MEASURINGTHEHALF-LIFEOFO-26 By ThomasRedpath ADISSERTATION Submittedto MichiganStateUniversity inpartialoftherequirements forthedegreeof PhysicsŒDoctorofPhilosophy 2019 ABSTRACT MEASURINGTHEHALF-LIFEOFO-26 By ThomasRedpath Aninterestingpropertyofsomeneutron-unboundsystemsistruetwo-neutronemissionwherethe neutronsareemittedsimultaneouslyasopposedtoasequentialdecaythroughanintermediatestate. Sinceneutronsareonlya ectedbytheangularmomentumbarrier,thetimescaleforthisprocess ismuchshorterthanfortwoprotonemissionwhichisdominatedbytheCoulombbarrier.One suchcaseis 26 Owhereaverylowdecayenergywasmeasuredandthetwovalenceneutronsare expectedtooccupy d -waveorbitals.Also,thegroundstateof 25 Oislocated700keVhigher.Using thedatafromapreviousmeasurementofthedecayenergy,theMoNAcollaborationextracteda lifetimeof4 : 5 + 1 : 1 1 : 5 ( stat ) 3 ( syst ) pswithalevelof82%(1).Resultsfromarecent measurementgive T 1 = 2 = 5 : 0 + 2 : 0 1 : 6 (stat) 1 : 7(syst)psandsupporttheprevious Measurementsofneutron-unboundsystemsusinginvariantmassspectroscopyareoftenper- formedusinglow-intensityradioactiveionbeams.Lowreactionyieldscanbecounteredbyusing athickertargetbutattheexpenseoflargeruncertaintiesinthereconstructedinvariantmass.Anew segmentedtargetwasdesignedtoaddressthistrade-o ,anditwasusedinthisexperimentto re-measurethegroundstatehalf-lifeof 26 O. Copyrightby THOMASREDPATH 2019 ACKNOWLEDGEMENTS IwouldliketothankGeorgePerdikakisforintroducingmetothefascinatingworldofnuclear physics.Withouthismentorship,Iwouldnothavehadthisopportunity.Iwouldliketothankmy advisorMichaelThoennessenforallofhispatience,knowledge,andwisdom.Ithasbeenanhonor andapleasuretoworkwithhim.Iwouldalsoliketothankthemembersofmythesiscommittee, AlexGade,JaideepSingh,MortenHjorth-Jensen,andElizabethSimmonsfortheiradviceand supportduringourmeetings. IthasbeenaprivilegetoworkwiththeMoNACollaboration.Thecollectiveknowledgeof thisgroupisextensive,andtheirkindandinclusiveattitudefostersaproductiveandsupportive atmosphere.ManythankstoAnthonyKuchera,ThomasBaumann,PaulGueye,NathanFrank, PaulDeYoung,JimBrown,WarrenRogers,andSharonStephensonfortheirsupport,advice,hu- mor,andguidance.IwouldalsoliketothankColePerschandCharlieBairdfortheirworkonthis project.Finally,ahugethankstomyfellowgraduatestudentsHanLiu,DanielVotawandDayah Chrismanfortheirhelpwitheverythingfromcourseworktorunningexperiments. iv TABLEOFCONTENTS LISTOFTABLES ....................................... viii LISTOFFIGURES ....................................... x CHAPTER1INTRODUCTION ............................... 1 1.1TheShellModel....................................2 1.2NuclearStability...................................5 1.2.1BindingEnergy................................5 1.2.2Half-life....................................6 1.2.3SeparationEnergies..............................8 1.3NuclearDecays....................................9 1.3.1AlphaDecay.................................9 1.3.2BetaDecay..................................11 1.3.3OtherDecayProcesses............................13 1.3.4NeutronandProtonEmission.........................14 1.3.4.1One-andTwo-ProtonRadioactivity................14 1.3.4.2ProspectsforTwo-NeutronRadioactivity.............15 1.4DissertationOverview.................................16 CHAPTER2BACKGROUNDANDMOTIVATION .................... 17 2.1PreviousExperiments.................................17 2.2TheoreticalBackground................................18 2.2.1Two-bodydecay...............................18 2.2.2Three-bodydecay...............................18 2.2.32 n Decayof 26 O...............................19 2.3TheUnbinnedLoglikelihood.............................21 2.4Fragmentmomentumreconstructionanddecayenergyresolution..........23 CHAPTER3EXPERIMENTALTECHNIQUE ....................... 25 3.1ExperimentalSetup..................................25 3.2BeamProduction...................................26 3.3A1900andTargetScintillators............................27 3.4SegmentedTarget...................................28 3.4.1SiliconDetectorsandBerylliumTargets...................28 3.4.2Signalreadoutandelectronics........................30 3.5Sweepermagnet....................................32 3.6ChargedParticleDetectors..............................32 3.6.1CRDCs....................................32 3.6.2IonizationChamber..............................34 3.6.3ThinTimingScintillator...........................34 3.7MoNALISA......................................34 v 3.8ElectronicsandDAQ.................................36 3.9InvariantMassSpectroscopy.............................40 3.10TheDecayinTargetTechnique............................42 CHAPTER4DATAANALYSIS ............................... 46 4.1CalibrationsandCorrections.............................46 4.1.1SegmentedTarget...............................46 4.1.1.1Energylosscalibration.......................47 4.1.1.2ReactionTarget...................50 4.1.1.3Positioncalibration.........................53 4.1.2Chargedparticlecalibrationsandcorrections................54 4.1.2.1A1900andTargetscintillators...................54 4.1.2.2CRDCs...............................56 4.1.2.3IonizationChamber........................61 4.1.2.4Thinscintillator..........................64 4.1.3MoNA-LISA.................................71 4.1.3.1ChargeCalibration(QDC).....................72 4.1.3.2Timingand x positioncalibration.................74 4.1.3.3Timingo sets...........................77 4.2EventSelection....................................79 4.2.1Beam..............................81 4.2.2Element............................81 4.2.3Isotope.............................82 4.2.4Two-NeutronSelection............................92 4.2.4.1CausalityCuts...........................93 4.2.4.2DecisionForest...........................94 4.3FragmentReconstruction...............................97 4.4ModelingandSimulation...............................102 4.4.1IncomingBeamParameters..........................104 4.4.2EnergyLossinSiliconDetectors.......................104 4.4.3ReactionParameters.............................105 4.4.4Additionalparameters............................106 4.4.5Cuts......................................107 4.4.6DecayModel.................................107 4.4.6.1Oneneutrondecays........................109 4.4.6.2Twoneutrondecays........................109 4.5Extracting T 1 = 2 ....................................109 CHAPTER5RESULTSANDDISCUSSION ........................ 113 5.1Half-lifeMeasurement................................113 5.1.1Results....................................114 5.1.2Implications..................................117 5.2SegmentedTargetEvaluation.............................117 5.2.1TargetThickness-Simulation........................118 5.2.2Improveddecayenergyresolution......................120 vi 5.2.3ResolutionImprovementsandtheHalf-lifeMeasurement..........121 CHAPTER6SUMMARYANDCONCLUSIONS ..................... 124 6.1Half-lifemeasurement.................................124 6.2SegmentedTarget...................................125 BIBLIOGRAPHY ........................................ 126 vii LISTOFTABLES Table3.1:Thicknessesforthesilicondetectorsandberylliumtargets.Theberyllium targetsweremeasureddirectlyusingcaliperswithadialindicatorandthe associatedmeasurementuncertaintiesare 4 m( 0 : 7mg / cm 2 Be).The siliconwaferthicknesseswerereportedbythemanufacturerwithuncertainties of 1 m( 0 : 2mg / cm 2 Si).............................29 Table4.1:Energylossofeightdi erentfragmentsineachsilicondetectorcalculated usingtheATIMAenergylosscalculatorincludedwithinthe LISE++ software package(85).Calculationofthesevaluesaccountsfortheenergylossinthe berylliumsegmentssincethetargetswerealwaysinthebeamline.Variations inthematerialthicknessescorrespondtovariationsinthecalculated dE less than 0 : 008MeV,so / 0 : 05%,whichissmallerthantheresolutionofthedetectors.49 Table4.2:Thesecondandthirdcolumnslistparametervaluesextractedfromthe dE cal = p 0 + p 1 dE raw ,seeFigure4.2.Theerrorsare < 2 : 0%intherange from10MeVto35MeV...............................50 Table4.3:CRDCcalibrationparametersappliedtorunsbefore(1038-1121)andaf- ter(1125-1179)theattemptedelectronicsrepairs(seetext).Badpadsare cathodepadsthatareremovedfromtheanalysis.Thepedestalsub- tractionissummarizedbytheaverageofallpedestalvalues.Similarly,the gainscalingfactorisaveragedoverallgoodpads.Notethatthesetwovalues arenotactualcalibrationparameters.The x y slopesando setswereextracted fromspectratakenwiththetungstenmaskinplace.................62 Table4.4:Finaltimeo setsforeachMoNA / LISAtable....................79 Table4.5:Velocities,timesandfractionoftotaleventsinthedE-ToFspectrumfor thefourmostintensebeamfragments.Thevelocitiesarecalculatedbasedon thecentralrigidity(4.5798Tm)ofthelastdipolemagnetbeforethetargetand thetimesarebasedonthe983.8cmpathbetweentheA1900and targetscintillators.Theremaining23%ofeventseitherfalloutsidethestrict 2Dgraphicalcutsorresultedfromlow Z beamfragments.............80 Table4.6:Iterativecorrectionstofragmentusedtoachieveparticleidenti- fortheleft,middle,andrightregionsinFigure4.25.Note thattwoparametersaregivenforiterationthree,thecorrectionviafocus x ; thisisbecausethatcorrectionhadtheform t corr3 = t corr2 ( p 2 x 2 + p 1 x ) so theandsecondvalueslistedcorrespondto p 2 and p 1 respectively.......87 viii Table4.7:Energyaddbackusedtoreconstructtheenergyofthe 24 Ofragmentsproduced from 27 Finoneofthe 9 Betargets..........................102 ix LISTOFFIGURES Figure1.1:Thenuclearchartwhereneutronnumberisplottedonthe x axisandproton numberonthe y axis.Tilecolorindicatesdecaymode:lightblue(red) indicates ( + ) decay,neutron(proton)emissionisplottedinblue(red), alphadecayisplottedingreen,andspontaneousinviolet.Blacktiles indicatestableorlong-livednuclideswith T 1 = 2 & 10 8 years.Thevertical (horizontal)blueboxesindicatemagicneutron(proton)numbers.Datais fromRef.(14)....................................2 Figure1.2:Contributionstothetotalpotential(redcurve)forad-waveneutronin 16 O. Theblacksolidcurveplotsthecentral,spin-independentcomponentparam- eterizedbyaWoods-Saxonshape.Thebluedottedcurveisthespin-orbit componentandthegreentripledot-dashedcurveplotsthecontributionfrom thecentrifugalbarrier.TheparametersfortheWoods-Saxonare V 0 = 53 MeV, R 0 = 3 : 15fm, a 0 = 0 : 65fm.........................3 Figure1.3:Neutronsingleparticleenergiesin 208 Pbforthreedi erentpotentialmod- els:harmonicoscillator(left),Woods-Saxonwithnospin-orbit(center),and Woods-Saxonwithspin-orbit(right).Theshelloccupanciesareindicatedby thenumbersinsquarebrackets.Thetotaloccupancysummedoveralllower shells(2,8,20,...)intheWoods-Saxonplusspin-orbitmodelisindicated inthespacesbetweengroupsofstates.ImagefromRef.(21)...........4 Figure1.4:Bindingenergypernucleonasafunctionof Z fornuclideswith A = 21.The half-livesfor 21 C, 21 O,and 21 Mgaregivenintheboxes.............6 Figure1.5:Theradialforasphericallysymmetricpotentialthatapproximatesthe interactionbetweenthe particleandthedaughternucleus.Theinteraction isattractiveatshortrange(0 < r < a )duetothenuclearinteractionand repulsivefor r > a duetotheCoulombrepulsionbetweentheprotonsin 4 He andinthedaughternucleus.............................10 Figure1.6:Anillustrationofthethree decayprocesses: decayisdepictedinthe toppanelwhile + decayandelectroncapturearedepictedinthebottomleft andbottomrightpanels,respectively........................12 Figure1.7:Calculatedlifetimesforaneutronemitter, 16 B,andtwoprotonemitters, 16 F and 151 Lu,asafunctionofdecayenergyforangularmomenta L = 0(solid lines), L = 1(dashedlines), L = 2(dash-dottedlines), L = 5(dottedlines). Imagefrom(22)...................................15 x Figure2.1:Illustrationsoftheenergyconditionscharacteristicofsequential(toppanel) andsimultaneous(bottompanel)threebodydecays................19 Figure2.2:Decaywidth / half-lifeasafunctionofdecayenergyfor2 n emissionfrom 26 O.Thegraylineassumesapureorbital[ d 2 ]coupledtothe totalangularmomentum L = 0.Thesolidblackcurveshowstheresultsfor thenoFSI, 24 Omass.Thebluedashedlineplotstheresultsforthe noFSIcalculation.Thereddottedlineisthecalculationwiththe n n FSI scaledby0.25,andthepurpleshort-dashedcurveisthefull n n FSIresults. Theverticalredlinesroughlyindicatetheexperimentalresultsfrom(48). Imageadaptedfrom(58)..............................20 Figure3.1:DetectorlayoutintheN2vault...........................26 Figure3.2:BeamproductionattheCoupledCyclotronFacilitystartsbyheatingasam- pleoftheprimarybeammaterialinanionsource.TheK500andK1200cy- clotronsacceleratethebeamwhichissubsequentlydirectedontoaBetarget. Fragmentsresultingfromnuclearinteractionswithinthetargetareby theA1900toprovidethedesiredsecondarybeam.Imagesource:(76)......27 Figure3.3:Eachdetectoris11cm 11cm 0.32cmincludingtheframehousingthe siliconwafer.Thethicknessesofthesiliconwafersare140 m,135 m, 138 m,and142 mfordetectors0,1,2,and3,respectively.Theberyllium targetsare2.8cmtallwiththicknessesof0.41cm,0.37cm,and0.33cmfor targets1,2,and3,respectively.Thespacingbetweeneachdetector / target is0.84cm(0.33inches)sointotaltheapparatusextends5.04cmalongthe beamaxis......................................29 Figure3.4:Drawingofthesegmentedtargetmountedinthebeamline:(a)beamviewer plateusedtoimagethebeamduringtuning(b)baseonwhichalldetectors aremounted(c)silicondetectorframe(d)berylliumtarget(e)baseonwhich alltargetsaremounted.Theviewerismountedtothetargetbase.Boththe detectorandtargetmountbasesareattachedtopneumaticdrivessotheycan beindividuallyinsertedintoandretractedfromthebeamline...........30 Figure3.5:Wiringdiagramshowingthesignalpathsfortheanodeandcornersignals fromoneofthesilicondetectors.Theanode(blackarrows)andcorner(blue arrows)signalswereroutedthroughseparatepreampandshaping Allshapedsignalswereprocessedbythesameanalog-to-digitalconverter....31 Figure3.6:SchematicofaCathodeReadoutDriftChamber(CRDC)expandedinthe z -direction.Theshapingwiresareomittedforvisibility...........33 Figure3.7:SchematicdrawingofasingleMoNA / LISAplasticscintillatorbar.Image source:Ref.(81)..................................35 xi Figure3.8:AdiagramofthespatialorderingoftheMoNA / LISAbarsastheywerecon- forthisexperiment.Neutronsfromthetargettravelfromlefttoright. Thespacingbetweengroupsoflayersisnotdrawntoscale............36 Figure3.9:SchematicdiagramoftheMoNA / LISA-Sweeperelectronicsfrom Reference(81).Start,stopandgatesignalsaredepictedwithgreen,red andbluearrowsrespectively.Thesignalusedasthecommonstopforthe MoNA / LISATDCsisindicatedbythereddashedline.Shaping areomittedforclarity................................37 Figure3.10:An(abbreviated)diagramoftheMoNA / LISAtriggerlogicfrom Reference(81).EachLevel1XLMisconnectedto32CFDstocountthe numberoftimestheleftandrightPMTsonthesamebarThenineLevel 1modules(layersJ-R)fortheLISAarrayandtheeightmodules(layersA- H)fortheMoNAarrayareconnectedtotwoseparateLevel2modules.The TDCsandQDCsareomittedforclarity......................39 Figure3.11:Thetoppanelillustratesthecaseofanextremelyshort T 1 = 2 ˘ 10 21 sand thebottompaneldepictsalonger T 1 = 2 ˘ 10 12 s.Thelonger 26 Oexiststhe moreenergyitlosesasittravelsthroughthetargetmaterial.Thismeansthe neutronsareemittedatalowervelocitythanif 26 Odecaysinstantaneously....43 Figure3.12:Relativespeeddistributionssimulatedwiththreedi erent 26 Ohalf-liveswhere T 1 = 2 = 0ps,4psand8psarethereddashed,blacksolidandbluedotted curvesrespectively.Therelativespeediscalculatedas j ~ v n jj ~ v f j where ~ v n is theneutronvelocityand ~ v f isthefragmentvelocity.Thecentroidsofthedis- tributionswith T 1 = 2 = 4ps,8psareshiftedtotheleftrelativetothe T 1 = 2 = 0 pscase.Thereaction 27 F ( 1 p ) ! 26 O ! 24 O + 2 n wassimulated.......43 Figure3.13:Averagevalueoftherelativespeeddistributionsasafunctionofhalf-life usingthereaction 17 C ( 1 p ) ! 16 B ! 15 B + n at80MeV / u(left)and250 MeV / u(right).Imagefrom(55)..........................44 Figure3.14:Averagevalueoftherelativespeeddistributionasafunctionofhalf-lifefor 17 C ( 1 p ) ! 16 B ! 15 B + n andthreedi erentcombinationsofbeam energyandtargetthicknesses.Thenumberinthelegendcorresponds tothebeamenergyinunitsofMeV / uandthesecondnumberisthetarget thicknessing / cm 2 .Thebeamenergy / targetthicknesscombinationswere selectedtogiveapproximatelythesameasymptoticvalueforverylonghalf- lives.ImagefromRef.(55).............................45 xii Figure4.1:Theupperpanel(a)illustratesthesiliconlatticestructurewithcovalentelec- tronbondsdepictedbytheblacklines.Thebottompanel(b)sketchesthe electronenergybandstructure.Insemiconductors, E g ˇ 1eV.Bothillustra- tionsareadaptedfromRef.(28)..........................48 Figure4.2:Blackpointsplotcalculatedenergyloss(y-axis)asafunctionofcen- troidoftheuncalibrated dE spectrum(x-axis)foreightdi erentfragments. Thefourpanelsshowtheresultsforthefoursilicondetectors.Thebluelines showtheextractedlinearThe x errorbarsfortheerrorsandthe y errorbarsforthemeasurement / calculationuncertaintiesaresmallerthanthe points.TheparametersfromthearelistedinTable4.2.............50 Figure4.3:Exampletargetplotsfor 27 F ( 1 p ) ! A Omeaningthatall eventsplottedhereenterthesegmentedtargetas 27 Fandleaveasanoxygen isotope.Thetoprowofplotsshowthemeasuredenergylossineachsilicon detector.Theleftpanelinthebottomrowofplotsshowsthemeasuredenergy lossinthesecondsilicondetectorvs.themeasuredenergylossinthe silicon.Themiddlepanelplotsthethirdsiliconenergylossvs.thesecond andtherightpanelshowsthefourthsiliconenergylossvs.thethird.......52 Figure4.4:Schematicofasilicondetectorandthecoordinateconventionusedforthe positioncalibration.Thesizeofthearrowsillustratesthesignalsizeateach ofthefourcornersforanioninteractingatthelocationoftheredcross......53 Figure4.5:ThetoppanelshowsanexampleTDCspectrummeasuredusinganOrtec timecalibratorsettodeliverstartandstopsignalsseparatedbyintegermulti- plesof40ns.Thebottompanelplotsthestart-stoptimeintervalsversusthe peaklocationsfromthetoppanel.Theredlineisalinearusedtoextract theconversionfactorfromTDCchannelnumbertotimeinnanoseconds.The slopeofthelineis0.0625ns / chandtheerrorisnegligible...........55 Figure4.6:Chargecollected(blackpoints)asafunctionofCRDC2padnumberfora singleevent.TheredcurveisaGaussianandtheredverticallineisthe centroidextractedfromthe..........................57 Figure4.7:CRDCpedestalsubtraction.TherawpadsignalsforCRDC1(left)and CRDC2(right)areshowninthetoprow.ThecentroidofaGaussianto thechargedistributionofeachpadissubtractedtoshiftthecenterofeach distributiontozero.Theresultofthepedestalsubtractionisshowninthe bottomrow:CRDC1(left)andCRDC2(right)..................58 Figure4.8:ExamplespectrashowingthetotalchargereadoutbytwopadsinCRDC2. Pad68(right)isanexampleofacorrectlyfunctioningpad;pad24(left)is ......................................59 xiii Figure4.9:Calibratedpositionspectrumwiththemaskpatternoverlaid.Thesame y o setextractedfromtheunreactedbeampositionisappliedtothemaskpattern.61 Figure4.10:Uncorrected(toprow)andcorrected(bottomrow)drifttimesforCRDC1. Thediscontinuityatrun1125isduetothetwoseparatecalibrationparame- tersdescribedinthebeginningofSection4.1.2.2.................63 Figure4.11:Thechargecollectedbytheionizationchamberpadsforunreacted 27 Fevents before(left)andafter(right)gainmatching....................64 Figure4.12:Theaverageoftheionchamberpadsignalsbefore(left)andafter(right) thecorrection.A 27 Fbeamgatehasbeenapplied.Thehorizontalbands correspondtoreactionproductswithdi erent Z .Thebandbetweenthered horizontallinescorrespondstooxygenreactionproducts.Thecorrectionwas madetostraightenthisbandinordertomorecleanlyselectoxygenreaction products.......................................64 Figure4.13:DiagramofthethinscintillatorwithitsfourPMTs................65 Figure4.14:Theraw,uncorrectedToFbetweenthetargetandthinPMT0isplottedversus chargedepositedinthetargetscintillator.Thereisnocorrelationbetweenthe twomeasurements.ThespectrafortheotherthreethinscintillatorPMTsare similartothisone..................................66 Figure4.15:Leftcolumn:rawenergylossversusToFbetweentargetandthinscintillators; ascalefactorwasappliedtothex-axestoconvertTDCchannelnumberto ns.Rightcolumn:resultsofcorrectingeachToFmeasurementagainstthe signalsizeineachPMT.Therawenergylosssignalsdidnotneedtobegain matchedsinceindependentcorrectionswereextractedseparatelyforeachPMT.68 Figure4.16:Leftcolumn:q-correctedToFversus x positionmeasuredinCRDC2.Right column:resultsofthe x positioncorrection....................69 Figure4.17:Leftcolumn:qx-correctedToFversus y positionmeasuredinCRDC2.Right column:resultsofthe y positioncorrection....................70 