SPECTROSCOPYOFNEUTRONUNBOUNDSTATESIN 24 OAND 23 N. By MichaelDavidJones ADISSERTATION Submittedto MichiganStateUniversity inpartialentoftherequirements forthedegreeof Physics{DoctorofPhilosophy. 2015 ABSTRACT SPECTROSCOPYOFNEUTRONUNBOUNDSTATESIN 24 OAND 23 N. By MichaelDavidJones Unboundstatesin 24 Oand 23 Nwerepopulatedfroman 24 Obeamat83.4MeV/uvia inelasticexcitationandprotonknockoutonaliquiddeuteriumtarget.UsingtheMoNA- LISA-Sweepersetup,thedecayofeachnucleuscouldbefullyreconstructed.Thetwo-body decayenergyof 23 Nexhibitstwoprominentpeaksat E =100 20keVand E =960 30keV withrespecttotheneutronseparationenergy.However,duetothelackof -raydetection, adeestatementonthestructureof 23 Ncouldnotbemade.Shellmodelcalculations withtheWBPandWBTinteractionsleadtoseveralinterpretationsofthespectrum.Both asinglestateat2.9MeVin 23 N,ortwostatesat2.9MeVand2.75MeVareconsistentwith theshellmodelanddata. Inaddition,atwo-neutronunboundexcitedstateof 24 O,populatedby( d;d 0 ),wasob- servedwithathree-bodyexcitationof E =715 110(stat) 45(sys)keV,placingitat E =7 : 65 0 : 2MeVwithrespecttothegroundstate.Three-bodycorrelationsforthede- cayof 24 O ! 22 O+2nshowclearevidenceforasequentialdecaythroughanintermediate statein 23 O.Neitheradi-neutronnorphase-spacemodelwereabletodescribetheobserved correlations.Thismeasurementconstitutestheobservationofatwo-neutronsequential decaythroughthree-bodyenergyandangularcorrelations,andprovidesvaluableinsight intofew-bodyphysicsattheneutrondripline. ACKNOWLEDGMENTS DuringmytimeatMSU,Ireceivedatremendousamountofsupportfrommyadvisor,the MoNAgroup,theNSCLandmyfellowgraduatestudents.Withoutthem,Icouldnot havecompletedmydegree.Firstandforemost,Iwouldliketothankmyadvisor,Michael Theonnessen,forhiswisdom,guidance,andmentorship.Hewasalwaystheretoanswermy questionsanddiscussideas.Ithasbeenagreatopportunityandapleasuretoworkwith him.Ialsothanktheothermembersofmycommittee,RemcoZegers,AlexBrown,Norman Birge,andPhillipDuxburyfortheirtime,advice,andguidanceduringourmeetings. WorkingwiththeMoNACollaborationhasbeenatremendouslyjoyfulexperience.The entirecollaborationhasbeenincrediblysupportiveandreceptivetomyquestionssincethe dayIjoined.Ialwayslookedforwardtotheyearlymeetingandwillcertainlymissit.Many thankstoNathanFrank,PaulDeYoung,andThomasBaumannforhelpingwitheverything fromsimplecalibrationquestionstotheDAQinthemiddleofthenight,todiscussing andassistingintheanalysisofourdata. Iwouldalsoliketothanktheothergraduatestudentsandmembersofthelocalgroup: GregChristian,SheaMosby,JennaSmith,JesseSnyder,KrystinStiefel,ZachKohley,An- thonyKuchera,andArtemisSpyrouforalltheirhelpandsupport.They,andmanyothers, providedinsightfulandvaluablediscussionsthatmadethesepastfewyearsproductiveand veryenjoyable. TheLiquidHydrogenTargetwascriticaltothesuccessofthisexperimentandIwould liketothankRemcoZegers,AndyThulin,andJorgePereira,aswellasPaulZeller,fortheir supportandassistanceininstallingandoperatingthetarget.Withouttheirsupport,the supportoftheMoNACollaboration,andtheoftheCoupledCyclotronFacilityaswell iii astheA1900group,theexperimentinthisworkwouldnothavebeenasuccess. ManythankstoMikeBennet,VincentBader,DanielWinklehner,DanAlt,andEric Deleeuw{mytesandfellowmembersoftheTechnicalInstituteofProcrastination inthePhysicalSciences(TIPPS).Fridayeveningwon'tbethesamewithouttheblaring dubstepofMuseandnerf-gunts.IwouldliketothankthetrioofChrises:ChrisMorse, ChrisSulivan,andChrisProkop,forputtingupwithmystupidquestions.Iwouldalso liketothankSamLipschutzforbeinganoutstandingRockportsalesman,andKenneth Whitmoreforneverbeingwrong. Iwouldlikethankmyfamilyfortheircontinuouslovingsupport,andinparticular,my grandfather,whosparkedmyinterestinphysicsatanearlyage. Finally,Iamgratefulfortheopportunitytoworkwiththegreatfaculty,post-docs, andgraduatestudentsoftheNSCL,withoutwhomthisworkwouldnothavebeenpossible. iv TABLEOFCONTENTS LISTOFTABLES .................................... viii LISTOFFIGURES ................................... x Chapter1Introduction ................................ 1 1.1ChartoftheNuclides............................... 1 1.2ShellModel.................................... 3 1.2.1IslandsofInversion............................ 7 1.2.2TensorForce................................ 8 1.3Three-bodyCorrelationsandDecays...................... 10 1.3.1Two-ProtonDecay............................ 11 1.3.2Two-NeutronDecay........................... 14 1.3.3Transitionfrom3-bodyto2-body.................... 16 1.3.4PreviousExperiments.......................... 20 Chapter2TheoreticalBackground ......................... 22 2.1OneNeutronDecay................................ 22 2.1.1R-Matrixderiviation........................... 22 2.2TwoNeutronDecay............................... 27 2.2.1Phasespacedecay............................ 28 2.2.2Sequentialdecay............................. 30 2.2.3Di-neutrondecay............................. 34 Chapter3ExperimentalTechnique ........................ 38 3.1ExperimentalSetup................................ 38 3.2BeamProduction(K500,K1200)........................ 40 3.3A1900&TargetScintillator........................... 41 3.4LiquidDeuterium(LD 2 )Target......................... 41 3.4.1TargetCell................................ 44 3.4.2TemperatureControlSystem...................... 45 3.4.3GasHandlingSystem........................... 46 3.4.4VacuumChamber............................. 46 3.5SweeperMagnet.................................. 48 3.6ChargedParticleDetectors............................ 49 3.6.1CRDCs.................................. 49 3.6.2IonChamber............................... 51 3.6.3TimingScintillators............................ 52 3.6.4Hodoscope................................. 54 3.7MoNALISA.................................... 55 v 3.8ElectronicsandDAQ............................... 57 3.9InvariantMassSpectroscopy........................... 60 3.10JacobiCoordinates................................ 61 Chapter4DataAnalysis ............................... 65 4.1CalibrationsandCorrections........................... 65 4.1.1ChargedParticleDetectors........................ 65 4.1.1.1CRDCs............................. 65 4.1.1.2IonChamber.......................... 72 4.1.1.3ThinScintillator........................ 78 4.1.1.4TargetandA1900TimingScintillator............ 84 4.1.1.5Hodoscope........................... 85 4.1.2LD 2 Target................................ 88 4.1.3MoNA-LISA................................ 96 4.1.3.1ChargeCalibration(QDC)................... 96 4.1.3.2PositionCalibration(TDC).................. 98 4.1.3.3Timecalibration(tmean+global).............. 99 4.2EventSelection.................................. 101 4.2.1BeamSelection.............................. 102 4.2.2EventQualityGates........................... 103 4.2.3ElementandIsotopeIden................... 105 4.2.4NeutronSelection............................. 113 4.2.4.1Two-NeutronSelection..................... 115 4.3InverseTracking.................................. 116 4.3.1V................................ 119 4.3.1.1 22 O+1n............................ 119 4.3.1.2 21 N+1n............................ 121 4.3.1.3 23 O+1n............................ 122 4.3.1.4 18 C+1n............................ 122 4.4ModelingandSimulation............................. 124 4.4.1IncomingBeamParameters....................... 125 4.4.2ReactionParameters........................... 126 4.4.2.11nKnockout,V 22 O................. 128 4.4.2.21pKnockout,V 22 N................. 130 4.4.2.3(d,d'),InelasticExcitation................... 131 4.4.3NeutronInteractionandMENATE R.................. 133 4.4.4OtherParameters............................. 135 4.4.5Cuts.................................... 135 4.4.6DecayModels............................... 135 4.4.6.1Oneneutrondecays...................... 135 4.4.6.2Twoneutrondecays...................... 136 4.4.7FittingandLikelihoodRatio....................... 138 vi Chapter5ResultsandDiscussion ......................... 140 5.1 22 O+2n..................................... 140 5.2 22 N+1n..................................... 154 5.2.1Interpretation............................... 155 5.2.1.1GroundStateDecay...................... 158 5.2.1.2SingleState........................... 159 5.2.1.3TwoState............................ 161 5.2.1.4BackgroundDiscussion-Searchfor 24 N........... 162 5.2.2Conclusions................................ 164 Chapter6SummaryandConclusions ....................... 165 BIBLIOGRAPHY .................................... 167 vii LISTOFTABLES Table3.1:Boilingandtriplepointsofhydrogen,deuterium,andneon...... 43 Table3.2:Listofcomponentsinthegashandlingsystem.Tableadoptedfrom Ref.[ 89 ].................................. 47 Table4.1:BadpadsintheCRDCswhichareremovedfromanalysis....... 66 Table4.2:SlopesandsetsfortheCRDCcalibration.............. 70 Table4.3:DriftcorrectionparametersfortheIonChamber............ 78 Table4.4:Cotsforpositioncorrection(left)anddriftcorrection(right) oftheThinscintillator.......................... 81 Table4.5:TimefortheThinscintillator.................. 83 Table4.6:TimingfortheTargetandA1900Scintillators......... 85 Table4.7:GlobaltmeaninnanosecondsforeachtableinMoNA-LISA.. 101 Table4.8:htcorrectioncotsforisotopeseparation...... 110 Table4.9:Simulationparametersfortheincomingbeamdistribution.Deter- minedbymatchingunreacted 24 Ointhefocalplane......... 127 Table4.10:ParallelandperpendicularglauberkicksusedtoreproducetheCRDC distributions................................ 127 Table5.1:Errorbudgetforthethree-bodydecayenergy.Thedominantcontri- butionisthedriftlength,d L ....................... 147 Table5.2:Spectroscopicoverlapsforvariousstatesin 23 Nwiththegroundstate of 24 O. E decay iscalculatedassumingthestatedecaysdirectlytothe groundstateof 22 N............................ 159 Table5.3:Ratioofintensitiesinthesinglestateinterpretationcomparedto theequivalentratioformedfromspectroscopicoverlapsforpossible initialstatesin 23 Nwithstatesin 22 NusingtheWBPandWBT interactions................................ 160 viii Table5.4:Spectroscopicoverlapsforpossibleinitialstatesin 23 Nwithnal statesin 22 NusingtheWBPandWBTinteractions.Forcomparison aretheintensitiesforthebest-ofthedata............. 161 ix LISTOFFIGURES Figure1.1:ChartoftheNuclides.Onthecoloraxisistheneutronseparation energy S n inMeV.DatatakenfromAME2012[ 8 ].......... 3 Figure1.2:(Top)inbindingenergybetweentheliquiddropmodeland experimentalobservation.Ontheleftistheenceasafunction ofneutronnumber N .Ontherightisasafunctionoftheproton number Z .(Bottom)ElectronIonizationenergyforallelementswith Z< 104ontheperiodictable.Notethesimilarityinclosedelectron shellsandclosedneutron/protonshells.Imagesources:[ 10 ],[ 11 ].. 5 Figure1.3:Singleparticleorbitsinthenuclearshellmodel.Energiesareshown foraharmonicoscillatorpotential,woods-saxon,andwoods-saxon withspinorbit.ImageSource:[ 10 ].................. 6 Figure1.4:(a)Diagramforwavefunctionofrelativemotionforcollisionofa pair.Notethatthiscaseisdeuteron-likeandattractive. (b)Inthenonspin-casethewavefunctionofrelativemotionis stretchedperpendiculartothespin(denotedbyblackarrows),and isrepulsive................................ 9 Figure1.5:Schematicforillustratingshellevolutionduetothetensorforce.The widthofthearrowsdenotesthestrengthoftheinteraction.(Left) Ainnucleinearstabilitywith N =20beingmagic. (Right)Asprotonsareremovedtheattractionwiththe 0 d 3 = 2 orbital weakenscausingittoriseinenergyrelativetothe 1 s 1 = 2 orbital creatinganewshellgapat N =16.................. 10 Figure1.6:(Top)TheoreticalpredictionfortheJacobi T (left)and Y (right) systemrelativeenergyandangularcorrelationsinthebreakupof 6 Bebasedonafullthree-bodycalculation[ 44 ]Thedataareshownon thebottompanels.ImageSource:[ 43 ]................. 13 Figure1.7:Jacobirelativeenergycorrelationsinthe T systemfortheunbound systems(Left) 5 H[ 51 ],(Right) 13 Li ,and 16 Be[ 60 ].Peaksatlow E x =E T indicatethattheneutron-neutronenergyislowrelativeto thetotalthree-bodyenergyandisinterpretedaseitheradi-neutron emissionorstateinteraction.................... 15 x Figure1.8:(Top)Levelstructureforthethree-bodydecayof 6 Be[ 67 ].(Bot- tom)RelativeenergyplotsintheJacobi Y system(proton-core)for tslicesinthetotalthree-bodyenergy.Ontheleft,theen- ergyregionisslightlyabovethe2 + stateandthecorrelationindicate athree-bodydecay.Ontheright,theenergyismuchgreaterthan theintermediatestateandcorrelationsindicatingsequentialemission begintoemerge[ 43 ]........................... 17 Figure1.9:(Top)Thetransitionfromthethree-bodytothetwo-bodyregimeas studiedwithinGrigorenko'smodelfor 12 Oand 19 Mg.Plottedare therelativeenergiesintheJacobi Y system[ 2 ].(Bottom)Level structureforthedecaysof 12 O[ 27 ]and 19 Mg.[ 29 ].Notethewidths oftheintermediatestates........................ 19 Figure1.10:Levelstructureofthemostneutronrichboundoxygenisotopes. Hatchedareasindicateapproximatewidths............... 21 Figure2.1:(Left)Schematicforphase-spacebreakup.(Right)Ahypothetical levelschemewhereonewouldexpecttoobserveaphase-spacedecay. E I denotesthethree-bodyenergy, E V theintermediatestate,and E F =0,thestate.Thehatchedareasindicatewidths.Note thattheintermediatestateisbroad................... 28 Figure2.2:(Top)Schematicforasequentialdecay.(Bottom)Ahypothetical levelschemewhereonewouldexpecttoobserveasequentialdecay. Thehatchedareasindicatewidths.Notethattheintermediatestate isnarrowandwellseparatedfrom E 1 .................. 31 Figure2.3:(Left)Schematicfordineutronemission.(Right)Ahypotheticallevel schemewhereonewouldexpecttoobserveadineutron.Thehatched areasindicatewidths........................... 34 Figure3.1:LayoutofthedetectorsintheN2Vault................. 39 Figure3.2:BeamproductionattheCoupledCyclotronFacility[ 84 ]. 48 Cais heatedupinanion-sourceandacceleratedbytheK500andK1200 cyclotrons.AfterimpingingonaBetarget,thefragmentsare bytheA1900toprovidethedesiredbeam............... 40 Figure3.3:SchematicoftheUrsinusCollegeLiquidHydrogenTarget.(Left)A headonviewalongthebeamaxis.(Right)Sideview......... 42 xi Figure3.4:Phasediagramfordeuterium.Thesolidlinemarkstheliquid/solid transition,whilethebluedottedlinedenotestheliquid/gastransi- tion.Theplusmarksaredataobtainedduringtheinitialof thetarget................................. 43 Figure3.5:(Left)PhotoofthetargetcellusedtocontaintheLiquidDeuterium. Theliquiddripsdownthroughthecenterholeseenonthetarget Theiridiumsealcanbeseensurroundingitbeforebeing pressed.(Right)DrawingoftheinnerringwheretheKaptonfoilis glued.Theringisthenclampedtothetargetbyanouterringvisible intheleftphoto.............................. 45 Figure3.6:Adiagramofthegashandlingsystemusedtocontrolthewof deuteriuminandoutofthetargetcell.Figureadoptedfrom[ 89 ].. 48 Figure3.7:SchematicofaCathodeReadoutDriftChamber(CRDC)wherethe z -directionhasbeenexpanded.Theshapingwiresarenotdrawn forvisibility.Notethattheelectronavalanceisdoesnotbeginuntil theelectronsencountertheFrischgrid................. 50 Figure3.8:(Left)Head-onview(lookingintothebeam)ofthethintimingscintil- lator.(Right)Anexamplelevelschemefortransitionsintheorganic materialwitha ˇ -electronstructure.Source:[ 91 ]........... 52 Figure3.9:EmissionspectrumforEJ-204.Thepeakwavelengthisaround410 nm[ 92 ].................................. 53 Figure3.10:SchematicoftheHodoscope,whichisanarrayofCsI(Na)crystals arrangedina5x5.................... 54 Figure3.11: 12 C(n,*)crosssectionsforneutronenergiesrangingfrom1-100MeV forseveralentreactions.Eachofthecrosssectionslistedhere areincludedinMENATE Randusedtomodeltheneutroninterac- tionsinMoNA-LISA.Imagesource:[ 93 ]............... 56 Figure3.12:SchematicoftheelectronicsforMoNA-LISAandtheSweeper.Dashed linesencompaseseachsubsystem.Greenarrowsindicatestartsignals, redstopsignals,andbluearrowsindicatetheQDC/ADCgateforin- tegration.Uponreceivingthesystemtriggerfromlevel3,allTDCs, QDCs,andADCsreadoutandareprocessed............. 59 Figure3.13:Jacobi T and Y coordinatesforthethree-bodysystem 22 O+2n.. 62 xii Figure3.14:SimulatedJacobi T and Y coordinatesforthethree-bodysystem 22 O +2nwith E T =750keV.fromneutronscatteringhavebeen removed.Acceptancesandareapplied.Thetcol- orsindicatetdecaymechanisms.Allspectraarenormalized to2 10 6 events.............................. 64 Figure4.1:PedestalsubtractionintheCRDCs.TherawpadsforCRDC1(left) andCRDC2(right)areshownintheandsecondpanelsfromthe left,whilethesubtractionisshowninthethirdandfourthpanels.. 67 Figure4.2:Before(left)andafter(right)gainmatchingofCRDC1andCRDC2. CRDC1isshownintheandthirdpanel,andCRDC2inthe secondandfourth............................. 69 Figure4.3:(Fromlefttoright)Maskruns3023,3078,and3142forCRDC2. Particlesthatilluminateanareaabovethemaskindicatethatthe maskdidnotfullyinsert......................... 71 Figure4.4:DriftcorrectionforCRDC1Yposition.(Topleft),UncorrectedY distributioninCRDC1asafunctionofRunNumber.(Topright), UncorrectedcentroidsofCRDC1YpositionasafunctionofRun Number.(Bottomleft).CorrectedYdistributioninCRDC1.(Bot- tomright)CentroidsofcorrectedYdistributioninCRDC1asafunc- tionofrunnumber............................ 72 Figure4.5:DriftcorrectionforCRDC2Yposition.(Topleft),UncorrectedY distributioninCRDC2asafunctionofRunNumber.(Topright), UncorrectedcentroidsofCRDC2YpositionasafunctionofRun Number.(Bottomleft).CorrectedYdistributioninCRDC2.(Bot- tomright)CentroidsofcorrectedYdistributioninCRDC2asafunc- tionofrunnumber............................ 73 Figure4.6:ExampleofgaussianprocedureforgainmatchingoftheIC pads.Pad4isshownontheleft,andthereferencepad,pad9onthe right.................................... 74 Figure4.7:ResultsofapplyingthegainmatchingcalibrationfortheIC.Rawpads onshownontheleftandcalibratedpadsontheright......... 75 Figure4.8:Exampleofpositioncorrectionintheionchamberforpad15.The rawsignalasafunctionofXpositionisshownintheleftpanel,with thebsuper-imposed.Therightpanelshowstheresultofthe correction................................. 76 xiii Figure4.9:DriftcorrectionforICsum.(Topleft),Uncorrectedchargedistri- butionintheICasafunctionofRunNumber.(Topright),Un- correctedcentroidsofthechargedistributionasafunctionofRun Number.(Bottomleft).DriftCorrectedchargedistributioninthe IC.(Bottomright)Centroidsofthedriftcorrectedchargedistribution. 77 Figure4.10:Before(left)andafter(right)thegainmatchingofthePMTsinthe thinscintillator.............................. 79 Figure4.11:Correctionofthepositiondependenceinthethinscintillator.(Left) therawchargesignalasafunctionofXpositionwiththeb superimposed.(Right)Resultofpositioncorrection.......... 80 Figure4.12:Driftcorrectionforchargecollectioninthethinscintillator.(Top left),UncorrectedchargedistributioninthethinasafunctionofRun Number.(Topright),Uncorrectedcentroidsofthechargedistribu- tionasafunctionofRunNumber.(Bottomleft).DriftCorrected chargedistributioninthethin.(Bottomright)Centroidsofthedrift correctedchargedistribution....................... 82 Figure4.13:Driftcorrectionforthetimingofthethinscintillator.(Top-left) Theaveragetime t thin isshownasafunctionofrunnumber.(Top- right)Thecentroidofthetimingsignalasafunctionofrunnumber. (Bottom-left)Driftcorrectedthintimingasafunctionofrunnumber. (Bottom-right)Correctedtimingcentroidsusingthemethod. 84 Figure4.14:Driftcorrectionforthetimingofthetargetscintillator.(Top-left) Thetimingsignalisshownasafunctionofrunnumber.(Top- right)Thecentroidofthetimingsignalasafunctionofrunnumber. (Bottom-left)Driftcorrectedtargetscintillatortimingasafunction ofrunnumber.(Bottom-right)Correctedtimingcentroidsusingthe method............................... 86 Figure4.15:DriftcorrectionforthetimingoftheA1900scintillator.(Top-left) Thetimingsignalisshownasafunctionofrunnumber.(Top- right)Thecentroidofthetimingsignalasafunctionofrunnum- ber.(Bottom-left)DriftcorrectedA1900timingasafunctionofrun number.(Bottom-right)Correctedtimingcentroidsusingtheo method.................................. 87 Figure4.16:LightcollectioninthemiddlerowoftheHodoscopeforRun3002 wherean 20 Obeamwassweptacrossthefocalplane.Idealbehaviour wouldbeauniformresponse.OntheXandYaxisaretheXand Ypositionsrelativetoeachcrystal,thecoloraxisisthetotalenergy deposited................................. 88 xiv Figure4.17:TemperatureandPressurecalibrationoftheLD 2 target.(Left)Raw voltagesfromtheEPICSreadoutfromthetemperaturecontroller andmanometer.Thedataareinblack,theregionusedtothe phasetransitionishighlightedinred.(Right)Calibrationtophase- transition(blueline)indeuterium.................... 91 Figure4.18:Measuredphasetransitionwithaneontestgasusingthecalibra- tionparametersfromthedeuteriumtransition.Theredline correspondstodatafromRef[ 100 ],andthedashedlinesaretheun- certaintybandsgivena 0 : 5 K Theredarrowsindicate datatakenduringinitialcooling,liquefaction,andwarmingofthe target.Theeventsaround26Kand1200Torrarearesultofa sensorerror................................ 92 Figure4.19:TemperatureandPressuretuationsoverthecourseoftheexper- imentstartingfrom4:00AM3/19/14.Timeismeasuredinhours.. 93 Figure4.20:Measuredkineticenergyofthe 24 Obeamafterpassingthroughthe fullLD 2 target.............................. 94 Figure4.21:A3.4mmbulgeintheLD 2 drawntoscale............... 95 Figure4.22:ExamplespectrausedforQDCcalibrationinMoNA-LISA.(Left.) RawQDCchannelsfordatatakenwithcosmicrays.Theredcurve isagaussiantothecosmic-raypeak.(Right)Pedestalsubtracted andcalibratedchargespectrum.Thecosmicraypeakappearsaround 20-30MeVee............................... 97 Figure4.23:TDCandX-positioncalibrationspectra.(Left)RawTDCchannels foraruntakenwithatimecalibratorwithaintervalof40 ns,thisdeterminestheTDCslope.(Middle)Rawtim spectrumusedtocalculatetheX-position.Theredlinesindicatethe physicalendsofthebar.(Right)Conversionoftime-dinto X-position.Datatakenwithcosmicswhichfullyilluminatethearray. 98 Figure4.24:Schematicofacosmicraytracksusedtodeterminetherelative betweeneachbar.withinatablearewithrespecttothe bottombarinthefrontlayer. J 0 and K 15 areexamplebarsinthe andsecondlayerofLISA...................... 100 Figure4.25:Reconstructedvelocityof -rayscomingfromthetargetincoinci- dencewith 22 Ofragments.Theglobaltimingisvariedforeach tabletoalignthecentroidsat v =29 : 97cm/ns............. 101 xv Figure4.26:(Left)Separationofthe 24 Obeambytime-otfromtheother beamcontaminates.(Right)Energylossvs.tspectrum forallbeamsandreactionproducts.Linesaredrawntoguidethe eyeforeachelement........................... 103 Figure4.27: ˙ vs.totalintegratedchargeintheCRDCs.Qualitygatesaredrawn inred................................... 104 Figure4.28:TotalintegratedchargeofCRDC1(Y-axis)vs.CRDC2(X-axis).A gate,showninred,isdrawnaroundeventsthatdepositeasimilar chargeinbothdetectors.Agateisappliedtoselectthe 24 Obeam.. 104 Figure4.29: dE vs.ToFforreactionproductscomingfromthe 24 Obeam.Coin- cidencewithaneutronisnotrequired................. 106 Figure4.30:3Dcorrelationsfordispersiveposition,angle,andtshow- ingisotopeseparationforthecarbonisotopes............. 107 Figure4.31:ProjectionofFig. 4.30 ontothe2DplaneofToFvs.dispersive positionfortheoxygenisotopes.Thecontouroftis showninblack.Thiscorrelation,whenplottedagainstthetime-of- tshowsseparationfortheoxygenisotopes(Right),andcanbe rotatedinthisplaneforthepurposesofmakinga1Dgate...... 109 Figure4.32:Matrixof A 2 vs. A=Z withtheconditionthat Z A uptomass ˘ 25.Eachpointisaseparatenucleus(someunphysical).Thered linesindicatecurvesofconstant Z ................... 111 Figure4.33:Energyloss dE intheionchambervs.Correctedtshow- ingisotopeseparation.Acoincidencegatewithaneutronisre- quiredtoreducedbackgroundfromunreactedbeam.Averticalline at A=Z =3isdrawntoguidetheeye.................. 111 Figure4.34:One-dimensionalparticleidenfortheOxygenisotopes.A gateisdrawnattheverticallinetoselect 22 O.Neutroncoincidence withMoNA-LISAisrequiredtoreducethebackgroundfromunre- actedbeam................................ 112 Figure4.35:One-dimensionalparticleidenfortheNitrogenisotopes.A gateisdrawnattheverticallinetoselect 22 N.Neutroncoincidence withMoNA-LISAisrequiredtoidentifycandidatesforreconstruction. 112 xvi Figure4.36:NeutrontspectruminMoNA-LISAwith 22 Ocoincidence. Thesplittingisduetothenarrowresonancein 23 Oandthefactthat theMoNA-LISAtablesarephysicallyseparated.Theinsertshows -rayscomingfromthetarget...................... 114 Figure4.37:Neutrontspectrumvs.ChargedepositedinMoNA-LISA with 22 Ocoincidence. -raystypicallydeposits < 5MeVee,whilereal neutronscandeposituptothebeamenergy.Nobackgroundfrom cosmicraysisevident........................... 114 Figure4.38:Relativedistance D 12 andvelocity V 12 for1n(Left)and2n(Right) simulations.Theselectionfor2neventsisshowninbytheshaded blueregion................................. 