430.300"[1/c2]0.9480.400a[MeV]203.92173.71b[MeV]65.1868.23#1.48381.6541K[MeV]210230m*/m0.70.75Table2.1ParametersusedinthepreviousandnewmomentumdependentMFs.Ineithercase,theparameterswereadjustedtoyieldsensiblenuclearincompressibilityKandnucleoneffectivemassm*.14Figure2.2PionmultiplicityincentralAu+Aucollisions,asafunctionofbeamenergy.SymbolsrepresentdataoftheFOPIcollaboration[1],,whilelinesrepresentthepBUUcalculationswiththeN!-adjustedmomentum-dependentMF.Theexperimentalerrorbarsareaboutthesizeofsymbols.Inthecontextofthediscrepancy,weexploreddifferentpossibilitiesforthemomentumdepen-denceofthemeanÞeldbymodifyingtheunderlyingparametrizationforthelocalparticlevelocity.Wetesteddifferentdensity-dependenciesofmomentum-dependenciesforthemeanÞeld,byre-placingthefactorinEq.(2.4),linearin",bydifferentfunctionsof"thatreducedto1atsaturationdensity,i.e.at"!##0=1.However,wefoundthesensitivityofpionyieldstothatreplace-menttobetoomeagertoeliminatethediscrepancybetweenthemeasuredandcalculatedpionyields.Ontheotherhand,wefoundthatamereadjustmentoftheparametervaluesintheoriginalparametrizationofEq.(2.4)couldreducesubstantiallythediscrepancybetweenthecalculatedandmeasurednetpionyields,withoutoverlycompromisingthedescriptionofmeasuredbaryonicßow15bythemodel.Inwhatfollows,werefertothemomentum-dependentmeanÞeldwiththenewparametersasN!-adjustedMF.ParametervaluesfortheN!-adjustedandpreviousv2-optimizedMFarelistedinTable2.1.Inthetable,nucleoneffectivemassisusedconventionallytorepresentthemomentum-dependentmeanÞeld;Cand"aretheparametersthatdictatethemomentumdependence.ThenetpionyieldsfortheN!-adjustedMFaredisplayed,togetherwiththedata,inFig.2.2.2.3OpticalpotentialcomparisonPropertiesofnuclearmatterandnucleonopticalpotentialshavebeenalsoafocusformicroscopiccalculationsstartingwithelementarynucleon-nucleoninteractions.Varioustheoriessuchasvari-ationalmethodofFriedmanandPandharipande[47]andBrucknerapproach[48,49,50,51,52],havebeendevelopedtoexplain,inparticular,themicroscopicoriginofthemomentumdepen-denceinnuclearmeanÞeld.Indataanalysis,themomentumdependencehasbeenreßectedintheneedtoreadjusttheopticalpotentialneededtodescribenucleon-nucleusscatteringatdifferentincidentenergies.IntestingthecharacteristicsoftheN!-adjustedMF,weexaminethemomentumdependenceofopticalpotentialsinzero-temperaturematter.FortheopticalpotentialUopt(p),weemploy,intherelativisticcontext,thefollowingoperationaldeÞnition:Uopt(#,p)=$(#,p)!!p2+m2.(2.12)Intheequation,$(p)isthesingleparticleenergycorrespondingtomomentump.OtherdeÞnitionshavebeenproposedintheliterature.Onceadopted,theyjustneedtobeusedconsistently.16Figure2.3Opticalpotentialinnuclearmatteratdifferentindicateddensities,asafunctionofmomentum.Dashedandsolidlinesrepresent,respectively,thev2-optimizedandN!-adjustedMFs.InFig.2.3weplottheopticalpotentialsforourtwoparametrizations,asafunctionofmo-mentum,withdifferentlinesrepresentingdifferentindicateddensities.Thedashedandsolidlinesrepresent,respectively,opticalpotentialsfromthev2-optimizedandN!-adjustedMFs.Themo-mentumdependenceinN!-adjustedMFisindeedsoftened,consistentwiththeexpectationdevel-opedonthebasisofFig.2.3.17Figure2.4Opticalpotentialinnuclearmatteratdifferentindicateddensities,asafunctionofnu-cleonenergy.Dashedandsolidlinesrepresent,respectively,UV14+UVIIvariationalcalculationsandourN!-adjustedMF.18Figure2.5Opticalpotentialinnuclearmatteratdifferentindicateddensities,asafunctionofnu-cleonenergy.Dashedandsolidlinesrepresent,respectively,AV14+UVIIvariationalcalculationsandourN!-adjustedMF.19Figure2.6Opticalpotentialinnuclearmatteratdifferentindicateddensities,asafunctionofnucleonmomentum.Dashedandsolidlinesrepresent,respectively,Dirac-Brueckner-Hartree-FockcalculationsandourN!-adjustedMF.20Figure2.7Opticalpotentialinnuclearmatteratdifferentindicateddensities,asafunctionofnucleonenergy.Dashedandsolidlinesrepresent,respectively,UV14+TNIvariationalcalculationsandourN!-adjustedMF.In[29],themomentumdependenceoftheopticalpotentialfromv2-optimizedmeanÞeldwascomparedtothatfoundforpotentialsfrommicroscopiccalculationsincludingthoserely-ingontheUrbanaV14two-bodyinteractioncombinedwithmodelVIIthree-bodyinteraction,i.e.UV14+UVII[47],AV14+UVII[47],aswellasDBHF[50,52],BBGandUV14+TNI[53,54].TheN!-adjustedMFproducesopticalpotentialsthatareclosestintheformandvaluestoUV14+UVII[47],withtherespectivecomparisonillustratedinFig.2.4.Similarcomparisonstotheothermi-croscopiccalculationscanbefoundinÞgures2.5-2.7.InFig.2.4,wecomparethesingle-particleenergy,"(p,#)!m,toUV14.Themomentumdependenceinthisrepresentationisimplicit.212.4EllipticßowFlowsignalsthemultipleinteractionsthatparticlesexperiencethroughoutthereaction.Alargernumberofinteractionsleadsthesystemclosertothermalequilibrium.EllipticßowisameasureofanisotropyofparticleemissioninHICinazimuthaldirectionsaroundthebeam-line.Inthepast,anisotropiesofcollectiveßowand,inparticular,theellipticßow,wereusedtotestthecharacteris-ticsofMFmomentum-dependenceincollisions[55,56]..Fig.2.8showsthebasicgeometryforaHIC[57].Theprojectilebeamisdirectedalongzaxis,xaxisisparalleltotheimpactparameterdirection,x-zplaneisdeÞnedasthereactionplane.Theyaxisisperpendiculartothereactionplane,orpointsoutofthereactionplane.Whentheimpactparameterbisnotzero,therewillbeanalmondshapedregionformedwherethetwonucleioverlapwitheachotherwhenpassingbyandwhereviolentinteractionstakeplace.Thisanisotropyinthespacewilltranslatetotheanisotropyinmomentumspaceforemittedparticles.Figure2.8SchematicdrawingofthegeometryinaHIC.Thebeamlineisalongzaxis,xaxisisparalleltotheimpactparameterdirection.x-zplaneisdeÞnedasthereactionplane,andyaxisisperpendiculartothereactionplane.Inexperiments,theanisotropiescanbeobservedthroughstudyinghowcorrelatedisparticle22emissioninazimuthaldirections.AschematicdescriptionoftheparticleemissionprocessisshowninÞgure2.9.Particleswithinthealmondshapedregionarecalledparticipants,andtherestoftheparticlesarespectators,astheymostlycontinuealongthebeamaxis,withoutexperiencingcollisionswithparticlesfromtheopposingnucleus.Figure2.9Particleemissionprocessesareshownwithrespecttothereactionplane.Theblockageofspectatorparticlesleadstotheout-of-planeemissionintheearlystageofthereactions.AconvenientwayofquantifyingtheemissionanisotropiesinthetheoryandexperimentistouseaFourierexpansionoftheparticledistributionswithrespecttoazimuthalangle:dNd!![1+2v1cos(!)+2v2cos(2!)+...],(2.13)wherev1andv2areknownascoefÞcientsofdirectedßowandellipticßow,respectively.ThevalueofdirectedßowandellipticßowcoefÞcientsareobtainedusingthefollowingexpressions:v1=<cos(!)>=!pxpy",(2.14)23v2=<cos(2!)>=!px2!py2px2+py2".(2.15)Mostcommonly,theellipticßowv2isstudiedatmidrapidity,i.e.y=0.Non-relativistically,therapidityyreducestoparticlespeedinunitsofc,butismoreconvenientinrelativisticcontext.TherapidityisdeÞnedwithy=12lnE+pzcE!pzc,(2.16)whereEandpxareinthesystemc.m.ThecoefÞcientv2>0correspondstoinplaneparticleemissionandv2<0correspondstooutofreactionplaneemission.ObviouslywhenmoredemandsareplacedonthenuclearmeanÞeld,suchastheproperde-scriptionoftotalpionyields,thedescriptionofthemeasuredellipticßowcannotgenerallystayasgoodasthatachievablewithoutthoseadditionalconstraints.Fig.2.10showstheout-oftoin-reaction-planeratio,R=1!v21+v2,forprotonsemittedatmidrapidityfrommid-peripheralBi+Bicollisionsat400AMeV,asafunctionofprotontransversemomentum.Thestrongertheellipticßow,thelargerthedeviationofRfrom1.TheÞlledtrianglesinFig.2.10representthedataoftheKaoScollaboration[2],whilethedashedandsolidlinesrepresent,respectively,thepBUUcalcu-lationswithv2-optimizedandN"-adjustedMF(denotedasMFv2andMFN",respectively,intheÞgures).ThetwocalculationsdescribeaboutequallywelltheKaoSdataatintermediatemomenta,butthev2-optimizedMFisfarsuperiorathighmomenta.24Figure2.10Ratioofoutofreactionplanetoin-planeprotonyields,asafunctionoftransversemomentum.SymbolsrepresentdatafromthemeasurementsoftheKaoSCollaborationofmid-peripheralBi+Bicollisionsatthebeamenergyof400AMeV(b!8.7fm)[2].SolidlinerepresentspBUUcalculationswiththeN!-adjustedmomentum-dependentMFanddashedlinerepresentscalculationswithv2-optimizedmomentum-dependentMF.Theindicatedtheoreticalerrorsaresta-tistical,associatedwiththeMonte-Carlosamplinginthetransportcalculations.25Figure2.11EllipticßowofparticlemassA=1particles,asafunctionoftransversemomen-tum.SymbolsrepresentdatafromthemeasurementsoftheFOPICollaborationofmid-peripheralAu+Aucollisionsatthebeamenergyof400AMeV(b!2.0"5.3fm).Theshadedregionrepre-sentspBUUcalculationswiththeN!-adjustedmomentum-dependentMF.Theindicatedtheoreti-calerrorsarestatistical,associatedwiththeMonte-Carlosamplinginthetransportcalculations.FurthercomparisonsweremadewithmorerecentexperimentaldatafromFOPIcollaborationdisplayedinFig.2.11-12[58].EllipticßowisplottedthereagainstthetransversemomentumoftheparticlesforAu+Aucollisionsatenergies400AMeVand600AMeV.Intheexperiment,thecentralityofthenuclearreactionsisdeterminedthroughmultiplicitymeasurementsofchargedpar-ticles,andM4intheÞgurescorrespondstotheimpactparameterwithintherangeof2.0-5.3fm.Thetransversemomentumisobtainedfrompt=!px2+py2.ItisapparentthatthepBUUsimu-lationsfailtoreproducetheexperimentaldataforellipticßowat600AMeV.ThedifÞcultyinthe26simultaneousdescriptionofhigh-momentumv2andnear-thresholdpionyieldsshowsthatchang-ingthemomentumdependenceinsuchasimplefashionisnotenoughtodescribevariousobserv-ables.Onepossibilityise.g.thelackofanisotropyinthemomentumdependence,foranisotropicmomentumdistributionsf,whenemployingEq.(2.4)-(2.6).WhileourimplementationEq.