Figure4.18:Theuncorrected(blackdottedhistogram)andcorrected(solidredhistogram) ToFdistributionsforunreacted 27 Fcalculatedastheevent-by-eventaverage ofthethinPMTsignals.Ano sethasbeenappliedtocenterthevelocity distributiononthevaluedeterminedfromenergylosscalculationsofthe 27 F beamthroughthesegmentedtarget(seetext)...................71 xiv Figure4.19:Exampleraw(left)andcalibrated(right)QDCspectra.Thepedestalvisible intheleftspectrumissuppressedusingahardwarethresholdduringtheex- perimentandwhilerecordingasubsequentcosmicrunusedtogeneratethe rightplot.TheredcurveisaGaussiantothecosmicpeak.TheQDC channelnumbersofthepedestalandthecosmicpeakdeterminedascaling toconvertQDCchannelnumbertolightdepositedinunitsofMeVee.The muonpeakappears ˘ 20MeVeeinthecalibratedspectrum............73 Figure4.20:AnexampleofanuncalibratedTDCspectrum(toppanel)generatedusing thetimecalibrator.Thenarrowpeaksrepresentrecordedeventswherethe timebetweenthestartandstoppulsesisanintegermultipleof40ns.The knowntimeintervalsareplottedagainstthepeaklocations(bottompanel)to extractascalefactorthatconvertsTDCchannelnumbertoatime.Theright paneldisplayshistogramsofthecalculatedTDCslopes..............75 Figure4.21:Intheleftpanel,theblackhistogramplotsthedi erencebetweencalibrated times t left t right ,thebluecurvesareFermifunctiontotheleftandright edgesoftimedi erencedistributionandtheredverticallinesaretheedges extractedfromtheTherightpanelshowstheresulting x positionspec- trum.Eventsplottedinthesespectraarerequiredtohaveacalibratedlight depositedsignal > 4MeVee............................76 Figure4.22:A ˜ 2 minimizationroutinewasusedtoGaussianfunctionstothefragment andneutronvelocitydistributions;thecentroidsanderrorsareplottedfor data(blackcircles)andsimulation(openbluediamonds).Theresultsfor thefragmentvelocitiesareplottedinthetoprowandtheneutronvelocitiesin thebottomrow.Theleft,middleandrightcolumnscorrespondtoresults wherethedistributionsaremadefromeventswherethereactionoccurredin thesecondandthirdberylliumtarget,respectively..............78 Figure4.23:SpectrumusedforbeamfragmentThefasterfragments(see Table4.5)haveashorterToF(x-axis)andfragmentswithahigher Z deposit moreenergyinthesilicondetector(y-axis)..................80 Figure4.24:Energyloss( dE )measuredintheionizationchamberversusthetime-of- fromthetargettothethinscintillator.Onlyeventsthatfallinsidethe 27 Fbeamgate(seeFigure4.23)areplottedhere.Themostintenseregion correspondstounreacted 27 Fbeamfragments( Z = 9);thebandimmediately belowcorrespondstooxygen( Z = 8)reactionproducts 27 F ( 1 p ) ! A O....82 Figure4.25:CRDC1 x positionversusCRDC2 x position;Twofunctioningdetectors woulddisplayasmooth,positivecorrelation.Theredlinesoutline2Dgates thatattempttoselecteventswithagoodCRDC1positionmeasurement.....84 xv Figure4.26:ThecorrelationbetweenToFdispersiveangleandpositionisshownforoxy- genbeamfragmentsontheleftandoxygenreactionfragmentsfrom 27 Fon theright.......................................85 Figure4.27:Thetoppanelisaprojectionontothe2Ddispersivepositionversusdispersive angleplane.Thecontourisshowninblack.Inthebottom panelthedispersiveplaneemittanceparameterdisplaysalinearcorrelation withmeasuredToFandcanbeprojectedontoanaxisperpendiculartothe redlineforthepurposeofmakinga1Dgate....................86 Figure4.28:ThetopthreepanelsplotthecorrectedToFsforthethreedi erentregions. Foreachregiona1Dgateisindicatedbytheredlinesandarrows,andis appliedtotheCRDC2 x distributionsplottedinthebottompanel.Thered, black,andbluepointsplottheCRDC2 x distributionsforeventsunderthe left,middle,andrightregiongatesrespectively.Thesegatesselectevents wheretheoxygenreactionproductshavealongToFandaredetectedonthe highrigidity( + x )sideofCRDC2.........................89 Figure4.29:ThetopthreepanelsplotthecorrectedToFsforthethreedi erentregions. Foreachregiona1Dgateisindicatedbytheredlinesandarrows,andis appliedtotheCRDC2 x distributionsplottedinthebottompanel.Thered, black,andbluepointsplottheCRDC2 x distributionsforeventsunderthe left,middle,andrightregiongatesrespectively.Thegatesshownhereselect eventswithaslightlyshortertimeandCRDC2 x distributionsthatare shiftedtotheleftcomparedtothegatesshowninFigure4.28...........90 Figure4.30:ThetopthreepanelsplotthecorrectedToFsforthethreedi erentregions. Foreachregiona1Dgateisindicatedbytheredlinesandarrows,andis appliedtotheCRDC2 x distributionsplottedinthebottompanel.Thered, black,andbluepointsplottheCRDC2 x distributionsforeventsunderthe left,middle,andblueregiongatesrespectively.Thegatesappliedtothese plotsselecteventswiththeshortestToFsforoxygenreactionproducts.Note thattheCRDC2 x distributionsareshiftedfurthertotheleftthantheother twosetsofgates...................................91 Figure4.31:Anillustrationof A 2 versus A = Z for10 A 30and Z A .Eachpoint isaseparatenucleus(someunphysical).Theredcurvesindicatecurvesof constant Z .Threenucleiwith A = Z = 3thatareintheSweeperacceptance arehighlightedwithgraycircles..........................92 Figure4.32:Ionchamber E versusglobalcorrectedToFforalleventsfromthe 27 F beamthatfallinsideoneoftheregiongatesshowninFigure4.25.Thered linecorrespondstothe A = Z = 3lineinFigure4.31................93 xvi Figure4.33:Onedimensionalparticlefortheoxygenreactionproducts. Eventsplottedherearerequiredtofallinsidetheoxygenreactionproduct gateshowninFigure4.24andinsideoneofthefigoodflCRDC1gatesshown inFigure4.25.Inaddition,coincidencewithoneoftheMoNA / LISAbarsis required.......................................94 Figure4.34:Anillustrationoftherelevantpositionvectorscalculatedinatwo-neutron event.Thepositionvector ~ d 0 ( ~ d 1 )isfromthetargettothelocationofthe (second)interactioninMoNA / LISA; ~ d 01 = ~ d 1 ~ d 0 .............95 Figure4.35:Schematicviewofadecisiontree.Ateachnodeabinarycutismadeonone oftheparameters x i ; j ; k ;::: .Imagesource:(95)...................96 Figure4.36:Theoutputisplottedtoexaminethereliabilityofthedecisionfor- est.Inthetoppanel,theoutputdistributionsforsimulated,labeled truetwo-neutron(blue)andone-neutronscattering(orange)eventsareplot- ted.Theblackcurveisthesumofthetwodistributions.Inthebottom panel,thedecisionforestiscomparedtothecausalitycuts;thegreenhis- togramplotstheoutputforeventsthatthecausalitycutsidentifyas truetwo-neutronevents.Thegrayhistogramshowstheoutputfor eventsthatthecausalitycutsidentifyasone-neutronscatteringevents.The redlineindicatesthecutontheoutputusedintheanalysis....98 Figure4.37:ThecorrelationbetweenpathlengthasafunctionofCRDC2 x isplottedin theupperrightpanel.TheabscissaforeachpointisthemeanofaGaussian totheCRDC2 x distribution.Theordinateofeachpointisthemeanof aGaussiantothecalibratedToFdistributionmultipliedbythevelocity of 27 Fafterthesegmentedtarget;theerrorbarsrepresenttheerrorand theuncertaintyinthevelocitycalculationintroducedbytheuncertaintiesin thesiliconandberylliumthicknesses, 1 mand 4 mrespectively.The leftandbottomrightpanelsareexamplesoftheToFandCRDC2 x spectra, respectively.RedcurvesplottheGaussian..................100 Figure4.38:Theleftpanelplotsthesimulateddistributionofangles betweenthelab- framefragmentmomentumvectorandthebeamaxis.Themiddleandright panelsplotthethree-bodydecayenergiesreconstructedfromthesimulated detectorresponseswherethedecayenergyavailableforeverysimulated eventwas50keV(middle)and1MeV(right).Theredcurvesplotthede- cayenergiescalculatedwiththefragmentangle = 0andtheblackcurves includetheangleinformation............................101 xvii Figure4.39:The ( x ; y ; x ; y ) forthesimulatedbeamwassetthroughcompar- isonstotheexperimentaldistributionsplottedfromtheunreacted 27 Fdata set.Thesilicon x ; y positiondistributionsareshowninthetoprowand theCRDC2 x ; y distributionareplottedinthebottomrow.Orangepointsare dataandbluelinesaresimulation.........................105 Figure4.40:Energylossof 27 Fbeamfragmentsmeasuredinthefoursilicondetectorsis plottedbytheblackdots,redtriangles,bluediamonds,andbrowncrosses; solidcurvesplotthesimulationoutput.Theresolutionofthesilicondetectors issimulatedusingaGaussianwith ˙ = 0 : 9MeV.................106 Figure4.41:Fragmentandneutronpositionspectrafor 24 Oeventsincoincidencewith twohitsinMoNA / LISA.Simulatedpositionspectra(bluecurves)areover- laidonthecorrespondingexperimentalspectra(blackpoints).Thetopleft andrightpanelsplottheCRDC2 x and y spectrarespectively.Themiddleleft andrightpanelsshowthe x and y distributionsforthetime-sortedhitin MoNA / LISA.Thebottomleftandrightpanelsshowthe x and y distributions forthesecondhitinMoNA / LISA.........................108 Figure4.42:NegativeLnLcurvesextractedfromapseudodatasetwith T 1 = 2 = 2ps (left)andasecondpseudodatasetwith T 1 = 2 = 4ps(right).Inbothpanels, rangevaluesof0.005cm / ns,0.050cm / ns,0.100cm / ns,and0.500cm / ns correspondtotheblackdots,redtriangles,bluediamonds,andgraycrosses respectively.....................................112 Figure5.1:Half-lifeextractedfrom 23 O ! 22 O + n data.Theleftpanelshowsthe relativespeeddistributionfordata(blackpoints)andthreesimulationswith T 1 = 2 = 0ps,4ps,7ps.Therightpanelshowsthenegativeloglikelihood asafunctionofhalf-life.Theredlineindicatestheupperlimitof1.7ps correspondingtoa95%level......................113 Figure5.2:TherelativespeeddistributionsandLnLcurvesfortwosetsofanalysisgates usedtoisolateeventscorrespondingtothe 26 O ! 24 O + 2 n decay.Thetop rowshowstheresultsforeventsextractedusingthecausalitycutsandthe bottomrowshowstheresultsusingthedecisionforest.Intherelativespeed plots,theblackpointsrepresentdataandthecurvesarefromsimulations with T 1 = 2 = 0ps,4ps,and8ps.TheverticalredlinesontopoftheLnL curvesdenotethe1 ˙ statisticaluncertainties....................115 Figure5.3:Thegrayverticallinesindicatetheupperandlowerlimitsonthehalf-lifeob- tainedfromadding,inquadrature,thestatisticalandsystematicuncertainties quotedinRef.(1).Theredverticallinesdenotetheupperandlowerlimits andtheblackpointsshowthenegativeloglikelihoodcurvefromthecurrent analysisusingthecausalitycuts..........................116 xviii Figure5.4:Decaywidth / half-lifeasafunctionofdecayenergyfor2 n emissionfrom 26 O.Thegraylineassumesapureorbital[ d 2 ]coupledtothe totalangularmomentum L = 0.Thesolidblackcurveshowstheresults whennostateinteraction(FSI)andan 24 Omassissimulated. ThebluedashedlineplotstheresultswhennoFSIandthecorrect 24 Omass issimulated.Thereddottedlineshowsthecalculationresultswhenthe n n FSIisscaledby0.25,andthepurpleshort-dashedcurveincludesthefull n n FSI.Theverticalredlinesroughlyindicatetheexperimentalresults from(48),andthegreenshadedareaindicateshalf-lifelimitsobtainedin thiswork.Imageadaptedfrom(58)........................118 Figure5.5:Resultsfromasimulationrunwithasingle11.1mmtarget(graycurves) andwithasegmentedtarget(redcurves).The11.1mmtargetisthesumof thethicknessesofthethreeberylliumtargetsusedinthesegmentedtarget. Forthesegmentedtargetsimulation,thedetectorandtargetthicknessesin thesimulationarethesameasthoselistedinTable3.1.Theleftpanelplots thedi erenceinspeed j ~ v n jj ~ v f j betweenthedetectedneutronand thechargedfragmentandtherightpanelplotsthethreebodydecayenergy reconstructedfromthesimulation.Deviationfromthe50keVinputdecay energyisduetotheenergyaddback.Neutronresolutionsareturnedo .....119 Figure5.6:Measuredrelativespeedandthreebodydecayenergyspectrafor 27 F ( 1 p ) ! 26 O ! 24 O + 2 n aredrawnasblacktrianglesandblackcircles,respectively. Thesolidcurvesarespectrareconstructedfromsimulation.Twodi erent decaychannelsweresimulated(seeinsetoftoprightpanel):(1)directpop- ulationofthe 26 Ogroundstatefollowedbytwo-neutronemission(redsolid line)and(2)populationofthe 26 Oexcitedstatefollowedbyasequential neutronemissionthroughthe 25 Ogroundstate(greendashedline).Inthe toprow,theenergyaddbackischosenevent-by-eventbasedonwhichberyl- liumsegmentwasasthereactiontargetusingthemethoddescribed inSection4.1.1.2.Inthebottomrow,theaddbackforthemiddleberyllium segmentisappliedtoallevents...........................121 Figure5.7:Thetopleftplotshowstherelativespeedreconstructedusingthetargetiden- (Section4.1.1.2)toinformtheenergyaddback.Theblackpoints aredataandthecurvesrepresentrelativespeeddistributionssimulatedas- sumingvarious T 1 = 2 for 26 O-reddashedis0ps,blacksolidis4psandblue dottedcorrespondsto8ps.Inthebottomleftasinglevaluefortheaddback (correspondingtotheenergylossthroughhalfofasinglethicktarget)isused forallmeasuredandsimulatedevents.Therightpanelsshowtheextracted negativeloglikelihoodcurvesforthesegmented(top)andsinglethick(bot- tom)targetreconstructions.............................122 xix CHAPTER1 INTRODUCTION Intheearlytwentiethcentury,remarkableadvanceswerereshapingandexpandinghu- manity'sunderstandingofthephysicaluniverse.Newexperimentalcoupledwitharevo- lutionarynewtheoreticalframeworksetthestageforthedevelopmentofmodernnuclearphysics. AntoineHenriBecquerel'saccidentaldiscovery 1 ofradioactivityin1896(4;5;6;7;8;9;10)fol- lowedbyMarieandPierreCurie'sworkwithradioactiveelementsprovidedthehintsforthe existenceofadynamicsysteminsidetheatomthatisstillbeingexploredtoday.ErnestRutherford reportedthediscoveryoftheatomicnucleusin1911basedontheresultsofanexperimentcarried outbyhisstudentErnestMarsden(11),andJamesChadwickreportedhisdiscoveryoftheneutron in1932(12).Thesediscoveries,togetherwithquantummechanicsandErnestLawrence'sinven- tionofthecyclotronin1929(13)helpedshapeanewfocusedonunderstandingtheatomic nucleus. Atomicnucleiarecomposedofprotonsandneutrons,andeachspeciesofnucleusisuniquely byitsnumberofprotons( Z )andnumberofneutrons( N ).The > 3000currentlyknown nucleicanbeorganizedgraphicallybyplotting Z asafunctionof N asshowninFigure1.1.The totalnumberofnucleons(protonsandneutrons) A = Z + N and Z arecommonlyusedtorefer toanucleus.Aconvenientshorthandnotationanucleusas A Z X N ormorecompactly A X where X istheatomicsymbolthatthenumberofprotons.Di erentnucleiwiththesame numberofprotonsanddi erent N arereferredtoasisotopes.Theisotopesofasingleelement (constant Z )occupyonerowinFigure1.1.Nucleiwiththesamenumberofneutronsanddi erent Z arereferredtoasisotones.Thetermnuclidereferstoanatomcontainingthenucleus A X. Atomicnucleiaccountfor99.9%ofthemassofvisiblematter,andtheyinvolvelengthscales 1 Itshouldbenotedthatthesamee ectthatBecquerelobservedwasalsoobservedbyphotog- rapherAbelNiepcedeSaintVictor40yearsearlier,buttherewerenofollow-upinvestigationsof thephenomenonuntilafterBecquerel'sreport(2).Becquerelsharedthe1903NobelPrizewith MarieandPierreCurie(3). 1 Figure1.1:Thenuclearchartwhereneutronnumberisplottedonthe x axisandprotonnumber onthe y axis.Tilecolorindicatesdecaymode:lightblue(red)indicates ( + ) decay,neutron (proton)emissionisplottedinblue(red),alphadecayisplottedingreen,andspontaneous inviolet.Blacktilesindicatestableorlong-livednuclideswith T 1 = 2 & 10 8 years.Thevertical (horizontal)blueboxesindicatemagicneutron(proton)numbers.DataisfromRef.(14). ˘ 10 15 m,densitiesoforder2 : 3 10 17 kg / m 3 andenergiesrangingfromafewkeVtoafewMeV. Thetimescalefornucleonmotioninsidethenucleusis ˘ 10 22 s.Timescalesfornucleardecays varyoveranenormousrangefrompicosecondstobillionsofyears.Furthermore,sinceanucleus canbecomprisedofanywherefrom1to ˇ 300nucleons,itasadynamicmany-body quantumsystem. 1.1TheShellModel Duetothemathematicalcomplexitiesassociatedwithsolvingthemany-bodyproblem,sim- modelswereconstructedtointerpretempiricalobservations.Onesuchmodelsoughtto describehownucleiwithcertainfimagicflnumbers(2,8,20,28,50,82,and126)ofneutronsor 2 Figure1.2:Contributionstothetotalpotential(redcurve)forad-waveneutronin 16 O.Theblack solidcurveplotsthecentral,spin-independentcomponentparameterizedbyaWoods-Saxonshape. Thebluedottedcurveisthespin-orbitcomponentandthegreentripledot-dashedcurveplotsthe contributionfromthecentrifugalbarrier.TheparametersfortheWoods-Saxonare V 0 = 53MeV, R 0 = 3 : 15fm, a 0 = 0 : 65fm. protonsareparticularlystable(15;16).Asimilare ectisobservedforatomicelectronswhere moreenergyisrequiredtoremoveanelectronfromatomsofthenoblegassesthanfromatomsof otherelements.Thissimilarityinspiredtheideaofnucleonorbitalsorshellsinanalogytothoseof atomicelectrons. Thestepinbuildinganuclearshellmodelistochooseaformforthepotentialthatbinds nucleonstogether.Sincetheforcebetweennucleonsisshort-rangedandthenucleardensityis constant(17),anucleoninthecenterofthenucleuswillinteractwiththesamenumberofneighbors regardlessofitsposition.Therefore,thepotentialshouldberoughlyconstantandnegativeinside thenucleus.Atthesurfaceofthenucleus,therearefewerneighborswithwhichanucleoncan interact,thereforethepotentialshoulddecreasetowardstheedge.Thisqualitativebehavioris oftenparameterizedusingaWoods-Saxonpotential,seetheblackcurveinFigure1.2, f ( r ) = V 0 1 + exp[ ( r R 0 ) = a 0 ] 3 Figure1.3:Neutronsingleparticleenergiesin 208 Pbforthreedi erentpotentialmodels:har- monicoscillator(left),Woods-Saxonwithnospin-orbit(center),andWoods-Saxonwithspin-orbit (right).Theshelloccupanciesareindicatedbythenumbersinsquarebrackets.Thetotaloccu- pancysummedoveralllowershells(2,8,20,...)intheWoods-Saxonplusspin-orbitmodelis indicatedinthespacesbetweengroupsofstates.ImagefromRef.(21). withparameters V 0 , R 0 ,and a 0 edthroughcomparisontoexperimentaldata.Addingaspin- orbittermresultsinenergylevelsspacedsuchthattheiroccupanciesreproducetheobservedmagic numbers(18;19;20),seeFigure1.3. Theresultingmodeldescribesasingleparticlemovingina V ( r ) .Theeigenstates ofthesingle-particleHamiltonianthatincludesthepotentialdescribedabovearecharacterized bytheirenergiesandasetofquantumnumbers,seeFigure1.3.Propertiesofthenucleusare 4 determinedbythesingle-particlelevelsaccordingtothePauliexclusionprinciple.Modern versionsoftheshellmodelaHamiltonianmatrixasthesumofone-andtwo-bodyoperators thatcanbediagonalizedtotheeigenvaluesofaparticularfew-bodysystem.Sincethenumber ofbasisstatesgrowsfactoriallywiththenumberoforbitalsincludedinthecalculation,nucleons inshellsareoftenapproximatedasasinglecoreandtruncationsofthebasisspacearemade tocircumventcomputationallimitations. 1.2NuclearStability 1.2.1BindingEnergy Theatomicmassisabasicquantitythatcanbemeasuredforthegroundstatesofnuclei.Themass energyofanucleus A Z X N isthemassenergyoftheatomminusthemassenergyof Z electrons minusthetotalelectronicbindingenergy.Thedecayandreactionenergiestypicallydealtwithin nuclearphysicsinvolvedi erencesinmassenergiessotheelectronbindingenergies(whichare ˘ 10 6 timesthetypicalatomicmassenergies)tendtocancel(17).Thenuclearbindingenergyis asthemassdi erencebetweenanucleusanditsconstituent Z protonsand N neutrons B f Zm ( 1 H ) + Nm n m ( A X ) g c 2 wheretheprotonandelectronmassesaregroupedinto Z neutralhydrogenatoms.Theconversion factor c 2 = 931 : 494273MeV / ucanbeconvenientwhenmassesaregiveninatomicmassunits(u). Thebindingenergy, B ,istheenergyneededtobreakanucleusintoitsconstituentnucleons,andit representsaquantitativemeasurefornuclearstability. Theconceptofstabilitycanbevisualizedbyplottingthebindingenergyasafunctionof Z for nuclideswithagivenmassnumber, A .TheblackpointsinFigure1.4plotthebindingenergyper nucleonforninenuclideswith A = 21anddi erent ( N ; Z ) .Alocalmaximumisclearlyseenat Z = 10correspondingto 21 Newhichisthemosttightlyboundofthisgroup. 5 Figure1.4:Bindingenergypernucleonasafunctionof Z fornuclideswith A = 21.Thehalf-lives for 21 C, 21 O,and 21 Mgaregivenintheboxes. 