116 Figure4.39:Comparisonofneutron(blue)andfragment(black)kineticenergyfor thedecayof 23 O.Therelativevelocityisshownontheright..... 120 Figure4.40:Comparisonofneutron(blue)andfragment(black)anglesatthe target.Thelargebinningin T Y isduetothediscretizationofthe MoNAbars................................ 120 Figure4.41:Comparisonofmeasuredneutron-fragmentopeningangle n f and fragmentangleinsphericalcoordinatesforthedecayof 23 O ! 22 O+ 1n.Inblackisthecurrentexperiment,showninblueisameasure- mentofthesamedecayusingthesameexperimentalapparatus[ 73 ] forcomparison.............................. 121 Figure4.42:1n E decay spectraforthedecayof 23 O.(Left)spectrumobtainedfrom thecurrentexperiment.(Right)Apreviousmeasurementperformed onthesameexperimentalapparatusforcomparisonfromRef.[ 72 ]. 121 Figure4.43: E decay spectrumforthedecayof 22 N ! 21 N+1n.Ontheleftisthe spectrumobtainedfromthecurrentexperiment.(Right)Aprevious measurementofthesameresonancealsoperformedwithMoNA[ 110 ]. 122 Figure4.44: E decay spectrumforthedecayof 24 O ! 23 O+1n.(Left)spectrum obtainedfromthecurrentexperiment.(Right)Apreviousmeasure- mentofthesameresonancesusingMoNAforcomparison.[ 111 ]... 123 Figure4.45:(Top) E decay spectrumforthedecayof 19 C ! 18 C+1nobtained fromthecurrentexperiment.(Bottom)Apreviousmeasurementof thesameresonanceusingMoNA[ 112 , 113 ]............... 123 xvii Figure4.46:Comparisonbetweensimulation(blue)anddata(black)fortheun- reacted 24 Obeaminthefocalplane.................. 128 Figure4.47:Reconstructedanglesandtargetpositionusinga4-parametermap. Thesimulation(blue)iscomparedtodatafortheunreacted 24 O beam(black)............................... 129 Figure4.48:Comparisonofreconstructedkineticenergydistributionsbetween simulation(blue)anddata(black)fortheunreacted 24 Obeam.The reconstructedenergyisafterpassingthroughthefullLD 2 target... 129 Figure4.49:Comparisonoffocalplanepositionandanglesbetweensimulation (blue)anddata(black)for1nknockoutto 23 O,whichthendecays to 22 O................................... 130 Figure4.50:Reconstructedkineticenergyforthe 22 Ofragmentscomingfromthe 1nknockoutreaction.Simulationresultsareshowninblueandthe datainblack............................... 131 Figure4.51:Comparisonoffocalplanepositionandanglesbetweensimulation (blue)anddata(black)for1pknockoutto 23 N,whichthendecays to 22 N................................... 132 Figure4.52:Reconstructedkineticenergyforthe 22 Nfragmentscomingfromthe protonknockoutreaction.Simulationisinblueanddataareinblack. 133 Figure4.53:Centerofmassangulardistributionforinelasticscatteringof 24 Oon d at82.5MeV/u(lab),estimatedwithFRESCO,aglobaloptical model,andthedeformationlengthof 12 C............... 133 Figure4.54:Breakdownof 12 C(n,*)cross-sectioninMENATE R.(Left.)MENATE R crosssectionswithoutanyadjustment.Thebluecurveisthesumof allcross-sectionintheenergyrange40-100MeV.(Right)Thetotal MENATE Rcross-sectionsaftermocomparedtotheDel Guerracompilation[ 122 ]andtheENDF[ 123 ]evaluation....... 134 Figure4.55:CRDC1Xvs.CRDC1 X forall 22 Ofragmentscoincidentwitha neutron.Agate,calledanXTXcut,isdrawnaroundthedataand appliedtosimulation.Thisistorejectfragmentswhichmayhavea strangeemittanceinthesimulation................... 136 Figure5.1:(Left)AcceptanceofMoNA-LISAfora1ndecayasafunctionof E decay .(Right)Acceptancefora2ndecaywithcausalitycutsapplied. 141 xviii Figure5.2:(a)1ndecayenergyspectrafor 23 Owithcontributionsfromneutron- knockoutandinelasticexcitation.(b)Thethree-bodydecayenergy for 22 O+2nforallmultiplicities 2.(c)Thethree-bodydecay energywithcausalitycutsapplied.Theinsertsshowalogarithmic view.................................... 142 Figure5.3:Jacobirelativeenergyandanglespectrainthe T and Y systemsfor thedecayof 24 O ! 22 O+2nwiththecausalitycutsappliedandthe requirementthat E decay < 4MeV.................... 144 Figure5.4:Levelschemeforthepopulationofunboundstatesin 23 Oand 24 O fromneutron-knockoutandinelasticexcitation.Hatchedareasindi- cateapproximatewidths......................... 145 Figure5.5:(a)1ndecayenergyspectrafor 23 Owithcontributionsfromneutron- knockoutandinelasticexcitation.(b)Thethree-bodydecayenergy for 22 O+2nforallmultiplicities 2.(c)Thethree-bodydecay energywithcausalitycutsapplied.Directpopulationofthe5 = 2 + stateand3 = 2 + statein 23 Oareshownindashed-redanddashed- greenrespectively.The2ncomponentcomingfromthesequential decayof 24 Oisshownindashed-blueanddecaysthroughthe5 = 2 + state.Thesumofallcomponentsisshowninblack.Theinsertsshow alogarithmicview............................ 146 Figure5.6:Jacobirelativeenergyandanglespectrainthe T systemforthedecay of 24 O ! 22 O+2nwiththecausalitycutsappliedandtherequire- mentthat E decay < 4MeV.Shownforcomparisonaresimulationsof severalthree-bodydecaymodes.Asequentialdecay(b),adi-neutron decaywith a = 18 : 7fm(c),and(d)aphase-spacedecay.Theam- plitudesaresetbytwicetheintegralofthethree-bodyspectrumwith causalitycuts............................... 149 Figure5.7:Neutroncforthetwo-neutronsequentialdecay.Filled circlerepresentparticlesandfadedcirclesrepresentholes.The2 + or 4 + of 24 Oisshownonthefarrightandisaparticle- holeexcitation.The5 = 2 + statein 23 Oisaholestatewhichdecays tothetwo-particletwo-holeof 22 Owhichisasmall componentofthegroundstatewavefunction............. 151 Figure5.8:Comparisonofexperimentallymeasuredstatesin 24 OwithUSDB shellmodelpredictions.DatatakenfromRefs.[ 70 , 71 ].Thestate observedinthepresentworkisshowninblue............. 152 xix Figure5.9:Jacobirelativeenergyandanglespectrainthe T and Y systems forthedecayof 24 O ! 22 O+2nwiththecausalitycutsapplied andtherequirementthat E decay < 4MeV.Indashed-blueisthe sequentialdecaythroughthe5 = 2 + statein 23 O.Theremainingfalse 2ncomponentsfromthe1ndecayof 23 Oareshowninshaded-grey. Thesumofbothcomponentsisshowninsolid-black......... 153 Figure5.10:Two-bodydecayenergyfor 22 N+1n.................. 154 Figure5.11:Shellmodelpredictionsfor 23 NwiththeWBPandWBTHamiltoni- ansaswellastheContinuumShellModel............... 155 Figure5.12:Possiblelevelschemesthatcouldgiverisetotheobservedtwo-body decayenergyfor 23 N.Case(a)isthe\GroundStateDecay,"while the\TwoState"scenarioiscase(b),andthe\SingleState"case(e). Thesinglestateinterpretationcouldalsobetwostatesclosetogether. Cases(c),(d),and(f)areexcludedduetotheweakintensityofthe 100keVtransition............................ 157 Figure5.13:Bestofthetwo-bodydecayenergyfor 23 Nbasedonthe\Ground StateDecay"interpretation.Thepurplecurveindicatesthe100keV transition,thedahsedredisthe960keVtransition.The2nback- groundisshowninshadedgray..................... 158 Figure5.14:Bestofthetwo-bodydecayenergyfor 23 Nbasedonthe\Single State"interpretation.Thepurplecurveindicatesthe100keVtransi- tion,thedahsedredisthe960keVtransition,andthedashedblueis the1100keVtransition.The2nbackgroundisshowninshadedgray. 160 Figure5.15:Bestofthetwo-bodydecayenergyfor 23 Nbasedonthe\Two State"interpretation.Thepurplecurveindicatesthe100keVtransi- tion,thedahsedredisthe960keVtransition,andthedashedblueis the1100keVtransition.The2nbackgroundisshowninshadedgray. 162 Figure5.16:Comparisonbetweena1nthermalbackground(Top)anda2nback- ground(Bottom).Ontheleftisthetwo-bodydecayenergy.The middlepanelsshownthethree-bodyspectrumfor 24 Nwithcausality cuts,andthefarrightpanelsshowthemultiplicity.Thepurpleline isthe E 1 =100keVBreit-Wigner,thedashedredisthe E 2 =960 keVresonance.Thebackgroundcontributionisshowninshadedgray. 163 xx Chapter1 Introduction 1.1ChartoftheNuclides Theatomicnucleusconsistsoftwotypesofcompositeparticleswithsimilarmass:protons andneutrons.Protonscarryoneunitofpositivecharge e andneutronsarecharge-neutral. Anuclearspeciesischaracterizedbythenumberofprotonsitcontains, Z ,andamass number A sosuchthatthemassofasinglenucleonisnearlyone.Inthenucleus weencounterlengthscalesontheorderof10 15 m,densitiesontheorderof ˘ 2 : 3 10 17 kg/m 3 ,andenergiestypicallyinthekeVtoMeVrange.Thetime-scalesfornuclearprocesses varyoveranenormousrange. and decaycantakeplaceontheorderofmillisecondsto hours,oreventhousandsormillionsofyears.Electromagneticdecaystypicallyoccurwithin lifetimesof10 15 s,andthebreakupofparticle-unboundresonances,like 25 O,areextremely short{occurringontimescalesof10 21 s[ 1 , 2 ]. Similartotheperiodictable,whicharrangeselements,thechartofnuclidesarranges allknownnuclearspeciesbythenumberofprotons(Y-axis),andnumberofneutrons (X-axis),showninFig. 1.1 .Forexample,plottingtheneutronseparationenergy S n ,the energyrequiredtoremoveasingleneutron,immediatelyrevealsaglobaltrendevidenced byastaggeringpatterninFig. 1.1 .Thiseven-oddoscillationbetweenseparationenergies canbeexplainedbytwo-bodypairing( n n )whichcanincrease/decreasethebinding. Thesametrendisobservedforprotons.Thechartofnuclidesatpresentisfarfrombeing 1 completelyexplored,andthelimitsforexistenceareunknown.Oftheapproximately3000 knownnucleithereisanestimated4000morethatmayexist[ 3 ].Theprotondriplineis unknownfortheheaviestelementsandtheneutrondriplineisonlyknownupto 24 O[ 4 ]. Therearealsopredictionsforan\islandofstability,"aregioninthesuperheaviesofstable nucleiyetunobservedbutexpectedtooccuraround Z =114, N =184[ 5 ]. Theover-archinggoalofnuclearphysicsistounderstand,withpredictivepower,the interactionsbetweenneutronsandprotonsandthepropertiesofthecomplexmany-body systemstheycancreatewhenarrangedtogether.Sostrictlyspeaking,nuclearphysicsis stronginteractionphysicsandbasedupontheStandardModel,weshouldbeconcerned withgluonexchangebetweentheconstituentquarks.Howeverthetypicalenergyscale characteristicofnuclearphysicsisinthelow-energyregimeofQCDwherethetheoryis non-perturbative,sodirectcalculationisAlthoughtheproblemcanbeapproached onadiscretizedlattice(latticeQCD),thismethodisstilllimitedbycomputationalpower andhassofaronlybeenimplementedinthelightestnuclei( 3 He, 4 He)[ 6 ].Inaddition, in1979,Weinberg[ 7 ]pointedoutthatanyetheoryfornucleonsandmesonsthat obeysthesamesymmetriesasQCDisequivalenttoQCD,whichallowsustoconsider nucleonsandmesonsasourstartingpointinsteadofquarks.Eventhisreductioncanbe untractablecomputationally.Ifonestartswithabaretwo-nucleon NN interaction,then anynuclearmany-bodyproblemcanbeexactlybuttheproblemgrowsfactorially andthesemicroscopiccalculationsarelimitedtoonlythelightestnuclei.Anucleuslike 238 U issimplyimpossibletocomputefromabare NN potential,andwilllikelybeforsometime. Atpresentthereisnosingletheoreticalformulationthatallowsustointerpretallnuclear phenomenainafundamentalway,andsonuclearphysicsisapproachedphenomenologically. Theappropriatemodeldependsonwhereoneisinthenuclearchartandwhatphenomenon 2 isbeingdiscussed.Itisfromthesemodelsthatwegaininsightintonuclearstructure. Figure1.1:ChartoftheNuclides.Onthecoloraxisistheneutronseparationenergy S n in MeV.DatatakenfromAME2012[ 8 ] 1.2ShellModel OnesuccessfulmodelistheShellModel.EarlyelectronandRutherfordscatteringexper- imentsshowed,surprisingly,thatthechargeandmatterradiiofnuclei arenearlyequalto withinabout0.1fm [ 1 , 9 ]{implyingthatthenuclearforceisthesamebetweenneutrons andprotons.(Althoughnotquitetrueasthesymmetryisbrokenbytheslightin quarkmasses).Bothhavean A dependenceas: r ˘ r 0 A 1 = 3 3 with r 0 =1 : 2fm.Fromthesemeasurements,thedensityeasafunctionof r was foundtobelargelyconstantwithinthenucleus,withasmoothasoneapproachesthe surface.Becauseoftheshortrangeofthenuclearinteraction,nucleonsmainlyinteractwith theirneighborsandsaturationoccurswherenucleonsonthesurfacedonotinteractstrongly withthoseinthecore.Thisobservationleadtotheliquid-dropmodel,proposedby Gamov,wherethenucleuswasthoughtoflikeadropletofliquid.Thebindingenergycan beexpressedinthefollowingway: BE ( N;Z )= 1 A 2 A 2 = 3 3 Z 2 A 1 = 3 4 ( N Z ) 2 A Wherethefourtermsrefertothevolume,surface,Coulomb,andsymmetryterms,respec- tively.Theisaresultofconstantdensity,whilethesecondaccountsforthefactthatthe nucleonsonthesurfacehavelessneighborstointeractwith.ThethirdresultsfromCoulomb repulsionbetweentheprotons,andthefourthtermcanbeunderstoodasanofthe Pauliprinciple. Whileagoodestimatoroftheaveragebindingenergy,theliquiddropmodeldoesnot accountforanyshellorpairingSimilartonobelgases,whichhavefullelectronshells, therearecertainnumbersofneutronsandprotonsthataremoretightlyboundthanothers. Fig. 1.2 showsthebetweentheliquiddropmodelpredictionforbindingenergy andwhatisobservedinexperiment.Largepeaksappearatwhatarecalledthe\magic numbers,"28,50,82,126,whereoneobservesmorebindinginnaturethanpredictedby theliquiddrop-model.Thisisstrikinglysimilartospikesintheelectronionizationenergy occurringinthenobelgasses.Thusinspiredbytheatomicshellmodel,MariaGoppert Mayer,Haxel,Jensen,andSuessallsoughttoexplaintheenhancedbindingwithasimilar 4 Figure1.2:(Top)inbindingenergybetweentheliquiddropmodelandexperi- mentalobservation.Ontheleftistheasafunctionofneutronnumber N .Onthe rightisasafunctionoftheprotonnumber Z .(Bottom)ElectronIonizationenergyforall elementswith Z< 104ontheperiodictable.Notethesimilarityinclosedelectronshells andclosedneutron/protonshells.Imagesources:[ 10 ],[ 11 ] approach[ 12 , 13 ].Thisideaisparticularlyattractivebecausenowspatialorbitsfornucleons canbediscussedinanalogytoelectronorbitals. Thestepindevelopingashellmodelistochooseapotential.Duetotheshort-range interactionandconstantdensity,anucleoninthemiddleofthenucleuswillinteractwith approximatelythesamenumberofnucleonsregardlessofit'sposition.Thusthepotential 5 shouldbeinsidethenucleus,andnegativesothatthenucleusisbound, V ( r ) < 0. Asthenucleonmovestowardsthesurfacethenumberofneighborsitcaninteractwith decreasesleadingtoashallowerpotential.Finally,outsidethenucleus,weexceedtheshort- interactionrange,andso V ( r )shouldapproachzero.Thisbehaviorisoftenparameterized withaWoods-Saxonshape,whichcanbefurtherapproximatedbyaharmonicoscillator. Byaddingaspin-orbitterm,MayerandJensenwereabletoreproducetheobservedgapsin orbitalenergy,correspondingtothemagicnumbers,forwhichtheywereawardedtheNobel Prize(1963). Figure1.3:Singleparticleorbitsinthenuclearshellmodel.Energiesareshownfora harmonicoscillatorpotential,woods-saxon,andwoods-saxonwithspinorbit.ImageSource: [ 10 ] 6 Thenuclearshellmodelsharessomepropertieswiththeatomicshellmodel.Asingle particleisplacedina V ( r ),andtheeigenstatesarecharacterizedbyorbitals withquantumnumbers n , l , j ,andanenergy E .Likeatomicorbitals,theeigenstatesare separatelywithneutronsandprotonsbeginningwiththelowestenergywhileobeying thePauliprinciple.However,unliketheatomicshellmodel,thespin-orbittermisgreater andofoppositesign,sotheorderingoftheorbitalsandtheplacementofshellgaps(magic numbers)ist. SinceMayerandJensen'swork,theShellModelhasbecomemuchmoresophisticated. AmoremoderncalculationwillwritetheHamiltonianasasumofone-andtwo-bodyop- erators,withanetwo-bodyinteractiontypicallyderivedphenomenologicallyfrom experimentalmeasurementsinaparticularmass-rangeofinterest.Theproblemthenbe- comesoneofmatrix-diagonalizationtodeterminetheeigenvaluesofaparticularfew-body system.Itisagoodtoassumethatattheshell-closuresthenucleonsina shellcanbeapproximatedasasinglecore.Thisapproachalsohascomputationallimi- tations,asthenumberofbasisstatesincreasesfactoriallywiththeadditionofmoreorbitals and/orparticlesandtruncationsareoftenmadetosimplifythecalculation. 1.2.1IslandsofInversion IthasbeenshownthatMayerandJensen'smagicnumbersbreakdownasonemovestowards neutronrichnuclei.Theconventionalmagicnumbersfornucleiinthevalleyofstabilityare notnecessarilymagicfornucleiwithextremeN/Zratios.Forexample 24 Oisdoublymagic withtheappearanceofanewshellclosureat N =16,asevidencedbytrendsin E (2 + ) energies[ 14 , 15 , 16 ].Onestrikingexampleisthe\IslandofInversion,"locatedaroundthe massregionof A ˘ 32[ 17 , 18 ].Here,the N =20shellgapisquenchedandnucleithat 7 shouldoccupygroundstatesinthe sd shellinsteadoccupyorbitalsinthe pf shell.This shifthasbeenattributedtothe NN tensorforce[ 19 ],three-bodyforces[ 20 ],andcontinuum incaseswherenucleiapproachthedriplines[ 21 ].Thediscoveryoftheislandof inversioncausedaparadigmshift,asitwaspreviouslythoughtthatthemagicnumberswere immutable[ 20 ].Ithassincebeenfoundthattherearemultiplesuchislandsofinversion, andtheofnuclearforcesontheshellstructureofnuclei,particularinneutron-rich regions,isanongoingareaofresearch. 1.2.2TensorForce Thetensorforceistheresultofone-pionexchangeandisthemostprominentspin-isospin interactionbetweennucleons.Inrecentyearsithasbeenshowntohavesystematic onthesingle-particleenergiesofexoticnuclei[ 19 , 22 ].Dependingontheangularmomenta oftheparticlesinvolved,thetensorforcecanbeeitherattractiveorrepulsive.Thetensor forceiswrittenas: V T =( ~˝ 1 ~˝ 2 ) S 12 V ( r ) where ~˝ i denotestheisospinoperatorsofnucleons1and2, V ( r ) > 0isafunctionofthe distance r betweenthenucleons,and S 12 is: S 12 =3( ~s 1 ^ r )( ~s 2 ^ r ) ( ~s 1 ~s 2 ) where ~s i arethespinofthenucleons.Forsimplicity,take ~s 1 = ~s 2 =+^ z ,andlet ` and ` 0 denotetheangularmomentumofaprotonandneutronrespectively.Theirtotalangular momentumwillbe j = ` 1 = 2,and j 0 = ` 0 1 = 2. 8 Forpartners,( j ;j 0 )thetensorforceisattractive.Whenthenucleonscollide, theywillhavelargerelativemomentumcausingthespatialwavefunctiontobenarrowly distributedinthedirectionofthecollisionlikethatillustratedintheleftofFig. 1.4 . Thisresultsinawavefunctionsimilartothedeuteron.Inthiscase, ~s 1 ~s 2 =+1,and ~s 1 ^ r = ~s 2 ^ r =1,thus S 12 =2.Since( ~˝ 1 ~˝ 2 )=2( ^ T 2 3 = 2)= 3foroppositeisospins, ( ^ T =^ ˝ 1 +^ ˝ 2 =0), V T becomesnegativeandthustheinteractionisattractive. Intheoppositecasewith( j ;j 0 ),thewavefunctionisstretchedalongthedirectionof motionasillustratedintherightofFig. 1.4 .Inthiscase ~s i ^ r =0andweobtain S 12 = 1 makingtheinteractionrepulsive. Figure1.4:(a)Diagramforwavefunctionofrelativemotionforcollisionofa pair.Notethatthiscaseisdeuteron-likeandattractive.(b)Inthenoncasethe wavefunctionofrelativemotionisstretchedperpendiculartothespin(denotedbyblack arrows),andisrepulsive. TheroleofthetensorforceindrivingshellevolutionisillustratedinFigure 1.5 .For stablenucleinear N = Z =20,theproton ˇ 0 d 5 = 2 orbital( j + )isfullandhasanattraction withtheneutronsinthe 0 d 3 = 2 orbital( j 0 ),aswellasarepulsionwith 0 f 7 = 2 orbital( j 0 + ). Thisresultsinthenormalshell-orderingatstability,asthe 0 f 7 = 2 orbitalisraisedandthe 9 0 d 3 = 2 orbitalisloweredcreatingalargegapat N =20.Ifoneremovesprotonsandtravels downtheisotones,the ˇ 0 d 5 = 2 0 d 3 = 2 attractionisweakenedcausingthethe 0 d 3 = 2 orbital toriseinenergyrelativetonucleiatstability.Thiscreatesagapbetweenthe 0 d 3 = 2 and 1 s 1 = 2 orbitalsleadingtoaquenchingofthe N =20gap,andtheappearanceofanew magicnumberat N =16. Figure1.5:Schematicforillustratingshellevolutionduetothetensorforce.Thewidth ofthearrowsdenotesthestrengthoftheinteraction.(Left)Ainnucleinear stabilitywith N =20beingmagic.(Right)Asprotonsareremovedtheattractionwiththe 0 d 3 = 2 orbitalweakenscausingittoriseinenergyrelativetothe 1 s 1 = 2 orbitalcreatinga newshellgapat N =16 1.3Three-bodyCorrelationsandDecays Inathree-bodydecaytherearethreeparticlesinthestate.Nucleiwhichundergo2nor 2pemission,suchas 10 He(2n)[ 23 , 24 , 25 ], 13 Li(2n)[ 23 , 26 ], 12 O(2p)[ 27 , 28 ], 19 Mg(2p)[ 29 , 30 , 31 ]),andmanyothersdiscussedinthissection,fallintothiscategory.Correlationsbetween thetwonucleonsaswellastheheavycorecanprovideinsightintothedecaymechanism. Ingeneral,thereareseveralbroadcategoriesusedtoclassifythetdecaymodes:(1) Di-neutron/protonemission,(2)asequentialdecay,or(3)athree-bodydecaywhereinit 10 isnecessarytosolvethethree-bodyschrodingerequation.Thelatterisatruethree-body processesandcanbecomplex.Itisalsousefultointroducetheconceptofaphase-space decay(whichisdistinctfromathree-bodyinteraction),wherethephasevolume( M 2 fn vs. M 2 n n )isuniformlypopulated.Thedineutron,phase-space,andsequentialdecaymodesare discussedindetailinSection2.2. Inthetwo-bodydecay,aresonancecanbecharacterizedbyjustanenergyandawidth. Theadditionofathirdparticlehoweverallowsfor9degreesoffreedominthestateifwe neglectspin.Threeofthemdescribethecenter-of-massmotion,andanotherthreedescribe theEulerrotationsthatthedecayplane.Thusforagiventhree-bodyenergy E T , therearetwofreeparametersleft.IntheJacobicoordinatesystems,discussedindetailin section3.10,itisconvenienttochoosetherelativeenergy andangle k betweentheJacobi momenta.Thesecorrelationsaresensitivetothedecaymechanism,andcaninprinciplebe reproducedwiththethree-bodywavefunction.Thustheyareapowerfultooltodiscernthe decaymechanism,aswellasconnectdirectlywithfew-bodykinematics.Measurementof2p and2ndecaysandtheirthree-bodycorrelationsnearthedriplinesprovideanopportunityto benchmarkourtheoreticalunderstandingoffew-bodyquantummechanicsaswellpotentially observenewphenomena. 1.3.1Two-ProtonDecay Thementionoftruetwo-protonemissionisintheworkofZeldovich[ 32 ],withamore explicitanddetaileddescriptiongivenbyV.I.Goldanskysoonafter[ 33 ].Although predictedbyGoldanskyin1960,ittooknearly40yearstotheexistenceoftwo- protonradioactivityin 45 Fe[ 34 ],andsincethenmanyothernucleihavebeenfoundtohave lifetimeslong-enoughtofallintotheregimeofradioactivity( ˝> 10 14 s)including 17 Ne 11 [ 35 ], 19 Mg[ 36 ], 48 Ni[ 37 ],and 54 Zn[ 38 ].Whileastringentlimitforradioactivitydoesnot exist,onesuggestedbyPfutzner[ 2 ]is\Aprocessofemissionofparticlesbyan atomicnucleuswhichoccurswithcharacteristictime(half-life)muchlongerthanthe K -shell vacancyhalf-lifeinacarbonatom(2 10 14 s)."Henceanuclearprocesswhoseduration exceedsthislimitcouldbeconsideredradioactive,includingthedecayofunboundground states. Experimentalattemptstosearchfor2pemissionstartedinthelightestnucleiastheyare relativelyeasiertoaccessexperimentally,withthesystemsstudiedbeing 6 Be[ 39 , 40 ], 12 O[ 28 , 41 ]and 16 Ne[ 42 ]. Thedecayof 6 Befallsintotheclassof\democraticdecays",wherethereisnostrong energyfocusingoftheparticlesandtheirmomentaaresmoothlydistributed.Evenin Geesaman'swork[ 39 ]itwasevidentthatasimplephase-spacedecay,di-protondecay,or evensimultaneousemissionoftwo p -waveprotonscouldnotdescribetheenergydistribu- tionof particlestheyobserved.Itwasconcludedthatafullthree-bodycalculationwas necessary.Morerecently,thefullpictureofthethree-bodycorrelationsin 6 Bewasexperi- mentallymeasuredandshowntobeinverygoodagreementwithathree-bodycalculation byGrigorenko.Fig. 1.6 showsthethe T and Y Jacobicorrelationsforthebreakupof 6 Be ! p + p + fromRef.[ 43 ],comparedtothetheoreticalmodel[ 44 ].Afullthree-body approachwasalsoshowntobenecessarytounderstandthe p p correlationsin 16 Ne[ 30 ], 19 Mg[ 30 ],and 45 Fe[ 44 ]. Inthecaseof 12 Otheopeninganglebetweentheemittedprotonswasmeasuredandwas inconsistentwithmodelsfordi-protonemission[ 28 ].Itwaslaterfoundthattheintermediate nucleus, 11 N,hasagroundstatebelow 12 O.Thestateisbroad{implyingthat 12 Ocannotbe aGoldansky-liketrue2pemitter.Initialmeasurementsinterpretedthedecayassequential 12 Figure1.6:(Top)TheoreticalpredictionfortheJacobi T (left)and Y (right)systemrel- ativeenergyandangularcorrelationsinthebreakupof 6 Bebasedonafullthree-body calculation[ 44 ]Thedataareshownonthebottompanels.ImageSource:[ 43 ] [ 28 ],howevermorerecentworkhasshownthat 12 Omoreappropriatelyfallsinthecategory of 6 Be,asthethree-bodyenergy E T iscomparabletothewidthoftheintermediatestate [ 27 ].Astudyof 14 O[ 45 ]alsofoundevidenceforsequentialemission,andalthoughnota2p decay,thereisevidenceforsequentialdecayinthethree-bodyexitchannelof 9 B ! p +2 whichpassesthroughanintermediatestatein 5 Li[ 46 ]. Thereareseveralcasesof -delayed2pemission,furtherdiscussedbyBlankandBorge in2008[ 47 ].Themoststudiedcase, 31 Ar[ 48 ]appearstodecayonlybysequentialemission, anditiscurrentlybelievedthatallstudied 2 p processesaresequential[ 2 ]. Ingeneral,for2pemittingnuclei,themechanismiseitheratruethree-bodyprocess orsequential,althoughtherearesomecaseswithevidencefordi-protonemission.For 13 example,thebreakupof 8 C ! 2 p + 6 Be ! 4 p + wasmeasuredrecentlyattheNational SuperconductingCyclotronLaboratory[ 46 ].Inthisstudy,itwasfoundthatthisdecaycan bedescribedastwo2pdecayswiththestep 8 C ! 2 p + 6 Beshowingsomedi-proton characteristics,andthesecondbeingathree-bodydecay.Inaddition,therearestudies onthe1 resonancein 18 Nepopulatedby 17 F+ 1 H[ 49 ]andbyCoulombexcitation[ 50 ] wherethe p p correlationspectraareexplainedwithadi-protoncomponent.Howeverthe statisticalclaimsinbothcasesareweak. 