PRECISIONMEASUREMENTOFISOSPINDIFFUSIONINPERIPHERALSN+SN COLLISIONSAT70MEV/U By JackRobertWinkelbauer ADISSERTATION Submittedto MichiganStateUniversity inpartialentoftherequirements forthedegreeof Physics-DoctorofPhilosophy 2015 ABSTRACT PRECISIONMEASUREMENTOFISOSPINDIFFUSIONINPERIPHERAL SN+SNCOLLISIONSAT 70 MEV/U By JackRobertWinkelbauer Muchhasbeenundertakenrecentlytoimproveconstraintsonthesymmetryenergy termintheNuclearEquationofState(EOS).Sp,thebehaviorofthesymmetry energyaboveandbelownuclearsaturationdensityplaysatroleintheproperties ofneutronstars,thestructureofheavynuclei,andthedynamicsofnuclearreactions.The tendencyforneutronstodriftfromaneutron-richregiontoatregionduring aperipheralcollisionofheavynucleiisknownasisospinandhasbeenpreviously showntobeasensitiveobservableforthestudyofthesymmetryenergyatsub-saturation densities. Projectilefragmentationreactionswithbeamsof 112 ; 118 ; 124 Snat70MeV/uontargetsof 112 ; 118 ; 124 SnhavebeenmeasuredatMichiganStateUniversity,inordertounderstandthe oftheisospinasymmetryonthereactiondynamics.Heavyfragmentswith Z> 20 weredetectedandisotopicallyidenusingtheS800Spectrometer,andthemomentum distributionsofthesefragmentswerereconstructed.Additionally,lightchargedparticlesand intermediatemassfragmentsweredetectedinanarrayofSi-CsItelescopestosimultaneously determinetheisotopicdistributionsoffragmentswith Z< 8.Theimpactparameterofthe collisionwascharacterizedbyameasurementofthechargedparticlemultiplicityina4 ˇ scintillatorarray.Thesedataprovideadetailedpictureoftheevolutionoftheprojectile-like residueoverarangeofisospinasymmetriesandimpactparameter. Themeasuredyieldratioshavebeenusedtoextractinformationabouttheof neutronsbetweentheprojectileandtargetduringperipheralcollisions.Thevalidityofusing isotopicyieldratiosasasurrogatefortheisospinasymmetryofthecompoundsystemare discussed,andtheassociatedisospinresultsarepresented. ACKNOWLEDGMENTS IwouldliketothankmyadvisorBettyTsangforallofheradvice,support,andinsight overthepastsixyears.ShewasalwaysavailabletoaskforhelpandIcouldalwayscount onhertoproofreadanabstractoreditaPowerpointpresentationatamomentsnotice.I amgratefulthatshehashadsuchpatiencewithsuchastubborngraduatestudent.Betty hasalwaysprovidedcriticalfeedbackaboutgraduateschoolandmycareerpath.Iwasalso fortunatetohaveBillLynchasanadditionalsourceofadviceandinsight.Ihavelearned agreatdealfromBillovertheyearswhileworkingonmanyexperiments,analyzingdata, andinmanyhoursofgroupmeetings.Ialsoenjoyedthemanybarbecuesandgroupdinners thatBillandBettyhosted,whichoftenseemedmorelikeagatheringwithfamilythanwith co-workers. Iamthankfultotheothermembersofmycommittee:EdBrown,PengpengZhang, andPawelDanielewicz.Iappreciatedalltheiradviceandpatiencetryingtounderstandmy complexthesisproject. Ourgrouphadmanygraduatestudentswhohelpedmealongtheway.RachelShowalter's organization,discipline,andattentiontodetailwasalwaysabenchmarktostriveforinmy ownwork.Rachelwasalsoindispensableinrunningmyexperiment,andprovideduseful feedbackduringourweeklyanalysismeetings.DanCouplandtaughtmeagreatdealabout dataacquisitionandprogramming.MikeYoungstaughtmealotaboutelectronics,soldering, andhealwaysknewwhototalktoifhedidn'tknowtheanswer.Ialsoenjoyedworking withAndyRogers,JennyLee,AlisherSanetullaev,andMichaKilburn,albeitforashort time.Ourgroupstwopostdocs,ZibiChajeckiandBecShanewereinvaluablewhenrunning myexperiment;theexperimentwouldhavebeenimpossiblewithoutthem. iv TheNSCLmembersalsodeservealotofcredit.Amongthemanypeoplewhomade thisexperimentpossible,afewthatstandoutareCraigSnow,RonFox,AndyThulin,Tom Ginter,DanielBazin,JohnYurkonandDaveSanderson. Whilemanypeoplehelpedmedowork,Iamalsogratefultothepeoplewhohelpedme notwork.AdamFritsch,ZachMeisel,JaydaMeisel,MikeScott,CurtRogers,andAdam Saloishelpedmemakesomegreatmemoriesincludingbowling,tailgating,pontoonboating, camping,andhunting.TripsupnorthwithmyMomandDadwereamuchneededbreak fromclasses,experiments,andthesiswriting. IamalsotrulythankfultomywifeJulieforherpatienceandunderstandingoverthe pastsixyearsofmidnightshifts,extendedtravel,andall-nighters.Juliewassupportiveand encouraging,andalwaysprovidedmewithasoundingboardforcomplicatedproblems.I reallyappreciateallofyourhelpforalloftheseyears. v TABLEOFCONTENTS LISTOFTABLES .................................... viii LISTOFFIGURES ................................... ix Chapter1Introduction ............................... 1 1.1TheNuclearEquationofState..........................3 1.2OutlineofDissertation..............................4 Chapter2Motivation ................................ 6 2.1DensityDependenceoftheSymmetryEnergy.................6 2.2PhysicalConsequencesoftheSymmetryEnergy................9 2.2.1NuclearAstrophysics...........................9 2.2.2NuclearStructure.............................11 2.2.3NuclearReactions.............................13 2.3Isospin.................................15 2.3.1IsoscalinginStatisticalProcesses....................18 2.3.2PreviousmeasurementofIsospinatNSCL..........21 2.3.3ANewMeasurementofIsospin................23 2.4TransportCalculations..............................24 Chapter3ExperimentalSetup ........................... 30 3.1OverviewofExperiment.............................30 3.2S800Spectrometer................................34 3.3LargeAreaSiliconStripArray(LASSA)....................38 3.4Miniball/MiniwallArray.............................43 Chapter4DataAnalysis .............................. 47 4.1DetectorCalibrations...............................47 4.1.1IonizationChamber............................47 4.1.2Hodoscope.................................49 4.1.3CathodeReadoutDriftChambers(CRDC's)..............50 4.1.4InverseMapping.............................57 4.2ParticleIden...............................59 4.2.1EmpiricalCorrections..........................60 4.2.2AbsoluteDeterminationofN,Z.....................71 4.3ChargeStateAnalysis..............................84 4.3.1CalculationofContaminationusingGLOBAL.............86 4.4ReconstructingMomentumDistributions....................91 4.4.1CombiningSpectrometerSettings....................91 vi 4.4.2FittingFunction.............................93 4.4.3SimultaneousFittingofIsobaricVelocityDistributions........94 4.5MiniballCentralitySelection...........................99 Chapter5Results ................................... 110 5.1SystematicsofVelocityDistributions......................110 5.2IsotopicDistributionsofResidueFragments..................121 5.3YieldRatiosandIsoscaling............................124 5.4IsospinTransportRatio.............................136 5.4.1LinearityofIsoscalingObservables...................139 5.4.2IsospinResults.........................146 Chapter6Conclusion ................................ 159 APPENDICES ...................................... 162 AppendixAParticleIdenSpectra......................163 AppendixBHodoscopeAnalysis...........................171 AppendixCAcceptanceCorrection..........................179 REFERENCES ...................................... 191 vii LISTOFTABLES Table2.1Neutronisoscalingparameter andisospintransportratiomeasured inthepreviousNSCLIsospinexperiment..........22 Table2.2ephysicalquantitiesresultingfromtheparametersetsused inthisstudy.Thesequantitiesarecalculatedfromtheinteractionat nuclearsaturationdensity........................29 Table3.1Targetthicknesses.............................30 Table3.2Statisticsobtainedfromeachreactionsystem,combiningthethree magneticsettingsoftheS800spectrometer...............33 Table3.3GeometricparametersoftheMiniball-Miniwallarrayusedinthis experiment................................46 Table4.1DescriptionsofthevariablesobtainedfromtheS800orcalculated usingtheinversemaps..........................58 Table4.2Maximumimpactparameterandtotalintegratedcrosssectionmea- suredintheforthesixmeasuredbeam-targetcombinations.....102 Table4.3Meanimpactparameterdeducedfromtheminiballmultiplicitytabu- latedbyatomicnumber Z .Theerrorsarethewidth( ˙ )ofagaussian totheimpactparameterdistributionforeach Z ..........109 Table5.1Fittingparametersfromthesystematicsofthevelocitydistributions.120 Table5.2IsospinTransportRatioandMoIsospinTransportRatioresults, tabulatedasafunctionofatomicnumber Z .Thesedataareplotted inFigures5.20to5.22..........................150 TableB.1ApproximatethicknessesoftheS800focalplanedetectors......172 TableB.2ApproximateenergylossasafunctionofAtomicNumberZ,Starting from60 MeV = u (near Bˆ =2 : 51)....................172 viii LISTOFFIGURES Figure3.1Schematicofthescatteringchamber,showingtheMiniball-Miniwall andtheLASSAarray.Thebeamentersfromtheleft,andtheheavy residuesexittotherightintotheS800spectrometer..........31 Figure3.2SchematicdrawingoftheS800focalplane.Figureistakenfrom[57].35 Figure3.3SchematicdrawingoftheCRDC'sintheS800focalplane.Figureis takenfrom[58]..............................36 Figure3.4SchematicdrawingoftheLASSAarray.Figureistakenfrom[61]..39 Figure3.5PhotooftheLASSAarray.TheforwardringsoftheMiniballand Miniwallareshownontheleftside,andtheeightLASSAtelescopes areshownontherightside........................40 Figure3.6AnglesofeachpixelfortheeightLASSAtelescopes,extrapolated frommeasurementswithROMERArm.Anglesarecalculatedwith respecttothemeasuredbeamline....................42 Figure3.7ForwardpartoftheMiniball-Miniwallarray,intheconused forthisexperiment.ThetwomostforwardringsoftheMiniwallare notused..................................44 Figure3.8FullyassembledMiniball-Miniwallarray,showingthebackwardrings. Thebeaminincidentfromtheleft,andthetargetisatthegeometric centeroftheminiball...........................45 Figure3.9Left:SchematicofaMiniballdetector[65].Right:Schematicshowing thetimestructureofaMiniballsignal[67].Thethreeshadedregions representthetimewindowswhereintheFast,Slow,andTailsignals areintegratedbytheChargeADC's..................45 Figure4.1GaspressureintheS800ionizationchamber,asafunctionoftime. Thepressuredeviatedfromthesetpressureof300Torrthroughout mostoftheexperiment.........................48 Figure4.2LeftPanel:Uncorrectedionizationchamberenergylossvs.gaspres- sure,gatedon 75 Br.Severaltreactionsat Bˆ =2 : 6areshown tosampletherangeofpressurefrom300-335Torr.RightPanel:Same asleft,butwiththecorrectionfromEquation4.3applied.......49 ix Figure4.3TopPanel:Rawhodoscopesignalsforeachcrystal,whenscanning the 124 Snbeamacrossalldetectors.BottomPanel:Sameastop, withallcrystalsgainmatched.....................50 Figure4.4TopPanel:Maximumrawpadsignalsvs.padnumberforonerun inthe 112 Sn+ 112 Snreactionsystemat Bˆ =2 : 4Tm.BottomPanel: Sameastop,butpadsaregainmatchedandcorrectedfornon- linearity.Slopedatlowchannelscorrespondstoathreshold intheE1scintillator,whichisusedastheS800DAQtrigger....52 Figure4.5Inducedpadsignalvs.padnumberforone(typical)eventinthe 112 Sn+ 112 Snreactionsystemat Bˆ =2 : 4Tm.Agaussiantothe dataisshown,andthedottedlineindicatestheextractedcentroidof theevent..................................53 Figure4.6SchematicdrawingoftheCRDCmask.Thelongedgecorresponds tothedispersivedirection........................55 Figure4.7MaskspectrumforCRDC1.......................55 Figure4.8TopPanel:RawCRDC1drifttimevs.runnumber,foralldata runs.BottomPanel:CalibratedandmatchedCRDC1non-dispersive positionvs.runnumber.........................56 Figure4.9a.)Uncorrected TOF vs.uncorrectedenergyloss,measuredinthe 112 Sn+ 112 Snreactionsystemat Bˆ =2 : 4Tm.Solidlineshowsthe twodimensionalgateusedtoselectanapproximate Z b.)Uncor- rected TOF vs.dispersivepositionatthefocalplane,requiringthe gateshowninpanela.Solidlineshowsthecorrelationbetween TOF anddispersiveposition.c.)Uncorrected TOF vs.dispersiveangleat thefocalplane,requiringthegateshowninpanela.Solidlineshows thecorrelationbetween TOF anddispersiveangle.d.)Corrected TOF iteration)vs.uncorrectedenergyloss...........62 Figure4.10a.) TOF iteration)vs.uncorrectedenergyloss,measuredin the 112 Sn+ 112 Snreactionsystemat Bˆ =2 : 4Tm.b.) TOF iteration)vs.dispersivepositionatthefocalplane,requiringthegate showninpanela.c.) TOF iteration)vs.dispersiveangleatthe focalplane,requiringthegateshowninpanela.d.) TOF (second iteration)vs.uncorrectedenergyloss..................64 x Figure4.11a.) TOF (seconditeration)vs.uncorrectedenergyloss,measuredin the 112 Sn+ 112 Snreactionsystemat Bˆ =2 : 4Tm.b.)Energyloss (uncorrected)vs.dispersivepositionatthefocalplane,requiringthe gateshowninpanela.c.)Energyloss(uncorrected)vs.dispersive angleatthefocalplane,requiringthegateshowninpanela.d.) TOF (seconditeration)vs.energylossiteration)........65 Figure4.12a.) TOF (seconditeration)vs.energylossiteration),measured inthe 112 Sn+ 112 Snreactionsystemat Bˆ =2 : 4Tm.b.)Energyloss iteration)vs.dispersivepositionatthefocalplane,requiringthe gateshowninpanela.c.)Energylossiteration)vs.dispersive angleatthefocalplane,requiringthegateshowninpanela.d.) TOF (seconditeration)vs.energyloss(seconditeration).......66 Figure4.13a.) TOF vs. E ,measuredinthe 112 Sn+ 112 Snreactionsystemat Bˆ =2 : 4Tm,showinggatesfora Z and N Z .b.)Projection ontothe TOF axisofthespectruminpanela,gatedononeisotope. c.)Projectionontothe E axisofthespectruminpanela,gated ononeisotope.d.) TOF vs. E ,showinganexampleofagate resultingfromthisprocedure.Thegateisanellipsewiththe semimajoraxesequalto2 ˙ ,extractedfromtheinpanelsbandc.68 Figure4.14Spectrumof TOF vs. E ,withtheparticleidengates adoptedinthe 112 Sn+ 112 Snreactionsystemat Bˆ =2 : 4Tm.....69 Figure4.15 a 1 cot,plottedasafunctionof TOF and E ,measuredin the 112 Sn+ 112 Snreactionsystemat Bˆ =2 : 4Tm.The a 1 cot representshowstronglythe TOF dependsonthequantity x 1+ x 2. Theblackcrossesrepresentthelocationsoftheisotopesofinter- est.Thevaluesofthecorrectioncotsareextrapolatedoutside thisrangeof TOF andareusedforthestepofthecorrection, whichisapproximate...........................72 Figure4.16Thecentroidpositionsin TOF and E foreachisotopeiden inthe 112 Sn+ 112 Snreactionsystemat Bˆ =2 : 4Tm.Measured centroidsforeachisotopeareshownassolidcircles,whileextrapo- latedpositionsareshownasopencircles.Averticallineisdrawnto indicatethelocationofthe N Z =0isotopes............74 Figure4.17Isotopicspacing(ns/nucleon)vs. Z ,measuredinthe 112 Sn+ 112 Sn reactionsystemat Bˆ =2 : 4Tm.AofEquation4.16,with Z =0isshowninthemainpanel.Intheinset,the ˜ 2 dis- tributionforvarying Z isshown..................76 xi Figure4.18ParticleIdencentroids,calculatedusingthecyclotronRF asthestarttime,measuredinthe 112 Sn+ 112 Snreactionsystemat Bˆ =2 : 51Tm.Thesolidcirclesrepresentacentroidextractedfrom data,whiletheopencirclesareextrapolatedcentroids.Inaddition, theparticleidenspectrumforthedegradedbeamrunis shownonthesameplot,withitscentroidshownasasolidcircle.The solidlinecorrespondstoanextrapolationofthelocationofthe Z =50 isotopes,andthedottedlinescorrespondtotheextrapolationsfor Z =49and Z =51............................78 Figure4.19Odd-EvenStaggeringratio,measuredinthe 112 Sn+ 112 Snreaction system,usingtheintegratedyields.Leftpanelshowseven- A (even N Z )fragments.Rightpanelshowsodd- A (odd N Z )fragments.79 Figure4.20Uncorrectedenergylossvs.magneticrigidity,measuredinthe 112 Sn+ 112 Sn reactionsystem,withallthreerigiditysettingscombined.Theyields arescaledbythelive-timeandthenormalizationofthebeamfrom theMiniball.Thesefragmentsshownhave N Z =5,andareference lineisdrawnat Z =30.........................80 Figure4.21Particleidencentroidsforthethreesymmetricreactions 112 Sn+ 112 Sn, 118 Sn+ 118 Sn,and 124 Sn+ 124 Sn,for Bˆ =2 : 4Tm...81 Figure4.22LeftPanel:Finalparticleidenmeasuredinthe 112 Sn+ 112 Sn reactionsystemat Bˆ =2 : 4Tm.RightPanel:Rectangularized particleidentivariables.Thesharparepresentbecause interpolationisused,andnoattempttoextrapolateismade.....83 Figure4.23Calculatedparticleidenlocationsforasubsetofthefrag- mentsofinterestinthe 112 Sn+ 112 Snreactionsystem.Thefully strippedfragmentsarerepresentedbytheblackcrossesandthepri- marycontaminantfromhydrogen-likefragmentsarerepresentedby redcrosses.Theerrorbarsrepresentonesigmaresolutioninthere- spectiveexperimentalquantities.The Z valuesforthecontaminant arescaledbytheratioofthe(slightlyt) m q ofthetwooverlap- pingfragments,whichapproximatestheintheirenergy-loss measurements...............................86 Figure4.24Chargestatecontribution(ratioofhydrogen-likeionstofullystripped ions)shownasafunctionofatomicnumber Z ,ascalculatedby GLOBAL .Isotopesshownhave N Z =6................88 Figure4.25Chargestatecorrection(inverseofleftsideofEquation4.21)shown asafunctionofatomicnumber Z .Isotopesshownhave N Z =6 ; 7 ; 8.90 xii Figure4.26Distributioninmagneticrigidity(momentum),for 88 Zr,measuredin the 124 Sn+ 124 Snreaction.Theloweraxisshowstherigidity,while theupperaxisshowsthelinearmomentum.Thedataarescaledby thelivetimeandtheminiballbeamnormalization..........92 Figure4.27Distributioninmagneticrigidity(momentum),for 88 Zr,measured inthe 124 Sn+ 124 Snreaction.Onlydatameasuredinthecentral regionoftheS800spectrometerwheretheacceptanceisconstantare includedinthisandintheprocedure.Theal resultforthisisotopeisshownasadottedline...........92 Figure4.28Fitresultfordistributionsinvelocity,for A =77,measuredinthe 124 Sn+ 124 Snreaction.Therangeofvelocitythatiscoveredbya rangeof Bˆ istforeachisobar,andisshownbythe dottedarrows...............................94 Figure4.29Fitresultsfordistributionsinmomentum,for A =77,measuredin the 124 Sn+ 124 Snreaction.Thefunctionisanormalgaus- sianfunction,withthreefreeparameters,andtheplottedcurvesare equallynormalized.Eachdistributionisindividually.t isobarsresultinditbecauseoftheasymmetryofthedistri- butioncombinedwiththelimitedacceptancerange..........95 Figure4.30Fitresultsfordistributionsinmomentum,for A =77,measuredin the 124 Sn+ 124 Snreaction.Thefunctionisaskewedgaussian function(Equation4.23,withfourfreeparameters,andtheplotted curvesareequallynormalized.Thethreedistributionsareindi- vidually..................................96 Figure4.31Fitresultfordistributionsinvelocity,for A =77,measuredinthe 124 Sn+ 124 Snreaction.Thethreespectraaresimultaneouslywith Equation4.24,sothattheshapeisidenticalineachspectrum....97 Figure4.32Fitresultfordistributionsinvelocity,for A =77,measuredinthe 118 Sn+ 118 Snreaction.Thethreespectraaresimultaneouslywith Equation4.24,sothattheshapeisidenticalineachspectrum....97 Figure4.33Fitresultfordistributionsinvelocity,for A =77,measuredinthe 112 Sn+ 112 Snreaction.Thethreespectraaresimultaneouslywith Equation4.25.ContrarytoFigures4.31and4.32,thecentroidsof thedistributionsareallowedtovarybetweenisobars.........98 xiii Figure4.34Totalreactioncrosssectionvs.miniballmultiplicity(left-sidepanels) andcalculatedimpactparametervs.miniballmultiplicity(right-side panels).Shownfortwotargetseachfor 112 Sn(Top), 118 Sn(Middle), and 124 Sn(Bottom)...........................101 Figure4.35Reducedimpactparametervs.miniballmultiplicity(right-sidepan- els).Shownforthereactions 112 Sn+ 112 Snand 124 Sn+ 124 Sn.....103 Figure4.36Two-dimensionalspectrumshowingtheimpactparameterextracted fromtheMiniballmultiplicityvs.theatomicnumberZofthefrag- mentmeasuredintheS800,plottedforthe 112 Sn+ 112 Snreaction data.Thesolidcirclesshowtheaverageimpactparameterforeach Z,wheretheerrorsaresmallerthatthedatapoints..........104 Figure4.37Two-dimensionalspectrumshowingtheimpactparameterextracted fromtheMiniballmultiplicityvs.theatomicnumberZofthefrag- mentmeasuredintheS800,plottedforthe 124 Sn+ 124 Snreaction data.Thesolidcirclesshowtheaverageimpactparameterforeach Z,wheretheerrorsaresmallerthatthedatapoints..........105 Figure4.38erencebetweentheimpactparameterdeterminedusingamean andwithagaussianfunctionvs.theatomicnumberZofthe fragmentmeasuredintheS800,plottedforthe 112 Sn+ 112 Snreaction data....................................106 Figure4.39MeasuredcorrelationbetweentheaverageZoftheheavyresidue measuredintheS800andtheimpactparameterextractedfromthe chargedparticlemultiplicity.Datapointsareshownforthe 112 Sn+ 112 Sn, 112 Sn+ 124 Sn, 124 Sn+ 112 Sn,and 124 Sn+ 124 Snsystems.Theimpact parameteriscalculatedasthemeanimpactparameterforevents whichyieldafragmentwithatomicnumberZ.............107 Figure5.1Skewparametervs.MassLoss( A proj A ).Top,middle,andbottom panelsarethe 112 Sn, 118 Sn,and 124 Snprojectiles,respectively.The 112 Sn, 118 Sn,and 124 Sntargetsarerepresentedbytheblue,green, andredpoints,respectively.Ahorizontaldashedlineisdrawnfor noskewing,orasymmetricgaussian.Anegativevaluerepresentsa longertailextendingtolowervelocity.................112 xiv Figure5.2Isobarvelocityslopeparametervs.MassLoss( A proj A ).Top,mid- dle,andbottompanelsarethe 112 Sn, 118 Sn,and 124 Snprojectiles, respectively.The 112 Sn, 118 Sn,and 124 Sntargetsarerepresentedby theblue,green,andredpoints,respectively.Ahorizontaldashedline isdrawnforzeroslope,whichmeansthatallisobarshaveidentical velocitydistributions,whereapositivevaluemeansthathigherZ isobarshavehighervelocity.......................114 Figure5.3Isobarvelocityslopeparametervs.MassLoss( A proj A ),for 40 ; 48 Ca and 58 ; 64 Niprojectileswith 9 Beand 181 Tatargets,at140 MeV u .Data takenfrom[75]..............................115 Figure5.4Most-probablevelocityvs.MassLoss( A proj A ).Top,middle,and bottompanelsarethe 112 Sn, 118 Sn,and 124 Snprojectiles,respec- tively.The 112 Sn, 118 Sn,and 124 Sntargetsarerepresentedbythe blue,green,andredpoints,respectively.Thedataarecorrectedfor theenergylossinthetargetandthetimingscintillator.Thebeam velocityforallthreeprojectilesis = : 367..............116 Figure5.5Gaussianwidthparametervs.MassLoss( A proj A ).Top,middle, andbottompanelsarethe 112 Sn, 118 Sn,and 124 Snprojectiles,re- spectively.The 112 Sn, 118 Sn,and 124 Sntargetsarerepresentedby theblue,green,andredpoints,respectively.Thedashedlinesindicate theresultofthemoGoldhaberdescriptiondiscussedin thetext..................................117 Figure5.6Isotopicyielddistributionsforereactionsystems,forZ=30to Z=35.Thedataarecorrectedforthechargestatecontamination, butarenotcorrectedforacceptance.Solidcirclesrepre- sentsymmetricreactions,anddottedlines/opencirclesrepresentthe mixedreactions..............................122 Figure5.7Isotopicyielddistributionsforereactionsystems,forZ=35to Z=41.Thedataarecorrectedforthechargestatecontamination, butarenotcorrectedforacceptance.Solidcirclesrepre- sentsymmetricreactions,anddottedlines/opencirclesrepresentthe mixedreactions..............................123 Figure5.8Neutronisoscalingratiofor 124 Sn+ 124 Snwithrespectto 112 Sn+ 112 Sn forZ=27toZ=43.Thelinesshownarelineartothedata.Odd-Z areshowninblue,andeven-Zareshowninredforclarity.......125 xv Figure5.9Neutronisoscalingratiofor 124 Sn+ 118 Snwithrespectto 112 Sn+ 112 Sn forZ=27toZ=43.Thelinesshownarelineartothedata.Odd-Z areshowninblue,andeven-Zareshowninredforclarity.......126 Figure5.10Neutronisoscalingratiofor 124 Sn+ 112 Snwithrespectto 112 Sn+ 112 Sn forZ=27toZ=43.Thelinesshownarelineartothedata.Odd-Z areshowninblue,andeven-Zareshowninredforclarity.......127 