1.2.2Half-life Thestabilityofanucleuscanalsobeintermsofitshalf-life,meanlifetimeordecay rate.Consideracollectionofidenticalunstablenuclei A Xthatdecayatsomerate disintegrations perunittime.Thetotalnumberofdecaysperunittimewillbeproportionaltothenumberof A X nucleiinthesamplewhichisitselfafunctionoftime.Thissituationismodeledusingaorder ordinarydi erentialequation(Eq.1.1)forwhichthesolutionmaybewrittenasinEq.1.2. dN ( t ) = N ( t ) dt (1.1) N ( t ) = N ( 0 ) e t (1.2) Thehalf-lifeisastheamountoftimeforhalfofthenucleitodecay.Themeanlifetimeis theaveragetimethatanucleussurvivesbeforedecaying.Therelationshipbetweenhalf-life( T 1 = 2 ), meanlifetime( ˝ )anddecayrate( )isgivenby T 1 = 2 = ln2 = ˝ ln2 : 6 Morestablenucleiwilltypicallyhavelongerhalf-lives(seeFigure1.4).Nuclidesthathavenot beenobservedtodecayareconsideredstable,andallothernucleiareconvertedintothesestable nucleithroughvariousdecayprocesses(seeSection1.3)withhalf-livesthatrangefrompicosec- ondstobillionsofyears. Anotherwaytoquantifythestabilityofanuclearstatecomesfromconsideringthetime- dependentwavefunction j ( r ; t ) j 2 = j ( r ; 0 ) j 2 e t anditsFouriertransform P ( E ) = k Z + 1 1 exp[ iE 0 t = ~ t = 2 + iEt = ~ ](1.3) where isthesamedecayconstantabove.Taking t = 0tobethetimeatwhichtheinitial unstablenucleuswascreated,theintegralinEq.1.3canbeevaluated.Thenormalizationconstant k isfoundtobe i = ( 2 ˇ ) andtheprobabilityofthenucleuswithenergy E isgivenbythe distribution j P ( E ) j 2 = 1 4 ˇ 2 1 ( E E 0 ) 2 + ~ 2 2 4 (1.4) whichhasafullwidthathalfmaximum = ~ .Eq.1.4impliesthatmultipleenergymeasure- mentsmadeonidenticallypreparedtime-evolvingstateswillresultinadistributionofenergies. Furthermore,alargerdecayratecorrespondstoawiderrangeinthemeasuredenergies.Thewidth canberelatedtothehalf-life / lifetime / decayrateaccordingto = ~ ln2 T 1 = 2 = ~ ˝ = ~ (1.5) Forastatewith T 1 = 2 = 1ps,Eq.1.5gives ˇ 10 4 eV. 7 1.2.3SeparationEnergies Twoquantitiesrelatedtothebindingenergyaretheneutronandprotonseparationenergiesdenoted as S n and S p ,respectively.Thesevaluesareasdi erencesbetweenthebindingenergiesof twonuclei S n B A Z X N B A 1 Z X N 1 S p B A Z X N B A 1 Z 1 X N where S n ( S p )determinestheamountofenergyneededtoremoveaneutron(proton)fromnucleus A Z X N .Theseseparationenergiesdeterminewhetherornotaparticularcollectionof Z protonsand N neutronscanformaboundnucleus.Anegativeseparationenergyimpliesthatthenucleus A Z X N isunboundwithrespecttoonenucleonemission.Thedistinctionbetweenboundandunbound systemsleadstotheconceptofneutronandprotondriplines.Thereissomedebateastotheexact ofthedriplines(seeSection2.1ofRef.(22)),butthisdissertationadoptstheconvention thattheneutron(proton)driplineisbythelimitwhere S n ( S p )crosseszero.Theneutron andprotondriplinesshouldthenbediscussedintermsofisotonesandisotopesrespectively(22). Withthisconvention,theneutrondriplinehasbeenmeasuredupto N = 24assuming 33 Fis unbound.Theprotondriplinehasbeenmappedforodd Z nucleiupto Z = 91butonlyupto Z = 12foreven Z . Theseconceptsofunboundnucleianddriplinesraisethequestionfiatwhatpointisacollection ofnucleonsconsideredanucleus?flTherearenoestablishedcriteriabywhichthisquestionis typicallyevaluated.Oneconventionsuggestscomparingthemeasuredhalf-lifeofanucleustothe typicaltimescalesfornucleonmotion(10 22 s)(22).Anotherconventionre-caststhequestionas fiwhatconstitutestheexistenceofanuclide?flThisquestionincorporatesthetimescaleoverwhich anucleusacquireselectrons.Fromthisstandpoint,theauthorsin(23)radioactivitytobeany nucleardecayprocessfimuchslowerthantheKvacancy,whoseduration,inprinciple,can bemeasureddirectly,andwithawidthmuchsmallerthanthethermalenergyatroomtemperature.fl 8 Bythisconventionunstablenuclidesfiexistfliftheydecaywitha T 1 = 2 & 10 14 s.Thediscussion inthisdissertationdoesnotrequireadoptingaconventionfortheexistenceofanucleusor nuclide,butitdoesadopttheofradioactivityasasubsetofnucleardecayprocessesfor which T 1 = 2 & 10 14 s. Atthispointitisimportanttohighlightadistinctionbetween(1)stabilityforboundnuclei and(2)stabilityintermsofboundversusunboundsystems.Inthecase,unstablenucleiwill undergosomedecayprocessthatresultsinoneormorenucleiwithahigherbindingenergythan theoriginalnucleus.Inthesecondcase,theunboundsystemwillemitoneormorenucleonsto produceaboundsystem. 1.3NuclearDecays Anucleusissaidtobestableifithasnotbeenobservedtoundergoanyprocessesbywhich its N and Z spontaneouslychangeorbywhichitspontaneouslydisintegratesintosmallernuclei. Thereare254stablenuclides(24)thatformthevalleyofstabilityonthenuclearchart(Figure1.1). Currentlymorethan3000nucleihavebeenobservedexperimentallywithasmanyas4000more predictedtoexist(25).Mostnucleiareunstableandundergosomedecayprocessthatemitsa characteristictypeofradiation.Thedecayingnucleusissometimescalledtheparentnucleuswhile thedecayproductsarereferredtoasdaughters. 1.3.1AlphaDecay Alphadecayisaprocessthroughwhichaheliumnucleus( A = 4and Z = 2)isemittedfroma largernucleus, A Z ! A 4 ( Z 2 ) + 4 He : Thenetenergyreleasedinthisprocess(the Q value)isgivenbythemassdi erencebetweenthe parentnucleusandthedaughters, 9 Figure1.5:Theradialforasphericallysymmetricpotentialthatapproximatestheinterac- tionbetweenthe particleandthedaughternucleus.Theinteractionisattractiveatshortrange (0 < r < a )duetothenuclearinteractionandrepulsivefor r > a duetotheCoulombrepulsion betweentheprotonsin 4 Heandinthedaughternucleus. Q = ( m Z m Z 2 m ) c 2 : Spontaneousemissionofan particlecanoccurincaseswhere Q ispositive.Withtheexception of 8 Be,thecondition Q > 0holdsmainlyfornuclideswith Z & 50and N & 60. Astraightforwardquantummechanicaltheorydescribingalphadecaywaspresentedin1928 (26;27).Inthistheorythealphaparticleisapproximatedasaparticleinaspherically symmetricpotentialwiththeradialsketchedinFigure1.5.Theattractivepotentialcreated bytheaveragenuclearforceintheregion0 < r < a andtherepulsiveCoulombpotentialinthe region r > a createabarrier,andthereisanon-zeroprobabilitythatthe particlewilltunnel throughthebarrier.Thetransmissionprobabilitycanbecalculatedanalyticallytreatingthebarrier asasequenceofrectangularbarrierswithheights / 1 = r andwidths dr .Then,aftermakingthe assumptionthat ( Q = B ) ˝ 1where B istheheightofthebarrierat r = a ,onecanarriveata formulaforthehalf-lifeofthe decayintermsof Q , B ,thedepthoftheattractivewell( V 0 )and thechargeofthedaughternucleus(seeEquation8.18ofRef.(17)).Thisapproximationisableto reproducethequalitativetrendin119experimentallymeasured decayhalf-livesspreadover25 10 ordersofmagnitude(21).Thecalculatedhalf-livesaresystematicallyshorterbyroughly2orders ofmagnitudeduethevariousassumptionsmade,includingthesphericalshapeandmeannuclear radius1 : 25 A 1 = 3 ofthedaughternucleus.Nevertheless,thisapproximationhighlightsseveralkey aspectsofthedecay-by-particle-emissionprocess.First,thehalf-lifeisextremelysensitivetothe amountofenergyavailableinthedecay, Q .Themeasured(calculated)half-lifeforthe decay of 222 This2 : 8 10 3 s(6 : 3 10 5 s);themeasured(calculated)half-lifeforthe decayof 232 This4 : 4 10 17 s(2 : 6 10 16 s).The Q valuesare8.13MeVand4.08MeVforthe 222 Thand 232 Thdecay,respectively.Adecreasein Q byafactorof2resultsinahalf-lifethatis20orders ofmagnitudelonger(17).Second,onecanseethatallowingthe particletohavesomenon-zero angularmomentum l willincreasethepotentialenergyintheregion r > a inFigure1.5dueto thecentrifugalpotential l ( l + 1 ) ~ 2 = 2 mr 2 .Theresultisathickerbarrier,adecreasedprobabilityfor the particletotunnelthroughand,therefore,alongerhalf-life.So,qualitatively,longerhalf-lives correspondtolowerdecayenergiesandlargerorbitalangularmomenta. 1.3.2BetaDecay Therearethreetypesofbetadecaythatinvolveconvertingaprotonintoaneutronorviceversa. TheyareillustratedinFigure1.6andcanberepresentedbythefollowingequations: + decay: A Z ! A ( Z 1 ) + e + + e electroncapture: e + A Z ! A ( Z 1 ) + e decay: A Z ! A ( Z + 1 ) + e + ¯ e In + decay,aprotoninsidethenucleusisconvertedintoaneutronthatremainsinsidethe nucleusandapositronandanelectronneutrinoareemitted.Asimilarprocess,referredtoas electroncapture,occurswhenanelectroninoneofthelow-lyingatomicorbitalsiscapturedby thenucleusandconvertsaprotonintoaneutronandanelectronneutrino.Themirrorprocess, decay,involvesspontaneousconversionofaneutronintoaprotonandemitsanelectronandan electronanti-neutrino.Betadecayprocessesrepresenttheprimarymethodsbywhichmostofthe 11 Figure1.6:Anillustrationofthethree decayprocesses: decayisdepictedinthetoppanel while + decayandelectroncapturearedepictedinthebottomleftandbottomrightpanels, respectively. known,unstablenucleiareconvertedintostableones.Unstablenucleiontheproton-richsideof thevalleyofstabilityundergo + decayandelectroncapturewhileneutron-richnucleiundergo decay.Theelectronsorpositronsrepresenttheradiationthatiseasiesttodetectfrombeta decaysandwerereferredtohistoricallyasbetarays. Thebetadecayprocessesmentionedinthelastparagraphrepresentasubsetofalargerclassof transformationsthatinvolvequarksandleptons(21).Theelementaryprocessesinvolvedinnuclear betadecayare 12 u ! d + W + ! d + e + + e (1.6) d ! u + W ! u + e + ¯ e (1.7) where u and d aretheupanddownquarks,respectivelyand W arethegaugebosonsthatmediate theweakinteraction.Sincenucleonsconsistofthreevalencequarks, ( u ; u ; d ) fortheprotonand ( u ; d ; d ) fortheneutron,thetransformationofaprotonintoaneutronresultsfromtheprocess describedbyeq.1.6.Similarly,eq.1.7describestheunderlyingtransformationthatconvertsa neutronintoaproton.Intermsofthestandardmodel,thecharacteristicradiationassociatedwith betadecayisduetothedecayofthe W bosons. 1.3.3OtherDecayProcesses Gamma-decayreferstotheprocessbywhichanucleusinanexcitedstatetransitionstoaloweren- ergylevelandemitsaphotonwithenergyroughlyequaltotheenergydi erencebetweentheinitial andlevels.Thedecayenergyispartitionedbetweenthegammaandtherecoilofthenucleus, buttherecoilenergyis ˘ 5eVfora1MeVtransitioninanucleuswith A = 100.Thisissmaller thantheenergyresolutionforahigh-puritygermaniumdetector(seeFigure12-9ofRef.(28)) soitisneglected.Thede-excitationprocesscaninvolvesingleparticlestates,multiplenucleon excitations,rotationalstatesorvibrationalstates(29).Theoriginoftheelectromagneticradiation producedinthesetransitionscanbeunderstoodbypicturingthetransitionsasre-organizations ofchargeandcurrentdistributionsinsidethenucleus.Intermsofclassicalelectrodynamics,the temporaldynamicsofthesedistributionswillproduceradiation.Itisusefultowritethecharge andcurrentdistributionsintermsofamultipoleexpansion.Aquantummechanicalmodelcanbe builtfromthisbasicpicturebyreplacingthemultipolemomentswithoperators.Variousassump- tionscanbemadeabouttheinitialandstatewavefunctionsinordertocalculateadecayrate correspondingtoeachterminthemultipoleexpansionforagiventransitionenergy. Nuclearisaprocessinwhichanucleussplitsintotwoormorelighternuclei.Oneor 13 moreneutronscanalsobeemittedinthisprocess.Fissioncanbespontaneousorinducedbysome reaction,forexample,absorptionofanincidentneutron.Alphadecay(Section1.3.1)isaspecial caseofwhereonedaughterproductis 4 He.Fissioncanbeunderstoodusingthesamemodel ofCoulombbarriertunnelingthatwasintroducedtostudyalphadecay.Thedaughternucleiare assumedtoresideinapotentialwellliketheoneshowninFigure1.5.Thereissomeprobability oftunnelingthroughtheCoulombbarrier,anditisextremelysensitivetotheheightofthebarrier. 1.3.4NeutronandProtonEmission Thedecayprocessesmentionedsofararetheprimarymethodsbywhichbound,unstablenuclei areconvertedintostablenuclei.Beyondthedriplinesand / orathighenoughexcitationenergies, neutronorprotonemissionarepossibledecaychannels.Thedecayenergyisdeterminedbythe absolutevalueoftheseparationenergy E decay = j S n ; p j because S n ; p isnegativeforunboundnuclei. Figure1.7plotsthecalculatedlifetimeasafunctionofdecayenergyforaneutron-unboundnucleus andtwoproton-unboundnuclei.Qualitatively,thehalf-lifefortheneutron / protonemissionwillbe determinedbythebarriercreatedbytheinteractionbetweentheemittedparticleandthedaughter nucleus.Theheightofthebarrierisdeterminedbythedecayenergy,theangularmomentumofthe neutron / proton,and,forprotonemission,theCoulombinteraction.Neutronemissionisinhibited onlybytheangularmomentumbarriersohalf-livesareordersofmagnitudeshorterthanproton emissionfordecayenergies . 100keV.TosummarizethediscussionfromSection1.2.3,special casesofneutron / protonemissionforwhich T 1 = 2 & 10 14 swillbereferredtoasradioactivity. 1.3.4.1One-andTwo-ProtonRadioactivity ThecombinationofCoulombandangularmomentumbarriersresultsinpotentiallylonghalf-lives forprotonemission(seeFigure1.7),andthisprocesscansuccessfullycompetewithotherdecay processes( + ; )incertaincases.Inthesecases,thehalf-lifeislongenoughtoqualifyasproton radioactivity.Thisprocessoccursinodd- Z nucleibeyondtheprotondripline.Itwasobserved fromanisomericstatein 53 Coin1970(30).Protonradioactivityfromthegroundstateof 151 Lu 14 Figure1.7:Calculatedlifetimesforaneutronemitter, 16 B,andtwoprotonemitters, 16 Fand 151 Lu, asafunctionofdecayenergyforangularmomenta L = 0(solidlines), L = 1(dashedlines), L = 2 (dash-dottedlines), L = 5(dottedlines).Imagefrom(22). wasreportedin1982(31).Therearemorethan40knownprotonemittersincludingemission fromlong-livedisomericstates. Two-proton(2 p )emissionwassuggestedbyZeldovichandGoldanskyin1960(32;33). Experimentalstudiesoflight2 p unboundsystems 6 Be(34;35), 12 O(36;37),and 16 Ne(36) uncoveredstateswithbroadwidthscorrespondingto2 p decayhalf-livestooshorttobe asradioactivity.Theshorthalf-livesinthesecasesresultfromthelowCoulombbarrierforthese systems.Theobservationof2 p radioactivitywasreportedfor 45 Fewith T 1 = 2 = 3 : 2 + 2 : 6 1 : 0 ms(38).SeeRef.(23)andreferencesthereinfordetailedreviewsofone-protonandtwo-proton radioactivity. 1.3.4.2ProspectsforTwo-NeutronRadioactivity One-ormultiple-neutronemissionfromaneutron-unboundnucleushasbeenobservedfortwenty di erentcases,seeTable16.1inRef.(24).Inparticular,two-neutronemissionhasbeenobserved from 10 He(seeChapter2ofRef.(39)forasummaryofthe 10 Hemeasurements), 5 H,(40;41;42), 15 13 Li(43;44), 16 Be(45),and 26 O(46;47;48).Awidth ˘ 1MeVforthe 5 Hgroundstatehasbeen extractedfromvariousmeasurementsof H H,seeTableIinRef.(49).Thiscorrespondsto T 1 = 2 ˘ 10 22 swhichistooshorttobeconsideredradioactivity.Extractingawidthfromtheground statemeasurementsof 13 Liand 16 Beisprecludedbytheexperimentalresolution.Theoretical calculationspresentedinRef.(50)suggestedthat 26 Oiscurrentlythebestcandidateforobserving 2 n radioactivity,seethediscussioninSection2.2.3,andtheexperimenttoattemptthisobservation isthesubjectofthisdissertation. 1.4DissertationOverview Theexperimentdescribedhereutilizedanewsegmentedtargetsystemtoproducetwo-neutron- unbound 26 Ofroma 27 FbeamattheNationalSuperconductingCyclotronLaboratory.Thehalf- lifeforthegroundstateof 26 Owasextractedfrommeasurementsofthedecayproducts,andthe performanceofthesegmentedtargetsystemwasevaluated.Thisdocumentisorganizedasfollows: Chapter2presentsthemotivationsforthisexperiment,Chapter3describesthedetectorsystems usedtocollectdata,Chapter4discussesthemethodsandproceduresusedtoanalyzethedata, Chapter5presentstheresults,andremarksandconclusionsaregiveninChapter6. 16 CHAPTER2 BACKGROUNDANDMOTIVATION Thepurposeofthisexperimentwastwo-fold.First,ameasurementofthehalf-lifeforthedecay ofthe 26 Ogroundstatecoulddetermineifthisdecayastwo-neutronradioactivity.The secondgoalwasanevaluationofthenewsegmentedtargetsystem.Thischapterdiscussesthese twomotivatingfactors.Firstabriefsummaryofprevious 26 Omeasurementsisgivenfollowed byanoverviewofthetheoreticalmotivationsforahalf-lifemeasurementonthisneutron-unbound system.Asectiondiscussesthemotivationsforbuildingthesegmentedtarget. 2.1PreviousExperiments Thenon-observationof 26 Oinexperiments(51;52;52;53;54)datingbackto1990suggested thatthisoxygenisotopeisunbound.TheMoNACollaborationreportedtheobservation(46) oftheunbound 26 Ogroundstate.Theexperimentusedinvariantmassspectroscopytomeasure thedecayenergyofthe 24 O + 2 n systemandprovidedveevidenceestablishingtheparticle- instabilityof 26 O.Thebesttothedecayenergyspectrumincludedaresonanceforthe 26 O groundstateat150 + 50 150 keVabovethe2 n separationenergyfor 24 O.Thewasinsensitivetothe widthofthisresonance.Thedistributionofrelativespeedsbetweenneutronsand 24 Ofragments j ~ v n jj ~ v f j wasanalyzedtoextractahalf-lifeof4 + 1 : 1 1 : 5 (stat) 3(syst)psatan82%level, andtheseresultswerereportedinasubsequentpublication,Ref.(1).Adescriptionofthemethod usedforthehalf-lifeanalysisisgiveninRef.(55)andSection3.10. The 24 O + 2 n systemhasalsobeenmeasuredusingtheR3B-LANDsetupatGSI.Theresults reportedinRef.(47)giveanupperlimitof120keVforthe 26 Ogroundstateandanupperlimit of5.7nsforitslifetime,bothata95%level.AnexperimentusingtheSAMURAI spectrometerandNEBULAatRIKENmeasured 26 Owiththehigheststatisticstodate(48).The groundstatewasfoundtolie18 3(stat) 4(syst)keVabovethe2 n separationenergyof 24 O.In addition,the2 + statewasobservedat1 : 28 + 0 : 11 0 : 08 MeVabovethreshold. 17 2.2TheoreticalBackground 2.2.1Two-bodydecay Thedecayofaparentsystemintotwodaughterparticles M ! m 1 + m 2 isasimpleprocesswhere theenergysharingbetweenthedaughtersisuniquelydetermined.Initsrestframe,theparenthas four-momentum ( M ; ~ 0 ) andthedaughtershave P 1 ; 2 = ( E 1 ; 2 ; ~ p 1 ; 2 ) ; E 2 1 ; 2 = m 2 1 ; 2 + p 2 1 ; 2 (2.1) Momentumconservationinthe M restframeimpliesthedaughters'3-momentaareequaland opposite. ~ p 1 = ~ p 2 (2.2) Squaringbothsidesof2.2andusingtherelation p 2 = E 2 m 2 , E 2 1 m 2 1 = E 2 2 m 2 2 (2.3) Thesecondequationneededtosolvefor E 1 ; E 2 istheconservationofenergy M = E 1 + E 2 . Withthesetworelations,thedaughterenergiescanbeexpressedintermsoftherestmassesofthe particles, E 1 = M 2 + m 2 1 m 2 2 2 M ; E 2 = M 2 + m 2 2 m 2 1 2 M (2.4) 2.2.2Three-bodydecay Incontrasttoatwo-bodydecay,theenergypartitioningamongdaughterparticlesinathree-body decayisnotuniquelydetermined.Forexample,inthecaseofnuclear -decay,adistributionof -particleenergieswillbeobservedinsteadofasinglevaluelikeinthecaseof -decay.Such anobservationpromptedWolfgangPaulitopostulatetheexistenceoftheelectronneutrino,thus 18 Figure2.1:Illustrationsoftheenergyconditionscharacteristicofsequential(toppanel)andsimul- taneous(bottompanel)threebodydecays. describing -decayasathree-bodyprocess(56).Nevertheless,theenergiesandrestmassesofthe parentanddaughterparticlesdorestricttheallowedphasespaceofthekinematicalvariables.For example,themaximumallowedmomentumforeachdaughterparticlecanbecalculatedintherest frameoftheparent p i max = 1 2 M q [ M 2 ( m i + m j + m k ) 2 ][ M 2 ( m j + m k m i ) 2 ] where M istherestmassoftheparentand m ijk aretherestmassesofthedaughters.Acyclic permutationoftheindices i ; j ; k gives p max fortheotherparticles j ; k .Foracompletederivation ofthisandotherkinematicallimitsforthree-bodydecaythereaderisreferredtoRef.(57). 2.2.3 2 n Decayof 26 O Threebodydecayscanbecharacterizedbyeithersimultaneousorsequentialparticleemission dependingontherelativespacingofenergylevelsintheparent,intermediateanddaughternuclei, seeFigure2.1.Theenergyconditionsforsimultaneous(alsocalledtrue)two-particledecaymake 19 Figure2.2:Decaywidth / half-lifeasafunctionofdecayenergyfor2 n emissionfrom 26 O.The graylineassumesapureorbital[ d 2 ]coupledtothetotalangularmomentum L = 0. ThesolidblackcurveshowstheresultsforthenoFSI, 24 Omass.Thebluedashedline plotstheresultsforthenoFSIcalculation.Thereddottedlineisthecalculationwiththe n n FSIscaledby0.25,andthepurpleshort-dashedcurveisthefull n n FSIresults.Theverticalred linesroughlyindicatetheexperimentalresultsfrom(48).Imageadaptedfrom(58). theemissionofasingleparticleenergeticallyimpossible.Therequirementthattwoparticlesbe emittedsimultaneouslyresultsinlongerlifetimescomparedtoasequentialdecay(50). The 26 Ogroundstatewasasapromisingcandidateforobserving2 n radioactivity basedonthreecriteria: 1. asequentialdecaythrough 25 Oisenergeticallyunfavorable 2. thedecayenergyislow 3. theangularmomentumbarrierismaximizedsincethetwovalenceneutronsareexpectedto occupythe0 d 3 = 2 orbital(59;60) Allofthesefactorscontributetothee ectivebarrierthroughwhichtheneutronshavetotunnel whichincreasesthelifetimeofthenucleus. 20 Resultsfromatheoreticalmodelforthe2 n decayof 26 Owerepresentedin(58).Themodelwas adaptedfromthethree-bodyhypersphericalharmonicsclustermodelusedtostudy2 p radioactivity (61),anditincorporatedtheexperimentalinformationabout 26 O(46;47)and 25 O(62)thatwas availableatthetime.ThemodelcalculatesadecaywidthbysolvingSchrödinger'sequationfora three-bodyHamiltonianthatdescribesthe 24 Oclusterandtwoneutrons.AGaussianformforthe n n potentialwasused,andaWoods-Saxonpotentialwasusedforthe 24 O n potential.The decaywidthwascalculatedassuming 1. no n n stateinteraction(FSI)andanmassfor 24 O 2. no n n FSIandthecorrectmassfor 24 O 3. the n n FSIscaledby0.25 4. thefull n n FSI TheresultsofthesecalculationsareshowninFigure2.2.Eachofthesesuccessiveapproximations decreasestheestimatedhalf-lifebyordersofmagnitude.Reducingtheuncertaintyintheexper- imentalmeasurementwouldhelpinformthetheoreticalmodelusedtointerpretthemicroscopic dynamicsofthisdecayprocess. 2.3TheUnbinnedLoglikelihood Thissectionsummarizesthemethodofmaximumloglikelihood(LnL)forparameterestimation andstatisticaluncertaintycalculationfollowingthetreatmentgiveninRef.