1.3.2Two-NeutronDecay Comparedto2pdecaysalongtheprotondripline,theneutrondriplineislessstudied.Analo- goustothe2pemitters,severaltwo-neutronunboundsystemshavebeenmeasuredincluding 5 H[ 51 ], 10 He[ 52 , 25 , 23 , 24 ], 13 Li[ 23 , 26 ], 14 Be[ 53 ], 16 Be[ 54 ],and 26 O[ 55 , 56 ].Several ofthethree-bodycorrelationsinthesesystemhavebeeninterpretedasdi-neutronemission [ 26 , 54 ].Inaddition, 26 Ohaspotentialtoexhibittwo-neutronradioactivity[ 57 ].Despite observing2pradioactivityapproximatelyadecadeago,upuntilrecentlytherehasbeen\no theoreticaltreatise"(Grigorenko,2011)[ 58 ]forneutronradioactivity.Thisispartlydueto thefactthatneutronradioactivityisyetunobserved,butalsothat2nunboundsystemsare challengingfrombothanexperimentalandtheoreticalpointofview.TheCoulombinterac- tionplaysamajorroleinunderstandingthedynamicsof2pemission,especiallyinheavier systemslike 45 Fe,andextendingthesemodelstotheneutrondriplineisnotassimpleasre- movingCoulomb[ 59 ].Inaddition,itistoidentifytwoindependentneutrons withinasingleeventexperimentally. Contrarytothe2pdecays,manyofthethree-bodycorrelationsin2nemittersindicate di-neutronemission,orastrong n n stateinteraction(FSI).Fig. 1.7 showstherelative 14 n n energy, E x comparedtothetotalthree-bodyenergy E T intheJacobi T systemfor 5 H [ 51 ], 13 Li[ 26 ],and 16 Be[ 60 ].Allthreespectraaresimilarandpeakatlow n n energywhich canbeinterpretedasthewavefunctionhavingadi-neutroncomponent,or n n FSI Correlationsfor 26 Ohavealsobeenmeasuredandpotentiallyshowdi-neutronlikecharacter aswell[ 61 ].However,inthiscasethestaticsarelowandthedataareindistinguishablefrom amodelwheretheneutronsareemittedback-to-backratherthanacluster[ 62 ].Inthese nuclei,theintermediatenucleusisenergeticallyinaccessibleandthelevelstructuremimics a\Goldanskytrue2p"emitterillustratedinFig. 2.3 . Figure1.7:Jacobirelativeenergycorrelationsinthe T systemfortheunboundsystems (Left) 5 H[ 51 ],(Right) 13 Li ,and 16 Be[ 60 ].Peaksatlow E x =E T indicatethattheneutron- neutronenergyislowrelativetothetotalthree-bodyenergyandisinterpretedaseithera di-neutronemissionorstateinteraction. Sequentialcorrelationsalongtheneutrondrip-linehavebeenleastobserved.Thereisa measurementofthedecayofhighlyexcitedstatesin 14 Be[ 53 ]wherethethree-bodyenergy correlationsshowsomeevidencefordecaysthroughintermediatestates.Howeverthiswork didnotpublishangularcorrelations.Thereisalsoindicationthatexcitedstatesin 24 Ocan decaybysequentialemission,asevidencedby[ 63 ]butthestatisticsweret toextractthree-bodycorrelations. 15 -delayedmultineutronemission,whiledominantintheveryneutron-richnuclei,isnot wellstudiedandcorrelationshavenotbeenmeasuredinthesesystems.Thisisinpartdueto theyofneutrondetection,andthefactthatfewcasesareknown.The -delayed neutrondecayswerediscoveredin1979[ 64 ]and1980[ 65 ],both 2 n and 3 n processeswere observedin 11 Li.Inaddition,thereisareportof 4 n emissionin 17 B[ 66 ],howeverthis workis 1.3.3Transitionfrom3-bodyto2-body Thelevelstructureofthenucleiinvolvedineither2por2nemissionstrongly althoughdonotappeartocompletelydetermine,themechanismofthedecay.Justbecause asequentialdecayisenergeticallyviable,doesnotguaranteeitwilloccur.Inthedemocratic decayof 6 Bethetransitionfromthree-bodytosequentialwasexaminedbylookingatthe p energyintheJacobi Y systemforentslicesofthetotalthree-bodyenergy E T [ 43 ].Fig. 1.8 showsthelevelstructureof 6 Beandtheintermediatenucleus 5 Li.Becausethe intermediatestatein 5 Liisbroad,thesequentialdecaymechanismis suppressed infavour ofthree-bodydynamics. Inthisstudyitwasfoundthatsequentialcorrelationswerenotvisibleuntilthedecay energy E T wasgreaterthantwicetheintermediatestateplusitswidth.Thisisshownin Fig. 1.8 bythedouble-humpstructureinthe E p =E T spectrum,indicatingahigh-energy andalow-energyprotoncomingfromdiscretestates.Contrastthiswiththethree-body bell-curveatlower E T ,whichpeaksat1 = 2andissymmetric.Thesymmetryisaresult ofthetwoprotonsbeingindistinguishable,andthemaximumat1 = 2canbeunderstood asaresultofthemaximumprobabilityforbarrierpenetrationoccurringwhentheproton energiesareequal.Goldanksyproposedanexponentialfactor w ( )astheproductoftwo 16 Figure1.8:(Top)Levelstructureforthethree-bodydecayof 6 Be[ 67 ].(Bottom)Relative energyplotsintheJacobi Y system(proton-core)fortslicesinthetotalthree-body energy.Ontheleft,theenergyregionisslightlyabovethe2 + stateandthecorrelation indicateathree-bodydecay.Ontheright,theenergyismuchgreaterthantheintermediate stateandcorrelationsindicatingsequentialemissionbegintoemerge[ 43 ] usualbarrierfactors[ 33 ]: w ( )= exp " 2 ˇ ( Z 2) p M p E T 1 p + 1 p 1 # Here E T isthetotalthree-bodyenergy,and and(1- )thefractionofenergygiventoeach 17 protonand = e 2 = ~ .Thisexpressionismaximumwhenthetwoprotonenergiesarethe same( =0 : 5). However,sequentialcorrelationsin 19 Mgwereobservedinthedecayofexcitedstatesonce theywereenergeticallyavailablebecausetheintermediate1 statein 18 Naisnarrowgiving awellprotonenergyandlongerlife-time[ 30 , 29 ].Thetransitionfromthethree- bodytothetwo-bodyregimewasexaminedfor 12 O(democratic),and 19 Mg(democratic g.s.sequentialexcited)withinthecontextofafullthree-bodycalculation[ 2 ].Plottedin Fig. 1.9 arethepredictionsforthe p core energyintheJacobi Y systemfromGrigorenko's three-bodymodelfor 12 Oand 19 Mgwithincreasingenergy E T .Inthecaseof 12 O,which issimilarto 6 Be,eventhoughtheenergyishighenoughtoundergoasequentialdecaya fullthree-bodyprocessisfavouredduetothelargewidthoftheintermediatestate.This isevidencedbythethree-bodybell-curve,whichsimplybecomesnarrower.Incontrast,as theenergyinthe 19 Mgsystemisincreasedlarge\horns"begintoappeararound0and1 indicatingtwodiscreteprotonenergies.Astheenergyincreasesfurther,thehornsbeginto dominatemoreofthespectrum.Theexactlimitsfortransitioningfromathree-bodydecay toasequentialdecayarenotknown.EmpiricalestimatesfromGrigorenko'sthree-body modelgivetherequirement: 0 E T 2 E 2 body + 2 body iswellifthedecayproceedsfromthe7.5MeVstatethroughthe5 = 2 + state.First tentativeevidencethatthisresonancesdecaysbysequentialemissionoftwoneutronswas deducedfromameasurementoftwodiscreteneutronenergiesincoincidencesimilartoa -raycascade[ 63 ].Sincethethree-bodystateisatroughly ˘ 600keVrelativetotheground stateof 22 Oandtheintermediatestatein 23 Olow-lyingandnarrow,thisstateistheoptimum caseforobservingsequentialcorrelations.Thisisthegoalofthecurrentexperiment,asfull three-bodycorrelationsdemonstratingasequentialdecayhavenotyetbeenobservedonthe neutrondrip-line. 20 Figure1.10:Levelstructureofthemostneutronrichboundoxygenisotopes.Hatchedareas indicateapproximatewidths. 21 Chapter2 TheoreticalBackground 2.1OneNeutronDecay Thissectiongivesaderivationoftheenergy-dependentBreit-Wignerlineshapethatisused tomodelanunboundresonanceinthecaseof1ndecayusingthe R matrixformalism. Modelsfortwo-neutrondecaysuselineshapesthatdependonthedecaymechanismandare discussedinlatersections. 2.1.1R-Matrixderiviation One-neutronunboundstatescanbepopulatedinanumberofways.Inthisexperiment unboundstatesin 23 Nand 23 Owerepopulatedviaone-proton/neutronremovalfroman 24 Obeam.Theresultingnucleithenproceededtodecaybyemittinganeutronanda chargedfragment.Thisprocessistakentobeadirectreactionwhichpopulatestheunbound stateandthenpromptlydecays.Thedecayoftheunboundnucleuscanbeinterpretedas aninelasticscatteringprobleminthecontextofR-matrixtheory.However,ratherthan makepredictionsbasedontheHamiltonian,hereweuseR-matrixphenomenlogytoderivea resonancelineshapethatwillbetothedatabasedonapoleenergy e p andwidthamplitude .Onlyasummaryispresentedhere,additionaldetailscanbefoundinThompsonand Nunes[ 75 ]andLaneandThomas[ 76 ] 22 Theproblemwewanttoconsiderinvolvestentranceandexitchannelsaswe wanttodescribetheunboundresonance,notjustelasticscattering.Inthiscontextthis canbeconsideredasinelasticlyscatteringfromchannel toanother 0 .Themethodfor themulti-channelproblemusesasbasisstates,theeigenfunctionsoftherealpartofthe diagonalpotentialineachchannel w ,where denotesthechannel.Itcanbeshownthat thegeneralizationinthiscaseforchannel andpole p ,the R matrixis:[ 75 ] R 0 = P X p =1 0 e p E Where E istheincidentparticleenergy, e p isthepoleenergy(resonance),and arethe reducedwidthamplitudes.Ingeneralthe areconstructedfromanexpansionof w [ 75 ]. Wewishtodescribethecrosssection ˙ ,whichisproportionaltotheabsolutesqureofthe S matrix.Intermsofthe R matrix,the S matrixcanbewrittenas: S =( t 1 = 2 H + ) 1 a R ( H 0 = H ) 1 a R ( H + 0 = H + ) Where H arediagonalmatricieswhoseelementsaretheHankelfunctionswhicharecon- structedfromtheregularCoulombfunctions, H l = G l iF l .Thematrix t isdiagonalwith elements t ~ 2 = 2 and isthelogarithmicderivativeofthe R matrixevalulatedatthe channelradius a ,whichisanarbiratrypointpastwhichonlylong-rangeinteractions playarole. isthereducedmass.Itisusefultodea\logarithmic"matrix L : L = H + 0 = H + = 1 a ( S + iP ) Wherewehaveintroducedthepenetrability P andshiftfunction S ,whicharediagonal 23 matriceswithelements: P = k a F 2 + G 2 (2.1) S =( _ F F + _ G G ) P (2.2) Thedotdenotesaderivativewithrespectto ˆ = kR ,where k isthequantummechanical wavenumber,and R theradialcoordinate.Since H 0 = H = L ,thescatteringmatrix S canbeputinthefollowingform: S = ( tH H + ) 1 = 2 1 a RL 1 a RL ( tH H + ) 1 = 2 Wherewehaveintroducedyetanotherdiagonalmatrixwithelements = e i˚ with ˚ beinghard-spherephaseshifts.Thematrixproduct tH H + isdiagonalwithelements: H H + t = ~ v a 2 P Where v = ~ k isthechannelvelocity. Thecrosssectionisproportionaltotheabsolutesquareofthesymmetricmatrix e S ,which isconstructedfrom S viaasimilaritytransformation e S v 1 = 2 Sv 1 = 2 andcanbewritten as: e S = [1+2 iP 1 = 2 (1 a RL ) 1 R P 1 = 2 ] Whichispowerful,aswenowhaveanexpressionforthe S matrixfromthe R matrixin 24 termsofthepenetrabilitiesandshiftfunctions(Eqs. 2.1 and 2.2 ).Thisexpressioncanbe greatlybymakingseveralassumptions.Firstsupposethatthereareonly two channels,theelasticandinelasticresonancewherethemassispartitonedtly.We alsoassumethatthereisonlyoneenergylevelintheunboundnucleus,i.e.asinglepole e p . Notethat ~ S 12 = ~ S 21 ,weobtaininthiscase: ~ S 12 = e i˚ 1 2 4 2 iP 1 = 2 0 P 1 = 2 0 ( e p E )(1 aR 11 L 1 aR 22 L 2 ) 3 5 e i˚ 2 = e i˚ 1 2 4 2 iP 1 = 2 0 P 1 = 2 0 e p E 2 1 ( S 1 ) 2 1 P 1 2 2 ( S 2 ) 2 2 P 2 3 5 e i˚ 2 (2.3) theformalwidth : =2 2 P (2.4) Aswellas S 0 ,theenergyshift ,andtotalenergyshift T inadditiontothetotalformal width T : S 0 = S = 2 S 0 T = X = 2 1 S 0 1 2 2 S 0 2 T = X =2 2 1 P 1 +2 2 2 P 2 Thevalueof canbesettoanyconstant.LaneandThomas[ 76 ]suggestusing = S ( e 0 ).SubstitutionoftheseintoEq. 2.3 reducestheformof e S 12 considerably. 25 ItalreadybeginstotakeontheshapeofaBreit-Wignerdistribution: ~ S 12 = e i˚ 1 2 4 i 1 = 2 1 1 = 2 2 ( e p E + T )+ i T = 2 3 5 e i˚ 2 Recallthatthecross-sectionisrelatedtothesymmetric S matrixbythefollowingrelation: ˙ 0 ( E )= ˇ k 2 i gJ tot j e S 0 j 2 Where k i isthewavenumberoftheentrancechannelandthespinweightingfactor gJ tot is: gJ tot 2 J tot +1 (2 I p i +1)(2 I t i +1) J i isthetotalspinofthepopulatedstateand I p i , I t i arethespinsoftheprojectileand target-likefragmentsrespectively.Substitutionof e S 12 intothisexpressionyields: ˙ 12 = ˇ k i gJ tot 1 2 ( E e p + T ) 2 + 2 T = 4 (2.5) Weareonlyinterestedindescribingthedecayofanunboundstate.Itisassumedthat thepopulationmechanismisunimportant.Eq. 2.5 canbefactoredintotwoexpressions: ˙ 12 = ˇ k i gJ tot 1 2 ( E e p + T ) 2 + 2 T = 4 ! Thedescribesthepopulationoftheunboundstatewhichweareunconcernedwith.We onlywishtoknowthe E dependenceof ˙ fortheunboundstate,andsothistermistreated asaconstant.Inaddition,theprobabilitytodecaythroughtheentrancechannelissmall, 26 2 >> 1 ,thus T ˘ 2 and T ˘ 2 .Thelineshapefortheneutrondecayreducesto: ˙ l ( E ; e p ; 0 ) / l ( E ; e p ; 0 ) e p E + l ( E ; e p ; 0 ) 2 + 1 4 l ( E ; e p ; 0 ) 2 (2.6) Herewehavedroppedthesubscriptsforthechannelsandexplicitlystatetheangularmo- mentumdependences. 0 ,thewidthofthedecayat e p ,isusedasasubstituteforthepartial width 2 (Eq. 2.4 ): 0 =2 2 P l ( e p ) Equation 2.6 isusedtotheprobabilitydistributionforthedecayenergy E decay in simulation. 2.2TwoNeutronDecay Intwo-bodykinematics,themassesoftheparticlesandthedecayenergycompletelydeter- minethesystembyconservationofenergyandmomentum.Theadditionofathirdparticle increasesthenumberofdegreesoffreedomandaddscomplexity.Theenergiesarenolonger monochromatic,but E and P conservationstillplacekinematicboundariesonwhatmo- mentaarepossible,andallthreeparticlesareemittedinthesameplane.Inmodellingthe two-neutrondecayofanunboundstatethreesimplemodelsareused:(1)Aphasespacede- cay,(2)adi-neutronmodel,and(3)asequentialdecay.Thechoiceofmodeldeterminesthe energyofeachneutronaswellaswhetherthedecayproceedsasatruethree-bodybreakup, ormultipletwo-bodyprocesses. 27 2.2.1Phasespacedecay Thephasespacemodelassumesnocorrelationsbetweentheoutgoingparticlesbyuniformly samplingthephasespaceoftheinvariantmasspairs m 2 12 and m 2 23 whileapplyingkine- maticconstraintstoconserveenergyandmomentum.Thismodelisusedasabaselinefor simulatingthedetectorresponseforobservingnothree-bodycorrelations. Figure2.1:(Left)Schematicforphase-spacebreakup.(Right)Ahypotheticallevelscheme whereonewouldexpecttoobserveaphase-spacedecay. E I denotesthethree-bodyenergy, E V theintermediatestate,and E F =0,thestate.Thehatchedareasindicatewidths. Notethattheintermediatestateisbroad. Considerthedecayofaparticleofmass M andmomentum P intothreeproductsdenoted by m i , p i ,andenergy E i asillustratedinFig. 2.1 .Take c =1,and p ij = p i + p j ,and m 2 ij = p 2 ij .Thenweobtaintherelations: m 2 12 + m 2 23 + m 2 13 = M 2 + m 2 1 + m 2 2 + m 2 3 And m 2 12 =( P p 3 ) 2 = M 2 2 ME 3 + m 2 3 Where E 3 istheenergyofthethirdparticleintherestframeof M .Inthisframeall particlesliewithinaplaneandtheirrelativeorientationcanbeiftheirenergiesare 28 known.thequantities: E 2 = ( m 2 12 m 2 1 + m 2 2 ) 2 m 12 E 3 = ( M 2 m 2 12 m 2 3 ) 2 m 12 Foragivenvalueof m 2 12 , m 2 23 ismaximumorminimumwhen p 2 isparalleloranti-parallel to p 3 .Setting ~p 2 = ~p 3 weobtainthelimits: ( m 2 23 ) max =( E 2 + E 3 ) 2 q E 2 2 m 2 2 q E 2 3 m 2 3 2 ( m 2 23 ) min =( E 2 + E 3 ) 2 q E 2 2 m 2 2 + q E 2 3 m 2 3 2 Thephase-spacemechanismcanthenbesimulatedbyuniformlysampling m 2 12 and m 2 23 undertheconstraintthat( m 2 23 ) min 0,thenthe levelstructureislikethatillustratedinFig. 2.3 .For S< 0,theintermediatestateisnot 31 classicallyforbiddenandwehaveaheirarchylikethatillustratedFig. 2.2 .Eq. 2.7 cannow berewrittenas: A T ( 1 ; 2 )= 1 p 2 A 1 ( 1 ) A 2 ( 2 )[ S + 2 i 2 2 ( 1 )]+ A 1 ( 1 ) A 2 ( 2 )[ S + 1 i 2 2 ( 2 )] [ S + 1 i 2 2 ( 2 )][ S + 2 i 2 2 ( 1 )] (2.8) Thesingleparticledecayamplitudesarerelatedtotheirwidthsbythefollowingrelation: i =2 ˇ j A i ( ) j 2 Whichcanbeequatedwiththesingle-particledecaywidth l ( )multipliedbyaspectroscopic factor S i : i =2 ˇ j A i ( ) j 2 = l ( ) S i Thesingle-particlewidthcanbeestimatedforaneutroninasquarewellusingtheexpression derrivedinBohr-MottlesonVol.I[ 79 ]: l = 2 ~ 2 ( kR ) 2 l 1 2 l +1 T l ( kR ) Here T l ( x )isthetranmissionprobabilitythroughthecentrifugalbarrier.For s and p waves, T 0 =1and T 1 = x 2 = (1+ x 2 ). R ˘ 1 : 3( A +1) 1 = 3 isthenuclearradiusinfm,and k = p 2 . isthereducedmassand theincidentneutronenergy.Togettoadecayrateandacross section,weneedtoutilizetheFermiGoldenRulewhichgivesthepartialdecaywidthas: d E ) 1 2 =2 ˇ ( E 1 2 ) j A T ( 1 ; 2 ) j 2 (2.9) 32 Puttingthedecayamplitudesintermsofthesingle-particlewidthsandapplyingFermi's Rule 2.9 ,weobtainthefollowingexpressionfortheerentialwidthintermsoftherelative energy E r = 1 2 : d E ) dE r = 1 8 ˇ l ( 1 ) l ( 2 ) " E +2 S i 2 2 ( 1 )+ 2 ( 2 )] [ S + 1 i 2 2 ( 2 )][ S + 2 i 2 2 ( 1 )] # 2 Thecross-sectionthenfollowsthefamiliarBreit-Wignerformwiththetialwritten intermsoftherelativeenergy: d˙ dE r / 1 ( E E 1 ) 2 + 2 T ( E ) = 4 d E ) dE r Wherethetotalwidth T ( E )isobtainedfrom: T = Z dE r d E ) dE r Inthismanner,therelativeenergydistributionsforthetwoneutronsarecalculated dependingontheenergyandwidthsofthestatesinvolved.Oncethedistributionforeach neutroniscalculated,theprocessistreatedastwotwo-bodydecays.Incaseswhere S> 0, theintermediatestateishigherinenergythanthethree-bodystateandthedecayproceeds throughit'swidth. Itshouldbenoted,thatthisformalismassumesthatthetwoneutronscomefromthe samesingle-particleorbitalandarecoupledto J ˇ =0 + .Inaddition,althoughtheBreit- Wignerlineshapefortherelativeenergiesdependsonthe ` valueofthethree-bodyand intermediatestate,theangulardistributionsoftheneutronsareassummedtobeisotropic intherestframeofeachtwo-bodydecay. 33 2.2.3Di-neutrondecay Thedi-neutronmodelusedinthisanalysisisalsobasedontheworkofA.Volya[ 77 , 78 ]. Inthismodelthedi-neutronclusterbreaksawayfromthecorebeforedecayingintotwo separateneutrons. Figure2.3:(Left)Schematicfordineutronemission.(Right)Ahypotheticallevelscheme whereonewouldexpecttoobserveadineutron.Thehatchedareasindicatewidths. Letthedineutronhavemass m D =2 m ,andreducedmass = m= 2. K asthe kineticenergyofthedineutron.Let I betheintrinsicenergyofthedineutron-orneutron- neutronenergy,and E 2 betheenergyofthe\intermediatestate"whichmaybeclassically forbidden.Thetotalthree-bodyenergyisdenotedby E 1 = K + I .Forsimplicitytake E 3 =0.Thedecayistreatedasatwo-stepprocesswherethedineutronseparatesfrom thecoreandthendecaysintotwoneutrons.Theamplitudeforthedecayisgivenby: A T ( K ; I )= A 1 ( K ) A 2 ( I ) I ( E 2 i 2 2 ( I )) Where, A 1 istheamplitudeforthedi-neutronemissionand A 2 istheamplitudeforthe di-neutronbreakup.SubstitutingthisexpressionintoFermi'sGoldenRule 2.9 ,weobtain thefollowingforthetialwidth: d K I = 1 2 ˇ ( E 1 K I ) 1 ( K 2 ( I ) [( I E 2 ) 2 + 2 2 ( I ) = 4] (2.10) 34 Next,itisassumedthatboththeemissionofthedineutron,andthesubsequentbreakup, canbeparameterizedasans-wavedecay.Inthiscase,thedecaywidthfor 2 ( I )becomes: 2 ( I )= 2 ~ 2 0 k I Where isthereducedmassofthedineutronand r 0 isthechannelradius,approximated as r 0 =1 : 4(2 1 = 3 ) ˘ 1 : 7fm.Likewise,thedecaywidthfor 1 ( K )becomes: 1 ( K )= 2 ~ 2 m D R k K Here m D isthemassofthedineutron,andthechannelradius R includesthesizeofthe core, R =1 : 4[( A 2) 1 = 3 +2 1 = 3 ].Todeterminetheintermediateenergy E 2 ,orvirtualstate, werequireconsistencywiththeerangeapproximation.Recallthatinthes-wave decaythedecaywidthisproportionalto p I .As I ! 0thedenominatorofthescattering amplitude A T mustbehavelike1 = a s + ik I where a s isthe n n scatteringlength,[ 78 ](Note: ~ 2 0 = 2 = 2 k I )thus: lim I ! 0 I E 2 i 2 2 ( I ) = ~ 2 0 ( 1 a s + ik I ) E 2 + i 2 2 ( I )= 2 ( I ) 2 k I 1 a s + ˆ ˆ ik I E 2 = 2 ( I ) 2 k I a s (2.11) Substituting 2 ( I )= 2 ~ 2 0 k I ,itfollowsthat: E 2 = ~ 2 0 a s 35 Foragivenscatteringlength,wecanassociateanenergy: 0 = ~ 2 2 2 s Sowemaywrite: E 2 = 0 2 a s r 0 Recallthat I and K areasthefollowing: K = ~ 2 k 2 K 2 m D I = ~ 2 k 2 I 2 Andwemayexpressthewidthsas: 1 ( K )=2 j a s j R p 0 K 2 ( I )=4 j as j r 0 p 0 I Settingthemassesto = m= 2and m D =2 m ,substitutingthewidthsintoEq. 2.10 and integratingover K weobtain: d E 1 ) 1 = 1 ˇ p ( E 1 I ) I 1+ r 0 2 a s I 0 2 0 + I r 0 R (2.12) Whichthedistributionfortheintrinsicenergyofthedineutron.Wecanmakean approximationif I << j 2 a s 0 =r 0 j .Then-nscatteringlengthis a s = 18 : 7fm[ 80 ].Using r 0 ˘ 1 : 7fm,and 0 ˘ 0 : 15MeVgivesus I << 3MeV.Inthisregime,wecanapproximate 36 theintegralofEq. 2.12 withtheexpression: E 1 )= r 0 R E 1 2 + 0 p 0 ( 0 + E 1 ) However,integrationofEq. 2.12 isstillperformednumericallyinsimulationtodetermine theenergydistributionsofthedineutronanditsintrinsicenergy. 37 Chapter3 ExperimentalTechnique 3.1ExperimentalSetup ThissectiongivesanoverviewoftheexperimentalsetupattheNationalSuperconducting CyclotronLaboratory(NSCL)usingtheModularNeutronArray(MoNA),theLarge-area multi-InstitutionalScintillatorArray(LISA),andtheSweepermagnet.Theoverallsetupis describedherewhilethedetailsofeachdetectorarediscussedinthefollowingsections. TheexperimentwasperformedattheNSCL,wherea140MeV/u 48 Cabeamimpinged upona1363mg/cm 29 Betargettoproducean 24 Obeamat83.3MeV/uwithapurityof ˘ 30%.TheA1900fragmentseparatorwasusedtoselectthe 24 Obeamfromtheother fragmentsinthesecondarybeam.The 24 Obeamcontinuedontotheexperimentalarea whereitimpingedupontheUrsinusCollegeLiquidHydrogenTarget[ 81 ],whichwas withliquiddeuterium(LD 2 ).Theaveragebeamratewasapproximately30particlesper second. Afterthebeamreactedinthetargettheresultingchargedfragmentswereswept43 : 3 by a4-Tmsuperconductingsweepermagnet[ 82 ]intoaseriesofpositionandenergy-sensitive chargedparticledetectors.Insidethesweeperfocalplaneweretwocathode-readoutdrift chambers(CRDCs),separatedby1.55m,anion-chamber,athintimingscintillator,andan arrayofCsI(Na)crystalscalledthehodoscope.Fig. 3.1 showsadiagramoftheexperimental setup. 38 Figure3.1:LayoutofthedetectorsintheN2Vault. Theneutronsproducedfromthedecayofunboundstatestraveledundisturbed8mto- wardsMoNAandLISA[ 83 ].MoNAandLISAeachcontain9verticallayerswith16bars perlayerandthecombinedarraywasintothreeblocksofdetectorbars.LISA wassplitintotwotables4and5layersthick,withthe4layertableplacedat0 infront ofMoNA,whiletheremainingportionofLISAwasplacedcenteredat22 .The resultingangularcoverageinthelabfromewasfrom0 10 forthedetectorsplaced at0 ,and15 32 fortheportion. Together,MoNA-LISAandthesweepermagnetprovideakinematicallycompletemea- surementoftheneutronsandthechargedparticlesfromwhichthedecaysofunboundstates canbereconstructed.Withthefull4-vectorinformationofeachparticle,three-bodycorre- lationscanalsobeexaminedincaseswherenucleidecaybyemissionoftwo-neutrons. 39 3.2BeamProduction(K500,K1200) An 24 ObeamwasprovidedbytheCoupledCyclotronFacility(CCF)[ 84 ]andA1900Frag- mentSeparatorattheNSCL[ 85 ].Thefacilityprovidesintenseheavyionbeams,bothstable andunstableviafastfragmentation[ 86 ].Since 24 Ohasahalf-lifeof64ms[ 87 ]itcannotbe accelerateddirectly.AdiagramofthebeamproductionprocessisshowninFigure 3.2 . Figure3.2:BeamproductionattheCoupledCyclotronFacility[ 84 ]. 48 Caisheatedupin anion-sourceandacceleratedbytheK500andK1200cyclotrons.AfterimpingingonaBe target,thefragmentsarebytheA1900toprovidethedesiredbeam. Abeamofstable 48 Cawasacceleratedto140MeV/uinthecoupledK500and K1200cyclotrons.AfteremergingfromtheK1200,thebeamimpingedupona1363mg/cm 2 9 Betargetwherethefragmentationprocessoccured.Awidevarietyofnucleiaremade simultaneouslybythisprocess.Inordertoisolatethe 24 ObeamtheA1900separatorwas used.TheA1900hasfourdipoleswithfocusingelementsinbetween,tolterthefragmenta- tionproductsbytheirrigidty Bˆ = p=q toselectaspmomentumtochargeratio.An achromaticaluminumwedgewiththickness1050mg/cm 2 wasplacedinbetweenthesecond andthirddipolestoimproveseparation.Theenergylossthroughthewedgeisproportional tosquareofthenuclearcharge, Z 2 .Hencetelementswiththesamerigidityentering thewedgewillhavetrigiditiesuponexit.The 24 Owasdeliveredtotheexperimental 40 vaultwithanenergyof83 : 4MeV/ucorrespondingtoarigidityof4.0344Tmatarateof ˘ 0 : 6pps/pnA.Thelargestcontaminantwas 27 Newhicharrivedwiththesamerigidity atanenergyof122 : 4MeV/u.Inadditiontothe 24 Obeam,an 20 Obeamofmuchhigher intensity( ˘ 700pps/pnA)wasalsoprovidedforcalibrationanddiagnostics. 3.3A1900&TargetScintillator AttheendoftheA1900isaplastictimingscintillatorlocated10.579mupstreamfromthe liquiddeuterium(LD 2 )targetintheN2vault.Itismadeof0.125mmthickBC-404andis opticallycoupledtoaPMT.Thetargetscintillator,alsomadeofBC-404,is0.254mmthick andwasplaced1.0492mupstreamfromtheLD 2 targetandwascoupledtoaPMT.When aparticlepassesthroughtheplastic,itcreateselectron-holepairsthatrecombineemitting photons.ThesephotonsarethencollectedinthePMTsandconvertedtoanelectronicsignal. Thesignalsfromthesedetectorscanbeusedtohelpseparatebeamcontaminantsbytime- tmeasurement.Inadditiontothetimingscintillators,thecyclotronradio-frequency (RF)isalsorecordedtoprovideadditionalseparation. 3.4LiquidDeuterium(LD 2 )Target Thisexperimentoptedtouseacryogenicdeuteriumtargetoverdeuteratedplasticssuchas CD 2 fortwomainreasons:(1)areducedcarbonbackground,and(2)itprovideshigher densityofD 2 forthesameoveralltargetthickness.TheUrsinus-NSCLLiquidHydro- gen/DeuteriumTarget(LD 2 )ahigh-density,low-backgrounddeuterontargetforava- rietyofexperimentsincludingelasticscattering,secondaryfragmentation,chargeexchange, andnucleontransfer.ThebasicprincipleoftheLD 2 targetissimple:avolumewithD 2 41 gasandcoolitsotheliquidcollectsatthebottomofatargetcell.TheLD 2 targetconsists ofemaincomponents,describedinthefollowingsub-sections: 1.) TargetCell:Holdstheliquiddeuterium.Cylindricalinshape. 2.) RefrigeratorSystem:ASumitomo205DCryocooler. 3.) VacuumChamber:Housesthetargetcell,coldrefrigeratorandheatshield. 4.) TemperatureControlSystem:Monitorsthetemperatureofthetargetcellandregulates thetemperatureoftheheaterblock. 5.) Gashandlingsystem:Regulatesthewofneon,hydrogenordeuteriumtotherefrig- erator. Figure3.3:SchematicoftheUrsinusCollegeLiquidHydrogenTarget.