Figure5.11Neutronisoscalingratiofor 118 Sn+ 118 Snwithrespectto 112 Sn+ 112 Sn forZ=27toZ=43.Thelinesshownarelineartothedata.Odd-Z areshowninblue,andeven-Zareshowninredforclarity.......128 Figure5.12Neutronisoscalingratiofor 112 Sn+ 124 Snwithrespectto 112 Sn+ 112 Sn forZ=27toZ=43.Thelinesshownarelineartothedata.Odd-Z areshowninblueinthelowerpanel,andeven-Zareshowninredin theupperpanel..............................129 Figure5.13Neutronisoscalingparameter for 112 ; 124 Sn+ 112 ; 124 Snand 118 Sn+ 118 Sn forZ=27toZ=43.Nocorrectionforchargestatecontaminationis included..................................130 Figure5.14Neutronisoscalingparameter for 112 ; 124 Sn+ 112 ; 124 Snand 118 Sn+ 118 Sn forZ=27toZ=43.Thecalculatedcorrectionforchargestatecon- taminationhasbeenappliedtothedata................131 Figure5.15Neutronisoscalingparameter, for 124 Sn+ 124 Snand 118 Sn+ 118 Sn withrespectto 112 Sn+ 112 Sn,forZ=27toZ=43,showingthe ofthechoiceofrange.Solidsymbolsshowthetwosymmetricre- actionswiththefullrange,whiletheopensymbolsshowtheresult whenoneisotopeisexcludedateitherendoftherange.Excluding thelowestisotopeincreasesthevalueof ,andexcludingthehighest isotopereducesthevalueof .Onlythetwosystemsareshownfor clarity...................................133 Figure5.16Neutronisoscalingparameter, for 124 Sn+ 124 Sn, 118 Sn+ 118 Sn, 124 Sn+ 112 Sn, and 112 Sn+ 124 Snwithrespectto 112 Sn+ 112 Sn,forZ=27toZ=43..134 Figure5.17Correctionforchargestatecontaminationtotheisospintransportra- tio,calculatedusingtheisoscalingparameter forthe 118 Sn+ 118 Sn reaction.Thecirclesrepresentthebetweenthecor- rectedandtheuncorrectedresult....................138 xvi Figure5.18Isospintransportratio,calculatedusingtheisoscalingparameter forthe 118 Sn+ 118 Snreaction.Threetversionsoftheresults areshown,withthreetrangesusedfortheisoscalingratio Thegreendatapointsshowstheresultforthefullrange, whiletheredandbluelinesindicatetheresultforremovingthehigh- estandlowestisotopesintherespectively.Thedashedlineshows thevalueexpectedfor R i fromthestandardisoscalingrelationships.140 Figure5.19Primaryresidueasymmetry,calculatedusingImQMD-Skywiththe fourparametersetsdetailedin2.4,plottedagainsttheasymmetry oftheprojectile,forthethreesymmetricreactions 112 Sn+ 112 Sn, 118 Sn+ 118 Sn, 124 Sn+ 124 Sn,forimpactparameter b =10fm.The calculatedasymmetryistheasymmetryforfragmentswith Z> 20, averagedoverallevents.........................144 Figure5.20Isospintransportratio,calculatedusingtheisoscalingparameter using5measuredreactions........................147 Figure5.21Moisospintransportratio,calculatedusingtheisoscalingpa- rameter using5measuredreactions.Theblackopensquaresare theaverageofthetwomixedsystems..................148 Figure5.22MoIsospinTransportRatiovs.AtomicNumberZ,forthetwo mixedreactions, 124 Sn+ 112 Snand 112 Sn+ 124 Sn............149 Figure5.23Primaryresidueasymmetry,calculatedusingImQMD-Skywiththe SLy4parameterset(softsymmetryenergy),plottedagainsttheav- erageZoftheresultingprojectile-likefragment,forthereactions 112 Sn+ 112 Sn, 112 Sn+ 124 Sn, 118 Sn+ 118 Sn, 124 Sn+ 112 Sn, 124 Sn+ 124 Sn, forimpactparameters b =10fmand b =6fm..............153 Figure5.24IsospinTransportRatio,calculatedusingImQMD-SkywiththeSLy4 parameterset(softsymmetryenergy),plottedagainsttheimpact parameter.Thereactions 112 Sn+ 112 Sn, 112 Sn+ 124 Sn, 124 Sn+ 112 Sn, 124 Sn+ 124 Snareshown.Thesolidlinesarecalculatedusingthe averagevaluesof forad b ,andthedashedlinesarecalculated usingtheaveragevaluesof fora Z ...............154 Figure5.25IsospinTransportRatio,calculatedusingImQMD-SkywithfourSkyrme parametersets,plottedagainsttheimpactparameter,forthe 124 Sn+ 112 Sn reaction,forimpactparameters b =10fmand b =6fm.........155 xvii Figure5.26IsospinTransportRatio,calculatedusingImQMD05for =0 : 5 ; 1 : 0 ; 2 : 0, plottedagainsttheimpactparameter,forthe 112 Sn+ 124 Snreaction, forimpactparameters b =6fm, b =9fm,and b =10fm.Onlyone mixedsystemisshownhereandtheverticalaxisisfocusedonthe calculatedsystem.............................157 Figure5.27MoIsospinTransportRatiovs.AtomicNumberZforthe 112 Sn+ 124 Snsystem.Thesolidregionshowstheexperimentalre- sult,andtheblue,green,andredcrossescorrespondto =0 : 5,1.0, and2.0,respectively.Thecalculationsaredoneat b =6fm, b =9 fm,and b =10fm.............................158 FigureA.1ParticleIdenforthe 112 Sn+ 112 Snreaction.Thethreepanels showthethreetmomentumsettingsthatweremeasured.See Section4.2fordetails...........................164 FigureA.2ParticleIdenforthe 112 Sn+ 124 Snreaction.Thethreepanels showthethreetmomentumsettingsthatweremeasured.See Section4.2fordetails...........................165 FigureA.3ParticleIdenforthe 118 Sn+ 112 Snreaction.Thethreepanels showthethreetmomentumsettingsthatweremeasured.See Section4.2fordetails...........................166 FigureA.4ParticleIdenforthe 118 Sn+ 118 Snreaction.Thethreepanels showthethreetmomentumsettingsthatweremeasured.See Section4.2fordetails...........................167 FigureA.5ParticleIdenforthe 124 Sn+ 112 Snreaction.Thethreepanels showthethreetmomentumsettingsthatweremeasured.See Section4.2fordetails...........................168 FigureA.6ParticleIdenforthe 124 Sn+ 118 Snreaction.Thethreepanels showthethreetmomentumsettingsthatweremeasured.See Section4.2fordetails...........................169 FigureA.7ParticleIdenforthe 124 Sn+ 124 Snreaction.Thethreepanels showthethreetmomentumsettingsthatweremeasured.See Section4.2fordetails...........................170 FigureB.1OveralldetectionoftheCsIhodoscopeforthe 112 Sn+ 112 Sn reactionsystem,at Bˆ =2 : 4 ; 2 : 51 ; and2 : 6Tm.Theissim- plythefractionofparticleswhichareotherwisedetectedandiden- inthefocalplanethatalsoleaveasignalinanyhodoscope crystal...................................173 xviii FigureB.2Energymeasuredincrystal#9,shownvs. Z ,forthe 112 Sn+ 112 Sn reactionsystem,at Bˆ =2 : 51Tm.Thisrequiresthatcrystal #9isthecrystalwiththelargestsignalamplitudeforthatevent...174 FigureB.3Non-dispersivepositionspectraatthehodoscope,shownforthetwo tsegmentsoftheexperiment.Thedottedlineshowstheap- proximatepositionofthegapbetweenthemiddletwocolumnsof hodoscopeelements.Thespectraarearbitrarilynormalizedtobeon thesamescale.Becausetheacceptancewaslimitedtotheregionnear thegap,mostofthedataiscorruptedbythepositiondependenceof thecrystalresponse...........................175 FigureB.4HodoscopeEnergyfromcrystal#9vs.Dispersivepositionspectra atthehodoscope,forthe 112 Sn+ 112 Snreactionsystem,at Bˆ =2 : 4. Thedottedlinesshowtheapproximatepositionsoftheedgeofcrystal #9....................................176 FigureB.5HodoscopeEnergyfromcrystal#9vs.Dispersivepositionspectra atthehodoscope,forthe 112 Sn+ 112 Snreactionsystem,at Bˆ = 2 : 4.Thedottedlinesshowtheapproximatepositionsoftheedgeof crystal#9.Onlyfragmentsidenas Z> 30usingthenormal particleidenspectrumareincludedthiswhichshould notreachthehodoscope.Asaresult,thesesignalscomefromeither coincidentlightfragmentsorbackgroundfrommultiplehits.....177 FigureB.6Hodoscopeenergy,measuredinthe 112 Sn+ 112 Snreactionsystem, at Bˆ =2 : 6Tm,empiricallycorrectedformomentum,for 46 Ti 22+ . Thesolidlineshowstheapproximatecentroidofthispeak,andthe dottedlineshowstheestimatedpositionoftheprimarycontaminant, 44 Ti 21+ ,whenaccountingforenergylossesandassumingtheenergy scaleislinear...............................178 FigureC.1SchematicdiagramshowingmagneticelementsoftheS800.Onlythe OpticsintheNon-DispersiveDirectionareshown.TheS800was tunedassumingthereactiontargetwasatthenormaltargetposition (thepivotpoint),whichresultedinover-Focusingthefragmentsfrom theactualtargetposition.Thedashedgreencurverepresentsthe trackofafragmentemittedatthemaximumscatteringanglefrom theoptimaltargetposition.Thesolidredcurverepresentsthetrack ofafragmentemittedatthemaximumscatteringanglefromthe actualtargetposition,upstreamofthepivotpoint..........180 xix FigureC.2Scatteringanglesatthetargetposition.Theleftpanelshowsthe 124 Sn+ 124 Snreactionatthe2 : 6Tmmagneticrigiditysettingasmea- suredinJune2011withtheproperbeamtuning.Therightpanel showsthe 124 Sn+ 124 Snreactionatthe2 : 6Tmmagneticrigidityset- tingasmeasuredinOctober2011withtheincorrectbeamtuning. Neitherhistogramrequiresaparticleidengate,butbothre- quireatimingsignalinthetimingstartscintillator,whichcausesthe squarecutoutinthecenter.(SeeFigureC.3)............181 FigureC.3Scintillatorusedtomeasurethestarttimeoftheheavyfragments thataredetectedintheS800focalplane.Noticethesquarehole cutout,whichallowstheunreactedbeamtopassthrough.......181 FigureC.4Scatteringanglesatthetargetposition.Theleftpaneshowsthe 112 Sn+ 112 Snreactionatthe2 : 4Tmmagneticrigiditysetting.The rightpaneshowsthe 124 Sn+ 124 Snreactionatthe2 : 4Tmmagnetic rigiditysetting.Thetverticalintheleftpanelisdue tomiscalibrationofthemechanicaltargetdriveusedtomovethe scintillator.................................182 FigureC.5Dispersiveangleatthetargetpositionversusfragmentenergy( dta ) measuredinthe 124 Sn+ 124 Snreactionatthe2 : 51Tmmagneticrigid- itysetting.NoPIDgatesorotherrequirementsareapplied,thesharp aresimplyduetotheacceptanceoftheS800Spectrometer. Thedottedlineshowsanexampleofthe ata dta gateusedtocal- culatetheacceptance...........................184 FigureC.6Non-dispersivepositionatthetargetversusNon-dispersiveangle, measuredinthe 124 Sn+ 124 Snreactionatthe2 : 51Tmmagneticrigid- itysetting,withagaterequiring : 03 20),approxi- mationsmustbemadewherepartofthenucleusisrepresentedbyaninertcorewithmagic numbersofprotonsorneutrons.Then,thecalculationislimitedtooneorafewnucleons thatltheorbitalsjustoutsideofthisinertcore.Becauseeachnucleoncaninteractwith allothernucleonsincludingthoseinthecore,precisecalculationsarelimitedtotheregion ofthenuclearchartclosetoshellclosures. Whendescribingmediumandheavymassnuclei,EnergyDensityFunctional(EDF) methodsareoftenused[8,9].EDFmethods,alsocalledSelf-ConsistentMeanField 1 meth- ods,useaneinteractiontoapproximatetheinteractionbetweennucleons.The tiveinteractionisnormallytunedtoreproducerelevantnuclearpropertiessuchasmasses, bindingenergies,ortheenergylevelsofexcitedstatesoveralargerangeofnuclei.Commonly usedinteractionsaretheSkyrmeinteraction,theGognyinteraction,andtheRelativistic 1 Ina\self-consistent"meanthepotentialiscalculatedfromtheactualpositions(andmomenta)of thenucleons 2 MeanFieldTheory[10,11,12].EDFmethodshavetheadvantagethattheycanbeutilized acrossthenuclearchart. 1.1 TheNuclearEquationofState Thegoalofthisworkistounderstandtheequationofstateforcoldnuclearmatter,which isimportantfornuclearastrophysicsandnuclearstructure.Inthiscontexttheinteresting relationshipishowtheenergyorpressurechangeswithdensity.Thenuclearequationof stateisdtoprobeonearth,becauseitisimpossibletocreatebulknuclearmatterin alaboratory.Thebestapproximationistostudythedynamicsincollisionsofheavy nucleiatintermediateenergy,wheredensitiestfromnormalnucleardensitycanbe created.Datafromtheseheavyioncollisionscanthenbecomparedtoatransportmodel thatincludestheimportantaspectsoftheEoS. Incentralcollisionsabovethefermienergy(theaverageenergyofanucleoninanucleus), densitieshigherthannuclearsaturationdensitycanbereached.Lightparticlesemitted duringtheexpansionofthecompressednuclearsystemcanprovideacharacterizationof theexpansionofthisdensematter[13].Insomecases,theexpansionresultsinacomplete breakupofthesystemwheremanyfragmentsofvaryingsizes(alsoknownasIntermediate MassFragments,with Z =3to Z =20)areemitted,aprocesscalledmultifragmentation. Inmid-peripheralcollisions,alow-densityneckcanformbetweenthetargetandprojectile nuclei,whichcanalsobeasourceofintermediatemassfragments.Inallcases,thegoalis tounderstandthedynamicsofnuclearcollisions,whicharecontrolledbytheEquationof State. 3 MuchhasbeenspentonconstrainingtheEoS,becauseofitsimportanceformany physicalsystems.Thebehaviorofnuclearmatterthatis symmetric ,i.e.thedensities ofprotonsandneutronsareequal,haslargelybeenconstrainedinthelaboratoryusinga varietyoftexperimentaltechniquesusingbeamsofheavy-ions[13].Thefrontier ofthisresearchisintheunderstandingof asymmetric nuclearmatter,spinthe understandingofthedensitydependenceofthesymmetryenergy. InformationabouttheEoSofasymmetricnuclearmatterisessentialtodescribingphys- icalpropertiesofaneutronstar.Onepossibleendpointforastarwithtinitial mass,neutronstarsarecompactobjectsthathavecollapsedgravitationallytothepoint wherenucleardegeneracypressureisinhydrostaticequilibriumwiththegravitationalforce. Tounderstandanyofthepropertiesofaneutronstar,itsEquationofState(EoS)mustbe known. 1.2 OutlineofDissertation Thisdissertationisastudyofthedensitydependenceofthesymmetryenergyterminthe nuclearEoSusingheavyioncollisionsatintermediateenergy.First,inChapter2,the motivationforthisexperimentisdescribed,aswellastheexperimentallandscapeleading uptothisexperiment.Thetheoreticalframeworkthatisusedtointerpretthisdataisalso introducedanddescribedhere.Chapter3describesthephysicalsetupoftheexperiment andthedetectorsusedinthisstudy.Chapter4presentsallthecriticalstepsinprocessing therawdataintophysicalobservables.Chapter5discussesthephysicalobservablesthat areextractedinthisstudy.Thesephysicalobservablesarealsocomparedtotheoretical predictions.Chapter6talksaboutthephysicsinsightsdrawnfromtheanalysisofthedata 4 fromthisexperiment,andopenissuesregardingthedensitydependenceofthesymmetry energy.IntheAppendices,severalaspectsoftheexperimentthatwereinvestigatedindetail duringthecourseofthedataanalysisarediscussed. 5 Chapter2 Motivation 2.1 DensityDependenceoftheSymmetryEnergy Theliquiddropmodelandthesemi-empiricalmassformulaprovidesomephysicalintuition regardingthenuclearEoSandthesymmetryenergy.Thesemi-empiricalmassformulais usedtodescribethebindingenergy(mass)ofnucleiusingonlythenumberofprotonsand neutrons[14,15].Itiscomposedofseveraltermsmotivatedbyphysics,andthecots ofeachtermarenormallyobtainedbyEquation2.1totheknownmassesofnuclei. Astandardformforthebindingenergypernucleonis E A = a V |{z} Volume a S A 1 = 3 | {z } Surface a C Z 2 A 4 = 3 | {z } Coulomb a sym ( N Z ) 2 A 2 | {z } Symmetry ( A;Z ) | {z } Pairing (2.1) Thethreetermsareeasilyvisualizedintermsoftheliquiddropmodel.Thevolume termissimplythebindingenergyduetothecohesionofthenucleonsinthenucleus.The nuclearforcehasalimitedrange,soeachnucleoninteractsonlywiththeadjacentnucleons. Thusthevolumebindingenergypernucleon, a V ,isconstant.Foranitenucleus,the surfacenucleonsareincontactwithfewernucleonsthanthecorenucleons,sothebinding energyisreduced.Thesurfacetermaccountsforthisreductionbysubtractingabinding energyproportionaltothesurfaceareaofthesphere.Thecoulombforcefurtherreducesthe bindingenergyduetotherepulsionofthepositivelychargedprotons,andtheapproximate 6 magnitudecanbecalculatedanalytically.Thelasttermiscalledthe\pairing"energyand accountsfortheextrastabilityforeven-Z,even-Nnucleiandthereductionofstabilityfor nucleiwithanunpairedprotonorneutron.Theremainingtermisknownasthe\symmetry" energy.Thesymmetrytermdecreasesthebindingenergyfornucleiwithtnumbers ofprotons(Z)andneutrons(N).with N = Z minimizethebindingenergy becauseprotonsandneutronstwoseparatesetsoforbitals.Stableheavynucleihave N>Z duetothecompetitionbetweenthecoulombtermandthesymmetryterm.The symmetrytermisparametrizedbytheisospinasymmetry, = ( N Z ) ( N + Z ) (2.2) Thesymmetrytermisproportionaltothesquareoftheasymmetry,becausethestrong nuclearforceshouldbeequivalentforneutronsorprotons,soonlyevenpowersof are allowed.Theisospinasymmetryisasmallnumber( <: 25)formostnuclei,andhigher powers(e.g. 4 )cannormallybeneglected.Usingjustthissimplepictureandafewfree parameters,Equation2.1reproducesthemassofknownnuclei,andprovidesestimatesofthe unmeasuredmassesofexoticnuclei.Infact,inmanynuclearreactionsimulations,variations oftheliquiddropmodelformulahavebeenusedtocalculatethemassesforextremelyproton- richorneutron-richnuclei. Thesemi-empiricalmassformulaisasuitableconceptualstartingpointforrealisticEoS, butitassumesthatthesystemisatzerotemperatureandatdensity.Tofurther developtheEoS,thenuclearpotentialmustbeparametrizedasafunctionaloftheproton andneutrondensity,whichisoneoftheprimarygoalsofnuclearphysicsresearchtoday. Thefocusofthisdissertationisonimprovingtheunderstandingthesymmetryenergypart 7 ofthisenergyfunctional.Thesymmetryenergypartofthenuclearenergydensitycanbe separatedfromthesymmetricmatterequationofstateinthefollowingway ( ˆ;T; )= ( ˆ;T; =0)+ S ( ˆ ) 2 (2.3) wherethesymmetryinteraction S ( ˆ )isafunctionofthenucleondensity. S ( ˆ ) 2 iscalledthe \symmetryenergy".Thisexpressionisanapproximationfor ˝ 1.Whilethetemperature canplayarole,themoreimportantunknownbehavioristhedependenceofthesymmetry energyonthedensity.Typically,formulationsof S ( ˆ )ignoreanyexplicittemperaturede- pendence.Whendescribingtheinteractionintermsofthenucleondensity,theasymmetry isby: = ( ˆ n ˆ p ) ( ˆ n + ˆ p ) (2.4) where ˆ p isthevolumedensityofprotons, ˆ n isthevolumedensityofneutrons,and ˆ is thetotalnucleondensity ˆ n + ˆ p .Thesymmetryenergyatsaturationdensityisrelatedto thesymmetryterm a sym ˇ 24MeVinthesemi-empiricalmassformula,but S ( ˆ )includes explicitlythedensitydependence. Thesymmetryenergyismodestlyconstrainedatsubsaturationdensities,andmostly unconstrainedforsupersaturationdensities.Forexample,thesymmetryenergynearnuclear saturationdensityhasbeenconstrainedfromthebindingenergyofstablenuclei[16].A commonenergyfunctionalusedforcalculationsofnuclearmatterandnucleiisthe Skyrme-Hartree-Fockmodel,[10]whichisdescribedinfurtherdetailinSection2.4.The Skyrmeforceiscontrolledbyasetparametersthatcanbevariedtooptimizethemodel's 8 abilitytoreproducespexperimentalobservablesfornuclei.Withinthismodel,the densitydependenceofthesymmetryenergyisvalidmainlyaroundnuclearsaturationdensity whichdescribesthecentraldnesityofmostheavynuclei.Thenuclearinteractionsthatare usedinthisworkarederivedfromthegeneralSkyrme-typeinteraction. 2.2 PhysicalConsequencesoftheSymmetryEnergy Thesymmetryenergyhasalargeininmanyareasofnuclearphysicsandastro- physics.Itdictatesmanypropertiesofneutronstars,becauseneutronstarsaredense,cold, andextremelyasymmetricnuclearmatter.Italsonuclearstructure;asthedensity decreasesnearthesurfaceofanucleus,agreaterasymmetryofneutronscanbesupported, creatingaso-calledneutronskin[16].Furthermore,thesymmetryenergythedy- namicsofnuclearreactionswithneutron-richheavynuclei.Thesetareasofnuclear physicscanprovideinformationaboutthesymmetryenergywhichcanlikewiseassistinthe interpretationofthesephysicalprocesses. 2.2.1 NuclearAstrophysics Aneutronstarhascollapsedtothepointwherethenuclearforceisinhydrostaticequilibrium withthegravitationalforce.Theintensepressurecauseselectroncapturereactionstobecome energeticallyfavorableandthemattercanbecomedominatedbyneutrons.Thesituation isknownas\betaequilibrium",becausetheenergygainedbyanelectroncapturereaction isequaltotheenergycostimposedbythesymmetryenergy,whichisafunctionofthe density.Inthisway,theprotonfractioninaneutronstariscompletelydeterminedby thedensitydependenceofthesymmetryenergy.Iftheprotonfractionishighenough,the 9 \directUrcaprocess"canoccur,whichcoolsneutronstarsbyallowingenergytobecarried awaybyneutrinos[17].IfthedirectUrcaprocesscantakeplaceatsomedepth(density)in aneutronstar,itwilldominatethecoolingoftheneutronstarandimpacttheevolutionof theneutronstar. Thebulkpropertiesofaneutronstararealsodictatedbythesymmetryenergy,inthe formofthemass-radiusrelationship.Whenthecompositionandequationofstateareknown, themassandradiusofasphericalbodyinhydrostaticequilibriumarerelatedby: dP ( r ) dr = GM ( r ) ˆ ( r ) r 2 (2.5) ThisismowithgeneralrelativitytoarriveattheTolman-Oppenheimer-Volkovequa- tion[18]: dP ( r ) dr = G r 2 ˆ ( r )+ P ( r ) c 2 M ( r )+4 ˇr 3 P ( r ) c 2 1 2 GM ( r ) c 2 r 1 (2.6) Oncetheequationofstate(therelationshipbetween ˆ ( r )and P ( r ))isspthemass- radiusrelationshipcanbedetermined.Conversely,ifthemassandradiusofaneutronstar canbesimultaneouslyandpreciselymeasured,aconstraintontheEoScanbedetermined. Themaximummassinthemass-radiusrelationshipallowsobservationsofthemassalone toprovideconstraintsontheneutronstarEoS.Tworecentmeasurementsofneutronstars closeto2solarmasses[19,20]areveryinformativeforthestudyoftheneutronstarequation ofstate,becausetheyruleoutEoS'swithsmallsymmetryenergyathighdensity.Onlyan EoSwithaasymmetrytermathigherdensityisabletosustainamassiveneutron starbyprovidingthepressureneededtoresistgravitationalcollapse.Thelocationofthe 10 transitionbetweentheheterogeneous,solidcrustandthehomogeneous,liquidoutercore isalsosensitivetothesymmetryenergy[21].Thedensitydependenceofthesymmetry energydeterminestheenergycostofseparatinguniform,isospin-asymmetricnuclearmatter intoregionsofhigherandlowerdensity.Forasymmetryenergyterm,thecore-crust transitionoccursatalowerdensity. 2.2.2 NuclearStructure Analogoustotheexampleofneutronstars,thesymmetryenergyalsoplaysaroleinthe structureofheavynuclei.Althoughheavynucleihave < 0 : 25,thesymmetryenergyisa crucialuncertaintyinmeanmodelsofthenuclearinteraction.Whendescribingnuclear matter, S ( ˆ )isoftendescribedwithaquadraticTaylorseries,expandedabout ˆ = ˆ 0 : S ( ˆ )= S 0 + L ˆ ˆ 0 3 ˆ 0 + K sym 2 ˆ ˆ 0 3 ˆ 0 2 ::: (2.7) Where S 0 isthesymmetryenergyatsaturationdensity, L istheslopeofthesymmetry energyatsaturationdensity,and K sym isthecurvatureofthesymmetryenergyatsaturation density.Thelineartermin2.7isthesourceofthe\symmetrypressure"whichcanbe obtainedfromtheHelmholtzfreeenergyfor T ˇ 0: P = ˆ 2 @ @ˆ (2.8) andsoatnuclearsaturationdensity ˆ 0 thepressureofpureneutronmatterisapproximately: P 0 = ˆ 0 L 3 (2.9) 11 Oneexampleoftheofthesymmetryenergyinnuclearstructureistheexistenceofa \neutronskin"inneutronrichnuclei.Quantitatively,theneutronskinisdescribedbythe betweentheroot-mean-squareneutronandprotonradii: np = q r 2 n q r 2 p (2.10) Theneutronskinof 208 PbcalculatedintheSkyrme-Hartree-Fockmodelhasbeenshownto bedirectlyandtightlycorrelatedwiththeslopeofthesymmetryenergy L [16].A symmetryenergy(higher L )resultsinalowervalueofthesymmetryenergyatsubsaturation densities.Therefore,theenergycostofanasymmetricneutronskin(lowdensity)islower thantheenergycostofspreadingtheasymmetryoverthevolumeofthenucleus(normal nucleardensity)resultinalargerneutronskinthickness.Varioustechniqueshavebeen employedtomeasuretheneutrondistributionsofheavynuclei[22,23,24,25].Interpretation ofthesedataisbecausetheresultsfromhadronicprobesarehighlymodeldependent. Thus,largeuncertaintiesexistintheneutrondistributionswhilethechargedistributionsare preciselyknown. Measuringtheneutronskinthicknessisausefultestfortheisospindependentpartofthe nuclearenergydensityfunctional.Themostmodelindependentmethodisameasurement oftheneutronradiusof 208 Pbusingparity-violatingelectronscatteringonapolarizedPb target.Thisexperimentwascompleted,buthadlargeerrorbarsbecauseitdidnotacquire enoughstatistics[26].Themeasurementfor 208 Pbwillberepeatedafterinstallingradiation- hardelectronicsandvacuumsystems,whichwerealargesourceofdown-timeduringthe experiment[27].Ameasurementfor 48 Cahasbeenapprovedaswell[28]. 