(63).Avariationof thismethodwasusedinthepresentanalysis.Notethatthediscussionherefollowstheconvention ofstandardstatisticalanalysistextsandreferstomaximizingtheLnLfunction.However,tobe consistentwiththepreviouspublishedwork,theanalysisdiscussedlaterusestheconventionof minimizingthenegativeLnL.Themethodsarethesameforbothconventionsanddi eronlybya sign. Considerarandomvariable x thathasbeenmeasured n timesinsomeexperiment.Suppose that x isknowntobedistributedaccordingtosomeprobabilitydensityfunction(p.d.f.) f ( x ; ) 21 withknownformbut(forsimplicity)oneunknownparameter .Thentheprobabilitythateach measurement x i liesintherange[ x i ; x i + dx i ]canbewritten P = n Y i = 1 f ( x i ; ) dx i and P willbeatamaximumforthecorrectchoiceoftheparameter .Furthermore,theparameters areindependentofeach dx i whichpermitsofthelikelihoodfunction L ( ) = n Y i = 1 f ( x i ; ) When L ( ) hasasimpleanalyticform,theusualprescriptionformaximizingafunctioncanbe followed,namely ‹ suchthat dL d ! = ‹ = 0 : Thissolutionisreferredtoasanestimator,denotedas ‹ ,forthetrueparameter .Whenthereis noanalyticform,numericalmethodsmustbeusedtocalculate L ( ) anditsmaximum. Itisconvenienttotakethelogarithmofthelikelihoodfunctionasthisconvertstheproductof termsintoasumandexponentsintomultiplicativefactors. ln L ( ) = n X i = 0 ln f ( x i ; ) (2.5) Sincethelogarithmisamonotonicallyincreasingfunction,thevalueof thatmaximizes L ( ) willalsomaximizeln L ( ) . Thesecondderivativeofln L ( ) canbeusedtodeterminethevarianceofanestimator ‹ : c ˙ 2 ‹ = 0 B @ 1 , @ 2 ln L ( ) @ 2 1 C A = ‹ : (2.6) Incaseswherethelikelihoodfunctionhasnoanalyticformitcanbeusefultoemployagraphical methodforobtainingthevarianceofanestimator.TounderstandhowthisworksconsideraTaylor expansionofln L ( ) abouttheminimum = ‹ 22 ln L ( ) = ln L ( ‹ ) + @ ln L ( ) @ ! = ‹ ( ‹ ) + 1 2! 0 B @ @ 2 ln L ( ) @ 2 1 C A = ‹ ( ‹ ) 2 + ::: Since ‹ correspondstoamaximum,ln L ( ‹ ) = ln L max andthesecondtermis0.Neglectingterms higherthansecondorder,eq.2.6canbeusedtore-writetheexpansionas ln L ( ) = ln L max ( ‹ ) 2 2 c ˙ 2 ‹ (2.7) Changingtheargument by N standarddeviationscanbeexpressedas ! ‹ N ‹ ˙ ‹ andeq.2.7 becomes ln L ( ‹ ‹ ˙ ‹ ) = ln L max N 2 2 2.4Fragmentmomentumreconstructionanddecayenergyresolution Invariantmassspectroscopy(Section3.9)experimentstostudyneutronunboundstatesare basedonradioactiveionbeams(RIBs)producedviaprojectilefragmentation.Theinten- sityofthefragmentbeamcanbe ˘ 10 1000particles / s(64;65).Thebeamisthendirectedonto thereactiontargettoinduceaone-ortwo-protonknockoutreaction.Typicalcross-sectionsfor theseknockoutreactionsare ˘ 1mband ˘ 0 : 1mb,respectively(66).Thuscertainexperimentsat thelimitoffeasibilitywouldgreatlyfromanymethodthatincreasesreactionyield. Onewaytoincreasereactionyieldistouseathickertargettoincreasethenumberdensity oftargetnuclei.However,thisapproachnegativelyimpactstheresolutionofthedecayenergy measurement.Akeycomponentthatthedecayenergyresolutionisthereconstruction ofthefragmentmomentumatthedecayvertex.Thedecayprocessoccursinsidethetargetmaterial duetoits ˘ 10 22 stimescale.Thechargeddaughterfragmentlosesenergyasittravelsthrough therestofthetarget,thereforeanysubsequentmeasurementofitsmomentumwillyieldalower valuethanatthedecayvertex.Torecoverthevertexmomentum,theenergylosttothetarget materialmustbeaddedback.Thedecayenergyresolutionisdirectlyimpactedbyhowwellthe 23 energyaddbackcanbeestimatedwhichinturndependsonhowwellthelocationofthedecayis measured. Byusinganactivetargetinwhichthedecaypositionand / orenergylostbythefragmentis measured,onecoulddirectlymeasuretheenergyaddbackevent-by-event.Incurrentactivetarget systemsthegasactsasboththetargetanddetectormaterial(67;68;69;70).However,to producethesamereactionrateasa1cmthicksolidberylliumtarget,anactivetargetwithan idealgaswouldneedtobe10 7 mlongandoperateat500kPa( ˘ 5atm).Thisestimateassumes identicalreactioncross-sectionsforberylliumandtheidealgas. Acompactsolutionisasystemcomposedofmultiplesolid,thintargetsinterleavedwithenergy lossdetectors.Thissystemlocalizesthedecaypositiontooneofthetargetsectionswhichallows forabetterapproximationoftheenergylossthroughthedetectorsystemthanforasinglethick target.Whensilicondetectorsareusedtomeasuretheenergyloss,theincidentbeamrateislimited to ˘ 1000particlespersecondtoavoidpileupandexcessiveradiationdamage. Insummary,segmentingasinglethicktargeto ersimprovedenergyresolutionbecausethe decaypositioncanbebetterlocalizedsothattheenergyaddbackcanbebetterapproximated. 24 CHAPTER3 EXPERIMENTALTECHNIQUE TheexperimentdescribedherewasperformedattheNationalSuperconductingCyclotronLabo- ratory(NSCL)between30Juneand7July2016usingthenewlydevelopedsegmentedtarget,the ModularNeutronArray(MoNA),theLarge-areamulti-InstitutionalScintillatorArray(LISA)and theSweepermagnet.Initssection,thischapterprovidesanoverviewoftheexperimental setup,thenitdescribesthebeamproductionanddownstreamdetectorsindetailinthefollowing sections. 3.1ExperimentalSetup A140MeV / u 48 Cabeamimpingedona775mg / cm 2 Betargettoproduceasecondarybeam of 27 Fviaprojectilefragmentation.TheA1900fragmentseparatorwasusedtoselect 27 Fatan energyof106.2MeV / ufromotherproductsofthefragmentationreaction.Thissecondarybeam wasdeliveredtotheexperimentalareaintheN2 / N3vaultwhereitwasdirectedontoaseriesof positionsensitivesilicondetectorsandberylliumtargets,thesegmentedtarget.Theaverage 27 F beamrateonthetargetwas17particlespersecond. Thereactionofinterestforthisexperimentwasaone-protonremovalfrom 27 Ftoproduce unbound 26 Owhichthendecaysthroughtwo-neutronemission.TheSweepermagnetsatbehind thesegmentedtargetandbentthecharged 24 Ofragmentsintoasuiteofchargedparticledetectors insidetheSweeperfocalplanebox.Twocathode-readoutdriftchambers(CRDCs),anionization chamberandathintimingscintillatorwerelocatedinsidethefocalplanebox. Theneutronscontinuedinthegeneraldirectionofthebeam.Una ectedbythemagnetic theyexitedtheSweepervacuumchamberandtraveledthroughairuntiltheyinteractwiththe plasticscintillatorsthatcomprisetheMoNA / LISAarrays.Thesearrayswerearrangedin17layers witheachlayerconsistingof16barsstackedontopofoneanother.Thelayerswerearrangedin groupsoftwoorthreeacrossthreesupporttables.Alllayerswerecenteredonthebeamaxis.Prior 25 Figure3.1:DetectorlayoutintheN2vault. totheexperiment,theeighteenthlayerwasremovedfromtheexperimentalareatobeusedinan experimentattheLosAlamosNeutronScienceCenter(71). TheSweeperfocalplanedetectorsandtheMoNA / LISAarraysallowforakinematicallycom- pletemeasurementofthechargedfragmentsandneutronstowhichthemethodofinvariantmass spectroscopy(Section3.9)canbeappliedtorecoverinformationabouttheunbound 26 Onucleus. Forthisexperiment,thedi erenceinmagnitudebetweentheneutronandfragmentvelocities j ~ v n jj ~ v f j wasanalyzedtolookforasignaturethatthe 26 Oexistedforsomemeasurableamount oftimebeforethetwo-neutronemission. 3.2BeamProduction TheCoupledCyclotronFacility(72)andtheA1900fragmentSeparator(73)providedthe 27 F beamviafastprojectilefragmentation(74).Since 27 Fhasahalf-lifeof5.0(2)ms(75)itcannotbe accelerateddirectlyandmustbeproducedininthiscase,viafragmentationof 48 Caon 9 Be. Astable 48 Cabeamwasacceleratedto140MeV / uinthecoupledK500andK1200cyclotrons thendirectedontoa775mg / cm 2 Betargetwherethefragmentationoccurred.Awidevarietyof nucleiwereproducedbythisprocess.TheA1900wasusedtoextract 27 Ffromthefragmentation products.TheA1900hasfourdipolemagnetswithfocusingelementsinbetweenthemto nucleibytheirmagneticrigidityandselectamomentumtochargeratio.Anachromatic 26 Figure3.2:BeamproductionattheCoupledCyclotronFacilitystartsbyheatingasampleofthe primarybeammaterialinanionsource.TheK500andK1200cyclotronsacceleratethebeam whichissubsequentlydirectedontoaBetarget.Fragmentsresultingfromnuclearinteractions withinthetargetarebytheA1900toprovidethedesiredsecondarybeam.Imagesource: (76). aluminumwedgewiththickness450mg / cm 2 wasplacedatthedispersivemid-planebetweenthe secondandthirddipolestoimproveseparation;theenergylossinthewedgeisproportionalto Z 2 sodi erentelementsenteringthewedgewiththesamerigidityexitwithdi erentrigidities.The 27 Fwasthendeliveredtotheexperimentalvaultwithanenergyof106.2MeV / ucorrespondingtoa rigidityof4.58Tm.Threemajorcontaminants(1) 28 Ne( E = 121 : 5MeV / u)(2) 29 Ne( E = 113 : 6 MeV / u)(3) 30 Na( E = 127 : 7MeV / u)arrivedwiththesamerigidity.Itshouldbenotedthat thebeamwasdeliveredatahigherrigiditythantheSweepercanaccommodate.However,the increased(relativetopreviousMoNAexperiments)targetthicknessmeansthatunreactedbeam andreactionproductsentertheSweeperbelowits4Tmlimit. 3.3A1900andTargetScintillators AplastictimingscintillatorislocatedatthefocalplaneoftheA1900fragmentseparator10.9 mupstreamfromthesegmentedtarget.Forthisexperiment,theA1900scintillatorwasa144 mthickBC-400scintillatoropticallycoupledtoaPMT.Thetargetscintillator(seeFigure3.1) waslocated1.03mupstreamfromthesegmentedtargetandconsistedofa420 mthickBC-404 scintillatorcoupledtoaPMT.Inbothcases,thechargedionspassingthroughtheplasticexcite 27 moleculesthatde-exciteproducingphotons.ThephotonsthatreachthePMTareconvertedinto anelectricalsignal.Thesignalrisetimesare ˘ 1ns,sothetimeatwhichtheionpassesthrough theplasticcanbemarkedwitharesolution ˘ 1ns.Withtheinformationfromthetwotiming scintillators,anion'stimeoffromtheA1900totheN2vaultwasmeasured. 3.4SegmentedTarget Section2.4discussedhowsegmentingasinglethicksolidtargeto ersimprovedenergyreso- lutionbecausethedecaypositioncanbebetterlocalizedsothattheenergyaddbackcanbebetter determined.Thefollowingsubsectionsdetailthesetupandelectronicsofthesegmentedtarget systemusedduringthisexperiment. 3.4.1SiliconDetectorsandBerylliumTargets Thesegmentedtargetconsistsoffourposition-sensitivesilicondetectorsandthreeberylliumtar- getsarrangedasshowninFigure3.3.Eachsilicondetectorisa62mm 62mm ˘ 140 m phosphorus-dopedn-typesiliconwafer.Thefrontfaceisaboron-implantedp-typelayerresistive anodeandisborderedby0.5mmresistiveion-implantedstrips.Theresistancesbetweenadjacent cornersis ˘ 5 : 6k .Aluminumcontactsateachcornerprovideelectricalconnectionsthatreadout thefoursignalswhichcanbeusedtoreconstructtheinteractionposition.Asignalistaken fromtherearfacenon-resistivelayerviaanaluminum-evaporatedcontacttoprovideanindepen- dentmeasurementofthetotalenergydepositedwithbetterresolutionthancanbeobtainedfrom thesumofthecornersignals. Thethicknessesofthethreeberylliumtargetswerechosentooptimizetheproductionrateof thenucleusofinterestanddecayenergyresolution.Thelinearthicknessesandareadensitiesof theberylliumandsilicontargetsarelistedinTable3.1. 28 SegmentThickness[ m]AreaDensity[mg / cm 2 ] Si014032.5 Be14100758.5 Si113531.3 Be23736691.2 Si213832.0 Be33302610.9 Si314233.0 Table3.1:Thicknessesforthesilicondetectorsandberylliumtargets.Theberylliumtargetswere measureddirectlyusingcaliperswithadialindicatorandtheassociatedmeasurementuncertainties are 4 m( 0 : 7mg / cm 2 Be).Thesiliconwaferthicknesseswerereportedbythemanufacturer withuncertaintiesof 1 m( 0 : 2mg / cm 2 Si). Figure3.3:Eachdetectoris11cm 11cm 0.32cmincludingtheframehousingthesilicon wafer.Thethicknessesofthesiliconwafersare140 m,135 m,138 m,and142 mfor detectors0,1,2,and3,respectively.Theberylliumtargetsare2.8cmtallwiththicknessesof 0.41cm,0.37cm,and0.33cmfortargets1,2,and3,respectively.Thespacingbetweeneach detector / targetis0.84cm(0.33inches)sointotaltheapparatusextends5.04cmalongthebeam axis. 29 Figure3.4:Drawingofthesegmentedtargetmountedinthebeamline:(a)beamviewerplateused toimagethebeamduringtuning(b)baseonwhichalldetectorsaremounted(c)silicondetector frame(d)berylliumtarget(e)baseonwhichalltargetsaremounted.Theviewerismountedtothe targetbase.Boththedetectorandtargetmountbasesareattachedtopneumaticdrivessotheycan beindividuallyinsertedintoandretractedfromthebeamline. 3.4.2Signalreadoutandelectronics Thefoursilicondetectorsgenerate20signalsconsistingof16cornersignalsand4anodesignals. ThesignalsfromthealuminumcontactsmentionedpreviouslyarecarriedbyAWG28wireto acustom-madePCBboard(oneforeachdetector)thatsubsequentlyroutesthevesignalsfrom eachdetectortoasectionofa60-conductorribboncable. Eachanodesignalisroutedtoathathasatypicaloutputsignalaround30mV witha0.05 srisetimeanda500 sfalltimewhena5.5MeV sourcewasplaced7.1cmin frontofthedetectorundervacuum.TheanodesignalsarethenroutedtoaTennelec241Sshaping withashapingtimeof6 s.TheshapedanodesignalsareroutedtoaMesytecMADC-32 analog-to-digitalconverterwitha4Vrange. Eachcornersignalisabovegroundbya10k resistoronthePCB.Thecornersignals areroutedtooneof16MesytecMMPR1Thecornersignalsaresomefractionof theanodesignaldependingontheinteractionlocationofthepassingion.Theoutputs areconnectedtoasingle16-channelMesytecMSCF-16shaping.Theshapingtimeis 2 sforallcornerchannels.Theshapedsignalsareroutedtoabankof16inputsonthesame 30 Figure3.5:Wiringdiagramshowingthesignalpathsfortheanodeandcornersignalsfromone ofthesilicondetectors.Theanode(blackarrows)andcorner(bluearrows)signalswererouted throughseparatepreampandshapingAllshapedsignalswereprocessedbythesame analog-to-digitalconverter. 31 MADC-32thatreceivestheanodesignals. 3.5Sweepermagnet TheSweepermagnetisalarge-gapsuperconductingdipolemagnetwithabendingradiusof approximately1meterandabendingangleof43.3 (77).Themagneticismonitoredusing aHallprobeandhasbeenmappedinpreviouswork(78).Forthisexperiment,themagnetwas settoacurrentof306Awhichcorrespondedtoacentralrigidityof3.445Tm,whichwasthe expectedrigidityof 24 Ofragmentsproducedfromthe 27 Fbeam.Ingeneral,unreactedbeam, reactionfragmentsandneutronsallexitthetargetinaforward-focusedcone.TheSweepermagnet bendsthechargedparticlesawayfromtheneutronpathandintoasuiteofchargedparticle detectors(describedinthenextsection).Theneutronscontinuealongthebeamaxisthrougha verticalgapof14cmandintotheneutrondetectors(describedinSection3.7). 3.6ChargedParticleDetectors ImmediatelydownstreamfromtheSweepermagnetwasacollectionofchargedparticledetec- torscontainedinalargevacuumbox.Thepositionsofreactionproductsbythemagnet weremeasuredintwoCathodeReadoutDriftChambers(CRDCs)separatedby1.54m.Behind theCRDCssatanionizationchambertomeasureenergylossfollowedbyalargeareaplastic scintillatortodetermineparticletiming. 3.6.1CRDCs ThedistancefromthetargettotheCRDCwas1.86mmeasuredalongthearcofthecentral trajectorythroughthemagnet.ThesecondCRDCwaspositioned z = 1 : 54mdownstreamfrom theTheCRDCsmeasurethe x and y positionsofparticlespassingthroughtheirvolumes. Theangleinthe xz planebetweentheparticle'strajectoryandthe z -axiscanbecalculatedfrom thetwo x measurementsandtheknowndistancebetweentheCRDCstan x = ( x 2 x 1 ) = z .A similarcalculationusingthetwo y measurementsgivesanangle y withrespecttothe z -axisin 32 Figure3.6:SchematicofaCathodeReadoutDriftChamber(CRDC)expandedinthe z -direction. Theshapingwiresareomittedforvisibility. the y z plane.Theseanglescanbeusedtotracetheparticle'spathbackthroughthemagnettothe target.AschematicofthedetectorisshowninFigure3.6. ACRDCfunctionsinamannersimilartothatofanionizationchamber.Itiswitha1:4 mixtureofisobutaneandCF 4 gasatapressureof40Torr(0.05atm)andsealedwithtwowindows. Whenaparticlepassesthroughitionizesthegasandcreatesionizationpairsthatareseparateddue toauniformappliedelectricThisdriftisproducedbya1000Vpotentialdi erence betweenaplateatthetopofthedetectorandaFrischgridnearthebottom.Fieldshapingwires paralleltothe x -axisareplacedatintervalsin y alongeachfaceofthedetector.Belowthe Frischgridisananodewireandaseriesof116aluminumcathodepadswithapitchof2.54mm inwidththatrunparalleltothe x direction. ElectronsdriftfromtheinteractionpointofthepassingparticletowardstheFrischgrid.As theypassthroughtheFrischgridtheyenterastrongelectriccreatedbytheanodewireand produceanelectronavalanchewhichiscollectedattheanode.Aninducedchargeisdistributed acrossthecathodepads,andthe x positionisextractedfromthedistributionofthisinducedcharge. The y positionisdeterminedfromthedrifttimeoftheelectronwhichismeasuredasthedi erence betweenasignalinthethintimingscintillatorandasignalontheanodewire.Theactiveareaof 33 eachCRDCis30 30cm 2 inthe x y plane. 3.6.2IonizationChamber TheionizationchamberisagdetectorsimilartoaCRDCbutwith16chargecollecting padssegmentedalongthe z direction.Theactivevolumeofthedetectoris40 40 65cm 3 .It iswithP-10gas(10%CH 4 and90%Ar)andheldat400Torr(0.53atm).Thewindowsare thintoallowparticlestopassthroughwithnegligibleenergyloss;theyaremadeofKevlar and12 mPPTAandaremountedwithepoxy.Theupstreamwindowis30 30cm 2 tomatchthe acceptanceoftheCRDCandthedownstreamwindowis40 40cm 2 toallowfordispersionof thebeam. Theionchamberhasaplateontopand16chargecollectingpadsonthebottom.Acharged particlepassingthroughthegascreatesionizationpairsandadriftvoltageof1500Visapplied tocollecttheionizationpairs.Theelectronsarecollectedonthe16padsatthebottom.Theenergy lostinthegasisproportionaltothetotalchargecollectedonthepads. 3.6.3ThinTimingScintillator Thethintimingscintillatorismountedjustdownstreamfromtheionchamber.Itisaplastic scintillator(EJ-204)withdimensions55 55 5cm 3 andhasfourphotomultipliertubesattached vialightguides-twoalongthetopedgeandtwoalongthebottomedge.Itmeasurestimeof andtriggersthedataacquistionsystem.Thelightguidesaretrapezoidalandopticallyconnected tothePMTs.AdiagramofthethinscintillatorisshowninFigure4.13. 3.7MoNALISA Neutronpositionand(ToF)measurementsweremadewiththeModularNeutron Array(MoNA)andtheLarge-areamulti-InstitutionalScintillatorArray(LISA)(79;80;66).The MoNAarrayconsistedof128Bicron-manufacturedBC-408plasticscintillatorbarsmeasuring 200cm 10cm 10cm.TheLISAarrayconsistedof144EljenEJ-200plasticscintillatorbars 34 Figure3.7:SchematicdrawingofasingleMoNA / LISAplasticscintillatorbar.Imagesource: Ref.(81). withthesamedimensions.Thetwomaterialsarephysicallyandchemicallyequivalent.Eachbar iswrappedinvematerialtoreducelightlossthencoveredwithblackplastictoprevent ambientlightfrominducingsignals.AschematicofasinglebarisshowninFigure3.7. Neutronsaredetectedinthescintillatorbarswhentheyinteractwiththeatomicnucleiinthe plastic.Theneutron-to-hydrogenmassratio( ˘ 1)isroughlyanorderofmagnitudelargerthanthe neutron-to-carbonmassratio( ˘ 0 : 08)sotheprobabilityislargerforneutron-hydrogeninteractions. Thereforemostofthescintillationlightisproducedinneutron-hydrogenscatteringprocesses;the recoilingprotonperturbstheelectronicstructureoftheatomsinthescintillatormaterialresulting inasdescribedinSection3.6.3.Thelightpropagatestobothendsofthebarandis collectedinphotomultipliertubes(PMTs).MoNAwaswithPhotonisXP2262 / BPMTs andLISAwasconstructedwithHamamatsuR329-02PMTs.ThePMTsconvertthescintillation lighttoanelectronicsignal.EachPMThousinghasthreeconnectorstoaccommodatetwosignal outputsandonehighvoltageconnection.Thetwooutputsignalsareusedtomeasure(1)thetime atwhichaninteractionoccursinthebarrelativetosometriggersignaland(2)theamountof scintillationlightproducedbytheinteraction. MoNAandLISAaremodularsystemsthatcanbearrangedinavarietyofways.Forthis experiment,16barswerestackedoneontopofanothertoformalayer.Seventeentotal16-bar layerswereasshowninFigure3.8.Layerswerearrangedingroupsoftwoorthree acrossthreetables.IndividualbarswerelabeledbytheirTable(LISA-2,LISA-1,MoNA),Layer 35 Figure3.8:AdiagramofthespatialorderingoftheMoNA / LISAbarsastheywerefor thisexperiment.Neutronsfromthetargettravelfromlefttoright.Thespacingbetweengroupsof layersisnotdrawntoscale. (A,B,C,etc.)andnumber(0,1,...,14,15). PlacementofthebarsrelativetothetargetisshowninFigure3.1.Thecenterofthefrontlayer onthemost-forward-positionedLISAtablewas596cmfromthetarget;thecenterofthefront layeronthenextLISAtablewas812cmfromthetarget.ThecenterofthefrontMoNAlayer was1041cmfromthetarget.Inthe x y plane,eachfrontlayer'scenterwaswithin3cmofthe theoreticalbeamaxis. 3.8ElectronicsandDAQ Theelectronicsanddataacquisition(DAQ)systemsaredescribedindetailinReferences(78; 82;83).Thissectiongivesabriefoverviewofthesesystemsanddescribestheirsettingsforthis experiment. MoNA,LISAandtheSweeperdetectorsoperatedasthreeindependentsubsystems.These 36 Figure3.9:SchematicdiagramoftheMoNA / LISA-SweeperelectronicsfromRefer- ence(81).Start,stopandgatesignalsaredepictedwithgreen,redandbluearrowsrespectively. ThesignalusedasthecommonstopfortheMoNA / LISATDCsisindicatedbythereddashedline. Shapingareomittedforclarity. 