(Left)Aheadon viewalongthebeamaxis.(Right)Sideview. 42 GasBoilingPointTriplePoint Hydrgoen(20.35K,760Torr)(13.85K,54Torr) Deuterium(23.50K,760Torr)(18.70K,128Torr) Neon(27.07K,760Torr)(24.56K,323Torr) Table3.1:Boilingandtriplepointsofhydrogen,deuterium,andneon. Includingtheresevoirinthegashandlingsystem,atotalof100Lofhydrogenordeu- teriumatSTPiscontainedintheentireapparatus.Thetargetoperatesbyingthetarget cellwithanappropriateamountofgasandcoolingittonearthetriplepoint,wherethegas condensestotheliquidstate.Table 3.1 documentsthetriplepointsforneon,hydrogen,and deuterium.Fordeuterium,thetriplepointisatapproximately18.7Kand128Torr.During operation,thetargetwasheldatroughly20Kand ˘ 850torr.Thisliesinthewindow betweenthesolid-liquid,andliquid-gasphasetransitionsofD 2 asseeninFig. 3.4 . Figure3.4:Phasediagramfordeuterium.Thesolidlinemarkstheliquid/solidtransition, whilethebluedottedlinedenotestheliquid/gastransition.Theplusmarksaredataobtained duringtheinitialofthetarget. AsseenbytheschematicofthetargetinFig 3.3 ,thedeuteriumtheentirevolumeof thegaslinewhichisconnectedtothetargetcellrestinginthebeampath.Thecryocooler, whichsitsontop,coolsthedeuteriumgasuntilitcondensesanddripsdownthegasline 43 intothetargetcell.However,sincethecoolerisoperatedbyaclosedliquidHecycle,it willcooltheD 2 to4K,whichwouldcausetheliquidtofreezeandclogthegasline.For thisreasonthetargetcelliscoupledtoaheaterblockwhichcounter-actsthecryocoolerto maintainthetemperatureoftheliquidatadesiredpoint.Thetemperatureisreadoutby twosilicondiodes,oneattachedtothetargetcell,andanotherattachedtotheheaterblock. Thepressureisreadoutbyamonometerinthegashandlingsystem,whichisconnectedto thetargetcellline. 3.4.1TargetCell Thetargetcelliscylindricalwithadiameterof5.4cmandalengthofabout3.0cm,and consistsofanaluminumframewithKaptonwindows125 mthickonbothends.Itis designedtoholdtheliquiddeuteriuminplace.Forthecellusedinthisexperiment,the designthicknessfordeuteriumwas400mg/cm 2 ,or200mg/cm 2 forhydrogen.Thecell couplestoacommercialrefrigerator,andhasadiodeattachedtotheframeforreadingout thetemperature.AphotoofthecellalongsideamechanicaldrawingcanbefoundinFig. 3.5 Thecellisdesignedtowithstandanoutwardpressuregradientofabout2atmasitsitsin avacuum.Whenthetargetcellis,thekaptonfoilsbulgeoutwardincreasingtheactual thicknessofthedeuterium.Thecellissurroundedbyaheatshieldtoreducethethermal loadandenablecoolingfromroomtemperaturetothetriplepoint.Toensuretemperature stabilityduringtheexperiment,theopeningsintheheatshieldwerecoveredwith5 mthick aluminizedemylar 44 Figure3.5:(Left)PhotoofthetargetcellusedtocontaintheLiquidDeuterium.Theliquid dripsdownthroughthecenterholeseenonthetargetTheiridiumsealcanbeseen surroundingitbeforebeingpressed.(Right)DrawingoftheinnerringwheretheKapton foilisglued.Theringisthenclampedtothetargetbyanouterringvisibleintheleftphoto. 3.4.2TemperatureControlSystem Thetemperatureofthetargetcellisregulatedbytheheaterblock.Sincethetargetcell sitsinavacuumduringoperation,heatcanonlybetransferedbythermalradiationor conduction.TheheaterblockiscontrolledbyaLakeshore331Stemperaturecontrollerunit whichisoperatatedmanuallybyaPCrunningLabView.TheLakeshore331Scontroller reportsthediodetemperatureasavoltagewhichisthenreadoutbytheExperimental PhysicsandIndustrialControlSystem(EPICS)[ 88 ]andlaterconvertedtoKelvin.Details oftheLabViewprogrammingandoperationofthetemperaturecontrolsystemcanbefound intheLiquidDeuteriumTargetusersmanual[ 89 ]. Thetemperaturecontrollerunitreliesonafeedbacklooptomaintainadesiredequillib- rium.Becausetheheaterblockisnotindirectcontactwiththetargetcell,thereisthermal lagbetweenitandthetargetcell.Thetimeittakesfortheheaterblocktothecell istoolongtousethediodeatthetargetcellinafeedbackloop.Insteadthetemperature ofthecellmustbemonitored,andcontrolledfromaseparatediodeattachedtotheheater block.Thisoftenmeansthatthe\setpoint",ordesiredtemperature,oftheheaterblock 45 istfromthedesiredtemperatureinthecell.However,duringthecell temperatureiscloselymonitoredtodetermineanappropriatesetpoint. 3.4.3GasHandlingSystem Thegashandlingsystemcontrolsthewofair,deuteriumanddrynitrogenwhenevacuating andthetarget.Itisdesignedwithsafetyasapriority,anditspurposeistoregulate thewofthesegasestoinsurethatthemixtureofdeuteriumandnitrogen/airstayswell belowtheylimit,andthatthereisneveranover-pressureonthetargetcellwhich maycauseittoimplode.AschematicofthegashandlingsystemisshowninFig. 3.6 .Table 3.2 describeseachofthecomponents.ThefulloperationofthissystemandtheLD 2 targetis describtedintheLHDTUsersManual[ 89 ].Thetargetpressureisreadoutbyamanometer attachedtothetargetcellline.Thismanometerreportsavoltagewhichisthenreadoutby EPICS. 3.4.4VacuumChamber Thevacuumchamberholdsthetargetassemblyinthebeamlineandinterfaceswiththe existingbeam-linestructure.Itwasconnecteddirectlytotheofthesweepermagnet, andtothetarget-scintillator.Duetotheheightofthetarget,andthegeometryofthe sweepermagnet,theentiretargethadtooperatedata30 angle.Thisresultedinonlya portionofthetargetcellbeingHoweverthetargetitselfwasmuchlargerthanthe beam-spot,ensuringthatthetilthadno Thebellowsbeneaththecryocooler,illustratedinFig. 3.3 ,allowforadjustingthetarget cellpositioninthebeamline.Usingalaseralignment,thecellwascenteredonthebeam- 46 ComponentFunction TargetCellCellcontainingLH 2 orLD 2 . Reservoir100LtankwhichservesasH 2 orD 2 reservoir. H 2 PressurizedgasbottlewithH 2 orD 2 . T(B141TP)/B141GVTurbo-pump/Turbogatevalve. PPascalC2series2015C2rotarypump. V 1 ; 3 ; 4 ; 5 ; 10 Manualvalves. V 2 Needlevalve. V 6 ; 7 ManualneedlevalvesonwmeterF 1 andF 2 . V 8 (B141FV)Forevalve. V 11 (B141VV)Ventingvalve. F 1 FlowmeterforH 2 /D 2 gas(alwaysopen). F 2 Flowmeterfordrynitrogen(alwaysopen). R 1 Regulatorforgasbottle. M 1 ; 3 Manometerswitharangeof1-5000Torr. M 1 readsthecellpressure,andM 3 thepressureinthereservoir. M 2 Manometerforthegascellwitharangeof0.001-1Torr. M 4 (B142PG)Piranigaugeandiongaugeforbeam-chambervacuum. N2ExhaustLineExhaustlinefordischargingtheH 2 /D 2 gas. DryNitrogen(DN)DryNitrogenSupply. DryNitrogen/VentVentlineforthetargetsystemwithdrynitrogen. Table3.2:Listofcomponentsinthegashandlingsystem.TableadoptedfromRef.[ 89 ]. 47 Figure3.6:Adiagramofthegashandlingsystemusedtocontrolthewofdeuteriumin andoutofthetargetcell.Figureadoptedfrom[ 89 ] axisandthefoilswerealignedperpendiculartothebeambylininguptheofthe laserthebackofthesecondfoilwiththeimpingingbeam-spot. 3.5SweeperMagnet Thesweepermagnetisalarge-gapsuperconductingdipolemagnetwithabendingangleof 43.3 andaradiusof1meter[ 82 ].It'spurposeistosweepchargedparticlesandunreacted beamawayfromMoNA-LISAandtowardsasuiteofcharged-particledetectorsdescribedin thefollowingsections.Thechargedreactionproductsexitingthetargetaresweptawaywhile 48 theneutronscontinuestraightthroughaverticalgapof14cmtowardsMoNA-LISA.The maximumrigidityofthesweeperis4Tm.Themagneticinthesweeperismonitered withaHallprobeandthedipolehasbeenmappedinpreviouswork[ 90 ].Forreactions withthe 24 Obeam,themagnetwassettoacurrentof320A,correspondingtoacentral rigidityof3.524Tm.Tomeasurethebackgroundinducedbythekaptonfoilsofthetarget, aswellasdiagnosethesweeper,theLD 2 targetwasputinagaseousstatebywarmingitto 50K.Thisreducesthethicknessofthedeuteriumto ˘ 1mg/cm 2 .Forthesettingswiththe warmgaseoustarget,thecurrentwasraisedto360Atoaccountforthemorerigidbeam. Thecentralrigidityonthosesettingwas3.85Tm. 3.6ChargedParticleDetectors Immediatelyfollowingthesweepermagnetwasacollectionofchargedparticledetectors residinginavacuumbox.Thepositionofthereactionproductsbythemagnet wasmeasuredwithtwoCathodeReadoutDriftChambers(CRDCs)separatedby1.55m. FollowingtheCRDCswasanion-chamberwhichprovidedameasurementofenergyloss, andathin(5mm) dE plasticscintillatorwhichtriggeredthesystemreadoutandgavea tmeasurement.Finally,anarrayofCsI(Na)detectorsconsistingof25crystals, each80x80x25mm 3 wasinstalledbehindthethinscintillatorandstoppedthefragments whilemeasuringtheirtotalenergy. 3.6.1CRDCs TheCRDCwasplacedapproximately1.56mfromthetarget,wherethedistanceis measuredalongthecentraltrackofthedipole,andthesecondCRDCwasplaced1.55m 49 downstreamoftheTheCRDCsmeasuretheXandYposition,fromwhichtheangleof theparticlescanbedeterminedandtheirpathtracedbackwardtothetarget.Aschematic ofthedetectorcanbefoundinFigure 3.7 . Figure3.7:SchematicofaCathodeReadoutDriftChamber(CRDC)wherethe z -direction hasbeenexpanded.Theshapingwiresarenotdrawnforvisibility.Notethatthe electronavalanceisdoesnotbeginuntiltheelectronsencountertheFrischgrid. TheCRDCissimilartoanion-chamber.Itiswitha1:4mixtureofisobutane andCF 4 gasatapressureof40torrandsealedwithtwowindows.Whenaparticlepasses throughthegasitcreatesionizationpairswhichdriftapartduetoauniformappliedelectric Thisiscreatedbytheapplicationofadriftvoltageof1000Vbetweenaplateat thetopofthedetector,andaFrishgridnearthebottomofthedetector.Fieldshapingwires areplacedatspintervalsalongeachfaceofthedetectorparrallelwiththeXdirection. 50 UndertheFrischgridisananodewire(parallelwiththeXdirection)andacollectionof116 aluminumcathodepadswithapitchof2.54mminwidthsegmentedalongtheXdirection. WhenelectronsdriftintotheFrischgridtheyenterastrongelectriccreatedbythe anodewirecausinganavalancheofelectrons.TheY-positionoftheinteractionisdetermined bythedrifttimeoftheelectron,whichisasthebetweenasignalinthe thintimingscintillatorandasignalontheanodewire.TheX-positionisdetermedfromthe inducedchargedistributiononthecathodepadcausedbyanavalancheofelectrons.The peakofthisdistributiongivestheX-positionoftheparticle.TheZ-positionisassumedto bethecenterofthedetectingvolumealongthebeamaxissincethereisnosegmentationin thisdirection.TheactiveareaofeachCRDCis30x30cm 2 intheXYplane. 3.6.2IonChamber Theion-chamberisaeddetectorsimilartotheCRDCsbutsegmentedintheZ- directionwith16pads.Theactivevolumeofthedetectoris40x40x65cm 3 .Theion- chamberiswithP-10gas(10%CH4and90%Ar 2 )andheldat300torr.Thewindows aremadeofKevlarentand12 mPPTAandaremountedwithepoxy.Theyallow particlestopassthroughwithneglibleenergyloss.Theupstreamwindowhasanactivearea of30x30cm 2 tomatchtheacceptanceoftheCRDC,whilethedownstreamwindowis40 x40cm 2 toallowfordispersionofthebeam. Theion-chamberhasaplateontopand16chargecollectingpadsonthebottombiased tocreateadriftvoltageof800V.Whenachargedparticleentersthegaseousvolumeofthe detector,ionizationpairsarecreatedandtheelectronsarecollectedonthe16pads.The energylossinthegascanbecalculatedfromthetotalchargecollectedonall16pads. 51 Figure3.8:(Left)Head-onview(lookingintothebeam)ofthethintimingscintillator. (Right)Anexamplelevelschemefortransitionsintheorganicmaterialwitha ˇ -electron structure.Source:[ 91 ] 3.6.3TimingScintillators Thethintimingdetectorrestsinbetweentheion-chamberandthehodoscope.Itisaplastic scintillator(EJ-204)withdimensionsof55cmx55cmx5mmandhaspairsofphotomulti- pliertubesattachedvialightguidesonthetopandbottomofthedetector.Itmeasuresthe tandtriggersthedataaquisitionsystem.Aschematicofthedetectorisshown inFig. 3.8 .ThelightguidesaretrapezoidalandopticallyconnectedtothePMTs. Whenachargedparticlepassesthroughtheorganicscintillatoritdepositsenergyinto thematerial.Asmallportionofthiskineticenergyisconvertedintotlight,while themajorityisdissapatednon-radiativelythroughlatticevibrationsorheat.The cencearisesfromtransitionsintheenergylevelstructureofaspmolecule,inthiscase Polyvinyl-toluenedopedwithantracene.Plasticscintillatorssuchasthesetakeadvantage ofthe ˇ -electronstructureofthesemolecules.Thetypicalspacingofvibrationalstatesin organicscintillatorsisontheorderof0.15eV,whichismuchgreaterthantheaveragether- 52 malenergyatroomtemperature kT =0 : 025eV,meaningnearlyallthemoleculesareinthe groundstate.ThetypicalspacingofthesingletstatesisontheorderofafeweV.Fig. 3.8 showsalevelschemeforanorganicmoleculewitha ˇ -electronstructure.Whenacharged particlepassesby,itexcitesthemoleculetooneofthesingletstates.Theprinciplesource ofrescencecomesfromthede-excitationofthesingletstate S 1 byinternal conversiontooneofthevibrationalstatesofthegroundstate S 0 x .Anystatewithexcess vibrationalenergyquicklylosesthatenergyasitisnolongerinequillibriumwithitsneigh- bors.Thisdecayiscalledpromptandistypicallyontheorderofnanoseconds. ForEJ-204,thedecayconstantis1.8ns. Itisalsopossibletodecaytoatripletstatewhichcanbelongerlived,withlifetimes upto10 3 s,causingadelayedemissionoflightwithalonger-wavelengthsincethetriplet stateisalowerenergythanthesinglet.This,alongwithmultipleavailableexcitedstates, causesaspectrumoflighttobeemitted.TheemissionspectrumforEJ-204isshowninFig. 3.9 ,andpeaksaround410nm[ 92 ].Thelightemittedinthede-excitationoftheorganic moleculesscatterswithintheplasticuntilitiscollectedinthePMTs[ 91 ]. Figure3.9:EmissionspectrumforEJ-204.Thepeakwavelengthisaround410nm[ 92 ]. 53 3.6.4Hodoscope Thelastdetectorinthesweeperfocalplaneisthehodoscope.Thehodoscopeisanarray ofCsI(Na)detectorsconsistingof253.25"x3.25"x2.16"crystalsorientedina5x5 square.AmechanicaldesignofthedetectorisshowninFigure 3.10 .Thearrayisassembled with5rowswhicheachcontain5crystalsandiscenteredonthecentraltrackthroughthe sweeper.Eachcrystaliswrappedinematerial0.2mmthick,andopticallycoupled toaHamamatsuPMTR1307withmagneticshielding.Thereisnolightguidebetweenthe PMTandthecrystal.Thehodoscopefullystopsthechargedparticlesduetoitsthickness andmeasurestheirremainingenergy. Figure3.10:SchematicoftheHodoscope,whichisanarrayofCsI(Na)crystalsarrangedin a5x5tion. Thescintillationmechanismdependsontheenergystatesdeterminedbythecrystal latticeofCsI.Electronsonlyhaveavailablediscretebandsofenergyininsulatorsorsemi- conductors.Whenachargedparticlepassesthroughthelattice,electronsareexcitedfrom 54 thevalencebandacrosstheforbiddenregionintotheconductancebandwhichleavesan electronhole.Areturnofanelectrontothevalencebandcancausetheemissionofapho- ton.Howeverforpurecrystalsthisisantprocesssooftenadopantorimpurity isintroduced.Theaddedimpuritycancreateavailablestatesintheforbiddenbandofthe crystalwhichincreasestheprobabilityofpopulatingandde-excitatingfromthesestates sincetheirenergiesarelessthanthefullforbiddengap.CsI(Na)typicallyhasaslowdecay timewhichconsistsoftwocomponentswithmeanlivesof0.46 and4.18 s.Inaddition, CsI(Na)ishydroscopicsocaremustbetakentoavoidexposuretotheambientatmosphere. Afour-sidedgascoversurroundsthecrystalsandaninletallowsdrynitrogentoowover thefaceofthecrystalswheverthedetectorboxisnotundervacuum. 3.7MoNALISA TheModularNeutronArray(MoNA),andtheLarge-areamulti-InstitutionalScintillator Array(LISA)eachconsistof144200x10x10cm 3 plasticscintillatorbars.Eachbarin MoNAismadeofBC-408,whileLISAismadeofEJ-200.Thebarsarewrappedine materialandblackplastictoreducelightlossandpreventambientlightfromleakingin. EachbarisalsocoupledtotwoPMTsoneitherendbylightguides.TheMoNA-LISAbars servetomeasurethepositionandtofneutronsthatinteractintheplastic.When aneutronscattershydrogenorcarbonnucleiinthebar,scintillationlightisproduced andthephotonsinternallyuntiltheyarecollectedbythePMTs.Thepositionalong thebarisdeterminedbythetimeofthetwoPMTs,andtheinteractiontimeis determinedfromtheiraverage.TheYandZcoordinatesaregivenbythediscretizationof thebars. 55 Figure3.11: 12 C(n,*)crosssectionsforneutronenergiesrangingfrom1-100MeVforseveral treactions.EachofthecrosssectionslistedhereareincludedinMENATE Rand usedtomodeltheneutroninteractionsinMoNA-LISA.Imagesource:[ 93 ] Whenaneutronenterstheplasticitinteractswithcarbonorhydrogennucleicausing themtoscatterwhichexcitestheplasticcausinginamannerdescribedinsection 3.6.3 .Figure 3.11 showstcrosssectionsasafunctionofneutronkineticenergy.At typicalbeamenergies,itismostlikelyforneutronstointeractinelasticllywithcarbonsince theelasticcrosssectionforH(n,p)dropsastheneutronenergyincreases.However,sincethe massofacarbonnucleusismuchlargerthanthatofhydrogen,therecoilismuchsmaller, andsolesslightisproducedintheplasticcomparedtoscatteringonhydrogen.Thusthe dominantsignalcomesfromscatteringonhydrogen.Theemittedlightscatterstowardsthe endswhereitiscollectedbyaPMT.MoNAandLISAusePhotonisXP2262/BPMTsand HamamatsuR329-02PMTsrespectively. 56 3.8ElectronicsandDAQ Thedataacquisition(DAQ)systemforMoNA,LISA,andthesweepermagnetisdescribed indetailinReferences[ 90 , 94 , 95 , 96 ].Anoverviewispresentedhere,withanabbreviated schematicinFig. 3.12 .Eachdetectorsubsystemrunsanindependentacquisitionsystem connectedbya\Level3"systemwhichgeneratesasystemtriggerandatimestamptobe relayedtoeachdetector.Afterthedataarewrittentodisk,theseperateforeach subsystemaremergedbymatchingtimestampsevent-by-event.Thesystemtriggerinthis experimentwastheleft-upperPMTofthethinscintillator(PMT0),andthetimestamp wasa64-bitwordgeneratedbytheclockoftheLevel3system. ThetriggerlogicwashandledbyXilinxLogicModules(XLMs),dividedinto3levels. TheLevel1andLevel2XLMsdetermineweatherornotaneventinMoNAorLISAisvalid, withavalideventbeingdbyagoodtimingsignalintheCFDchannelsforbothPMTs inasinglebar.Level3containsaclockwhichrunswhenthesystemisnotbusyprocessing anevent,andhandlesthecoincidencetriggerlogicbetweenMoNA-LISAandtheSweeper. Uponrecievingasignalfromthesystemtrigger,theLevel3systemopensacoincidencegate of35nsandwaitsforavalidsignalfromeitherMoNAorLISA.Ifoneisrecieveda\system trigger"signalissenttoeachsubsystemandtheeventisprocessed.Ifnosuchsignalis recieved,thenthecoincidencegatewillcloseandthesystemwillfastclear.IfMoNA-LISA triggerbutnotthesweeper,thentheywillsendatriggerandbusysignaltoLevel3causing ittogobusyandrejectsignalsfromthesweeper.Withoutasignalfromthesweeper,the systemtriggercannotbeproduced,andthecoincidencegateisneveropened.SinceMoNA andLISAdonotrecieveasystemtriggerinthiscase,theyfastclear. MoNAandLISAeachconsistofidenticalbutindependentelectronicset-ups.EachPMT 57 hastwooutputs,ananodeandadynode.Theanodesignalisusedforlocaltriggeringand fortiming,whilethedynodeisusedtomeasurethechargecollectedinthePMT.Thetiming signalproceedstoaconstantfractiondiscriminator(CFD).Thetimingsignalsfromboth PMTsthengofromtheCFDtoatime-to-digitalconverter(TDC)andanXLMmodule. TheTDCmodulesarerunincommonstopmode,meaningatriggerinMoNA/LISAwill signalthestart,withthestopcomingfromtheLevel3systemtrigger.Finally,thecharge signalfromthedynodegoestoacharge-to-digitalconverter(QDC)forintegration.This processisduplicatedforeverybarinMoNAandLISA. Inthesweeper,theelectronicsaresetupforthreetimingscintillators,twoCRDCs, anion-chamber,andaCsI(Na)array.EachtimingsignalfromthethinPMTs,target scintillator,andA1900scintillatorgoestoaCFDandthenaTDC.Inthecaseofthethin, thetimingsignalfromPMT0issenttoLevel3toactasthesystemtrigger.Wherecharge signalsareavailable,theyaresenttoamptlitude-to-digitalconverters(ADCs).AllTDCsin thesweeperoperatedincommonstartmode.TheLevel3triggerbeganthemeasurement, andthetimingsignalitselfprovidedthestop.ThepadsoftheCRDCsweredigitizedby Front-End-Electronics(FEE)modulesthatsampledthepulseandsentittoanXLM.For theionchamber,thesignalsfromeachpadweresentdirectlytoashaperandthentoan ADC.Finally,thePMTsignalsfromtheCsI(Na)array,orHodoscope,wentthroughshaping andthenADCs.EverytimethesystemrecievedatriggerfromLevel3,allTDC, ADC,QDCandXLMswerereadoutandprocessed. 58 Figure3.12:SchematicoftheelectronicsforMoNA-LISAandtheSweeper.Dashedlines encompaseseachsubsystem.Greenarrowsindicatestartsignals,redstopsignals,andblue arrowsindicatetheQDC/ADCgateforintegration.Uponreceivingthesystemtriggerfrom level3,allTDCs,QDCs,andADCsreadoutandareprocessed. 59 3.9InvariantMassSpectroscopy Considerthedecayofaparticlewithmass M intoalargefragment A and N neutrons,with masses M A and m n respectively.Energyandmomentumareconserved,thus: E M = E A + i = N X i =1 E n and P initial = P final Thequantity M 2 =( P ) 2 islorentzinvariant,andthereforeindependentofreference frame.Thedecayenergyofanunboundnucleusisastheenergybetween theinitialnucleusandit'sdecayproducts: E decay = M A + n M A i = N X i =1 m n (3.1) where M A + n istheinvariantmassoftheinitialnucleus, M A therestmassofthefragment, and m n therestmassofaneutron.The4-vectoroftheinitialnucleusisobtainedbysumming the4-vectorsofalldaughterproducts,whichdetermines M A + n .Inthecaseofaone-neutron decaythisexpressionbecomes: E decay = q M 2 A + M 2 n +2( E A E n ~p A ~p n ) M A m n (3.2) Fortwo-neutronemission,theexpressionbecomesslightlymorecomplicated: E decay = q M 2 T +2( E T ~p T 2 ) M A 2 m n 60 Where M 2 T = M 2 A +2 m 2 n E 2 T = E A E n 1 + E A E n 2 + E n 1 E n 2 and ~p T 2 = ~p A ~p n 1 + ~p A ~p n 2 + ~p n 1 ~p n 2 Whilethedecayenergytoasinglealgebraicexpressioninthecaseofone,oreven twoneutrons,anexpressionlike 3.1 canbecomecumbersomeincasesof3or4neutron emissionduetoanabundanceofcross-terms.Itismucheasierinthosecasestohandlethe 4-vectorsofeachparticlenumericallytocalculatedotproducts. 3.10JacobiCoordinates Inthecaseofthree-bodydecays,ortwo-neutronemission,onecanutilizeJacobicoordinates toexaminecorrelationsbetweentheneutronpairandthecore.TheJacobicoordinate systemhastheadvantageofremovingthecenter-of-massmotionsothattherelativemotion isexposed.Giventhreeparticles,therearethreeuniquecoordinatesystemsthatcanbe chosen.However,sincetheneutronsareindistinguishablethisreducestotwochoices:the T andthe Y systemsillustratedinFig. 3.13 .Thevector ~ k x isdrawnfromthesecond particletotheandthesecondvector ~ k y fromthethirdparticletothecenter-of-mass ofthetwo-bodysubsystem.Therelativemotionofthethree-bodysystemcanbedescribed byanenergy ,andangle k as[ 2 ]: = E x =E T 61 cos ( k )= ~ k x ~ k y = ( k x k y ) where istheenergyofthetwo-bodysubsystemrelativetothetotalthree-bodyenergy,and cos ( k )theanglebetweenthevectors k x and k y .Moreexplicitly: E T = E x + Ey = ( m 1 + m 2 ) k 2 x 2 m 1 m 2 + ( m 1 + m 2 + m 3 ) k 2 y 2( m 1 + m 2 ) m 3 where ~ k x and ~ k y areas: ~ k x = m 2 ~ k 1 m 1 ~ k 2 m 1 + m 2 ~ k y = m 3 ( ~ k 1 + ~ k 2 ) ( m 1 + m 2 ) ~ k 3 m 1 + m 2 + m 3 And m i and k i denotethemassandmomentumofeachparticleinthethree-bodysystem. Inthe T system istheenergyoftheneutron-neutronpairrelativetothetotalthree- bodyenergy,whereasinthe Y systemitistheneutron-coreenergy.Additionalinformation onJacobicoordinatescanbefoundinRef.[ 2 ]andRef.[ 75 ].TheJacobiCoordinatesarea Figure3.13:Jacobi T and Y coordinatesforthethree-bodysystem 22 O+2n. 62 powerfulexperimentaltoolastheydistinguishbetweenthetdecaymodesatwo- neutron(ortwo-proton)unboundsystemcanundergo.Forexample,inadi-neutron-like correlationthetwo-neutronsareclusteredtogetherandsotheangle k iscloseto ˇ inthe Y systemand cos ( k )peaksat-1.Inaddition,theneutron-neutronenergyislowrelative tothetotalthree-bodyenergycausing topeakat0inthe T system. Forasequentialdecayitisneccessarytomakethedistinctionbetweenthelimitingcases of\even"and\uneven",astheyhaveverytcorrelations.Inanevensequentialdecay, theintermediatestateisathalfthetotalthree-bodyenergy,andsobothneutronsdecay withsimilarenergy.Whereasinanunevendecay,theintermediatestateiseithercloseto thethree-bodystateorthestate,resultinginahighandalowenergyneutron.Dueto thevaryingenergiesbetweenthetwocases,thecorrespondingthree-bodycorrelationsare dramaticallyt,asillustratedinFig. 3.14 . Sinceameasurementofthedecayenergyiskinematicallycomplete,theJacobispectra canalsobeconstructedfromthemeasured4-vectorsofthetwoneutronsandthecore. Measuringthethree-bodycorrelationsallowonetoconnecttothethree-bodywavefunction asthefew-bodySchrodingerequationcanbesolvedinthesamecoordinatesystemand predictionsmadeforthethree-bodydecay.Three-bodycorrelationsinboththe T and Y areshowninFig. 3.14 forthetcasesofsequential(uneven/even),di-neutron,and phasespacedecays.fromneutronscatteringhavebeenremoved,illustratingthe starkbetweenthetdecaymodes. 63 Figure3.14:SimulatedJacobi T and Y coordinatesforthethree-bodysystem 22 O+2n with E T =750keV.fromneutronscatteringhavebeenremoved.Acceptancesand areapplied.Thetcolorsindicatetdecaymechanisms.Allspectra arenormalizedto2 10 6 events. 64 Chapter4 DataAnalysis ThischapterdetailstheproceduresandmethodsusedtocalibrateMoNA,LISA,andthe detectorsintheSweepersetupandhowmeaningfuldataareobtainedfromtherawsignals ineachdetector.Afterthecalibrationsarecomplete,eventselectionisdiscussedinaddition tomodelingandsimulation. 4.1CalibrationsandCorrections 4.1.1ChargedParticleDetectors Followingthesweepermagnetareseveralcharged-particledetectors:twoCRDCs,anion- chamber,athinscintillatorandahodoscope.Thefollowingsectionsdetailthecalibration proceduresforthesedetectorsincludingthetimingscintillatorsupstreamofthetarget. 