Studies in the statistical and thermal properties

arXiv:nucl-th/9609061v126Sep1996RU9687StudiesinthestatisticalandthermalpropertiesofhadronicmatterundersomeextremeconditionsK.C.Chase,A.Z.MekjianandP.MeenakshisundaramDepartmentofPhysics,RutgersUniversityPiscataway,NewJersey08854(February9,2008)AbstractThethermalandstatisticalpropertiesofhadronicmatterundersomeex-tremeconditionsareinvestigatedusinganexactlysolvablecanonicalensemblemodel.Aunifiedmodeldescribingboththefragmentationofnucleiandthethermalpropertiesofhadronicmatterisdeveloped.Simpleexpressionsareobtainedforquantitiessuchasthehadronicequationofstate,specificheat,compressibility,entropy,andexcitationenergyasafunctionoftemperatureanddensity.Theseexpressionsencompassthefermionicaspectofnucleons,suchasdegeneracypressureandFermienergyatlowtemperaturesandtheidealgaslawsathightemperaturesandlowdensity.Expressionsaredevel-opedwhichconnectthesetwoextremeswithbehaviorthatresemblesanidealBosegaswithitsassociatedBosecondensation.Inthethermodynamiclimit,aninfiniteclusterexistsbelowacertaincriticalconditioninamannersimilartothesuddenappearanceoftheinfiniteclusterinpercolationtheory.TheimportanceofmultiplicityfluctuationsisdiscussedandsomerecentdatafromtheEOScollaborationoncriticalpointbehaviorofnucleicanbeaccounted1forusingsimpleexpressionsobtainedfromthemodel.25.70.Pq,21.65.+f,05.70.Ce,05.70.JkTypesetusingREVTEX2I.INTRODUCTIONOneofthegoalsofnuclearphysicsistocharacterizethethermalpropertiesofnuclearmatter,describedbyitsequationofstate.Investigatingheavyioncollisionsoffersoneexperimentalmethodofprobingnucleiawayfromthetypicallowtemperatureanddensity,andanumberofexperimentsareunderwaystudyingthebreakupofnuclei,contributinggreatlytoourunderstanding[1–6].Severaltheoreticalinvestigationsofthesepropertieshavebeeninstigated,thermal[7–10],statistical[11–15]transport[16–20]andpercolative[21–26]modelsallofferinginsightsintothepropertiesoffragmentingnuclei.Inaprevioussetofpapers[27–30]anexactlysolvablecanonicalensemblemodelwasdevelopedforstudyingthemultifragmentationofnucleiinducedbyhighenergycollisions.Detailedpropertiesoftheclusterdistributionssuchasinclusiveyieldsandcorrelationswereexplored.Inthispaper,westudythestatisticalandthermalpropertiesofnucleiusingthesamemodel,extendingresultspresentedin[31].Thismodelgivesaunifieddescriptionofmultifragmentationphenomenaandthethermodynamicpropertiesofhadronicmatter,suchasitsequationofstateandthenuclearcompressibility.Simpleexpressionsforthesequanti-tiesareobtainedwhichextrapolateorconnectbetweenalowtemperaturenearlydegenerateFermigastoahightemperatureidealMaxwell-Boltzmanngasofnucleons.Importantly,theeffectsofthefragmentationdegreesoffreedomonthethermodynamicsismanifestlyincluded.Questionsrelatedtoveryhightemperaturesandparticleandanti-particlepro-ductionwillbedevelopedinasubsequentpaper.II.MODELSTUDIESGiventhepartitionfunctionZA(V,T)forasystemofAnucleonsatagivenvolumeVandtemperatureTwecandeterminethethermodynamicfunctionsbytakingvariouspartialderivatives,i.e.theinternalenergyandpressurearegivenbyU=kBT2∂∂TlnZAV,P=kBT∂∂VlnZAT.Additionalpartialderivativesleadtosuchimportantquantities3asthespecificheatandisothermalcompressibility,CV=∂U∂TV,1κT=−V∂P∂VT,andthermodynamicpotentialssuchastheentropyandGibbsfreeenergycanberelatedtothesequantities,S=U/T+kBlnZA,G=−kBTlnZA+PV.Ourinterestofcourseisnotinthepartitionfunctionitself,butratherinthesederivedfunctions,andassuchthepartitionfunctionismerelyameanstothatend.Thissectionconstructsapartitionfunctionforfragmentingnuclearmatter.Itdoessobyinvestigatingthethermodynamicfunctionsatbothlowandhightemperaturesanddensi-ties.Atlowtemperature/highdensity,thenucleonscoalesceintoasinglenucleuswhichcanbemodeledasadegenerateFermigaswithSkyrmeinteractions.Athightemperature/lowdensity,thenucleonsevaporateintoaMaxwell-Boltzmanngasofindividualnucleons.Withtheseresultsfirmlyinmind,astatisticalfragmentationmodelisintroducedwhoseparame-tersdependonthethermodynamicsinawayconsistentwiththehighandlowtemperaturelimits,yetallowingmorecomplicatedfragmentationpatternsatintermediatetemperatures.RHSalwaysdivergesintheinfinitelimitTheremainingfreedominthemodeliselimi-natedbyassumingthattheclusteryieldsfollowapowerlaw.Interestinglytheseminimalassumptionsleadtoamodelsurprisinglyrichinphysicsasweshallseeinthenextsection.A.LowtemperaturebehaviorandidealFermigaslawsAtzerotemperature,thenucleuscanbetreatedasadegenerateFermigas,withFermimomentumpFgivenbygS,I43πp3F=h3ρ,(1)wheregS,I=4isthespin-isospindegeneracyfactorandρ=A/Visthedensity.Atnormaldensityρ0=A/V0=0.17nucleonsfm−3,nucleonsarenon-relativisticandtheFermienergyandmomentumarerelatedbyεF=p2F/2m.FromtheaboveexpressionfortheFermimomentum,weseethatεF∝ρ2/3.Atlowtemperatures,thetotalkineticenergyforsuchaFermigasisgivenby[32]4EK=35AεF1+5π212kBTεF!2.(2)However,thenucleonsdointeract.InteractionsbetweenthenucleonscanbetakenintoaccountbyusingadensitydependentSkyrmeinteraction[33,34].Inthisapproach,theinternalenergyofanucleusisgivenbyULT=EK−Aa0ρρ0−a3ρρ0!1+σ,(3)wherea0anda3areSkyrmeparameters(forsimplicity,temperatureindependent)det