37 subsystemswereconnectedtoafiLevel3fllogicsystemthatgeneratedasystemtriggerandan eventtag.TheLevel3systemgeneratedatriggerwhenitreceivedasignalfromthechannel0 PMTofthethinscintillator.Eacheventtagwasaunique64-bitnumbergeneratedbytheLevel3 logicanddistributedtoeachofthethreesubsystems.Eachsubsystemindependentlyreadoutits dataandtheeventtaguponreceivingthesystemtrigger.Thethreedatasetsweremergedo ine bymatchingeventtags.Forthisexperiment,receivingasignalfromthethinscintillatorwasthe onlyconditionforgeneratingaLevel3trigger.Therefore,datawererecordedeveniftherewasno signalfromtheneutrondetectors. TheMoNAandLISAelectronicssetupswereindependentbutidentical.EachPMToutput twosignals,anfianodeflandafidynodeflsignal.ThedynodesignalwasroutedfromthePMT throughaninverterintoacharge-to-digitalconverter(QDC)thatintegratedthechargeproduced bythePMTtomeasuretheamountofscintillationlightthatthePMTregistered.Theanodesignal wassenttoaconstantfractiondiscriminator(CFD).TheCFDoutputtwocopiesofalogicpulse: onetriggeredthestartofatime-to-digitalconverter(TDC)andtheotherwasdeliveredtoalogic module(describedinthenextparagraph).TheTDCmodulesoperatedincommonstopmode whereasignalfromtheCFDstartedthetimerandasignalfromtheLevel3logicsystemstopped thetimer.Thissignalpathwasreplicatedforeachofthe288LISAPMTsand256MoNAPMTs. ChargeandtiminginformationforeachPMTwasreadoutfromtheQDCsandTDCs,respectively, bytheDAQcomputerthroughaVMEinterface. AnoverviewoftheMoNA-LISA-SweeperelectronicssetupandsignalroutingfortheLevel3 logicisdepictedinFigure3.9.ProgrammableXilinxLogicModules(XLMs)handledthetrigger logicwhichwasdividedintothreelevels.Inthelevel,oneXLMforeachlayercountedthe numberofcoincidentleft-rightCFDsignalsinthatlayerandpassedthisinformationuptoLevel 2.TheLevel1modulealsosenttheintegrationgatetothelayer'sQDCsassoonasthesignal arrivedfromtheCFD.ThereweretwoLevel2XLMmodules-oneforeacharray.ALevel2 modulecollectedtheleft-rightcoincidenceinformationandifatleastonelayerhadgoodtiming signalsinbothPMTsonasinglebar(theofavalidevent)thesystemwaspreparedto 38 Figure3.10:An(abbreviated)diagramoftheMoNA / LISAtriggerlogicfromRefer- ence(81).EachLevel1XLMisconnectedto32CFDstocountthenumberoftimestheleftand rightPMTsonthesamebarThenineLevel1modules(layersJ-R)fortheLISAarrayandthe eightmodules(layersA-H)fortheMoNAarrayareconnectedtotwoseparateLevel2modules. TheTDCsandQDCsareomittedforclarity. 39 readoutdata.WhenLevel3receivedasignalfromthechannel0PMTonthethinscintillatorit generatedaneventtagandopenedacoincidencegate.IfLevel2registeredavalideventduringthis window,MoNA / LISAandSweeperdatawerereadout.IfLevel2didnotregisteravalidevent duringthecoincidencewindow,onlySweeperdatawerereadout.IfMoNA / LISAregistereda valideventbutnoLevel3triggerwasgeneratedLevel2generatedafastclearoftheMoNA / LISA TDCsandQDCs.Figures3.9and3.10areschematicsoftheMoNA-LISA-Sweeperelectronics andMoNA-LISAelectronicssetupsrespectively. TheelectronicsfortheSweeperdetectorsweresetuptoreadoutsignalsfromthreetiming scintillators,twoCRDCs,theionizationchamberandthesilicondetectorsinthesegmentedtarget. TimingsignalsfromthePMTsontheA1900,targetandthinscintillatorswereroutedthrough CFDstoTDCs.AllSweeperTDCsoperatedincommonstartmodewheretheLevel3trigger beganthetimerandtheindividualCFDsignalsstoppedthetimer.SeparatesignalsfromthePMTs onthetargetandthinscintillatorsweredeliveredtoQDCstomeasurethetotallightcollected. ThepadsignalsfromtheCRDCsweredigitizedbyFront-End-Electronics(FEE)moduleswhich sampledthepad'ssignalandsenttheinformationtoanXLMforreadout.Eachionchamberpad signalwasprocessedthroughashapingintoananalog-to-digitalconverter(ADC).The signalpathforthesilicondetectorswasdescribedinSection3.4.2. 3.9InvariantMassSpectroscopy Invariantmassspectroscopycanprovideinformationaboutsystemsthatdecayonextremely shorttimescales.Neutronemissionfromanunboundsystemtypicallyoccursontheorderof 10 21 s,soadirectmeasurementoftheparentsystemisimpossible.Invariantmassspectroscopy reconstructstheenergydi erence( E decay )betweentheinitialnucleusandthedecayproducts. Energyandmomentumareconservedinthedecayprocessthatconvertsaninitialnucleuswith A + n nucleonsintoadaughterfragment A and n neutrons.Therestmassesoftheinitialsystem,daughter fragmentandneutronare M , M A ,and m N respectively.Energyandmomentumconservationis expressedas 40 E initial = E parent = E = E A + n X i E N P initial = P where P initial isthetotalfour-momentumoftheunboundnucleusbeforethedecayand P is thesumofthefour-momentaofthedecayproducts.TheLorentzinvariantquantity P P = M 2 isindependentofreferenceframe.Squaringbothsidesofthemomentumconservationequation gives P P initial = M 2 P P = 0 B B @ n + 1 X i P i 1 C C A 0 B B @ n + 1 X j P j 1 C C A = M A + n X i m 2 N + 2 n X i n + 1 X j = i + 1 ( E i E j ~ p i ~ p j ) wherethetotal n + 1termscorrespondtothe n neutronsplusonefragment M A ,with i and j inthe doublesumindexingoverallofthedaughterparticles.Theconservationofmomentumgivesan expressionfortheinvariantmassoftheinitialsystem M . M 2 = M 2 A + n X i m 2 i + 2 n X i n + 1 X j = i + 1 ( E i E j ~ p i ~ p j ) (3.1) Thedecayenergyisasthedi erencebetweentheenergyoftheinitialnucleusandthe energiesofthedecayproducts E decay M M A n X i m N : (3.2) Forthecaseofsingleneutronemission,therearetwoparticlesemitted:afragmentwith mass M A andaneutron m N .ThereforeEq.3.1reducesto 41 M 2 = M 2 A + m 2 N + 2 ( E A E N ~ p A ~ p N ) : Takingthesquarerootofthisexpressionandsubstitutingtheresultinfor M inEq.3.2gives E decay = q M 2 A + m 2 N + 2 ( E A E N ~ p A ~ p N ) M A m N Fortwoneutronemission,Eq.3.1iswritten M 2 = M 2 A + 2 m 2 N + 2 ( E 2 2 ~ p 2 2 ) where E 2 2 = E A E n 1 + E A E n 2 + E n 1 E n 2 ~ p 2 2 = ~ p A ~ p n 1 + ~ p A ~ p n 2 + ~ p n 1 ~ p n 2 sothedecayenergyis E decay = q M 2 A + 2 m 2 N + 2 ( E 2 2 ~ p 2 2 ) M A 2 m N (3.3) Thekeypointisthatthedecayenergycanbeexpressedintermsoftheenergiesandmomenta ofthedaughterproducts,andthesequantitiescanbemeasuredinthelab. 3.10TheDecayinTargetTechnique ThedecayintargettechniqueisillustratedinFigure3.11anddescribedindetailinRef.(55). Ithasbeenusedtomeasurethehalf-lifeoftwo-neutronunbound 26 O(1).Ifthedecayof 26 O doesnotproceedinstantaneously,thenthenucleuswillslowdownasittravelsthroughthetarget material.Asaresult,theneutronsareemittedfromanucleustravelingataslowerspeedthanif thedecaydidhappeninstantaneously.Thedaughterfragmentsandneutronsaredetectedafterthe target,andintheanalysis,thefragmentspeedatthecenterofthetargetisreconstructed.Theslower neutronspeedresultingitsdelayedemissionshiftsthecentroidoftherelativespeeddistribution j ~ v n jj ~ v f j belowzero.Thelongerthehalf-lifeforthedecaythelargerthespeeddi erencebetween theneutronsandthedecayfragment,seeFigure3.12. 42 Figure3.11:Thetoppanelillustratesthecaseofanextremelyshort T 1 = 2 ˘ 10 21 sandthebottom paneldepictsalonger T 1 = 2 ˘ 10 12 s.Thelonger 26 Oexiststhemoreenergyitlosesasittravels throughthetargetmaterial.Thismeanstheneutronsareemittedatalowervelocitythanif 26 O decaysinstantaneously. Figure3.12:Relativespeeddistributionssimulatedwiththreedi erent 26 Ohalf-liveswhere T 1 = 2 = 0ps,4psand8psarethereddashed,blacksolidandbluedottedcurvesrespectively.Therelative speediscalculatedas j ~ v n jj ~ v f j where ~ v n istheneutronvelocityand ~ v f isthefragmentvelocity. Thecentroidsofthedistributionswith T 1 = 2 = 4ps,8psareshiftedtotheleftrelativetothe T 1 = 2 = 0pscase.Thereaction 27 F ( 1 p ) ! 26 O ! 24 O + 2 n wassimulated. 43 Figure3.13:Averagevalueoftherelativespeeddistributionsasafunctionofhalf-lifeusingthe reaction 17 C ( 1 p ) ! 16 B ! 15 B + n at80MeV / u(left)and250MeV / u(right).Imagefrom(55). ThesensitivityofthemethodwasevaluatedinRef.(55)byexaminingtheaveragevalueof therelativespeeddistributionasafunctionofhalf-lifeforthesimulatedone-neutrondecayof 16 B producedfrom 17 C: 17 C ( 1 p ) ! 16 B ! 15 B + n .Thesimulatedrelativespeeddistributions foldedintheresolutionofthevelocitymeasurements.Figure3.13plotstheaverageoftherelative speeddistributionfortwodi erentbeamenergiesandfourdi erenttargetthicknesses.Larger neutron-fragmentspeeddi erencescorrespondtoahighersensitivity.Basedonthesesimulations thedecayintargetmethodissensitiveto T 1 = 2 > 1ps.Thickertargetsaremoresensitivebecause thespeeddi erencedependsontheenergylostinthetargetbythedecayingneutron-unbound system.Thisisalsowhythemethodismoree ectiveatlowerbeamenergies.Ultimately,the choiceoftargetthicknessforagivenbeamenergyislimitedbytherequirementthatthefragment exitthetargetwithenoughenergytoallowforcleanofthefragmentandagood energymeasurement.Althoughahigherbeamenergywillaccommodateathickertarget,abeam energy ˘ 80MeV / uandtheappropriatelyselectedtargetthicknesswaspredictedtoyieldthebest sensitivityinthe T 1 = 2 ˇ 4psregionforthe 16 Bdecay(seeFigure3.14). 44 Figure3.14:Averagevalueoftherelativespeeddistributionasafunctionofhalf-lifefor 17 C ( 1 p ) ! 16 B ! 15 B + n andthreedi erentcombinationsofbeamenergyandtargetthick- nesses.ThenumberinthelegendcorrespondstothebeamenergyinunitsofMeV / uandthe secondnumberisthetargetthicknessing / cm 2 .Thebeamenergy / targetthicknesscombinations wereselectedtogiveapproximatelythesameasymptoticvalueforverylonghalf-lives.Image fromRef.(55). 45 CHAPTER4 DATAANALYSIS Thischapterdescribesthecalibrationmethodsforproducingmeaningfuldatafromtherawde- tectoroutputsofMoNA,LISA,andtheSweeperdetectors.Aftercalibrations,theeventselection procedureisdiscussedfollowedbymodelingandsimulation. 4.1CalibrationsandCorrections Informationfromeachdetectorchannelisdigitizedandwrittentodisk.Inordertoextract meaningfulphysics,thisinformationmustbeconvertedfromrawdetectorvaluestophysicalquan- tities.Forexample,TDCchannelnumbersneedtobeconvertedtotimes.Eachtypeofdetector hasacalibrationprocedure. 4.1.1SegmentedTarget ThesegmentedtargetwasdescribedinSection3.4andwasusedinthisexperimenttoincrease reactionyieldwithoutdecayenergyresolution,seethediscussioninSection2.4.Energy lossinformationfromthesilicondetectorswasusedtodetermineinwhichberylliumtargetthe reactiontoproduce 26 Oandthesubsequent2 n decayoccurred.Thissectiondescribeshowthe energylosssignalswerecalibratedandhowthereactiontargetwasNotethatasurvival timeof10psinthelabframecorrespondstoadistanceof1mmforadecayingparticletraveling at10cm / ns.Therefore,any 26 Oproducedinaberylliumtargetwilldecayinsideorwithin1mm oftheBesegment.Forreference,theberylliumtargetsare ˘ 4mmthickandthedistancebetween elementsofthesegmentedtargetis8.4mm.Theprobabilityforproducing 26 Oinasilicondetector isnegligible. 46 4.1.1.1Energylosscalibration TheCoulombforcebetweenelectronsandheavychargedparticlesistheprimaryinteractionby whichthekineticenergyoftheheavyionsistransferredtothematerialthroughwhichtheyare traveling(28).Themaximumenergytransferredinasingleinteractionis4 Em 0 = m where E isthe kineticenergyofachargedparticlewithmass m and m 0 istheelectronmass.Sincetheelectron- to-nucleonmassratio m 0 = m np ˇ 1 = 2000,themaximumenergytransferforasingleinteractionis about1 = 500ofthe KE = A oftheheavyparticle.Throughmanyoftheseinteractionstheparticle continuouslylosesenergyintheabsorbingmaterial.Theenergylossperunitpathlength,or stoppingpower, dE = dx isproportionaltothesquareofthechargeoftheparticledividedbyits velocitysquared dE dx / Z 2 2 (4.1) where isthespeedoftheparticledividedby c and Z istheatomicnumberoftheparticle.The particlesarefullyionizedinthecyclotrons,andtheymaintainahighenoughvelocitythattheir chargestatedoesnotchangeduringthecourseoftheexperiment.Thefullformoftheenergyloss (Bethe)equationforachargedparticleinanabsorberisgiveninEquation(2-2)ofRef.(28)and Equation(1)inRef.(84). Inthecaseofsilicon,itssemiconductorpropertiesallowameasurementoftheenergytrans- ferredtotheelectronstobemade.Thebriefexplanationofthisprocessgivenhereissummarized fromChapter11inRef.(28).Theperiodiclatticeintowhichsiliconatomsorganizeestablishes allowedenergybandsfortheelectronsinthesolid;seeFigure4.1.Thesebandsareseparated bygapsorrangesofforbiddenenergies.Thelowerenergybandiscalledthevalencebandand correspondstoelectronsthatarepartofcovalentbondsbetweenadjacentatomsinthecrystalline structure.Ahigherenergyficonductionflbandcorrespondstoelectronsthatarefreetomigrate throughthecrystal.TheCoulombinteractionsdescribedinthepreviousparagraphcanexciteelec- tronsfromthevalencebandintotheconductionbandwheretheycanmovefreelythroughthe crystal.Theseexcitedelectronsleavebehindavacancyinthevalencebandwhich,togetherwith 47 Figure4.1:Theupperpanel(a)illustratesthesiliconlatticestructurewithcovalentelectronbonds depictedbytheblacklines.Thebottompanel(b)sketchestheelectronenergybandstructure.In semiconductors, E g ˇ 1eV.BothillustrationsareadaptedfromRef.(28). thefreedelectron,arereferredtoaselectron-holepairs.Applyinganelectrictothecrystal causestheelectronsandholestomigrateinoppositedirectionstotheedgesofthematerialwhere theycanbecollectedonelectricalcontactstoproduceasignalthatisproportionaltotheenergy transferredfromthepassingchargedparticle. Thesilicondetector dE signalscanbeusedtoidentifythereactiontargetwithoutanabsolute calibration,however,onewasperformedinordertocheckforconsistencybetweenmeasurement andsimulation.Fromthedatarecordedduringtheexperiment,atotalofeightdi erentbeam fragmentswerewithenergylossesinthesamerange(5MeV-30MeV)astheenergy lossesoftheoxygenreactionproductsfrom 27 F.TheyarelistedinTable4.1alongwiththeir kineticenergiesdeterminedbythesettingsoftheA1900fragmentseparator(73)usedto anddeliverthebeam.Theeightbeamfragmentswereusedtocalibratetheenergylossmeasured byeachofthefoursilicondetectorsandquantifytheuncertaintyassociatedwiththecalibration accordingtothefollowingfourstepprocedure. 1. Foreachfragment,eventswereselectedwherethefragmentchargedidnotchangeinthe segmentedtarget.Thiswasachievedusingtwoanalysisgates:(1)onthesiliconenergy lossasafunctionofA1900-to-target-scintillator(ToF)toidentifyeachincom- ingfragmentand(2)agateontheenergylostintheionchamberasafunctionoftheToF 48 Energyloss[MeV] FragmentKE[MeV / u] Si0Si1Si2Si3 27 F105.3 14.83815.91918.56822.649 28 Ne120.2 16.66717.73220.39324.315 29 Ne112.8 17.47018.74721.93926.785 30 Na124.9 19.62621.00624.42729.619 21 O136.8 9.74810.00010.94612.122 22 O125.3 10.34510.71911.86313.316 23 O115.2 10.97811.48512.88914.707 24 O106.3 11.65012.29814.00416.358 Table4.1:Energylossofeightdi erentfragmentsineachsilicondetectorcalculatedusingthe ATIMAenergylosscalculatorincludedwithinthe LISE++ softwarepackage(85).Calculationof thesevaluesaccountsfortheenergylossintheberylliumsegmentssincethetargetswerealways inthebeamline.Variationsinthematerialthicknessescorrespondtovariationsinthecalculated dE lessthan 0 : 008MeV,so / 0 : 05%,whichissmallerthantheresolutionofthedetectors. betweenthetargetandthinscintillatorstoidentify Z afterthetarget. 2. Usinga ˜ 2 minimizationroutine,GaussianwereperformedontheuncalibratedADC spectratodeterminethecentroidlocationsfortheenergyloss( dE )distributionsofeach detector. 3. CalculationsusingtheATIMAenergylosscalculatorembeddedinLISE ++ (85)determined theexpectedenergylossforeachfragment.Thethicknessesofthetargetsanddetectors werevariedby 4 m(targets)and 1 m(detectors)toquantifyanuncertaintyinthe calculationsintroducedbytheuncertaintyinthemeasuredthicknessesofthematerials.The thicknessvariationscorrespondto dE variationsof 0 : 05%. 4. Calibrationcurves(seeFigure4.2)wereextractedtoconvertrawADCchannels( dE raw )to unitsofMeV.Thecalibrationshadtheform dE cal = p 0 + p 1 ( dE raw ) andwereextracted fromtoplotsofthecalculatedenergylossfromstep(3)asafunctionofthecentroids extractedinstep(2).Theuncertaintyfromthecalibrationcurvesis < 2 : 0%inthe rangeofenergylossesbetween10MeVand35MeV. 49 Figure4.2:Blackpointsplotcalculatedenergyloss(y-axis)asafunctionofcentroidofthe uncalibrated dE spectrum(x-axis)foreightdi erentfragments.Thefourpanelsshowtheresults forthefoursilicondetectors.ThebluelinesshowtheextractedlinearThe x errorbarsforthe errorsandthe y errorbarsforthemeasurement / calculationuncertaintiesaresmallerthanthe points.TheparametersfromthearelistedinTable4.2. SiliconSlope ( p 0 ) O set ( p 1 ) Detector[MeV / ADCch][MeV] 00 : 0104 ( 1 ) 0 : 86 ( 2 ) 10 : 0104 ( 1 ) 0 : 99 ( 2 ) 20 : 0120 ( 1 ) 1 : 30 ( 2 ) 30 : 0117 ( 1 ) 1 : 52 ( 2 ) Table4.2:Thesecondandthirdcolumnslistparametervaluesextractedfromthe dE cal = p 0 + p 1 dE raw ,seeFigure4.2.Theerrorsare < 2 : 0%intherangefrom10MeVto35MeV. 4.1.1.2ReactionTarget ThesegmentedtargetsystemdescribedinSection3.4wasusedinthisexperimenttoincreasethe reactionyieldbyplacingthreeseparateberylliumtargetsinthebeamline,thusincreasingthe numberdensityoftargetatoms.Energylossmeasurementsfromthesilicondetectorswereused toidentifytheberylliumtargetinwhichthe 26 Owasproducedevent-by-event.Eventswherethe incomingfragmentwas 27 Fareviaenergylossmeasuredinthesilicondetectorand (ToF)fromtheA1900tothetargetscintillator.Eventswhereareactioninoneof theberylliumtargetsproducedanoxygenisotopearefromtheenergylossmeasuredin theionizationchamberaftertheSweepermagnetandtheToFfromthetargetscintillatortothe 50 thinscintillator.Thissetofanalysisgatesselectsasubsetofeventswhereaone-protonknockout reactionoccurred.Todetermineifaparticularberylliumtargetinducedtheprotonknockout, theenergylossmeasuredintheupstreamsilicondetectormustbecomparedtotheenergyloss measuredinthedownstreamsilicondetector.Considertwocaseswhere(1)anuclearinteraction somewhereintheberylliumtargetknocksaprotonoutof 27 Fand(2)noprotonisknockedout:in bothcasestheenergylossregisteredbytheupstreamdetectoristhesamewhilethedownstream detectorwillrecordasmallerenergylossincase(1)thanitwillincase(2)because Z changes fromninetoeightandenergylossisproportionalto Z 2 . Figure4.3illustrateshowthereactiontargetisusingtheenergylossmeasurements fromthesilicondetectors.AlleventsplottedinFigure4.3areselectedsothattheincomingfrag- mentwas 27 Fandtheoutgoingfragment(afterthesegmentedtarget)wasanoxygenisotope.The toprowplotsthe dE spectraforeachsilicondetector.Thebottomrowplotstheenergylossmea- suredinonedetectorversustheenergylossmeasuredinthepreviousdetector;spectrainthetop roware1Dprojectionsontothe x and y axesofthebottomrowplots.The1DspectratitledfiSi 0dEflisthex-axisoftheleft-mostplotinthebottomrow.Si1 dE isthey-axisontheleft-most andthex-axisonthemiddleplot.Si2 dE isthey-axisonthemiddleplotandthex-axisonthe right-mostplot.Si3 dE isthey-axisontheright-mostplot. Intheleft-mostpanelsofthetopandbottomrowsinFigure4.3,thesilicondetector (Si0)registeredroughlythesameenergylossforallevents.Thespreadisintroducedbythe energyspreadofthebeamandtheresolutionofthedetectoritself.ThedEmeasurementfromSi 1separateseventsintotwodistinguishablegroups.Eventsintheuppergrouponthe2Dspectrum (red-highlightedpeakinthe1Dspectrum)losemoreenergyinSi1becausenoprotonknockout occurredbetweenthetwodetectors.Therearethreeclustersinthemiddlepanelcorresponding tounreacted 27 F(top),oxygenreactionproductsfromBe1(bottomleft)andoxygenreaction productsfromBe2(bottomright);thetwolowerclustersbothcontributetotheblue-highlighted peakontheleftofthefiSi2dEfl1Dspectrum.Thetwogroupsintheright-bottompanelcorrespond tooxygenreactionproductsfromBe1orBe2(left)andoxygenreactionproductsfromBe3 51 Figure4.3:Exampletargetplotsfor 27 F ( 1 p ) ! A Omeaningthatalleventsplotted hereenterthesegmentedtargetas 27 Fandleaveasanoxygenisotope.Thetoprowofplotsshow themeasuredenergylossineachsilicondetector.Theleftpanelinthebottomrowofplotsshows themeasuredenergylossinthesecondsilicondetectorvs.themeasuredenergylossinthe silicon.Themiddlepanelplotsthethirdsiliconenergylossvs.thesecondandtherightpanel showsthefourthsiliconenergylossvs.thethird. 52 Figure4.4:Schematicofasilicondetectorandthecoordinateconventionusedfortheposition calibration.Thesizeofthearrowsillustratesthesignalsizeateachofthefourcornersforanion interactingatthelocationoftheredcross. (right). 4.1.1.3Positioncalibration The ( x ; y ) coordinateswherethechargedparticlepassedthroughasilicondetectorcanberecon- structedbasedontherelativepulseheightsofthecornersignalsrecordedbytheanalog-to-digital converter(ADC).Thisinformationwasusedtosetthebeam ( x ; y ) forthesimulation;see Section4.4.1.Themethodforcalculating x and y event-by-eventreliesonresistivechargedivision duetothesurfaceresistanceoftheboron-implantedfrontlayerwherethealuminumevaporated contactsaremadeateachcorner;seeSection3.4.1.Roughlyequal-sizedsignalswillbegenerated atthefourcornercontactswhenachargedparticlepassesthroughtheexactcenterofdetector. However,achargedparticlepassingthroughthedetectorclosetooneofthecornerswillresultin alargersignalgeneratedatthecontactonthatcorner. The x and y coordinatesfortheinteractionpointofachargedparticleinoneofthesilicon detectorsmaybecalculatedinthefollowingway.Witharighthandedcoordinatesystem(i.e. 53 lookingintothebeam),labelthechargecollectedattheupperleft,upperright,lowerrightand lowerleftcornercontacts A ; B ; C ; and D respectively,seeFigure4.4.thefollowing relationshipsforthesignalamplitudesgeneratedateachcontact: A + B + C + D L A + D R B + C U A + B D C + D The x and y coordinatesarethengivenby x = S R L y = S U D where S isthesidelengthofthedetector'sactivearea,31mminthiscase. 4.1.2Chargedparticlecalibrationsandcorrections Upstreamfromthetargetaretwotimingscintillatorsusedtoidentifythebeam.Followingthe Sweepermagnetareseveralcharged-particledetectors:twoCRDCs,anionizationchamberanda thinscintillator. 