4.1.1.1CRDCs TheCathodeReadoutDriftChambers(CRDCs)provideameasurementoftheXandY positionofchargedparticlespassingthroughthedetector.Theyareusedtotrackthe reactionproductsforisotopeseparationandtodeterminetheirenergyandmomentumat thetarget.Hence,goodtrackingisessentialforasuccessfulreconstruction.Therearetwo CRDCsinthesweeperfocalplane.CRDC1wasplacedroughly1.56mfromthecenterof theLD 2 target(alongthecentertrack),andCRDC2wasplaced1.55mbehindCRDC1. 65 CRDC1CRDC2 BadPad96-10024 104121 106 Table4.1:BadpadsintheCRDCswhichareremovedfromanalysis. TheXpositionisdeterminedbychargecollectionalong116segmentedpadswithapitch of2.54mminwidth.First,thequalityofeachpadmustbeexamined.Badpadsthat showpoorchargecollectionandgiveerroneoussignalsmustberemoved.Smallleakage currentsintheCRDCscanbereadoutintheelectronics,evenwithnobeam.Thissignalis calledthepedestalandmustbesubtractedtoproperlydeterminethetotalchargecollected. ThepedestalsubtractionforCRDC1andCRDC2isshowninFigure 4.1 ,agaussian algorithmisusedtodeterminethepedestalforeachpad.Fromthesubtractionseveral pathologicalpadscanbeidentheyarelistedinTable 4.1 . Afterthepedestal-subtractionthepadsneedtobegainmatched.Thisisaccomplished usinga\continuoussweep"runwherethebeamismovedacrossthedetectortoilluminateall padswithinacceptance.An 20 Obeamwith99%puritywasusedforthispurpose.Thepad withthemaximumchargedepositedwasselectedevent-by-event.Thechargedistribution ofeachpad,whenitregisteredasthemaximumpad,wasthengainmatchedtoareference (pad64)withthefollowingrelation: m = ref i where i isthecentroidofthechargedistributionontheithpad.Figure 4.2 showsa comparisonbeforeandafterthegainmatchingprocedure.Thetotalchargecollectedonthe 66 Figure4.1:PedestalsubtractionintheCRDCs.TherawpadsforCRDC1(left)andCRDC2 (right)areshownintheandsecondpanelsfromtheleft,whilethesubtractionisshown inthethirdandfourthpanels. 67 padisdeterminedbyaRiemann-sum: Q pad = 1 n n X i =0 q i q pedestal where n isthenumberofsamplesand q i istheamountofchargepersample.Thegain- matchedsignalisthen: Q cal = m Q pad TheX-positionisthendeterminedbythechargedistributionacrossallpads.Agaussian line-shapeistothisdistributiontodeterminethecentralvalue. TheY-positionisdeterminedbythedrift-timeofthechargecarriersintheactivevolume ofthedetector.Thus,thedrifttimeandpad-distributionneedtobeconvertedintophysical XandYpositionsinthelab-frame.Thisisaccomplishedusingtungstenmaskswithsp hole-patternsdrilledintothem.Themaskisplacedinfrontofthedetectorandtheknown hole-patterncanbeusedtodeterminealineartransformationfromcharge/timetoX/Y position. ItshouldbenotedthattheCRDCsareplacedinoppositeorientationsinthexdirection. ForCRDC1thepadnumberincreasesinthe+ x directioninthelabframe,whileforCRDC2 increasingpadnumbersareinthe x direction.ThisresultsintheirXslopeshavingopposite sign. Thetungstenmasksareputinplacebyahydraulicdrive.Duringtheexperiment,the driveforCRDC2wasunabletofullyliftthemaskintopositionthusnotcoveringthefull faceofthedetector.TheX-slopeisdeterminedbythepitchofthepads,andthefound fromthemask.TheY-slopeisfoundfromthespacingoftheverticalholes,andlikewisethe fromtheabsolutepositionoftheholes.However,sincethemaskdidnotfullyinsert, 68 Figure4.2:Before(left)andafter(right)gainmatchingofCRDC1andCRDC2.CRDC1is shownintheandthirdpanel,andCRDC2inthesecondandfourth. 69 X slope [mm/pad]X offset [mm]Y slope [mm/ns]Y offset [mm] CRDC12.54-171.3-0.195398.5 CRDC2-2.54187.3-0.2043103.7 Table4.2:SlopesandfortheCRDCcalibration. thismeansthattheforCRDC2cannotbedeterminedbythismethod.Figure 4.3 shows3tmask-calibrationrunstakenattpointsduringtheexperiment,the upperedgeofthemaskbecomesvisible,anditisevidentthemaskdidnotfullyinsert. TherewasnoindicationofthisoccurringforthemaskofCRDC1. ThesetofCRDC2wasdeterminedbyexaminingthedecay-kinematicsof 23 O ! 22 O+1n,sincethewillnotpreventisotopeseparationbutwillthefragment energy.Thiswasconstrainedbyliningupthereconstructedfragmentangleswiththe neutronconeandmatchingthefragmentandneutronenergies.Thisprocedureisiterative sinceanmustbeguessed.TheparametersforthisdecayareshowninSections 4.3 and 4.4 .TheslopesandforbothCRDCscanbefoundinTable 4.2 . TheCRDCscanexhibitadriftinthemeasuredYposition.TheapparentY-positioncan ifthegas-pressurechanges,orifthedriftvoltageThisiscorrectedby gain-matchingtherawdrifttimetoareferencerun(Run1035),andperformingarun-by-run correction.Thecorrectionfactorisdeterminedby: m = TACref i where TACref and i arethecentroidsoftheTACsignalintheCRDCforthereference runandanarbitraryrunrespectively.Thecorrecteddrift-timeissimply: t corr = m t drift 70 Figure4.3:(Fromlefttoright)Maskruns3023,3078,and3142forCRDC2.Particlesthat illuminateanareaabovethemaskindicatethatthemaskdidnotfullyinsert. 71 Figure 4.4 and 4.5 showsthiscorrectionforCRDC1andCRDC2.ThedriftintheCRDCs iscorrelatedbecausethetwodetectorsareconnectedtothesamegas-handlingsystem. Figure4.4:DriftcorrectionforCRDC1Yposition.(Topleft),UncorrectedYdistribution inCRDC1asafunctionofRunNumber.(Topright),UncorrectedcentroidsofCRDC1Y positionasafunctionofRunNumber.(Bottomleft).CorrectedYdistributioninCRDC1. (Bottomright)CentroidsofcorrectedYdistributioninCRDC1asafunctionofrunnumber. 4.1.1.2IonChamber Theionchamberissegmentedinto16padsalongthez-axis,orbeamdirection.Theraw chargecollectedoneachpadcanbeusedforelementseparation,howevereachpadneeds tobegainmatchedandanyXorY-positiondependenciesremoved.Togainmatchtheion chamberabeamwassentdownthecenterofthedetector.Forthisexperiment,an 20 O 72 Figure4.5:DriftcorrectionforCRDC2Yposition.(Topleft),UncorrectedYdistribution inCRDC2asafunctionofRunNumber.(Topright),UncorrectedcentroidsofCRDC2Y positionasafunctionofRunNumber.(Bottomleft).CorrectedYdistributioninCRDC2. (Bottomright)CentroidsofcorrectedYdistributioninCRDC2asafunctionofrunnumber. beamwasusedforthispurposesincethe 24 Obeamwasnotintenseenough. Inordertoensurethateachpadgivesthesamesignalforthesameamountofcharge collecteditisnecessarytogainmatchthem.Forthisprocedure,theincoming 20 Obeam isselectedtoremoveimpurities.Inaddition,agateisplacedonagoodresponseinthe CRDCstoremoveeventswithstrangetrajectories.Itisalsoassumedthatthebeamloses anegligibleamountofenergyinthehalfofthedetectorcomparedtothesecondhalf. Ifweapproximatethedetectoras65cmofpureArgas( ˆ =1 : 66 10 3 g=cm 3 ),thetotal energylossfor 20 Oat118MeV/uinthehalfofthedetectoris16MeV,and16.1MeV 73 inthesecondhalf. Figure4.6:ExampleofgaussianprocedureforgainmatchingoftheICpads.Pad4 isshownontheleft,andthereferencepad,pad9ontheright. Thegainmatchingisperformedbydeterminingaslopeforeachpad: q cal = m i q raw Where q raw istherawchargecollectedoneachpad,and m i istheslopeas: m i = c ref =c i Here c ref isthecentroidofthereferencepad,and c i isthecentroidoftheithpaddetermined byagaussianExampleareshowninFig. 4.6 forpad4andpad9.Pad9waschosen asthereferencepad,asitisinthemiddleofthedetectoranddisplayedasignalthatwas roughlyinthemiddleofthevariationfromallotherpads. TheresultsofthegainmatchingprocedureareshowninFig. 4.7 .Itisapparentthatpad 1and8showabnormallylowchargecollection,andthusareexcludedfromanalysis.For 74 theremaininggoodpads,thepeakslineupandthewidthsareapproximatelythesame. Figure4.7:ResultsofapplyingthegainmatchingcalibrationfortheIC.Rawpadsonshown ontheleftandcalibratedpadsontheright. ThereissomevariationinthechargecollectedineachpadasafunctionoftheXposition duetotchargecollection.Eachpadhasatpositiondependence,andso eachpadmustbecorrectedindependentlytoachievethebestresolution.Toaccomplish this,a\sweeprun"wasusedwherethebeamwassweptbackandforthacrossthefocal planetoilluminatethedetectorattX-positions.Sincemagneticdonowork, thebeamhasthesameenergydespitethevaryingsweepersettingandsothesameamount ofchargeisbeingdepositedateachXposition.Using5mmwideslicesinX,thecentroidof thechargedepositedoneachpadisdeterminedbyagaussianandplottedasafunction oftheXposition.Thedependenceisthenwithapolynomialofappropriateorderand removedbythefollowingexpression: q pos = q cal P a i x i where q cal isthegainmatchedcharge,and a i representthepolynomialcots. 75 Figure4.8:Exampleofpositioncorrectionintheionchamberforpad15.Therawsignal asafunctionofXpositionisshownintheleftpanel,withthebsuper-imposed.The rightpanelshowstheresultofthecorrection. ThismethodwasonlyappliedfortheXpositioncorrection,astherewasnonoticeableY- dependence.Aftereachpadhasbeengainmatchedandposition-corrected,thetotalenergy lossisdeterminedbythesumofallpads: Q tot = X q pos Theresultsforthepositioncorrectionareshownforpad15asanexampleinFig. 4.8 . Theionchamberexhibitedaslowdriftoverthecourseoftheexperiment.Thisisshown inFig. 4.9 forthereactionproductsfromthe 24 Obeam.Thedriftwaswiththefunctional form: dE ( t )= p 0 + p 1 t + p 2 76 Thedrift-correctedenergylossthenbecomes: dE corr = Q tot =dE ( t ) Todeterminethecotsforthedriftcorrection,agateisplacedontheselectionof 24 O beam,goodCRDCs,andonoxygenintheionchamber(denotedbytheredlinesinFig. 4.9 ).ThecocientsforthedriftcorrectionaresummarizedinTable 4.3 . Figure4.9:DriftcorrectionforICsum.(Topleft),UncorrectedchargedistributionintheIC asafunctionofRunNumber.(Topright),Uncorrectedcentroidsofthechargedistribution asafunctionofRunNumber.(Bottomleft).DriftCorrectedchargedistributionintheIC. (Bottomright)Centroidsofthedriftcorrectedchargedistribution. 77 DriftCorrection p 0 154.8 p 1 210.9 p 2 -2971.2 Table4.3:DriftcorrectionparametersfortheIonChamber. 4.1.1.3ThinScintillator Thethinscintillatorislocatedaftertheionchamberinthesweeperfocalplaneandprovides anadditionalmeasurementoftheenergylossinadditiontotime-of-tinformation.The detectorhasfourPMTsmountedonlight-guidesasillustratedinFig. 3.8 ,andarelabeled 0through3.ThesignalfromeachPMTgivesatimeandachargemeasurementwhichare combinedtogiveatotalenergylossandtheinteractiontime.Duetoinhomogeneitiesand attenuationintheplastic,thereisvariationinthecharge-collectionforeachPMT basedontheinteractionpositioninthedetector.Thus,eachPMTneedstobegainmatched andpositioncorrected. An 20 ObeamwassentdownthecenterofthefocalplaneforgainmatchingofthePMTs. Thisensuredthatthemiddleofthethinscintillatorwasilluminatedandthatthedistance fromtheinteractionpointtoeachPMTwasroughlyequal.Thesweeperwassetto I =350A withacentralrigidityof Bˆ 0 =3 : 767Tmforthisrun.Duetothelogisticsofwarmingand coolingtheliquiddeuteriumtarget,itwasmorenttousea670mg/cm 2 Be degraderwhilethetargetwaskeptinagaseousstate(50K)insteadofwaitingforthetarget toliquefy.Thechoiceof670mg/cm 2 ofberylliumwastomimictheexpectedenergy-loss throughthedeuteriumtarget.Inaddition,aselectionwasmadeonoxygenisotopesinthe ionchamber,andeacheventwasrequiredtofallwithina20mmx20mmsquarecentered onthethinscintillator.Thepositiononthethinscintillatorwasdeterminedbyprojection 78 fromtheCRDCs. Figure4.10:Before(left)andafter(right)thegainmatchingofthePMTsinthethinscin- tillator. ThechargesignalsfromeachPMTarecalculatedinasimilarmannertotheionchamber, andgainmatchedinthesameway: q cal = m i q raw + q 0 Wheretheslopearedeterminedbyaratioofgaussianwidthswithrespecttoareference PMT(PMT0,seeFig. 3.8 ),andtheissettolineupthepeaks: m i = ˙ ref ˙ i Fig. 4.10 showsthechargecollectedineachPMTbeforeandaftergainmatching.Once allfourPMTSaregain-matched,theirsignalsarecombinedtoprovideameasurementof 79 thetotalchargedepositedintheplastic,whichisdasfollows: q top = q LU + q RU 2 q bot = q LD + q RD 2 q tot = q q 2 top + q 2 bot 2 Where q LU;LD;RU;RD arethegainmatchedsignalsfromtheleft-upper(lower)andright- upper(lower)detectorsrespectively.Thetotalchargesignal q tot exhibitedbothapositional dependenceinherentineachPMTsignalaswellasanoveralltimedependenceasthede- tectordriftedoverthecourseoftheexperiment.Thesewerecorrectedbyusingthe samemethodastheionchamber.Thepositiondependencewasremovedbyasixth-order polynomial: q poscorr = q tot P a i x i Figure4.11:Correctionofthepositiondependenceinthethinscintillator.(Left)theraw chargesignalasafunctionofXpositionwiththebsuperimposed.(Right)Resultof positioncorrection. 80 PositionCorrectionDriftCorrection a 0 651.7 p 0 742.7 a 1 0.2001 p 1 -43364 a 2 -4.71*10 3 p 2 -2374 a 3 -2.31*10 5 a 4 7.55*10 8 a 5 3.08*10 10 a 6 4.63*10 15 Table4.4:Cotsforpositioncorrection(left)anddriftcorrection(right)oftheThin scintillator. Figure 4.11 showstheenergy-lossinthethinscintillatorasafunctionoftheXposition determinedbytheCRDCs.Tomapoutthisdependence,a\sweeprun"wasusedasdescribed inprevioussections.NotYdependencewasobserved,andthuswasnotcorrected. Thedetectordriftwashandledthesamewayastheion-chamber: dE ( t ) thin = p 0 + p 1 t + p 2 Withtheenergysignalbeing: dE corr = q tot =dE ( t ) thin Thedrift-correctedenergysignalcanbeseeninFig. 4.12 .Theslightcorrelationisremoved. Thecotsforthepositionandenergy-driftcorrectioncanbefoundinTable 4.4 . OnlyanisusedtocalibratethetimingsignalofeachPMT.TheslopeoftheTDCis assumedtobe0.1ns/chsincetherangeoftheTDCisInaddition,thereisabouta20 nsjitterintroducedtheFieldProgrammableGateArray(FPGA)inthesweeperelectronics. SincetheTDCsreceiveastartfromtheindividualPMTSandthestopisgeneratedby theFPGA,theamountofjittercanvaryevent-by-event.However,thesamestopsignalis 81 Figure4.12:Driftcorrectionforchargecollectioninthethinscintillator.(Topleft),Uncor- rectedchargedistributioninthethinasafunctionofRunNumber.(Topright),Uncorrected centroidsofthechargedistributionasafunctionofRunNumber.(Bottomleft).DriftCor- rectedchargedistributioninthethin.(Bottomright)Centroidsofthedriftcorrectedcharge distribution. usedforallPMTssothejitteristhesameforeachtimingsignal.Thejitteriseliminated bysubtractingonetimingsignalfromtheremainingothersafterapplyingtheslopeof0.1 ns/ch.InthepastthereferencesignalhasbeenThinPMT0,howeverinthisexperiment thissignalwoulddropoutintermittentlyandsoamorestablecopyofthesignalpassing throughtelectronicswasused,called\sweepertrigger".Thisdidnotctthe q tot , asthechargesignalwasalwayspresent. TheindividualforeachPMTweredeterminedusingthesamemethodasthe gainmatching.Eventswhichhitthecenterofthedetectorwereselectedandthecentroid 82 PMT[ns] 031.3 1-24.0 2-74.9 3-5.94 Table4.5:TimefortheThinscintillator. ofeachPMTwasalignedwiththatofThinPMT0.TheoverallforThinPMT0was determinedbytheexpectedtforthebeam.Forthiscalibration,thebeamwas 20 Oat118MeV/uimpingingupona670mg/cm 2 Bedegraderwithatpathof417.86 cmfromthetargettothethin.Thisgivesacalculatedtof t 0 =31 : 28ns,and determinestheofThinPMT,andhencetheoftheotherPMTs.These arelistedinTable 4.5 .Oncethearedetermined,thecalibratedtimingsignalforthe entiredetectorisformedfromtheaverageofeachPMT: t thin = 1 n n X i =0 t i Where n isthenumberofPMTsthatforagiveneventand t i arethetimingsignals ofeachPMT.TheindividualtimingsignalsofeachPMTdriftedoverthecourseofthe experimentwiththeworstcasebeingabouta1nsdriftintiming.Becausethetrendsfor eachPMTarettheywerecorrectedindividually.Insteadofaglobalfunctionforthe drift,likeinthecaseoftheion-chamber,theforeachPMTwerevariedrun-by-run toeliminatethisdrift.Thiswasdonebyselectingareferencerun,Run3030,and thechangeintimingrelativetothereference, t = t t ref ,andsubtracting t fromthe observedtime.Alistof t wasthenstoredinahash-tablesothatthecalibrationcouldbe performedrun-by-run.Figure 4.13 showsthedriftinthecombinedsignalandtheof thecorrection.Thegradualshiftintimingisremoved. 83 Figure4.13:Driftcorrectionforthetimingofthethinscintillator.(Top-left)Theaverage time t thin isshownasafunctionofrunnumber.(Top-right)Thecentroidofthetiming signalasafunctionofrunnumber.(Bottom-left)Driftcorrectedthintimingasafunction ofrunnumber.(Bottom-right)Correctedtimingcentroidsusingthesetmethod. 4.1.1.4TargetandA1900TimingScintillator Therearetwoothertimingscintillatorsinthebeam-line.TheA1900scintillatorwasplaced immediatelyaftertheA1900,andthetargetscintillatorwasplaced105.92cmupstreamfrom thecenteroftheLD 2 target.ThetargetscintillatorhasasinglePMTattachedtoitthat recordsbothtimeandcharge.TheA1900scintillatorwaslocated10.579mupstreamand onlythetimingsignalwasrecorded.BothTDCsforthescintillatorshaveaslopeof 0.1ns/ch,andtheircalibrationfollowsthesamemethodastheindividualthinPMTs.After theTDCchannelisconvertedtons,theFPGAjitterissubtracted.Thisleavesaglobal 84 A1900andTargetScint.[ns] A1900-123.6 Target52.61 Table4.6:TimingfortheTargetandA1900Scintillators. tobedetermined.Tothea\beamdowncenter"runwasusedwiththe an 20 Obeamanda 9 Bedegrader.Thesetsaresetsothatthevelocityoftheunreacted beamisproperlyreproducedbetweenthetwoscintillators.TheycanbefoundinTable 4.6 . BoththetargetscintillatorandtheA1900exhibitedagradualtimedependence.This iscorrectedinthesamemannerasthethinPMTs.Areferenceischosenanda t calculatedrelativetothatonarun-by-runbasis.Thecalibratedtimeisthenshifted t toremovethedrift.ThedriftcorrectionforthetargetscintillatorisshowninFig. 4.14 , andfortheA1900scintillatorinFig. 4.15 . 4.1.1.5Hodoscope TheCsI(Na)array,orHodoscope,didnotfunctionproperlyduringthisexperimentdue toasmoothingofthecrystalsurface,likelyduetotheformationofwaterdropletswhich causedthewrappingtocomeincontactwiththecrystalface.Thedetectorsrelyon totalinternaltodirectthelightproducedfromaneventtoaPMTattheendof theCsI(Na)crystal.NormallythesurfaceoftheCsIissandedsothatthesurfaceisrough leavinggapsbetweenitandthecovering.Fortotalinternaltooccur,the interfaceofthecrystalandthemusthaveagapsothatthewhenthelightisincident onthemediuminbetweenthecrystalanditswrappingitisnottransmitted.Whenthe angleofincidenceexceedsacriticalangle,theamplitudeofthetransmittedwavebecomes zero.Thecriticalangledependsontheindexofrefractionofthematerials: 85 Figure4.14:Driftcorrectionforthetimingofthetargetscintillator.(Top-left)Thetiming signalisshownasafunctionofrunnumber.(Top-right)Thecentroidofthetimingsignal asafunctionofrunnumber.(Bottom-left)Driftcorrectedtargetscintillatortimingasa functionofrunnumber.(Bottom-right)Correctedtimingcentroidsusingthemethod. sin ( c )= n medium =n CsI However,ifthesizeofgapbetweentheandCsIbecomescomparabletothewave- lengthofthelight,itcanbetransmittedthroughtheresultinginpoorresolution. CsI(Na)crystalsarehydroscopic.Ifthecrystalswereexposedtoahumidenvironmentdur- ingtheirmanufacturethenwaterdropletscouldhaveformedinbetweentheandthe surfaceofthecrystal.Theresultingcondensationcancausethesurfacetobecomesmooth allowingtosticktothecrystalleadingtolightleaks. 86 Figure4.15:DriftcorrectionforthetimingoftheA1900scintillator.(Top-left)Thetiming signalisshownasafunctionofrunnumber.(Top-right)Thecentroidofthetimingsignal asafunctionofrunnumber.(Bottom-left)DriftcorrectedA1900timingasafunctionof runnumber.(Bottom-right)Correctedtimingcentroidsusingthemethod. A 20 Obeamwasusedtomap-outthepositiondependenceofthecrystals.Bysweeping thebeambackandforththemiddlerow(modules10-14)ofthehodoscopewasilluminated. ThepositiononthecrystalfacecanbedeterminedwiththeCRDCsto ˘ 1mmresolution. Plottingthetotalchargecollectedonthez-axisandtheXandYpositioninthecrystalon theXandYaxisshowsuniquepatternsforeachcrystal(Figure 4.16 .) Itisevidentthatthesignalqualityisdegradedinareaswithlowlightcollection,causing theoveralresolutionofthearraytobentlyworsened.Largeportionsofthecrystals areun-usable.Uponremovingthemiddle-rowaftertheexperiment,aninspectioncon 87 Figure4.16:LightcollectioninthemiddlerowoftheHodoscopeforRun3002wherean 20 Obeamwassweptacrossthefocalplane.Idealbehaviourwouldbeauniformresponse. OntheXandYaxisaretheXandYpositionsrelativetoeachcrystal,thecoloraxisisthe totalenergydeposited. thatthewasstickingtothesurface.Duetoatdegradationofthehodoscope resolution,thisdetectorremainsunusedinthisanalysis. 4.1.2LD 2 Target TheUrsinusCollegeLiquidDeuteriumTargethastwoquantitiesthatweremonitoredduring theexperiment:pressureandtemperature.Itisimportanttomonitorthesequantitiesduring operationofthetargettoensurethesafetyofthetargetaswelltrackanythat mayoccur.Thepressurewasmonitoredbyamanometerinthegas-handlingsystemthat providedameasurementofthetargetcellpressure.Themanometerwascontrolledbya MKSPDR200unit[ 97 ],whichoutputsarawvoltagesignalwitharesolutionof1mVthat 88 mustbeconvertedintoapressure.Thetemperatureismonitoredbyasilicondiodewhichis read-outbyaLakeShoreModel331Stemperaturecontrollerthatoutputsananalogsignal witharesolutionof < 1mV.Theuncertaintyonthissystemisaround ˘ 0.25Kaccordingto themanufacturer[ 98 ].(TheIEEE-488analoginterfacehasanaccuracyof 2.5mV,which correspondsto0.75K.Factorysettingsfordiodesarebetween0.25-1Kuncertainty). Tocalibratethetemperatureandpressureofthecell,therawsignalisconvertedto theappropriatequantityassumingthefollowingfunctionalforms(asrecommendedbythe manufacturers).Forpressuretherelationis: P = p 0 10 2 V [Torr] Andfortemperature: T = t 0 V [K] Thecots p 0 and t 0 canbeconstrainedsimultaneouslybyttingthephase-transition ofaparticulargas.Forthisexperiment,twotypesofgaswereanddatarecorded forthephasetransition.Aneontestgaswasusedtoinitiallycheckthetargetsystemwhile deuteriumwasusedfortheactualexperiment.Thisprovidesamethodofcrosschecking, asacalibrationtothephasetransitionofonegasmustreproducethephasetransitionof theother.Thephasetransitionsofneonanddeuteriumhavebeenmeasuredandcanbe parameterizedwiththefollowingforms. Forneon,datacanbeobtainedfromNIST[ 99 , 100 ]: log 10 ( P )=3 : 75641 95 : 599 T 1 : 503 89 wheretheunitsfor P arebar,andKelvinfor T .Thevaporpressurefordeuteriumhasalso beenmeasured[ 101 ]: log 10 ( P )=5 : 8404 70 : 044 =T + 4 : 59 10 4 ( T 23) 2 where P isinmmofHgand T inKelvin. Theseparameterizationsofthevaporpressureareonlyvalidforasptemperature regimeastheyareanapproximation.Fortheneondatathisintervalisfrom15.9-27K, andfordeuteriumitis14-24.5K.Thecots p 0 and t 0 canbedeterminedby directlytoaknownphase-transition.Figure 4.17 showsthetotherawphasediagramfor D 2 gasobservedduringliquefaction.FromthiscalibrationweseethattheD 2 gasdidnot crossintoaregionwhereitwouldhavefrozen.Thebestparametersare: p 0 =9 : 621 10 6 [Torr] t 0 =30 : 34[K/V] Thecalibrationcanbevbyapplyingittothedatatheneonphase-transition,and isshowninFig. 4.18 .Thecalibrateddataareshowninblack,whiletheNISTdataare showninred.Thedashed-linesgiveanerrorband.Thisisdeterminedbysystematically shiftingthetemperaturebytheuncertaintyreportedintheneonmeasurement( T =0 : 5 K [ 100 ])usedtoconstraintheNISTparameterization.Thedatafallwithintheuncertaintyof thepreviousmeasurementcothecalibration. Oncethetemperatureandpressurehavebeencalibratedthetargetcanbecheckedfor anydriftsduringitsoperation.Figures 4.19 showsthetemperatureandpressureoverthe 90 Figure4.17:TemperatureandPressurecalibrationoftheLD 2 target.(Left)Rawvoltages fromtheEPICSreadoutfromthetemperaturecontrollerandmanometer.Thedataarein black,theregionusedtothephasetransitionishighlightedinred.(Right)Calibration tophase-transition(blueline)indeuterium. courseoftheexperiment.Thetemperatureislessthan0.15Kwhichiswellbelow theuncertaintyofthetemperaturecontroller.Inaddition,thepressureisverystableexcept foraslowdropofabout5Torrinthemiddleoftheexperiment.Thepressureequalizedfor theremainingdurationandnofurtherchangewasdetectable.Itisnotlikelyaleakora failureofthetargetcell.Sincethischangeislessthan1%ofthepressureitisneglected. Itiscrucialtodeterminethetargetthickness.Althoughthecellisdesignedtocontain 200mg/cm 2 ofLH 2 ,theKaptonwindowscandeformunderpressurecausingthenominal thicknessofthetargettoincrease.Inaddition,theheat-shieldoftheLD 2 targetwaswrapped in5 ofaluminizedmylartohelpkeepthetemperaturestable,whichaddsaditionalenergy loss,albeitsmall.Thetargetthicknessisdeterminedbymeasuringthekineticenergyofthe unreacted 24 Obeaminthefocalplaneanddeterminingthetotalenergyloss.Tomeasurethe kineticenergyoftheunreactedbeam,itisnecessarytocalibratetheCRDCsandperform theinversetracking.Detailsonthosecalibrationsandtheinversetrackingcanbefoundin sections 4.1.1.1 and 4.3 . 91 Figure4.18:Measuredphasetransitionwithaneontestgasusingthecalibrationparameters fromthedeuteriumtransition.TheredlinecorrespondstodatafromRef[ 100 ],and thedashedlinesaretheuncertaintybandsgivena 0 : 5 K Theredarrows indicatedatatakenduringinitialcooling,liquefaction,andwarmingofthetarget.The eventsaround26Kand1200Torrarearesultofasensorerror. Theincomingbeamenergyisknownsincethefocusingquadrupoletripletbeforethe targetwassettoa Bˆ of4.03146Tmthusgivinganenergyof E beam =83 : 25 1MeV/u forthe 24 Obeam.Ameasurementonanemptycellgivesareconstructedbeamenergyof E beam =83.4MeV/u,whichagreeswellwiththetripletsetting. Thekineticenergyoftheunreacted 24 OisshownaftertheinversereconstructioninFig. 