(2.4)-(2.6)ofthemeanÞeldmomentumdependence,withoutanisotropy,allowsinpracticeforahigherprecisionofcalculationsthanothermeanÞeldparametrizations[59],thatimplementationmayturnouttobeahandicaphere.Wealreadyundertooksteps,cf.theworkofSimonandDanielewicz[59],towardsimplementinganisotropywithoutcompromisingcalculationalprecisionorspeed.However,inchapter3,wewillshowanotherresolution,whichinsteadofonlyfocusingonthemomentumdependenceofthemeanÞeld,weinvestigatethecompetingeffectsofthemomentumandthedensitydependence.Weareabletoresolvethepuzzlebystudyingawiderrangeofthemo-mentumanddensitydependenceofthemeanÞeld,andusingpionandßowobservabletoconstraintheEOS.27Figure2.12Ellipticßowofproton,asafunctionoftransversemomentum.SymbolsrepresentdatafromthemeasurementsoftheFOPICollaborationofmid-peripheralAu+Aucollisionsatthebeamenergyof600AMeV(b!2.0"5.3fm).TheshadedregionrepresentspBUUcalculationswiththeN!-adjustedmomentum-dependentMF.Theindicatedtheoreticalerrorsarestatistical,associatedwiththeMonte-Carlosamplinginthetransportcalculations.28Figure2.13Transverserapiditydistributionofprotons.Rapidityvaluesyxarescaledwiththeprojectilerapidityinthecenter-of-massframe:yx0=yx/yp.Thetransverserapiditydistributionwithrespecttoyxm0isobtainedwithamidrapiditycutof|yz0|<0.5.TrianglesrepresentdatafromthemeasurementsoftheFOPICollaborationofcentralAu+Aucollisionsatthebeamenergyof400AMeV(b=1fm).ThesquaresrepresentpBUUcalculationswiththeN!-adjustedmomentum-dependentMF.29Figure2.14Transverserapiditydistributionoftritons.Rapidityvaluesyxarescaledwiththeprojectilerapidityinthecenter-of-massframe:yx0=yx/yp.Thetransverserapiditydistributionwithrespecttoyxm0isobtainedwithamidrapiditycutof|yz0|<0.5.TrianglesrepresentdatafromthemeasurementsoftheFOPICollaborationofcentralAu+Aucollisionsatthebeamenergyof400AMeV(b=1fm).ThesquaresrepresentpBUUcalculationswiththeN!-adjustedmomentum-dependentMF.30Figure2.15TransverserapiditydistributionofHelium3.Rapidityvaluesyxarescaledwiththeprojectilerapidityinthecenter-of-massframe:yx0=yx/yp.Thetransverserapiditydistributionwithrespecttoyxm0isobtainedwithamidrapiditycutof|yz0|<0.5.TrianglesrepresentdatafromthemeasurementsoftheFOPICollaborationofcentralAu+Aucollisionsatthebeamenergyof400AMeV(b=1fm).ThesquaresrepresentpBUUcalculationswiththeN!-adjustedmomentum-dependentMF.Asacomplementtoellipticßow,wehaveexaminedthetransverserapiditydistributionsofdifferentparticles.CalculationswerecarriedoutforAu+Aucentralcollisionat400AMeV,withtheN!-adjustedmomentum-dependentMF.Figs.2.13-2.15showthetransverserapiditydistribu-tionforproton,tritonandHelium3,respectively.TrianglesrepresentmeasurementsfromFOPICollaboration[1],andsquaresrepresenttheoreticalpredictions.Rapidityvaluesyxandyzarescaledwiththeprojectilerapidityinthecenter-of-massframe:yx0=yx/yp,yz0=yz/yp,wheresubscript31pstandsforprojectile.Thetransverserapiditydistributionwithrespecttoyxm0areobtainedwithamidrapiditycutof|yz0|<0.5.WefoundagoodagreementbetweenpBUUcalculationsanddataontransverserapiditydistributionsforalltheparticles.2.5ConclusionsInthischapter,werevisitedthemomentum-dependenceofthenuclearmeanÞeld.Previously,themeanÞeldimplementedinpBUUmodelhasbeentestedagainstßowdataonly.Wefoundherethatthemomentumdependencededucedfromßowalonefailedtodescribethepionmultiplicitiesnearpionproductionthresholdenergies.WemodiÞedthemomentumdependencetoresolvethediscrepancybetweentheoreticalpredictionandexperimentalmeasurements.Theimprovementinpionmultiplicities,however,gaverisetoasigniÞcantlyinferiordescriptionofellipticßow,andinspiredfurtherstudyofbothmomentumanddensitydependenceofthenuclearmeanÞeldpresentedinthenextchapter.32CHAPTER3CONSTRAINTSONNUCLEARINCOMPRESSIBILITY3.1IntroductionEquationofstate(EOS)ofinÞnitesymmetricnuclearmatter,typicallyconsideredatzerotempera-tureandexpressedintermsofenergypernucleonasafunctionofdensity,E(!),isoneofthemostimportantcharacteristicsofnuclearmatter.Astheenergyatthesaturationdensityreachesalocalminimum,theÞrstderivativethereiszero:dEd!|!0=0.Inconsequence,inordertodescribetheequationofstatewhenmovingawayfromsaturationpoint,informationonthecurvatureofE(!)isdesired.Expandingtheenergyaroundsaturationdensity,onegetsE(!)=E(!0)+118K!(!!!0!0)2+...(3.1)Here,theincompressibilitycoefÞcientK!ofsymmetricnuclearmatterisdeÞnedasthescaledcurvatureofEA(!):K!=9!2"2E/A"!2|!0.(3.2)HICdynamics,giantcollectiveoscillationsofnucleiandsupernovaeexplosionsareallsensi-tive,directlyorindirectly,tothenuclearincompressibility.Acomparisonofdataonisoscalargiantmonopoleanddipoleresonancestononrelativisticrandom-phase-approximationcalculationssug-gestedtheincompressibilityKvalueintherangeof220-233MeV[60,61,62]whilecomparisonstorelativisticcalculationsproducedastifferEOSwithKvalueintherange250-270MeV[8].Thesummaryconclusionfromthosecomparisons,givensomelimitationsinbothtypesofcalculations,wasofvalueforK=240±20MeV[63].33However,arecentreanalysisofcomparisonstogiantmonopoleresonanceenergiesshiftedtherangeK!to250<K!<315MeV[64].Withthis,thequestionaboutthevalueofK!remainssomewhatopen.Interestingly,ifoneforcedaparabolicÞttoenergytopassthroughzeroenergyatzerodensity,theresultingincompressibilitywouldhavebeenK!=18!16MeV=288MeV.Laterinthischapterwearriveatconstraintsonnuclearincompressibilitybasedonanalysisofellipticßowinheavy-ioncollisions.3.2IncompressibilityandisoscalarGiantMonopoleResonanceThecompressionalmodethatismostdirectlyrelatedtothenuclearincompressibilityistheisoscalargiantmonopoleresonance(IGMR).TheÞrstobservationofIGMRwasmadein1970s,establishingthecentroidexcitationenergyof208Pbat13.7MeV[65].Itwassubsequentlyfoundthatforheavynuclei,suchasSnandPb,theIGMRstrengthispeakedaround80A"1/3MeV[66].Incompress-ibilityforaspeciÞcnucleuscanbededucedfromtheIGMRenergyfollowingtherelation[66,67]:EIGMR=!øh2KAm<r2>.(3.3)AnalogoustotheenergyforaÞnitenucleus,theincompressibilityKAmayberepresentedintermsoffourmaincontributions:thevolumetermK!,thesurfacetermKsurf,thesymmetrytermKsymandtheCoulombtermKCoul[68]:KA=K!+KsurfA"1/3+Ksym(N"Z)2/A2+KCoulZ2A"4/3,(3.4)OnecanusetheaboverelationtoarriveattheincompressibilityforinÞnitenuclearmatter.Inthefollowingcontext,weomitthesubscript!inK!whenthereisnoambiguity.3.3EllipticßowInintermediate-energyHIC,mid-rapidityparticlesarepreferentiallyemittedout-ofratherthaninthereactionplane.Thisistermedsqueeze-outandistiedtonegativeellipticßow.Themagni-34tudeofthatellipticßowteststhemomentumdependenceofMFactingontheparticlesinthecollisionaswasdemonstratedinthepreviouschapter.Besides,itisalsoanimportantobserv-ableprobingthedensitydependenceofthenuclearMF,whichistiedtonuclearincompressibil-ity.Figs.3.1-3.3demonstratetheeffectsofemphasizedmomentum-dependenceanddensity-dependenceofnuclearmeanÞeldonellipticßowv2atmidrapidity.Themidrapiditywindowischosenas|y(0)|<0.1,wheretheparticlerapidityisscaledwithprojectilerapidityatcenterofmassframe:y(0)=(y/yp)cm.v2isplottedthereagainstthescaledtransversemomentumpernu-cleonp(0)t=(pt/A)/(pcmp/Ap),wherept=!p2x+p2y.Datausedforreference,intheÞgures,arefromtheFOPImeasurementsofthemid-peripheralAu+Aucollisionsat400AMeV[58],withtheimpactparametersrangingfrom2.0to5.3fm.Asthedetectionsystemdoesnotprovideiso-topedetermination,theclustermassnumberwasassumedtobeA=2ZforallelementsheavierthanH.Theoreticalresultswerecalculatedwitheitherv2-optimizedMFormomentumindepen-dentMF,andthecorrespondingincompressibilityof,either210MeVor380MeV.Inparticular,thev2-optimizedMFhasbeenoptimizedtotheimpactparameterdependenceofdirectedßow[69,42]andtomomentumdependenceofellipticßow[29].ThetwomeanÞelds[70]yieldaboutthesamedirectedßowinsemicentralcollisions,whenthatßowisintegratedovertransversemo-menta.However,ellipticßowcomparisonsofsimulationanddataforA=1-3particles,inFigs.3.1-3.3showsaclearpreferenceforthev2-optimizedMF,withalowerincompressibilityinpBUUmodel,thanmomentum-independentMFwithK=380MeV.Still,theÞguresdemonstratethatastrongdensitydependenceinEOSandMFmayhaveasigniÞcantimpactontheellipticßow.EvenwithnomomentumdependenceinMF,thestiffEOSyieldsellipticßowabouttwiceaslargeinmagnitudeasthesoftEOS.35Figure3.1EllipticßowofA=1(proton)particles,asafunctionofscaledtransversemomentum.CloseddotsrepresentdatafromthemeasurementsoftheFOPICollaborationofmid-peripheralAu+Aucollisionsatthebeamenergyof400AMeV(b!2.0"5.3fm).TheopensquaresrepresentpBUUcalculationswiththesoftequationofstate(K=210MeV),whichwasadjustedtoKaoSßowdata.ThetrianglesrepresentpBUUcalculationswithstiffequationofstate(K=380MeV)andnomomentumdependenceinMF.36Figure3.2EllipticßowofA=2(deuteron)particles,asafunctionofscaledtransversemomentum.CloseddotsrepresentdatafromthemeasurementsoftheFOPICollaborationofmid-peripheralAu+Aucollisionsatthebeamenergyof400AMeV(b!2.0"5.3fm).TheopensquaresrepresentpBUUcalculationswiththesoftequationofstate(K=210MeV),whichwasadjustedtoKaoSßowdata.ThetrianglesrepresentpBUUcalculationswithstiffequationofstate(K=380MeV)andnomomentumdependenceinMF.37Figure3.3EllipticßowofA=3(tritonandHelium3)particles,asafunctionofscaledtransversemomentum.CloseddotsrepresentdatafromthemeasurementsoftheFOPICollaborationofmid-peripheralAu+Aucollisionsatthebeamenergyof400AMeV(b!2.0"5.3fm).TheopensquaresrepresentpBUUcalculationswiththesoftequationofstate(K=210MeV),whichwasadjustedtoKaoSßowdata.ThetrianglesrepresentpBUUcalculationswithstiffequationofstate(K=380MeV)andnomomentumdependenceinMF.Fig.2.10-2.12intheprecedingchaptercomparedtheellipticßowfromFOPImeasurementstotheßowfrompBUUsimulationsutilizingthenewmomentum-dependentmeanÞeld(K=230MeV),whichwasÞttedtoobservedpionmultiplicities.Afterincludingthemomentumdepen-dence,theK=230MeVsimulationscouldnotreproduceanymoreadequatelytheellipticßow38data.However,observationsfromFigs.3.1-3.3suggestthatfurtherinvestigationsofthedensity-dependenceandmomentum-dependenceofthenuclearMF,mightresultinsomereasonablecom-binationofthosedependenciesthatwouldallowforasimultaneousdescriptionofpionandellipticßow.3.3.1EllipticßowandimpactparameterBeforeturningtooptimizationofanypotentialdetailsindensityandmomentumdependenceofMF,weexaminetheeffectofimpactparameteronellipticßowatmid-rapidity,tounderstandthepotentialsourceofuncertaintiesindrawnconclusions.Differentimpactparametersleadtodifferentgeometriesinnuclearreactions.Inacentralnuclearcollision,theoverlappingregionofthetwonucleiisclosetoasphere,whileinmid-peripheralcollisions,theoverlappingregionhasanalmondshape.Ontheotherhand,largerimpactparameterimplyfewerparticipantnucleonsandreducedmaximaldensitiescomparedtomorecentralcollisions.Thesefactorscontributetoproducingdifferentellipticßowsundervariousconditions.InFig.3.4,weplottheellipticßowfrompBUUcalculations,witheffectivemassm*/m=0.782andincompressibilityK=270MeV,asafunctionofimpactparameter,forAu+Aucollisionsat1.2AGeV.Thesmoothlineapproximatingtheresultsservestoguidetheeye.Themagnitudeofellipticßowrisesfrom0atb=0anditsaturatesathigherimpactparameters.39Figure3.4Protonellipticßowasafunctionofimpactparameter,forAu+Aucollisionsat1.2AGeV.DotsarecalculationpointsforsoftmomentumdependentMFwithm*/m=0.782andK=270MeV,thelineisforguidancepurpose.Inabsenceofdeformationforcollidingnuclei,duetosymmetry,theellipticßowneedstovanishatb=0.Asgeometryofthereactionbecomesmoreasymmetricwithincreasingimpactparameter,sodoesthemagnitudeofellipticßow.Athigherimpactparameters,theßowsaturatesforamomentum-dependentMF.Inanexperiment,theimpactparameterisdeterminedthroughacorrelationofreactionobserv-ables,suchasnetparticlemultiplicity,withreactioncentrality.SincesuchacorrelationalwayshasaÞnitewidth,theexperimentendsupselectingarangeofimpactparameters.Giventhatcrosssectionforselectinganimpactparametershrinksto0asimpactparameterapproaches0,theexperimentisnevercapableofselectingverylowimpactparameters.Guidedbycrosssectionconsiderations,foragivenrangeofimpactparameters,weselecttheimpactparameterequaltothequadraticmeanofthelargestandsmallestvalueofimpactparameterinexperimentforsimulations.40Thisapproachisassumedtobegenerallyvalid.However,weapproachtheissuewithmorecau-tionwhenaparticularobservable,suchasellipticßow,changesrapidly,particularlynonlinearly,withintheconsideredregionofimpactparametersandwetestthesensitivityoftheconclusionstothedecisionsontheimpactparameterinourcalculations.3.3.2EllipticßowandeffectivemassItisbelievedthatthemomentumdependenceofnuclearmeanÞeldplaysadominantroleinde-terminationoftheßowobservables.IthasbeenmentionedinChapter2thateffectivemassattheFermisurfaceisusuallycalculatedtolabelthedifferentmomentumdependentmeanÞeld.Toillustratetheimpactofeffectivemassonsqueeze-out,weshowinFig.3.5theanisotropyofprotontransversemomentumdistributionatmidrapiditycalculatedwithdifferenteffectivemassvalues.SpeciÞcallyshowntherearefortheratioofoutofthereactionplanetoin-planeprotonyieldscalcu-latedforBi+Bicollisionatthebeamenergyof400AMeVandimpactparameterofb=7.6fm.Thesolid,dot-dashedanddashedlinesrepresentpBUUcalculationswitheffectivemassm*/m=0.8,0.7and0.6,respectively.TheincompressibilityKforthosecalculationswassetto270MeV.TheÞlledtrianglesinFig.3.5represent,forreference,thedataoftheKaoSCollaboration[2].Cal-culationwithm*=0.7mproducesastrongerellipticßowthanothereffectivemassvalues,anditispreferredbytheexperimentaldata.Themomentum-dependencewithm*/m=0.8yieldssimilarellipticßowasthemomentum-dependencewithm*/m=0.6.Thenon-monotonicbehaviorofellip-ticßowwithchangesineffectivemasscanbeunderstoodasaresultofcompetingeffectsofthemomentum-anddensity-dependenceofMF.Forstrongmomentumdependence(e.g.m*/m=0.6),thedensitydependenceinMF,fromadjustingparameterstothesamenuclearincompressibility,isweaker.Duringthereaction,moreover,onaccountoftheenhancedrepulsionbetweennucleonswithlargerelativemomenta,nuclearmattergetslesscompressed.Lowermaximaldensitiesimplyfewercollisionsbetweennucleonsandslowerequilibration.Withlessequilibrationatmaximalcompression,thesqueeze-outsignalmaydropratherthanincrease,againstnaiveexpectationsfordroppingeffectivemass.41Figure3.5Ratioofout-of-the-reaction-planetoin-planeprotonyields,asafunctionoftransversemomentum,inmid-peripheralBi+Bicollisionsatthebeamenergyof400AMeVandimpactpa-rameterb!7.6fm.SymbolsrepresentdatafromthemeasurementsoftheKaoSCollaboration[2].Solid,dot-dashedanddashedlinesrepresentpBUUcalculationswithMFscharacterizedbyK=270MeVandeffectivemassm*/m=0.8,0.7and0.6,respectively.Withintheanalysisoftheprecedingchapter,wefoundaspeciÞcmomentum-dependentMFthatbestdescribedthepionproductioninHIC,forK=210MeV,butnotßow.InordertoseekaMFwithinpBUUmodelthatoptimallydescribesavarietyofobservables,wehavesubsequentlyengagedinextensiveinvestigationsoftheimpactofthemomentumdependenceintheMF.423.3.3EllipticßowandincompressibilityInthissubsectionweexplorethedependenceoftheellipticßowonnuclearincompressibility.InFig.3.6wepresentagaintheratioofoutofthereactionplanetoin-planeprotonyieldsforBi+Bicollisionat400AMeV,butnowemphasizingsensitivitytoincompressibility.TheÞlledtrianglesrepresentthedataoftheKaoSCollaboration[2].ThelinesrepresentpBUUcalculationscarriedoutwithaneffectivemassofm*/m=0.7.Thedot-dashed,dashed,solidanddottedlineshavebeenobtainedfortheincompressibilityvaluesofK=300,270,240and210MeV,respectively.Wecanseequiteastrongimpactofincompressibilityontheellipticßow.43Figure3.6Ratioofout-of-the-reaction-planetoin-planeprotonyields,asafunctionoftransversemomentum,inmid-peripheralBi+Bicollisionsatthebeamenergyof400AMeVandimpactpa-rameterb!7.6fm.SymbolsrepresentdatafromthemeasurementsoftheKaoSCollaboration[2].Dot-dashed,dashed,solidanddottedlinesrepresentpBUUcalculationswithincompressibilityK=300,270,240and210MeV,respectively,witheffectivemassm*=0.7m.ThecalculationwiththehighestemployedincompressibilityofK=300MeVgivesthestrongestßowasexpectedandtheonewiththelowestincompressibilityofK=210MeVgivestheweakest.However,similarlytothesituationwhenchangingtheeffectivemassonly,weobserveanon-monotonicbehaviorwhenchangingincompressibilityonlywithintherangeofvaluesK=240-270MeV.ThatbehaviorcanagainbeattributedtothecompetingeffectsofmomentumanddensitydependenceinthenuclearMF.SpeciÞcally,foralowerincompressibilitysuchasK=240MeV,thenuclearmatteriscompressedtohigherdensityduringcollisions,thanforahigherincom-44pressibility,suchasK=270MeV.Theeffectivemassgenerallydropswithincreasingdensity.Athigherdensitythenucleonsmoveathigherspeedsandundergomorecollisionsthatleadtoafasterequilibration.Intheend,thelowerincompressibilitymayleadtoastrongersqueeze-outsignal,particularlyathighmomenta.Wehavetestedthat,forincompressibilityhigherthan300MeVandlowerthan210MeV,theellipticßowmonotonicallyincreaseswithincreasingK.Fig.3.6demonstratedthesensitivityoftheellipticßowtothenuclearincompressibility,i.e.density-dependenceofnuclearMF.ObservationofthesigniÞcantsensitivityisimportantforusasopeningthepossibilityofsimultaneouslydescribingpionproductionandellipticßowinHIC.Clearlythedensity-dependenceofnuclearMFcanimpactthecollectivemotionofparticlesemerg-ingathightransversemomenta.Inthepast,thecollectivemotionwasconsideredwithacoarseinsertionorremovalofmomentumdependenceintoMF,butmoresubtlepointsoftheinterplaybetweenthemomentumanddensitydependenciesinMFwerenotstudied.3.4Constraintsonnuclearincompressibilityfromßowandpionobserv-ablesGiventhestrongeffectofincompressibilityonellipticßow,werevisitedtheunsolvedproblem,formulatedattheendofchapter2,ofdifÞcultyinsimultaneouslydescribingobservedpionmul-tiplicitiesandellipticßowwithinnucleartransporttheory.ThesensitivityofpionyieldstothemomentumdependenceofnuclearMFwasdemonstratedinchapter2.Therewefocusedonadjustingthemomentum-dependencetoreproducetheexperimentaldata.Inthefollowing,weexploretheconsequencesofmomentum-dependenceanddensity-dependenceofMFatthesametime,simultaneouslyvaryingtheparametrizationoflocalparticlevelocityandvalueofnuclearincompressibility.Givenlimitsonhowaccurateasemiclassicaltransporttheorycanbe,thevarietyofchallengesbeforeexperimentsstudyingmultiparticleÞnalstatesandageneralexperienceintheÞeld,itisnotgoingtomakesensetorequiremorethan!20%inaccuracyofdescribingdatawiththetransport45theory.Whenallowingforupto20%differencebetweenellipticßowandpionmultiplicitypre-dictionscomparedtoexperimentaldata,andallowingforsimultaneousvariationsineffectivemassandincompressibility,wearrivedatarangeofMFandEOSparametrizationsmeetingthecondi-tions.InFig.3.7,weshowresultsforprotonellipticßowfromthetwosetsofcalculationsthatbrackettherangeofEOSasfarasincompressibilityisconcerned,K=240MeVwithm*/m=0.782,andK=300MeVwithm*/m=0.582.Thetheoreticalresultsaredisplayedasafunctionoftrans-versemomentumandcomparedtodatafromtheKaoSCollaborationfromBi+Bicollisionsatb=7.6fmandeitherbeamenergyof400AMeV(triangles)or700AMeV(circles).Withinthe20%accuracyeitherEOSdescribesadequatelydataathigherpt,whereellipticßowissigniÞcant.46Figure3.7Ratioofoutofreactionplanetoin-planeprotonyields,asafunctionoftransversemomentum.SymbolsrepresentdatafromthemeasurementsoftheKaoSCollaborationofmid-peripheralBi+Bicollisionsatthebeamenergiesof400AMeVand700MeV(b=7.6fm)[2].ShadedregionsrepresentpBUUcalculationswithanoptimalEOSforincompressibilitybetweenK=240-300MeV.Next,inFig.3.8wecomparethepBUUpredictionsforyieldsofpions,positiveandnegative,respectively,inAu+Aucollisionsatb=1.4fm,obtainedfordifferentMFs,tothemeasurementsoftheFOPICollaborationat400AMeVand800AMeV.ThedataarerepresentedtherewithÞlledcircles.TheoptimalrangeofEOS,asfarasreproducingbothpionmultiplicityandellipticßow,isrepresentedwithshadowedregions.Subsequently,weshowinFigs.3.9-3.11calculationsfromtwoEOS,onewithK=210MeVandanotherwithK=270MeV(m*/m=0.75inbothcases).TheformerEOSisabletoreproduceonlypionmultiplicitiesbutnotellipticßow;thelatterEOS47reasonablydescribestheellipticßowbutnotpionmultiplicities.Thesearetwoexamplesshowingthatoutsideofourconstrainedrange,onecandescribeonlyoneoftheobservablesatthebest.Figure3.8PionmultiplicityincentralAu+Aucollisionsvsbeamenergy.SymbolsrepresentdataoftheFOPICollaboration[1].TheshadedregionrepresentpBUUcalculationswiththerangeofnuclearequationofstateÞttedtoellipticßowandpionyieldssimultaneously.48Figure3.9!