4.1.2.1A1900andTargetscintillators TheA1900scintillatorwaspositionedattheA1900focalplaneandthetargetscintillatorwas placed1.03mupstreamfromthesegmentedtarget.Bothtimingscintillatorswerewith asinglePMTthatmeasuredtimeandlightoutput.TheA1900scintillatorwaslocated10.9m 54 Figure4.5:ThetoppanelshowsanexampleTDCspectrummeasuredusinganOrtectimecalibra- torsettodeliverstartandstopsignalsseparatedbyintegermultiplesof40ns.Thebottompanel plotsthestart-stoptimeintervalsversusthepeaklocationsfromthetoppanel.Theredlineisa linearusedtoextracttheconversionfactorfromTDCchannelnumbertotimeinnanoseconds. Theslopeofthelineis0.0625ns / chandtheerrorisnegligible. upstreamfromthesegmentedtargetandonlyatimingsignalfromitsPMTwasrecorded.Timing signalsfrombothPMTswereprocessedbyseparatetime-to-digitalconverters(TDCs)inthesame MesytecTDCmodule.TheseTDCshavemanufacturerslopesof0.0625ns / chandthis wasusinganOrtectimecalibrator.TDCsmeasurethetimebetweenstartandstop signalsandthetimecalibratorprovidesstartandstopsignalsatpreciseintervals.Thetoppanelof Figure4.5showsanexampleTDCspectrumtakenwiththetimecalibratorsettodeliverstartand stopsignalsseparatedby40ns,80ns...320ns.Thismeanspeaksinthespectrumareseparatedby 55 40ns.ThebottompanelinFigure4.5plotsthestart-stoptimeintervalssetbythetimecalibrator versusthepeakpositionintheTDCspectrum.Theslopeofalineartothesepointsgivesthe conversionfactor(0 : 0625 6 10 16 ns / ch)fromTDCchannelnumbertotimeinnanoseconds. InadditiontotheA1900andtargetscintillatortimingsignals,fourtimingsignalsfromthethin scintillator(discussedinSection4.1.2.4)wereprocessedbytheMesytecTDCmodule.Theslopes foreachofthesesixTDCsweretobe0.0625ns / ch,andthisconversionfactorwasused toconvertallsixtimingspectratounitsofnanoseconds.Afterthisconversionthedi erencebe- tweentheA1900andtargetscintillatortimingsignalscorrespondedtothetimeforbeamfragments totraversethe9.87mdistancebetweenthetwoscintillators. Thelightoutputsignalfromthetargetscintillatorwasleftuncalibratedsinceanabsolutemea- sureofthelightproducedinthisdetectorwasnotnecessaryfortheanalysis.However,thetime di erence(e.g.betweenA1900andtargetscintillators)wasplottedagainsttherawlightoutput signalforthetargetscintillatortocheckfordeviationsinthemeasuredtimedue,forexample, tolowlightoutputinthetargetscintillator.Nodeviationswereobserved. 4.1.2.2CRDCs TheCRDCsmeasurethe ( x ; y ) positionofchargedparticlespassingthroughtheiractivevolumes; thesedetectorsweredescribedinSection3.6.1.TheCRDCwasplaced1.86mfromthe segmentedtarget(measuredalongthecentralarc).ThesecondCRDCwaspositioned1.54m downstreamfromthe Duringtheexperimentsomeoftheintegratedcircuit(IC)chipsthatprocesssignalsfromthe cathodepadsonCRDC1overheated.Itwasdeterminedthatreplacingthesechipsandrepairingthe electronicsboardwithinthetimeallottedfortheexperimentwasnotpossible.Furthermore,even ifrepairswerecompletedthesystemwouldlikelyoverheatagainanddestroythenewchips.This malfunctioncorruptedthe x positioninformationinsectionsofCRDC1.Thisinformation waslostbecausetherewasnosystematicpatterntothedatacorruptionthatwouldhaveallowedit tobecorrected.DuringtheattemptedrepairofCRDC1thecorrespondingcircuitboardforCRDC 56 Figure4.6:Chargecollected(blackpoints)asafunctionofCRDC2padnumberforasingleevent. TheredcurveisaGaussianandtheredverticallineisthecentroidextractedfromthe 2wasreplacedbecauseitwasbeginningtoshowsignsofasimilarfailure.Thesechangestothe electronicsnecessitatedtwoseparatesetsofcalibrations.Theslopesando setsfortheposition calibrationdidnotchangebecausethephysicallocationofthepadsandthegaspressureand driftvoltageswerethesame.However,thepedestalsandgainmatchingscalefactorswerea ected bytheelectronicschanges.ThefollowingparagraphsdescribetheCRDCcalibrationprocedures andTable4.3summarizesthecalibrationparameters. TheCRDC x positionforoneeventisdeterminedbymeasuringthechargecollectedon116 2.54mmwidepads.ThedistributionofchargeacrossthepadsisapproximatelyGaussian(see Figure4.6).Inordertoextractareliableposition,thepadsmustbepedestal-subtracted,badpads mustberemovedandthepadsmustbegainmatched.Thephysicalsizeofthedetectorandits measuredpositioninspacedeterminetheconversionfrompadnumberto x relativetothe beamaxis. Asmallleakagecurrentproducessignalsthatvaryfrompadtopad.Thisfipedestalflsignal mustbesubtractedtoaccuratelydeterminethetotalchargecollected.Datawererecordedwhen theelectronicswereonandthebeamwaso .Thechargedistributionforeachpadareplottedin thetoprowofFigure4.7.AGaussiantothechargedistributionofeachpadprovidesacentroid 57 Figure4.7:CRDCpedestalsubtraction.TherawpadsignalsforCRDC1(left)andCRDC2(right) areshowninthetoprow.ThecentroidofaGaussiantothechargedistributionofeachpadis subtractedtoshiftthecenterofeachdistributiontozero.Theresultofthepedestalsubtractionis showninthebottomrow:CRDC1(left)andCRDC2(right). whichisusedasano settocentereachpaddistributionaroundzero.Theresultofthispedestal subtractionisshowninthebottomrowofFigure4.7. Beforecalculatingpositions,thequalityofeachpadisexamined.Padsthatshowpoorcharge collection(duetonoise,forexample)areremovedfromtheanalysis,seeFigure4.8. Aftersubtractingthepedestals,CRDCpadsmustbegainmatchedtoaccountfordi erences inchargecollectionandbetweenpads.TheSweepercurrentiscontinuouslyramped upanddowninordertosweepthebeamacrossthechargedparticleacceptance.Sincethebeam particleshaveroughlythesameenergytheenergydepositedinatrackthatpassesbypadAshould 58 Figure4.8:ExamplespectrashowingthetotalchargereadoutbytwopadsinCRDC2.Pad68 (right)isanexampleofacorrectlyfunctioningpad;pad24(left)is bethesameastheenergydepositedinatrackthatpassespadB.Thismeansthatadi erence betweenthesignalsrecordedbypadAandpadBisduetosomeartifactofthedetectorortothe electronicsthatprocessthepadsignals.Gainmatchingthepadsremovesthesetypesofsystematic di erencesbetweenpads. Unreacted 27 FbeamwasusedtogainmatchtheCRDCpadsbecausethisbeamwascentered intheA1900soithadthemostnearlyuniformkineticenergydistribution.Foreveryevent,the padthatregisteredthemostchargeoutofall116padsonthedetectorwasAcharge distributionforeachpadwasbuiltoutofeventswherethatpadregisteredthemostcharge.These distributionswerethenscaledtoareference: m i = ref i where i isthecentroidofthechargedistributiononpad i and m i isthescalefactorforpad i . ThetotalchargeinducedonapadisestimatedbyaRiemannsumoverfoursamplestaken duringthecourseoftheevent.Thetotalchargeiscalculatedas Q i = 1 n n X i = 0 q i q pedestal 59 where n canvaryfromonetofour.Thegainmatchedpadchargeisthencalculatedas Q ( cal ) i = m i Q i Thecalibratedcharge Q ( cal ) i foreachpadisplottedforasingleeventinFigure4.6.AGaussian tothedistributionofchargeacrossthepadsdeterminesthe x positionintermsofpadnumber. The y positionisdeterminedbythetimeittakeselectronstodriftfromtheinteractiontrack totheanodewire.The ( x ; y ) positioninthelabframerequiresaconversionfrompadnum- ber / timetodistance.Thisisdoneusingatungstenmaskwithaknownholepattern.Themask isplacedinfrontofthedetectorandstopsbeamparticlesbeforetheypassthroughtheCRDC.A positionspectrumtakenwiththemaskinplaceprovidesalineartransformationfrom(padnumber, drifttime)to ( x ; y ) . Thepitchofthepadsis2.54mmandthisdeterminestheslopeofaorderpolynomial functionthatconvertspadnumbertolabframe x .Ano settoorientthedetectorrelativetothe beamaxisisdeterminedfromthepositionspectrumtakenwiththemaskinplace.Thescalefactor forconvertingdrifttimeto y positionisdeterminedfromtheknownholespacingonthemask. The y o setisdeterminedbythemeasuredpositionofunreacted 27 F-thisalignsthe xz -plane withthebeamtrajectory.Ultimately, y positionmeasurementsdidnotcontributetothefragment reconstructionbecauseofthefaultyCRDC1,therefore,ahigh-accuracycalibrationforabsolute y positionwasnotcrucial. TheCRDCsareplacedinoppositeorientationsrelativetothe + x direction.Thepadnumber forCRDC1increasesinthe + x direction;thepadnumberforCRDC2increasesinthe x direction. Thisiswhythe x slopesdi erbyasign. The y positionmeasuredbyaCRDCcandriftovertimeduetointhegaspressure andthedriftvoltage.Thesee ectsarecorrectedbyscalingtherawdrifttimetoareference runandperformingarun-by-runcorrection.Thecorrectionfactoriscomputedas m = ref TAC i 60 Figure4.9:Calibratedpositionspectrumwiththemaskpatternoverlaid.Thesame y o setex- tractedfromtheunreactedbeampositionisappliedtothemaskpattern. where ref TAC and i arethecentroidsofthetimingsignaldistributionsinthereferencerunandthe i th runrespectively.TheresultsofthiscorrectionareshowninFigure4.10.Asimilarcorrection wasappliedtoCRDC2. 4.1.2.3IonizationChamber TheionizationchamberwaspositionedimmediatelybehindthesecondCRDC.Itissegmented into16padsalongthe z direction.Duringaneventeachpadcollectsachargeproportionaltothe energylostbyanionpassingthroughthedetector.Theaverageofthepadsignalsisproportional tothetotalenergydepositedandcanbeusedtoidentifyparticle Z event-by-event.Thepadsignals mustbeinspectedtoidentifymalfunctioningpadsandremovethemfromtheanalysis.Good padsaregainmatchedtonormalizethepadsignals.Finally,anydriftinthepadresponsesover timeareremovedbynormalizingtoareferencerun.Examinationofthepadresponsesshowed 61 Runs1038-1121 BadpadsAvg.pedestalAvg.gainscalefactor CRDC1 0-56,99-1151211.01 CRDC2 24120.51.01 x slope[mm / pad] x o set[mm] y slope[mm / ns] y o set[mm] CRDC1 2.54-177.9-0.075106.0 CRDC2 -2.54187.0-0.074105.7 Runs1125-1179 BadpadsAvg.pedestalAvg.gainscalefactor CRDC1 0-53,54,56,63,64,95-115110.41.02 CRDC2 23,24,30,31,32116.51.05 x slope[mm / pad] x o set[mm] y slope[mm / ns] y o set[mm] CRDC1 2.54-177.9-0.075106.0 CRDC2 -2.54186.9-0.074110.3 Table4.3:CRDCcalibrationparametersappliedtorunsbefore(1038-1121)andafter(1125- 1179)theattemptedelectronicsrepairs(seetext).Badpadsarecathodepadsthatare removedfromtheanalysis.Thepedestalsubtractionissummarizedbytheaverageofallpedestal values.Similarly,thegainscalingfactorisaveragedoverallgoodpads.Notethatthesetwovalues arenotactualcalibrationparameters.The x y slopesando setswereextractedfromspectrataken withthetungstenmaskinplace. thatchannels0,4,7and15collectedtentimeslesschargeornoneatall;seetheleftpanelin Figure4.11.Thesepadswereremovedfromtheanalysis. Thegainmatchingprocedureensuresthatthesignalfromeachpadisthesameforthesame amountofenergydeposited.Thisrequiresselectionofasubsetofeventswherethekineticenergy andtypeofparticlepassingthroughthedetectorarethesame.Unreacted 27 Feventswereusedto gainmatchtheionizationchamberpads. Thegainmatchingisimplementedbyascalefactor m i foreachpadsuchthat m i = c ref c i where c ref and c i arethecentroidsofareference(pad12)pad'sandthe i th pad'scharge-collected distributionrespectively.Thegainmatchedchargeforpad i isthen q ( cal ) i = m i q ( raw ) i 62 Figure4.10:Uncorrected(toprow)andcorrected(bottomrow)drifttimesforCRDC1.Thedis- continuityatrun1125isduetothetwoseparatecalibrationparametersdescribedinthebeginning ofSection4.1.2.2. TheresultsofthegainmatchingprocedurefortheionchamberpadsareshowninFigure4.11. Theaverageofthe12signalsfromthegoodpadsistakenevent-by-eventtomeasuretheenergy depositedbythechargedfragmentsintheionchamber. Atime-dependentdriftinthechargecollectedbyeachionchamberpadwasobservedoverthe courseoftheexperiment.Thiswascorrectedusingarun-dependento settoalignthecollected- chargedistributionofeachpadfromruntorun.Withthisshiftthecalibratedpadsignalis givenby q ( cal ) i = m i q ( raw ) i + b i ( r ) 63 Figure4.11:Thechargecollectedbytheionizationchamberpadsforunreacted 27 Feventsbefore (left)andafter(right)gainmatching. Figure4.12:Theaverageoftheionchamberpadsignalsbefore(left)andafter(right)thecorrec- tion.A 27 Fbeamgatehasbeenapplied.Thehorizontalbandscorrespondtoreactionproductswith di erent Z .Thebandbetweentheredhorizontallinescorrespondstooxygenreactionproducts. Thecorrectionwasmadetostraightenthisbandinordertomorecleanlyselectoxygenreaction products. where m i isthegainmatchingscalefactorand b i ( r ) istheo setforrun r .Theresultofthedrift correctionisshowninFigure4.12. 4.1.2.4Thinscintillator Thethinscintillatorwaslocated9.5cmbehindtheionizationchamberandmeasuresionenergy lossandprovidesatimingsignalfora(ToF)measurement.Thedetectorhasfour 64 Figure4.13:DiagramofthethinscintillatorwithitsfourPMTs. PMTstwoconnectedtothetopedgeofthescintillatorandtwoconnectedtothebottomedge. SignalsfromeachPMTaresplittoprovidetimingandenergylossmeasurements.TheToFmea- surementbetweenthetargetscintillatorandthethinscintillatorwascrucialfortheanalysis.Inho- mogeneityandattenuationintheplasticcanintroducesmallvariationsinthetimingsignalfrom eachPMT;this,inturn,a ectstheToFmeasurement.Aseriesofthreecorrectionsweremadeto theToFsmeasuredbetweenthetargetscintillatorandeachofthefourthinscintillatorPMTs.The fourmeasurementswerecorrectedagainsttheenergylossmeasurementsandthe x and y positions measuredbyCRDC2.Afterthesecorrectionsthetimingsignalswereaveragedtogetherandthe di erencebetweenthisaverageandthetargetscintillatorwastakenasthemeasuredToF. TherawToFbetweenthetargetPMTandeachofthethinPMTswascheckedagainstthe charge-collectedsignalfromthetargetPMT.Therewasnocorrelationbetweenthemeasureden- 65 Figure4.14:Theraw,uncorrectedToFbetweenthetargetandthinPMT0isplottedversuscharge depositedinthetargetscintillator.Thereisnocorrelationbetweenthetwomeasurements.The spectrafortheotherthreethinscintillatorPMTsaresimilartothisone. ergylossinthetargetscintillatorandthetarget-thinToF,seeFigure4.14.Thereforethemeasured ToFdoesnotdependonthesizeofthesignalproducedbytheincomingbeaminthetargetscintilla- tor.Thisisbecausethebeamspotwassmallandlocalizedonthesmaller( ˘ 7cm)targetscintillator comparedtothelargespotsizeoftheunreactedbeamafterthedispersiveSweepermagnetonthe largerthinscintillator.Thee ectsofinhomogeneityandattenuationaremuchsmallerinthesmall scintillatorwhenthelightisalwaysproducedinroughlythesameplace. Next,therawToFs(targettoeachthinPMT)wereplottedagainsttheenergylosssignalsfrom eachofthethin'sPMTstocorrectforvariationsintheToFduetothesignalsize.Theenergyloss versusToFforeachPMTwasplottedforasubsetofeventsthatwereasunreacted 27 F (seeSection4.2.2)andpassedthrougha10 10cm 2 windowatthecenteroftheunreactedbeam spotasmeasuredonCRDC2.Thenarrowpositiongateensuresthateventsusedforthecorrection havesimilarfragmenttrajectories;otherwisethecorrectioncouldreducethesensitivityoftheToF measurement.ThecorrelationsshownintheleftcolumnofFigure4.15werefoundtobeidentical 66 forfourotherpositiongates.Theresultsoftheq-correctionareshownintherightcolumnof Figure4.15. Theq-correctedToFswerethencorrectedagainstCRDC2 x positiontocompensateforvaria- tionsinthetimingsignalsgeneratedbylightproductionindi erentregionsofthescintillator.The datausedforthiscorrectionwererecordedwhiletheSweepermagneticwasrampedupand downsothattheunreacted 27 F(ataknownvelocityof10 : 300 0 : 008cm / ns)issweptacrossthe chargedparticleacceptance.The ˘ 3nsvariationbetweentheaverageToFsmeasuredforevents passingthroughtheedgesofCRDC2(seeFigure4.16)isassumedtobedueentirelytovariations inthetimingsignalinducedbynonuniformlightcollection.VariationintheToFduetobeam energyspreadanddi erentpathlengthsaswellastheToFresolutioncontributetothewidthofthe distribution.Asimilarproceduregeneratedacorrectionfromtheqx-correctedToFsversus CRDC2 y (seeFigure4.17). ThecalibratedToFmeasurementsforunreacted 27 Fisshownastheredhistogramin Figure4.18.Ano setappliedtotheqxy-correcteddistributionwasdeterminedinordertocenter thevelocitydistributiononthe10.3cm / nsvaluedeterminedfromenergylosscalculations.Starting fromthebeamenergydeterminedbytheA1900settings,theunreacted 27 Fbeamvelocityafter thesegmentedtargetisdeterminedtobe10.3cm / nsbycomputingtheenergylossthroughthe materialsinthesegmentedtarget.Thepathlengthalongthebeamaxisfromthesegmentedtarget throughtheSweepertothethinscintillatorwasmeasuredtobe429cm.Calculatingthevelocities v = 429cm t foreventsthatverynearlyfollowthebeamaxis(thisisaccomplishedbygatingona10 10cm 2 centralregionintheCRDC2 x y spectrum)producesaroughlyGaussiandistribution.Findingthe o set(41.7ns)neededtocenterthedistributionon10.3cm / nsisthestepincalibratingthe ToFmeasurement. Itisimportanttonotethatthiso setfoldsinasecondadjustment.RecallthattheToFisthe measuredtimedi erencebetweentimingsignalsfromthetargetandthinscintillators.Thismea- surementincludestheadditionaltimeittakesthebeamtotravel103cmfromthetargetscintillator 67 Figure4.15:Leftcolumn:rawenergylossversusToFbetweentargetandthinscintillators;ascale factorwasappliedtothex-axestoconvertTDCchannelnumbertons.Rightcolumn:resultsof correctingeachToFmeasurementagainstthesignalsizeineachPMT.Therawenergylosssignals didnotneedtobegainmatchedsinceindependentcorrectionswereextractedseparatelyforeach PMT. 68 Figure4.16:Leftcolumn:q-correctedToFversus x positionmeasuredinCRDC2.Rightcolumn: resultsofthe x positioncorrection. 69 Figure4.17:Leftcolumn:qx-correctedToFversus y positionmeasuredinCRDC2.Rightcolumn: resultsofthe y positioncorrection. 70 Figure4.18:Theuncorrected(blackdottedhistogram)andcorrected(solidredhistogram)ToF distributionsforunreacted 27 Fcalculatedastheevent-by-eventaverageofthethinPMTsignals. Ano sethasbeenappliedtocenterthevelocitydistributiononthevaluedeterminedfromenergy losscalculationsofthe 27 Fbeamthroughthesegmentedtarget(seetext). tothesegmentedtarget.Bysettingtheo settoreproducethevelocityofunreacted 27 Fafterthe segmentedtarget,thetimefromscintillatortotargetisremoved.Thismeansthattheo set inthecalibrationisonlyvalidforeventsfromthe 27 Fbeam. 4.1.3MoNA-LISA ThesignaloutputfromeachPMTontheMoNA / LISAbarsprovidesachargeandatimemeasure- ment.ThePMTconvertsscintillationphotonsintoanelectricalsignal.Onecopyofthesignal isprocessedbyaconstantfractiondiscriminator(CFD)andusedtomeasurethearrivaltimeof thesignal.Anothercopyisintegratedtomeasurethetotalamountofscintillationlightproduced bytheinteractionofaphoton,neutron,orchargedparticlewiththebar.Thetermsfichargefland filightflarebothusedtorefertotheamountofscintillationlightproduced.Aseriesofcalibrations 71 areneededtoconvertthesemeasurementsintodepositedcharge,interactiontimeandposition( x ) alongthebar.TheorientationofthelabframecoordinateaxesareshowninFigure3.1;the y ; z coordinatesaredeterminedbythepositioninganddiscretizationofthebars. First,eachPMTwasgainmatchedandthechargemeasurementscalibrated.ThentheTDCfor eachPMTwascalibrated,andsubsequently,aconversionfromtimedi erencetopositionalong thebarwasextracted.Atimingo setwasthendeterminedtoplaceeachbarintimerelativetothe layeroneachtableandtoplaceeachtablerelativetothetarget.Allofthesecalibrations (exceptforthetimingo set)wereperformedusingcosmicraymuonmeasurements.Muons producedintheupperatmospherebycosmicraysuniformlyilluminatethearraysandprovidea meansofcalibratingthepositionsandtimingofindividualbars.TheMoNA / LISAacquisition systemsweresettooperateinstandalonemodeforthesemeasurementssonocoincidencewiththe Sweeperdetectorswasrequired. Cosmicraymuondataweretakenbeforeandaftertheexperimentusingindividuallayersof MoNAandLISAandusingtheentirearray.Muonsfromcosmicraysdepositroughly2MeV / cm (84)astheypassthroughabout10cmofscintillatingmaterialdepositingapproximately20.5MeV electronequivalent(MeVee)oflightineachbar.Sincethelightyieldinorganicscintillatorsis dependentonthetypeofparticle,unitsofelectronequivalentenergydepositedareusedtoquantify theabsoluteamountoflightproduced.OneMeVeeisequaltothelightproducedbyanelectron with1MeVofkineticenergy.Sincethemuonstravelatclosetothespeedoflight(86)measuring themprovidesanidealmetricforcalibratingtherelativetimingbetweenbars. 4.1.3.1ChargeCalibration(QDC) Thestepincalibratingthecharge / lightdepositedwastoroughlygainmatchthePMTsby adjustingtheappliedvoltage.Thisiterativeprocessinvolved 1. Takingonehourofcosmicraymuondata 2. ExtractingthemuonpeakfromtheuncalibratedQDCspectraforeachPMT 72 Figure4.19:Exampleraw(left)andcalibrated(right)QDCspectra.Thepedestalvisibleinthe leftspectrumissuppressedusingahardwarethresholdduringtheexperimentandwhilerecording asubsequentcosmicrunusedtogeneratetherightplot.TheredcurveisaGaussiantothe cosmicpeak.TheQDCchannelnumbersofthepedestalandthecosmicpeakdeterminedascaling toconvertQDCchannelnumbertolightdepositedinunitsofMeVee.Themuonpeakappears ˘ 20 MeVeeinthecalibratedspectrum. 