12 2.2.3 NuclearReactions Thesymmetryenergyalsothedynamicsofmanytypesofnuclearreactions.For example,multifragmentationistheprocessbywhichanexcitedheavynucleusdecayswhen theexcitationenergyistoolargeforthenucleustodeexcitebyemittingonlynucleonsand gammarays.Theprocesscanbepicturedastheformationofaregionofhotnuclearmatter shortlyaftercollisionsoftwonuclei,whichthenexpandstosubsaturationdensityandbreaks upintomanysmallerfragments.Thefragmentsaredistributedinmassandchargeaccording tothefreeenergyofapartitionoffragments.Thefragmentscanstillhavet excitationenergy,andrealisticpredictionsrequirefurtherevaporationoflightparticles. TheStatisticalMultifragmentationModel(SMM)treatsthisprocessinasimple,statistical, andsemi-empiricalway,andprovidesreasonablepredictionsofmultifragmentationyields. [29]Thesymmetryenergyintheformofthechemicalpotentialsplaysacriticalrolein multifragmentation,inparticulartheisotopicdistributions.AswillbediscussedinSection 2.3.1,howtheisotopicdistributionschangefromonesystemtoanotheriscorrelatedtothe strengthofthesymmetryenergy.SMM-typemodelsdonottreatthenuclearinteraction directly,insteadtheyapproximatetheprocessintoseveralinstantaneoussteps.Nonetheless, SMMrequiresassumptionsabouttheEoSwhichcanbeinformedbyexperimentalconstraints onthesymmetryenergy. AnotherclassofnuclearprocessthatispartiallytiedtothesymmetryenergyandtheEoS ofnuclearmatterareknownasGiantResonances.ImportantexamplesaretheGiantDipole Resonance(GDR)andtheGiantMonopoleResonance(GMR).TheGMRisalsoknownas a breathingmode ,andcanbeunderstoodasaradialoscillationoftheprotonandneutron densities.TheresonanceenergyoftheGMRisdirectlyrelatedtotheincompressibility 13 ofnuclearmatter,becausetheincompressibilityprovidestherestoringforceagainstthe vibration.Theincompressibilityofbulknuclearmatterisnedtobe K 0 =9 ˆ 2 d 2 E = A dˆ 2 ˆ = ˆ 0 ; (2.11) butfornuclei,surfaceandcoulombmustbeincluded.Anucleusincom- pressibilitycanbeextractedfromtheexperimentalenergyoftheGMRandtheRMSnuclear radiusby E GMR = s ~ 2 K A m : (2.12) K A canbedeterminedfromexperiment,buttherelationshipbetween K A and K 0 isuncer- tain,whichmakesthedeterminationof K 0 TheGDRisadipoleoscillationofprotonandneutrondensitiesalongoneaxisofthe nucleus.Whentheprotonandneutrondensitiesoscillateinphase,theexcitationiscalled isoscalar,andwhentheprotonandneutrondensitiesoscillateoutofphaseitiscalledisovec- tor.TheisovectorGDRisanoscillationwheretherestoringforceisprovidedbythesym- metryenergy.Thus,studyingtheGDRprovidesinformationaboutthesymmetryenergy. Theseoscillationsofitenucleionlyprovideinformationaboutdensitiesclosetonormal nucleardensity.Incontrast,thePygmyDipoleResonance(PDR)isproposedtobean oscillationoftheisospinsymmetriccoreofthenucleusinsidetheasymmetricneutronskin. [30,31]Becausethiswouldpreferentiallyprobethesurfaceofthenucleus,itcouldprovide informationaboutthelowdensitybehaviorofthesymmetryenergy.Therelativestrength ofthePDRhasbeenshowntoberelatedtothedensitydependenceofthesymmetryenergy. 14 2.3 Isospin Oneconceptthatwasdevelopedinthestudyofthesymmetryenergywithheavyioncollisions iscalledisospinThissimpleconceptisanaturalstepinreactiondynamics:put aneutron-richnucleusveryclosetoaneutron-poornucleusandseehowfastthesystem reachesisospinequilibrium.Thesymmetryenergywoulddrivethetwonucleitoexchange protonsorneutrons,untiltheisospinasymmetryofthetwonucleibecameequalized.In practice,thesituationismorecomplicated. Toovercomethecoulombrepulsion,theprojectileandtargetnucleimusthavelargerel- ativevelocity.Forisospinsiontooccur,thereactiontimescaleshouldbecomparableto thenucleon-nucleoncollisiontimescaleinsidethenucleus.Forthisreason,thephysicsmust beinterpretedusingadynamicalreactionmodelthatdescribestheevolutionofindividual nucleonsthroughthecollision.Todescribethismanybodyproblem,manyapproximations mustbemade.Nonetheless,bycarefullychoosingreliableobservablesandcontinuouslyim- provingthenucleartransportcalculations,constraintsonthesymmetryenergyhavebeen obtained. Forthepurposesofthisdissertation,isospinwillbediscussedinthecontextof isotopesofSn.Thisisanaturalchoice,asSnhasstableisotopesrangingfrom 112 Sn( = : 1071)to 124 Sn( = : 1935),andtheSn+Snsystemislargeenoughtoapplyamacroscopic approachtothephysics.Withhigh-intensityradioactivebeamsattheFacilityforRare IsotopeBeams(FRIB,underconstruction),therangeofasymmetrycanbeincreasedusing radioactiveisotopesfrom 108 Snto 132 Sn. When 112 Snand 124 Sncollideinaperipheralcollision(largeimpactparameter b ),an overlapregionformsbetweenthecollidingnucleiandnucleonscanbeexchanged.Onaverage, 15 neutronswouldfromtheneutron-rich 124 Sntotheneutron-poor 112 Snthroughthe neckregion.Thedensitiesofprotonsandneutronscanbedescribedwithaequation, j n = D ˆ n ˆ D n (2.13) j p = D ˆ p ˆ D p ; (2.14) wherethe D n;p arethecotsforneutronsandprotonsfordensityandisospin. CombiningEquation2.13and2.14gives j = j n j p = D n D p | {z } Isospin D ˆ n D ˆ p ˆ | {z } IsospinMigration (2.15) whichindicatesthattwomechanismsdrivetheofisospinbetweentheprojectile andthetarget.Theisospingradientdrivesaofisospindirectly.Aseparate knownas\isospinmigration"alsotheequilibration.Isospinmigrationisanet ofisospinbecausethedensitycot D ˆ n forneutronsisrentfromthe densitycot D ˆ p forprotons.Becausethesymmetryenergydecreaseswith decreasingdensity,alowdensityregioncansupportabiggerisospinasymmetrysoneutrons fasterintotheneckregionthanprotons[32].Isospinmigrationwouldbepresentin boththesymmetricandmixedcollisions. Theobservablethatwouldconstituteadirectmeasurementofisospinwould beameasurementoftheisospinasymmetryoftheexcitedprojectile-likefragmentatthe momentoftheseparationbetweenprojectileandtarget,whichhappensonthetimescaleof ˇ 10 22 s .Unfortunatelythereactionproductsaremeasuredonatimescalethatismany 16 ordersofmagnitudelonger,andthedetectedfragmentshavedeexcitedbyparticleemission. Becauseofthisdeexcitation,thecompositionoftheprimaryexcitedfragmentisobscured. Whensimulatingthereactionwithadynamicaltransportmodel,theasymmetryofthe primarysourcecanbecalculateddirectly.Inanexperiment,themeasuredobservablesmust berelatedtothisobservableinaquantitativeway.Forthepurpose,the\isospintransport ratio"hasbeenconstructed(inthiscaseforthemixedreaction 112 Sn+ 124 Sn 1 )toquantify theamountof R I ( X 112+124 )= 2 X 112+124 ( X 124+124 + X 112+112 ) X 124+124 X 112+112 (2.16) whereXisanobservablethatisrelatedtotheisospinasymmetryachievedinthereaction. Thisobservablehasseveralusefulproperties.Ifanexperimentalobservable X isiden where = aX + b ,then R I ( )= R I ( X ).Thisiscriticalbecausethetheoreticalcalculations producetheisospinasymmetry ,whichcannotbemeasureddirectlyintheexperiment.If X 112+124 = X 112+112 then R I = 1,andif X 124+112 = X 124+124 then R I =+1. R I = 1 indicatesthatnooccurred.Ifdrivesthesystemtoisospinequilibrium, then R I ( X )=0ineithermixedreaction. 2 Ifanobservablewhichmeetsthelinearityconditionisidenthenthemeasured observablecanbecomparedtothecalculatedisospinasymmetry.Thisisanexperimental questionindependentfromthemeasurementofisospinPreviously[33]andforthis purposesofthisstudy,theobservablethatisexploredistheisotopicscaling(denotedby ) offragmentswhichisthetopicofthenextsection.Itwasshownin[34]that = + b for centralcollisions,butthislinearityshouldbemeasuredforperipheralcollisionsaswell. 1 Theconventionforreactionnotationthroughoutthedissertationwillbe:(projectile)+(target) 2 Becauseofthemassasymmetryoftheprojectiles, R I ( )=0.0507atequilibrium,seeSection5.4. 17 2.3.1 IsoscalinginStatisticalProcesses \Isoscaling"describesthewaythatanisotopicyieldratiooffragmentsemittedinastatistical processfollowsanexponentialfunctionoftheneutronandprotonnumberofthefragment. Whenplottedasalogarithm,thetrendislinear.Thisbehaviorwasrstdescribedin[35], andwasideninmanyphysicalsystemsincludingmultifragmentation,evaporation,and deeplyinelasticreactionmechanisms.Sincethattime,isoscalinghasbeenobservedinmany otherphysicalsystemssuchasandprojectilefragmentation.[36,37]Theisoscaling trendisgenerallydescribedwithathreeparameterrelation: R 21 ( N;Z )= Y 2 ( N;Z ) Y 1 ( N;Z ) = Ce ( N + Z ) (2.17) where Y i ( N;Z )istheyieldoftheisotopewithatomicnumberZandneutronnumberN, measuredinthereaction i .Byconventionthemoreneutron-richsystemisinthenumer- ator,whichresultsin > 1and < 1.Thisisausefulwaytodescribehowtheisotopic distributionchangeswhentheisospincontentoftheemittingsystemischanged.Inprac- tice,formingyieldratiosbetweentreactionseliminatesthedependenceonabsolute normalizations,detectoracceptance,andthestructureoftheemittedfragments,aswellas reducestheofsequentialdecays.Isoscalingisfundamentallyrelatedtothesymmetry energy,andvariousstatisticalmodelspredictthatthemagnitudeoftheisoscalingslope (or )shouldbeproportionaltothesymmetryenergypartofthenuclearbindingenergy C sym T .[38]Thedescriptionmostoftenused,whichkeepstheleadingordertermsinNand 18 Z,andassumesthetemperaturetobethesameinthetwosystemsbeingcompared,is: = n T ˇ 4 C sym T " Z 1 A 1 2 Z 2 A 2 2 # (2.18) = p T ˇ 4 C sym T " N 1 A 1 2 N 2 A 2 2 # (2.19) n p )referstotheoftheneutron(proton)chemicalpotentialbetweenthe twoemittingsources,andareapproximatedbytheneutronandprotonseparationenergies. Theseexpressionsareproducedinthecontextofseveralstatisticalmodels[35],butexper- imentally,someambiguityremains.Equations2.18and2.19showthat and donot dependlinearlyontheisospinasymmetryoftheemittingsource.Inthepreviousstudyof isospin[33],itwasassumedthatthisexpressionmightbelinearwiththeasym- metryofthecompositesystemasitistodeterminetheasymmetryoftheemitting system. Analternativepicturecanbedescribedintermsofscalingwiththemassandneutron excessasopposedtotheneutronandprotonnumber.Itisasimpletransformation,buthas someusefulfeatures.Inthiscase and arereplacedwith 0 and 0 ,whicharethescaling cotsformassandneutronexcess,respectively: R 0 21 ( A;N Z )= Y 2 ( A;N Z ) Y 1 ( A;N Z ) = C 0 e 0 A + 0 ( N Z ) (2.20) where 0 = ( + ) 2 (2.21) 0 = ( ) 2 (2.22) 19 andinthisalternativepointofview,theresultsarerelatedtotheisospincompositionby: 0 = n + p ) 2 T ˇ C sym T 2 2 2 1 (2.23) 0 = n p ) 2 T ˇ 2 C sym T ( 2 1 )(2.24) Thisalternativeisoscalingwassuggestedoriginallyin[39]whereisoscaling(oflightfrag- ments)wasstudiedwithmultifragmentationreactionsinducedbylightions.Inthatstudy, itwasnotedthattheabsolutemagnitudesof and weresimilar,so 0 wassmallalthough wasmeasureablypositive.In[40],\isobaricscaling"wasstudied,wheretheyieldfora constantAwasasafunctionof N Z whenmeasuringheavyresiduesfromprojectile fragmentation.Inthisstudy, 0 wasshowntohavesimilarbehaviorto ,but 0 wasas- sumedtobenegligible.Thechoicemayseemsimplynotational,butonemethodmayhave advantages.Forinstance,whenstudyingprojectilefragmentationthetemperaturecouldbe approximatedbyalinearfunctionofthefragmentmass:[41] T ( A )= T o + T 1 A A proj (2.25) Fromthepointofviewofmeasuringisoscalingofheavyresidueyieldsfromprojectilefrag- mentation,especiallyoveralargerangeofZandA,avaryingtemperaturewouldresultin asystematicallyvarying and .Whentheisobaricyieldratiosasafunctionof N Z ,becauseisotopesofamasslikelycomefromasingletemperature,theisoscal- ingtrendmaybemorereliable.Itispossibletoextractaslopeforeachfragmentmass, 0 ( A ),andaslopeforforeachneutronexcess, 0 ( N Z ).Withthe\normal"isoscaling 20 variables, ( Z )and ( N )canbeextracted.Intheidealcase,theformulationsareidentical andinterchangeable.Intheexperiment,botharepotentiallyuseful. Anotherconsiderationisthatofthepossibleisoscalingvariables,only 0 = ( ) = 2 has alineardependenceon ,whichmakesitappealingforusewiththeisospintransportratio. Also,oneveryinterestingrelationshipappearswhentakingtheratioofthe 0 and 0 from 2.23and2.24: 0 0 = + = 1 + 2 2 = (2.26) Sincethe T and C sym cancel,thisratiowouldprobetheaverageasymmetrybetweenthe tworeactionsdirectly,withoutrelyingonacalibrationofthetemperature.Sinceinthis experimentthereferencereaction(reaction\1")is 112 Sn+ 112 Sn,theaverageasymmetryis ameasureoftheasymmetryofthecomparedreaction(reaction\2").Theisospintransport ratiocalculatedwiththisobservableisusefultounderstandthesystematicsoftheisoscaling behaviorinprojectilefragmentation. Thisdissertationseekstouseisoscalingasanobservablefortheisospintransportratio tomeasureisospinbetween 112 Snand 124 Sn.Inaddition,asecondarygoalisto verifytherelationswith and with . 2.3.2 PreviousmeasurementofIsospinatNSCL ThepresentstudyisacontinuationofalongprogramatNSCLtoconstrainthesymmetry energypartofthenuclearEoS.[42]Isospinwasusedsuccessfullyforthispurposein [33].Inthatexperiment,thefourreactions 112 Sn+ 112 Sn, 112 Sn+ 124 Sn, 124 Sn+ 112 Sn,and 124 Sn+ 124 Snat50MeV/uwerestudied.TheexperimentcombinedtheMiniball-Miniwall 21 Reaction R I ( ) 112 Sn+ 112 Sn 112 Sn+ 112 Sn 0 -1 112 Sn+ 124 Sn 112 Sn+ 112 Sn 0 : 16 0 : 02 0 : 45 0 : 05 124 Sn+ 112 Sn 112 Sn+ 112 Sn 0 : 42 0 : 02 0 : 47 0 : 05 124 Sn+ 124 Sn 112 Sn+ 112 Sn 0 : 57 0 : 02 +1 Table2.1Neutronisoscalingparameter andisospintransportratiomeasuredinthe previousNSCLIsospinexperiment. arraywiththeLargeAreaSiliconStripArray(LASSA)(bothdescribedinChapter3)and anadditionalannularSilicon-CsIdetectoratveryforwardangles,andthepurposewasto measureisoscalingofintermediatemassfragments(IMF's)fromthetheSn+Sncollisions. Thedatathatwasusedtoformtheseisoscalingratioswasselectedtobe\mid-peripheral", byrequiring b = b max > 0 : 8and y = y beam > 0 : 7where b isthescatteringimpactparameterand y istherapidity.Thepurposeofthisimpactparameterselectionistoselectcollisions wherethenucleonscanbeexchangedbetweenthetargetandtheprojectile.Ifmorecentral collisionswereincluded,thedatawouldcontainfragmentsemittedfrommultifragmentation oftheentirecombinedSn+Snsystem,whichwouldrepresentisospinequilibrium. IMFyieldsweremeasuredfor3 Z 8,andthefragmentswereusingthestandard threeparameterisoscalingformula.Resultsarepublishedin[34,33].Forexample,the isoscalingparameter thatwasextractedislistedinTable2.1.Theresultobtainedwas R I ( )=+0 : 47 0 : 05forthe 124 Sn+ 112 Snsystemand R I ( )= 0 : 45 0 : 05forthe 112 Sn+ 124 Snsystem.Theinterpretationofthisresultisthatthesystemevolvedroughly halfwaytoisospinequilibriumduringthecollision. 22 2.3.3 ANewMeasurementofIsospin Thefocusofthisdissertationisanewmeasurementofisospinusingtheisoscaling parametersextractedfromheavyprojectile-likefragmentyields.Theexperimentisdescribed indetailinChapter3. Therewereseveralshortcomingsinthepreviousisospinexperimentthatare meanttobeaddressedinthisstudy.OneissuewithmeasuringisoscalingofIMF'sisthat theyieldsforthesefragmentsarerelativelylow.BecausetheLASSAarrayonlycoversasmall portionofthepossiblescatteringangles,thefordetectingthesefragmentsisquite lowaswell.Asanalternativewayofprobingtheisospinasymmetryoftheprojectilelike fragment,heavyprojectile-likefragmentscanbemeasuredatveryforwardangles.Projectile fragmentationcrosssectionsarehigh;asemi-peripheralcollisionwillverylikelyresultin producingaheavyfragment.Highercrosssections(aswellasmanymoredatapoints) willallowformuchhigherstatistics.Thepropertiesoftheheavyfragmentsarealsomore directlyconnectedtothoseoftheexcitedprimarysource.TheyieldsofIMF'scanbe byvariousreactionmechanismswhichcanobscuretheisospinsignal. Variousmeasurements[37,43,44]ofisoscalingfromheavyprojectile-likefragmentsalso providesomethatthistypeofstudycanbeusedwithgoodprecision.Inaddition tologisticaladvantages,theoreticalcalculationssuggestthattheheavyresiduesmayhavea tsensitivitytothesymmetryenergythantheIMF's[45]. Anothernicefeatureofthemeasurementofheavyresiduesisthatheavyfragmentsonly resultfromperipheralcollisions.Thiswillallowforanadditionalmeasureofthecentrality ofthecollision:higherZfragmentscomefrommoreperipheralcollisions.Inaddition,the measurementoftheimpactparameterwillbemorereliable,withanimprovedmethodfor 23 normalizingthebeamcurrentandtotalreactioncrosssections.Areliableextractionof theimpactparameterbeingprobedintheexperimentisimportantinthecomparisonto theoreticalcalculations. Finally,thisexperimentwillattempttoverifytherelationshipsoftheisoscalingparame- ters and withthe N Z or oftheemittingsource.Thiswillbeaccomplishedbymeasuring adatasetforthe 118 Sn+ 118 Sninadditiontothe 112 Sn+ 112 Snand 124 Sn+ 124 Snreactions. The 118 Sn+ 118 Snsystemrepresentsadatapointapproximatelymidwaybetweentheother twosymmetricsystems,andshouldgiveanindependentmeasureof\isospinequilibrium". Inotherwords,byexaminingthetrendoftheisoscalingobservableswiththeincreasing asymmetryofthetSnisotopes,theisoscalingrelationshipsdescribedinthischapter canbetested.Measuringadditionalcombinationswiththe 118 Sn isotopemayalsohelpto constrainotherectsthattheisospincontent,suchasisospinmigration. 2.4 TransportCalculations Whenstudyingthenuclearequationofstate,physicalquantitiesarerarelyprobeddirectly. Normally,anobservableismeasuredinthelaboratoryandtheresultsarethencomparedto arealisticsimulation.Theinputparametersofthemodelcalculationarevariedtothe bestagreementwiththedatabeforedrawinganyphysicalconclusions.Thisrequiresamodel thatcannotonlyreliablyreproduceexperimentalquantities,buthasphysicallymotivated inputparameters.Aphenomenologicalmodelislessusefulbecausesomemodelparameters maynotbephysicallymeaningful,andthemodelsmaynotcontainallofthephysicsinvolved inthecollisions.WhenusingheavyioncollisionstostudytheEoS,thecollisionprocesshas alargenumberofdegreesoffreedomsothecalculationsarecomputationallydemanding. 24 Nonetheless,themodelsmustbetlycomplexsothatthemicroscopicphysicsis treatedrealistically. Forthisstudy,theImQMDmodelisused[46,47,48].Thisisavariationintheclassof QuantumMolecularDynamics(QMD)models.MolecularDynamicsmodelsweredeveloped todescribemolecularsystems,butthemethodhasbeenadoptedinnuclearphysicsaswell. InQMD,eachnucleonisrepresentedbyagaussianwavepacketwhichmovessemi-classically subjecttoaself-consistentmeanThewidthofthegaussianwavepacketisanimportant parameterinthesimulations,anditsvalueisduringthereaction.Forsystemwith tsizes,aphenomenologicalformulawasproposedtoparametrizethewidthofwave packet: ˙ 2 r = ˙ 2 r;A p roj + ˙ 2 r;A target 2 (2.27) where ˙ 2 r;A = 0 : 16 A 1 = 3 +0 : 49 2 fm 2 : (2.28) QMDkeepstrackofthecorrelationsbetweenNnucleons,comparedtoBUU-typemodels [49].Asaresult,itismorecomputationallydemandingthanBUU.Thesingleparticlephase spacedistributionfunctionisgivenby: f i ( ~r;~p )= 1 ( ˇ ~ ) 3 exp ( ~r ~r i ) 2 2 ˙ 2 r ( ~p ~p i ) 2 2 ˙ 2 p ! (2.29) 25 where ˙ r and ˙ p arethepositionandmomentumwavepacketwidths.Thespdetails oftheQMDmodelarewelldocumentedelsewhere[46,47,48]andarenotcriticaltothis study. Becausethemodelsaresemi-classical,thePauliexclusionprinciplehastobeincluded atleastinanapproximateway,whichisreferredtoas\Pauliblocking".Therearemany tversionsbasedontheQMDmodel,whichusuallybythemethodthatthey handle\Pauliblocking".InAntisymmetrizedMolecularDynamics[50,51],eachnuclear systemisrepresentedbyaSlaterdeterminantofgaussianwavepackets,which thePauliprinciple.Thisleadstoamuchmorecomputationallyintensivemodel,soAMD calculationsforverylargesystemsarenotcommonlyperformed.TheCoMDformulation [52]addressesPauliblockingbyplacingalimitonthephasespaceoccupationdensity;if aparticletravelsintooccupiedphasespace,theparticleisscattered.IfPauliblockingis ignoredinthemodel,thesystemwillevolvetowardsaclassicalthermodynamiclimit,which isnotapplicabletonuclei. TheImprovedQuantumMolecularDynamicsModel(ImQMD)wasoriginallydeveloped forthedescriptionoffusionprocessesnearthecoulombbarrier[46].ImQMDusesisospin dependentnucleon-nucleonscatteringcrosssections;thecrosssectionforprotonsist thanforneutrons.ThecriticalinputtoImQMDistheformofthemeanpotential thatdictatestheevolutionofthenucleonicwavepackets.Recently,ImQMD05[47]hasbeen motostudytheNuclearEoSusingheavyioncollisions.InImQMD05,thesymmetric partofthenuclearpotentialcanbetakenfromtheSkyrmeforce[10]: V loc = 2 ˆ 2 ˆ 0 + +1 ˆ +1 ˆ 0 + g sur 2 ˆ 0 ( r ˆ ) 2 (2.30) 26 where , , , ˆ 0 ,and g sur areparametersofthemodelandarerelatedtothestandard Skyrmeparameters.InthecaseofImQMD05,thegoalistostudythedensitydependenceof thesymmetryenergy,soasimplepowerlawdependenceisadoptedforthepotentialenergy partofthesymmetryenergydensity: sym = C sym 2 ˆ ˆ 0 2 (2.31) whichallowsforsimplecontrolofthesymmetryenergy.Highvaluesof ( > 1)arelabeled "andlowervalues( < 1)arelabeled\soft". ImQMD05wasusedsuccessfullyindeterminingconstraintsonthesymmetryenergyfrom heavyioncollisions.Measurementsofisospinusingisoscalingofintermediatemass fragmentsand 7 Li = 7 Be yieldratioswerereproducedbyImQMDcalculationsoftheaverage isospinasymmetryincollisionsatseveralimpactparameters[34].The\double-ratio"of neutronstoprotonsincentralcollisionsof 112 Sn+ 112 Snand 124 Sn+ 124 Snpresentedin[53] wasalsoreproducedbycalculationswithImQMD05in[54].Theextractedconstraintson thedensitydependenceofthenuclearsymmetryenergyaresummarizedin[42]. AnewerversionoftheImQMDmodel,ImQMD-Sky,wasdevelopedtoutilizeaneven morerealisticnuclearinteraction[48].AmoredetailedSkyrmepotentialisused,butthe spin-dependentpartofthepotentialisneglectedbecausethisisthoughttobeaminor contribution.Themomentumdependenceofthenuclearmeanisincludedas: V md = Z d 3 pd 3 p 0 h C o f ( ~r;~p ) f ( ~r;~p 0 ) ~p ~p 0 2 i (2.32) + Z d 3 pd 3 p 0 h D o f n ( ~r;~p ) f n ( ~r;~p 0 ) ~p ~p 0 2 + f p ( ~r;~p ) f p ( ~r;~p 0 ) ~p ~p 0 2 27 where C 0 and D 0 areeachdirectlyafunctionoftheparametersoftheSkyrmeforceandthe localpartofthenucleonicenergydensityis V loc = 2 ˆ 2 ˆ 0 + +1 ˆ +1 ˆ 0 + g sur 2 ˆ 0 ( r ˆ ) 2 | {z } SymmetricMatter (2.33) + g sur;iso ˆ 0 r ˆ n ˆ p 2 + A sym ˆ 2 + B sym ˆ +1 2 | {z } AsymmetricMatter andtheparameters ;;;;g sur ;g sur;iso ;A sym ;B sym canbeobtainedfromthenormal Skyrmeinteractionparameters.ThebofusingtheSkyrmeinteractionisthatthe Skyrmepotentialhasbeendevelopedtodescribenuclearstructureproperties,whichprovides somethatthephysicsisproperlytreated.Inaddition,theSkyrmeparametersets areeasilyinterchangedandatinteractioncanbeused.UsingtheSkyrmeinteraction allowsmoredegreesoffreedomindescribingtheinteraction,butthesecanbehard tounderstandintuitively.Onewaytopicturethemomentumdependentinteractioniswith theconceptofanemass.Theemasscanbeintroducedtodescribehow thepotentialenergydependsonthemomentum.Thiscanbepicturedbywritingdown Hamilton'sequation: _ x = @H @p = @T @p + @V @p (2.34) = p m + @U @p (2.35) = p m 1+ m p @U @p p m 28 Name ˆ 0 ( fm 3 ) E 0 K 0 S 0 L K sym m = m m n = m m p = m SLy4 0.160 -15.97 230 32 46 -120 0.69 0.68 0.71 SkI2 0.158 -15.78 241 33 104 71 0.68 0.66 0.71 SkM 0.160 -15.77 217 30 46 -156 0.79 0.82 0.76 Gs 0.158 -15.59 237 31 93 14 0.78 0.81 0.76 Table2.2ephysicalquantitiesresultingfromtheparametersetsusedinthisstudy. Thesequantitiesarecalculatedfromtheinteractionatnuclearsaturationdensity. where m isthetivemass. m m representstheoverallofthemomentumdependence ofthenuclearpotential. m n and m p arenotfreeparametersoftheinteraction,butresult fromofthechoiceoftheSkyrmeparameterset.Itisalsoafunctionofthelocalnucleon density.Thetivemassatsaturationdensityisapproximately30%lowerthanthefree nucleonmass.Thisectivemasssplitting"isameasureofhowthemomentumdependence canthesymmetryenergypartofthenuclearinteraction,andthereforecan isospin-sensitiveobservables. Inthisdissertation,calculationswithbothversionsImQMD05andImQMD-Skyhave beenperformed.ForImQMD05, 'sfrom0.5to2.0wereused.ForImQMD-Sky,the fourparametersetsthatwillbeshownare SLy4 , SkI2 , SkM ,and Gs .Thesaturation parametersforthefourSkyrmeparametersetsareshowninTable2.2. Althoughisospinismainlybythedensitydependenceofthesymme- tryenergy,themomentumdependenceofthenuclearinteractionintheee masssplitting)canalsotheisospintransportratio[45].Byexaminingcalculations withavaryingsymmetryenergydensitydependence(ImQMD05),aswellascalculations withseveralacceptedSkyrmeparametersets(ImQMD-Sky),thebestpictureoftheisospin dynamicscanbeobtained.TheresultsofthecalculationswillbeshowninChapter5. 