+multiplicityincentralAu+Aucollisionsvsbeamenergy.SymbolsrepresentdataoftheFOPICollaboration[1].Solidanddashedlinesrepresentcalculationscarriedoutwithincom-pressibilityK=210and270MeV,respectively,andtheeffectivemassm*/m=0.75forboth.49Figure3.10!!multiplicityincentralAu+Aucollisions.SymbolsrepresentdataoftheFOPICollaboration[1].SolidanddashedlinesrepresentcalculationswithincompressibilityK=210and270MeV,respectively,bothwiththeeffectivemassm*/m=0.75.50Figure3.11Ratioofoutofreactionplanetoin-planeprotonyields,asafunctionoftransversemomentum.SymbolsrepresentdatafromthemeasurementsoftheKaoSCollaborationofmid-peripheralBi+Bicollisionsatthebeamenergyof400AMeV(b=8.7fm)[2].DashedandsolidlinesrepresentpBUUcalculationswithincompressibilityK=210and270MeV,respectively,bothwiththeeffectivemassm*/m=0.75.NextweturntoconstraintsonEOSthatfollowfromrequiringthattheassociatedMFsproduceasensiblesimultaneousagreementoftheoreticalpredictionswithbothdataonpionmultiplicityincentralcollisionsandonprotonellipticßow.IntheplaneofenergypernucleonincoldmattervsdensitywecrossouttheregioncoveredbytheEOSthatyieldsimultaneousacceptableagreementwithbothpionandellipticßowdata.Incollisionstowhichthedatapertain,thematterisexcited.Sinceweneedtoextrapolatetozerotemperature,webroadentheregionbyuncertaintyinextrap-olation.Wecomplementthatuncertaintybyoneduetothefactthatimpactofdifferentdensitiesis51averagedoverspaceandtime.OurÞnalconstraintsareshowninFig.3.12andarecomparedtheretotheconstraintsarrivedrecentlybytheFOPI-IQMDPartnershipanalyzingellipticßowfromtheFOPImeasurements.Asisapparent,ourconstraintsyieldasomewhatstifferEOSintheregionof!>1.5!0thantheFOPI-IQMDPartnership.Figure3.12Energypernucleonforsymmetricnuclearmatterasafunctionofscaleddensity.Thesolidlinesrepresentconstraints,upperandlower,ontheenergyarrivedatbytheFOPI-IQMDPartnership.Theshadedregionrepresentsourconclusions,withthepionyieldsandellipticßowtestingthesupranormalregion.Theverticaldashedlinesshowtheroughdensityregionthatgetsprobedbytheobservablesinthecalculations.52Figure3.13Nuclearequationofstateplottedaspressureintermsofscaleddensity.Patternedshadedareasrepresentconstraintsdeducebycomparingtransporttheorytodataonkaonmulti-plicityandondirectedandellipticßow.Thesolidlineontheleftrepresentstheequationofstatepreferredbyanalysisoftherecentgiantmonopoleresonance(GMR)experiment.DotsrepresentsthenoninteractingFermigasandtrianglesrepresenttherelativisticmeanÞeld(RMF)modelNL3.Theshadedregions,pinkandblue,representrespectivepressureconstraintsthatcanbededucedwhenrelyingoneitheranEOSwithK=240MeVorK=300MeVwithinthepBUUtransportmodel.Theverticaldashedlinesshowtheroughdensityregionthatgetsprobedbytheobservablesinthecalculations.Inasimilarmannerasinextractingenergypernucleon,weextractpressureasafunctionofdensityinnuclearmatter.OurconclusionsonpressurearerepresentedinFig.3.13andcompared53theretootherconclusionsdrawnintheliterature.EachofourboundaryEOSyieldsanuncertaintyregiontiedtotheextrapolationtozerotemperature.Bycomparingpressuredensityalongbeamaxisandtransversedirections,weestimatedtheerrorinthepressurefromthedifference.Thenetuncertainty,whenvaryingK,roughlycorrespondstothecombinationofthetworegions.Ifeventuallyanindependentpreferenceemergesforoneoranotherendoftheincompressibilityrange,theuncertaintyregionfortheEOSmayshrink.Inadditiontoourownconstraints,weshow,withpatternedregions,theconstraintsarriveinthepastintheliteraturewhenanalyzingcombineddirectedandellipticßowdataandwhenanalyzingkaonyields.ThesolidlinerepresentstheEOSpreferredbyarecentGMRanalysis.ForreferenceweshowfurtherthepressurefortherelativisticmeanÞeldmodelNL3andthepressureforanoninteractingFermigas,withÞlledtrianglesandcircles,respectively.Theverticaldashedlinesindicatethedensities,fromsimulations,thatgetprobedbytheobservablesweconcentrateon.Asonecansee,thereisgooddegreeofoverlapbetweendifferentconstraints.Ourconstraintsseemconsistentwiththosefromcombineddirectedandellipticßowanalysis,whenextrapolatedtohigherdensities.3.5ConclusionInthischapterwetestedthesensitivityofellipticßowtonuclearincompressibility,inadditiontothesensitivitytothemomentumdependenceofMF.WereexaminedtheparametrizationsofEOSandMFinpBUUmodelaimingatasimultaneousdescriptionofpionyieldsandprotonellipticßow,at20%level.WehavedemonstratedthatitispossibletodescribebothsetsofdatawhenassumingtheincompressibilitytobewithintherangeK=(240-300)MeV.Eventhoughtherangeofincompressibilitiesiswide,therangeofenergiespernucleonandpressuresisrelativelynarrowforhigherdensities,evenwhenaccountingforvariousuncertaintiesindrawingtheconclusions.54CHAPTER4CONSTRAINTSONSYMMETRYENERGYATSUPRANORMALDENSITIES4.1IntroductiontosymmetryenergyIsospinisaquantumnumberintroducedtoprovidemathematicalframeworkforthesymmetryofstronginteractionsassociatedwiththefactthatupanddownquarkhavenearlythesamemassonthescaleofenergiesrelevantforstronglyinteractingsystems.Proton(p)andneutron(n)areconsideredtobedifferentdirectionsforthesameparticleinisospinspace.Fortwonucleons,theexistenceofstabledeuterondemonstratesthatnetisospinT=0npinteractionisstrongerthantheT=1nnandppinteractions.Propertiesofstronglyinteractingnuclearmatterarelargelyaffectedbytheisospinstructureofthesystem.Therefore,thesymmetryenergy,whichisrelatedtothen-pimbalanceinanuclearsystem,hasbeenextensivelystudiedbynuclearphysicists.Asweshowedinthepreviouschapter,energypernucleoninanuclearmattercanbeexpandedinpowersoftheneutron-protonasymmetry!ofthesystem:EA(",!)=EA(",0)+S(")!2+O(!4).(4.1)ThecoefÞcientS(")istermedasthesymmetryenergy.Itisusuallyassumedthatcontributionsfromfourth-ordertermaresmall.ItiseasytoseetheeffectofsymmetryenergyintheBethe-Weizsackerformulaaswell.Thissemi-empiricalmassformulagivesagoodpredictionforthenuclearbindingenergies.LiquiddropconceptsareusedforjustiÞcationoftheformula,andthebindingenergyisexpressedintermsofÞvemainterms:EB=aVA!asurfA2/3!asym(N!Z)2/A!aCZ2/A1/3+Epair.(4.2)55Above,Aisthetotalnumberofnucleons,NisthenumberofneutronsandZisthenumberofprotons.ThecoefÞcientsaV,asurf,asym,aCrepresentsthestrengthofthevolumeterm,surfaceterm,symmetryterm,andCoulombterm,respectively.Thelasttermrepresentstheempiricallyparametrizedpairinginteraction.Thesymmetryenergytermaccountsfortheimbalanceofthepro-tonandneutronnumbersinanucleus.Duetoitsdensitydependence,thesymmetryenergypushestheexcessprotonsandneutronstothesurfaceofthenucleus,whilehelpingtobringthesystemtothelowestenergystate.Sucheffectisobservedintheexistenceofneutronskininasymmet-ricnuclei[71].Theneutronskinthicknessisdirectlycorrelatedtotheslopeofsymmetryenergywithdensity[13,72,73,74].Otherphysicalproperties,forinstance,nuclearmasses,isovectorGi-antDipoleResonance[12],andmulti-fragmentationinheavyioncollisionsareallaffectedbythenuclearsymmetryenergy[75,7,76,77,78,79,80,81].Inastrophysicalscenarios,supernovady-namics,proton-neutronstarevolution,neutronstarstabilityagainstgravitationalimpulsion,stellarradii,momentofinertia,ect.aredependentonthesymmetryenergyaswell[3,82,83].56Figure4.1Themultifacetedinßuenceofthenuclearsymmetryenergy.[3]Instablenuclei,thedensityinthecenterisaboutthenormaldensity!=0.16fm!3anditde-creasestozerowithinthesurface.ThetemperatureofgroundstatenucleiisbydeÞnitionzero.Inintermediateenergyheavyioncollisions,weareprobingthedensityregionaroundtwicethenormaldensityandthenuclearsystemsareathightemperatures.Inneutronstarcase,thedensityinthecentercanreachashighasninetimesthenormaldensity,whilethetemperatureisrelativelylow.Todescribepropertiesofnuclearsystemsinsuchdifferentregionsofthephasediagram,thebehaviorofsymmetryenergyasafunctionofdensityismuchneededinordertounderstandthepropertiesofvariousnuclearsystems.Figure4.1istakenfromref.[3].Itliststhephysicsob-servablesandtheoreticalmethodsthatareimportantforextractingandstudyingtheinformationaboutthedensitydependenceofthesymmetryenergyaswellasitsmagnitude.Inthischapter,weexploresensitiveobservablestoconstrainthehighdensitybehaviorofthesymmetryenergy.574.2MotivationAswepointedout,nuclearsymmetryenergyisanimportantquantitythatdirectlyrelatestomostofthephysicalpropertiesofneutronstars,isospindynamicsinHIC,etc.However,todate,densitydependenceofsymmetryenergyathigherthannormaldensityisnotwellconstrained.Figure4.2showsavarietyoftheoreticalexpectationsregardingthenuclearsymmetryenergy[4].Amongthe21setsofSkyrmeinteractionsshown,allhavebeenchosentoÞttothebasicnuclearpropertiesatsaturationdensity.However,theypredictverydifferentbehaviorofsymmetryenergybeyondthesaturationdensity.Forsomeoftheinteractionsthesymmetryenergymonotonicallyincreaseswithdensity(stiffsymmetryenergy),whileforsomeinteractionsthesymmetryenergystartstodecreaseathigherdensities(softsymmetryenergy).Figure4.2Densitydependenceofnuclearsymmetryenergyfor21setsofSkyrmeinteractionparameters.Symbolsrepresentmomentum-dependentinteractionsinIBUU04[4].58Withsuchlargeuncertaintiesinthetheoreticalexpectations,itisimportanttoÞndasensitiveobservableforexperimentstoconstrainthebehaviorofsymmetryenergyatsupranormaldensities.PionsproducedinHICgenerallyoriginatefromhigherthannormaldensityregions,sopionsmightserveasagoodprobeofthehighdensitybehaviorofsymmetryenergy.4.3ChargedpionratiosDuringintermediatestagesofheavy-ioncollisions,densityintheoverlapregionofthetwonucleicaneasilyreachvaluestwiceashighasthesaturationdensity.Pionsareproducedinthatregionthroughproductionanddecayofdeltaresonances,whentheinter-nucleonenergyexceedsthepionthreshold.InthepBUUtransportmodelparametrization,apartfromCoulombinteractions,pionsalsofeelapionpotentialthatdependsonisospinandsymmetryenergy.TheisospincontributiontotheenergyETintheparametrizationoftotalenergyforpBUUisET=4!d!rS(!)!2T!,(4.3)where!T=!X!Xt3Xandt3XisthethirdcomponentofisospinforspeciesX.Thesymmetry-energyfactorSabovecanbeconvenientlydecomposedasS(!)