3. Calculatingandapplyinganewvoltagesettingtoadjustthepositionofthemuonpeak Thisprocesswasrepeateduntilthecosmicpeakwaspositionedaroundchannel1000inallQDC spectra.Finalvoltagesettingsrangedfrom1400Vto1950V. Thecharge-to-digitalconverters(QDC)usedtointegratethesignalcurrentanddetermine chargehaveasmallinherentbiascurrentreferredtoasapedestal.Thepedestalwassuppressed duringtheexperimentusingahardwarethreshold.Withoutthisthreshold,everyQDCchannel wouldbereadoutforeveryeventthuscausingdeadtime.However,thisthresholdwas turnedo whentakingcosmicraydatafortheQDCcalibrationsothattheuncalibratedQDCbin numberofthepedestalcouldbeandusedtosetthethresholds.Thepedestalcorresponds toachargesignalofzeroandwasusedasonepointinalinearcalibration,alongwiththebin numberofthecosmicpeak,toconvertuncalibratedQDCspectratounitsofMeVee. TheQDCcalibrationisextractedfromacosmicraydatasettakenafterallPMTshavebeengain matched.ThelinearconversionfromQDCchannelnumbertoMeVeeforeachPMTisdetermined 73 fromthepedestalandthetothecosmicpeak q cal = m q ( q raw q ped ) where m q = 20 : 5 = ( q ( raw ) q ( raw ) ped ) inunitsofMeVee / chistheQDCslopethatcalibratesthemuon peakto20.5MeVee; q ( raw ) istheQDCchannelnumbercorrespondingtothecentroidofthe muonpeak.Thequantity q raw q ped subtractstheintegratedchargedduetothepedestalfrom measurement.Thehardwarethresholdiscalculatedas q thresh = q ped 16 + 2 Thefactorof16convertsthepedestalchannelfrom12bitsto8bitssincetheQDCmodules recordmeasurementsas12bitnumbersandstorethethresholdsas8bitnumbers.The2assures thatthethresholdisplacedabovethepedestal.Thresholdsrangedfrom0.5to1.0MeVeeanda post-experimentsoftwarethresholdof1.0MeVeewasappliedtoallPMTsandalsotosimulation output.AnexampleoftheQDCcalibrationisshowninFigure4.19fortherightPMTofbarJ-14. 4.1.3.2Timingand x positioncalibration ThesignalfromeachPMTisprocessedbyaconstantfractiondiscriminator(CFD)togenerate alogicpulsethatprovidesthestartsignalforatime-to-digitalconverter(TDC)tomeasurethe temporalseparationbetweenparticleinteractionsinthearrays.ATDCoperateslikeastopwatch: whenitreceivesastartsignalfromtheCFDitbeginschargingacapacitoruntilitreceivesastop signalfromthelogicsystem.Theamountofchargeonthecapacitorcorrespondstotheamount oftimebetweenthestartandstopsignals.Therearevariationsbetweenthecapacitorsindi erent TDCssoaslopemustbecalculatedtoconvertfromTDCchannelnumbertotimeforeachTDC. ThisprocessusedanOrtecNIMTimeCalibratormodule(Module462)whichprovidesstartand stoppulsesseparatedbyintervals.Theintervalwassetto40nsandtherangeto350nsso thatthestartandstopsignalswereseparatedbyanintegermultipleof40nsnogreaterthan350 ns(40ns,80ns...320ns).ThetoppanelinFigure4.20showsanexampleofanuncalibrated 74 Figure4.20:AnexampleofanuncalibratedTDCspectrum(toppanel)generatedusingthetime calibrator.Thenarrowpeaksrepresentrecordedeventswherethetimebetweenthestartandstop pulsesisanintegermultipleof40ns.Theknowntimeintervalsareplottedagainstthepeak locations(bottompanel)toextractascalefactorthatconvertsTDCchannelnumbertoatime.The rightpaneldisplayshistogramsofthecalculatedTDCslopes. TDCspectrumproducedusingthetimecalibrator.Thestructureresultsfromthe t = n 40ns intervals( n = 1 ; 2 ;:::; 8)betweenstartandstoppulses.Thetimeintervalssetbythetimecalibrator areplottedagainstpeaklocations(seethebottompanelofFigure4.20)toextractaconversionfrom TDCchannelnumbertotime.AseparateconversionfactoriscalculatedforeveryTDCchannelin MoNAandLISAandtheyareplottedintherightpanelofFigure4.20. OncethePMTtimingmeasurementsarecalibrated,thetimedi erencebetweenleftandright PMTscanbeusedtodeterminethepositionalongthebarwhereaninteractionproducedscintilla- tionlight.Cosmicraymuonsilluminatetheentirelengthofabarsothetimedi erencespectrum fromacosmicmuondatasetisusedtocalibratethe x positionsusingtheleftandrightedgesof thedetector.Anexampleofaleft-righttimedi erencespectrumisplottedintheleftpanelof Figure4.21.Separateoftheform 75 Figure4.21:Intheleftpanel,theblackhistogramplotsthedi erencebetweencalibratedtimes t left t right ,thebluecurvesareFermifunctiontotheleftandrightedgesoftimedi erence distributionandtheredverticallinesaretheedgesextractedfromtheTherightpanelshows theresulting x positionspectrum.Eventsplottedinthesespectraarerequiredtohaveacalibrated lightdepositedsignal > 4MeVee. f ( x ) = a 1 + exp[ b ( x c ) ] ; areusedtoextracttheleftandrightedgesofthedistribution( a , b , c areparameters).The edgesarethenusedtocalculateascalefactorthatconvertstimedi erencetopositionandano set thatcentersthedistributioninthebar'sreferenceframe.Anexampleofacalibrated x position spectrumisshownintherightpanelofFigure4.21. Oncetheleft-righttimedi erenceforeverybarhasbeencalibratedtoan x position,theco- ordinatesystemforeachbarmustbeconvertedtothelabcoordinatesystem.Thiswasachieved usingtheresultsfromasurveythatmeasuredtheplanesofthefrontlayersoneachtable(seeSec- tion3.7)relativetothetargetlocation.Thismeasurementwasusedtodeterminetheorientation ofthefitablecoordinatesystemsflbyavectorconnectingthetargetlocationtothecenter ofeachofthefrontlayers.Ultimately,thealignmentofthesevectorswiththetheoreticalbeam axiswasbetterthan2 .Thistranslatestoanadjustment ˘ 0 : 1cmwhichisroughlyanorderof magnitudesmallerthanthe ˘ 7cmpositionresolutionofthebars.Therefore,thelayersweretaken tobeperpendiculartothebeamaxis.Thecoordinatetransformationsfromeachbarsystemtothe 76 labsystemhavetheform x = x 0 + x 0 y = y 0 + y 0 z = z 0 + z 0 wheretheunprimedcoordinatescorrespondtothelabsystem,theprimedcoordinatescorre- spondtothebarsystemand ( x 0 ; y 0 ; z 0 ) specifythelabcoordinatesofthebarcenters. 4.1.3.3Timingo sets TherelativetimingbetweensignalsineachMoNA / LISAbarmustbeknowninordertoaccurately measuretheneutrontime.Thetimeofaparticleinteractioninsideabarisdeterminedbythe averageofthetwoPMTtimes,butano setmustbedeterminedtocorrectlycalibratethemeasured timerelativetothetarget.Theprocedurefordeterminingthiso setinvolveso setsthat correctlysetthetiming(1)betweenbarsand(2)relativetothetarget.O setsbetweenbars(1) canbedeterminedusingcosmicraydatawhilex-raysand -raysfromthetargetcanbeusedto determineo set(2).Thecosmicraymuonvelocityisapproximatedas29 : 8cm / nsandisusedto setthetimingforeventswherethemuonpassedthroughall16barsinalayerorthroughmultiple barsonatable. First,o setsarecalculatedtosetthetimingofallbarsinalayerrelativetothetopbar.For muonstravellingthroughalayerthetraveltimeis t = d v (4.2) where d = q ( x 1 x 0 ) 2 + ( y 1 y 0 ) 2 isthedistancebetweeninteractionsdeterminedbythecali- brated x positionandthephysical y locationofthebars; v = 29 : 8cm / nsistakentobethemuon speed.Thedi erencebetweentheobservedandtheexpectedtimeistheo set. 77 Figure4.22:A ˜ 2 minimizationroutinewasusedtoGaussianfunctionstothefragmentand neutronvelocitydistributions;thecentroidsanderrorsareplottedfordata(blackcircles)and simulation(openbluediamonds).Theresultsforthefragmentvelocitiesareplottedinthetop rowandtheneutronvelocitiesinthebottomrow.Theleft,middleandrightcolumnscorrespond toresultswherethedistributionsaremadefromeventswherethereactionoccurredinthe secondandthirdberylliumtarget,respectively. Next,o setsforeachlayeraredeterminedtosettherelativetimingbetweenlayersonasingle table.Equation4.2isagainusedtocalculatetheexpectedtimewherenow d = q ( x 1 x 0 ) 2 + ( y 1 y 0 ) 2 + ( z 1 z 0 ) 2 : where z isdeterminedbythephysicallocationofeachlayer.Thedi erencebetweenthemeasured andexpectedtimesistheo setthatsetsthetimingbetweenlayersonatable. Thestepistodetermineano setforeachofthethreetablesthatcorrectlysetsthetiming relativetothetarget.Sincetherateofmuonspassingthroughbarsonseparatetableswasnegli- gible, -raysfromthetargetwereusedtodeterminethiso set.Justpriortothestartofthe experiment,a6.35mmaluminumblockwasplacedinthetargetpositionanda140MeV / u 48 Ca beamwasdirectedontoit.Neutrons,lightchargedparticlesand -raysfromthefragmentationof 78 MoNAO set[ns]LISA-1O set[ns]LISA-2O set[ns] 453.1443.5442.5 Table4.4:Finaltimeo setsforeachMoNA / LISAtable. 48 CaweremeasuredinMoNAandLISA.Thesemeasurementswereusedtodetermineonetim- ingo setforeachofthethreetablessuchthat(1)thegammapeakinthevelocityspectrumwas locatedat29.979cm / nsand(2)theneutron / lightchargedparticlepeakinthevelocityspectrum wasbetween10.0cm / nsand14.5cm / nscorrespondingtothebeamvelocitiesatthefrontandback edgesofthealuminumblock.Theglobaltimingo setsextractedforeachtableinthiswayneeded oneadjustmenttoaccountforthedi erenceintimesforthe 48 Ca( v = 14 : 8cm / ns)and 27 F ( v = 13 : 2cm / ns)beamparticlesoverthe103cmdistancefromthetargetscintillatortothetarget position.Thiso setwascalculatedas t o set = 103 v ( F ) 103 v ( Ca ) = 0 : 8ns : Thetimingo setswerecheckedbycomparingthemeasuredandsimulatedneutronveloc- itiesforsixdi erentspeciesofreactionfragmentsproducedfromthe 27 Fbeamfragments;see Figure4.22.Thetimingo setvaluesarelistedinTable4.4. 4.2EventSelection Duringtheexperiment,morethan50millioneventswererecorded.Includedinthisdataset areeventsresultingfrombackgroundphysicsprocesses,eventswhereoneormoreofthedecay productswerenotdetectedandeventswithincompleteinformationinoneormoredetectors.This sectiondescribeshowthedatasetwasficleanedfltoextracteventsresultingfromacompleteand reliablemeasurementofthe 27 F ( 1 p ) ! 26 O ! 24 O + 2 n processofinterest. 79 BeamFragmentVelocity[cm / ns]Flighttime[ns]Fraction[%] 27 F13.2174.515 28 Ne13.9670.536 29 Ne13.5872.58 30 Na14.2569.018 Table4.5:Velocities,timesandfractionoftotaleventsinthedE-ToFspectrumforthefour mostintensebeamfragments.Thevelocitiesarecalculatedbasedonthecentralrigidity(4.5798 Tm)ofthelastdipolemagnetbeforethetargetandthetimesarebasedonthe983.8cm pathbetweentheA1900andtargetscintillators.Theremaining23%ofeventseitherfalloutside thestrict2Dgraphicalcutsorresultedfromlow Z beamfragments. Figure4.23:SpectrumusedforbeamfragmentThefasterfragments(seeTable4.5) haveashorterToF(x-axis)andfragmentswithahigher Z depositmoreenergyinthesilicon detector(y-axis). 80 4.2.1Beam TheA1900deliveredthe 27 Fsecondarybeamalongwiththreeadditionalspeciesofbeamfrag- ments: 28 Ne, 29 Ne, 30 Naandanumberofotherlow Z fragmentsresultingfromreactionsinthe aluminumwedge.Thecentralrigidityofthemagnetbeforethetargetwassetto4.5798Tmin ordertotransportthebeamfragmentsfromtheA1900intotheexperimentalarea.Theveloci- tiesatwhichthebeamfragmentstraveledfromtheendoftheA1900tothetargetarelistedin Table4.5.The(ToF)betweentheA1900andthetargetscintillatoralongwiththe energylostbythebeamfragmentsinthesegmentedtarget'ssilicondetector(Si0)wereused toidentifybeamfragmentsevent-by-event.TheToFmeasurementwassu cienttoseparate 27 F fromtheotherbeamfragmentsbecausenucleiwiththesame B ˆ butdi erent A and Z willhave di erentvelocities.ThelightfragmentswereoutusingtheSi0energyloss( dE )measure- mentbecausetheenergylostinthesiliconisproportionalto Z 2 = v 2 .A2Dgraphicalcutshown inFigure4.23wasgeneratedtoplaceatightgateonthe 27 Fevents.Usingthesemeasurements thefractionofeventsas 27 Finthe dE -ToFspectrumbeamwascalculatedtobe15%. Table4.5summarizesthefractionofeventsaseachofthefourmainbeamfragments. 4.2.2Element Manyreactionproductswereexpectedfromthefragmentationof 27 Fintheberylliumtargets, butonlythosereactionproductsthatproducedasignalinthethinscintillatorwererecorded.The Sweepermagnetwassettoacentral B ˆ = 3 : 445Tm(306A)correspondingtotheexpectedenergy of 24 Oreactionfragmentsafterthesegmentedtarget.Theacceptanceofthesweeperis 8%in rigiditysoonlythosereactionfragmentswithmomentainthisrangeafterthesegmentedtarget reachedthethinscintillator.Forthisexperiment,thefragmentsofinterestwere 22 Oand 24 O. Identifyingthecharge( q = Ze )ofareactionfragmentutilizesthecorrelationbetweentheen- ergyloss( dE )measuredintheionizationchamberandthe(ToF)fromthetargetto thethinscintillator.Figure4.24plotsthe dE measuredintheionizationchamberagainsttheToF fromthetargettothethinscintillatorrequiringthesilicon dE andA1900-to-target-scintilator 81 Figure4.24:Energyloss( dE )measuredintheionizationchamberversusthefrom thetargettothethinscintillator.Onlyeventsthatfallinsidethe 27 Fbeamgate(seeFigure4.23) areplottedhere.Themostintenseregioncorrespondstounreacted 27 Fbeamfragments( Z = 9); thebandimmediatelybelowcorrespondstooxygen( Z = 8)reactionproducts 27 F ( 1 p ) ! A O. ToFmeasurementstofallinsidethe 27 Fbeamgate.Groupswith Z = 9,8,7,6,5,and4aredis- cernableanda2Dgateontheoxygenbandwasusedtoselecteventswhereaone-protonknockout reactiontookplace.Forsomecalibrationproceduresa2Dgatewasplacedontheintenseregionin thebandtoselecteventswerenoreactionoccurredinthesegmentedtarget. 4.2.3Isotope Oncethecharge( q = Ze )ofthereactionfragmentisthemassnumber( A )mustbe determined.Themagneticrigidityofanon-relativisticchargedparticlecanbewrittenas B ˆ = m v q ; 82 anditsvelocity v = L = t .Withthesetworelations,itcanbeshownthatthe throughamagneticisproportionaltothefragmentmass Am u v = L t = B ˆ q m = B ˆ Ze Am u / 1 A (4.3) where q = Ze isthecharge, L isthepathlengthofthetrajectorythroughthemagnetic t is the A isthenumberofnucleonsand m u isthenucleonmass.Thusdi erentisotopes withconstant B ˆ (constantmomentum)can,inprinciple,bemass-separatedbymeasuringtime- Inreality,the(ToF)distributionsfordi erentisotopesarebroadandoverlapdue tovariationsinboth L and B ˆ .Thesevariationsarisefromfactorsincludingtheemittanceof thebeam,momentumacceptanceofthefragmentseparator,stragglinginthetargetsandsilicon detectors,thenucleardynamicsassociatedwiththeknockoutreactionandthemomentumkick fromtheneutrondecay.Additionally,themagneticoftheSweeperisnotuniformduetoits 14cmverticalgap.Nevertheless,themagneticrigidityand L ofthechargedparticlesarerelated totheiremittancemeasuredaftertheSweepermagnet.Thecorrelationbetween dispersiveangleandpositionneedstobeuntangledtoproduceaficorrectedflToFparameterthat canseparatethedi erentisotopespresentintheoxygenbandshowninFigure4.24.Examplesof theToF-dispersiveangleandpositioncorrelationisshowninFigure4.26. Calculationofthedispersiveanglerequirestwopositionmeasurementsafterthemagnetic SincetheCRDC1positionmeasurementisunreliableforcertainregions,thedeconvolutionwas carriedoutseparatelyforthelowerleft,middleandupperrightregionsoutlinedinredinFig- ure4.25.Theprocedureisillustratedfortheupperrightregionandtheresultingmass-separating ficorrectedflToFparameterisplottedinFigure4.33. Theadditionalbeamfragmentsthatarrivedwiththe 27 Fincludedthesameoxygenisotopes 22 24 Oasthereactionfragmentsofinterest.Eventswheretheseoxygenbeamfragmentswere measuredareselectedbygatingontheoxygenbandintheionizationchamber dE -ToFspectrum andontheSi0andA1900-to-target dE -ToFspectrum.TheToFdistributionsfromtheseunreacted 83 Figure4.25:CRDC1 x positionversusCRDC2 x position;Twofunctioningdetectorswoulddis- playasmooth,positivecorrelation.Theredlinesoutline2Dgatesthatattempttoselectevents withagoodCRDC1positionmeasurement. oxygenbeameventsdonotexhibitthesamebroadeningresultingfromthenuclearreactiondy- namicsandthemomentumkicksincetheydonotundergoareaction.Asaresulttheycanalready bemass-separatedusingtheuncorrectedToFmeasurement.Furthermore,theoxygenbeamfrag- mentsloselessenergyinthesegmentedtargetthanthe 27 Fbeam,thereforetheyentertheSweeper withahigher B ˆ thantheoxygenreactionfragmentswhichmeanstheyaredetectedonthe + x sideofCRDC2(50 x 150mm)whilemostoftheoxygenreactionfragmentsaredetected withCRDC2 x < 100mm.Comparedtotheoxygenreactionfragments,thebeamfragmentspro- videacleanerstartingpointfordeconvolvingtheToF-dispersiveangleandpositionrelations(see Figure4.26)aswellashigherstatisticsespeciallyintheupperrightregionofFigure4.25. ThestepingeneratingthecorrectedToFistoconstructasingleparameterthatdescribes thedispersiveplaneemittance,bothangleandposition.Thisisaccomplishedbyprojectinga3D scatterplotofToFversusdispersiveangleversusdispersivepositionontothedispersiveangleand 84 Figure4.26:ThecorrelationbetweenToFdispersiveangleandpositionisshownforoxygenbeam fragmentsontheleftandoxygenreactionfragmentsfrom 27 Fontheright. positionplane.Theprojectionoftheleft3DscatterplotinFigure4.26isshowninthetoppanelof Figure4.27wherethecolorofeachboxinthegridindicatesthemeanofthedistributionformed whenthecontentsofthe3DcellareprojectedontotheToFaxis.Breaksincolorindicatecontours ofiso-ToFwhicharewithasecondorderpolynomial f ( x ) = p 0 + p 1 x + p 2 x 2 indicatedbytheblackcurveinthetoppanelofFigure4.27.Theresultisusedtoconstructa parameterdescribingbothpositionandangleforconstantToF: g ( x ; x ) = x f ( x ) PlottingthisparameteragainstToFseparatesthedi erentisotopesasshowninthebottompanel ofFigure4.27.AcorrectedToFisconstructedbyrotatingtheToFaxistolieperpendiculartothe redlineinthebottompanelofFigure4.27whichisas 85 Figure4.27:Thetoppanelisaprojectionontothe2Ddispersivepositionversusdispersiveangle plane.Thecontourisshowninblack.Inthebottompanelthedispersiveplane emittanceparameterdisplaysalinearcorrelationwithmeasuredToFandcanbeprojectedontoan axisperpendiculartotheredlineforthepurposeofmakinga1Dgate. 86 IterationParameterLeftMiddleRight 1 g ( x ; x ) 0 : 130 : 080 : 08 2Si0 x 1 : 590 : 860 : 88 3focus x 0 : 01 ; 0 : 94 0 : 01 ; 0 : 01 0 : 01 ; 0 : 11 4focus x 0 : 010 : 030 : 01 Table4.6:Iterativecorrectionstofragmentusedtoachieveparticlefor theleft,middle,andrightregionsinFigure4.25.Notethattwoparametersaregiven foriterationthree,thecorrectionviafocus x ;thisisbecausethatcorrectionhadtheform t corr3 = t corr2 ( p 2 x 2 + p 1 x ) sotheandsecondvalueslistedcorrespondto p 2 and p 1 respectively. g ( x ; x ) = m 0 t target ! thin + b where m 0 istheslopeand b isanarbitraryo set.ThecorrectedToFisthen t corr = t target ! thin m 1 0 g ( x ; x ) TheseparationisthenimprovedbyiterativelyplottingotherparametersagainstcorrectedToF andremovinganycorrelationsaccordingtothesameprocedure: 1. PlotaparameterorcombinationofparametersversuscorrectedToF 2. ExtracttheformforalinearcorrelationbetweenthecorrectedToFandtheparameter(com- bination) 3. GenerateanewiterationoftheToFcorrection( t i )accordingto t i = m 1 i 1 y where y isthe parameterorcombinationofparameters 4. Compareplotsof y versus t i and y versus t i 1 toensurethatthenewcorrectionremovesthe correlation Theprocedureforextractingadispersiveplaneemittanceparameterandtheniterativelycor- rectingtheToFwasappliedseparatelytoeachofthethreeregionsoutlinedinFigure4.25using oxygenbeamand / orreactionfragmentsdependingonwhichdatasethadthebeststatisticsfora givenregion.TheparametersusedforthecorrectionsarelistedinTable4.6.TheToFcorrections 87 werethenappliedtotheoxygenreactionfragmentsfromthe 27 Fbeamand1Dgatesforeachof thethreeregionsweregeneratedtoselectthedi erentisotopes.TheCRDC2 x distributionsunder theisotopegateswerethencheckedforthecorrecttrendbetweenToFanddispersiveposition. Reactionproductswithdi erentmassesproducedfromthesamebeamshouldhaveroughly thesamevelocities.Thisimpliesthatthemomentum( B ˆ )foraheavierreactionproductwillbe largerthanthatofalightermassone.UponpassingthroughtheSweepermagnet,theheavier reactionproductswillbebentlessandtakeaslightlylongerpaththroughthemagnetresultingin alongertime.Therefore,theheaviermassreactionproductsshouldhavealongerToFand morepositiveCRDC2 x distributionscomparedtothelighterreactionproducts.Figures4.28- 4.30illustratethetrendbetweenToFandCRDC2 x position. ThecorrectedToFissu cienttoseparateisotopesbutitdoesnotidentifymasses.Recallfrom Eq.4.3thattheenergylossisproportionalto Z 2 = v 2 .Assumingnon-relativistickinematicsand that B ˆ remainsconstant,recallalsothat B ˆ = p = q = m v = Ze fromwhichitfollowsthat v 2 = ( B ˆ Ze ) 2 m 2 / Z 2 A 2 ; E / Z 2 v 2 / Z 2 ( B ˆ Z ) 2 A 2 / A 2 SinceToFisinverselyproportionaltovelocity,measurementsof E andToFcanseparateevents accordingto A 2 and A = Z .Figure4.31illustrateshownucleiarearrangedaccordingtothesetwo parameters.Notethatnucleiwithintegervaluesof A = Z liealongaverticallineperpendicularto the A = Z axis.ItisthenstraightforwardtoidentifygroupsofeventsplottedinFigure4.32.The groupatthetopareunreacted 27 Fandthe 24 Oreactionfragmentsliedirectlybeneaththem. TheonedimensionalcorrectedToFisplottedinFigure4.33fortheoxygenreactionproducts. Aregion-dependento setwasappliedtothecorrectedToFparameterfromeachofthethree separateregionsindicatedinFigure4.25toalignthethreespectra.Thisparameterisreferred toastheglobalcorrectedToF. 88 Figure4.28:ThetopthreepanelsplotthecorrectedToFsforthethreedi erentregions.For eachregiona1Dgateisindicatedbytheredlinesandarrows,andisappliedtotheCRDC2 x distributionsplottedinthebottompanel.Thered,black,andbluepointsplottheCRDC2 x distributionsforeventsundertheleft,middle,andrightregiongatesrespectively.Thesegates selecteventswheretheoxygenreactionproductshavealongToFandaredetectedonthehigh rigidity( + x )sideofCRDC2. 89 Figure4.29:ThetopthreepanelsplotthecorrectedToFsforthethreedi erentregions.For eachregiona1Dgateisindicatedbytheredlinesandarrows,andisappliedtotheCRDC2 x distributionsplottedinthebottompanel.Thered,black,andbluepointsplottheCRDC2 x distributionsforeventsundertheleft,middle,andrightregiongatesrespectively.Thegatesshown hereselecteventswithaslightlyshortertimeandCRDC2 x distributionsthatareshiftedto theleftcomparedtothegatesshowninFigure4.28. 