4.20 .Thepeakofthedistributionisat E 0 =66 : 4MeV/ugivingatotalenergylossof E loss =17 : 1MeV/u.Thetargetthicknesscanbedeterminedbyestimatingtheenergy 92 Figure4.19:TemperatureandPressureoverthecourseoftheexperimentstart- ingfrom4:00AM3/19/14.Timeismeasuredinhours. losswithLISE++[ 102 ].However,thisrequiresknowingthedensityofliquiddeuterium whichchangeswiththetemperature.Aparametrizationofthedensityasafunctionofthe temperaturecanbefoundfromdatatakenatNIST[ 103 ]: ˆ ( T )=0 : 1596+3 : 395 10 3 T 1 : 4086 10 4 T 2 93 Giventheobservedtemperaturetheaveragedensityis ˆ =0 : 1712g/cm 3 withanuncertaintyofabout 1 : 4 10 3 g/cm 3 .Thisresultsinanuncertaintyinthetarget thicknessofabout10mg/cm 2 ,correspondingtoabout0.2MeV/uintheenergylossofthe beam.Atthisdensity,thenominalthicknessofthetargetis514mg/cm 2 . Figure4.20:Measuredkineticenergyofthe 24 ObeamafterpassingthroughthefullLD 2 target. Usingadensityof ˆ =0 : 1712g/cm 3 forLD 2 ,theresultingenergyafterpassingthrough themylar,kapton,anddeuteriumcanbecalculatedandcomparedtoexperiment.The targetthicknesswasdeterminedtobe t =630 +45 40 mg/cm 2 ,wheretheerrorarisesfrom theuncertaintyinthebeamenergy( ˙ beam =1MeV/u).Thisisantlylargerthan thenominalthickness.Repeatingthisprocesswiththe 27 Necontaminantbeamyieldsa thicknessof t =650 30mg/cm 2 .Thereisapproximatelya40mg/cm 2 uncertaintyfrom thebeamenergy,5mg/cm 2 systematicuncertaintyinthechoiceofbeam,and10mg/cm 2 duetothedensityofthetarget,givingacombinedthicknessof t =640 45 94 mg/cm 2 . Additionally,thisuncertaintyinthetargetthicknessresultsinapproximatelya0 : 5 1 MeV/uuncertaintyintheenergylosswhichultimatelybroadensanydecayenergymeasure- ment. Figure4.21:A3.4mmbulgeintheLD 2 drawntoscale. TheexcessthicknessofthetargetisaresultofthebulgingoftheKaptonwindows.To accountfortheobservedthickness,thisbulgewouldhavetobeapproximately b =3 : 4mm whichisroughly10%thelengthofthetargetcell.Figure 4.21 showsadiagramofthebulge drawntoscale.Thesizeofthebulgecanbeestimatedusinganempiricalresultderived forclampedwindows.FromtheBNLOSHAsafetyguide(June7,1999)\Glassandplastic windowdesignforpressurevessels,"[ 104 ],thebulge b canbeexpressedintermsofthe pressuregradient P ,theYoung'sModulusofKapton, E ,thewindowthickness t =125 m,andwindowdiameter(d=38mm): Pd 4 Et 4 = K 1 b t + K 2 b t 3 Theconstants K 1 and K 2 arederivedforwindowswhichareclampedalongtheiredge,and are K 1 =23and K 2 =55respectively.Usingthemeasured850Torrpressureerential givesavalueof b =2 : 5 mm whichislessthanthebulgeneededtoexplaintheobserved 95 thickness,howeverthisexpressionisonlyanestimate. 4.1.3MoNA-LISA TherawoutputfromthePMTsinMoNAandLISAprovideachargeandatimemeasure- mentthatthetotallightcollectedbythePMTandthetimeofitsarrival.Several calibrationsareneededtoconvertthesemeasurementsintodepositedcharge,position,and interactiontime.First,eachPMTmustbegainmatchedandtheQDCchannelscalibrated. ThenthecorrespondingTDCscalibrated,andaconversionfromtoposition withinabardetermined.Eachbarmustthenbeplacedintimerelativetothebar ineachtable.Finallyeachtableisplacedintimerelativetothetarget.Themajorityof calibrationscanbedonewithcosmicrays.Cosmicdatawastakenbothbeforeandafterthe experiment.Cosmicmuonsdepositroughly2.05MeV/cm[ 105 ]ineachdetectorastheypass throughwithavelocityclosetothespeedoflight.Thusapproximately20MeVelectron- equivalent(MeVee)oflightisdepositedintoeachbar.Becauseofthedependenceofthe lightyieldinorganicscintillatorsonthetypeofparticle,theMeVeeisusedtoquantifythe absoluteamountoflightproduced.1MeVeeisastheamountoflightproducedby anelectronwith1MeVofkineticenergy.Sincethespeedofthemuonsiscloseto c they canbeusedtodeterminetherelativetimingofthebars. 4.1.3.1ChargeCalibration(QDC) EachPMTwasgainmatchedbychangingthevoltagesuntilthepeakfromcosmicmuons appearedinroughlythesamechannelforallPMTs.Thisprocesswasrepeateduntilthe cosmicpeakwasatroughlychannel900,anduntiltheindividualinaPMTs voltagewerebelow10V.Typicalvoltagesrangefrom1300-1950VinMoNAandLISA. 96 Figure4.22:ExamplespectrausedforQDCcalibrationinMoNA-LISA.(Left.)RawQDC channelsfordatatakenwithcosmicrays.Theredcurveisagaussiantothecosmic-ray peak.(Right)Pedestalsubtractedandcalibratedchargespectrum.Thecosmicraypeak appearsaround20-30MeVee. TheQDCcalibrationisdonebytakingcosmicdataafterhavinggainmatchedallPMTs anddeterminingalinearrelationbetweenrawchannelsandMeVeebythepedestal peakandthecosmicpeak: q cal =( q raw q ped ) m q Where m q istheQDCslopeinMeVee/chand q raw istherawQDCchanneland q ped the pedestal(ch).Athresholdisplacedabovethepedestal,determinedby: Q thresh = q ped = 16+2 Theslopeisgivenbythebetweenthecosmicpeakandthepedestal.Thefactor of16isnecessarytoconvertthepedestalchannelfrom12bitsto8bits,asthepedestaland thresholdarestoredas12and8bitnumbersrespectively.The2assuresthatthethreshold isplacedabovethepedestal. 97 Figure4.23:TDCandX-positioncalibrationspectra.(Left)RawTDCchannelsforarun takenwithatimecalibratorwithaintervalof40ns,thisdeterminestheTDCslope. (Middle)RawspectrumusedtocalculatetheX-position.Theredlines indicatethephysicalendsofthebar.(Right)ConversionofintoX-position. Datatakenwithcosmicswhichfullyilluminatethearray. Thisprocessisautomatedwithaprocedurethatthelocationofthepedestaland cosmicpeakviagaussianAnexampleisshowninFigure 4.22 . 4.1.3.2PositionCalibration(TDC) MoNAandLISAusetime-to-digitalconverters(TDCs)tomeasuretimingof eventswithinthearray.WhenaTDCchannelreceivesapulsefromtheanodeofaPMT itbeginschargingacapacitoruntilitreceivesadelayedstopsignalfromthelogicofthe electronics.TheamountofchargeonthecapacitorcorrespondstothetimetheTDCwas charging.ThereisslightvariationinthecapacitorsofeachTDCandsoaslopemustbe determinedforeachTDC.ThisisdonebypulsingthesystemwithanOrtecNIMTime Calibratormodule(Model462),whichprovidespulsesatspintervalstotheTDCs.For thisexperiment,thepulseratewassetto40nsintervalsandtheTDCrangewas350ns. Figure 4.23 showsanexampleTDCspectraforthiscalibration.The\picket-fence"isspaced in40nsintervals,andcanbeusedtocalculateaslopefortheTDC.Aslopeisdetermined foreveryTDCchannelinMonAandLISA.Theyaretypicallyaround0.09ns/ch. OncetheslopeofeachTDCiscalibrated,thetimebetweentheleftandright 98 PMTsofabarcanbeusedtodeterminethepositionofanevent.Usingcosmicrays,thefull lengthofeachMoNA/LISAbarwasilluminatedandspectraofleft-righttimed weregenerated.Afermi-functionisusedtotheedgeofeachbar,fromwhichthe isconvertedtoapositionviaalinearrelationship.Figure 4.23 showsan examplerawbar-positioninaLISAandthecorrespondingcalibratedposition.Theslopeis determinedbytheedgeoftheandtheissuchthatthebariscentered initsownreferenceframe. 4.1.3.3Timecalibration(tmean+global) Itisimportanttoknowtherelativetimingbetweeneachdetectorinthearray,sothatthe neutrontimtcanbeaccuratelydetermined.Whilethetimeofaneventisdetermined bytheaverageofthePMTs,atimingneedstobedeterminedtoplaceeachbarrelative toareferencebar.Finally,thereferencebarneedsanabsolutetoplaceitrelativeto thetarget.Thetimingbetweenbarsisdeterminedusingcosmicraydata,while -rays fromthetargetareusedtodeterminetherelativetothetarget.Theknownvelocityof cosmic-raymuonscanbeusedtodeterminethetimingbetweeneventswhichpassthrough all16barsinalayer,eitherverticallyordiagonally.Onlyeventswhichtriggeredamajority ofthebarsinalayeranddepositedapproximately20MeVineachbarwereused. Exceptforthelayer,thetopbarofeachlayerisusedasareferenceandeach subsequentbarinagivenlayerisplacedrelativeintimetothetopbar.Foreventswhich passverticallythroughthearraythetraveltimeis: t = d v 99 Where d thedistancebetweeninteractions,and v =29 : 8cm/nsisthespeedofthemuon. ThebetweentheexpectedandobservedtimedeterminestheOnceeach verticallayerisplacedintime,thelayersthemselvesneedtobeplacedrelativetothebottom barofthefrontlayerineach\table".Thisisdoneusingdiagonaltrackswhichtravelfrom thetopofeachlayertothebottombarofthefrontlayer.Figure 4.24 illustratedthemuon tracksusedtoperformthiscalibration. Figure4.24:Schematicofacosmicraytracksusedtodeterminetherelativebetween eachbar.withinatablearewithrespecttothebottombarinthefrontlayer. J 0 and K 15 areexamplebarsintherstandsecondlayerofLISA. Onceeachtable(1forMoNA,2forLISA)isplacedintimerelativetothebottombar ofthefrontlayer,amustbedeterminedtoplacethatbarwithrespecttothe target.Toaccomplishthis, -raysemittedfrominteractionsinthetargetareusedsince theirvelocityiswandthepositionsofthebarsarewell-known.Theisset suchthatthereconstructedspeedofthe -raysisthespeedoflight. Duetothelowbeamrate( < 200pps),theentiresetofproductiondatawasnecessary inordertogainenoughstatistics.Inaddition,agatewasplacedon 22 Ofragmentscoming fromthe 24 Obeambecausetheunreacted 24 Oproducesalargebackgroundwhenitstopsat 100 LISA=0)LISA=22 )MoNA[ns] 411.66410.53434.54 Table4.7:GlobaltmeaninnanosecondsforeachtableinMoNA-LISA. theendofthesweeperfocalplane.Fig. 4.25 showstheresultsofthecalibration,the -ray peaksshowupat29.97cm/nsforeachtable.Table 4.7 liststhetimeforMoNA andLISA. Figure4.25:Reconstructedvelocityof -rayscomingfromthetargetincoincidencewith 22 Ofragments.Theglobaltimingisvariedforeachtabletoalignthecentroidsat v =29 : 97cm/ns. 4.2EventSelection Thissectiondetailshowthedataarereducedtothephysicseventsofinterest.Overthe courseoftheexperiment,therearemanyeventsthatarerecordedthatareunrelatedtothe physicalprocessonemaywishtostudy.Thismayincludebackground,contaminantbeam, oreventsthatdonotcreateabigenoughsignaltobeusefulandareofpoorquality.For example,someoftheCRDCorionchamberpadsmalfunctionedandproduceddatathat arenotuseful.Thephysicalprocessesofinterestare 24 O(d,d') 24 O ! 22 O+2nand 24 O(- 1p) 23 N ! 22 N+1n,whichrequireselectionofthebeamandreactionproductsincoincidence 101 withneutrons. 4.2.1BeamSelection Thecoupledcyclotronsprovidedan 24 Obeamwith32%purityatanintensityof0.6pps/pnA at83 1MeV/u,withthemajorcontaminantbeing 27 Ne.Therearehoweverat amountofcontaminantscreatedinthewedgeoftheA1900.Themagnetbeforethetarget wassettoacentralrigidityof Bˆ =4 : 03146Tmcorrespondingtoabeamvelocityof11.89 cm/ns.Ideally,toidentifythebeamcomponents,onewouldsendthebeamintothefocal planewithoutatarget.HoweverduetothenatureofthesetupoftheLiquidHydrogen Target,itwasimpossibletohaveadatasetwithnomaterialinthebeam-pipe.Theclosest approximationthatcouldbeachievedwastowarmthetargetto50Kwhereitwouldbein agaseousless-densestateandsendthebeamthroughthefoils,whichcauseanenergyloss ofaround1MeV/u.Usingawarm-targetrun,thebeamwassentintothefocalplaneand elementidenwasachievedbylookingat dE vs. ToF target ! thin . Onecanremovethewedgefragmentsbycorrelatingthetfromtheendof theA1900tothetargetwiththetimebetweentheRFsignalfromthecyclotrons andtheA1900.Fragmentswiththesame Bˆ ,butt Z and A willhavet velocities,andsoseparationcanbeachieved.Here,theRFsignalfromtheK1200cyclotron isused.Theselectionof 24 OisshowninFigure 4.26 ,alongwiththeZidenofthe beam-componentsintheionchamber. 102 Figure4.26:(Left)Separationofthe 24 Obeambytfromtheotherbeamcontam- inates.(Right)Energylossvs.tspectrumforallbeamsandreactionproducts. Linesaredrawntoguidetheeyeforeachelement. 4.2.2EventQualityGates Toseparateisotopesandreconstructtheirenergiesandmomenta,itisnecessarytohave accuratepositioninformationintheCRDCs.OccasionallytheCRDCsfailedtocollectallthe chargedepositedbyaneventresultinginanunreliablepositiondetermination.Theseevents canberemovedbyapplyingqualitygatestotheCRDCs,aseventsthathaveapathological chargedistributionwillgiveanunreliableposition.Theseeventscanbeidenbylooking atthe ˙ ofthegaussianalgorithmfortheXpositionasafunctionofthepadsum{ orintegratedtotalcharge.TheCRDCqualitygatesforCRDC1andCRDC2areshownin Figure 4.27 .AnadditionalqualitygateismadebetweentheCRDCs.Therearesomeevents thatdepositlowchargeinonedetector,buthighchargeintheother.Thetrackingofthese eventscanbeunreliable.Agateisplacedaroundeventsthatbehavelinearlyinthecharge depositionbetweenthetwodetectors.ThiscutisshowninFigure 4.28 . SincetheY-positionintheCRDCsisdeterminedbyatimebetweenaninter- actioninthethinscintillatorandthecollectionofchargecarriersintheanodeoftheCRDC, therearesomeeventswherethechargecarrierswereunabletobefullycollectedresulting 103 Figure4.27: ˙ vs.totalintegratedchargeintheCRDCs.Qualitygatesaredrawninred. inapoordeterminationoftheY-position.Theseeventsareremovedbyrequiringavalid Y-positioninthecalibrationalgorithm. Figure4.28:TotalintegratedchargeofCRDC1(Y-axis)vs.CRDC2(X-axis).Agate, showninred,isdrawnaroundeventsthatdepositeasimilarchargeinbothdetectors.A gateisappliedtoselectthe 24 Obeam. AdditionalcutsaremadeonproperchargecollectioninthePMTsofalltimingscintilla- tors.AgoodsignalisrequiredintheRF,A1900,targetscintillator,andthethinscintillator. Thisensuresthattheeventsofinteresthaveagoodsignalthroughouttheentiresystem. 104 4.2.3ElementandIsotopeIden Severalreactionproductswereexpectedfromthe 24 ObeamontheLD 2 target.Thesweeper wassettoacentral Bˆ =3 : 524Tm(320A),correspondingtotheexpectedenergyof(d,p) reactionproducts.Theacceptanceofthesweeperisapproximately 8%inrigidity,andso onlyafractionoftheisotopesproducedmadeitintothefocalplane.Forthisexperiment, thereactionproductsinclude 22 24 O, 18 22 N, 16 18 C, 13 15 B,and 10 12 Be. Elementseparationisachievedbythecorrelationbetweentheenergyloss dE intheion chamberandthetfromthetargettothethinscintillator.TheBethe-Bloch relationgivestheenergylossasafunctionofseveralparameters,includingthedensityof thematerial,meanexcitationpotential,butmostimportantlythecharge Z andthevelocity : dE dx / Z 2 2 f ( ) ThefullexplicitformulacanbefoundinRef.[ 91 ](pg.31).Elementseparationcan beachievedbyplottingtheenergylossasafunction ,orthet.A dE E measurementcouldnotbecleanlymadeduetopoorresolutionofthehodoscope.Figure 4.29 showstheelementforproductsfromthe 24 Obeam. Thenextstepisisotopeseparation.Ideally,adipolemagnetwillseparateisotopesby theirrigidity.Givenasetofisotopeswiththesame Bˆ ,theirvelocitycanbewrittenas: v = L t = Bˆq m = Bˆq Am u / 1 A Where q isthecharge, L thetpathand t thet.Thusthetand massareproportional. 105 Figure4.29: dE vs.ToFforreactionproductscomingfromthe 24 Obeam.Coincidence withaneutronisnotrequired. Inpracticehowever,thedistributionsarebroadwhichmakesseparationsince thereisvariationinboth L andthe Bˆ .Thisvariationarisesfromseveralfactorsincluding theemittanceofthebeam,stragglingwithinthetarget,thenucleardynamicsofthereaction, andthemomentumkickfromneutronevaporation. Theenergyresolutionofthehodoscopeorthethinscintillatorisnotttoseparate isotopes.However,therigidityand L ofthechargedparticlesarerelatedtotheiremittancein thesweeperfocalplane.Tofullyuntangletheisotopes,itisnecessaryexaminethecorrelation betweenthet,dispersiveangleandposition.Theconvolutionbetweenthese parametersismostclearforthelightestisotopesandcanbeseenwhenplottedin3Dasin Figure 4.30 . 106 Figure4.30:3Dcorrelationsfordispersiveposition,angle,andtshowingisotope separationforthecarbonisotopes. Tountanglethiscorrelation,aprojectionontothedispersiveangleandpositionaxesis madeforagivenToFslice.Contoursoftarethentoaquadraticform: f ( x )= a 2 x 2 + a 1 x + a 0 AnexampleofthiscontourisshowninFig. 4.31 fortheoxygenisotopes.Fromthis quadraticexpressionaparameterdescribingbothpositionandangleforconstantToFis constructed: t ( x; x )= x f ( x ) Plottingthisparameteragainstthetshowsisotopeseparation(Fig. 4.31 ). Arotationwillgivea1-dimensionalparametertocutonforisotopeseparation.Thisis accomplishedbyalineartooneoftheisotopes: 107 t corr = t target ! thin + m 0 t ( x; x ) Inaddition,thelargegapofthesweepercreatessomedependenceonthey-positionand yangle.Althoughitisinthenon-dispersivedirection,thesweeperisnotcompletely uniform.Thecorrectioncanalsobetakentohigherordersthanquadraticmakingthemore generalform: t corr = m 1 0 t target ! thin + t ( x; x ;y; y ;::: ) Wherethefunction t ( x;tx;y;ty;::: )isalinearcombinationofallthetermslistedinTable 4.8 withtheirrespectivecots.Whilethismethodprovidesacorrectedt forisotopeseparation,itdoesnotidentifythemass.Asimplewaytodeterminethemassof theoxygenisotopesistoidentifythebeamspot,howeverthisluxurydoesnotexistforthe otherisotopes.Re-examiningtheBetherelationandassumingnon-relativistickinematicsas wellasaconstant Bˆ : E / Z 2 v 2 Recallthat Bˆ = p=q = mv=Z ,thus v 2 = Z 2 ( Bˆ 2 ) =m 2 .SubstitutingthisintotheBethe relationweobtain: E / Z 2 v 2 = Z 2 Z 2 ( Bˆ ) 2 m 2 / m 2 Hence E / A 2 .Inaddition,thetisinverseproportionaltothevelocitythus: ToF / A Z 108 Figure4.31:ProjectionofFig. 4.30 ontothe2DplaneofToFvs.dispersivepositionforthe oxygenisotopes.Thecontouroftisshowninblack.Thiscorrelation,when plottedagainstthetshowsseparationfortheoxygenisotopes(Right),andcan berotatedinthisplaneforthepurposesofmakinga1Dgate. 109 ParameterCoient t target ! thin 7.8231 x -0.5871 x 2 1 : 7182 10 3 x 0.98959 2 x 6 : 1376 10 4 3 x 1 : 51604 10 5 y 6 : 04954 10 2 y 2 5 : 44903 10 3 Table4.8:tcorrectioncotsforisotopeseparation. Aplotof E vs. ToF corr ,shouldthenproduceamatrixwhereeachnucleusisuniquely idensince A and Z arediscrete.Figure 4.32 showshownucleifallinthismatrixfor variouscombinationsof A and Z (somewiththeonlyconstraintbeing Z A . Whatisimportanttonotice,isthatforintegervaluesof A=Z ,telementsfall directlyaboveone-anotherinaverticalline.Makingaplotof E vs. ToF corr shows thisbehaviourandtheisotopesareeasilyidenImmediatelybelowthebeamspotwith thesame A=Z =3is 21 N,makingtheheaviestnitrogenisotopeintheacceptance 22 N. Continuingalong,weseethatotherheaviestelementsare 18 C, 15 B,and 12 Be.Aneutron coincidencegateisrequiredinMoNAtoreducethebackgroundcausedbytheunreacted beam.Nounboundstatesinthelithiumisotopescanbeseenasthestatisticsaretoolow. Theonlylithiumisotopeinacceptanceis 9 Li,howevertherearenoneutronsincoincidence withthisfragment. Theone-dimensionalprojectionsforisotopeseparationareshowninFigure 4.34 and Figure 4.35 foroxygenandnitrogenrespectively.Alineisdrawnwherethegateforthe selectionof 22 Oand 22 Nisplaced. 110 Figure4.32:Matrixof A 2 vs. A=Z withtheconditionthat Z A uptomass ˘ 25.Each pointisaseparatenucleus(someunphysical).Theredlinesindicatecurvesofconstant Z . Figure4.33:Energyloss dE intheionchambervs.Correctedtshowingisotope separation.Acoincidencegatewithaneutronisrequiredtoreducedbackgroundfrom unreactedbeam.Averticallineat A=Z =3isdrawntoguidetheeye. 111 Figure4.34:One-dimensionalparticleidenfortheOxygenisotopes.Agateisdrawn attheverticallinetoselect 22 O.NeutroncoincidencewithMoNA-LISAisrequiredtoreduce thebackgroundfromunreactedbeam. Figure4.35:One-dimensionalparticleidenfortheNitrogenisotopes.Agateis drawnattheverticallinetoselect 22 N.NeutroncoincidencewithMoNA-LISAisrequired toidentifycandidatesforreconstruction. 112 4.2.4NeutronSelection MoNA-LISAisdesignedtodetectneutrons,butitisalsosensitivetobackgroundradiation andothereventswhichcancausescintillation.Theprimarysourceofbackgroundarecosmic muonsand -raysproducedeitherfromthetargetorinthesurroundingenvironment.Events thatcorrespondtoaneutronneedtobecorrectlyidenToensurethattheanalysisis beingperformedonaneventwhichismostlikelytobeapromptneutron,eacheventwithin thearrayistime-sorted. Figure 4.36 showstheneutrontime-otincoincidencewith 22 Ointhesweeper.Fig 4.37 showsthecorrelationbetweentandtotalcharge-deposited.Twopeaksare visibleinthetimeoftspectrum.ThisisduetotheseparationoftheMoNA-LISA tables.ThepeakisfromtheportionofLISAat0degrees(z=7.5m),whilethesecond isfromMoNAwhichisfurtherback(z=8.8m).Thisseparationisonlyvisiblebecause 23 Ohasanunboundresonanceataverylowdecayenergyof E decay =45keV[ 73 , 68 , 63 ]. TheportionofLISAdoesnotseeanyeventsfromthisdecaybecausethe\neutron cone"fromthedecayof 23 Odoesnotintersectthedetector.Alow-energydecaylikethatin 23 Oisforwardfocusedinthelabframe.Theneutronvelocitydistributionisnarrowenough todistinguishthetables.Largerdecayswith E decay ˘ 1MeVwillsmearoutthetime-of- tdistribution,duetoalargerforward/backwardkickinthecenter-of-massframeofthe decayingnucleus. Prompt -raysfromthetargetcanbeseendistinctlyataround25ns,eachpeakcorre- spondstoatableinMoNA/LISA.Theyaregatedoutbyrequiringtheneutront tobegreaterthan50ns.Anadditionalgateisplacedat150ns,aseventsbeyondthisare dominatedbyrandombackground.Examiningthecharge-depositedcanhelpimprovethe 113 Figure4.36:NeutrontspectruminMoNA-LISAwith 22 Ocoincidence.The splittingisduetothenarrowresonancein 23 OandthefactthattheMoNA-LISAtablesare physicallyseparated.Theinsertshows -rayscomingfromthetarget. Figure4.37:Neutrontspectrumvs.ChargedepositedinMoNA-LISAwith 22 O coincidence. -raystypicallydeposits < 5MeVee,whilerealneutronscandeposituptothe beamenergy.Nobackgroundfromcosmicraysisevident. eventselectionsincetherandombackgroundfrom -raysdepositslittlecharge( < 5MeVee). Ahighthresholdof5MeVeeisusedtoexcludethevastmajorityofbackgroundevents.Inad- dition,amultiplicity,consistingofmultipleeventswithinMoNA-LISAabovethisthreshold, 114 isconstructed. Cosmicrayswilldepositaround20MeVeeofenergyinthedetectorandwouldbeun- correlatedintimewithafragment,thustheirdistributionwouldbeuniform.Nosuchband isvisibleinthechargevs.ToFspectraforneutronsincoincidencewith 22 Oor 22 N,andso thisbackgroundisnegligible. 4.2.4.1Two-NeutronSelection Whenconstructingathree-bodysystem,itiscrucialtocorrectlyidentifyeventswhich aretruetwo-neutronevents.ThefactthatMoNA-LISAdoesnotdistinguishbetweenone neutronscatteringtwicefromtwouniqueneutronsinteractingindependentlyintroducesa complication.Fromapure-detectionpointofview,thetwosituationsareidentical.However, thereisamethodforincreasingthelikelihoodthattheselectionofamultiplicity2(and greater)eventwillconsistoftruetwo-neutronscomparedtoasingleneutronscattering twice. First,ahighthresholdisrequiredoneveryhitinMoNA{theenergydepositedmustbe greaterthan5MeVee.Thisremoveseventsthatproduce -raysfrominelasticllyscattering carbon.Forexample,asingleneutroncouldundergoa 12 C(n, )reactionwithinabar andscatteroutofthedetectorvolumewithoutinteractingagain.Theresidual -raycan thenbedetected,andtheeventwillhavemultiplicity2despitethereonlybeingoneneutron interaction. Next,a\CausalityCut"ismade.Thisisacutontherelativevelocity V 12 anddistance D 12 ofthetwointeractionsinMoNA-LISA.Thistechniquehasbeenusedtoenhance thetwo-neutronsignalinseveralpreviousmeasurementsofthree-bodystates[ 26 , 54 , 63 , 25 , 106 , 55 , 107 ].Figure 4.38 showsasimulation,forcomparison,ofa1ndecayanda2ndecay 115 Figure4.38:Relativedistance D 12 andvelocity V 12 for1n(Left)and2n(Right)simulations. Theselectionfor2neventsisshowninbytheshadedblueregion. inthe D 12 vs V 12 phase-space.Eventswhichcomefromoneneutronscatteringtwiceoccur primarilyinbandsalongshort-distanceandhighrelativevelocityorlarge-distanceandlow relativevelocity. Byrequiringalargerelativedistance D 12 > 50cm,eventswhichhaveaclearspatial separationareselected.Whenaneutronscattersitwillloseenergyanditsrelativevelocity willbelessthanthatofaseparateneutronatbeamvelocity.Toremovetheseevents,acut isplacedon V 12 > 12cm/ns,whichisthebeamvelocity. 4.3InverseTracking Inordertomeasureatwo-orthree-bodydecayenergyitisnecessarytoknowthefull4- vectoroftherecoilingfragmentinadditiontotheneutron.The4-vectoroftheneutroncan easilybeobtainedwithknowledgeofitstandposition: n = d t 116 n = 1 p (1 2 ) fromwhichthetotalenergyandmomentumcanbeobtained.Inthecaseofthefragment however,thesequantitiesarenotdirectlymeasured.Instead,thepositionandangleof thefragmentafterexitingthesweeperaredeterminedbytheCRDCsandthesevariables ( x crdc ; crdc x ;y crdc ; crdc y )mustbetransformedintothetarget-frame.Theenergyisdeter- minedbythefragmentsdeviationfromthecentralpathofthemagnet,whichhasaknown rigidity.Oncetheenergyisknown,themomentumcanbecalculated,andthe componentsaredeterminedby T x and T y : p x;y =p 0 = sin ( x;y ) AfulldescriptionofthetechniqueisdescribedinRef.[ 108 , 90 ],onlyasummaryis presentedhere.Itispossibletocalculateion-opticalquantitiesforaparticleasitexitsthe magnet: 0 B B B B B B B B B B B B B @ x crdc crdc x y crdc crdc y L 1 C C C C C C C C C C C C C A = M f 0 B B B B B B B B B B B B B @ x T T x y T T y 1 C C C C C C C C C C C C C A where M f isaforwardtransformationmatrix.Thevariable L istheebetween thelengththeparticletraversed,andthelengthofthecentraltrackwhichisasthe distanceforwhichtheparticleisbyamagneticThisisclosetothedistance fromthetargettoCRDC1.Thedriftlength L 0 isthetheoreticallengthaparticlewould 117 travelifit's Bˆ matchedthesweeperexactly.Thevariable istherelativeenergydeviation giveby: = E E 0 E 0 Where E 0 isgivenbythecentraltrackandthe Bˆ settingofthemagnet.Ahallprobe isplacedinsidethesweeperchambertoprovideameasurementofthemagneticThe ofthedipolehasbeenmeasured[ 90 , 94 ],andtheisusedasaninputtothe ion-opticscodeCOSYINFINITY[ 109 ]. Thematrix M f isusefuliftheincomingdistributionofthebeamisknown.However, thequantitiesthataremeasuredexperimentallyare after thedipole.Thusitisnecessary toinvertthematrixsothattheCRDCvariablescanbeusedasaninputtocalculatethe appropriatequantitiesatthetarget: 0 B B B B B B B B B B B B B @ T x y T T y L 1 C C C C C C C C C C C C C A = M i 0 B B B B B B B B B B B B B @ x crdc crdc x y crdc crdc y x T 1 C C C C C C C C C C C C C A However,todoadirectinversionofthematrixthequantity L needstobeknown a priori .TheapproachtakenbyCOSYistocalculateaninversematrix M i assumingthat thex-distributionofthebeamisadeltafunctionat x =0atthetarget.Thisallowsfora partialinversionofthematrix M f andthecalculationof( x T ; T x ;y T ; T y )and atthetarget allowingthe4-vectorofthefragmenttobedetermined.Theassumptionof x =0,willworsen theresolutionofthedecayenergybutitshouldnotshiftthepeakofthedistribution.Ifthe 118 beamisnotcentered,thenthereconstructedenergyand x anglewillalsoshift.Thedrift lengthusedforreconstructionwas L 0 =1 : 56m.Theinversereconstructioncanbev byexaminingtheneutronandfragmentenergiesandangles,aswellasbycomparingto previousmeasurement. 