=Skin0"!!0#23+Sint(!),(4.4)wheretheÞrstr.h.s.term,withSkin0!12.3MeV,representsthesymmetryenergyinabsenceofinteractions,duetoPauliprinciple,andthesecondtermrepresentsinteractioncontribution.In[29]andthecalculationsheresofar,theinteractioncontributionwasofthesimplestpossiblelinearformSint0(!)=Sint0"!!0#.(4.5)However,thiscanbemodiÞedtoapowerparametrizationSint0(!)=Sint0"!!0#",(4.6)59formoregenerality.Largervaluesof!producesymmetryenergiesrisingquicklywithdensityaround"0.Suchsymmetryenergiesaregenerallytermedstiff.Lowvaluesof!yieldsymmetryenergieschangingslowlyaround"0.Thesearetermedsoft.DescriptionofnuclearmassesrequiresSint0!20MeV[84,85],bestaccompaniedbyapositivecorrelationbetweenSint0and!.Thepionpotential,asaresult,isU#±="8Sint0"T"!#1"!0.(4.7)Consistentlywiththe!/N$%n+#decay,theopticalpotentialfor!resonancessatisÞesU!=UN+U#,whereUNistheopticalpotentialfornucleons.PionproductionyieldsconsistentwithClebsch-GordancoefÞcientsleadtoasimplerelationbetweenprimordialchargedpionratioandneutron-protonratiooftheparticipantsfollowsfromconsiderationofthechainofprocesses[86]:##/#+&5N2+NZ5Z2+NZ'(N/Z)2.(4.8)Aboverelationshowsadirectcorrelationbetweenchargedpionratioandtheisospincontentinthesystems.Hence,pionobservablesinHICarealsoveryimportantforconstrainingthestiffnessofsymmetryenergy.LiwasÞrsttoproposethatchargedpionratioisasensitiveobservableforhighdensitybehaviorofsymmetryenergy[81].ThelinkbetweenthepionyieldratioandsymmetryenergyturnedoutsubsequentlytobelessstraightforwardthanÞrstproposed[81],though,withdifferenttransportmodelscontradictingeachother,asisinparticularillustratedinFig.4.3.60Figure4.3(Coloronline)PionratiosincentralAu+Aucollisions,asafunctionofbeamenergy.DataoftheFOPICollaborationarerepresentedbyÞlledtriangles.Theleftpanelcomparespre-dictionsfromIBUUandImIQMDmodelstothedata.TheIBUUcalculationsemployingstiffandsoftsymmetryenergiesarerepresentedtherebyÞlleddiamondsandÞlledcircles,respectively.TheImIQMDemployingstiffandsoftsymmetryenergiesare,ontheotherhand,representedtherebyÞlledsquaresandcrosses,respectively.TherightpanelcomparespredictionsfrompBUUmodeltothedata.Calculationsemployingv2-optimizedMFandN!-adjustedMFarerepresentedbyÞlledcirclesandÞlledsquares,respectively.Inourcalculationshere,thepotentialpartofthesymmetryenergyislinearindensity.Figure4.3displaysratiosofnetyieldsofchargedpionsstemmingfromcentralAu+Aucolli-sionsatdifferentbeamenergies.TheÞlledtrianglesrepresentmeasurementsoftheFOPICollab-oration[87].Othersymbolsrepresentresultsofdifferenttransportcalculations.Inthepanel(a)ofFig.4.3,itisseenthat,withinIBUUcalculations[88],astiffsymmetryenergygivesrisetoalower!!/!+ratiothandoesasoftenergy.However,theoppositeistruefortheImIQMDcalcu-lations[89],asseeninthesamepanel,whichisoneofthecurrentcontradictionsintheliterature,61mentionedbefore.Figure4.4(Coloronline)RatioofnetchargedpionyieldsincentralAu+Aucollisionsat400AMeVand200AMeV,asafunctionofthestiffnessofsymmetryenergy!,frompBUUcalculationsusingN"-adjustedMF.Thedashedregionrepresentsthe400AMeVFOPImeasurement.ThetheoreticalerrorsareduetostatisticalsamplinginthepBUUcalculations.Inourowncalculations,the"!/"+netyieldratioispracticallyindependentofthedetailsinthemomentumdependenceofMF,asillustratedinpanel(b)ofFig.4.3,whereweshowresultsutilizingbothv2-optimizedandN"-adjustedMF.TheresultsareobtainedforAu+Aucollisionsatb<2fm.WeuseherethelinearSint,Eq.(4.6),andeithersetofresultsagrees,withinstatis-ticaluncertainty,withtheFOPImeasurements.Importantly,wefurtherÞndthatthenetchargedpionratioandtheagreementwiththemeasurementsremainlargelyindependentofthestiffnessofsymmetryenergy.ThatisillustratedinFig.4.4,whereweshowpBUUresultsobtainedin62calculationsofcentralAu+Aucollisionsat200and400AMeV,whenchanging!inthesymmetryenergyEq.(4.7).4.4PionpotentialOnedetailinpBUUthatmaygiverisetodifferentsensitivitytothesymmetryenergyfornetpionyields,thaninothertransportcalculations,isthepresenceofastronginteractionpotentialactingonpionsanddrivenbyisospinimbalance.Thenon-zeropionpotentialisgiveninEq.4.7.InIBUUandImIQMD,suchstrong-interactionpotentialsactingonpionshavebeenlacking.Pion-nucleusopticalpotentialhasbeenusedtoexplaintheexistenceofpionicatoms.Pionicatomsaresystemsconsistingofanegativelychargedpionandpositivelychargedatomicnucleus.Becauseoftheheaviermass,pionhassmallerBohrradiusthanthatoftheelectron,providingbettertoolfortestingnuclearproperties.WhiletheCoulombinteractionattractsthepiontothenucleus,thestronginteractionrepelsthepionswhenthenucleushasmoreneutronsthanprotons.Tokietal.,inparticular,constructedapionpotentialthatsuccessfullydescribedthedeeplyboundstatesofpionicatoms[5].InFig.4.5,thepotentialinpBUU,forthreevaluesof!,iscomparedtothatofToki,for197Au.Giventhatourpotentialintheform(4.8)canonlyrepresentthesocalleds-wavecontributiontothe"-nucleuspotential,wedrop,inthecomparison,thesmallp-wavecontributiontothepotentialof[5].ThetailsaredifferentinourpotentialscomparedtoToki,duetoexcessivelyabruptchangesofdensityinthesemiclassicalThomas-Fermimodel(theT=0limitofourtransportmodel)inthesurfaceregion.ForpionsmovingacrossaHICzone,however,themostimportantisthemagnitudeofthepotentialoverregionswheredensitychangesslowly,includingnuclearinteriorinthegroundstate.Intheinterior,ourpotentialsfor!from1to2arewithin30%fromtheTokiÕspotential.63Figure4.5(Coloronline)S-wavecontributionto!!197Auopticalpotential.SolidlinerepresentstheworkofToki[5].Short-dash,dottedanddash-dottedlinesrepresentpionpotentialsfrompBUUparameterizationfor"=0.5,1.0,2.0,respectively,intheinteractionpartofthesymmetryenergy.LongdashedlinerepresentsthelackofcorrespondingpotentialsintheIBUUandImIQMDmodels.Thepotentialsofdifferentsignfor!+and!!,eachequalinmagnitudetothedifferencebetweenneutronandprotonmeanÞelds,andalsoadifferenceinthepotentialsfor!,mayproduceenoughdifferenceinthepropagationofchargedpionsinthepBUUrelativetoothermodelstoaffectpredictions.644.5DifferentialpionratiosWhilewefoundnosensitivityinpBUUofnetchargedpionyieldratios,aroundthreshold,toS(!),stillthegeneralidea[81]containsconvincingelements.Potentially,moredifferentialratiosofchargedpionyieldscouldprovideaccesstoS(!)atsupranormaldensities.InFig.4.6-4.8,weexplorethesensitivityofcharged-pionspectratothestiffnessofsymmetryenergy.TheÞrsttwoÞguresillustratethe"!/"+ratioasafunctionofpionc.m.energyandthethirdillustratestheaveragec.m.energiesforthechargedpions.Differenceintheaveragec.m.energies,between"+and"!,isadditionallyplottedinFig.4.8,asafunctionofthestiffness#ofthesymmetryenergy,forAu+Auat200AMeV.65Figure4.6(Coloronline)ChargedpionratioincentralAu+Aucollisionsat200AMeV,asafunc-tionofkineticenergyinthecenterofmassframe,fordifferentvaluesofthestiffness!ofthesymmetryenergy,from0.5to2.0.Thehorizontallinerepresentstheratioofnetchargedpionyields.66Figure4.7(Coloronline)Chargedpionratioincentral124Sn+132Sncollisionsat300AMeV,asafunctionofkineticenergyinthecenterofmassframe,fordifferentvaluesofthestiffness!ofthesymmetryenergy,from0.5to2.0.67Figure4.8(Coloronline)Averagecenter-of-masskineticenergyof!+and!!incentralAu+Aucollisionsat200AMeV,plottedagainststiffness"ofthesymmetryenergy.TheÞguresdisplaycompetingeffectsoftheisospincontentofthesystem,ofCoulombinterac-tionsandofthesymmetryenergy.Obviously,theneutronexcessgenerallymakesnegativepionsmoreabundantthanpositive,withtheeffectampliÞedbylargerisospinmagnitudeforthepionsthanforthenucleons.Thelong-rangeCoulombinteractionsplaytheprimaryroleinmakingthe!!/!+ratiodependentontheenergyoftheemittedpions.Thus,afterthepionsceasetointeractstronglyandmoveoutfromthereactionregion,describedthenbyprimordialspectrasharingtoadegreecharacteristicsbetween!+and!!(and!"),theCoulombinteractionsaccelerate!+anddecelerate!!.TherelativeCoulombpushbooststhe!!/!+ratiosatlowc.m.energies,abovetheoverallratioforthereactions,andlowerstheratiosathighc.m.energies,seeFigs.4.6and4.7.Thepushalsogivesrisetosubstantiallyhigheraveragec.m.energiesfor!+than!!,seeFigs.684.8.Contributionstomean-ÞeldpotentialsassociatedwiththesymmetryenergyprincipallyactoppositetoCoulombinteractions,buttheyactwhilepionscontinuetorescatter,infactwithlargecross-sectionsduetotheformationof!-resonance,downtolowdensities.Thescatteringtendstoerasetheimpactofdifferentaccelerationsfor!+and!!(andfornucleonsand!Õswithdifferentisospinaswell)duetotheisospin-dependenceofmeanÞelds.Withthescatteringratesbeinglinearindensity,themeanÞeldscanwinovertherescattering,inthelowdensityregion,iftheirdependenceondensityisslowerthanlinear.Thelow-energypartofthespectrumisgenerallydominatedbyparticlesemittedfromlowerdensityregions,lateinthehistoryofthereactions.InFigs.4.6and4.7,wecanseethatthesymmetryenergyisindeedeffectiveincounteringtheeffectsofCoulombenhancementofthelow-energy!!/!+ratio,when"<1andtheinteractionsymmetryenergyislargeatlowdensities.At">1,theeffectÞzzlesout.NotablyexcitationofthemediumsuppressestheroleofPauliprincipleandoftheassociatedkineticcontributiontothesymmetryenergy.InFig.4.9,wecanseethattheimpactofthestiffnessofsymmetryenergy,on!+!!!average-energydifference,weakenspast""1.69Figure4.9(Coloronline)Differencebetweenaveragec.m.kineticenergyof!+and!!incentralAu+Aucollisionsat200AMeV,plottedagainststiffness"ofthesymmetryenergy.Withregardtotheparticlesemittedathigherc.m.energies,thattendtostemfromearlystagesofthereactionandhigherdensities,anotherhigh-densityeffectofthesymmetryenergycomesintoplay.Namely,astiffsymmetryenergypushesawaytheneutron-protonasymmetryfromthehigh-densityregion[25],seeFig.4.10.Withthereductioninthehigh-densityasymmetry,the!!/!+ratiogetsreducedathighc.m.energies.Thus,qualitativelyastiffsymmetryenergyactsinthisenergyregionastherelativeCoulombboost,cf.Figs.4.6and4.7.Withthis,itbecomespossibletoaccessthestiffnessofhigh-densitysymmetry-energythroughthehigh-energy!!/!+yieldratio.70Figure4.10(Coloronline)Ratioofneutron-to-protonnumbersatsupranormalnetdensities,!>!0,incentralAu+Aucollisionsat200AMeV,asafunctionoftime.Atearlytimes,thenumbersintheratioaremarginal,andtheratio,thus,notverymeaningful.Intheearlierversion[90]ofthiswork,wealsoexploredthe"!