29 Chapter3 ExperimentalSetup 3.1 OverviewofExperiment ThisexperimentwasperformedattheCoupledCyclotronFacility(CCF)attheNational SuperconductingCyclotronLaboratory(NSCL)atMichiganStateUniversity(MSU).The experimentutilizedprimarybeamsof 112 Sn, 118 Sn,and 124 SnfromtheK1200Cyclotron at120MeV/u.TheenergyoftheprimarybeamwasdegradedusingtheA1900fragment separator,whichresultedinbeamswithanenergyof70MeV/uwithamomentumwidth of 0 : 125%.ThesebeamsweretransportedfromtheA1900tothetargetpositionintheS3 vaultatthepivotpointoftheS800Spectrometer.Thebeamswereimpingedonisotopically enriched( > 99 : 5%purity)targetsof 112 Sn, 118 Sn,and 124 Sn.Thethicknessesofthetargets usedareshowninTable3.1.UsingtheMiniball-Miniwallarrayasanindirectbeammonitor, thebeamrateontargetwasdeterminedtobebetween2 10 7 s 1 and6 10 7 s 1 . Target Thickness mg cm 2 NumberDensity 10 19 atoms cm 2 112 Sn 5.940 3.19 118 Sn 6.316 3.22 124 Sn 5.512 2.68 Table3.1Targetthicknesses. 30 Figure3.1Schematicofthescatteringchamber,showingtheMiniball-Miniwallandthe LASSAarray.Thebeamentersfromtheleft,andtheheavyresiduesexittotherightinto theS800spectrometer. Theexperimentalapparatusincludedthreeseparatedetectorsystems:theMSUMiniball- WashUMiniwallarray,theLASSAarray,andtheS800spectrometer.Aschematicdrawing ofthetargetchamberisshowninFigure3.1. ThereactiontargetswerelocatedatthecenteroftheMiniball,andchargedparticlesfrom reactionsinthetargetweredetectedintheMiniballandintheLASSAarray,whileheavy residuespassedthroughthescatteringchambertobedetectedintheS800spectrometer. Additionally,athinscintillator(madefromBC-408scintillatormaterial)waslocatedap- proximately0.75metersfromthetargetposition.Thisthinscintillatorwasusedtomeasure thet(TOF)ofheavyfragmentsthroughtheS800. Thecombinationofseveralcomplexdetectorsystemsposesuniquechallenges.Theex- perimentusesstablebeamsandthereactionsofinteresthavelargecrosssections,sothe 31 ratelimitingfactorwasthethroughputofthedataacquisitionsystem.Tohelplimitthe dead-timeassociatedwitheachevent,theelectronicswereseparatedintothreedistinctdata acquisitionsystems(LASSA,Miniball-Miniwall,S800).Eacheventwastime-stampedand theresultingdataweremergedThisallowedeachsystemtodigitizesignals inparallel,whichdecreasedthedead-time.Inthisation,thedead-timeisdictated bythedead-timeoftheslowestsystem.Theresultinglive-timewas70%orgreaterwhen readingupto500events/second. Torecordthetimestampforeachevent,eachofthethreesystemscontainedanXLM72 asa64-bitscaler.Thisscalerwasincrementedbyasignalfroma100MHzclock modulewhichwasresetatthebeginningofeachrun.Tocombinethedatathe timestampoftheeventrecordedineachdetectorsystemwerematchedupevent-by-event. Anegligiblenumberofeventswerelostinthismergingprocedure.Inordertoverifythat thethreesystemsweretimestampedcorrectlyonline,certainreferencesignalswererecorded ineachsystem,whichcouldbecompareddirectlywithoutmergingthedata. Duringtheexperiment,thetriggerforthedataacquisitionsystemwasacoincidence oftheS800spectrometer(derivedfromtheE1timingsignal),withthemultiplicitytrigger fromtheMiniball-Miniwallarray(multiplicity 2).TheLASSAarraywasnotpartof thetrigger,anditsdatawasonlytakenalongwiththeS800andtheMiniball-Miniwall.In theend,seventSn+Snreactionsystemsweremeasured,andtheamountofevents recordedineachsystemareshowninTable3.2. 32 ReactionSystem Events(Millions) 112 Sn+ 112 Sn 11.4 112 Sn+ 124 Sn 8.7 118 Sn+ 112 Sn 3.8 118 Sn+ 118 Sn 10.7 124 Sn+ 112 Sn 12.3 124 Sn+ 118 Sn 10.1 124 Sn+ 124 Sn 15.2 Table3.2Statisticsobtainedfromeachreactionsystem,combiningthethreemagnetic settingsoftheS800spectrometer. 33 3.2 S800Spectrometer TheS800spectrometer[55]wasusedinthisexperimenttomeasuretheheavyresidualfrag- mentsfrom 112 ; 118 ; 124 Sn+ 112 ; 118 ; 124 Sncollisions.TheS800consistsoftwodipolemagnets thatarecapableofbendingfragmentsupto4.0T m.Themomentumacceptanceis ˇ 5% andtheangularacceptanceisabout20msr.Inthisexperiment,theS800isoperatedin focusedmode,wherethebeamisfocusedatthetargetposition,andthefragmentsaredis- persedbytheirmagneticrigidityinthefocalplane.TheS800canidentifythemassand chargeofthesefragmentsbyusingthe E -TOF-B ˆ method,describedinSection4.2. Themeasurementoftheheavyfragmentsbeginswhenthefragmentspassthrougha 100 mplasticscintillatorlocatedabout0.75mdownstreamofthetarget.Arectangularhole wascutoutofthecenterofthescintillatortoallowthebeamtopassthrough,undegraded. Otherwise,the 112 SnbeamwouldhavebeentedintotheS800focalplane.Ifthefull beamintensityenteredthefocalplane,thefocalplanedetectorswouldhavebeendamaged. Thescintillatoritselfwouldalsodeterioratequicklyifthebeamwasimpingedonit.The ofthisholeontheresultsisdiscussedinmoredetailinAppendixC. Thefocalplaneconsistsofseveraldetectorsthatareusedtoprovidekinematicalinforma- tionaboutthedetectedfragments.AschematicdrawingofthefocalplaneisshowninFigure 3.2.First,thefragmentspassthroughtwoCathodeReadoutDriftChambers(CRDC's)[56]. Eachprovidesapositionmeasurementintwodimensions.EachCRDCisapproximately30 cmby59cmwide,andisabout1.5cmthickinthebeamdirection.Theyarewithagas mixturethatis80%CF 4 and20%C 4 H 10 at140Torr.Thegasvolumeisisolatedfromthe vacuumbythinwindowsmadefrom12 mthickPPTA,apolymersimilartokevlar.When aparticlepassesthroughthegasvolume,thegasisionizedandtheionizationelectronsare 34 Figure3.2SchematicdrawingoftheS800focalplane.Figureistakenfrom[57]. driftedbyanelectricTheyareavalanchedonanodewiresthatrunperpendiculartothe beamdirection,whichinducesachargeonthecathodepads.EachCRDChas224padsthat registertheinducedchargefromtheavalanchingelectrons.Theinducedchargeisspread overabout10padsinaroughlygaussiandistributionBythechargedistribution,the positioninthisdimension(the\dispersive"direction)canbedeterminedwitharesolutionof ˙<: 5mm.Bycomparingthetimethatthesignalarrivestoareference,theposition inthenon-dispersivedirectioncanbedetermined.AschematicdrawingoftheCRDC'sis showninFigure3.3. AftertheCRDC's,thefragmententerstheionizationchamber,wheretheyloseasig cantportionoftheirenergy.TheatomicchargeZoftheheavyfragmentisdeterminedusing theenergylossmeasuredinthisdetector.LiketheCRDC's,theS800ionizationchamber 35 Figure3.3SchematicdrawingoftheCRDC'sintheS800focalplane.Figureistakenfrom [58]. 36 isadetector,with300TorrofP10gas.(90%Ar,10%Methane).Itisseg- mentedinto16separatevolumes,andtheionizationineachvolumeismeasuredindividually. Theionizationchamberissegmentedtodecreasetheoverallnoiseinthecombinedsignal. Eachsegmenthas1/16 th oftheelectronicnoiseofthetotalvolume,butthenoiseaddsin quadratureandnotadditively. Thefragmentsthendepositmostoralloftheirremainingenergyina30cmby59cmwide, 1mmthickscintillator,calledtheE1scintillator.ThetimingsignalfromtheE1scintillator issubtractedfromthetimingsignalfromthescintillatorinthetargetchambertogivethe tthroughtheS800.Thetime-oftisusedtoidentifytheheavyfragmentsby theirmass-to-chargeratio.ThescintillatorhasanEMI9807Bphotomultipliertubeateach end.Inthisexperiment,onlyinformationfromthehigh-momentumsidewasused,because thissideprovidedtheexperimentaltrigger. Finally,thehighestenergyfragmentsdepositanyremainingenergyina4by8arrayof 32CsI(Na)scintillators,whichareeachreadoutwithaHamamatsumodelR1307photomul- tipliertube[59].ItcanbeseenattheendofthefocalplaneinFigure3.2.Thehodoscope wascommissionedshortlybeforethisexperiment.Itspurposeistoprovideanadditional measurementofthefragmentenergy.Thetotalenergyisneededtodistinguishfragments thatarenotfullystrippedofelectrons.Theanalysisofthedatafromthehodoscopeis describedseparatelyinAppendixB. Inordertoreconstructthefullmomentumdistributionoftheheavyfragments,three tmagneticsettingswereusedintheS800.Therangeofisotopesthataremeasured inthisexperimentwasdeterminedbythechoiceofmagneticrigidity.Foragivenvelocity, veryneutron-richfragmentshavehighmagneticrigidity,andneutron-poorfragmentshave lowmagneticrigidity.Theleastrigidbeamspeciesis 112 Sn,with Bˆ =2 : 745T m.To 37 preventanychargestatesoftheprimarybeamfromenteringthefocalplane,measurements weremadeat2.40,2.51,and2.60T m.Thisalsomeansthatthefragmentsthatwere measuredaremoreneutron-poorthan 112 Sn,becausethefragmentswillallhavesimilar velocity. 3.3 LargeAreaSiliconStripArray(LASSA) TheLargeAreaSiliconStripArray(LASSA)wasusedtodetectandidentifylightparticles andintermediatemassfragments(IMF's)uptoCarbon[60].TheLASSAarray,forthis experiment,wasmadeupofeighttelescopes,eachconsistingofa500 mdouble-sided siliconstripdetectorinfrontofagroupoffourCsI(Tl)crystals.AschematicoftheLASSA arrayisshowninFigure3.4.ThesiliconDSSDissegmentedinto16stripsoneachside,with thefrontstripsorthogonaltothebackstrips.Whenachargedparticlepassesthroughthe siliconwafer,asignalisinducedinonebackstripandonefrontstrip.Theoverlapbetween thetriggeredfrontandbackstripapixel,a3mmby3mmsquare.Thepositionof thepixelcanbeusedtodeterminethescatteringangleofthedetectedparticle.Theidentity oftheparticlecanbedeterminedbycomparingthesignalmeasuredinthesilicondetector withthesignalmeasuredintheCsIcrystal.Usingthismethod,isotopicidencanbe obtaineduptooxygenisotopes,butinthisexperimentthedynamicrangeoftheelectronics waschosentomeasurethroughCarbon. TheLASSAarray,with8telescopes,contains256individualsilicondetectorstrips.Each detectorstripwasconnectedtoachargesensitive(CSA),whichwaslocatednear thedetectorsinsidethevacuumchamber.TheCSA'swerebuiltspfortheLASSA array,andaredescribedindetailin[60].Theelectronicsusedtoprocessthesignalsfrom 38 Figure3.4SchematicdrawingoftheLASSAarray.Figureistakenfrom[61]. theLASSAsilicondetectorswereApplication-SpIntegratedCircuits(ASIC)[62],which werespdevelopedforusewithsiliconstripdetectorssuchasLASSA(orHiRA[63]). Theadvantagesofusingthesehigh-densityASICelectronicsaretheirrelativelylowcostand thelowspacerequirementsforinstrumentingthearray.BecausetheASIC'saredesigned forsparsereadout(onlychannelswhichareabovethresholdarerecorded),manychannels canbeinstrumentedwithoutdrasticallyincreasingthedead-timeofthesystem. ThedynamicrangeoftheASICelectronicsislimited,whichlimitstherangeofelements thatcouldbeisotopicallyidenTosolvethisproblem,thesignalswererecordedwith twogainstages.Thesignalfromthehigh-gaincharge-sensitivewasresistively splitandonesignalwasattenuatedbyafactoroffour.Thehighgainsettingwasoptimized fordistinguishingisotopesofhydrogenandhelium,andthelowgainsettingwaschosento identifyisotopesthroughcarbon.Thismethodhassincebeenusedinseveralexperiments 39 Figure3.5PhotooftheLASSAarray.TheforwardringsoftheMiniballandMiniwallare shownontheleftside,andtheeightLASSAtelescopesareshownontherightside. withtheHiRAarray[64].Thenumberofchannels(512)wasmanageablecomparedtothe fullsuiteofHiRAdetectorswhichrequiresnearly2000channels. TheLASSAarraytooktheplaceoftherightsideoftheforwardpartoftheMiniball- MiniwallArray.AphotooftheLASSAintheexperimentalisshowninFigure 3.5.Duringtheexperiment,thesilicondetectorswerecoveredwithathinMylarfoilto preventlightleaksaswellasseverallayersofSn-Pbfoiltoabsorbelectronsejectedfromthe target. AnimportantfeatureofDSSD'sisthepixelstructurethatresultsfromtheperpendicular frontandbackstrips.Thepositionofthepixeldeterminesthescatteringangleofaparticle ifthepositionofthedetectorrelativetothetargetismeasured.ThepositionoftheLASSA arraywasmeasuredusingaportablecoordinatemeasuringmachine(CMM)arm,commer- ciallyavailableunderthebrandnameROMER.Thisarmiscapableofmeasuringpositionsin 40 threedimensionsinanapproximately1mradiuswithaprecisionof < 100 m.TheROMER armmeasuresthepositionofaprobetippedwitha3mmrubysphere.Bymeasuringdata pointsalongasurface,3dimensionalfeatures(planes,spheres,cylinders,etc.)canbecon- structed.BecausetheROMERarmrequiresphysicalcontactwithasurface,theposition ofthesilicondetectorswasdeterminedbymeasuringthepositionofthealuminumhousing ofthetelescope.ThepixelpositionswerethenfoundusingtheCADdesignparametersof theLASSAtelescope.Thepositionsarerstmeasuredinthelocalcoordinatesystemof theROMERarm,andarethentransformedintotheNSCLglobalcoordinatesystem.The beamlineandS800arealignedtothisreferenceframe,sotheangleswithrespecttothe beamlinecanbedetermined.ThemeasuredpositionsareshowninFigure3.6. 41 Figure3.6AnglesofeachpixelfortheeightLASSAtelescopes,extrapolatedfrommeasure- mentswithROMERArm.Anglesarecalculatedwithrespecttothemeasuredbeamline. 42 3.4 Miniball/MiniwallArray TheMSU-Miniball[65]wasdevelopedtostudymultifragmentationinducedbyheavyion collisions.Initsoriginalitconsistedof188individualdetectors,covering89% of4 ˇ .Itconsistedof11azimuthallysymmetricringsofdetectors,andeachringismounted separatelyonapairofrailsalongthebeamdirection.Heavyioncollisionsatintermedi- ateenergiesrequirehighgranularityatforwardangles,becausethereactionsproducehigh chargedparticlemultiplicityatforwarddirections.Toaddressthis,theMiniwallarraywas builtatWashingtonUniversity[66].TheMiniwallreplacedtheforwardtworingsofthe Miniballwith6newrings.TheMiniwalldetectorsarefunctionallysimilartotheMiniball detectors.Asopposedtobeingmountedonindividualringstructures,theMiniwallde- tectorsareallmountedfromasinglemetalbase.TheMiniwalldetectorsaretly furtherfromthetarget,andaremoredenselypacked,whichallowsforatincrease ingranularity.TheMiniwallisshown,withthetwominiballrings,inFigure3.7,and thefullMiniball-MiniwallassemblyisshowninFigure3.8. Eachdetectorhastwoactiveelements:athin,fastplasticscintillatorinfrontofa 2cmthickCsI(Tl)scintillatorcrystal.Thisiscalledaphoswich(\phosphor sandwich"),andthelightproducedinthescintillatorsismeasuredbyaphotomultiplier tube.AschematicofasingledetectorisshownintheleftsideofFigure3.9.Thetwo scintillatingelementshavettimeresponses,sothatsignalsinthefastscintillator canbedisentangledfromtheCsIbycomparingthesignalattregionsintime.The rightsideofFigure3.9showsthestructureoftheoutputofaMiniball-Miniwalldetector. Sp,elementswithhigherZwilllosealargerportionoftheirenergyinthefast scintillatorcomparedtotheCsI,andsotheratioofFast:Slowwillbelarger.Inaddition, 43 Figure3.7ForwardpartoftheMiniball-Miniwallarray,intheusedforthis experiment.ThetwomostforwardringsoftheMiniwallarenotused. forlightparticles(HandHe),theshapeofthepulsedependsweaklyontheZandAofthe incidentparticle.BycomparingtheSlowandTailsignals,theshapecanbeusedtoidentify isotopesofHandHe. TheMiniball-Miniwallelectronics,forthisexperiment,consistedof12banksof16chan- nels(192channels),andforeachchannel,foursignalsneededtobedigitized.(Fast,Slow,Tail, andTime).DetaileddiagramsoftheMiniballelectronicsareshownin[61]and[68],and theelectronicsusedinthisexperimentweredesignedinthesameway.Table3.3showsthe geometriccoverageoftheMiniball-Miniwallarraythatwasusedduringthisexperiment. BecausemanydetectorshadtoberemovedtoallowtheLASSAarraytobemounted,the coverageoftheremainingarrayaddsupto77%of4 ˇ . Forthepurposesofthisdissertation,theMiniball-Miniwallarrayfunctionsasacharged particlecounter.Bysimplycountingthenumberofdetectorsthatareabovethreshold, 44 Figure3.8FullyassembledMiniball-Miniwallarray,showingthebackwardrings.Thebeam inincidentfromtheleft,andthetargetisatthegeometriccenteroftheminiball. Figure3.9Left:SchematicofaMiniballdetector[65].Right:Schematicshowingthetime structureofaMiniballsignal[67].Thethreeshadedregionsrepresentthetimewindows whereintheFast,Slow,andTailsignalsareintegratedbytheChargeADC's. 45 Ring# #ofDetectors (msr) 3(MW) 10/22 10 2.59 4(MW) 12/26 13 2.85 5(MW) 11/24 16.625 5.56 6(MW) 11/24 21.875 10.64 3' 13/28 28 11.02 4 10/24 35.5 22.9 5 13/24 45 30.8 6 15/20 57.5 64.8 7 20/20 72.5 74 8 16/18 90 113.3 9 14/14 110 135.1 10 12/12 130 128.3 11 8/8 150 125.7 Total 165/264 - 9712(77%of4 ˇ ) Table3.3GeometricparametersoftheMiniball-Miniwallarrayusedinthisexperiment. theapproximatechargedparticlemultiplicitycanbedetermined.Thechargedparticle multiplicitycanberelatedtotheimpactparameter,whichisdescribedindetailinSection 4.5.ThemorecomplexfunctionsoftheMiniball,liketheparticleidenonorenergy calibration,arenotusedinthisdissertation,butcanbeusedinthefutureforfurtheranalysis. Forexample,thetotaltransverseenergycanbeusedasameasureofthecentralityofthe collisionaswell[69]. 46 Chapter4 DataAnalysis 4.1 DetectorCalibrations 4.1.1 IonizationChamber ThepurposeoftheS800ionizationchamberistoprovideameasureoftheenergylossthat afragmentexperiences,whichgivesinformationabouttheatomicnumberofthefragment. (seeSection4.2)Theionizationchamberissegmented[56]intosixteenseparatesections, whichreducestheoverallsignalresolution.The16signalsaregainmatchedsothatthe signalfromtheprimarybeamisequalineachsegment.Tocombinethesignalsfrom16 anodes,thesignalsaresimplysummed(anddividedby16,tokeepthenumbersonthesame scale). E sum = 1 16 16 X i =0 E raw ;i (4.1) Themaincalibrationthatneededtobeappliedtotheionizationchamberwasacorrection forthedriftofthegain,duetoasystematicshiftofthegaspressureduringtheexperiment. ThepressureoftheP10gaswasnominallysetat300Torr,butduetoaprobleminthe gashandlingsystem,thegaspressuredriftedbetween300and330Torr.Thepressurewas 47 Figure4.1GaspressureintheS800ionizationchamber,asafunctionoftime.Thepressure deviatedfromthesetpressureof300Torrthroughoutmostoftheexperiment. recordedbythegashandlingsystemsoanempiricalcorrectionismaderun-by-run.The trendofthepressureasafunctionoftimeisshowninFigure4.1. Thedriftofthegaspressurewouldtheresolutionoftheparticleiden spectruminagivenbeam-target- Bˆ setting,sincethegainwoulddriftproportionally.The isparticularlyproblematicduringthedatatakenwiththe 112 Snbeam,sinceeach Bˆ settingwasmeasuredintwosegments,withappreciablytgaspressures.The correctionisbasedontheconceptthattheamountofionizationwilldependonthevolume numberdensityinthegas,whichisdirectlyproportionaltothegaspressure.Aorder correctioncanbeobtainedbysimplyscalingtheenergylosswiththefractionalchangein thepressure: E = E sum 1+ P 300 300 (4.2) butanempiricalscalingfactorforthepressurewasdetermined,sothattheresulting correctionwas 48 Figure4.2LeftPanel:Uncorrectedionizationchamberenergylossvs.gaspressure,gated on 75 Br.Severaltreactionsat Bˆ =2 : 6areshowntosampletherangeofpressure from300-335Torr.RightPanel:Sameasleft,butwiththecorrectionfromEquation4.3 applied. E = E sum 1+( : 00394) ( P 300) (4.3) Thecorrectionwasfoundbythetrendoftheenergylosswiththemeasured pressure.Sinceanyonesettingonlysampledasmallrangeofpressure,manysettingshad tobeused.TheresultofthecorrectionforpressureisshowninFigure4.2.Thisenergyloss, correctedforpressure,isthequantityusedforparticleideninSection4.2. 4.1.2 Hodoscope ThehodoscopeattheendoftheS800focalplaneconsistsof32CsI(Na)scintillatorcrystals, arrangedina4x8array,andisusedtomeasurethetotalkineticenergyofthefragment detectedinthefocalplane.Duringtheexperiment,thegaswasremovedfromtheCRDC's andtheionizationchamber,andthebeamwasscannedacrossthefaceofthedetectors. Thissuppliedamonoenergeticsignaltomatchthegainofallthecrystals.Sinceonlyone 49 Figure4.3TopPanel:Rawhodoscopesignalsforeachcrystal,whenscanningthe 124 Sn beamacrossalldetectors.BottomPanel:Sameastop,withallcrystalsgainmatched. calibrationpointwaspossible,andthelightoutputofCsIdependsonthespeciesofthe fragment,allchannelsaresimplyscaledtohavethesamepeakvalue.Thiscalibrationis showninFigure4.3.Whencalibratingtoatrueenergyscale,anon-linearfunctionmust beapplied,asdescribedin[70].Thehodoscopedidnotperformasexpectedduringthis experimentsothehodoscopedatawerenotused.Furtheranalysiswiththehodoscopeis describedinAppendixB. 4.1.3 CathodeReadoutDriftChambers(CRDC's) TheS800spectrometerhasadispersivefocalplane:afragment'spositioninthefocalplane determinesitsmagneticrigidity,ormomentum.TwoCRDC'sareusedtomeasurethe positionoftheincomingfragmentintwodimensions,attwoplanesseparatedbyabout1m. Thesedetectorsprovidecriticalinformationaboutthedetectedfragments,becausethese positionmeasurementsallowfortheparticlestrajectorytobetracedbacktothetarget 50 position.TheinversemappingisdescribedinSection4.1.4.Thissectiondescribesthe calibrationofthepositioninformationfromtheCRDC's. Thex(dispersive)positionsaremeasuredbyanalyzingthechargeinducedonanarrayof 224pads.Padswhicharemissing,havesaturatedsignals,orotherwisemarginalperformance areexcluded.Chargeisinducedonmanypads,andthecorrespondingpositioncanbe extractedwitharesolutionthatisbetterthanthepadpitch.Thispositioncanbeextracted eitherbyusingthecenter-of-gravityofthechargedistributionorby.Sinceboth ofthesemethodsinvolvecomparingtheamplitudesofneighboringpads,aprocedurewas implementedtomatchthegainofallthepads.Foreachpad,thechargeinducedonthepad wascomparedtotheenergylossintheionizationchamber.Eachpadsignalwasmatched totheenergylossintheionizationchamber,actuallyusinganon-linearfunctionto thesaturationofsomepads.Theresultofthis,fortheCRDC,isshowninFigure 4.4.Althoughthelinearitycorrectionisprobablyunnecessaryfortheprecisionneededin thisexperiment,thisproceduremaybeusefulforexperimentsrequiringveryhighprecision angularresolution. Fittingthechargedistributionasopposedtousingcenter-of-gravitywasfoundtobe muchmoreprecisewhencommissioningtheCRDC's,sothatmethodisusedinthisanalysis [56].Thisdistributionissampledforeachevent,andanexampleofthisforasingle eventisshowninFigure4.5.Theyieldsapadvalue,whichisconvertedintoanxposition usingthepadpitch(2 : 54mm/pad). Theexactcalibrationofthedispersivepositioncanbevusingdatafrommask runs,butingeneral,thecalibrationofthisdimensiondoesnotchange,sincetheCRDCpads areinspacefromoneexperimenttothenext.Thecalibrationofthenon-dispersive positiondoeschangefromoneexperimenttothenext,andeventhroughoutanexperiment. 51 Figure4.4TopPanel:Maximumrawpadsignalsvs.padnumberforoneruninthe 112 Sn+ 112 Snreactionsystemat Bˆ =2 : 4Tm.BottomPanel:Sameastop,butpadsare gainmatchedandcorrectedfornon-linearity.Slopedatlowchannelscorrespondsto athresholdintheE1scintillator,whichisusedastheS800DAQtrigger. 52 Figure4.5Inducedpadsignalvs.padnumberforone(typical)eventinthe 112 Sn+ 112 Sn reactionsystemat Bˆ =2 : 4Tm.Agaussiantothedataisshown,andthedottedline indicatestheextractedcentroidoftheevent. 