4.3.1V Theinversetrackingcanbevbyexaminingthereconstructedneutronandfragment anglesandenergies.Inaddition,wherepossible,severalunboundresonancescanbecom- paredtopreviousexperimentsalsoperformedwiththeMoNA-Sweepersetup. 4.3.1.1 22 O+1n Theinversetrackingisvedusingthewell-knownlowenergydecayof 23 O ! 22 O+1n. Thisresonancehasbeenmeasuredpreviouslymultipletimes,andisknowntobelow-lying at E =45keV.Thisisanidealcasetocheckthetracking,asMoNA-LISAhasthehighest detectionfordecays < 100keVandthefullneutron-conefallswithinthedetectors acceptance.Inadditionifthedecayisreconstructedcorrectly,theinversetrackedanglesof thefragmentwillmatchtheneutronangles.Thecentroidsofthereconstructedfragment energyandtheneutronkineticenergyshouldalsolineupand,therelativevelocity v n v f ,mustbesymmetricaboutzero.TheseareshowninFigures 4.39 and 4.40 ,where theneutronandfragmentkineticenergy,relativevelocity,andanglesarecompared.The neutronandfragmentdistributionsareshowninblueandblackrespectively. Asanadditionalcheck,theangulardistributionofthefragmentandtheneutron-fragment openinganglecanbecomparedwithapreviousmeasurementofthesameresonanceusing thesameexperimentalequipmentwithsimilarresolutions[ 73 ].Inthatexperiment 23 O 119 Figure4.39:Comparisonofneutron(blue)andfragment(black)kineticenergyforthedecay of 23 O.Therelativevelocityisshownontheright. Figure4.40:Comparisonofneutron(blue)andfragment(black)anglesatthetarget.The largebinningin T Y isduetothediscretizationoftheMoNAbars. waspopulatedviaknockoutfrom 26 Neandthelow-lyingstatewasobserved.Whilethe experimentalequipmentissimilar,therearesomeForexample,theadditionof moreMoNAbarsandtheiraswellassomeadjustmentsintheSweeper.The widthandoverallshapeofthedistribution-whicharedeterminedbythedecay-agreewell. Thetwo-bodydecayenergyfor 22 O+1nisshowninFig. 4.42 . 120 Figure4.41:Comparisonofmeasuredneutron-fragmentopeningangle n f andfragment angleinsphericalcoordinatesforthedecayof 23 O ! 22 O+1n.Inblackisthecurrent experiment,showninblueisameasurementofthesamedecayusingthesameexperimental apparatus[ 73 ]forcomparison. Figure4.42:1n E decay spectraforthedecayof 23 O.(Left)spectrumobtainedfromthe currentexperiment.(Right)Apreviousmeasurementperformedonthesameexperimental apparatusforcomparisonfromRef.[ 72 ] 4.3.1.2 21 N+1n Inadditionto 23 N,unboundstatesin 22 Nwerealsopopulatedvia1p1nremoval.Gatingon 21 NinthePIDandreconstructing 22 Nshowsa ˘ 600keVpeakingoodagreementwitha previousobservationbyM.J.Strongman[ 110 ]showninFig. 4.43 . 121 Figure4.43: E decay spectrumforthedecayof 22 N ! 21 N+1n.Ontheleftisthespec- trumobtainedfromthecurrentexperiment.(Right)Apreviousmeasurementofthesame resonancealsoperformedwithMoNA[ 110 ]. 4.3.1.3 23 O+1n Unboundstatesin 24 Owerepopulatedthroughinelasticscattering,(d,d').Previousinvariant- massmeasurementsobserveda2 + stateat4.70MeV,anda1 + at5.39MeV,withdecay energies0.51and1.2MeV,respectively[ 70 , 111 ].ShownforcomparisoninFigure 4.44 are theresultsfromthisexperiment,andthoseof etal. [ 111 ]whichwasalsomeasured usingMoNA.Weareunabletoresolvethetwostatesbecauseofthethickdeuteriumtarget. Uncertaintyinthereaction-pointcausesabroadeninginthereconstructionwhichworsens theresolutionofthedecayenergy.IntheworkofRogers etal. [ 70 ],itwasshownthata thinnertargetimprovedtheresolutionenoughtoseparatethetwostates. 4.3.1.4 18 C+1n Unboundstatesin 19 ChavealsobeenpreviouslymeasuredwiththeMoNA-Sweepersetup, [ 112 ]inadditiontoaRIKENmeasurement[ 114 ].A76keVresonancewasobservedinthe 18 C+1nsystem.Figure 4.45 .showstheresultsfromthisexperimentcomparedtoprevious 122 Figure4.44: E decay spectrumforthedecayof 24 O ! 23 O+1n.(Left)spectrumobtained fromthecurrentexperiment.(Right)Apreviousmeasurementofthesameresonancesusing MoNAforcomparison.[ 111 ] Figure4.45:(Top) E decay spectrumforthedecayof 19 C ! 18 C+1nobtainedfromthe currentexperiment.(Bottom)ApreviousmeasurementofthesameresonanceusingMoNA [ 112 , 113 ]. 123 [ 112 ].Thelow-lyingresonanceisapparent. Theinversetrackinginthisexperimentisabletoreproducedecayenergyspectraforat least4tunboundsystemsmeasuredonthesameapparatus,givingetothe calibrations. 4.4ModelingandSimulation Oncethedatahavebeencalibratedandthespectraofinterestgenerated,theparametersfor anobservedresonancehavetobeextracted.Thisisdonebycomparisontosimulation.An in-houseMonteCarlosimulationisusedtogeneratesimulateddatathatareconvolutedwith theexperimentalresolution,acceptanceand.Thesimulationstakeintoaccount theincomingbeamthegeometryofthedetectors{includingthesweeperaperture, aswellastheofMoNAandLISA. Thesimulationisdividedintomultiplesteps.Theinputbeamimpingesatargetofgiven thicknessandthereactionpointisrandomlychosenwithinthetarget.Afterdetermining theappropriateenergyloss,thereactionmechanismissimulated.Inthecaseofneutron emission,theneutron4-vectorsaredeterminedatthereactionpointandpassedtoGEANT [ 115 ].Theremainingchargedfragmentpassesthroughtherestofthetargetandthrougha forwardmapofthemagnet,whichdeterminesthedistributionsofpositionsattheCRDCs. TheCRDCdistributionsarethenfoldedwiththeirresolution. Theneutron4-vectorispassedtoGEANTwhereinteractionsinMoNAandLISAare modelled.UsingMENATE R[ 116 ],theinteractioncrosssectionsforneutronsoncarbon andhydrogenarereferencedtosimulatetheinteractionsintheplastic.Inthecaseswhere theangulardistributionofareactionisknown(e.g.elasticscattering),thisdistributionis 124 usedinthecenter-of-massframe.However,therearemanyinelasticprocessesforwhichthe angulardistributionisnotknownandisassumedtobeisotropic.Theenergydepositedfrom theneutroninteractionisdeterminedbymodelingthelightcollectedinthePMTs,andthe timeoftheinteractionisdeterminedthesamewayasinthedata. Inthecaseofmultipleneutronsimulations,GEANThandlestheneutronsindependently, buttheeventsaremixedafterward.Inatwo-neutrondecay,the4-vectorsofbothneutrons arehandledseparatelytodeterminedthenumberofinteractionsandtheirinteraction times.Tomakethiscomparablewithdata,theinteractionsfromtheneutronsaretime-sorted togiveasinglelistofinteractionsthatcomefromeitherneutron. Oncetheinteractionpositionoftheneutronisdeterminedanditstcalcu- lated,thesimulateddataaretakenandpassedthroughthesameanalysisprocedureasthe data.Inotherwords,theoutputsofthesimulationareusedasiftheyweredatatoconstruct spectrathatcanbedirectlycompared. Variouskindsofdataareusedtoconstrainsimulation.Datatakenonagaseoustarget isusedtothebeamenergywhilethebeamandtargetthicknessareconstrained byreproducingtheunreactedbeaminthefocalplane.Thereactionparametersarethen determinedbyexaminingthedistributionofeventsintheCRDCsforagivenreaction. Thesimulationmustreproducetheobservedneutronkineticenergy,andreconstructthe observedfragmentenergy.Finally,thedecayenergiesandthree-bodycorrelationscontrain theresonanceparameters. 4.4.1IncomingBeamParameters Theincomingbeamparametersweresetbymatchingthedistributionoftheunreacted beaminthefocalplane.Inaddition,theyalsohadtomatchtheinverse-reconstructed 125 distributions( T X ; T Y ;y T )usinganenergyfromthedataonagaseoustarget.Thenecessary targetthicknessinsimulationdeviatesslightlyfromwhatisobserved.Thisisbecause thesimulationdoesnotaccountfortheKaptonandMylarwrapping,orthecurvatureof thetarget.Becauseofthis,thethicknessisvariedinthesimulationuntilthefragment energyisproperlyreproduced.Thisensuresthatfragmentswiththecorrectenergyare beingtransportedthroughthemodeloftheexperimentalsetup.Thusthethicknessin thesimulationisanethickness.Athicknessof650mg/cm 2 ofLD 2 matchesthe incomingbeamdistributionsverywell 4.46 .Table 4.9 summarizesthenecessaryincoming beamdistributiontoreproducetheunreactedbeaminthefocalplane. TheA1900momentumslitsweresetto2% p=p whichresultedinanenergyspreadof 1.2%.Giventhatthemagneticquadrupolebeforethetargetwassettoa Bˆ =4 : 03146Tm, thisgivesabeamenergyfor 24 Oof E beam =83.4 1MeV/uwhichisusedinsimulation. 4.4.2ReactionParameters The1 p and1 n -knockoutmechanismsweresimulatedbyremovingnucleonsfromthebeam andgivingtheresultingsystemamomentumkickinboththeparallelandtransversedirec- tion.TheparallelmomentumkickwasparameterizedbasedontheGoldhaber[ 117 ]model whilethetransversekickwastakenfromVanBibber'smodel[ 118 ].Inthesemodels,the parallelandtransversekicksaregaussianandbywidths ˙ ? ,and ˙ k whicharefreepa- rameters.ThewidthsarebymatchingthedistributionoffragmentsintheCRDCsfor agivenreaction.Anadditionalmultiplicativefactorisappliedtoslowthebeamwithinthe targetandisattributedtodissipativeinteractionswithinthetarget.Table 4.10 summarizes thereactionsandtheparallelandtransversewidthsusedtoreproducethem. 126 ParamaterSimulationSetting eBeam83.4BeameneryinMeV/u. dTarget650LD 2 Thicknessmg/cm 2 . bSpotCx0 x centroidofincomingbeam. bSpotCtx-0.01 x centroidofincomingbeam. bSpotCy0.0 y centroidofincomingbeam. bSpotCty-0.001 y centroidofincomingbeam. bSpotDx0widthof x distribution. bSpotDy0.002widthof y distribution. bSpotDtx0.007widthof x distribution. bSpotDty0.008widthof y distribution. bSpotCx20 x centroid,2ndbeamcomponent. bSpotCtx20.005 x centroid,2ndbeamcomponent. bSpotCy20.003 y centroid,2ndbeamcomponent. bSpotCty20.006 y centroid,2ndbeamcomponent. bSpotDx20.00widthof x distribution,2ndcomponent. bSpotDy20.01widthof y distribution,2ndcomponent. bSpotDtx20.005widthof x distribution,2ndcomponent. bSpotDty20.008widthof y distribution,2ndcomponent. normscale10.70relativeintensityof1stcomponent(70%). normscale21relativeintensityof2ndcomponent(30%). crdc1MaskLeft0.15+ x edgeofCRDC1inm. crdc1MaskRight-0.15 x edgeofCRDC1inm. crdc2MaskLeft0.15+ x edgeofCRDC2inm. crdc2MaskRight-0.1305 x edgeofCRDC2inm. crdc2MaskTop0.15topedgeofCRDC2inm. crdc2MaskBot-0.15bottomedgeofCRDC2inm. crdc2dist1.55distancebetweenCRDCsinm. cosymap"m24O Jones320A"inverseandforwardmap Table4.9:Simulationparametersfortheincomingbeamdistribution.Determinedbymatch- ingunreacted 24 Ointhefocalplane. Reaction ˙ ? (MeV/c) ˙ k (MeV/c) v shift 24 O(-1n) 23 O92640.9875 24 O(-1p) 23 N2751020.9550 Table4.10:ParallelandperpendicularglauberkicksusedtoreproducetheCRDCdistribu- tions. 127 Figure4.46:Comparisonbetweensimulation(blue)anddata(black)fortheunreacted 24 O beaminthefocalplane. 4.4.2.11nKnockout,V 22 O Usingtheincomingbeamsettingsdeterminedbytheunreacted 24 Obeamsetting,theparallel andtransversekicksformodelingthe1nknockoutreactionto 23 Oareconstrainedbydata 128 Figure4.47:Reconstructedanglesandtargetpositionusinga4-parametermap.Thesimu- lation(blue)iscomparedtodatafortheunreacted 24 Obeam(black). Figure4.48:Comparisonofreconstructedkineticenergydistributionsbetweensimulation (blue)anddata(black)fortheunreacted 24 Obeam.Thereconstructedenergyisafter passingthroughthefullLD 2 target. with 22 OincoincidencewithneutronsasshowninFig. 4.49 .Settingsforthebeamcanbe foundinTable 4.9 ,andthereactionparametersinTable 4.10 .Inordertoreproducethe distributionsintheCRDCsitwasnecessarytothesignontheincoming x centroid, indicatingthatthe 22 Ofragmentscomeinatahighangle.Itisclearthatthemomentum distributionofthesefragmentsisnotfullyinacceptance. 129 Figure4.49:Comparisonoffocalplanepositionandanglesbetweensimulation(blue)and data(black)for1nknockoutto 23 O,whichthendecaysto 22 O. 4.4.2.21pKnockout,V 22 N Thedatafor 22 Nincoincidencewithneutronsconstraintheparametersformodeling1p knockoutinthesimulation.TheCRDCdistributionsforthisreactionareshowninFigure 130 Figure4.50:Reconstructedkineticenergyforthe 22 Ofragmentscomingfromthe1nknock- outreaction.Simulationresultsareshowninblueandthedatainblack. 4.51 .SettingsforthebeamcanbefoundinTable 4.9 andthereactionparametersinTable 4.10 . 4.4.2.3(d,d'),InelasticExcitation The( d;d 0 )reactioncanbeapproximatedusingtheglobalopticalmodelsthatareavailable. Theangulardistributionforinelasticscatteringof 24 OonD 2 wasestimatedusingFRESCO [ 119 ]andaglobalopticalpotentialfor 24 O+ d [ 120 , 121 ].Asthedeformationlengthisnot known,thedeformationlengthfor 12 Cwasused.Thetialcrosssection, d˙=d was calculatedforthe0 + ! 2 + transitionin 24 O.Theangulardistributionwasthenrandomly sampledfrominthesimulationandthereactionkinematicstreatedastwo-bodykinematics. Theexcited 24 Othendecayed,andtheresultingreactionproductspropagatedthroughthe restofthesimulation. Sincethethree-bodycorrelationsanddecayenergiesarerelativemeasurements,thedata arenotsensitivetothereactionmechanism.Identicalspectraareproducedwhetherornot thereactionmechanismismodelledappropriately.Thereisnodiscernablebetween 131 Figure4.51:Comparisonoffocalplanepositionandanglesbetweensimulation(blue)and data(black)for1pknockoutto 23 N,whichthendecaysto 22 N. usingtheFRESCOcalculationoraglauber-kickwithoutanystrippinginthedecayenergies andJacobispectra. 132 Figure4.52:Reconstructedkineticenergyforthe 22 Nfragmentscomingfromtheproton knockoutreaction.Simulationisinblueanddataareinblack. Figure4.53:Centerofmassangulardistributionforinelasticscatteringof 24 Oon d at82.5 MeV/u(lab),estimatedwithFRESCO,aglobalopticalmodel,andthedeformationlength of 12 C. 4.4.3NeutronInteractionandMENATE R TheneutroninteractionsaremodelledwithGEANT[ 115 ]andtheMENATE Rpackage [ 116 ].MENATE Rprovidescrosssectionsforinteractionsonhydrogen,carbon,iron,based onpreviousmeasurementsandthecode'sabilitytoreproducetheneutrondetection forplasticscintillators.MENATE Rcontainsseveralinelasticprocessesforneutronson 133 carbon,forexample 12 C( n;np ) 11 B, 12 C(n, )intheenergyrangeof0to100MeV.However, dataforthesereactionsisscarceandtheangulardistributionsarenotknownandtheyare crudelyapproximatedinsomecases.Forexamplethe 12 C( n; )interactionismodelledwith theemissionofonlyone -rayat4MeV,wheninreality 12 Ccanbeexcitedbeyondthe excitedstateandemitacascade.Figure 4.54 showsasummaryoftheinelasticprocesses includedinMENATE R. Figure4.54:Breakdownof 12 C(n,*)cross-sectioninMENATE R.(Left.)MENATE Rcross sectionswithoutanyadjustment.Thebluecurveisthesumofallcross-sectioninthe energyrange40-100MeV.(Right)ThetotalMENATE Rcross-sectionsaftermo comparedtotheDelGuerracompilation[ 122 ]andtheENDF[ 123 ]evaluation. The 12 C( n;np ) 11 Bcrosssectionisoverpredictedinthesimulation.Summingthecom- ponentsofeachinelasticprocess,thetotalinelasticcrosssectionisapproximately100mb toolargecomparedtothe1976compilationbyDelGuerra[ 122 ],andtheENDF[ 123 ] andJENDL-HE2007[ 124 ]databases.Inaddition,theonlymeasurementofthisreaction at90MeV[ 125 ]reportedthiscrosssectionasafactor2lowerthanwhatisincludedin MENATE R.Forthisreason,thiscross-sectionwasmoinMENATE Rtobetteragree withthetotalinelasticcrosssection.Fig. 4.54 . Thehighthresholdof5MeVeeremoves -raysthatareproducedby 12 C( n; ),which arenotappropriatelymodelledbyMENATE R.ThebestagreementwithMENATE Ris 134 achievedwhenthisthresholdisapplied. 4.4.4OtherParameters ThemapsfortheSweepermagnetarebymeasurementfromaHallprobeinserted intheofthemagnet.Thegeometricacceptancesofthedetectorsaredeterminedby theirgeometricallayoutinthesimulation,whichisidenticaltoexperiment.Thesteelvacuum chamberofthemagnetisincludedinthesimulation.ResolutionsforMoNAandLISAare includedasgaussiandistributionsandthequantizationofthebarsisalsotakenintoaccount. 4.4.5Cuts Additionalcutsweremadetothesimulationtoremovetracksthroughthemagnetthatare unphysical.Thesecutsweremadetotheemittance,showninFigure 4.55 .Thephysical apertureoftheCRDCsareincludedinthesimulationbuthardcutsaremadeontheir acceptanceasaredundancy.Inaddition,thesimulatedneutronsarerequiredtohavephysical tsandbeabovethe5MeVeethreshold.Identicalcausalitycutsarealsomade tothesimulation.Inthiswaythesimulationsaremadedirectlycomparabletodata. 4.4.6DecayModels 4.4.6.1Oneneutrondecays Thedecayenergyforasingleneutrondecayisgivenbyanenergy-dependentBreit-Wigner oftheform(describedinSection 2.1.1 : ˙ l ( E ; E 0 ; 0 ) / l ( E ; E 0 ) [ E 0 E + E ; 0 )] 2 + 1 4 l ( E ; 0 )] 2 135 Figure4.55:CRDC1Xvs.CRDC1 X forall 22 Ofragmentscoincidentwithaneutron.A gate,calledanXTXcut,isdrawnaroundthedataandappliedtosimulation.Thisisto rejectfragmentswhichmayhaveastrangeemittanceinthesimulation. Wherethenormalizationisarbitrary.Thismodelisusedtodeterminetheneutronenergy andangularmomentum ` relativetothefragmentinthecenter-of-massframe,andthusthe centroid E 0 andwidth 0 oftheunboundresonance.Bothparticlesarethenboostedback intothelabframeinthesimulation.Thislineshapewasusedformodelingthe1ndecayof 23 Oand 23 Npopulatedbyneutronandprotonknockoutfrom 24 O. 4.4.6.2Twoneutrondecays Thedecayoftheexcitedstatein 24 Owasmodelledasatwo-neutrondecayto 22 O.Thedecay wasmodelledasmultipletwo-bodydecaysusingVolya'sdescription[ 77 ]forasequential decay.Theintermediatestatein 23 Oiswellknownandhasbeenmeasuredmultipletimes, andtheenergyandwidthwerefortheminimization.Thetotalthree-bodyenergywas determinedbythe 22 O+2nspectrawithcausalitycuts.Theenergyandwidthwere leftasfreeparametersaswellastheamplitude.The ` valueisundetermined,astherewas noobservablechangeinthelineshapebetween ` =1,or ` =2onceresolutionswerefolded 136 withthesimulation.Inaddition,Volya'smodelassumesthatbothneutronsdecaywiththe same ` valueandcomefromthesamesingleparticleorbtial.Althoughthisisnotentirely thecaseinthepresentdecay,asdiscussedinChapter5,itisttodescribethedata. Sincethelow-lyingintermediatestatein 23 Odecaysbyemissionofa d 5 = 2 neutron,the ` valuewastakentobe2forbothdecays. Forthephase-spacemodel,theTGenPhaseSpaceclasswasusedasimplementedinROOT [ 126 ].Thismodeluniformlysamplesthephasespaceofinvariantmassofthefragment- neutronandneutron-neutronpairs( M 2 f n vs. M 2 n n )giventhemassesofeachparticleand total n -bodyenergyandwidthasaninput.Itisusedasabaselineforthescenariowithno correlationssincethismodelassumesnointeraction. Forthedi-neutrondecay,thetwoneutronsareassumedtobeina 1 S 0 withascatteringlengthof a = 18 : 7fm.Inthismodel,thedi-neutronclusterseparates fromthecoreandthenproceedstobreakupwiththearelativeenergy.Thedistribution forthetotalthree-bodyenergyaswellasthe n n relativeenergyisdeterminedbyVolya's di-neutronmodeldescribedinSection 2.2.2 .Inboththedi-neutronandsequentialmodels, thetwo-neutronsareemittedisotropicallyintherestframeoftheneutron-core/two-body- subsystem.Additionaldetailsontwo-neutrondecayscanbefoundinSection 2.2 . Afterfoldingwithresolutionsandacceptance,allexperimentalobservablesofinterestare modeledinthesimulationundertassumptionsandthebvaluesdeterminedby log-likelihoodratiodescribedinthenextsection. 137 4.4.7FittingandLikelihoodRatio Forpoissonerrors,thelog-likelihoodfunctioniswrittenas: L = Ln [ ]= X i Ln n i i e i n i ! ! = X i Ln ( n i !)+ i n i Ln ( i ) Sinceweareonlyconcernedwiththesetof i generatedbyahypothesisforwhichthe likelihoodismaximum,theabsolutemagnitudeisirrelevant.Thuswecanaddanyconstant tothisexpressionwithoutectingtheminimization.Considerthelikelihoodof n i given theexpectationof n i : Ln [ ]= X i Ln n i i e i n i ! ! + X i Ln n n i i e n i n i ! ! = Ln ( n i ; i ) ( n i ; n i ) Whichisthelikelihoodratio.Thisexpressionreducesto: Ln [ ]= X i i n i + n i Ln n i i Thisquantityisdistributedas ˜ 2 = 2inthelimitoflarge n .Inthiscasethenullhypothesis H 0 isthesetof i whoseprobabilitydistributionisdeterminedbytheory(resonanceparameters), andthealternativehypothesis H 1 isthesetofdatapoints n i undertheassumptionthat theytooaresampledfromaprobabilitydistribution(the\true"resonanceparameters).The dataareassumedtobepoissondistributed.Let i denotetheresonanceparameterswhich determinetheprobabilitydistributionof i .Theminimization: @L @ i =0 138 givestheparametersforwhichthelikelihoodismaximum(i.e.the\b Oncethesimulateddistributionsaremadecomparabletodatabyfoldingwithreso- lution,acceptances,andthelog-likelihoodratiocanbecalculatedforagiven setofenergiesandwidths,orundertmodels(e.g.di-neutron,phase-space).Plot- ting Ln ( )asafunctionoftheseparametersthenthestatisticalboundariesvia rejection/acceptancewithinacriticalregion.Inthecaseof ˜ 2 thisisoftenthe p value. Insomecasesitisusefultominimizeonalinearcombinationof Ln [ ]tobetter constrainthesimulation.Forexample,onthesumofthemult==1andmult==2 gatedspectrawillforcethemodeltoreproducetheratioofmultiplicitieswhereasona singlespectrumalonecancauseover-predictionoftheother(andviceversa).Theoptimum combinationofspectradependsontheproblemathandandcanvarydependingonwhat onewantstoaccomplish.Inthecaseofmultiplefreeparameters,thecriticalregioncanbe determinedby: i 2 ˆ ˜ 2 <˜ 2 min + ˜ 2 ( ) ˙ Where isnumberoffreeparametersintheand ˜ 2 ( )isthechangein ˜ 2 from theminimumsuchthatintegrationoverthe ˜ 2 probabilitydistributionfunctiongivesthe appropriate ˙ limits.UsingWilk'sTheorem,thestatistic Ln [ ]isdistributedas ˜ 2 = 2 inthelimitofsamplesize N approachingy.Forsimplicity,onecanapproximatethe criticalregionfor Ln [ ]withthecriticalregionof ˜ 2 = 2,althoughMonteCarlomethods willmoreaccuratelythisboundary. 139 Chapter5 ResultsandDiscussion 5.1 22 O+2n Thetwo-andthree-bodydecayenergyspectrafor 23 Oand 24 Oareshownrespectivelyin Fig. 5.2 .Alsoshownisthethree-bodydecayenergyspectrumfor 24 Owithcausalitycuts applied.Thespectrashownrequirethattheneutrontime-otfallbetween50and150 ns.However,therequirementofgoodtandmultiplicity 2isnott toseparatetwo-neutroneventsfromone-neutronevents.Toaccomplishthis,twocutsare necessary:(1)thecausalitycut,and(2)ahighthresholdof5MeVeeoneachinteraction. Thecausalitycutsarecutsonrelativevelocity V 12 anddistance D 12 betweenthe twointeractionsinMoNA.Byrequiringalargerelativedistance,nearbyscatterisremoved. Anadditionalcuton V 12 willalsoremovescatteringeventsasascatteredneutronwilllose energy.MoreinformationonthecausalitycutscanbefoundinSection 4.2.4.1 . Thereasonforthishighthresholdisbecause 22 Oispopulatedinmultiplewaysinthis experiment,withthedominantcontributioncomingfromone-neutronknockoutfrom 24 O whichdirectlypopulates 23 OasillustratedinFig. 5.4 .Thisresultsinanoverwhelming contributionof1neventswhichneedtobesuppressed.The5MeVeethresholdeliminates eventswhereaneutroninteractsintheplasticandproducesa -raywhichcaninteractin anotherlocationgivingtheappearanceoftwoneutroninteractions.Itwasobservedthrough simulationthatapplyingthisthresholdgreatlyreducedthecontributionof1nscatterevents 140 Figure5.1:(Left)AcceptanceofMoNA-LISAfora1ndecayasafunctionof E decay .(Right) Acceptancefora2ndecaywithcausalitycutsapplied. withinthecausalitycuts. Thelow-energypeakin 23 Oisevidentinthetwo-bodyspectrum,correspondingtothe decayofa5 = 2 + holestate.Itisconsistentwithpreviousmeasurement[ 63 ].Thethree-body spectrumfor 24 Owithcausalitycutsshowsanapparentbroadpeakat ˘ 1MeVwhichis tlyhigherthanthepreviouslyreportedvalueof ˘ 600keV.However,hereonemust becareful,astheacceptanceofthecausalitycutsdependsstronglyontheof MoNAandLISA.Unlikethe1nncy,the2nydoesnotpeakatzerodecay energyasshowninFig. 5.1 ,andsoaproperminimizationisnecessarytodeterminethe energy. 141 Figure5.2:(a)1ndecayenergyspectrafor 23 Owithcontributionsfromneutron-knockout andinelasticexcitation.(b)Thethree-bodydecayenergyfor 22 O+2nforallmultiplicities 2.(c)Thethree-bodydecayenergywithcausalitycutsapplied.Theinsertsshowa logarithmicview. 142 Figure 5.3 showsthethree-bodycorrelationsforthe 22 O+2nsysteminthe T and Y Jacobicoordinatesystems.Thespectrahavethecausalitycutsapplied,aswellasan additionalrequirementthat E 3 body < 4MeV. Evenwithoutathereareseveralfeaturesthatindicateasequentialdecay.Therelative energyinthe Y -system,whichrepresentstheneutron-coreenergy,peaksaround0and1. Thisisindicativeofanunevensequentialdecaywhereoneneutronishigh-energy,andthe otherlow.Inadditiontherelativeangleinthe T -systempeaksstronglyat-1and1with avalleyinbetween.Thisisalsotheresultoftwoneutronswithdisparateenergies.The T -systemisconstructedinthecenter-of-massframe.Sinceoneneutronimpartsabigger kickthantheother,whenthelessenergeticparticleisboostedintotheframeofthemore- energeticone,itsdirectionappearsbackward(eventhoughitisisotropicallyemittedinthe n-core+nframe).Thisresultsinpeaksat-1and1,or0and180degrees. Notealsothatthespectralackthefeaturesonewouldexpectfromadi-neutrondecay. Theratio E x =E T inthe T -systemdoesnotpeakat0,butratherat ˘ 1 = 2implyingthatthe neutron-neutronenergyislargerelativetothetotalavailableenergy.However,thespectra donotcontain100%true2nsignals,andonemustrelyonsimulationtoestimatetheamount ofcontaminationfrom1nscatter. Aspreviouslymentioned, 22 Ocanbepopulatedbymultiplepaths:(1)directpopulation of 23 Ovia1nknockoutfrom 24 O,and(2)inelasticexcitationto 24 O whichdecaysby two-neutronemission.Figure 5.4 illustratesthesetwopopulationpaths. Sincebothpathsmaycontribute,itisimportanttoconsiderboththeoneandtwo- neutrondecayenergyspectrainFig. 