/"+yieldratiointhedirectionoutofthereactionplaneasaprobeofthesymmetryenergyatsupranormaldensities.Inthatdirec-tionthehigh-densitymatterisdirectlyexposedtothevacuum.However,withahigherstatisticsinthecalculations,ourdirectionalsignal[90]forthesymmetryenergyhasweakened.4.6IsospinfractionationAstothesymmetryenergybelowsaturationdensity,andcomparingthesymmetryenergyatlowandhighdensity,oneusefulprobeistheisospinfractionation.Justasinnucleargroundstate,71theneutron-protonimbalanceinareactionislikelytomigrateincorrelationwiththebehaviorofthesymmetryenergywithdensity.Atlowdensity,asoftsymmetryenergyishigherthanastiffsymmetryenergy,andathighdensity,thestiffsymmetryenergyishigher.Withthis,inasystemsimulatedwithasoftsymmetryenergy,moreoftheneutron-protonimbalanceisexpectedtobepushedtohigherdensityandlesstolower,ascomparedtoasystemsimulatedwithastiffenergy.Theregionsofhighdensityaremorelikelytocontributetoemissionofparticleswithhighenergyandthoseoflowdensitytoemissionofparticleswithlowenergy.Thus,bystudyingrelativeyieldsofneutronsandprotonsasafunctionofparticleenergy,onemayassesshowsymmetryenergychangeswithdensity.Inthefollowing,weexaminethen/pratiosfrompBUUsimulationsasafunctionofparticlekineticenergy.Figure4.11Ratioofneutron-to-protonnumbers,incentralAu+Aucollisionsatthebeamenergyof200AMeVandimpactparameterb=1fm,asafunctionofkineticenergyincenterofmassframe.72Figure4.12Ratioofneutron-to-protonnumbers,incentral132Sn+124Sncollisionsatthebeamenergyof200AMeVandimpactparameterb=1fm,asafunctionofkineticenergyincenterofmassframe.WesimulatedthecentralnuclearcollisionsofAu+Au,aswellasof132Sn+124Sn,at200AMeVandimpactparameterofb=1fm.Asoftsymmetryenergywithparameter!=0.5andastiffsymmetryenergywithparameter!=1.75havebeenchosentotestthesensitivityofn/pratiostothestiffnessofsymmetryenergy.Theneutron-protonratiosasafunctionofthekineticenergyincenterofmassframeareplottedinFig.4.11fortheAu+AureactionsandinFig4.12for132Sn+124Sn.WeseeadeÞnitesystemdependenceofthen/pratioswhencomparingthetwoÞgures.Thenetneutron-protonasymmetry"atthestartofthereactionis!0.20forAu,and!0.22for124Sn+132Snsystem.Then-pratiosareindeedhigherforthesecondsystem,butalsotheyarehigherthannaivelyexpectedfromtheoverallneutron-protonimbalance.Thisisbecause73ofresidualregionsremainingattheendofareactionsimulation,persistingatmoderatesubnormaldensitythataregoingovertimescalesthatarelongcomparedtothoseforwhichsemiclassicaltransporttheoryisappropriate.Thoseregionstendtotrapprotonstoalargerextentthanneutronswhentheimbalanceispresent,enhancingthen-pratioforfreenucleons.Thetrappingitselfcandependonthesymmetryenergy.Alsonotethat,withdifferent!,n/pratiosaresmallcomparedtothepionratiosatthisenergy.Figure4.13Ratioofneutron-to-protonnumbers,incentral132Sn+124Sncollisionsatthebeamenergyof300AMeVandimpactparameterb=1fm,asafunctionofkineticenergyincenterofmassframe.74Figure4.14Ratioofneutron-to-protonnumbers,incentral132Sn+124Sncollisionsatthebeamenergyof300AMeVandimpactparameterb=3fm,asafunctionofkineticenergyincenterofmassframe.InFig.4.13,weshown/pratiocalculationsfor132Sn+124Sncollisionathigherbeamenergyof300MeV/nucleon,attheimpactparameterofb=1fm.Fig.4.12and4.13togetherdemonstratethedependenceofn/pratiosonthebeamenergy.Theoverallfallofthen-pratiowithbeamenergymaybeunderstoodinthefactthattheresidualremnantsaresmallerinsizeathigherbeamenergy,hencetheorderofmagnitudeoftheratiosforthereminderdriftstowardsn/pbalancefromtheoverallsystemasymmetry.Fig.4.14showsresultsfromacalculationof132Sn+124Sncollisionat300MeV/nucleon,butnowthecalculationisattheimpactparameterofb=3fm.Atahigherimpactparameter,theresidualregionincreasesinsizeandthen-pratioforfreenucleonsdriftsupagain.Irrespectivelyoftheimpactparameter,beamenergy,orsystemtype,thehigh-energyn-pratiois75alwaysßatterasafunctionofenergyforsoftsymmetryenergy,andthelow-energyratioisalwaysßatterforstiff,consistentlywiththeexpectationthatthedifferentialn-pyieldratiocanbeusedtoassessthesymmetryenergybothatthehighandlowdensity.4.7ConclusionsWithanewparameterizationformomentum-dependentMF,pBUUgivesareasonabledescrip-tionofpionmultiplicitiesinmoderate-energycentralHIC.ThepuzzlingÞndingisthatthesameparameterizationoftheMFmomentum-dependencecannotbesimultaneouslyusedindescribingthenetpionyieldsaroundthresholdandthehigh-momentumellipticßowofprotons.Onepoten-tialavenueforresolvingthispuzzleisintheadjustmentofnuclearincompressibilityasdescribedinChapter3.Wecomparedournewmomentumdependenceofnucleonicopticalpotentialwithseveralmicroscopiccalculations.ThemodiÞedpotentialiswithintherealmofuncertaintiesformicroscopicpredictions,justlikethepreviouspotential.Next,weusedpionratioobservablestostudythesymmetryenergybehaviorathigherdensitythannormal.WhileIBUUandImIQMDyieldopposingsensitivitiestothedensitydependenceofsymmetryenergy,for!!/!+netyieldratios,weÞndnosigniÞcantsensitivityofthatratiotoS(")inpBUU.OnefactoraffectingthatsensitivitymaybethepionopticalpotentialinpBUU,drivenbyisospinasymmetry.Weexaminedthedependenceofchargedpionratioonpionc.m.energyinpBUU.Toisolatetheeffectofsymmetryenergyatsupranormaldensities,welookedatthehighenergytailofthespectraÑwhereaclearsensitivityofpionratiotodifferentformsofsupranormalsymmetryenergyisseen.Additionally,thedifferenceofaveragec.m.kineticenergyofemitted!+and!!alsoshowsasensitivitytodifferentsymmetryenergies.InRef.[90],weappliedcombinedenergeticandangularcutstothepionratiosandproposeditasanewdifferentialobservableforfutureexperiments.Lastly,weexaminedtheimpactofthedensitydependenceofsymmetryenergyontheenergydependenceofn/pyieldratio.Dependingontheenergyregion,boththelow-andhigh-density76dependenceofthesymmetryenergycouldbetested.77CHAPTER5GAUSSIANQMCMETHOD5.1IntroductionSolvingstronglycorrelatedquantummany-bodyphysicsproblemshasbeenachallengeforthe-oreticalphysics.Becauseofthestrongimpactofinteractions,useofperturbationtheorycanbequestionableinsuchcases.Thestraightbruteforceapproachesarenotpracticalduetodimen-sionalityandtheintrinsiccomplexityofmany-bodywavefunctions.Forfermionicsystems,thenotorioussignproblemisencounteredinnumericalcalculations,thatinhibitsunderstandingofmany-bodyphysics.Thesignproblemisencounteredbecausefermionwavefunctionschangesignunderparticleinterchange.Innumericalcalculations,integrationsneedthentobecarriedoutoverfunctionsthatarehighlyoscillatorywithpositiveandnegativevaluesnearlycancelingeachother.Itbecomesnumericallyexpensivetoobtainaccurateresults,andsamplingerrorsbecomeverylarge.Thenumericalsignproblemisencounteredinnumericalcalculationsinmanyareasofphysics,includinglatticeQCDcalculationsofquarkmatterandcalculationsofultra-coldatomicFermigases.Incondensedmatterphysics,theproblemisencounteredwhentacklingsystemswithstronglycorrelatedelectrons.Innuclearphysics,wherenucleonsinteractwitheachotherthroughstrongforce,thesignproblemlimitsapplicationofab-initiomethodstolightnucleionly.TheGaussianphase-spacerepresentationmethodhasbeendevelopedinthecontextofdynamicandstaticproblemsinultra-coldatomicphysics[91,92,93].ItaimsatsimulatingBosonandFermionsystemsfromÞrst-principles.ThedensitymatrixofasystemisexpandedintheGaussianphase-spacebasisthat,bybeingovercomplete,allowsforexclusivelypositivedeÞniteexpansioncoefÞcients.ThequantummasterequationforthedensitymatrixisnextcastintotheformofaFokker-PlanckequationinthespaceofexpansioncoefÞcients.AMonte-CarlosamplingisusedinsolvingthestochasticItostochasticequationsequivalenttotheFokker-Planckequations.The78purposeofstudyingtheGaussianphase-spacerepresentationmethodinthisthesisistoanalyzethemethodologyandseekopportunitiesforitsapplicationtoproblemsinnuclearphysics.5.2PhasespacemethodsIn1932,Wignerbroughttheideaofanexpectationvalueinquantummechanicsthatcorrespondedtotheclassicalprobabilityfunctiondistributioninspatialcoordinatesandmomenta[94].Later,otherconstructionswereputforwardthataimedatprovidingsimultaneousinformationonspaceandmomentumforasystem.TheseincludedtheHusimiQ-function[95],P-representation[96,97],complexP-representation,positive-Prepresentation[98,99],squeezed-stateexpansion[100],etc.5.2.1Classicalphase-spacerepresentationsIn1963,Glauber[96]andSudarshan[97]independentlydevelopedanimportantclassicalphasespacemethodemployingthecoherentstatesasabasisforrepresentingthedensitymatrixofbosonicsystems:!!="P(!")|!"!"!"|d2M!".(5.1)Intheabove,Misdimension,!"isanM-modecoherentstate,P(!")canbeinterpretedastheprobabilitydensity.ThemethodwaslatercalledGlauber-SudarshanP-representation,anditwassuccessfullyemployedinthequantumlasertheories.However,sincethespeciÞcbasiscouldnotdescribeentangledstates,theapplicationofclassicalphasespaceexpansionbecamequitelimited.Inaddition,duetoincompletenessofthemapping,systemevolutioncouldresultsinnegativevaluesforP,incompatiblewithitsinterpretationasaprobability.5.2.2Quantumphase-spacerepresentationsIn1980s,apositiveP-representationwasproposed,modiÞedrelativetooriginaltoallowforquan-tumentanglementintheexpansionbasis:79!!="P(!",!#)|!#!"!"|"!"|!#!d2M!"d2M!#.(5.2)Nowthenumberofvariableshasbeendoubled,andoffdiagonalmatrixelementsgotincludedintheexpansion.ThecoefÞcientsinthenewbasiscanbechosenaspositiveandthemethodworkswellforBosonicsystems.However,complexityariseswithfermioniccoherentstatesinthatGrassmannnumbersgetemployedinthebasisdecomposition.Coherentstateshavebeen,inparticular,usedinpathintegralsforFermions.TheissueisthataGrassmannnumberisamathematicalconstructionmadetoobeyanti-commutationrelationsandspeciÞcalgebraicrulesanditcannotbesimplytreatedasprobability.5.3Gaussianphase-spacerepresentationGaussianphasespacerepresentationusesamoregeneralizedbasisthancoherentstates,forbothBosonsandFermions.TheFermionoperatorsarerepresentedinpairs,hencenoGrassmannnum-bersareneededwithinthealgebra.!!(t)="P(!$,t)ö!(!$)d!$(5.3)whereP(!