53 Thenon-dispersivepositionisobtainedbyusingthedrifttimeoftheelectronsfromthe ionizationtrackalonganelectricdinthenon-dispersivedirection.Thismeasurementis sensitivetovariablessuchasthegaspressureandthepurityofthedetectorgas,whichvary withtime.Thecalibrationisobtainedbyinsertingatungstenmaskintothepathofthe fragments.Themaskwaslocatedapproximately10cminfrontofeachCRDC.Themask wasmachinedwithpreciseholesandslotstoallowfragmentstopassthroughatsp positions.ACADdrawingoftheCRDCmaskisshowninFigure4.6.Whenthemaskisin place,onlyfragmentwhichtravelthroughtheholes/slotswillbedetected. Whencalibratingtheposition,therearetwominorcomplications.Onecomplication comesfromthespacingbetweentheCRDCanditsmask.Forfragmentswhichtransit atanappreciableangle,thepositionmeasuredinthespectrumwillbefromthedesign positionofthehole.Becausereactionfragmentsareusedtopopulatethemaskrun,there isalargespreadofangles,whichtlydecreasestheprecisionofthecalibration. Toaccountforthis,theinformationfrombothCRDCscanbecombinedtoutilizethis angleinformation.Thisprocedurecausesthesecondcomplication,becauseitcausesthe calibrationofCRDC1todependonthecalibrationofCRDC2.ThetwoCRDC'smustbe thencorrectediteratively,oneatatime,untilthecalibrationconverges.Theintermediate stepsarenotshownhere,buttheresultingCRDC1maskspectrumisshowninFigure4.7. ThelaststepincalibratingtheCRDC'sistomakeacorrectiontothecalibrationinthe non-dispersivedirection,toaccountforthechangingelectrondriftvelocityovertime.By simplyextractingthecentroidinthenon-dispersivedirectionrunbyrun,andthenscaling thatcalibration,thevariationfromruntoruniseliminated.Thematched,calibratednon- dispersivepositionisshowninFigure4.8 54 Figure4.6SchematicdrawingoftheCRDCmask.Thelongedgecorrespondstothe dispersivedirection. Figure4.7MaskspectrumforCRDC1. 55 Figure4.8TopPanel:RawCRDC1drifttimevs.runnumber,foralldataruns.Bottom Panel:CalibratedandmatchedCRDC1non-dispersivepositionvs.runnumber. 56 4.1.4 InverseMapping Themeasuredpositionsinfocalplanecanbetransformedintocorrespondingquantitiesat thetargetpositionbyuseofaninversemap[71].Theseinversemapsareavailablepublicly, andareutilizedinmostexperimentswiththeS800.ThemagneticoftheS800magnets wasmappedandthedatawereincorporatedintoasimulationusingCOSYINFINITY, whichisthenusedtoextractanempiricaltransformationbetweentheincoming(target) andoutgoing(focalplane)positionsandangles.Thistransformationistheninverted,and canbeusedintheanalysistoreconstructthereactionatthetargetposition. Theinversemapcanbedescribedwithamatrixequation: 0 B B B B B B B B B @ dta yta ata bta 1 C C C C C C C C C A = 0 B B B B B B B B B @ ( dta j xfp )( dta j yfp )( dta j afp )( dta j bfp ) ( yta j xfp )( yta j yfp )( yta j afp )( yta j bfp ) ( ata j xfp )( ata j yfp )( ata j afp )( ata j bfp ) ( bta j xfp )( bta j yfp )( bta j afp )( bta j bfp ) 1 C C C C C C C C C A | {z } InverseMapMatrix 0 B B B B B B B B B @ xfp yfp afp bfp 1 C C C C C C C C C A (4.4) wherethevariablesaredescribedinTable4.1.Tofullyaccountforthemeasuredmagnetic eachelementintheinversemapmatrixisanonlinearfunctionofthefourmea- suredparameters.Thenon-dispersivepositionatthetarget, yta ,canbeobtainedthrough theinversemap,whilethedispersivepositionatthetarget, xta ,cannot.Thisissimplydue tothefactthatthe x directionhasdispersioninthemagneticsothat ata and dta are convoluted. 57 Variable Description xfp Dispersivepositionatthefocalplane crdc 1 :x yfp Non-dispersivepositionatthefocalplane crdc 1 :y afp Dispersiveangleatthefocalplane crdc 2 :x crdc 1 :x 1 : 073m bfp Non-dispersiveangleatthefocalplane crdc 2 :y crdc 1 :y 1 : 073m dta FragmentEnergy(Fractional) OutputfromMap E E 0 E 0 yta Non-dispersivepositionatthetarget OutputfromMap ata Dispersiveangleatthetarget OutputfromMap bta Non-dispersiveangleatthetarget OutputfromMap E 0 Centralenergyofmagneticsetting Constantforanexperimentalsetting Table4.1DescriptionsofthevariablesobtainedfromtheS800orcalculatedusingtheinverse maps. Usingtheinversemapallowsforthedeterminationoftheenergy(withhighprecision)of theoutgoingfragment,aswellasprovidesangularinformation.Theangularacceptancein thisexperimentwastoolimitedtoprovideusefulinformationaboutthesystematicsofthe angulardistributionsproducedinprojectilefragmentation,althoughtheangularinformation canbeusedtomakecorrectionsforacceptance,whichisdescribedinmoredetailinAppendix C. Thelaststepinvolvesapplyingtothetargetangles( ata and bta )tocorrectfor theincomingangleofthebeam.Foreachbeam,thebeamwasdegradedintothefocalplane usingvaryingthicknessesofaluminumfoil.Thesecalibrationrunsprovidetheoutgoing targetanglesthatarethenaszeroangle.Iftherewereaninthedispersive positionatthetarget,thatwouldresultinamiscalibrationoftheenergyofthefragment detectedinthefocalplane.Thiscouldbeaddressedusingtheenergyoftheprimarybeam intotheS800focalplaneasacalibrationpoint,butaprecise,absoluteenergycalibrationis notcriticalforthephysicsinthisexperiment. 58 4.2 ParticleIden OneofthemainadvantagesoftheS800Spectrometerisitsabilitytoisotopicallyidentify heavyreactionproducts.Toanalyzethedata,the ˆ methodisused.First,the mass-to-chargeratiocanberelated(formotionthroughamagnetictotheTime-Of- Flight(TOF)andmagneticrigidity(B ˆ )by: m q / Bˆ / Bˆ TOF (4.5) Where m isthefragmentmass, q isthechargeofthefragment, Bˆ isthemagneticrigidityof theparticle, isvelocityofthefragmentinunitsofthespeedoflight,and istheLorentz factor.Theatomicnumbercanberelatedtotheenergylossintheionizationchamberby theBethe-Blochformula: dE dx = 4 ˇe 4 Z 2 m 0 c 2 2 Nz ln 2 m 0 c 2 2 I ln 1 2 2 (4.6) whichgivesthestoppingpower dE dx inamediumofatomicnumber z ,atomicnumberdensity N andionizationpotential I .Where e and m 0 arethechargeandmassofanelectron, Z istheatomicnumberofthefragment.Finally,sinceboththeenergyloss E )and TOF havesomenon-trivialdependenciesonthevelocity,thesenumbersarecorrectedempirically usingthemeasured Bˆ .Inaddition,dependenciesofthe E and TOF onthemeasured 59 trajectoriesinthefocalplanecanberemovedaswell.Finally,thecorrected E and TOF canbeconvertedinto Z and A .Thisprocessisdescribedindetailinthissection. 4.2.1 EmpiricalCorrections Thereareseveralquantitiesthatobviouslythe TOF and E .Forinstance,therewill beaspreadintheraw TOF associatedwiththespreadinthefragmentmomentum,because theS800hasa5%momentumacceptanceatagivenrigiditysetting.Asaconcreteexample, 75 Brinthe Bˆ =2 : 51rigiditysettingtravelsapproximately15mfromthetargettotheS800 focalplaneinbetween138 : 7and144 : 9ns(assumingonlythemomentumspreadis the TOF ),buttheseparationbetween 75 Brand 76 Brisonly1 : 65nsat Bˆ =2 : 51.Thisalso wouldassumethatthetpathhasaedlength,whichisonlytrueforfragmentsthat travelalongtheopticalaxisoftheS800.Forexample,fragmentsofamomentumthat arescatteredtotheinsideofthebendoftheS800dipoleswouldexperienceashortert paththanparticleswhichtravelontheoutsideofthebend.Inadditiontophysical in TOF ,therecanbeduetothedetectionsystems.Forexample,themeasured TOF wouldbebypropagationtimeofthelightcreatedintheE1timingscintillator,since itisaverylargescintillatorandthetimingsignaliscollectedattheends.Fragmentsthat hitthetimingscintillatorclosetothephotomultipliertubewillhaveashortermeasured TOF thanfragmentsthathitthemiddleofthescintillator.Insteadoftryingtounderstand eachoftheseseparately,the TOF and E aresimplycorrectedempiricallyusingthe measuredpositionsinthefocalplane.Becausethemethodforapplyingthesecorrectionsis ofgeneralinterest,thestepsforonecaseareshownindetailinthissection.Usually,the 60 TOF and E arecorrectedthefollowingway: TOF corr = TOF raw d ( TOF ) d ( x 1) crdc 1 :x d ( TOF ) d ( afp ) afp (4.7) d ( TOF ) d ( y 1) crdc 1 :y d ( TOF ) d ( bfp ) bfp E corr = E raw d E ) d ( x 1) crdc 1 :x d E ) d ( afp ) afp (4.8) d E ) d ( y 1) crdc 1 :y d E ) d ( bfp ) bfp wherethelinearcorrectionsareobtainedempiricallyfromthedata.Forabetterdescription ofthevariablesused,seeSection4.1.4. Thegeneralgoalofthecorrectionsistosimplysubtractthedependenceofthe TOF and E onthefocalplanecoordinates.Ideally,thisdependencewouldbeextractedafterthe particleidenisdone,butinpractice,thismustbedoneiteratively.InFigure4.9a, the TOF and E areshownwithnocorrections.Inthiscase,slantedlinescorresponding tofragmentsofagivenelementareseen,althoughwithinadequateresolution.Becausethe TOF changescontinuouslywith Z aswellaswith m = q ,itisnecessarytorequireaselection on Z toextractthecorrectionforthe TOF .ThetwodimensionalgatewhichselectsoneZ isshownasasolidlineonFigure4.9a.Figures4.9band4.9cshowthecorrelationbetween theraw TOF andthedispersivepositionatCRDC1(correspondsroughlytothefragment momentum)andthecorrelationbetweentheraw TOF andthedispersiveanglemeasured atthefocalplane,respectively.Toobtaintheapproximatetrend,alineisdrawnandthe linearfunctionisextracted.TheseapproximatelinearfunctionsareshowninFigures4.9b and4.9casasolidline.Thislineartrendwithbothvariablesissubtractedfromthe TOF 61 Figure4.9a.)Uncorrected TOF vs.uncorrectedenergyloss,measuredinthe 112 Sn+ 112 Sn reactionsystemat Bˆ =2 : 4Tm.Solidlineshowsthetwodimensionalgateusedtoselect anapproximate Z b.)Uncorrected TOF vs.dispersivepositionatthefocalplane,requiring thegateshowninpanela.Solidlineshowsthecorrelationbetween TOF anddispersive position.c.)Uncorrected TOF vs.dispersiveangleatthefocalplane,requiringthegate showninpanela.Solidlineshowsthecorrelationbetween TOF anddispersiveangle.d.) Corrected TOF iteration)vs.uncorrectedenergyloss. andtheresultisshowninFigure4.9d,andtheisotopicstructurestartstobecomeapparent. Asanote,Figure4.9showsaloweriencybetween 20 mrad 28,itprovidesreasonableresultsforlower Z aswell.Theversionof GLOBAL thatis packagedwithLISE++isused,whichhasminorimprovementsoverthepublishedalgorithm. Theresultsofacalculationforisotopeswith N Z at Bˆ areshowninFigure 4.24.Forthepurposesofthisstudy,theequilibriumchargestatedistributionisused,and thedistributioncomesfromthetimingscintillator,whichismodeledassimplycarbon.For theisotopesshown,thechargestatecontaminationdecreaseswithincreasingvelocity, asexpected.Toproperlyaccountforthevelocitydependence,alineartothreedata points(at Bˆ =2 : 4 ; 2 : 51 ; 2 : 6)isusedtocalculatethecorrectionasafunctionof Bˆ . Tosimplifytheprocessofcorrectingthedata,itisassumedthatonlythecharge statecontributes,whichshouldbereliableaslongasthiscontributionissmall( < 20%). Also,itisassumedthattheparticleidenpreviouslycontainsonlythefullystripped fragmentandtheonecontaminantisotope.Withthisassumption,thecorrectionforisotope ( N;Z )dependsonfourvariables:theyieldofisotope( N;Z ),thechargestatefractionof isotope( N;Z ),theyieldofisotope( N 2 ;Z ),andthechargestatefractionofisotope ( N 2 ;Z ).Sincethemeasuredyield Y exp isacombinationofthefullystrippedionsand thecontaminantions: Y exp ( N;Z )= Y ( N;Z;q = Z ) | {z } fullystripped + Y ( N 2 ;Z;q = Z 1) | {z } contaminant (4.17) 87 Figure4.24Chargestatecontribution(ratioofhydrogen-likeionstofullystrippedions) shownasafunctionofatomicnumber Z ,ascalculatedby GLOBAL .Isotopesshownhave N Z =6. andthedesiredyieldis: Y ( N;Z )= Y ( N;Z;q = Z )+ Y ( N;Z;q = Z 1)(4.18) whichmeansthatthemeasuredyieldisrelatedtothedesiredyieldby: Y exp ( N;Z )= Y ( N;Z )+ a Y ( N 2 ;Z ) b Y ( N;Z )(4.19) =(1 b ) Y ( N;Z )+ a Y ( N 2 ;Z ) 88 where a = Y ( N 2 ;Z;q = Z 1) Y ( N 2 ;Z ) (4.20) b = Y ( N;Z;q = Z 1) Y ( N;Z ) and a and b arethefractionscalculatedusing GLOBAL . a representsthecontaminationofthe isotopeofinterestfrom N 2,and b representsthelossoftheisotopeofinterestto N +2. Togetacorrectionfactor,justdivideby Y ( N;Z ): Y exp ( N;Z ) Y ( N;Z ) =(1 b )+ a Y ( N 2 ;Z ) Y ( N;Z ) (4.21) Buttheyieldratioontherightsideinvolvesthedesiredyields,whichareunknown.Since theresultsfrom GLOBAL areapproximate,theratiooftheuncorrectedexperimentalyields canbeused: Y exp ( N 2 ;Z ) Y exp ( N;Z ) ˇ Y ( N 2 ;Z ) Y ( N;Z ) (4.22) Theprocedureisslightlymorecomplicatedthanisdescribedabove,sincetheyieldsare broadlydistributedin Bˆ andthechargestatefractionsdependon Bˆ (velocity)aswell. Inthatcasetheyields, Y ( N;Z ),wouldbereplacedbydistributionsinrigidity, y ( N;Z;Bˆ ). Thisishandledappropriatelyintheanalysis,butforclarity,thedescriptionhereonlyrefers tothetotalintegratedyields.Sincethecorrectiondependsontheratio, Y ( N 2 ;Z ) Y ( N;Z ) ,the correctionmustbecalculatedseparatelyforeachsystem.Thecorrection,calculatedusing 89 theintegratedyieldsfor 112 Sn+ 112 Snand 124 Sn+ 124 Sn,isshowninFigure4.25.The thatthiscorrectionwouldhaveontheisoscalingobservablesisshowninSection5.3. Figure4.25Chargestatecorrection(inverseofleftsideofEquation4.21)shownasafunction ofatomicnumber Z .Isotopesshownhave N Z =6 ; 7 ; 8. 90 4.4 ReconstructingMomentumDistributions 4.4.1 CombiningSpectrometerSettings Inthisexperiment,thedataweremeasuredinseveralsettings,eachcoveringabout5%in magneticrigidityormomentum.Thethreerigiditysettingshadcentralvaluesof2 : 4,2 : 51, and2 : 6Tm.Thewidthofthedistributioninmomentumisontheorderof0 : 2Tm(FWHM), dependingonthe A and Z ofthespecies,soeachmomentumdistributionsspannedallthree momentumsettings.Thesesettingswerechosensuchthattherewouldbeatleastthree isotopeswithamajorityoftheirmomentumdistributionmeasured.Tocombinethethree tsettings,eachsettingwasscaledbythelivetime(measuredusingaclockscalerin theS800DAQ)andscaledbytherawminiballscaler(proportionaltothenumberofbeam particles).Thedistribution(nocorrectionforacceptanceorchargestates)foroneisotopeis showninFigure4.26,withthethreesettingsshownasseparatecolors.Itisapparentthat theacceptancedropsneartheminimumandmaximummomentummeasuredineach setting.ThisisdiscussedmoreinAppendixC. Althoughtherearesomeintheacceptancefromonesettingtoanother,this wasmostpronouncedforsystemsusingthe 112 Snbeam,atthebeginningofthe experiment.Duringthesystemsusingthe 124 Snbeam,theacceptancewasmorestable betweenrigiditysettings.Assumingtheacceptanceisafunctionof Bˆ whichisnearthe centralrigidity,thecentralregioncanbeusedtoobtainarigidity(momentum)distribution thatisaccurateuptoascalingfactorrelatedtotheangularacceptance.Basedonthe informationinAppendixC,spFigureC.5,thecentralregionwithatacceptance correspondsto : 025 > > > > > < > > > > > > : a 34 exp( ( x a 1 ) 2 2( a 2 + a 3 ( x a 1 )) 2 ): A =34 a 35 exp( ( x a 1 ) 2 2( a 2 + a 3 ( x a 1 )) 2 ): A =35 a 36 exp( ( x a 1 ) 2 2( a 2 + a 3 ( x a 1 )) 2 ): A =36 (4.24) 96 wherethesixparameters a 1 ;a 2 ;a 3 ;a 34 ;a 35 ; and a 36 areoptimizedsimultaneously,and a 34 ;a 35 ; and a 36 aretherelativenormalizationsforeachisobar.Whenthissimultaneous isapplied,asinglevelocitydistributionreasonablydescribesthedatafromthe 124 Snbeam andforthe 118 Snbeam.Thisresultforthe A =77isobarsisshownfor 124 Sn+ 124 Snin Figure4.31and 118 Sn+ 118 SninFigure4.32. Figure4.31Fitresultfordistributionsinvelocity,for A =77,measuredinthe 124 Sn+ 124 Sn reaction.ThethreespectraaresimultaneouslywithEquation4.24,sothattheshapeis identicalineachspectrum. Figure4.32Fitresultfordistributionsinvelocity,for A =77,measuredinthe 118 Sn+ 118 Sn reaction.ThethreespectraaresimultaneouslywithEquation4.24,sothattheshapeis identicalineachspectrum. Toillustratetheadvantageoftheisobarssimultaneously,Figure4.28showsa distributionfromthe 124 Sn+ 124 Snreaction,withtherangeinrigiditycoveredbyeachisobar shownbydottedarrows.Fittingthedistributionssimultaneouslyallowsformoreconsistent fortheisobarswhereonlyonesideofthedistributionisgiven,whichisnecessaryfor extrapolationoftheintegratedyield. 97 Inthedatausingthe 112 Snbeam,thesituationissomewhatmorecomplicated.When thethreedistributionssimultaneously,thereisasystematicshiftinthepeakvalue ofthedistributionfortisobars,sothatgwithacommondistributionisnot possible.Tothesedistributions,anadditionalparametercanbeaddedsothatthepeak centroidhasalineardependenceonthe Z oftheisobars.Thiswasalsoseeninthe datafromRef.[75],whereasimilarlineardependencewasusedwhenisotopesatthe edgeoftheacceptance.MoredetailsofthesesystematicswillbediscussedinSection5.1. Toallowforthisadditionalparameter,thefunctionbecomes: f ( x )= 8 > > > > > > < > > > > > > : a 34 exp( ( x a 1 ) 2 2(( a 2 + a 4 i )+ a 3 ( x a 1 )) 2 ): A =34 a 35 exp( ( x a 1 ) 2 2(( a 2 + a 4 i )+ a 3 ( x a 1 )) 2 ): A =35 a 36 exp( ( x a 1 ) 2 2(( a 2 + a 4 i )+ a 3 ( x a 1 )) 2 ): A =36 (4.25) where i isanindexovertheisobars.Theparameter a 4 allowsthecentroidtoshiftacrossa chainofisobars,andforthisexamplechangesthepeakofthevelocitydistributionbyabout 0.5%betweeneachisobar.Theresultforthisnewfunctionforthe 112 Sn+ 112 Sn systemisshowninFigure4.33.Forconsistency,allsystemsareallowingthisparameter tovary.ThesystematictrendsofthevelocitydistributionsarepresentedinSection5.1. Figure4.33Fitresultfordistributionsinvelocity,for A =77,measuredinthe 112 Sn+ 112 Sn reaction.ThethreespectraaresimultaneouslywithEquation4.25.ContrarytoFigures 4.31and4.32,thecentroidsofthedistributionsareallowedtovarybetweenisobars. 98 4.5 MiniballCentralitySelection TheMiniballallowsforcharacterizationoftheimpactparametersprobedinthemeasured dataset.TheparametersoftheMiniballaredescribedinSection3.4.Theminiballmultiplic- ity,whichisclosetothetotalchargedparticlemultiplicity,iscalculatedsimplybycounting thenumberofminiballdetectorswithsignalsabovethresholdinanevent.Althougheach miniballdetectormeasures4quantities(fast,slow,tail,andtime),onlytheenergyinthe slowgateisusedforthiscalculation.Thischoiceismadebothforconvenience,andbecause theelectronicsfortheslowbranchwerethemostreliable. Oncethemultiplicityisextracted,thisinformationcanbetransformedintoameasure- mentoftheimpactparameter.Thisprocedurehasbeenusedinthepreviouslypublished resultswithheavyioncollisions,andisdetailedin[76].Thecalculationreliesonthesimple conceptofageometricalcrosssectionforacollisionwithaspherewithradius r : ˙ = ˇr 2 (4.26) Thiscrosssectioncanbeparametrizedusingthescatteringimpactparameterb,sothatthe crosssectionofacollisionthathasanimpactparameterof b orlessis: Z b 0 ˙ ( b 0 ) db 0 = ˇb 2 (4.27) 99 andtheminiballmultiplicity( N c )shouldbeanticorrelatedwiththeimpactparameter: Z b 0 ˙ ( b 0 ) db 0 = Z 1 N c ˙ ( N 0 c ) dN 0 c (4.28) where ˙ ( N c )isaquantitythatisactuallymeasurableinexperiment.Animportantcon- siderationinthismeasurementisthattheexperimentalapparatusmeasures any reaction betweenthebeamandtarget,andsothedataacquisitionsystemistriggeredusingonlythe Miniball,requiringamultiplicityof N c > =1.Thisistfromtheexperimentaltrig- gercondition,sothesenormalizationrunsweremeasuredforeachbeam-targetcombination separatelyfromthefragmentproductiondata.Intheseruns,thebeamwasattenuatedand thedownstreamtimingscintillatorwasmovedtothenormalizationpositionsothatthebeam ratecanbecounteddirectly.Thisprocedurewasdonewiththetargetintomeasurethe totalreactioncrosssection,aswellaswiththetargetout(throughablanktargetframe)to measuretheamountofbackgroundnotcomingfromthereactiontarget.Thespectraineach arenormalizedbythenumberofincidentbeamparticles(measureddirectlyby thescintillator),aswellasthearealatomicnumberdensityoftherespectivetarget.Then, thetarget-outspectraaresubtractedfromthetarget-inspectra.Theresultingspectraare shownintheleft-handpanelsofFigure4.34. Thetotalreactioncrosssection,andinturn b max areby: ˇb 2 max = Z 1 1 ˙ ( N 0 c ) dN 0 c (4.29) andtheseresultsareshowninTable4.2 100 Figure4.34Totalreactioncrosssectionvs.miniballmultiplicity(left-sidepanels)and calculatedimpactparametervs.miniballmultiplicity(right-sidepanels).Shownfortwo targetseachfor 112 Sn(Top), 118 Sn(Middle),and 124 Sn(Bottom). Theextracted b max forthe 124 Sn+ 124 Snreactionissmallerthanforthe 112 Sn+ 112 Sn reaction,eventhoughtheradiioftheneutronrich 124 Snwouldbelarger.Thissuggeststhat themultiplicityisbytheneutron-richnessofthesystemaswellastheimpact parameter,sinceemittedneutronsarenotdetected. Whencomparingsystemswhichhavetsize,itisconvenienttousethereduced impactparameter: ^ b = b b max (4.30) 101 Reaction ˙ total (barn) b max (fm) 112 Sn+ 112 Sn 4.04 11.34 112 Sn+ 124 Sn 3.79 10.98 118 Sn+ 112 Sn 4.07 11.38 118 Sn+ 118 Sn 3.71 10.86 124 Sn+ 112 Sn 4.07 11.38 124 Sn+ 124 Sn 3.62 10.73 Table4.2Maximumimpactparameterandtotalintegratedcrosssectionmeasuredinthe forthesixmeasuredbeam-targetcombinations. whichgivesadimensionlessvariablewhichscaleswiththecentralityofthecollision,event byevent.Therelationshipbetween ^ b and N c isshowninFigure4.35. Theexperimentaldataisaveragedoverarangeofimpactparameters,whiletheoretical calculationshaveaimpactparameter.Theamountofisospinthatwilloccur iscorrelatedwithtimewhentheprojectileandtargetareincontact.Thiscontacttime increaseswithincreasingcentralityofthecollision.Thus,theimpactparametermustbe characterizedtocomparethesimulationtothedata.Theexperimentalobservable(isoscal- ing)ismostconvenientlymeasuredasafunctionof Z ,soarelationshipbetweenthemeasured Zandtheimpactparametermustbedetermined. Becausethedataaremeasuredinthreeerentmomentumsettings,itisimpossible tohavea\minimumbias"normalizationaswasdonewiththeMiniball.Therelationship betweenthemeasuredZandthededucedimpactparameterisshowninFigures4.36and 4.37forthe 112 Sn+ 112 Snand 124 Sn+ 124 Snreactions,respectively.Thesearemade bycombiningallthedatafromthethree Bˆ settings,withtheappropriatenormalizations foreachsetting.ThesespectrahaveaduetotheacceptanceatlowZ,becausethe gainoftheS800focalplanedetectorswasnothighenoughtodetectlowZfragments.At 102 Figure4.35Reducedimpactparametervs.miniballmultiplicity(right-sidepanels).Shown forthereactions 112 Sn+ 112 Snand 124 Sn+ 124 Sn. highZ,therearelowstatisticsnearZ=50becausetheentireisotopicdistributionwasnot measured,onlythemoreneutron-poorfragmentsweremeasured. InFigures4.36and4.37,theblackcirclesrepresentthemeanimpactparameterfor alleventsthatresultinaheavyfragmentofcharge Z .Alternatively,themostprobable impactparametercouldbeused.Figure4.38showstheinthededucedimpact parameterwhencalculatingthemeancomparedtoextractingthepeakfromagaussian tothedistribution.Theislessthan0 : 1fm,whichisnegligible.Sincetheisospin dataismeasuredasanaverageoverimpactparameterforaZ,themean impactparameterwillbeusedinallfurtheranalysis. 103 Figure4.36Two-dimensionalspectrumshowingtheimpactparameterextractedfromthe Miniballmultiplicityvs.theatomicnumberZofthefragmentmeasuredintheS800,plotted forthe 112 Sn+ 112 Snreactiondata.Thesolidcirclesshowtheaverageimpactparameterfor eachZ,wheretheerrorsaresmallerthatthedatapoints. 104 Figure4.37Two-dimensionalspectrumshowingtheimpactparameterextractedfromthe Miniballmultiplicityvs.theatomicnumberZofthefragmentmeasuredintheS800,plotted forthe 124 Sn+ 124 Snreactiondata.Thesolidcirclesshowtheaverageimpactparameterfor eachZ,wheretheerrorsaresmallerthatthedatapoints. 105 Figure4.38betweentheimpactparameterdeterminedusingameanand withagaussianfunctionvs.theatomicnumberZofthefragmentmeasuredintheS800, plottedforthe 112 Sn+ 112 Snreactiondata. 106 Figure4.39MeasuredcorrelationbetweentheaverageZoftheheavyresiduemeasured intheS800andtheimpactparameterextractedfromthechargedparticlemultiplicity. Datapointsareshownforthe 112 Sn+ 112 Sn, 112 Sn+ 124 Sn, 124 Sn+ 112 Sn,and 124 Sn+ 124 Sn systems.Theimpactparameteriscalculatedasthemeanimpactparameterforeventswhich yieldafragmentwithatomicnumberZ. 