5.2 simultaneously.Thisisdonebyasimultaneous minimizationofthelog-likelihoodratio, Ln on(a)the 22 O+1 n decayenergy,(b) the 22 O+2ndecayenergy,(c)andthe 22 O+2ndecayenergywithcausalitycuts.This 143 Figure5.3:Jacobirelativeenergyandanglespectrainthe T and Y systemsforthedecay of 24 O ! 22 O+2nwiththecausalitycutsappliedandtherequirementthat E decay < 4 MeV. methodprovidesadditionalconstraintsovereachspectraindependentlybyrequiring themodeltoappropriatelyreproducetheratioof1to2multiplicityevents.Forexamplea pure1nmodelwillthelow-lyingpeakin 23 Obutwillunabletoalsothecausalitycuts providedareal2nsignalispresent. Toallowforthedirectpopulationof 23 Othedecayoftwopreviouslyreportedstates wasincluded:thelow-lyingsharpresonanceat45keV[ 73 , 68 , 72 , 63 ]andthexcited 144 Figure5.4:Levelschemeforthepopulationofunboundstatesin 23 Oand 24 Ofromneutron- knockoutandinelasticexcitation.Hatchedareasindicateapproximatewidths. stateat1.3MeV[ 74 ].Theone-neutrondecaysusetheBreit-Wignerlineshapediscussed inSection 2.1.1 .Thetwo-neutrondecaywasmodelledusingtheformalismofVolya[ 127 ], detailedinSection 2.2.2 .Thebestforthedecayof 23 Oand 24 OisshowninFig. 5.5 . 145 Figure5.5:(a)1ndecayenergyspectrafor 23 Owithcontributionsfromneutron-knockout andinelasticexcitation.(b)Thethree-bodydecayenergyfor 22 O+2nforallmultiplicities 2.(c)Thethree-bodydecayenergywithcausalitycutsapplied.Directpopulationofthe 5 = 2 + stateand3 = 2 + statein 23 Oareshownindashed-redanddashed-greenrespectively. The2ncomponentcomingfromthesequentialdecayof 24 Oisshownindashed-blueand decaysthroughthe5 = 2 + state.Thesumofallcomponentsisshowninblack.Theinserts showalogarithmicview. 146 ParameterDeviationin E 3 body ˙ = p V [keV] InputBeam x inSim.( 15[mrad])14 Target E loss ( 5% ˘ 10[MeV])10 DriftLengthdL(1.56vs.1.7[m])40 CRDC1X( 1[mm])2 CRDC2X( 1[mm])8 Global t mean forMoNA-LISA( 0 : 2[ns])6 Total45 Table5.1:Errorbudgetforthethree-bodydecayenergy.Thedominantcontributionisthe driftlength,d L . Usingpreviouslyreportedvaluesforthestatesin 23 O[ 73 , 68 , 72 , 63 , 74 ]weobtain goodagreementwiththedata.Thebforthethree-bodydecaygivesanenergyof E =715 110( stat: ) 45( sys: )keV,whichagreeswithpreviousmeasurement[ 63 ],although thecentralvalueisapproximately100keVhigher.Thesystematicerrorwasdetermined byadjustingseveralcalibrationparametersandrepeatingtheminimizationtoobtainthe intheminimumenergy.TheerrorbudgetisgiveninTable 5.1 . Onlyanupper-limitcanbeplacedonthewidthwith < 2MeV,asthewidthis dominatedbytheexperimentalresolutionandtheBreit-Wignerlineshapesaturatesatlarge WithinVolya'sdescription,evenaninputwidthof=6MeVproducedaresonancewith aFWHMof ˘ 400keV.Thebestisshownusingthesingle-particlewidth spdw =120 keV.Using S 2 n =6.93 0 : 12MeV[ 8 ],weobtainanexcitationenergyforthethree-body stateat7 : 65 0 : 2MeVwithrespecttothegroundstateof 24 O.Nobranchingthrough the3 = 2 + statein 23 Owasnecessarytofullydescribethedata.AllthreespectrainFigure 5.5 arewelldescribedbyasinglestatein 24 O.Eventhoughthedataaredominatedby1n contributionsfromdirectpopulationof 23 O,aone-neutronhypothesiscannotdescribethe causalitycutsasevidencedbyitslowamplitudeinthecausalitycutspectrumofFig. 5.5 (redline). 147 Itispossibletothethree-bodyenergywithanyofthetwo-neutronmodelsconsideredin thisanalysis.Sincethatspectrumonlycontainsrelativeenergyinformation,theorientation ofthenucleonsisunimportantsolongastheirenergyaddsuptothethree-bodystate.The Jacobicorrelationshowever,apowerfultoolfordiscriminatingbetweenthedit possibledecaymodes.Figure 5.6 showstheexperimentaldatainthe T systemnextto thepredictionsforeachthree-bodymodel.ShownherearethesamedatainFigure 5.3 howevernowonecanseehowtherelativeenegy E x =E T iscorrelatedwithanglecos( ).The amplitudesofthesimulationaresettotwicetheintegralofthethree-bodyspectrumwith causalitycuts. ItisimmediatelyevidentbyFigure 5.6 thatadi-neutronorphase-spacemodelwillbe unabletodescribethedata.Bothmodelshavetheincorrectincos( ),anddo notreproducethebellshapein E x =E T around ˘ 1 = 2.However,comparisontothephase- spacemodelisstilluseful,asitservestoillustratewhatthe\base-line"lookslikeinthe casewherenocorrelationsarepresent.Thedi-neutronisshownaswell.Eventhough itisnotexpectedbasedonthestructureoftheenergylevels,inprinciplethedecayis aninterferenceofbothadi-neutronandsequentialemission.It'sinclusiondemonstrates thattheobservedcorrelationsaredistinctfromdi-neutronemissionwhichother2nunbound nucleiareinterpretedasemitting[ 51 , 26 , 54 ].Whilethesequentialmodeldescribesthevalley incos( ),andbell-shapein E x =E T ,itpredictsthevalleytobedeeperthanisobserved.This canbeexplainedbycontaminationfrom1neventsfromthedecayof 23 O. Figure 5.9 showsthepredictionofthebofthedecayenergyspectraontopofthe observedJacobispectra.Thesespectraarereproducedandnot.Hereweassumethe decayisasequentialprocesspassingthroughthelow-lyingintermediatestatein 23 O.The agreementisverygood.Shownindashed-blueisthecontributionfromthesequentialdecay, 148 Figure5.6:Jacobirelativeenergyandanglespectrainthe T systemforthedecayof 24 O ! 22 O+2nwiththecausalitycutsappliedandtherequirementthat E decay < 4MeV.Shown forcomparisonaresimulationsofseveralthree-bodydecaymodes.Asequentialdecay(b), adi-neutrondecaywith a = 18 : 7fm(c),and(d)aphase-spacedecay.Theamplitudesare setbytwicetheintegralofthethree-bodyspectrumwithcausalitycuts. 149 andingrayisthecontributionfromthe1ndecayof 23 O.TheobservedshapeoftheJacobi spectraarewhatweexpectgivethattheintermediatestatein 23 Oislow-lyingandnarrow. Inthetwo-protondecayof 6 Be[ 43 ]itwasobservedthatthesequentialdecaymechanism wassuppresseduntilthedecayenergythefollowingrelation: E 3 body > 2 E 2 body + 2 body whichiscertainlyhere.Itisinterestinghowever,thatafullthree-bodycalculation isnotnecessarytodescribethedecayobservedhere.Ontheprotondrip-line,severaltwo- protonemittingnucleishowsignsofsequentialemissioncompetingwithtruethree-body processes[ 30 , 43 , 44 ]. Thesequentialdecayisalsosupportedwithintheshellmodel.The( d;d 0 )reactionmech- anismwillpopulateparticle-holestatesin 24 O.SuchaisillustratedinFigure 5.7 andwouldbeaparticle-holestatewithspin-parity2 + or4 + .TheUSDBhamiltonian predictsa2 + and4 + roughly300keVapartabovethetwo-neutronseparationenergy.Itis possiblethattheobservedresonanceisasuperpositionofboththe2 + and4 + states,how- everasingleresonanceisttodescribethedata.Iftheobservedpeakcorresponds tothe4 + ,thentheagreementwiththeUSDBinteractionwouldbeverygoodasshownin Figure. 5.8 .A0 + isalsopredictedtobenearbyinenergyat7.5MeV.However,the0 + wouldbeatwo-particletwo-holeexcitationwhichcannotbemadeby( d;d 0 ).Inaddition, anotherstatehasbeenobservedabovethetwo-neutronseparationenergyin 24 Oat7.3MeV. Thisstatehowever,wasobservedtoundergoaone-neutrondecayandwasconcludedtohave (1 s 1 = 2 ) 1 ( fp ) 1 withnegativeparity[ 71 ]. Inthispicture,theneutrondecaysfromthe 0 d 3 = 2 orbital,andthesecondfrom 150 Figure5.7:Neutronforthetwo-neutronsequentialdecay.Filledcirclerepre- sentparticlesandfadedcirclesrepresentholes.The2 + or4 + of 24 Oisshown onthefarrightandisaparticle-holeexcitation.The5 = 2 + statein 23 Oisaholestatewhich decaystothetwo-particletwo-holeof 22 Owhichisasmallcomponentofthe groundstatewavefunction. the 0 d 5 = 2 ,withthebeingadecaytoatwo-particletwo-holeofthe groundstateof 22 O.Althoughthespectroscopicfactorforthestepissmall,itis asthegroundstateof 22 Ocontainsamixtureofthe (1 s 1 = 2 ) 2 (0 d 5 = 2 ) 2 at roughly14%asillustratedinFigure 5.7 Itisexpectedthattheneutronshaveangulardistributionseofthefactthat theybothhave ` =2.ItisatthispointthatVolya'smodelbreaksdown,asitisassumed thattheneutronsoriginatefromthesameorbital,carrythesameangularmomentum,and areemittedisotropically.Atpresent,theangulardistributionsoftheneutronsareleftas isotropicandthe J + ofthestateistentativelyassignedto2 + or4 + .Thisdoesnot thedecayenergymeasurementasitisascalar.Afullthree-bodycalculationthatproperly includestheangulardistributionscouldprovidevaluableinsightintothedecaymechanism. 151 Figure5.8:Comparisonofexperimentallymeasuredstatesin 24 OwithUSDBshellmodel predictions.DatatakenfromRefs.[ 70 , 71 ].Thestateobservedinthepresentworkisshown inblue. 152 Figure5.9:Jacobirelativeenergyandanglespectrainthe T and Y systemsforthedecay of 24 O ! 22 O+2nwiththecausalitycutsappliedandtherequirementthat E decay < 4 MeV.Indashed-blueisthesequentialdecaythroughthe5 = 2 + statein 23 O.Theremaining false2ncomponentsfromthe1ndecayof 23 Oareshowninshaded-grey.Thesumofboth componentsisshowninsolid-black 153 5.2 22 N+1n Inadditiontoobservingthesequentialdecayof 24 O,unboundstatesin 23 Nwerealsopopu- latedviaone-protonknockout.Thetwo-bodydecayenergyfor 23 NisshowninFigure 5.10 afterselectionof 22 NinthefocalplaneandrequiringavalidneutroninMoNA-LISA. Figure5.10:Two-bodydecayenergyfor 22 N+1n. Asofpresent,therearenoreportsofunboundstatesin 23 Nalthoughitisknowntohave aboundgroundstate[ 128 ]withahalf-lifeofabout14ms[ 129 ].Inaddition,thereareno reportsofanyboundexcitedstate.Thedatashowtwopeaksaround ˘ 100keVand ˘ 1 MeV,bothofwhicharelessthanthe2nseparationthresholdin 23 N,whichisat S 2 n =3 : 07 MeV[ 8 ]andcorrespondsto1.3MeVinFigure 5.10 .Ifthe ˘ 1MeVpeakdecaystothe groundstateof 22 N,itwouldcorrespondtoastateatapproximately2.8MeVwithrespect tothegroundstateof 23 N.Duetothein(Fig. 5.1 ),theintensityofthe ˘ 1MeVpeakisnearly3 10timeshigherthanthelower-energypeak. 154 Figure5.11:Shellmodelpredictionsfor 23 NwiththeWBPandWBTHamiltoniansaswell astheContinuumShellModel. 5.2.1Interpretation Itisimportanttonotethat 22 Nhastwoboundexcitedstatesthattheneutrondecayof 23 Ncouldbranchto.Astudyofin-beam -rayspectroscopyof 22 Nobserveda183keV andan834keVtransitionincoincidence,whichhavebeeninterpretedasthelevelscheme inFigure 5.11 .Althoughthespinandparityofthegroundandexcitedstatesof 22 Nare tentative,boththeWBTMandWBTinteractionspredictanorderingof0 ; 1 ,and2 fortheground,andsecondexcitedstates,respectively.Itisthenpossible,thatthe observeddecayenergyisnotthetrueenergyofastatein 23 N,butratherthein energybetweenastatein 23 Nandanexcitedstatein 22 N.Inordertodistinguishthetwo, itwouldbenecessarytohave -raydetectionaroundthetarget.Suchdetectorshavebeen usedwithMoNAinthepast,e.g.CAESAR,howeverthismeasurementwasnotwithinthe originalscopeoftheexperiment.Thus,no -rayinformationispresentinthecurrentdata. 155 Becauseofthisambiguity,manylevelschemesanddegeneraciesarepossible.Atbest,one canonlyspeculatewhileguidedbytheoreticalcalculations. Itisnotlikelytopopulatea5 = 2 statein 23 Nbyprotonknockoutfrom 24 O,whichhasa thegroundstatecof0 + .Topopulatea5 = 2 staterequiresan ` =3component ofthewavefunctionandthegroundstategurationof 24 Ohasnone.A3 = 2 statecanbe populatedbyremovalofa ˇ 0 p 1 = 2 proton,inwhichcasethesingleprotonleftinthe ˇ 0 p 1 = 2 orbitcouplestoa2 + ofneutrons( 1 s 1 = 2 0 d 3 = 2 ).Additionally,a3 = 2 stateismoreeasilymadebysimplyremovinga ˇ 0 p 3 = 2 proton.Whilemoretightlybound, thereareagreaternumberofprotonsinthe p 3 = 2 orbitalcomparedtothe p 1 = 2 orbital,and populatingthe3 = 2 inthiswaywouldnotrequirere-arrangingtheneutrons.Calculations withtheWBPandWBThamiltonianaswellasthecontinuumshellmodel(CSM),predict thelowestexcitedstatesin 23 Ntobe3 = 2 and5 = 2 inthevicinityof2 ˘ 4MeVasshown inFig. 5.11 .Theexcited1 = 2 doesnotappearuntilaround5MeVofexcitation.Thus 3 = 2 isthemostlikelycandidateforthespinandparityofthestate(s)populatedin 23 N. Ifoneassumesthatthe E ˘ 1MeVpeakdoesnotoriginatefromatransitiontothe 2 statein 22 N,thenthesetofpossiblelevelschemesisreduced.Thisisareasonable assumption,sinceadecaytothisstatewouldimplyobservingaresonanceabovethetwo- neutronseparationthresholdin 23 Nthatundergoesaone-neutrondecay.However,simply beingabove S 2 n doesnotprevent1nemission.Forexample,suchadecayhasbeenobserved in 24 O[ 68 ]whereastateabovethetwo-neutronseparationenergyunderwenta1ndecayto 23 O.UnderthisassumptionthepossibleorderingforthesestatesisshowninFigure 5.12 . 156 Figure5.12:Possiblelevelschemesthatcouldgiverisetotheobservedtwo-bodydecay energyfor 23 N.Case(a)isthe\GroundStateDecay,"whilethe\TwoState"scenariois case(b),andthe\SingleState"case(e).Thesinglestateinterpretationcouldalsobetwo statesclosetogether.Cases(c),(d),and(f)areexcludedduetotheweakintensityofthe 100keVtransition. 157 5.2.1.1GroundStateDecay First,inordertodeterminetheenergy,considerthe\GroundStateDecay"whereeach peakconsistsofonlyasinglecontributionandrepresentsanindependentstatein 23 N.Ifwe assumethe1MeVpeakisnotanoverlapoftwostates,thenthebisshowninFigure 5.13 .Thebestdescriptionwasobtainedwithtwo ` =2Breit-Wignersat E 1 =100 20 and E 2 =960 30keV.Thewidthisdominatedbytheexperimentalresolutionanddoes notminimize.The ˜ 2 asymptotesaround500keVatavaluewithin1 ˙ .Forthisreason,the single-particlewidthsof 1 =2keVand 2 =115keVareusedfortheandsecond3 = 2 respectively.Thisscenario,aswellasmanyotherpossibilitiesinFig. 5.12 canbeexcluded byexaminingthespectroscopicoverlaps C 2 S between 23 Nand 24 O.Thesearesummarized inTable 5.2 fortheWBTandWBPhamiltonians. Figure5.13:Bestofthetwo-bodydecayenergyfor 23 Nbasedonthe\GroundState Decay"interpretation.Thepurplecurveindicatesthe100keVtransition,thedahsedredis the960keVtransition.The2nbackgroundisshowninshadedgray. AccordingtotheWBPinteraction,thelowest3 = 2 1 hasthestrongestoverlap,byabout afactorof ˘ 2comparedtothesecond3 = 2 .TheWBTinteractionpredictsslightlylarger overlapforthe3 = 2 2 .Ifweinterpretthe ˘ 100keVpeakascomingfromastatebelowthe 158 ˘ 1MeVpeak,thenwereachacontradictionbetweenthedataandshellmodel.Dueto our,the ˘ 1MeVpeakisapproximately3 ˘ 10timesmoreintensethanthelower energypeakimplying C 2 S (3 = 2 2 ) >C 2 S (3 = 2 1 ),whichisnotwhattheWBPinteraction predicts.AlthoughtheWBTinteractiondoespredicttheoverlaptothesecond3 = 2 to begreater,itisstillaboutafactorof4toosmall.Theintensityofthe ˘ 100keVpeak inanywillalwaysbesmallerthanthe ˘ 1MeVpeakduetotheacceptanceofMoNA. Tobeconsistentwiththespectroscopicfactorsthe ˘ 100keVtransitioncanoriginatefrom eitherthesamestateasthe ˘ 1MeVtransitionorastateaboveit.Thisleavesacouple possibilities,referredtoasthe\SingleState"and\TwoState"scenarios:(1)Asinglestate at1.1MeV(oranunresolveddoublet),and(2)astateat1.1MeV,andanotherbelowitat 960keV. 5.2.1.2SingleState Thedatacanalsobefullydescribedbyasinglestateat E =1 : 1MeV.Inthisscenario,the ˘ 100keVtransitionisan ` =0 ; 2decaytothe2 statein 22 N,andtheobservedpeak isasuperpositionofa960keVdecaytothe1 stateanda1100keVdecaytotheground stateof 22 N.Table 5.3 showstheratioofintensitiesofeachcontributioninthewhichis WBPWBT E calc E decay J ˇ h 23 N j 24 O i E calc E decay J ˇ h 23 N j 24 O i (MeV)(MeV) C 2 S (MeV)(MeV) C 2 S 0-1 = 2 1 1.9328 0-1 = 2 1 1.9529 4.9613.1611 = 2 2 0.0025 5.2573.4571 = 2 2 0 3.6101.8103 = 2 1 1.4645 3.6101.8103 = 2 1 0.6893 4.5252.7253 = 2 2 0.6480 4.7642.9643 = 2 2 1.0483 Table5.2:Spectroscopicoverlapsforvariousstatesin 23 Nwiththegroundstateof 24 O. E decay iscalculatedassumingthestatedecaysdirectlytothegroundstateof 22 N. 159 IntesntiyRatio WBP WBT SingleState I i =I 0 C 2 S i =C 2 S 0 C 2 S i =C 2 S 0 1100keV I 1 =I 0 ˘ 0 0.002 4.52 I 2 =I 0 0.326 0.058 0.821 Table5.3:Ratioofintensitiesinthesinglestateinterpretationcomparedtotheequivalent ratioformedfromspectroscopicoverlapsforpossibleinitialstatesin 23 Nwithstatesin 22 NusingtheWBPandWBTinteractions. proportionaltothepartialwidth.Theweakestintensityisthetransitiontothe2 ,while theothertwodecays( ` =0,and ` =2)sharetheirintensityinroughlya4:1ratioinfavor ofthe ` =0decay.TheWBPinteractionisconsistentwithsuchascenario.Alsocompared inTable 5.3 aretheratiosofspectroscopicoverlapsfor h 22 N j 23 N i forboththeWBPand WBTinteractions.Althoughtheypredictthe s -wavedecays(3 = 2 ! 1 )tohavealmost nooverlap,asmallspectroscopicfactordoesnoteliminatethepossibility.Case(e)ofFigure 5.12 illustratesthelevelschemein 23 Nforthe\SingleState"scenario. Figure5.14:Bestofthetwo-bodydecayenergyfor 23 Nbasedonthe\SingleState" interpretation.Thepurplecurveindicatesthe100keVtransition,thedahsedredisthe960 keVtransition,andthedashedblueisthe1100keVtransition.The2nbackgroundisshown inshadedgray. 160 Transition WBP WBT TwoState G.S.Decay J ˇ i ! J ˇ f C 2 S h 22 N j 23 N i C 2 S h 22 N j 23 N i 1100keV+950keV 100keV+950keV 3 = 2 1 ! 0 0.4792 0.6867 0.0209(degenerate) 0.0049 3 = 2 2 ! 0 0.2926 0.0564 0.0054 0.025 3 = 2 1 ! 1 ˘ 0 0.0013 - - 3 = 2 2 ! 1 0.0304 0.0220 - - 3 = 2 1 ! 2 0.1562 0.0401 - - 3 = 2 2 ! 2 0.0301 0.0492 0.0051 - Table5.4:Spectroscopicoverlapsforpossibleinitialstatesin 23 Nwithstatesin 22 N usingtheWBPandWBTinteractions.Forcomparisonaretheintensitiesfortheb ofthedata. 5.2.1.3TwoState Thereisanotherinterpretationconsistentwithshellmodel.Itispossiblethatthe100keV transitionoriginatesfromastateaboveaseparatestateat960keV.Inthiscaseweobserve thatthelowest3 = 2 ismoststronglypopulatedandthesecondoneisweaklypopulated. Thiswouldcreateanother960keVtransitionwhichwouldbedegeneratein ourspectrum,aswellasa ˘ 700keVtransition.However,thesewouldbe s -wavedecays. Figure 5.15 showsthebestinthisscenario,whichissimilartothe\GroundStateDecay" interpretationbutthereisasmallcontributionfromthe1.1MeVdecay.Thiscaseisreferred toasthe\TwoState"scenario.No s -wavecomponentisnecessary.Table 5.4 showsthe spectroscopicoverlapsforpossibleinitialstatesin 23 Nwithstatesin 22 Ncomparedto thebintensitiesofthe\GroundState"and\TwoState"scenarios.BoththeWBP andWBTinteractionpredictoverlapsforthe ` =0transitionstobeverysmall < 0 : 002. Thesmallamplitudeofthe1.1MeVtransitionisalsoconsistentwiththeWBTinteractions whichpredictsasmalloverlapforthedecayof3 = 2 2 ! 0 .Thestrongestintensityisfrom the960keVtransition,whichisinterpretedasadirectdecaytothegroundstatefromthe lowest3 = 2 1 andisconsistentwithboththeWBPandWBTinteractions. 161 Figure5.15:Bestofthetwo-bodydecayenergyfor 23 Nbasedonthe\TwoState" interpretation.Thepurplecurveindicatesthe100keVtransition,thedahsedredisthe960 keVtransition,andthedashedblueisthe1100keVtransition.The2nbackgroundisshown inshadedgray. 5.2.1.4BackgroundDiscussion-Searchfor 24 N Thereisahigh-energytailinthedata,andsomebackgroundcontributionisnecessary.This tailcannotbedescribedbyincreasingthewidthofthe E 2 =960keVpeak,asthewidth ofthisdistributionsaturates.Avarietyofline-shapescandescribethisbackgroundand provideanadequateTheinclusionofanotherstateat3MeV,abroadgaussianat10 MeV,orathermaldistributionwith T =4MeValldescribethehigh-energytail. Theprotonknockoutreactionmechanismshouldbeclean.Thenecessistyofsomeback- groundinanotherwisecleanreactionmechanismsuggeststhepresenceofanotherreaction channel.Inprinciple,charge-exchangefrom 24 Oto 24 Ncouldhaveoccurredinthisexper- iment. 24 N,beingunbound,woulddecayto 23 Nwhichwasunfortunatelynotwithinthe acceptanceofthesweeper.However,ifthecharge-exchangepopulatedatwo-neutronun- boundexcitedstatethen 22 Ncouldbeproducedcreatingabackgroundintheoneneutron spectrumof 23 N(Figure 5.10 ).Figure 5.16 showsthereconstructedtwo-andthree-body energiesusingeithera1nor2nthermalbackground.Themultiplicityisalsoshownfor 162 Figure5.16:Comparisonbetweena1nthermalbackground(Top)anda2nbackground (Bottom).Ontheleftisthetwo-bodydecayenergy.Themiddlepanelsshownthethree- bodyspectrumfor 24 Nwithcausalitycuts,andthefarrightpanelsshowthemultiplicity. Thepurplelineisthe E 1 =100keVBreit-Wigner,thedashedredisthe E 2 =960keV resonance.Thebackgroundcontributionisshowninshadedgray. comparison.Themultiplicityincombinationwiththetwo-andthree-bodydecayenergies hasbeenshowntobeausefulwayofdeterminingthenumberofneutronsemittedina decay.PreviousstudieswithMoNAdemonstratedthissensitivityinsearchesfor4nand3n emissioninthereactions 14 Be(-2p2n) 10 He[ 130 ],and 17 C(-2p) 15 Be[ 131 ]. Whilethe1nthermalbackgroundreproducesthethetwo-bodydecayenergy,itfailsto describetheobservedcountsinthethree-bodyspectrumwithcausalitycuts.Inaddition, thesimulationunderpredictsmultiplicities3and4.A2nthermalbackgroundcanbetter describeallthreespectrasimultaneously.Thereduced ˜ 2 forthecausality-cutspectrum isgoodat ˘ 7 : 7 = 5,comparedto ˜ 2 ˘ 27 = 5forthe1nhypothesis,andthemultiplicity isbetterreproduced.Avarietyoftwo-neutronlineshapeswereconsidered,includingphase- spaceanddi-neutrondecays,howevertheydonotreproducethetwo-bodyspectrum.The2n 163 thermalbackground,with T =3MeV,isusedinthisanalysisasitprovidesasimultaneous descriptionofallobservables. 5.2.2Conclusions Itisimportanttonotethattheinteractionsdiscussedheredonotreproducetheexperimen- tallyobservedenergiesinboth 22 Nand 23 N,althoughtheyarewithin1MeV.As 23 Nisat thedripline,continuumandthree-bodyforcesmaybegintoplayatrole, andtheseectswerenotconsideredinthisanalysis.Experimentally,wecannotdistinguish betweenanynumberofcasesordegeneraciesthatproducetheenergyobservedin thedecayenergy.Ultimatelyarepeatmeasurementwith detectionisnecessarytoclarify thestructureof 23 N. 164 Chapter6 SummaryandConclusions Inthisdissertation,measurementsofneutronunboundstatesin 24 Oand 23 Nhavebeen reported.Thesestateswerepopulatedbyreactionsfroman 24 Obeamprovidedbythe CoupledCyclotronFacilityattheNSCL.Unboundstatesin 24 Owerepopulatedbyinelastic excitationonaliquiddeuteriumtarget,andstatesin 23 Nwerepopulatedviaone-proton knockout.Thesestatesweremeasuredbyinvariantmassspectroscopy,whichrequireda kinematicallycompletemeasurementofboththechargedfragmentandtheneutronsemitted inthedecayoftheunboundstate.Theresultingchargedfragmentswerebentbythesweeper magnetintoasuiteofchargedparticledetectorswhichmeasuredtheirpositionandangle, providingisotopeseparation.Theneutrons,bythemagnetictravelled straighttowardMoNA-LISA{anarrayofplasticscintillatorsthatmeasuredtheneutron positionandtimeoft.WiththeinformationprovidedbyMoNA-LISAandthesweeper, theinvariantmassofthecompoundnucleusofinterestwasmeasured.Inthecaseoftwo- neutronemission,three-bodycorrelationswereexaminedtodeterminethedecaymechanism. Unboundstateswereobservedin 23 Nforthetime.Itisknownthatinthisregion ofthenuclearchart,MayerandJensen'smagicnumbersbreakdownandnewshell-closures appear.ThisshifthasbeenattributedtotheNNtensorforce,three-bodyforces,and continuum 24 Oisdoublymagicwith Z =8and N =16. 23 Nshouldalsoexhibitthe N =16shellgapalthoughoneexpectsittobereduced.Forexample,the N =14sub-shell gapin 22 Owasobservedtodisappearbythetimeonereaches 20 C.Thus,idenof 165 statesin 23 Ncouldhelpbetterconstrainthetensorforceandprovideabetterunderstanding ofshellevolution. Thetwo-bodydecayenergyfor 23 Nshowsprominentpeaksat E 1 =100 20keVand E 2 =960 30keV. 22 Nhastwoboundstatesthatthedecayof 23 Ncouldbranchto,implying thattheobserveddecaydoesnotnecessarilypopulatethegroundstateof 22 N.Ratherit couldbeatransitiontoanexcitedstatein.However,duetothelackof -raydetectionin theexperiment,onecannotmakeaestatementonthestructureof 23 Nwiththe currentdata.ShellmodelcalculationswiththeWBPandWBTinteractionsleadtoseveral interpretations.Asinglestateat1.1MeVabove S n ,or2.9MeVwithrespecttotheground stateof 23 N,isconsistentwithdataandtheoryaswellastwostatesat2.9MeVand2.75 MeV,respectively.Arepeatmeasurementwith -raydetectionisnecessarytoclarifythe structureof 23 N. Inadditionto 23 N,astateabovethetwo-neutronseparationenergywasobservedin 24 O. Itdecaysbyemissionoftwo-neutronswithathree-bodyenergyof E 3 body =715 110(stat) 45(sys)keV,placingitat E =7 : 65 0 : 2MeVwithrespecttothegroundstate.The three-bodycorrelationsinthe T and Y Jacobicoordinatesystemsshowclearevidenceof asequentialdecaythroughanarrowstatein 23 O.Aphase-spaceordi-neutronhypothesis wereunabletodescribetheobservedcorrelations.Thisconstitutesthemeasurementof asequentialdecay,exposedbyenergyandangularthree-bodycorrelationsina2 n unbound system.Themeasurementdemonstratestheabilitytodistinguishexperimentally,adi- neutronsignalfromatdecaymodeinadditiontoprovidingvaluableinformation aboutfew-bodyphysicsattheneutrondripline.Thethree-bodycorrelationswerenot sensitivetothe ` valueofthedecayleavingthespin-parityofthisstateundetermined.A separatemeasurementisnecessarytodeterminetheangularmomentumofthisstate. 166 BIBLIOGRAPHY 167 BIBLIOGRAPHY [1] KennethKrane. IntroductoryNuclearPhysics .JohnWiley&Sons,OregonState University,USA,1987. 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