$,t)istheprobabilitydistribution,ö!isamemberofthegeneralizedbasisand!$rep-resentsthephasespacecoordinates.Regardingaconnectiontotheclassicalphase-spacerepre-sentation,ö!(!$)correspondsto|"!""|inthefoundingformula,andinconnectiontothequantumphase-spacerepresentation,ö!(!$)correspondsto|#!""|.TherealtimeorimaginarytimeevolutionofdensitymatrixcanbecastintoaLiouvilleequationform:%%tö!(t)=öL(ö!(t)).(5.4)UponinsertionoftheexpansionofthedensitymatrixintotheLiouvilleequation,thepartialdifferentialequationacquiresanintegro-differentialform:80!dP(!!,t)dtö!(!!)d!!=!P(!!,t)öL[ö!(!!)]d!!.(5.5)ActionsintheLiouvilleoperatorcanbemappedontooperationsinvolvingexpansioncoefÞcients(probabilitydistribution)labeledintermsofphase-spacevariables,suchasderivativeswithrespecttothevariablesandmultiplicationbythevariables,i.e.ö!.Integratingbypartsandassumingvan-ishingofcoefÞcientsatinÞnityforaboundedsystem,wearriveatthefollowingintegro-differentialequationsforthedistributionfunctionsP(!!,t).!dP(!!,t)dtö!(!!)d!!=!L!P(!!,t)ö!(!!)]d!!.(5.6)Foranyarbitrarybasis,thedynamicalequationforthedistributionfunctioncanbeobtained:dP(!!,t)dt=L!P(!!,t).(5.7)Ifinteractionsinthesystemareofone-bodyandtwo-bodytypeonly,wearriveataFokker-PlanckequationforP(!!,t)containingonlyÞrst-orderandsecond-orderderivatives.dP(!!,t)dt=["p"a=0""!aAa(!!)+12p"a,b=0""!a""!bDab(!!)]P(!!,t).(5.8)TheformofmatricesAandDisdeterminedduringthemappingprocess,andtheybothturnouttobepositive-deÞnite.ItcanbeshownthatthereexistsanItostochasticequationequivalenttotheaboveFokker-Planckequation.Asaresult,thedifferentialequationforPcanbesolvedviaMonte-Carlosamplingofthestochasticequationwithinthephase-spacewith!!asacoordi-nate.ExpectationvaluesoftheobservablescanbeobtainedfromcalculatingmomentsofP.Inthefollowingsubsections,wedescribetheGaussianphase-spacerepresentationsforBosonsandFermionsseparately.5.3.1Gaussianphase-spacerepresentationsforBosonsForanM-modebosonicsystem,öarepresentsacolumnofannihilationoperators,andöa€arowofcreationoperators.ThoseoperatorssatisÞesthecommutationrelations:81[öai,öa€j]=!ij.(5.9)Forconvenienceinmanipulations,wenextintroduce2M-vectorsforcomplexnumbersandoperators"andöa,andadisplacementoperator!öa:!öa=öa!"=!"#öaöa€$%&!!"#ö"ö"€$%&(5.10)ThegeneralGaussianbasiscanbegivenaquadraticformintermsofthe2M-vectors:ö!(!#)="'|$|:exp[!!öa€$!1!öa/2]:(5.11)Intheaboveexpression,::representsnormalordering,introducedforalgebraicpurposes,inwhichallcreationoperatorsaremovedtotheleftoftheannihilationoperators.InthecaseofBosons,thereisnosignchangesassociatewithnormalordering::öa€öa:=öa€öa,(5.12):öaöa€:=öa€öa.(5.13)Normalorderingappliedtoanexponentialfunctionofoperatorsimpliesanexpansionoftheexpo-nentialintoseriesandapplicationofthenormalorderingtoeachtermintheseries::exp(#öa€öa):=#$n=0#nn!öa€nöan.(5.14)Thecomplexmatrixelementsin$playtheroleofphase-spacevariablesthatcanhaveprincipallymorephysicalcontentthanjusttheclassicalcoordinates(!r,!p):$=()*I+nmm+I+nT+,-.(5.15)Here,nandm,m+arecomplexMbyMmatrices,andmandm+aretwoindependentsymmetricmatrices.Withthosewecanputtogetherthephase-spacevariablesforBosons:82!!=(!,","+,n,m,m+).(5.16)Amongthevariables,!istheweightofdifferenttrajectories,ofuseincaseofimaginary-timeevolution,"representstheeigenvalueoftheannihilationoperatoröaassociatedwiththecoherentstate|"!,öa|"!="|"!."+istheconjugateto".Thematrixelementsofnrepresentthenormalcorrelationbetweendifferentpairsofmodes;matrixelementsofmarethecorrelationsofpairsofannihilationoperators,andm+arethecorrelationsofpairsofcreationoperators.ThegeneralityoftheGaussianbasismaybereducedand,inparticular,theGaussianbasisforthepositive-Prepresentationtakesonthefollowingform:ö"P(!,",#)=!|"!"##|"##|"!.(5.17)Matrices(n),(m)and(m)+arezerointhiscase.Inthiscase,themappingbetweentheoperatorsactionsandphase-spaceoperationsisasfollows:ö"=!$$!ö",(5.18)öaö"="ö",(5.19)öa€ö"=[#+$$"]ö",(5.20)ö"öa=["+$$#]ö",(5.21)ö"öa€=#ö".(5.22)835.3.2Gaussianphase-spacerepresentationsforFermionsForanMsingle-particlemodesfermionicsystems,creationandannihilationoperatorsforeachmodesatisfytheanticommuntationrelations:[öbk,öbj€]=!kj,(5.23)[öbk,öbj]+=0.(5.24)Thesubscriptskandjherespanvaluesfrom1toM.Forconvenienceinmanipulations,wecandeÞneanM-columnvectoröbconsistingofannihilationoperators,andanM-rowvectoröb€ofcreationoperators.WecanmoreoverdeÞneanextendedvectorwith2Moperatorsas:öb=!"#öböb€T$%&(5.25)Thephase-spacevariablescanbesimilarlycombinedintoanarrowvector!",andconsequentlyamemberoftheGaussianoperatorbasiscanbecastintothefollowinggeneralform:ö!(!")="1N:exp[!öb€#b/2]:.(5.26)Intheabove,NisthefactorthatcombineswiththetraceofthefollowingGaussianoperatorto1.Thedimensionofthecomplexmatrix#is2M"2M,andthatmatrixcanbeexpressedas#=(#!1!2I).(5.27)Thematrixelementsof#aresimilarlydeÞnedandhavesimilarphysicalcontentasinthecaseofBosons:#='()nT!Imm+I!n*+,(5.28)84ThematrixIisdeÞnedwithI=!"#!I00I$%&(5.29)Forfermionicsystems,thephase-spacecoordinatesare!!=(!,n,m,m+).(5.30)TocalculateanexpectationvalueofanoperatoröO,onecanusethefollowingexpression:"öO#=Tr[öOö"]/Tr[ö"]='P(!!,t)Tr[öOö"]d!!'P(!!,t)!d!!$"O(!!)#P.(5.31)5.4PropertiesofGaussianphase-spacemethodforFermionsSincetheGaussianphase-spacerepresentationiscloselytiedtothepositivePrepresentation,var-iousdesiredfeaturesofthebasisextendinanaturalmannerfromonetoanother.Fornuclearphysicsmoreimportantisthefermioniccaseandthosefeaturesarelessobviousinthatcase.Hence,weconcentratenowontheaspectsofthebasisinthatcase.SomedetailsrelevanttothederivationcanbefoundinthepaperbyJ.F.Corney[92].5.4.1Single-modeGaussianoperatorFirst,wearriveatexplicitexpressionsforsingle-modeGaussianoperators.SimilarlytothenormalorderingoftheexponentialfunctionofBosonicoperators,thesingle-modeGaussianoperatorforFermionscanberepresentedasasumofseries:ö"%1(µ)=:exp[!µöb€öb]:=:#$k=01k!(!µöb€öb)k:=1!µöb€öb.(5.32)TheGaussianoperatorisleftwithtwotermsonlybecauseofthePauliprincipleforFermionsandthenormalorderingappliedwithineachterm.Thetraceoftheoperatoraboveisthen2!µ.Upon85introducingn=(1!µ)/(2!µ),and÷n=1!n,thenormalizedGaussianoperatorcanbenextexpressedintermsofanewvariable:ö!1(n)=÷n:exp[!(2!1/µ)öb€öb]:=÷nöböb€+nöb€öb.(5.33)Equivalentlywecanwriteö!1(n)=÷n|0"#0|+n|1"#1|.(5.34)WecanseethattheGaussianbasisisacompletebasisforthenumber-conservingsingle-modeHilbertspace.WithinthatspaceanydensitymatrixcanbeexpandedintoGaussianoperatorswithpositive-deÞnitecoefÞcients.ö!=÷nö!1(0)+nö!1(1).(5.35)5.4.2CompletenessToprovethecompletenessofaGaussianphase-spacebasis,weÞrstexpandthedensitymatrixusingstateswithdeÞniteoccupations:ö!="!n"!m|!n"#!n|ö!|!m"#!m|.(5.36)Withinthebasisof|!n"#!m|,eachstatecanbeoccupiedbyzerooroneFermion.Intotal,thenumber-stateprojectorhas22Mindependentelements.Theproductsofstatesformacompletebasisforthedensityoperator.Infact,sincethatbasiscontainsnon-Hermitianmatrices,itisnecessarylyanovercompletebasis.OnecanÞndnon-uniqueexpressionsfortheexpansionofthedensitymatrix,ashasbeenshowninthesingle-modecase.865.5FreegasAnoninteractingFermigasinequilibriumcanbenaturallydescribedusingtheGaussianphase-spacerepresentation.Thus,theHamiltonianforfreegasonlycontainsdiagonalsingle-particleenergyterms:öH=öb€!öb.(5.37)Theevolutionequationforgrandcanonicaldensitymatrixininversetemperature/imaginarytimeisdd"ö#=!12[H!µN,ö#],(5.38)$$"ö#=!12(öb€!öbö#+ö#öb€!öb).(5.39)Here,"isthescaledinversetemperatureT=1/kB",andµisthechemicalpotentialassociatedwithparticlenumber.Thefollowingmappingrelationsapplyhereforthethermalstates:öb€öbö#"[nk!$$nk(1!nk)nk]P,(5.40)ö#öb€öb"[nk!$$nk(1!nk)nk]P,(5.41)ö#"!$$!!P.(5.42)Inconsequence,wearriveattheÞrst-orderFokker-Planckequation:$P$"="k!k[$$nk(1!nk)+$$!!]nkP.(5.43)Theaboveequationcanberecastintothedifferentialequationset:87ú!=!"k!k!nk,(5.44)únk=!!knk(1!nk).(5.45)TheFermi-Diracdistributionsreadilyfollowasasolutionofthesecondpartofthesetandyieldnk=1e!k"+1,(5.46)!=!0#ke!!knk".(5.47)5.6Fermi-BosemodelingFermi-Bosemodelhasbeenhistoricallyusedincondensedmatterphysicsforsimulatingthefor-mationoftwoelectronsintoabosonicCooperpair.Inthecontextofultra-coldatomicphysics,themodelrepresentsdissociationofaBosemoleculeintotwoatomicconstituents(twoBosonsortwoFermions)andtheirreassociation.WetestedtheutilityoftheGaussianoperatorrepresentationforstudyingtimeevolutionwithinamodelwhereasingle-modeBosondissociatesintotwotwo-modeFermions.TheHamiltonianofthesystemis:H=øh#öa€öa+øh!(öb€1öb1+öb€2öb2)+øhg(öa€öb1öb2+öaöb€2öb€1).(5.48)Intheabove,öa€(öa)isthecreation(annihilation)operatorforthequantaofthebosonicÞeld,andöb€i(öbi)arecreation(annihilation)operatorsofthefermionicsingle-particleenergylevelsin,e.g.differentspinstates.TheconservedcombinationoftheatomicparticlenumbersisN=2öa€öa+öb€1öb1+öb€2öb2.(5.49)88AstheHamiltonianisquadraticintheoperators,higher-orderproductsofannihilationandcreationoperatorswillbefactorizedtoproductsofthenormalandanomalousdensities:n1=öb€1öb1=n2=öb€2öb2,m=öb1öb2,m+=öb€1öb€2.WeconsiderauniformsysteminacubicboxofsideL,withatomicparticledensity!0,andthetotalnumberofatomsequaltoN=!0L3.ApplyingthemeanÞeldapproximation,wherethebosonicoperatora(t)isreplacedbyarealfunction"(t),wecanarriveatthedifferentialequationsforthenormalandanomalousdensities:dnidt=2g"0Re{m},(5.50)dmdt=!2i(#!$)m+g"0(1!2ni).(5.51)wherei=1,2.Thesolutiontotheaboveequationsare:ni=(g"0)2(g"0)2+(#!$)2sin2(!(g"0)2+(#!$)2t),(5.52)m=g"0"(g"0)2+(#!$)2cos(!(g"0)2+(#!$)2t)sin(!(g"0)2+(#!$)2t)!ig"0(#!$)"(g"0)2+(#!$)2sin2(!(g"0)2+(#!$)2t).(5.53)Theaveragedbosonicmoleculenumberscanbenextcalculatedexploitingconservationofatomicparticlenumbers(Eq.5.44):"(t)2=N/2!(g"0)2(g"0)2+(#!$)2sin2(!(g"0)2+(#!$)2t).(5.54)Ontheotherhand,whenemployingtheGaussianoperatorbasis,thedensitymatrixofthesystemmaybedescribedintermsofsixindependentcomplexvariables:(","+,n1,n2,m,m+).Theactionofdifferentoperatorscanbemappedonthedensityoperator.ApplyingthemappingidentitiesinEq.5.17-Eq.5.20,theequationsofmotionforthesevariablesfollow:ún1=ig("+m!"m+)!"in1(mW#1+m+W#2),(5.55)ún2=ig("+m!"m+)!"in2(mW#1+m+W#2),(5.56)89úm=!ig!