107 Therelationshipbetweenthemeanimpactparameterandthefragment Z isshownfor thefourmainreactionsystemsinFigure4.39.Overall,thetrendissimilarbetweenall foursystems.Forsmaller Z ,thededucedimpactparameterdoesnotstronglydependonthe system,butforlarger b thereisasmallbetweenthefoursystems.Thesimilarityof thesetrendsgivessomethatthechargedparticlemultiplicityisareliablemeasure oftheimpactparameter.Thediscrepancyathigh Z betweenthetsystemsmaybe causedbytheexperimentalacceptance,becauseonlythemoreneutron-poorfragmentsare measuredineachreaction. Becausetheimpactparameterdistributionissharplypeaked,nocutontheimpact parameterismadebeforeextractingtheisoscalinginformation.Thisallowstheuseofthe fullmeasuredstatistics,andtherelationshipbetweenZandbcanbeusedtomeasureisospin asafunctionofimpactparameter.Fromthemeasuredrelationship,themeasured datafromabout Z =27to Z =43spansarangeofimpactparametersbetween b ˇ 7 fmand b ˇ 10fm.TheresultsforthefourmainreactionsystemsareshowninTable4.3. Theerrorsquotedinthistableareobtainedfromagaussiantotheimpactparameter distributionforeach Z .TheseresultswillbeusedinSection5.4toconnectthemeasured datawiththeoreticalcalculations. 108 b(fm) b(fm) b(fm) b(fm) Z 112 Sn+ 112 Sn 112 Sn+ 124 Sn 124 Sn+ 112 Sn 124 Sn+ 124 Sn 27 7.1 1.5 7.4 1.4 7.1 1.5 7.1 1.3 28 7.2 1.5 7.5 1.4 7.2 1.5 7.2 1.3 29 7.3 1.4 7.6 1.4 7.3 1.5 7.3 1.3 30 7.4 1.4 7.7 1.3 7.4 1.4 7.4 1.3 31 7.6 1.3 7.9 1.3 7.6 1.4 7.6 1.2 32 7.7 1.4 8 1.2 7.7 1.4 7.7 1.2 33 7.9 1.4 8.1 1.2 7.8 1.4 7.8 1.2 34 8 1.3 8.3 1.2 7.9 1.3 7.9 1.2 35 8.2 1.3 8.4 1.1 8.1 1.3 8 1.1 36 8.3 1.3 8.6 1.1 8.2 1.3 8.2 1.1 37 8.5 1.2 8.7 1.1 8.3 1.3 8.3 1.1 38 8.7 1.1 8.9 1.1 8.5 1.2 8.4 1 39 8.9 1.1 9.1 0.93 8.6 1.2 8.6 1 40 9 1.1 9.2 0.91 8.8 1.1 8.7 1 41 9.2 1.1 9.4 0.82 8.9 1.1 8.9 0.93 42 9.4 0.97 9.5 0.8 9.1 1.1 9 0.91 43 9.6 0.88 9.7 0.76 9.3 1.1 9.2 0.83 Table4.3Meanimpactparameterdeducedfromtheminiballmultiplicitytabulatedby atomicnumber Z .Theerrorsarethewidth( ˙ )ofagaussiantotheimpactparameter distributionforeach Z . 109 Chapter5 Results 5.1 SystematicsofVelocityDistributions Inthissection,themeasuredfragmentvelocitydistributionsarediscussed.Bycomparing thevelocitydistributionstoestablishedtrends,thesystematicsmayprovideinformation aboutthereactiondynamicswhichmayexplaintheisoscalingandisospinresults inSections5.3and5.4.Manyexperimentsfocusonprovidingproductioncrosssectionsfor exoticnuclei,inordertoimprovemodelsusedtopredictpropertiesofexoticbeams.Accu- ratepredictionsofprojectilefragmentationcrosssectionsarepracticallyusefulbecausefast fragmentationistheprimarymethodforproducingveryexoticbeams.Thedatameasuredin thisexperimentmaybeusefultoimprovemodelsusedtopredictthepuritiesandintensities ofradioactivebeams.Inparticular,thisexperimentmeasureshowtheprojectileandtarget N = Z thefragmentationdynamics,becauseofthewiderangeofisospinasymmetry thatwasprobed. Sincethedynamicsofprojectilefragmentationaredictatedbythemasslossofthefrag- ment, A = A proj A ,soallparameterswillbeplottedinthischaptervs. A .Although secondarydecayaltersthemassofthedetectedfragments,themasslosscanberelatedto theimpactparameterandtheenergydissipationduringthecollision.Plottingavariable vs.themasslossthencangivesomeinsightintohowthatparameterisvaryingwiththe centralityanddissipationofthecollision,whichisusefulforcomparisontotheory. 110 Theparameterthataccountsforthe\skewing"ofthedistributions, skew ,isshownin Figure5.1.Themagnitudeofthisparameterhowasymmetricthevelocitydistribu- tionsis;asmallermagnitudeof skew meansclosertoanormalgaussiandistribution.Many studiesofprojectilefragmentation[75,77,78]haveshownthatthemomentumdistribution isasymmetric.Semi-empiricalmodelshavebeenputforwardtoexplainwhythismightbe, butisbeyondthescopeofthiswork.Thetrendofthemeasureddatashowsthat skew does notseemtovarywiththetargetmassatall.Thereisaweakdependenceonthemassof theprojectile.Forallsystems,themagnitudeof skew increaseswithincreasingmassloss. Iftheprocessproceededby\purefragmentation",thegaussiandistributionwouldbemore symmetricwith skew ˇ 0.Theexperimentalresultthat skew 6 =0isasignatureofthe increasingimportanceofnon-fragmentation Anothersystematictrendwhichisnotsimplyunderstoodishowthevelocityofisobars changes.Simplemodelsofprojectilefragmentationdonotpredictanisospindependencefor thevelocitydistributions[79,80].Mostofthemeasuredreactionsdemonstrateavelocity thatdependsonlyonthemasslossofthemeasuredfragment.However,theneutront reactions 112 Sn+ 112 Snand 112 Sn+ 124 Snhaveacleardependenceontheneutronexcessof theisobars.Whensimultaneouslythevelocitydistributions,(seeSection4.4)the centroid 0 oftheskewedgaussiandistributionisdescribedasalinearfunctionofthecharge oftheisobar: 0 ( A;Z )= 0 ( A;Z 0 ) 1+ 0 dZ ( Z Z 0 ) (5.1) 0 dZ isaparameteroftheandisplottedinFigure5.2.Forthereactionswiththe 112 Sn projectile,thectofchanging Z byonewhileholding A constantisanapproximately0 : 5% 111 Figure5.1Skewparametervs.MassLoss( A proj A ).Top,middle,andbottompanelsare the 112 Sn, 118 Sn,and 124 Snprojectiles,respectively.The 112 Sn, 118 Sn,and 124 Sntargets arerepresentedbytheblue,green,andredpoints,respectively.Ahorizontaldashedlineis drawnfornoskewing,orasymmetricgaussian.Anegativevaluerepresentsalongertail extendingtolowervelocity. 112 shiftinthevelocity.Forcomparison,theshiftfromthebeamvelocity( = : 367)is < 5%, andtheaveragevelocityonlyshiftsbyabout2%overthewholerangeof20 < A< 60. Thismeansthatforthemostneutron-poorprojectile,changingZbetweenisobarshasa tlylargerthandoeschangingAbetweenisotopes.Forthemoreneutron-rich projectiles, 0 dZ hasasmallermagnitudeandchangeswithtargetmass. Inthisexperiment,thereislimitedangularacceptance(seeAppendixC,andthereare othersuchaschargestatecontaminationwhichcouldpotentiallycreatethis sodatafrom[75]wasinvestigatedforthissameInthatexperimentperformedatthe NSCL,beamsof 40 ; 48 Caand 58 ; 64 NiwereimpingedonBeandTatargetsandthereaction productsweremeasuredandidenbytheA1900fragmentseparator.Fortheneutron- poorreactions, 0 dZ isagainmeasurablypositiveandhasasimilarmagnitudeasresults fromtheSn+Sncollisions.ThisresultisshowninFigure5.3.In[75],theneutron-rich projectilesshowtheshiftinthevelocitydistributions,butwith 0 dZ < 0.Fortheneutron richprojectiles, 0 dZ alsoshowsastrongdependenceonthetarget.Thesechangesinthe velocitydistributionfollowacleartrend,andsuggestsfurtherstudytoinvestigatethecause. Thepeakofthevelocitydistribution 0 alsoexhibitsaveryregulartrend,shownin Figure5.4.Forsmallmassloss, 0 decreaseswithincreasingmassloss.Asthemassloss becomeslarger,thetrendbecomesInthereactionswith 118 Snand 124 Snprojectiles thetrendsaresimplybecausemorenucleonsmustberemovedtoproducethesame isotopes.[81]showedasimilartrendfor 0 andsuggestedthat 0 canbeusedatracerof thedissipationinthereaction.Thatexperimentmeasuredreactionswitha 86 Krbeamand 112 ; 124 Snand 58 ; 64 Nitargetsat15 MeV u . Thistrendinthevelocityisveryusefulintheinterpretationofthereactiondynamics, becausetheisoscalingrelationsrequirethatthetemperaturebeequalinthesystemsbeing 113 Figure5.2Isobarvelocityslopeparametervs.MassLoss( A proj A ).Top,middle,and bottompanelsarethe 112 Sn, 118 Sn,and 124 Snprojectiles,respectively.The 112 Sn, 118 Sn, and 124 Sntargetsarerepresentedbytheblue,green,andredpoints,respectively.Ahorizon- taldashedlineisdrawnforzeroslope,whichmeansthatallisobarshaveidenticalvelocity distributions,whereapositivevaluemeansthathigherZisobarshavehighervelocity. 114 Figure5.3Isobarvelocityslopeparametervs.MassLoss( A proj A ),for 40 ; 48 Caand 58 ; 64 Ni projectileswith 9 Beand 181 Tatargets,at140 MeV u .Datatakenfrom[75] 115 Figure5.4Most-probablevelocityvs.MassLoss( A proj A ).Top,middle,andbottom panelsarethe 112 Sn, 118 Sn,and 124 Snprojectiles,respectively.The 112 Sn, 118 Sn,and 124 Sntargetsarerepresentedbytheblue,green,andredpoints,respectively.Thedataare correctedfortheenergylossinthetargetandthetimingscintillator.Thebeamvelocityfor allthreeprojectilesis = : 367. 116 Figure5.5Gaussianwidthparametervs.MassLoss( A proj A ).Top,middle,andbottom panelsarethe 112 Sn, 118 Sn,and 124 Snprojectiles,respectively.The 112 Sn, 118 Sn,and 124 Sn targetsarerepresentedbytheblue,green,andredpoints,respectively.Thedashedlines indicatetheingresultofthemoGoldhaberdescriptiondiscussedinthetext. compared.Thesystematictrendandthemagnitudeof 0 arequitesimilar,whichsuggest thatthesereactionshaveasimilardegreeofdissipation.Inaddition,becausethevelocity outwithincreasingmassloss,itcanbeassumedthatforthesefragments(i.e. A & 35for 112 Sn+ 112 Sn),theexcitationenergy(andtemperature)isnotchangingtly. Theparameterofinterestinthevelocitydistributionsisthewidthofthedistribu- tions.Goldhaberproposedamodelforthesystematicsofthewidth[80],andthiswasfurther mobyMorrissey[79].Itisstillanactiveareaofresearch,[57,82,75]becauseitis criticalintheproductionofradioactivebeamsviafastfragmentation.Goldhaberexpanded 117 ontheworkbyFeshbachandHuang[83]toexplainthetrendsinmanyfragmentationdata sets,andshowedthatthemomentumdistributionsfollowacommontrendregardlessof manyassumptionsmadeaboutthefragmentationprocess.TheGoldhaberformulationcan besimplyderivedbyassumingthatthesystemcomestoequilibriumwithatemperatureT. Becausetheprojectileseparatesintotwofragments,thesystemhasonedegreeoffreedom, andsotheaveragethermalkineticenergyis kT= 2.Assumingthattheprojectileisseparated intotwofragmentswithmasses A and A proj A whichhaveequalandoppositemomenta, theenergyinthecenterofmasscanbewrittenas kT 2 =

2 m n A +

2 m n ( A proj A ) (5.2) where m n isthenucleonmassand

isthemeansquaredmomentuminthecenterof massframe.Recognizingthat

= ˙ 2 0 ,Equation5.2canberearrangedtogive ˙ 2 = mkT A ( A proj A ) ( A proj ) : (5.3) Thesesimpleassumptionsdonotreproduceafactorof A proj = ( A proj 1)whichisseenin thedata,butthisisanegligiblefactorforlarge A proj .TheformofGoldhaber'ssystematics isusuallywrittenas: ˙ 2 = ˙ 2 0 A ( A proj A ) ( A proj 1) (5.4) forafragment A andaprojectile A proj ,where ˙ 0 isapproximately90MeV/candcanbe relatedtothenucleonfermienergy. 118 Becausethedatafromthisexperimentdoesnotincludefragmentsclosetotheprojectile, distinguishingbetweentmodelsisnotpossible.But,comparingthemeasuredsys- tematicstothesemodelscanindicatetherobustnessoftheprocedureforthevelocity distributions.Tothisend,thedataisonlycomparedtotheGoldhabersystematics. Theactualmeasuredwidth,showninFigure5.5,isaconvolutionofthefragmentation process,thestragglingintheenergylossinthetargetandthetimingscintillator,thebeam spotsizeinthedispersivedirection,andtheintrinsicmomentumwidthoftheincoming beam.Thebeamsusedinthisexperimentaredegradedfrom120 MeV u ,to70 MeV u using theA1900fragmentseparatortoselecttheappropriateportionofthebeam.Thisresults inanapproximately0.25%spreadintheincomingbeammomentum.Becausethisspread wasnotmeasureddirectlyforeachbeam,aconstantterm ˙ proj isaddedtotheGoldhaber formula,andisallowedtovaryintheprocedure.Sincevelocitydistributionsand notmomentumdistributionsareusedinthepresentresults,theGoldhaberformulamustbe describedintermsofvelocityandmassloss A : ˙ 2 = ˙ 2 0 A ) ( A proj A )( A proj 1) + ˙ 2 proj ( A proj A ) 2 (5.5) Theofthisfunctiontothedata(forthethreesymmetricsystems)areshowninFigure 5.5.TheparametersarelistedinTable5.1.Inordertothedata,theconstant additivefactormustbeontheorderof1%(ofthetotalmomentum),whichismuchlarger thanthe0.25%momentumwidthofthebeam.Thisdiscrepancyimpliesthatthereismore complicateddynamicsthanthesimpleGoldhaberdescription.Nonetheless,theconclusion ofthisprocedureisthatthereducedwidth ˙ 0 issimilartotheGoldhaberdescription, 119 ReactionSystem ˙ 0 (MeV/c) ˙ proj (MeV/c) 112 Sn+ 112 Sn 94.6 416 118 Sn+ 118 Sn 101 410 124 Sn+ 124 Sn 95.3 397 Table5.1Fittingparametersfromthesystematicsofthevelocitydistributions. whichgivessomeintheconsistencyofthedatafromonereactionsystemto another. 120 5.2 IsotopicDistributionsofResidueFragments Onegoalofthisexperimentistouseisoscalingobservablestoinferinformationaboutthe isospincontentoftheheavyprojectile-likefragmentdistributions.Thisrequiresmeasuring preciselythe shape oftheisotopicdistributions,whichareusedtomaketheisoscalingratios inthenextsection.Therelativeyields,foragivenelement,areshowninFigure5.6and 5.7,whereetreactionsystemsareshownforeachelement.Althoughonlyrelative yieldsaremeasuredinthisexperiment,theshapesoftheisotopicdistributionsshouldbe preserved. ComparingthedottedandsolidlinesinFigures5.6and5.7clearlyindicatesthatchanging thesystemasymmetrychangestheisotopicdistributions,albeitasmallForthehigher Zfragments,thepeakofthedistributionsareshiftedtotherightsimplybecausefragments ofalargermasscansupportalargerneutronexcess.Toisolatethatisoscalingratios areconstructedinSection5.3. 121 Figure5.6Isotopicyielddistributionsforereactionsystems,forZ=30toZ=35.Thedata arecorrectedforthechargestatecontamination,butarenotcorrectedforacceptance.Solid circlesrepresentsymmetricreactions,anddottedlines/opencirclesrepresentthe mixedreactions. 122 Figure5.7Isotopicyielddistributionsforereactionsystems,forZ=35toZ=41.Thedata arecorrectedforthechargestatecontamination,butarenotcorrectedforacceptance.Solid circlesrepresentsymmetricreactions,anddottedlines/opencirclesrepresentthe mixedreactions. 123 5.3 YieldRatiosandIsoscaling Toquantifytherencesintheisotopicdistributionsbetweenreactionsystems,isoscaling canbeusedtocondensetheinformationfrommanydatapointsintoasingle,physically meaningfulquantity.Inparticular,theparameterthatwillbeextractedforthisanalysisis theslopeoftheisotopicyieldratios,knownasthe\neutronisoscalingparameter". inmoredetailinSection2.3.1)Bymakingaratiooftheisotopicdistributions,microscopic suchasbindingenergiesandleveldensitiesarecancelled.Themeasuredisoscaling ratios,withatoextracttheslopeforeachindividualelementisshowninFigures5.8to 5.12.Thelastoftheseforthe 112 Sn+ 124 Snreaction,isplottedintwopanelsfor claritybecausetheslopeoftheyieldratiosaresmallsoneighboringelementswouldoverlap. Thequantitythatisimportanttoextractistheslopeofthelinesshownineach Toextractaprecisenumberitisimportanttoaddressthenon-linearitiesintheisoscaling ratio.Inallthesystems,whenalinetomorethanthreeisotopes,thereisanobvious curvaturesothattheslopeincreaseswithhigherN.Theseemstobemorepronounced forthelowermassfragmentsthanforthehighermassfragments.Otherstudiesofisoscaling haveseenasimilarThenon-linearitydoestheinterpretationofthevalueof theisoscalingparameter ,aswellasectingtheassociatederrorbars,buttheisospin transportratiomaynotbeasisdiscussedinthenextsection. Thetrendoftheisoscalingparameter withatomicnumberZisshowninFigure5.13. ThedataplottedinthisaretheparametersfromtheindividualinFigures5.8to 5.12.Thedataforthe 112 Sn+ 124 Snsystemisdividedintoevenandodd Z forclarity,because thevaluesof aresmall.Thesehavenotbeencorrectedforthecontaminationdue tochargestates.Toseethectofthechargestatecorrection,theisoscalingparameters 124 Figure5.8Neutronisoscalingratiofor 124 Sn+ 124 Snwithrespectto 112 Sn+ 112 SnforZ=27 toZ=43.Thelinesshownarelineartothedata.Odd-Zareshowninblue,andeven-Z areshowninredforclarity. 125 Figure5.9Neutronisoscalingratiofor 124 Sn+ 118 Snwithrespectto 112 Sn+ 112 SnforZ=27 toZ=43.Thelinesshownarelineartothedata.Odd-Zareshowninblue,andeven-Z areshowninredforclarity. 126 Figure5.10Neutronisoscalingratiofor 124 Sn+ 112 Snwithrespectto 112 Sn+ 112 SnforZ=27 toZ=43.Thelinesshownarelineartothedata.Odd-Zareshowninblue,andeven-Z areshowninredforclarity. 127 Figure5.11Neutronisoscalingratiofor 118 Sn+ 118 Snwithrespectto 112 Sn+ 112 SnforZ=27 toZ=43.Thelinesshownarelineartothedata.Odd-Zareshowninblue,andeven-Z areshowninredforclarity. 128 Figure5.12Neutronisoscalingratiofor 112 Sn+ 124 Snwithrespectto 112 Sn+ 112 SnforZ=27 toZ=43.Thelinesshownarelineartothedata.Odd-Zareshowninblueinthelower panel,andeven-Zareshowninredintheupperpanel. 129 Figure5.13Neutronisoscalingparameter for 112 ; 124 Sn+ 112 ; 124 Snand 118 Sn+ 118 Snfor Z=27toZ=43.Nocorrectionforchargestatecontaminationisincluded. obtainedwithcorrecteddataareshowninFigure5.14.Thechargestatecorrectioncausesan increaseof upto10%.Becausethechargestatecorrectionreliesonanempiricalcalculation ofthechargestatethecorrectionintroducessomesystematicerrorintotheabsolute valueof .Fortunately,thecorrectionhasverylittleontheisospintransportratio, whichisdiscussedinSection5.4. Themainsourceoferrorinthedeterminationof comesfromthenon-lineartrendin theisoscalingratios.Becauseofthenon-lineartrend,theslopeishighlycorrelatedwith thechoiceofrange.Sharpvariationsinthevalueof asafunctionofZaresimply causedbychangesintherangeofisotopesthataremeasured.Todemonstratethis 130 Figure5.14Neutronisoscalingparameter for 112 ; 124 Sn+ 112 ; 124 Snand 118 Sn+ 118 Snfor Z=27toZ=43.Thecalculatedcorrectionforchargestatecontaminationhasbeenapplied tothedata. 131 intermsof ,therangecanbemobyexcludingadatapointoneitherendofthe range.Figure5.15showshowthechoiceofrangethedeterminationofalpha. Removingonlyoneisotopechangestheextractedvalueofalphabyasmuchas5-10%.It isalsoclearfromthisthatthereisasimilarforbothreactionsystems.This suggeststhatthereissomesystematicerror(duetothenon-linearityoftheisoscalingratio) thatissimilarforeachsystem.Onewaythiscanbeinterpretedisthattheprimaryemitting sourcehasarangeofentasymmetriesthatarecorrelatedwiththeneutronexcess, whichisaprobableconsequenceofmeasuringasystemwhichhasnotreachedcomplete equilibrium.Anotherpossibilityisthatthectivetemperatureofthesesourcesisslightly t.Thisistheclaimmadein[85],whichsuggeststhatthesourcetemperatureshould belinearlydependentonthesourceasymmetry.Whetherornotthisisthecase,toa rangeofisotopeswouldchooseacertaintemperaturerange,andthischoiceshouldbe thesameforeachsystem.Section5.4willshowthatthischoicedoesnotntly theisospintransportratio. Althoughthereareseveralthatcancausesomechangetotheisoscalingparameter thatisextractedfromthedata,theisospinusionectcanbeclearlyseeninthemixed systems, 112 Sn+ 124 Snand 124 Sn+ 112 Sn,showninFigure5.16.Thegeneraltrendupward asafunctionofZcanbeinterpretedastheofadecreasingtemperatureformore peripheralcollisions,becauseitisobservedinthesymmetricsystemsaswell.Inthemore peripheralcollisions,lessnucleonsareabradedfromtheprojectile,resultinginalessexcited nucleus.Thestructure,especiallyforthe 124 Snprojectile,isaresultoftheofthe trangesfortZ.Itwillbeshownthatthisstructure,andtheoverall trendwithZismostlycanceledintheisospintransportratio. 132 Figure5.15Neutronisoscalingparameter, for 124 Sn+ 124 Snand 118 Sn+ 118 Snwithrespect to 112 Sn+ 112 Sn,forZ=27toZ=43,showingtheofthechoiceofrange.Solidsymbols showthetwosymmetricreactionswiththefullrange,whiletheopensymbolsshowthe resultwhenoneisotopeisexcludedateitherendoftherange.Excludingthelowestisotope increasesthevalueof ,andexcludingthehighestisotopereducesthevalueof .Onlythe twosystemsareshownforclarity. 133 Figure5.16Neutronisoscalingparameter, for 124 Sn+ 124 Sn, 118 Sn+ 118 Sn, 124 Sn+ 112 Sn, and 112 Sn+ 124 Snwithrespectto 112 Sn+ 112 Sn,forZ=27toZ=43. 134 Clearly,theisoscalingparameter issensitivetotheisospincontentofthesystem,butthe precisionof aloneisnotadequateforthisstudy.Althoughisoscalingisaphenomenonthat isreproducedbymanystatisticalmodels,thepresentstudyrequiresadynamicaltransport modeltounderstandtheBecauseofthecomputationalwithmicroscopic transportmodels,modelssuchasImQMDcannotbeusedtogenerateaccurateisotopic distributionsforheavynuclei,sotheisoscalingresultscannotbecomparedtotheory,directly. Thenextsectionwilldiscusshowtheparameter canbeusedtoformanisospintransport ratio,whichiscriticalforarealisticcomparisontotheory. 135 5.4 IsospinTransportRatio Theisospintransportratiocombinesinformation(fromisoscaling,inthiscase)fromtwo symmetricsystemsandonemixedsystemtodescribethedegreeofisospinequilibriumthat themixedsystemreachedduringthecollision.Inparticular,theendgoalistounderstand thedynamicsinthereactionbetween 112 Snand 124 Sn,andhowtheirisospin asymmetrythedynamics.Inthisexperiment,the 118 Sn+ 118 Snreactionismeasured aswell,whichisrelevantfortheinterpretationoftheisospindata.Usingtheisospin transportratioallowsforadirectconnectionbetweenphysicalobservablesandtheoretical quantities,andisasimple,intuitivequantity.Theisospintransportratioisforthe purposesofthisexperimentas: R I ( X (112+124or124+112) )= 2 X (112+124or124+112) ( X (124+124) + X (112+112) ) X (124+124) X (112+112) (5.6) wheretheobservableXisquantitythatiseithermeasuredexperimentallyorcalculated theoretically.Inthiscase,theobservableXistheisoscalingparameter .Becauseisoscaling isaquantitythatisdeterminedfromtworeactionsystems,thisexpressionmustbemo The 112 Sn+ 112 Snsystemisusedasthereference,sodividingnumeratoranddenominator by X (112+112) gives: R I ( X (112+124or124+112) )= 2 X (112+124or124+112) X (112+112) ( X (124+124) X (112+112) +1) X (124+124) X (112+112) 1 (5.7) 136 andrecallingthattheisoscalingparameterinvolvesthelogarithmoftheyieldratios,the fractionsontherighthandsidecanbereplacedwiththeisoscalingparameter(andlog(1)= 0),giving: R I ( X (112+124or124+112) )= 2 112+124or124+112 112+112 124+124 112+112 124+124 112+112 (5.8) whichgivestheexpectedvaluesof-1whenthemixedreactionbehaveslikethe 112 Sn+ 112 Sn reaction,and+1whenthemixedreactionbehaveslikethe 124 Sn+ 124 Sn.Practically,the isospintransportratioissimplyalineartransformationofanobservable.Thereareother advantagesthatwillbediscussedfurtherinthissection. Thereareseveraladvantagestousingtheisospintransportratio,thathelptominimize theofseveralexperimentalproblems.First,becausetheofchargestatecontam- inationhasbeencalculatedapproximately,itsontheisospintransportratiocanbe estimated.Asdiscussedintheprevioussection,thecontaminationamountstoacorrection to ofupto10%.Becausethecorrectionthetsystemsinalinearway,the correctionhaslittleontheisospintransportratio.Figure5.17showsthisfor the 118 Sn+ 118 Snreaction.Thecorrectiontotheisotopicyieldsisasmoothfunctionandis mostlycancelledinconstructingtheisospintransportratio.Becausethemainuncertainty inthechargestatecorrectionisthecalculationfromGLOBAL,whichisthesameforall systemsandelements,sothereisnoindicationthatanythingwouldchangethecancellation ofthecorrection.Thecorrectionislessthan0.02forall Z ,andbecauseofthelimited reliabilityofGLOBAL,thisresultissimplyincludedasasystematicuncertaintyof 0 : 02. Ingeneral,thebiggestsourceofuncertaintyinthedeterminationof isthenon-linearity oftheisoscalingratio.Therearevariousexplanationsforthisnon-linearitysuchassecondary 137 Figure5.17Correctionforchargestatecontaminationtotheisospintransportratio,cal- culatedusingtheisoscalingparameter forthe 118 Sn+ 118 Snreaction.Thecircles representthebetweenthecorrectedandtheuncorrectedresult. 