(1!n1!n2)+"i[!("!#)m2W#1+n1n2W#2],(5.57)úm+=ig!+(1!n1!n2)+"i[n1n2W#1!("!#)m2W#2],(5.58)ú!=!igm!"iW1,(5.59)ú!+=igm++"iW2.(5.60)Inthoseequations,W1(t),W2(t)representcomplexGaussiannoisesthatsatisfytherelations:$Wi(t)Wi%(t%)&=0,(5.61)$Wi(t)W#i%(t%)&=$ii%$(t!t%).(5.62)WehavesolvedthissetofLangevinequationsnumericallyandwedisplaytheaveragenumberofmoleculesasafunctionoftimeinFig.5.1-5.2.Forsimplicity,thevaluesfortheparametersinHamiltonianhavebeenchosenas:g=1,!0=!N/2,"!#=!0.1.Thedashedlinesrepre-sentanalyticsolutioninEq.5.49.ThedifferencebetweentheGaussianQMCmethodandtheanalyticsolutionisduetonon-uniformityofthebosonicÞeldandduetoÞnitenumberofbosonicmolecules.IntheÞgureonecanobservethat,uptosmallerrorsthatshrinkassamplingisin-creased,themethodproducestheexpectedgradualdecreaseofBosonmoleculenumbertowardsequilibrium.However,beyondacertaincriticaltime,theerrorincreasesdramatically.Foragivensamplingsize,suchadramaticgrowthoferroriseventuallyreachedaroundatimethatlittlede-pendsonsamplesize.Changingthesystemsizeortimestepinthecalculationhavelittleeffectonsamplingerror.90Figure5.1Averagenumberofbosonmoleculesasafunctionoftime,inasystemofsingle-modebosonmoleculesdissociatingintotwotwo-modefermions,fromasimulationwithintheGaussianoperatorrepresentation.DashedlinerepresentanalyticsolutionusingmeanÞeldapproximationinEq.5.49.Thesystemstartswith10moleculesandnofermions.InthespeciÞccase,thenumberbosonicmoleculesiscalculatedfromtheaverage4000trajectories.Thesamplingerrorisundercontrolforashorttimeandthengrowsdramatically.91Figure5.2Averagenumberofbosonmoleculesasafunctionoftime,inasystemofsingle-modebosonmoleculesdissociatingintotwotwo-modefermions,fromasimulationwithintheGaussianoperatorrepresentation.DashedlinerepresentanalyticsolutionusingmeanÞeldapproximationinEq.5.49.Thesystemstartswith20moleculesandnofermions.InthespeciÞccase,thenumberbosonicmoleculesiscalculatedfromtheaverage4000trajectories.Thesamplingerrorisundercontrolforashorttimeandthengrowsdramatically.Agoodcontrolofthesamplingerrorforacertaintime,followedbyadramaticgrowthintheerrorseemstobeagenericfeatureoftheGaussianrepresentationmethod.Notably,thegrowthintheparticularcaseisaroundthetimewherethesystemappearstoreachequilibriumandequilib-riumisexpectedbothfortheaveragenumberofmoleculesandßuctuationofthenumberaroundequilibrium.Qualitativelydifferentwayofcontrollingsamplingerrorforlongtimesmaybeputforward,exploitingßexibilityinthestochasticrealizationoftheevolutionequations.Inthiscon-92text,weexploredthreespeciÞcstochasticgaugesaimingatcontrolofthesamplingerror.FirstwastheFermigauge.ForFermionoperators,anytermwithmorethantwofermionicoperatorsvanishesonaccountofthePauliprinciple.Correspondingly,onecanaddsuchtermstotheHamil-tonianwithoutaffectingthephysics,whilealteringthestochasticstrategy.ThesecondgaugeweexploredwastheDiffusiongauge.Thefreedomofchoosingthatgaugefollowsfromthesquarerootofamatrixbeingnon-unique.InmappingtheFokker-PlanckequationsontotheItostochasticequations,onehastoevaluateasquarerootofthediffusionmatrix-theextrafreedominthediffu-sioncoefÞcientswillchangethestochasticnoise.Thethirdgaugeweexploredisthedriftgauge.Thefreedomofemployingthatgaugefollowsfromthefollowing.WeattemptedtousethelattertwogaugesinordertochangethenoiseinstochasticcalculationsandextendtheperiodoverwhichsolutionstotheevolutioninGaussianoperatorrepresentationscouldbeofutility.However,anyimprovementsincontrollingthenoiseforlongtimesturnedouttobenegligible.SimilardifÞcultieshavebeenencounteredelsewherewhenemployingtheGaussianrepresentations,nomatterwhetherasystemofFermionsorBosonswasinvestigated.5.7ConclusionInthischapter,wediscussedtheGaussianphase-spacerepresentationmethod,whichwasbeforediscussedintheliteratureonlyinthecontextofeithercondensedmatterphysicsorphysicsofultra-coldatomicsystems.ThemethodallowsforaÞrst-principleapproachinsolvingquantummany-bodyproblems,ofspecialbeneÞtforstronglycorrelatedsystems.ThemethodreliesonageneralovercompleteGaussianbasis,bothforFermionsandBosons,andonexpansionoftheden-sitymatrixofasystemwithinsuchbasishaspositivedeÞnitecoefÞcients.ThosecoefÞcientcanbeinterpretedaspositiveprobabilityweightsallowingtousetheMonte-Carlosamplingforsolvingthetimeevolution.ThisapproachinprinciplelargelyimprovestheefÞciencyinnumericalcalcu-lations,andisabletodealwithHamiltoniansoflargedimensionsthatmoretransitionalmethodscannotcopewith.Themethodcanbeemployedtosolveimaginarytimeaswellasrealtimeprob-93lems.Thelimitationofthemethodisthatthesystemshouldonlyinteractuptotwo-bodyforcetomaintaintheexpansioncoefÞcientspositive.Abruptdivergenceinthesamplingerrorappearstopermeatethereal-timeevolutionsintheGaussianrepresentationsandhasbeenillustratedherewithintheFermi-Bosemodel.94CHAPTER6CONCLUSIONSWhenparameterizationofmomentum-dependentMFischanged,pBUUcanprovideareasonabledescriptionofpionmultiplicitiesinmoderate-energycentralHIC.ThepuzzlingÞndingisthatthesameparameterizationoftheMFmomentum-dependencecannotbesimultaneouslyusedfordescribingthenetpionyieldsaroundthresholdandthehigh-momentumellipticßowofprotons.Wecomparedournewmomentumdependenceofnucleonicopticalpotentialwithseveralmicro-scopiccalculations.ThemodiÞedpotentialiswithintherealmofuncertaintiesformicroscopicpredictions,justlikethepreviouspotential.InspiredbytherecentellipticßowanalysisoftheFOPI-IQMDPartnership,westudiedthedensitydependenceofnuclearmeanÞeldtoarriveatconstraintsonequationofstateofsymmetricmatter.Byexploringawiderrangeofnuclearincompressibilityandmomentum-dependenciesthanbefore,wewereabletoreproducepionyieldsandellipticßowsimultaneously.OnthebasisofthatanalysiswecanconcludethattheincompressibilityKfornuclearmatterisintherangeof240-300MeV.ThededucedrangeofenergiespernucleonandpressurefornuclearmatteratmoderatelysupranormaldensitiesisfairlynarrowandrepresentssomewhatstifferEOSthanclaimedbytheFOPI-IQMDPartnership.Thededucedconstraintsshouldbeofutilityinastrophysicalmodelingofsupernovaexplosionsandneutronstars.Next,weusedpionratioobservablestostudythesymmetryenergybehaviorathigherdensitythannormal.WhileIBUUandImIQMDyieldopposingsensitivitiestothedensitydependenceofsymmetryenergy,for!!/!+netyieldratios,weÞndnosigniÞcantsensitivityforthatratiotoS(")inpBUU.OnefactoraffectingthatsensitivitymaybethepionopticalpotentialinpBUU,drivenbyisospinasymmetry.Weexaminedthedependenceofchargedpionratioonpionc.m.energy.Toisolatetheeffectofsymmetryenergyatsupranormaldensities,welookedatthehighenergytailofthespectraÑthereaclearsensitivityofpionratiotodifferentformsofsupranormal95symmetryenergyisseen.Additionally,thedifferenceofaveragec.m.kineticenergyofemitted!+and!!alsoshowsadistinguishingpowerfordifferentsymmetryenergies.InRef.[90],weappliedcombinedenergyandangularcutstothepionratiosandproposeditasanewdifferentialobservableforfutureexperiments.Finally,weexploredtheutilityoftheGaussianphase-spacerepresentationmethodsforsolvingquantummany-bodyproblems.WeexaminedthebasicpropertiesoftheGaussianoperatorbasis,andapplicationstothefreegasproblemand3-modeFermi-Bosemodel.Weexaminedtheuseofgaugetermsforcopingwithsamplingerrors.Themethodingeneralworksbetterformanyweakly-interactingparticlesthanforafewstrong-interactingparticles.Unfortunately,thepotentialadvantageoftheGaussianoperatormethodforHIC,overanyothermethod,wouldbeexpectedinthelatterlimit.96BIBLIOGRAPHY97BIBLIOGRAPHY[1]W.Reisdorfandetal.(FOPICollaboration).Nucl.Phys.A,848:366,2010.[2]D.Brillandetal.Z.Phys.A,355:61,1996.[3]A.Steinerandetal.Phys.Rep.,411:325,2005.[4]B.A.Liandetal.Phys.Rep.,464:113,2008.[5]H.Tokiandetal.Nucl.Phys.A,501:653,2009.[6]U.Garg.Nucl.Phys.A,791:3,2004.[7]M.B.Tsangandetal.Phys.Rev.C,76:034603,2007.[8]G.A.LalazissisandP.RingJ.Konig.Phys.Rev.C,55:540,1997.[9]A.LeFevre,Y.Leifels,W.Reisdorf,J.Aichelin,andCh.Hartnack.Nucl.Phys.A,945:112,2016.[10]M.B.Tsang,Y.Zhang,P.Danielewicz,M.Famiano,Z.Li,W.G.Lynch,andA.W.Steiner.Phys.Rev.Lett.,102:122701,2009.[11]A.Klimkiewiczandetal.Phys.Rev.C,76:051603,2007.[12]L.Trippa,G.Colo,andE.Vigezzi.Phys.Rev.C,77:061304,2008.[13]B.A.Brown.Phys.Rev.Lett.,85:5296,2000.[14]R.Serber.Phys.Rev.,72:1114,1947.[15]N.Metropolis,R.Bivins,M.Storm,J.Miller,G.Friedlander,andetal.Phys.Rev.,110:204,1958.[16]H.W.Bertini.Phys.Rev.,131:1801,1963.[17]J.Cugnon.Phys.Rev.C,22:1885,1980.[18]P.CarruthersandF.Zachariasen.Rev.Mod.Phys.,55:245,1983.[19]G.Bertsch,H.Kruse,andS.Gupta.Phys.Rev.C,29:673,1984.[20]H.StockerandW.Greiner.Phys.Rep.,137:277,1986.[21]W.Bauer,G.Bertsch,W.Cassing,andU.Mosel.Phys.Rev.C,34:2127,1986.[22]G.BertschandS.DasGupta.Phys.Rep.,160:189,1988.[23]P.DanielewiczandG.Bertsch.Nucl.Phys.A,533:712,1991.98[24]D.R.Hartree.Proc.CambridgePhil.Sco.,24:89,1928.[25]J.C.Slater.Phys.Rev.,81:385,1951.[26]M.Borromeo,D.Bonatsos,H.Muther,andA.Polls.Nucl.Phys.,A539:189,1992.[27]J.Cugnon,T.Mizutani,andJ.Vandermeulen.Nucl.Phys.A,352:505,1981.[28]L.P.KadanoffandG.Baym.QuantumStatisticalMechanics,1962.[29]P.Danielewicz.Nucl.Phys.A,673:375,2000.[30]L.Boltzmann.SitzungsberichteAkad.Wiss.,66:275,1872.[31]E.A.UehlingandG.E.Uhlenbeck.Phys.Rev.,43:552,1933.[32]W.Cassing,V.Metag,U.Mosel,andK.Niita.Phys.Rep.,188:363,1990.[33]S.Teis,W.Cassing,M.Effenberger,A.Hombach,U.Mosel,andG.Wolf.Z.Phys.A,356:421,1997.[34]J.Aichelin.Phys.Rep.,202:233,1991.[35]C.Hartnackandetal.Eur.Phys.J.A,1:151,1998.[36]Q.Li,J.Wu,andC.Ko.Phys.Rev.C,39:849,1989.[37]B.Blaettel,V.Koch,andU.Mosel.Rep.Prog.Phys.,56:1,1993.[38]S.Bass,M.Belkacem,M.Bleicher,M.Brandstetter,L.Bravina,andetal.Prog.Part.Nucl.Phys.,41:255,1998.[39]E.Santini,M.Cozma,A.Faessler,C.Fuchs,M.Krivoruchenko,andetal.Phys.Rev.C,78:034910,2008.[40]W.EhehaltandW.Cassing.Nucl.Phys.A,602:449,1996.[41]P.Danielewicz.Phys.Rev.C,51:716,1995.[42]P.Danielewicz.Nucl.Phys.A,661:82,1999.[43]C.Galeand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