138 decayorsintemperatureoftheemittingsource.Thenon-linearityoftheisoscaling ratiolog( R 21 )isactuallyminor,butbecauseitissystematicallynon-linear,therecanbe asystematiconthedeterminedvalueof .Althoughthesystematicerrorinthe determinationof islarge,(seeFigure5.15)theontheisospintransportratioismore subtle.Figure5.18showsthethattheisoscalingrangecanhaveontheisospin transportratio.Removingoneisotopefromthehasanontheextracted R i which issmallerthanthepropagatedstatisticalerrorbars.Onlytheresultfor 118 Sn+ 118 Snis shown,becausetheresultdoesnotdependstronglyontheatomicnumberofthefragments usedintheisoscalingTheontheisospintransportratioismuchsmallerthanthe ontheisoscalingparameteritself.Thethreeoptionsshownareforthefullrangeof isotopes,andforremovingoneisotopefromeachendoftherange.Thevariationbetween thethreechoicesislessthanthestatisticalerrorforeachelement,sothereisnoindication thatthechoiceofrangetheisospintransportratio,aslongastherangeiskept thesameforeachsystem.Thisisfortunate;itindicatesthatthisobservableisnotvery sensitivetotheselectionoffragmentsmeasuredinthetheexperiment.Morequantitatively, whentakinganaverageofthepoints,theofthefullrangegives R i =0 : 207 0 : 013,while thesub-rangesgive R i =0 : 214 0 : 015and R i =0 : 187 0 : 009.Thus,theuncertaintiesthat arisefromtherangeareneglible. 5.4.1 LinearityofIsoscalingObservables Themeasurementofthe 118 Sn+ 118 Snsystemgivesimportantinformationaboutthevalid- ityoftheisoscalingparameterasanisospinobservable.Inthesimplestassumption,the 118 Sn+ 118 Snsystemrepresentswhenthe 124 Sn+ 112 Snsystemhasreachedisospinequilib- rium;ifthe 112 Snand 124 Snnucleisimplyexchangedneutronsandnonucleonsarelost.This 139 Figure5.18Isospintransportratio,calculatedusingtheisoscalingparameter forthe 118 Sn+ 118 Snreaction.Threetversionsoftheresultsareshown,withthreet rangesusedfortheisoscalingratioThegreendatapointsshowstheresultforthe fullrange,whiletheredandbluelinesindicatetheresultforremovingthehighestand lowestisotopesintherespectively.Thedashedlineshowsthevalueexpectedfor R i from thestandardisoscalingrelationships. 140 measurementhelpstoverifytheimportantassumption: doestheisoscalingparameterhave alinearrelationshipwiththeisospinasymmetryoftheemittingsource ?Theisospinasym- metryoftheprimaryfragmentscanbecalculatedintheImQMDframework,butcannotbe measuredinexperiment.Theisoscalingparametercanbemeasuredintheexperiment,but cannotbeeasilycalculatedintheImQMDframework.Theisospintransportratioallows thesetwovariablestobecompareddirectly,butonlyiftheyarelinearlyrelatedtoeach other. OneconsiderationisthemassasymmetryofthetSnisotopes.Becauseexchanging neutronsbetweentheprojectileandthetargetalsochangesthemass,the 118 Sn+ 118 Sn systemisactuallynothalfwaybetween 112 Sn+ 112 Snand 124 Sn+ 124 Sn,intermsof = N Z A . Infact,whencalculating R i usingonlythe oftheSnisotopes,the 118 Sn+ 118 Snsystem wouldyieldavalueof R i = : 051[45].Thevaluefoundinthisstudyhasbeenshowntobe appreciablyhigher, R i =0 : 207 0 : 013.Actually,theisoscalingparameterhasbeenassumed tohavethefollowingform,asderivedinseveralstatistical-typemodelframeworks[38] 21 = 4 C sym T Z 1 A 1 2 Z 2 A 2 2 ! = 4 C sym T NZ (5.9) where T isthetemperatureoftheemittingsource,and C sym isthestrengthofthesymmetry energy.Noattemptismadeheretodeducethetemperatureandextract C sym ,butthis equationcangivesomeinsightintothedependenceof ontheisospincontentoftheemitting source.Thetemperatureconfusesthesituationuntiltheisospintransportratioisused,which cancelsthe T and C sym ,assumingthatthetemperaturesreachedinthetcollisions aresimilar.However, stilldependsontheofthesquareof Z=A betweenthetwo systems.Assumingthistobeexactlytrue,the R i thatisobtainedusingEquation5.9should 141 be R i =0 : 0762.ThisexpectedquantityisshownasadashedlineinFigure5.18.Although thediscrepancyisoutsideoftheerrorrangefortheexperimentalresult,therelationbetween and NZ isremarkablysimilartoexpectationsgiventhemyriadofassumptionsthatmust bemadetoderiveEquation5.9.Thisdiscrepancymustbeaddressedinordertocompare theisospinresultstoImQMDcalculations. OnepossibilityisthattheassumptionsmadewhenderivingEquation5.9donothold. Theassumptionwasthatthesymmetryenergyprovidedthemaincontributiontothe neutron(proton)separationenergybetweenthetwosystems.Assumingastatisti- calprocesscreatedthesefragments,theisoscalingparameter ( )isequaltotheerence intheneutron(proton)chemicalpotentialbetweenthetworeactions,dividedbythetem- perature.Whiletfragmentsdetectedinthisstudycouldcomefromttemper- ature,(removingmorenucleonsfromtheprojectileresultsinhigherexcitationenergy)the regularlineartrendoftheisoscalingratioslog( R 21 )impliesthatthetemperature betweenthetworeactionsissmall.Equation5.9wasderivedbytakingthederivativeof thebindingenergywithrespecttoneutron(proton)numberassumingthesymmetryenergy isthemaincontribution.Whencomparingthesymmetry,surface,andcoulombterms,the symmetrytermmakesup98%ofthisneutronseparationenergyMoreimpor- tantly,theseparationenergyisstilllinearwith NZ whenincludingthesurface andcoulombterms.Inthecaseoftheprotonisoscalingparameter ,thecontributionsfrom coulombandsurfaceareslightlylarger,butthetrendisstillmostlylinearaswell. Thispartoftheanalysisdoesnotseemtobethecauseofthisnon-linearity. Anotherpossibilityisthatthepreequilibriumemissionstronglytheasymmetry oftheprojectile-likefragmentintheearlypartofthereaction.Theisoscalingparameter theasymmetryoftheprimaryfragmentatthepointthatthereactionbecomes 142 astatisticalthermodynamicprocess.Nucleonsthatareemittedearlyduringthecollision wouldnotthisobservable.Becauseneutronsarenotmeasured,thisectcanonlybe predictedbycalculations. Athirdpossibilityisthatthe measuredfor 124 Sn+ 124 Snwithrespectto 112 Sn+ 112 Sn ismoedbysecondarydecayoffragmentsmorestronglythanthealphafor 118 Sn+ 118 Sn withrespectto 112 Sn+ 112 Sn.Thiswouldhappen,forinstance,ifsecondarydecay theyieldsfor 124 Sn+ 124 Snmorethantheothertworeactions.Becausesecondarydecay bringstheisotopicdistributionsclosertogether,itdecreasesthemagnitudeof and . ThiscanbeexaminedintheImQMDframeworkaswell.ImQMDdoesnotinclude theofsecondarydecay,theresultsonlygiveinsightintotheprimaryfragmentyields. Theisospinasymmetry,averagedoverallfragmentswith Z> 20,isplottedinFigure5.19. ThecalculationshowniswithImQMD-Skywithfourparametersets,andtheimpactparam- eteris b =10fm.Thisresultshowsthattheaverageasymmetryoftheprimaryfragmentsis predicted(inImQMD-Sky)tobelinearlycorrelatedwiththeprojectileasymmetryforthe symmetricreactions,regardlessoftheinteractionused.Therequirementof Z> 20does notmuchtheresultinthiscase,becauseforsuchaperipheralreaction,veryfewlarge fragmentsarecreatedwith Z< 30. WhiletheofsecondarydecayarenotincludedinImQMD,andarenotcalculated here,thecanbeunderstoodinaqualitativeway.Whenmeasuringisoscaling,theyield ofaspisotopeiscomparedtothesameisotopeinanotherreaction.Theeventsthat resultinthiscertainisotopebeingmeasuredhaveverythistoriesinonereactionver- susanother.Therearetwomainthatwouldcausethereactiontoprogressrently, butarriveatthesamemeasuredfragment.First,themassoftheSnprojectilesaret; toformaprimaryfragmentofmassnumber85requiresremoving27nucleonsfrom 112 Sn 143 Figure5.19Primaryresidueasymmetry,calculatedusingImQMD-Skywiththefourpa- rametersetsdetailedin2.4,plottedagainsttheasymmetryoftheprojectile,forthethree symmetricreactions 112 Sn+ 112 Sn, 118 Sn+ 118 Sn, 124 Sn+ 124 Sn,forimpactparameter b =10 fm.Thecalculatedasymmetryistheasymmetryforfragmentswith Z> 20,averagedover allevents. 144 butrequiresremoving39nucleonsfrom 124 Sn.Therefore,thehighermassprojectilewould createalargertemperatureandexcitationenergy.Then,thenucleuswithahighertemper- aturewouldrequiremoreparticleemissiontodecaytoitsstate.Thiswouldthe comparisonbetweenthe 118 Sn+ 118 Snand 112 Sn+ 112 Snreactionstoasimilardegreethat itwouldthecomparisonbetweenthe 124 Sn+ 124 Snand 118 Sn+ 118 Snreactions,sothe temperaturealonewouldnotleadtothisnon-linearity. Onceitisestablishedthatthedtsystemsrequiretamountsofsecondary decay,itfollowsthattomeasurefragmentsofthesame(N,Z)intheditreactions,they musthaveoriginatedfromtprimaryfragments.Ingeneral,evaporationofnucleons hasbeenshowntocausethesystemtoapproachan\EvaporationAttractorLine"(EAL), wherethedecaywidthsofprotonsandneutronsbecomeequal[86].Asthedecayprocess bringsthesystemclosertotheEAL,thedrivingforcedecreases,andthesystemapproaches theEALasymptotically.Ingeneral,howclosethesystemgetstotheEALisdirectlyrelated totheexcitationenergy.Becausethehigher-massprojectilewouldbeginwithhotter,more massiveprimaryfragmentstogettothesamenalisotope,theaverageasymmetrywould bepushedmoretowardstheEAL.Thiswillbeinvestigatedinthefuturebyusingadecay modeltostudythede-excitationoftheresidues. Theresultofthisexaminationoftheresultsfromthe 118 Sn+ 118 Snsystemshowsthat theinformationfromthethirdsymmetricsystemisveryhelpfultounderstandtheisoscaling parameterobtainedinthemixedsystems.The R i obtainedfromthe 118 Sn+ 118 Snreaction inbothexperimentandinthecalculationsisnonzero,sosomecorrectionshouldbemade beforecomparingtheisospintransportratios.Althoughtheprecisefunctionalformofthe relationshipbetween and isunknown,assumingtheissmoothandcontinuous,a 145 quadraticcorrectioncanbeused.Amoisospintransportratio R i canbeas: R i ( X )= R i ( X )+ R i ( X eq ) R i ( X ) 2 1 (5.10) where X eq istheobservablemeasuredinthe 118 Sn+ 118 Snreaction.Thisexpressionstill yields forthe 124 Sn+ 124 Snand 112 Sn+ 112 Snreactions,butusesthefreedominthe quadratictermtoput 118 Sn+ 118 Snat R i ( X eq )=0.Thepurposeofdevelopingtheisospin transportratiowastoallowforamoredirectcomparisonbetweenexperimentandtheory. Oneresultofthisexperiment,showninthissection,wastoinvestigatehowtheisoscaling parameterdependsontheasymmetryoftheemittingsource.Thisassumptionturnsoutto beapproximatelytrue,althoughthereisanon-linearontheorderof10%,whichis remarkableconsideringthemanyassumptionsmade.Toattempttomakeamoreprecise measurement,themoisospintransportratiowillbedescribedinthenextsection. 5.4.2 IsospinResults Onepurposeofthismeasurementwastounderstandhowthedynamicsduringcollisionsof heavyionsatintermediateenergiesarebychangingtheisospinasymmetryofthe reactionsystem.Inparticular, howdoestheisotopicdistributionofheavyfragmentschange? InSection5.2,theisotopicdistributionsareshown,andthereisaclearshiftinthepeaks oftheisotopicdistributionbetweenthesymmetricsystemsandthemixedsystems.This isshownmoreclearlyusingtheisospintransportratioinFigure5.20.Themagnitudeof thebetweenthesymmetricsystemandthemixedsystem,describeshowmuch hasoccurred.Asdiscussedintheprevioussection,theisoscalingparameterhas somenon-linearbehaviorwithrespecttotheisospinasymmetry.Forthesamereasonthat 146 Figure5.20Isospintransportratio,calculatedusingtheisoscalingparameter using5 measuredreactions. the 118 Sn+ 118 Snsystemgives R i > 0,thetwomixedsystemsinFigure5.20giveresults thatareasymmetricaboutzero. Thestrikingfeatureofthisresultisthattheabsolutevalueof R i ( )increasesforin- creasingZ,meaningthatthedecreasesforincreasingZ.Thisisanexpectedfea- ture,becauselargerZfragmentsresultfrommoreperipheralcollisionswheretheprojectile andtargetwillbeincontactforashortertime,resultinginlessTheImQMD simulationspredictthattheamountofshouldbeatleastsimilarinthetwomixed reactions,sobeforecomparingtothecalculations,thenonlinearityof R i ( )mustbere- moved.Usingtheaveragevalueof R i forthe 118 Sn+ 118 Snsystem,themoisospin 147 Figure5.21Moisospintransportratio,calculatedusingtheisoscalingparameter using5measuredreactions.Theblackopensquaresaretheaverageofthetwomixed systems. transportratioestablishestheexpectedsymmetry,showninFigure5.21.Thevaluesare transformedfromthestandardisospintransportratiousingtheaveragevalueof R i ( )for 118 Sn+ 118 Sn,sothegreensquaresarecenteredat R i ( )=0bydesign.Theaverageofthe twomixedsystemsisshownaswell.Theaverageiscloseto R i ( )=0,butismeasurably lower, R i ( )= 0 : 036 0 : 008.Thisiseitheranindicationthatthecorrectionbeingmade doesnothavethesimplequadraticformthatisassumedfor R i ,orthatthetwomixedsys- temsexhibitatamountofisospinEitherofthoseexplanationsispossible, sothisisameasureofthesystematicerrorintheresults. 148 Figure5.22MoIsospinTransportRatiovs.AtomicNumberZ,forthetwomixed reactions, 124 Sn+ 112 Snand 112 Sn+ 124 Sn. Theisospindresultusingthemoisospintransportratioisshownin Figure5.22.The 124 Sn+ 112 Snsystemshowsaslightlylargerextentofisospinthan the 112 Sn+ 124 Snsystem.Thisdependencecouldresultfromthesimplecorrectionapplied forthenonlinearityof .Thisasymmetrycouldalsoresultfromthetwosystemshaving slightlytimpactparameters. 149 150 R i R i R i R i R i R i Z 112 Sn+ 124 Sn 118 Sn+ 118 Sn 124 Sn+ 112 Sn 112 Sn+ 124 Sn 118 Sn+ 118 Sn 124 Sn+ 112 Sn 27 -0.57 0.059 0.2 0.097 0.78 0.12 -0.71 0.06 -0.0033 0.098 0.7 0.13 28 -0.62 0.05 0.15 0.066 0.73 0.072 -0.75 0.051 -0.057 0.066 0.64 0.074 29 -0.61 0.07 0.25 0.1 0.76 0.1 -0.74 0.072 0.051 0.1 0.67 0.1 30 -0.65 0.053 0.21 0.088 0.77 0.082 -0.77 0.054 0.012 0.088 0.68 0.084 31 -0.66 0.039 0.19 0.056 0.78 0.059 -0.78 0.039 -0.015 0.056 0.7 0.061 32 -0.62 0.04 0.17 0.06 0.79 0.051 -0.75 0.04 -0.027 0.06 0.71 0.053 33 -0.6 0.07 0.26 0.11 0.81 0.12 -0.73 0.071 0.067 0.11 0.74 0.13 34 -0.68 0.048 0.26 0.058 0.81 0.071 -0.79 0.049 0.061 0.059 0.74 0.073 35 -0.66 0.042 0.18 0.067 0.79 0.077 -0.78 0.043 -0.02 0.067 0.72 0.079 36 -0.68 0.035 0.22 0.053 0.87 0.063 -0.79 0.035 0.025 0.053 0.82 0.065 37 -0.69 0.02 0.2 0.029 0.81 0.021 -0.8 0.021 -0.00031 0.029 0.74 0.021 38 -0.78 0.037 0.2 0.036 0.83 0.057 -0.86 0.038 0.005 0.036 0.76 0.059 39 -0.75 0.03 0.24 0.041 0.81 0.062 -0.84 0.03 0.048 0.041 0.74 0.063 40 -0.83 0.032 0.22 0.039 0.83 0.017 -0.89 0.033 0.017 0.039 0.77 0.018 41 -0.83 0.034 0.18 0.042 0.89 0.065 -0.9 0.035 -0.019 0.042 0.85 0.067 42 -0.9 0.059 0.21 0.059 0.91 0.053 -0.94 0.061 0.015 0.059 0.87 0.055 43 -0.9 0.045 0.27 0.08 0.91 0.067 -0.94 0.046 0.078 0.08 0.87 0.069 Table5.2IsospinTransportRatioandMoIsospinTransportRatioresults,tabulatedasafunctionofatomicnumber Z . ThesedataareplottedinFigures5.20to5.22. WhenconstructingacomparableobservablefromtheImQMDcalculations,thebiggest uncertaintyistheambiguityinreproducingtheimpactparameterdependence.Whensimu- latingthesecollisionswithImQMD,computingresourcesarealimitingfactor.Asaresult, thesecalculationswereonlydoneforimpactparametersof b =6fm, b =9fm,and b =10 fm.Amorethoroughstudywouldbetogeneratecollisionswithaweighteddistributionof impactparameter.Thiswouldallowforamoredetailedcomparisonofhowtheobservables dependontheimpactparameterselection,orequivalently,whichrangeofZacalculation shouldbecomparedto.Morecalculationsarebeingdonetoestablishthisdependence,but arenotavailableforthisstudy. Sincetheexperimental R i isformedbycomparingresultswithaconstantZ,thecalcu- lationmustbeconstructedthesameway.Fortunately,aimpactparameterresultsin similarrangeofZineachsystem,becausetheprojectileshaveequalcharge.Figure5.23 showsanexampleoftheresultsfortheSLy4skyrmeparameterset,fortwoimpactparame- ters.Fora b ,themixedsystemshaveataverageZthanthesymmetricsystem ofthesameprojectilebyabout1unitofcharge.Thismaybeexplainedbythe intheradiiofthetisotopesofSn;abiggertargetabradesmorenucleons.Another importanttrendisthattheslopeoftheasymmetryinthemixedsystemswith Z ist fromthesymmetricsystems.Thisisalsoexpected,becausetheamountofshould increasewithdecreasingimpactparameter.Thecombinationofthesetwotrendsresultsin asystematicshrinkingoftheisospintransportratiosothat <> 6 = Z forbothmixed systemsandforanyZ.ThedatacomparethesameZintworeactions,whereasthecalcu- lationscomparesameimpactparameterintworeactions.Thisshouldbeaddressed whencomparingtheImQMDresultstothedata.Althoughthedetailedobservables,such 151 astheZdistributionmaynotbereproducedbyImQMD,therewillbeasimilartinthe realdata. Figure5.24showshowt R i resultfromusingeithertheaverageasymmetry, <> , foraimpactparameterorusingtheasymmetryforaZ, Z .Theis largest( ˇ 15%moretheSLy4parameterset,whichpredictsthelargestisospin signal.ToaccuratelyaccountforthismoreImQMDcalculationsareneeded atimpactparametersfrom7fmto12fm.Theresultusingonlythecalculationsat6fmand 10fmshowsthebehaviorqualitatively,butisnotrobustenoughtoformaconstraintonthe symmetryenergy. Figure5.25showstheresulting R i ( Z )whentheimpactparameterdependenceisin- cluded,forthefourskyrmeparametersetsusedinthisstudy.Onlythecalculationfor the 124 Sn+ 112 Snreactionisshownforclarity.Theofthedensitydependenceofthe symmetryenergyisclearlydemonstrated.SkI2hasa"densitydependenceofthesym- metryenergy,whichmeansthatthesymmetryenergyissmalleratsubsaturationdensity.A smallersymmetryenergyresultsinlessbecausethedrivingforceinthelowdensity neckregionwouldbeweaker.Askyrmesetwitha\soft"symmetryenergyproducesmore TheSLy4interactionhasasoftsymmetryenergybutasimilarmomentumde- pendenceastheSkI2interaction,andtheisaccordinglylargerforSLy4.The ofthetmomentumdependentinteractionsusedisshown,dbythet emasssplitting.Becausetheskyrmeparametersetshavemanyparameterswhich aresimultaneouslytodata,itistoisolatetheofoneaspectofthephysics. Thetemasssplittingisobscuredbythefactthattheisoscalaremass issimultaneouslychanging.Tounderstandtherelationshipsbetweenthevariousinteraction properties,thecovarianceanalysisisdescribedin[87]. 152 Figure5.23Primaryresidueasymmetry,calculatedusingImQMD-SkywiththeSLy4pa- rameterset(softsymmetryenergy),plottedagainsttheaverageZoftheresultingprojectile- likefragment,forthereactions 112 Sn+ 112 Sn, 112 Sn+ 124 Sn, 118 Sn+ 118 Sn, 124 Sn+ 112 Sn, 124 Sn+ 124 Sn,forimpactparameters b =10fmand b =6fm. 153 Figure5.24IsospinTransportRatio,calculatedusingImQMD-SkywiththeSLy4pa- rameterset(softsymmetryenergy),plottedagainsttheimpactparameter.Thereactions 112 Sn+ 112 Sn, 112 Sn+ 124 Sn, 124 Sn+ 112 Sn, 124 Sn+ 124 Snareshown.Thesolidlinesarecal- culatedusingtheaveragevaluesof fora b ,andthedashedlinesarecalculatedusing theaveragevaluesof fora Z . 154 Figure5.25IsospinTransportRatio,calculatedusingImQMD-SkywithfourSkyrmepa- rametersets,plottedagainsttheimpactparameter,forthe 124 Sn+ 112 Snreaction,forimpact parameters b =10fmand b =6fm. 155 TheImQMD-Skymodelprovidessomeinsightintothephysics,buttoisolatethe ofthedensitydependenceofthesymmetryenergy,ImQMD05isused.ForImQMD05,the interactionportionofthenuclearsymmetryenergyisdescribedbyasimplepowerlaw, S i ( ˆ )= S i; 0 ˆ ˆ 0 .Thesecalculationsweredonefor b =6fm, b =9fm,and b =10fm,but onlythe 112 Sn+ 124 Snmixedsystemwascalculatedfor b =6fm.Usingthislimitednumber ofimpactparameters,alineardependenceon Z for R i canbegeneratedfromthemodel calculations.ThisisshowninFigure5.26.Thereisadependenceof R i onthe ofthesymmetryenergyterm,sotheexperimental R i shouldallowforaconstraintonthe exponent . Thisresultcanbecomparedtothecalculations,althoughadetailedcomparison willbedonewhenmorecalculationsareavailable.Theavailablecalculations(shownin Figure5.26)areplottedwiththedataforthe 112 Sn+ 124 SnsysteminFigure5.27.The impactparametercalibrationforthe 112 Sn+ 112 Snsystemisused,andtheerrorregion correspondstoanerrorintheaverageZof 2 : 5units. Thepreliminarycomparisonsuggestsadensitydependenceof ˇ 1 : 0.Thepreviously measuredconstraintsfromisospinwere =0 : 75 0 : 25.Thepreliminaryresultsmay beconsistentwiththesepreviousstudies.Untilmoretheoreticalcalculationscanbedoneat theappropriateimpactparametersandtheresultsunderstood,noconclusionscanbemade. Also,workiscurrentlyunderwaytostudythecovariancerelationshipbetweentheextracted ,thestrengthofthesymmetryenergyatsaturationdensity S 0 ,andothertransportmodel inputparameters.Asimilar ˜ 2 analysisasin[42]willbedonewithadditionalImQMD05 calculations,inordertoobtainaenceintervalfor .Nonetheless,theisospin resultusingheavyresidueprojectilefragmentationisaviableobservableforconstraining thedensitydependenceofthesymmetryenergy. 156 Figure5.26IsospinTransportRatio,calculatedusingImQMD05for =0 : 5 ; 1 : 0 ; 2 : 0,plotted againsttheimpactparameter,forthe 112 Sn+ 124 Snreaction,forimpactparameters b =6fm, b =9fm,and b =10fm.Onlyonemixedsystemisshownhereandtheverticalaxisisfocused onthecalculatedsystem. 157 Figure5.27MoIsospinTransportRatiovs.AtomicNumberZforthe 112 Sn+ 124 Sn system.Thesolidregionshowstheexperimentalresult,andtheblue,green,andredcrosses correspondto =0 : 5,1.0,and2.0,respectively.Thecalculationsaredoneat b =6fm, b =9fm,and b =10fm. 158 Chapter6 Conclusion Inthisdissertation,thedynamicsofnuclearreactionsatintermediateenergieswereinvesti- gatedbymeasuringheavyresiduesfromcollisionsof 112 ; 118 ; 124 Snbeamswith 112 ; 118 ; 124 Sn targets.Isotopicyieldratiosofheavyfragmentswith Z> 25wereusedtodeterminethe extentofisospinthatoccurredinthesereactionsystems.Bymeasuringalarger matrixofreactionsthaninpreviousisospinstudies,anon-linearrelationshipbe- tweentheisoscalingparameter andtheisospinasymmetryoftheexcitedfragmentswas idenTheseresultswillbeusedtoimprovethecurrentexperimentalconstraintsonthe densitydependenceofthenuclearsymmetryenergy. TherelativeyieldsofheavyresiduesweremeasuredusingtheS800spectrometer.Isotopic andelementalidenwasdeterminedusingtandenergyloss,whichwere correctedevent-by-eventusingthemeasuredfragmenttrajectories.Anapproximatecor- rectionwasderivedforthecontaminationduetomultiplechargestatesoffragmentsbeing detectedintheS800.Seventreactionsweremeasuredusingthreemagneticrigidity settings,andthevelocitydistributionofeachisotopewasreconstructed.Therelativeyields wereobtainedbyintegratingthesevelocitydistributions. TheMSUMiniballwasusedtomeasuretheimpactparameterevent-by-eventusingthe chargedparticlemultiplicityandbymeasuringthetotalreactioncrosssection.Thereduced impactparameterextractedfromthechargedparticlemultiplicitywasshowntohavean approximatelylinearrelationshipwiththeatomicnumberofthefragmentmeasuredinthe 159 S800.Anapproximaterelationshipbetweentheimpactparameterandtheatomicnumber Zoftheheavyresiduewasdetermined. TherelativeisotopicdistributionsmeasuredbytheS800wereusedtoformisotopic yieldratiostocomparetheisospinasymmetryobtainedineachsystem.Thedatawere showntofollowestablishedisoscalingrelationships,andtheneutronisoscalingparameterwas extractedasafunctionofatomicnumberforallsystems.Theneutronisoscalingparameter wasshowntobepositivelycorrelatedwithatomicnumber,whichcanbeattributedtoan increasingtemperaturewithincreasingmassloss.Theneutronisoscalingparameterwasalso foundtovarywiththerangeofisotopesusedforthesothatmoreneutronrich fragmentsresultedinalarger . Theisoscalingparameterwasusedtoformanisospintransportratiotocomparethe 118 Sn+ 118 Snsystemtothe 112 Sn+ 112 Snand 124 Sn+ 124 Snsystems.Theisospintransport ratioforthissystemwasshowntobeinsensitivetothesystematicthatcausedvari- ationsintheisoscalingparameter,andwasapproximatelyconstantovertherangeofZ measuredinthisexperiment.Thissystemyieldedanisospintransportratiowhichislarger thanzero,indicatingthattheisoscalingparameterisnotexactlylinearlyrelatedtothe isospinasymmetryoftheinitialcompoundsystem. Theisospintransportratiowasalsousedtoquantifytheamountofisospinusion thatoccurredinthemixed 124 Sn+ 112 Snand 112 Sn+ 124 Snsystems.Theisospintransport ratioforthe 124 Sn+ 112 Snsystemhadalargerabsolutevaluethanforthe 112 Sn+ 124 Sn. Thisasymmetryisconsistentwiththepositiveisospintransportratiomeasuredforthe 118 Sn+ 118 Snsystem. Toaccountforthenon-linearityoftheisoscalingparameter,amoisospintransport ratiowasproposedwhichincorporatedthemeasurementofthe 118 Sn+ 118 Snsystem.This 160 correctionlargelyremovedtheasymmetryofthetwomixedreactions.Itwasfoundthatthe amountofincreasedastheatomicnumberZofthemeasuredfragmentdecreased, whichwasinterpretedasadependenceontheimpactparameterinthecollision. TheImprovedQuantumMolecularDynamics(ImQMD)transportsimulationwasused tointerprettheisospinresults.Thecollisionsweresimulatedatseveralimpact parameters,andwithseveraltformsofthedensitydependenceofthesymmetry energy.Theresultwasshowntobesensitivetotheimpactparameter,whichqualitatively agreeswiththeexperimentalresults. BoththeImQMD-SkyandImQMD05resultsshowacorrelationbetweentheisospin transportratiowiththedensitydependenceofthesymmetryenergy.Thedataandthe calculationshowastrongdependenceontheimpactparameter.Becauseofcomputational limitations,calculationswerecompletedforalimitedsetofimpactparameters.Tobetter accountfortheimpactparameterdependencemorecalculationsareneeded,soaprecise constraintontheimpactparameterisnotproducedatthistime.Calculationswithother transportmodelssuchastheBoltzmann-Uehling-Uhlenbeck(BUU)modelareunderwayas well. Theexperimentalsomeasuredtheyieldsoflightparticlesandintermediatemassfrag- mentsintheLASSAarray.Isoscalinginformationwillbeextractedfromtheintermediate massfragmentyields,andthisinformationwillbecomparedtotheresultspresentedhere. Theintermediatemassfragmentsshouldgiveasecondindependentmeasurementofisospin whichwillimprovethereliabilityoftheseresults. 161 APPENDICES 162 AppendixA ParticleIdenSpectra ThissectionshowsthecorrectedparticleidenationspectraobtainedusingtheS800 Spectrometer.Eachshowsthethreemagneticrigiditysettingsmeasuredforasingle reaction.Seventreactionsystemsareshown. 163 164 FigureA.1ParticleIdenforthe 112 Sn+ 112 Snreaction.Thethreepanelsshowthethreetmomentumsettings thatweremeasured.SeeSection4.2fordetails. 165 FigureA.2ParticleIdenforthe 112 Sn+ 124 Snreaction.Thethreepanelsshowthethreetmomentumsettings thatweremeasured.SeeSection4.2fordetails. 166 FigureA.3ParticleIdenforthe 118 Sn+ 112 Snreaction.Thethreepanelsshowthethreetmomentumsettings thatweremeasured.SeeSection4.2fordetails. 167 FigureA.4ParticleIdenforthe 118 Sn+ 118 Snreaction.Thethreepanelsshowthethreetmomentumsettings thatweremeasured.SeeSection4.2fordetails. 168 FigureA.5ParticleIdenforthe 124 Sn+ 112 Snreaction.Thethreepanelsshowthethreetmomentumsettings thatweremeasured.SeeSection4.2fordetails. 169 FigureA.6ParticleIdenforthe 124 Sn+ 118 Snreaction.Thethreepanelsshowthethreetmomentumsettings thatweremeasured.SeeSection4.2fordetails. 170 FigureA.7ParticleIdenforthe 124 Sn+ 124 Snreaction.Thethreepanelsshowthethreetmomentumsettings thatweremeasured.SeeSection4.2fordetails. AppendixB HodoscopeAnalysis TheS800spectrometermeasurest,energyloss,andmagneticrigidityinorder toextractthecharge,mass,andmomentumofheavyresiduefragments.Whenfragments entertheS800with Z 6 = q theywillbemisidenACsIhodoscopetomeasurethetotal kineticenergy(TKE)ofthefragmentswascommissionedjustpriortothisexperimentin ordertosolvethisproblem.Thehodoscopedidnotexistatthetimeoftheproposal,so theexperimentwasnotoptimizedtotakeadvantageofit.Theenergyoftheprojectilewas increasedfromtheproposedenergysothatfragmentscouldbedetectedinthehodoscope. Ultimately,thehodoscopedidnotprovideusefulinformationforthisexperiment,butthe attemptedanalysisisdescribedhere. TwooverlappingisotopesinFigure4.23,ata Bˆ , m q ,and Z ,wouldentertheS800 withttotalkineticenergy(TKE),andthehodoscopewouldbeabletodiscriminate betweenthesetwoisotopes.Inpractice,thereareseveralcomplications.Theproblem isthatthefragmentsmusttravelthroughseveraldetectors(thetwoCRDC's,theionization chamber,andthetimingscintillator)aswellasaTcoveringovertheCsIarray.The hodoscopewascommissionedusingabeamof 76 Ge(Z=32)at130 MeV = u ,or9870MeVTKE. [59]Inthatcase,aftertakingintoaccountallenergylosses,thecalculatedenergyatthe hodoscopewas8680MeV.FragmentationproductswerealsomeasuredusingbothAuand Betargets,andthehodoscopewascharacterizedovertheenergyrangeof3600MeVto7600 MeVTKEforelementsfrom Z =17to Z =33.[70]InthisexperimentwithSnbeamsof 70 MeV = u ,themeasuredenergiesoffragmentsfrom Z =20to Z =45aremuchlower.Table B.1showsthematerialsandestimatedthicknessesofthedetectorsintheS800focalplane. Toinvestigatetheofenergylossesforthisexperiment,calculationsusingLISE++ [88,89]areshowninTableB.2.Thecalculationsaredoneforanisotopeinthemiddleof themeasuredisotopicdistribution,andtheincomingenergyis60 MeV = u forallfragmentsfor simplicity.60 MeV = u ischosenbecauseitcorrespondsthe Bˆ =2 : 51setting,inthecenter oftheexperimentalacceptance.Becausetherearemanyuncertainvariablessuchasthegas pressure(whichvariedthroughouttheexperiment),theE1timingscintillatorthickness,or theexactthicknessofthecoveringofthehodoscope,thisisonlyanestimateofthe energylosses. BecauseenergylossincreaseswithZ,fragmentswith Z> 35donotreachthehodoscope. For Z =25,alreadyhalfofthefragmentenergyisestimatedtobelostbeforereachingthe hodoscope.Toseetherealoftheenergylosses,FigureB.1showstheoverall fordetectingfragmentsasafunctionof Z .Thisiscalculatedastheratioofevents withasignalabovethresholdinthehodoscopecomparedtothetotalnumbereventswitha fragmentideninthefocalplaneforeachgivenZ.FigureB.1agreesqualitativelywith thecalculationsfromTableB.2,butthecalculationsclearlyunderestimatetheenergyloss. 171 Layer Material EstimatedThickness CRDC1entrancewindow PPTA 12micron CRDC1gas 80%CF 4 -20%C 4 H 10 40torr CRDC1exitwindow PPTA 12micron CRDC2entrancewindow PPTA 12micron CRDC2gas 80%CF 4 -20%C 4 H 10 40torr CRDC2exitwindow PPTA 12micron ICentrancewindow PPTA+Kevlar 14.0 mg = cm 2 ICgas P10 300Torr ICplates PPTA 2 mg = cm 2 ICexitwindow PPTA+Kevlar 14.0 mg = cm 2 E1scintillator polyvinyltoluene 1.0mm Thodoscopecover T 300 m hodoscope CsI(Na) 5cm TableB.1ApproximatethicknessesoftheS800focalplanedetectors FragmentZ(N-Z) EAfterIC EAfterE1 EintoHodo 20(4) 54.1 MeV = u 42.5 MeV = u 35.3 MeV = u 25(5) 52.6 MeV = u 37.7 MeV = u 27.8 MeV = u 30(6) 51.2 MeV = u 32.9 MeV = u 19.7 MeV = u 35(7) 49.9 MeV = u 28.2 MeV = u 9.94 MeV = u 40(8) 48.7 MeV = u 23.4 MeV = u 0 MeV = u TableB.2ApproximateenergylossasafunctionofAtomicNumberZ,Startingfrom 60 MeV = u (near Bˆ =2 : 51) For Bˆ =2 : 51,whichshouldbedirectlycomparabletoTableB.2,thedropsby 50%by Z =28.Furthermore,therangewherechargestatesbecomeanappreciable isfor Z> 30. Tofurtherunderstandtheofenergylosswithincreasing Z ,FigureB.2showsthe hodoscopeenergyforonecrystalasafunctionof Z ,forthe 112 Sn+ 112 Snreactionsystem,at Bˆ =2 : 51Tm.Thehodoscopeenergyquicklydropstozeroas Z increasesbeyond Z =25, whichisconsistentwiththeenergylossdescribedabove.Anotherthatmay playaroleisthelightresponseoftheCsI(Na)fortheseveryhighlyionizingparticles.This wasdescribedin[70],butthatanalysisreliedontheabilitytoaccuratelycalculateall energylossesbeforethehodoscope.Withoutbettercalibrationsofthematerialthicknesses, thiswouldbefutileforthepresentexperiment. Anotherproblem,alsoobservedinthecommissioningofthehodoscope,istheposition dependenceoftheCsIsignals.Thearrayismadeupof32separatecrystalsarrangedina 4x8array.Itwasfoundthattheenergydeposited(bythemonoenergeticprimarybeam) hadanomalousbehaviorneartheedgesofthecrystals.Withinabout5mmofeachedgeof thecrystalthelightoutputisenhanced.Inthatexperiment,datafromtheedgesofthe crystalsweresimplyremoved.Inthisexperiment,theproblemisunavoidable.Asdiscussed 172 FigureB.1OveralldetectionoftheCsIhodoscopeforthe 112 Sn+ 112 Snreaction system,at Bˆ =2 : 4 ; 2 : 51 ; and2 : 6Tm.Thencyissimplythefractionofparticles whichareotherwisedetectedandideninthefocalplanethatalsoleaveasignalinany hodoscopecrystal. inAppendixC,duetoaprobleminthetuningoftheS800,theacceptanceoftheS800was verylimited,especiallyinthenon-dispersivedirection.Asaresult,insteadofilluminating theentiremiddletwocolumnsofcrystals,onlytheseambetweenthesetwocolumnsis illuminated.Duringthefewdaysoftheexperiment,inJune,thefragmentsarewell spreadoverthehodoscope,asexpected.ThisisshowninFigureB.3.Basedontheenergy- positioncorrelationnotedintheJunesegmentoftheexperiment,toremovetheedgeregion wouldlikelymeanremoving 5mm,whichwouldremovemostofthedata.Toanalyzethese datawouldrequireacarefulcalibrationofthepositiondependence,whichisnotdonehere. Tomakeacorrectionforthisthebestcasescenariowouldbetosendamonoenergetic beamintothefocalplanetoscanthecrystals.Inthiscase,becausetheexperimentuses beamsofSn,( Z =50),thebeamdoesnotreachthehodoscope,duetoenergylosses.When calibratingthehodoscopecrystals,thegaswasremovedfromtheionizationchamberandthe twoCRDC's,sothecalibrationdatahasnotrackinginformation.Thispositiondependence isevidentinthedatafromJune,althoughonlyqualitatively,soitisnotshownhere. 173 FigureB.2Energymeasuredincrystal#9,shownvs. Z ,forthe 112 Sn+ 112 Snreaction system,at Bˆ =2 : 51Tm.Thiserequiresthatcrystal#9isthecrystalwiththelargest signalamplitudeforthatevent. Anotherproblemthatmaytheresolutionofthehodoscopeisthepossiblepresence ofmultiplehitsinthehodoscopeinasingleevent.Sincehigher- Z fragmentsthatare triggeredinthetimingscintillatorarestoppedthereorintheTcoverofthehodoscope, therearemanyeventsthatdonotincludeacorrespondingTKEinthehodoscope.Inthese events,thereareoftensignalswithappreciableamplitude,whichmustcomefromeitherpile- upfromotherevents,multiplehitsinasingleevent,orsomeothersourceofbackground. Thesignatureoftheseisthesame:therearesignalsinhodoscopecrystalswhere thereshouldbenone.ThiscanbeseensimplyfromFigureB.1where,athighZ,allthree Bˆ settingsapproachanon-zerovalue,around4%.ByrequiringthattheTKEisextracted fromthehodoscopecrystalthatisindicatedbythepositiontracking,someofthesespurious signalscouldberemoved,buttheremaystillbeaproblemofmultiplehitsincrystalswith realevents. FigureB.4showsthecorrelationbetweenasignalinagivencrystalandthepositionof thatsignal.Thereareseveralfeaturesinthisworthnoting.First,asidefrommostof thecountsinsidethedimensionsofthecrystal,(shownbytwodottedlines)thereisalsoa 174 FigureB.3Non-dispersivepositionspectraatthehodoscope,shownforthetwot segmentsoftheexperiment.Thedottedlineshowstheapproximatepositionofthegap betweenthemiddletwocolumnsofhodoscopeelements.Thespectraarearbitrarilynor- malizedtobeonthesamescale.Becausetheacceptancewaslimitedtotheregionnearthe gap,mostofthedataiscorruptedbythepositiondependenceofthecrystalresponse. distributionofeventsthatareoutsidethosedimensions,althoughstillcorrelatedspatially. Thismaybeasignatureofangularstragglingforsomelow- Z particles.Thereisalsosome backgroundthatisnotspatiallycorrelated,whichcoversthesameenergyrangeasthesignals presumedtobefromtheheavyfragments.Thesecouldbepile-up,multiple-hitevents,or possiblylightparticlesemittedfromthestoppedheavyfragments.Tobettercharacterize thebackgroundduetothetheseFigureB.5showsthehodoscopespectrumforcrystal #9,requiringahigh- Z idenfromtheionizationchamber.Sinceahigh- Z fragment stopsbeforereachingthehodoscope,signalsinFigureB.5mustbespuriousevents. FigureB.5showsthattheTKEspectrumwouldcontaincontaminatedsignals,evenif theparticletrackingwereenforced.InthelowenergyportionofFigureB.4,thereisan excessofcountsastheenergygoestozero.Lookingcloseratthisregionshowsevidence ofthegainshiftingneartheedgesofthecrystal,whichrulesoutelectronicsnoiseorADC pedestalvalues.Sincethecountsdoappearspatiallycorrelatedwiththecrystal,itcanbe 175 FigureB.4HodoscopeEnergyfromcrystal#9vs.Dispersivepositionspectraattheho- doscope,forthe 112 Sn+ 112 Snreactionsystem,at Bˆ =2 : 4.Thedottedlinesshowthe approximatepositionsoftheedgeofcrystal#9. assumedthattheycomefromrealparticles.Iftheseparticlesarearrivingincoincidence withtheheavyfragmentinthefocalplane,theremaybeatfractionofevents whereaheavyfragmentdoesleavearealsignalinthecrystal,butiscoincidence-summed withotherdetectedparticles,sothehodoscopemisidentheTKEofthefragmentof interest. Afterthisaggregationofproblems,thebestcasescenarioforextractingusefulinformation isthehighestvelocity,lowest Z fragmentsthataremeasuredwithappreciablestatistics.This leavesonlythe Bˆ =2 : 6Tmsetting,anda Z of22.Focusingonthisisotope,the hodoscopeenergycanbecorrectedempiricallyforthemomentum(similartothe TOF and E ),whichallowsfora1-dprojectionontothehodoscopeenergy.ThisisshowninFigure B.6.Assumingthattheenergyscaleislinear,theresolutionforthiselementmaybesut toestimatethecontributionofchargestates,butnottophysicallyresolvethem. Becauseofthemanyproblemsanduncertaintiesassociatedwiththehodoscopedatafrom thisexperiment,nousefulinformationisextractedfromthehodoscope.Theconclusionis thatthehodoscopeisnotausefultoolformeasuringthechargestatesdistributionsfor 176 FigureB.5HodoscopeEnergyfromcrystal#9vs.Dispersivepositionspectraattheho- doscope,forthe 112 Sn+ 112 Snreactionsystem,at Bˆ =2 : 4.Thedottedlinesshowthe approximatepositionsoftheedgeofcrystal#9.Onlyfragmentsidentas Z> 30using thenormalparticleidenspectrumareincludedthiswhichshouldnotreach thehodoscope.Asaresult,thesesignalscomefromeithercoincidentlightfragmentsor backgroundfrommultiplehits. fragmentsof Z> 25withenergiesatorbelow60 MeV = u .Forthistypeofexperiment, somepossibleimprovementswouldbetodecreasethethicknessoftheE1scintillatorand tominimizeandmeasuretheexactthicknessofthematerialcoveringthefrontfaceofthe hodoscopecrystals.Todoprecisionworkwiththehodoscope,thenon-linearitynearthe edgesofthehodoscopewouldhavetobecarefullycharacterized.Inanysimilarfuture experiments,thequestionofmultiplehitsinthehodoscopewouldhavetobeaddressed. Forthisexperiment,thecontributionsfromchargestatesarecharacterizedusingempirical modelsasoriginallyplanned,describedintheSection4.3.1. 177 FigureB.6Hodoscopeenergy,measuredinthe 112 Sn+ 112 Snreactionsystem,at Bˆ =2 : 6 Tm,empiricallycorrectedformomentum,for 46 Ti 22+ .Thesolidlineshowstheapproximate centroidofthispeak,andthedottedlineshowstheestimatedpositionoftheprimary contaminant, 44 Ti 21+ ,whenaccountingforenergylossesandassumingtheenergyscaleis linear. 178 AppendixC AcceptanceCorrection Aswithmostdetectorsystems,theS800Spectrometeronlyacceptsalimitedrangeof kinematicalvariables.InthecaseoftheS800,thesituationiscomplicated,asthedetectors arelocatedinthefocalplane,afterpassingthroughseveralmagnets.Asaresult,particle trajectoriesmustbetracedbackthroughthemagnetstoreconstructthereactionatthe targetposition.ThisprocessisdescribedinmoredetailinSection4.1.4.Trackingthrough themagnetsalsointroducesacorrelationbetweentheangleofaparticleandthemagnetic rigidityofaparticle.Tocorrectforthistheacceptancemustbesimulatedusinga MonteCarloalgorithm.Thisexperimentwasdesignedtomeasureratiosofparticleyields, sptoavoidthistypeofproblemwithacceptance.Ifthespectrometeracceptance wasunchangedfromonereactionsystemtoanother,theacceptancecorrectionwouldcancel intheratio.Duringtheexperiment,therewereseveralproblemsthatledtohavingdit acceptancecorrectionsinthetreactionsystems,andtheseareaccountedforinthe calculationoftheacceptance. Thelargestproblemtheacceptancewastheresultofanerrorinthedevice tuningfortheS800itself.Toaccommodatethelargevolumeoftheminiballdetectors aroundthetargetposition,thetargetwasplacedabout50cmupstreamofthenormalfocal pointoftheS800.Thisrequiresasimplescalingoftheinthequadrupolemagnets afterthetarget,tooptimizetheacceptanceforthenewtargetposition.Whilethiswasdone correctlyduringtheportionoftheexperimentinJune2011,whentheexperimentwas resumedinOctober2011thisstepwasaccidentallyomittedfromthebeam-tuningchecklist. TheproblemisshownschematicallyinFigureC.1.WhileitwasobservedthattheS800was workingnormallyinthebeginningoftheexperimentinJune,thisproblemwasnotnoticedin Octoberandpersistedthroughallthreebeamsusedintheexperiment.Fortunately,because ofthepossibilityofchangesindetectorgainsorthresholds,the124Snbeammeasurements wererepeatedwhentheexperimentresumedinOctober,socomparabledatawastakenfor allbeams.Aftermappingtheanglestothetargetposition,thediareobvious,as showninFigureC.2.Theendresultisthattheangularacceptanceisabout25%ofthe nominalacceptance,andthisreducedotherchangesintheacceptance. FigureC.2showsthattheacceptanceatverysmallangles(near afp =0, bfp =0)is reducedbyacutoutinthescintillatorplacedafterthetarget.Thisscintillatorwasrequired tomeasurethestarttimeofthetfromthetargetpositiontothefocalplane. Becausetheexpectedbeamratewaslargerthan10MHz,andthestablebeamsusedwere highlyfocused,theplasticscintillatormaterialwouldberapidlydegradedifthefullbeam ratewasimpingedonit.Asquareholewascutintothescintillator,whichwouldallowthe unreactedbeamtopassthroughit.Thelight-guideforthescintillatorwasthendesigned withtwophotomultipliertubestocollectlightfromparticleshittingeithersideofthehole. 179 FigureC.1SchematicdiagramshowingmagneticelementsoftheS800.OnlytheOpticsin theNon-DispersiveDirectionareshown.TheS800wastunedassumingthereactiontarget wasatthenormaltargetposition(thepivotpoint),whichresultedinover-Focusingthe fragmentsfromtheactualtargetposition.Thedashedgreencurverepresentsthetrackof afragmentemittedatthemaximumscatteringanglefromtheoptimaltargetposition.The solidredcurverepresentsthetrackofafragmentemittedatthemaximumscatteringangle fromtheactualtargetposition,upstreamofthepivotpoint. FigureC.3showsanimageofthescintillatoraftertheexperiment,withoneofthetwo photomultipliertubesstillattached.Thisscintillatorholemotheacceptancebetween reactionsystemsbecauseofafaultymechanicaltargetdrivethatwasusedtoadjustthe positionofthestarttimingscintillator.Itwasdiscoveredpart-waythroughtheexperiment thatthedrivedidnotholditscalibrationreliably.Theuncertaintyinthescintillatorposition wasminimizedbyrecalibratingthedrivebeforeeachmovement.Thelargestmiscalibration occurredduringthe 112 Snbeam,whichisshown(fortheworstcase,whencomparingtothe 124 Snbeam)inFigureC.4.Becausethesechangesinacceptancecanmakeanimpactonthe resultingfragmentyields,acarefulcalculationoftheacceptancewasdone,andisdescribed infurtherdetailinthissection. 180 FigureC.2Scatteringanglesatthetargetposition.Theleftpanelshowsthe 124 Sn+ 124 Sn reactionatthe2 : 6TmmagneticrigiditysettingasmeasuredinJune2011withtheproper beamtuning.Therightpanelshowsthe 124 Sn+ 124 Snreactionatthe2 : 6Tmmagneticrigid- itysettingasmeasuredinOctober2011withtheincorrectbeamtuning.Neitherhistogram requiresaparticleidengate,butbothrequireatimingsignalinthetimingstart scintillator,whichcausesthesquarecutoutinthecenter.(SeeFigureC.3) FigureC.3Scintillatorusedtomeasurethestarttimeoftheheavyfragmentsthatare detectedintheS800focalplane.Noticethesquareholecutout,whichallowstheunreacted beamtopassthrough. 181 FigureC.4Scatteringanglesatthetargetposition.Theleftpaneshowsthe 112 Sn+ 112 Sn reactionatthe2 : 4Tmmagneticrigiditysetting.Therightpaneshowsthe 124 Sn+ 124 Sn reactionatthe2 : 4Tmmagneticrigiditysetting.Thetverticalintheleft panelisduetomiscalibrationofthemechanicaltargetdriveusedtomovethescintillator. 182 C.1 MonteCarloAcceptanceCalculation WhiletheS800hasacomplexgeometricacceptance,theabilitytotrackparticlesfrom thefocalplanetothetargetpositionallowsforanaccuratecalculationoftheacceptance correction.Acomparableprocedurehasbeenusedmultipletimestomakecorrectionsat largescatteringangleswhenusingthenormalacceptanceoftheS800.[90,91]Inthisexper- iment,thedistributionsinenergyandinscatteringanglearenecessarytomakecomparisons betweentbeam-targetcombinations.Thegoalofthecorrectionistoderiveaweight- ingfactorforagiven( ;dta )thatcanbeappliedonaneventbyeventbasis.Thebasicsteps todothisareasfollows:generatealargesetofsimulateddataevents,applytherequire- mentsoftheexperimentalconditions,andsimplycalculatewhatfractionoftheeventsata given( ;dta )wouldbedetected.Theprocedureassumesthat,insidethegateschosen,the ofdetectingaparticleis100%,whichmeansthatedgesmustbeexcludedinthe simulationaswellasthedata. Thestepistogeneratearandomdistributionof( ;˚;dta;yta ).For( ;dta ),a uniformdistributionisused,sincethecorrectioniscalculatedindependentlyforeach( ;dta ). For ˚ ,auniformdistributionisusedbecausethereactionshouldbecylindricallysymmetric. SincetheS800measuresscatteringangleintermsofrectilinearvariables ata and bta , and ˚ aretransformedaccordingly. yta ,whichisthenon-dispersivepositionatthereaction target,istakenfromthedataitself,anditsimplyrepresentsthespreadofthebeamspot onthetarget.Inthisexperiment,thebeamwaswellfocusedatthetargetposition,and isrepresentedinthesimulationbyagaussiandistribution.Thewidth(FWHM)ofthe yta distributionistypically3mmbutisextractedforeachbeam-target-rigiditysetting separately,toaccountforpossibleincalibrationsbetweensettings. Oncethepseudodataisgenerated,severaltwo-dimensionalgatescanbeapplied.The S800measuresfourparameters( ata , bta , yta , dta ),andthesefourvariablescanhavecorrela- tionsbetweenthem.Inpreviousstudies,2dimensionalboundariesinthe ata dta plane andthe yta bta planewereapplied.Inthisexperiment,becauseofthesquarecutoutin thetimingscintillator,thereisalsoacorrelationinthe ata bta plane.Also,asaresultof theoverfocusingoftheS800quadrupolemagnets,thereisanappreciablecorrelationinthe bta dta planeaswell. Thegate,whichistheeasiesttounderstand,isinthe ata dta plane.SincetheS800 dipolesfragmentsinthedispersivedirection,therangeof ata thatcanbedetected becomessmallerasthemagneticrigiditymovesawayfromthecentralvalue.Thiscorrelation isshowninFigureC.5.Theacceptanceinthe ata dta planechangesfromsystemtosystem onlybecausethethreerentbeamsimpingeonthetargetwithslightlytangles. Theincomingbeamangleisdeterminedbydegradingeachbeamintothefocalplaneusing varyingwidthsofaluminum. Thenextgatethatcanbeappliedtothepseudodataisinthe yta bta plane.Inthiscase, withtheS800infocusedmode,thereisonlyasmallspreadin yta .Nonetheless,when bta approachestheedgeoftheacceptance,thevalueof yta becomesimportant.Extractingthis behavioriscomplicated,becauseofthecorrelationsintheboundariesinthe ata bta plane aswellasthe bta dta plane.Sinceitisimpossibletovisuallydeterminetheseboundaries in3and4dimensions,thismultidimensionalcorrelationmustbeextractediteratively.To 183 FigureC.5Dispersiveangleatthetargetpositionversusfragmentenergy( dta )measuredin the 124 Sn+ 124 Snreactionatthe2 : 51Tmmagneticrigiditysetting.NoPIDgatesorother requirementsareapplied,thesharparesimplyduetotheacceptanceoftheS800 Spectrometer.Thedottedlineshowsanexampleofthe ata dta gateusedtocalculatethe acceptance. understandtheboundaryinthe yta bta plane,thedependenceof bta max on ata and dta shouldbesubtracted,andthismustbedoneseparatelyfor bta min (notshown).This correctionisshowninFigureC.6.Anothercorrelationthathastobeaccountedforinthe acceptancecalculationisbetween bta and dta .Again,thisisshownfor bta max inFigure C.7.And,thecorrelationbetween ata and bta isshowninFigureC.8. Aseparaterestrictionontheacceptancecomesfromthecutoutinthetimingscintillator. Thiscanbedeterminedbymakingthesamespectraasdescribedabove,butrequiring asignalfromthetimingscintillator.ThesecorrectionscanbeseeninFigureC.9,andthe sameproceduredescribedabovetodetermine bta max isusedtodetermine bta hole . Thepseudodataisbyalldescribedgates,andthefractionofthegeneratedpar- ticleswhichareacceptedateach dta and iscalculated.Anexampleoftheresultfromthis calculationisshowninFigureC.10. 184 FigureC.6Non-dispersivepositionatthetargetversusNon-dispersiveangle,measuredin the 124 Sn+ 124 Snreactionatthe2 : 51Tmmagneticrigiditysetting,withagaterequiring : 03