90 Figure4.30:ThetopthreepanelsplotthecorrectedToFsforthethreedi erentregions.For eachregiona1Dgateisindicatedbytheredlinesandarrows,andisappliedtotheCRDC2 x distributionsplottedinthebottompanel.Thered,black,andbluepointsplottheCRDC2 x distributionsforeventsundertheleft,middle,andblueregiongatesrespectively.Thegatesapplied totheseplotsselecteventswiththeshortestToFsforoxygenreactionproducts.Notethatthe CRDC2 x distributionsareshiftedfurthertotheleftthantheothertwosetsofgates. 91 Figure4.31:Anillustrationof A 2 versus A = Z for10 A 30and Z A .Eachpointisa separatenucleus(someunphysical).Theredcurvesindicatecurvesofconstant Z .Threenuclei with A = Z = 3thatareintheSweeperacceptancearehighlightedwithgraycircles. 4.2.4Two-NeutronSelection Thefour-momentumofbothneutronsfroma2 n decaymustbemeasuredinordertocorrectly calculatethethree-bodydecayenergy.Thedetectorsystemdoesnotdistinguishbetweenone neutroninteractingtwiceandtwoneutronsinteractingindependently.Therefore,identifyingevents whereMoNA / LISArecordedtwointeractions(alsoreferredtoashits)incoincidencewithan oxygenfragmentdoesnotguaranteethatthetwohitscorrespondtothetwoneutronsfromthe decay.Whenaneutroninteractswiththescintillatormaterial,itcantransferanyamountofenergy uptoitstotalkineticenergy.Therefore,itispossiblethatthesecondhitresultsfromasecond 92 Figure4.32:Ionchamber E versusglobalcorrectedToFforalleventsfromthe 27 Fbeamthat fallinsideoneoftheregiongatesshowninFigure4.25.Theredlinecorrespondstothe A = Z = 3 lineinFigure4.31. detectionofasingleneutronafteritwasscatteredduringitsinteraction.Forthepurposesof thisdocument,eventswhereasingleneutronisdetectedtwicewillbereferredtoasfione-neutron scatteringfleventsandeventswherebothneutronsaredetectedwillbereferredtoasfitruetwo- neutronflevents. 4.2.4.1CausalityCuts Contributionsfromone-neutronscatteringarereducedbyapplyingficausalitycuts.flTheseanalysis cutsplacerestrictionsonthespatialseparationandthehitspeedwhichis asthedistancebetweenhitsdividedbythetimebetweenhits;seeFigure4.34.Thistechnique hasbeenusedtoenhancethetwo-neutronsignalinpreviousmeasurementsofthree-bodystates (87;88;89;46;90;91;92;93;94).Forthisanalysisthecausalitycutswere d 12 25cmand 93 Figure4.33:Onedimensionalparticlefortheoxygenreactionproducts.Events plottedherearerequiredtofallinsidetheoxygenreactionproductgateshowninFigure4.24and insideoneofthefigoodflCRDC1gatesshowninFigure4.25.Inaddition,coincidencewithoneof theMoNA / LISAbarsisrequired. 7 v 12 30cm / ns. 4.2.4.2DecisionForest Amachinelearningapproachwasusedasanalternativemethodforselectingtruetwo-neutron eventsusingtheToolkitforMultivariateDataAnalysis(TMVA)(95)builtintotheROOTData AnalysisFramework(96).ThisapproachisdescribedinRef.(39)andabriefsummaryisprovided inthissection. TheTMVAtoolkitsuppliesthecomputationalimplementationforanumberofmachinelearn- ingtechniquesofwhichthedecisionforestwasfoundtobethemostsuitableforclassifyingevents aseithertruetwo-neutronorone-neutronscatteringevents.Adecisiontreemakesbinarycutson anumberofdi erentparametersinordertoclassifyaneventassignalorbackground.Inthis analysis,theparametersarethe x ; y ; z ,and t componentsoftheandsecondhitvectorsas 94 Figure4.34:Anillustrationoftherelevantpositionvectorscalculatedinatwo-neutronevent. Thepositionvector ~ d 0 ( ~ d 1 )isfromthetargettothelocationofthe(second)interactionin MoNA / LISA; ~ d 01 = ~ d 1 ~ d 0 . wellasthehitseparationdistance j ~ d 01 j ,relativespeed j ~ d 01 j = ( t 1 t 0 ) ,opening( )andscattering ( )angles;seeFigure4.34. AschematicviewofadecisiontreeisshowninFigure4.35.Theprocessfortraining / building adecisiontreemaybesummarizedasiterativelydividingthelabeledtrainingdataintosubsetsby theparameterandcorrespondingcutvaluethatmaximizestheseparationbetweensignal andbackground.AteachnodeinFigure4.35,thetrainingdataaredividedintotwogroupsbased onwhichparameter x i bestdistinguishessignalfrombackground.Theprocessstopsoncethe trainingdatasubsetsreachsomeminimumsize( ˘ 5%ofthetrainingsamplesize).Thefileaffl 95 Figure4.35:Schematicviewofadecisiontree.Ateachnodeabinarycutismadeononeofthe parameters x i ; j ; k ;::: .Imagesource:(95). nodesattheendofthetrainingprocessarelabeledfisignalflorfibackgroundfldependingonthe classtowhichthemajorityofeventsineachfileafflsubsetbelong.Thisprocessessentiallydivides theparameterspaceintomanyregionsthatarebasedonthemajorityoflabeledtest eventsintheleafnode.Theseparationcriterionusedinthisanalysiswasby p ( 1 p ) where p = 0 : 5foranevenlymixedsignal / backgroundsample. Asingledecisiontreemaybeunstablewithrespecttostatisticalinthetraining sample(95).Tocircumventthisinstability,multipletreesarebuiltandtheirare averagedtogethertogiveadimensionlessvalue 1 x 1foreveryevent.Asinglecutbasedon thisvariableisusedtodistinguishbetweentruetwo-neutronandone-neutronscattering events.Avalueof0 : 03waschosentooptimizestatisticsandsignalpurityintermsofboththe andthecausalitycutsdiscussedinSection4.2.4.1. Oncethedecisionforestisbuiltandtrainedtheexperimentalandsimulateddatasetsarepro- 96 cessedthroughthealgorithmtocomputethevariableforeveryevent.The toppanelinFigure4.36showstheoutputforasimulateddatasetthatwasnotused fortrainingandvalidation.Insimulationthetruetwo-neutron(signal)andone-neutronscattering (background)eventsarelabeledsothee cacyofthecanbeassessed;thepurityof thesimulatedtruetwo-neutronsignalabovethecut = 0 : 03,theredlineinFigure4.36) is90%. Buildingthedecisiontreesrequiresatrainingdatasetwithmorestatisticsthanareavailablein theexperimentaldata.Therefore,simulateddatawereusedtotrainthedecisiontree.Thismethod hastheadvantagethatthetrainingdataareeasilylabeled.However,thecripplingdisadvantageis thatthisreducesthereliabilityofthedecisiontreesincetheconstructionofthetreeisentirelybased onthesimulation.Anycorrelationsthatarepresentinthesimulation,whetherornotthey reality,willbeincorporatedintotheroutine.Trainingadecisiontreeonexperimental dataisnotwithoutitsowndrawbacksbecausethereisnowaytolabeltruetwo-neutronandone- neutronscatteringeventswith100%certainty.Theperformanceofthedecisionforestcanbe comparedtothatofthecausalitycuts.ThebottompanelinFigure4.36showstheoutput intermsofthesignal / backgrounddeterminationmadebythecausalitycuts.Accordingtothe causalitycuts,thesignalpurityabovethecutis80%. Thedecisionforesttoidentifytwo-neutroneventsresultsinslightlyhigherstatis- ticswhilethecausalitycutsareasimple,easy-to-implementsolution.Therefore,twoversionsof the 26 Ohalf-lifeanalysiswerecarriedout;oneusedthedecisionforesttheotherusedthecausality cutstoselecttwo-neutronevents. 4.3FragmentReconstruction Measurementofatwo-orthree-bodydecayenergyrequiresthatthefour-momentumvectorof therecoilingfragmentbeknownatthedecayvertex.Typicallythisisrecoveredfrommeasure- mentsofthefragmentpositions ( x ; y ) ( D ) andangles ( x ; y ) ( D ) aftertheSweepermagnet(see (97;78)).Thesemeasurementsarefedintoanion-opticalcalculationthatincludesinformation 97 Figure4.36:Theoutputisplottedtoexaminethereliabilityofthedecisionforest.Inthe toppanel,theoutputdistributionsforsimulated,labeledtruetwo-neutron(blue)andone- neutronscattering(orange)eventsareplotted.Theblackcurveisthesumofthetwodistributions. Inthebottompanel,thedecisionforestiscomparedtothecausalitycuts;thegreenhistogram plotstheoutputforeventsthatthecausalitycutsidentifyastruetwo-neutronevents.The grayhistogramshowstheoutputforeventsthatthecausalitycutsidentifyasone-neutron scatteringevents.Theredlineindicatesthecutontheoutputusedintheanalysis. 98 aboutthestrengthandshapeofthemagneticandoutputstheangles ( x ; y ) ( T ) andenergy ofthefragmentattheexitofthetarget.Thismethodofreconstructionwasnotviableforthis experimentbecausetheCRDC1malfunction(Section4.1.2.2)corruptedoneofthepositionmea- surementsneededtocalculatethedispersiveangleafterthemagnet ( D ) x .Therefore,thefragment energywasdeterminedusingthemeasuredbetweenthetargetandthinscintillators: = d t = 1 q ( 1 2 ) E = m P = m (4.4) where c = 1, d isthepathlengththatthefragmentwithrestmass m traversesinthemeasuredtime t ; and arethevelocityandLorentzfactorsrespectivelyand E and P areenergyandmomentum. Thepathlengthbetweenthetwotimingscintillatorsisnotdirectlymeasuredandvariesevent- by-eventbecausethedecaykinematicsandpositionandangularspreadsoftheincomingbeam resultinslightlydi erenttrajectoriesthroughthemagnet.Toorder,atrajectoryalonganinner arcthroughtheSweeperandendingonthe x sideofCRDC2willcoverashorterdistancethana largerarcalongtheouteredgeofthemagnetandendingonthe + x sideofCRDC2(seecoordinates inFigure3.1).Therefore,themeasured x positiononCRDC2canprovideanestimateof thepathlengthofthefragmenttrajectory. InordertoextractacorrelationbetweenCRDC2 x andpathlength,thecalibratedtime-of- (ToF)measuredforunreacted 27 Fbeamfragmentsbetweenthetargetandthinscintillators (seeSection4.1.2.4)wasusedalongwiththeknownbeamvelocitytoestimatepathlengthasa functionofCRDC2 x .DatawererecordedwheretheSweepermagneticwasadjustedtoplace theunreacted 27 Fbeamatdi erentpositionsacrossthechargedparticleacceptance.Thedispersive e ectofthemagneticintroducesacorrelationbetweenmomentum / velocityand x positionfor theunreactedbeam;soselectingeventsnearthecenterofthe x distributioncorrespondstoselecting 99 Figure4.37:ThecorrelationbetweenpathlengthasafunctionofCRDC2 x isplottedintheupper rightpanel.TheabscissaforeachpointisthemeanofaGaussiantotheCRDC2 x distribution. TheordinateofeachpointisthemeanofaGaussiantothecalibratedToFdistributionmultiplied bythevelocityof 27 Fafterthesegmentedtarget;theerrorbarsrepresenttheerrorandthe uncertaintyinthevelocitycalculationintroducedbytheuncertaintiesinthesiliconandberyllium thicknesses, 1 mand 4 mrespectively.Theleftandbottomrightpanelsareexamplesofthe ToFandCRDC2 x spectra,respectively.RedcurvesplottheGaussian eventsnearthecenterofthemomentumdistribution.Sincethecentralvelocity / momentumofthe 27 FenteringthemagnetisknownandunchangedbyadjustmentstotheSweepermagneticthe distancetodi erent x positionscanbeapproximatedbymultiplyingthevelocitybythemeasured ToF.TheupperrightpanelinFigure4.37plotsthecorrelationbetweenCRDC2 x and 27 Fvelocity ToFcentroid;theothertwopanelsshowtheCRDC2 x andToFdistributionsforoneofthedata setsduringwhichthecurrentintheSweepermagnetwassetto300A.Datawastakenwithfour di erentcurrentsettings:290,300,310and320A. Thecorrelationdescribedabovewasextractedfortheunreacted 27 Fbeamandsubsequently usedtoestimatethepathlengthevent-by-eventforoxygenreactionproducts.Thevelocityof oxygenreactionfragmentswasthencalculatedevent-by-eventbydividingthepathlengthbythe calibratedToFbetweenthetargetandthinscintillators;seeSection4.1.2.4.Theenergyandmo- 100 Figure4.38:Theleftpanelplotsthesimulateddistributionofangles betweenthelab-frame fragmentmomentumvectorandthebeamaxis.Themiddleandrightpanelsplotthethree-body decayenergiesreconstructedfromthesimulateddetectorresponseswherethedecayenergyavail- ableforeverysimulatedeventwas50keV(middle)and1MeV(right).Theredcurvesplotthe decayenergiescalculatedwiththefragmentangle = 0andtheblackcurvesincludetheangle information. mentummagnitudeforthefragmentswascalculatedaccordingtoeq.4.4.Themomentumvectors foreveryfragmentwereassumedtobealignedwiththebeamaxis.Three-bodydecayenergies reconstructedunderthisassumptionwill,onaverage,belowerthanifthefragmentangleisin- cluded.However,throughsimulation,thisshiftwasfoundtobe ˘ 1 : 5%for50keVdecayenergies and ˘ 3 : 8%for1MeVdecayenergies;seeFigure4.38. Finally,thekineticenergyofthefragmentatthedecayvertexisapproximatedbyaddingback anestimatefortheenergylostbythefragmentasittraveledfromthedecayvertextotheedgeof thetarget.Afteridentifyingtheberylliumsegmentinwhichthereactionanddecaytookplace, seeSection4.1.1.2,anestimatefortheenergyaddbackisselectedevent-by-event,seeTable4.7. Theaddbackestimateforreactionfragmentsproducedinaberylliumsegmentisdeterminedbythe energylosscalculatedforan 24 Ofragmentproducedinthemiddleoftheberylliumsegmentplus theenergylossthroughallsubsequentsegments.Thereareseventotalsegmentsinthesegmented target,indexingthemfromzerotosiximpliesthattheberylliumtargetshaveindices i = 1 ; 3 ; 5,so theaddbackestimateforthesesegmentsis dE addback i = dE ( 0 : 5 t i ) + 6 X j = i + 1 dE ( t j ) where t i ; t j arethethicknessesforthe i th ; j th segmentedgiveninTable3.1.Forexample,theBe 101 2addbackiscalculatedastheenergylossthrough1868 mofberyllium,138 mofsilicon,3302 mofberyllium,and142 mofsilicon. Target EnergyAddback[MeV] Be1 785.0 Be2 488.7 Be3 177.6 Table4.7:Energyaddbackusedtoreconstructtheenergyofthe 24 Ofragmentsproducedfrom 27 F inoneofthe 9 Betargets. 4.4ModelingandSimulation The 26 Ohalf-lifewasextractedbycomparingthemeasuredrelativespeeddistributiontosim- ulatedonesthatweregeneratedusingdi erentvaluesforthehalf-life.AMonteCarlosimulation wasusedtoproducesimulateddatasetsthatareconvolutedwiththeexperimentalresolution, acceptanceande ciencyandtakeintoaccountthebeamreactionanddecayprocesses, energylossesinthesegmentedtarget,andthehalf-lifeoftheneutron-unboundstate. Ingeneral,thesimulationconsistsoftwopartsthat(1)modeltheincomingbeamandthe dynamicsofthereactionandsubsequentneutrondecayaswellastheenergylossinthetarget materialandtransportationofthechargedfragmentsthroughthemagnetand(2)modeltheneutron interactionswiththeMoNA / LISAdetectorsusingGEANT4(98;99;100).Foreachsimulated event,abeamtrajectoryisdescribedbyrandomlygenerated ( x ; x ; y ; y ) basedonasetofuser- Gaussiandistributions.Thebeamenergyisrandomlyselectedfromauniformdistribution correspondingtothemomentumacceptanceoftheA1900fragmentseparatorintheexperiment. Thenarandompointinsidethereactiontargetisselectedasthelocationforthenucleonremoval. Next,anenergylossisdeterminedbythekineticenergyoftheparticleandtheamountofmaterial upstreamfromthereactionpoint.Thisenergylossissubtractedfromthestartingenergytoset theparticle'skineticenergyatthereactionpoint.Next,theknockoutreactionissimulated(see Section4.4.3);thisisaone-protonremovalforthecaseof 26 O.Ifthehalf-life, T 1 = 2 ,issetto benonzero,arandomvalueforthesurvivaltime, t s ,oftheunboundsystemisdrawnfroman 102 exponentialdistributioncharacterizedby T 1 = 2 .Thentheunboundnucleusispropagatedadistance d tothedecaypointdeterminedbyitsspeedand t s ;thepropagationincorporatestheenergyloss basedonthenucleus ( A ; Z ) andthematerial.Theneutrondecayissimulatedandthekinetic energyofthedaughterfragmentandthedistancebetweenthedecaypointandtheedgeofthetarget determinesanenergylossvaluethatissubtractedfromthekineticenergyofthedaughterfragment tosetthekineticenergyafterthetarget.If T 1 = 2 = 0ps,theneutrondecayoccursimmediatelyafter thenucleonremovalreaction(i.e.thereactionanddecaypointsarethesame).Finally,theenergy lossthroughtheremainderofthesegmentedtargetisdeterminedtosetthefragmentenergyand momentumgoingintotheSweepermagnet.Anion-opticalmatrixisusedtocalculatethefragment ( x ; x ; y ; y ) afterthemagneticThistransformationdeterminesthe ( x ; y ) positionsatthe CRDCs.The ( x ; y ) positionsarethenfoldedwiththedetectorresolutionswhicharesimulatedby addingarandomnumberdrawnfromaGaussiandistributionwithmean = 0mmand ˙ = 3 mm. Thefour-momentumvectorsoftheneutronsarepassedtoGEANT4whereinteractionswith MoNAandLISAaremodelled.TheMENATE_Rdatabase(101;102)suppliescross-sectionsfor neutron-carbonandneutron-hydrogeninteractions.Whentreatingamulti-neutrondecay,GEANT4 generatesseparatelistsofinteractionlocation,timeandlightoutputforeachneutron.Beforecom- paringtodata,theselistsarere-orderedtoproduceasinglelistsortedbyinteractiontime. ThechargedparticleandneutronsimulationoutputsarecombinedintoasingleROOT thenprocessedwiththesameanalysiscodeusedtomakespectrafromthedata.Thesimulationis designedsuchthatasinglereaction / decaychannelinoneoftheberylliumtargetsissimulatedat atime.Therefore,threesimulations,oneforeachtarget,arerunforeachreaction / decaychannel thencombinedbeforecomparingtodata.Analysiscutsontheexperimentaldataareusedto extractsetsofeventswithacertaincombinationofbeamandreactionfragments(e.g. 27 Fand 24 O orunreacted 27 F).Therelevantsimulationscanthenberunandcomparedtotheexperimentaldata set. 103 4.4.1IncomingBeamParameters Theincomingbeamwasdeterminedbycomparingthe ( x ; y ) distributionsfromthesil- icondetectorandCRDC2tothecorrespondingdistributionsfromtheunreacted 27 Fexperimental dataset.Thebeamenergyanduniformenergyspreadjustupstreamfromthesegmentedtarget weredeterminedtobe105 : 3MeV / uand2 : 75MeV / u,respectively,bythesettingsoftheA1900 fragmentseparatorandthethicknessofthetargetscintillator.The ( x ; y ) distributionsmeasured bythesilicondetector(Si0)determinedtheshapeofthebeamspotandthe ( x ; y ) param- etersweretunedtomatchthesimulatedCRDC2 ( x ; y ) distributionstotheexperimentaldata,see Figure4.39. ThediscrepancybetweendataandsimulationfortheCRDC2 y distribution(lowerrightpanel inFigure4.39)existsbecausethesimulatedbeamentersthemagnetico setfrom y = 0dueto theconstraintthatthesimulationreproducethe y positiondistributionmeasuredonthesilicon detector.ThemismatchbetweensimulatedandmeasuredCRDC2 y distributionsinFigure4.39 impliesthattheo setusedintheroughcalibrationofthesilicon y positionisincorrect.This o setcanbereducedtobringthesimulatedandmeasuredCRDC2 y distributionsintoagreement. However,theCRDC2 y measurementisnotusedincalculatinganyphysicsquantities(e.g.decay energyorrelativespeed)soithasnoimpactontheanalysis,thereforeanexactmatchbetweendata andsimulationforthisobservableisnotcritical. 4.4.2EnergyLossinSiliconDetectors Theunreacted 27 Fdatasetwasalsousedtovalidatethesimulationofionenergylossesinthe silicondetectors.Thesiliconenergylossmeasurementsinthisdatasetweremadeon 27 Fions comingfromabeamwithawell-knownenergy105 : 3 2 : 75MeV / uimpingingonsegments ofsiliconandberylliumwiththicknesses(measurementuncertainty < 1%),sotheenergydeposited bytheionscanbeeasilymodeled.ThestoppingpowertablesfromtheSRIMsoftwarepackage (103;104)determinetheenergylossperunitlength dE = dx forionswithmass A andcharge Z travelinginamaterial,inthiscaseeithersiliconorberyllium.The dE = dx ismultipliedby 104 Figure4.39:The ( x ; y ; x ; y ) forthesimulatedbeamwassetthroughcomparisonstothe experimentaldistributionsplottedfromtheunreacted 27 Fdataset.Thesilicon x ; y position distributionsareshowninthetoprowandtheCRDC2 x ; y distributionareplottedinthebottom row.Orangepointsaredataandbluelinesaresimulation. thethicknessofthematerialtogivetheenergydepositedbytheion.Theresolutionoftheenergy lossmeasurementisreproducedusingaGaussiandistributionwith ˙ = 0 : 9MeV.Figure4.40 comparestheenergylossmeasuredwiththesilicondetectorstothesimulatedenergyloss. 4.4.3ReactionParameters The1 p -knockoutreactionwassimulatedbyremovingnucleonsfromthebeamfragmentandim- partingamomentumkicktotheresultingsystem.Thecomponentofthemomentumkickparallel tothebeamaxiswasparameterizedaccordingtotheGoldhabermodel(105).Thecomponentof 105 Figure4.40:Energylossof 27 Fbeamfragmentsmeasuredinthefoursilicondetectorsisplotted bytheblackdots,redtriangles,bluediamonds,andbrowncrosses;solidcurvesplotthesimulation output.TheresolutionofthesilicondetectorsissimulatedusingaGaussianwith ˙ = 0 : 9MeV. thekicktransversetothebeamaxiswasparameterizedaccordingtothemodeldescribedin(106). BothmodelsdescribetheparallelandtransversemomentadistributionsasGaussiansby widths ˙ k = 120MeV / cand ˙ ? = 92MeV / crespectively.Thesewidthswereedthrough comparingthesimulatedandexperimentalCRDC2 x positiondistributions. 4.4.4Additionalparameters Theareadensitiesofthesilicondetectorswerebythemanufacturer,andtheareadensities oftheberylliumtargetsweredeterminedfromtheirmechanicallymeasuredthicknesses.Table3.1 summarizesthethicknessesofeachcomponentinthesegmentedtarget.Thematricesfortheion- opticscalculationsaregeneratedbasedonmeasurementsoftheSweepermagneticAlibrary ofmeasurementswasproducedinpreviouswork(78)andaHallprobeinsertedinthe 106 duringtheexperimentdeterminedthestrengthandthuswhichsetofmeasurements(along withthemassandchargeofthefragmenttobeanalyzed)touseforproducingtheion-optical matrices.Thegeometricacceptancesofthedetectorswerebasedonthemeasuredphysicalsizes ofthedevices.ResolutionsforMoNA / LISAincludedGaussianresolutionsfortime( ˙ = 0 : 18ns) andposition( ˙ = 3cm)measurementsalongthebar(82).Thelengthofthebarsaswellastheir discretizationinthe y and z directionswerealsoincorporatedintothesimulation. 4.4.5Cuts ThegraphicalcutsshowninFigure4.25werenecessaryforareliableparticle(e.g. Figure4.33).Thesesamecutswereappliedtothesimulationoutputinordertoreplicateany biasesintroducedbyacceptingdi erentregionsofthereactionfragmentmomentumdistributions inanon-uniformmanner.TheresultingmeasuredandsimulatedCRDC2 x distributionsareplotted inupperleftpanelofFigure4.41. Cutsonthe ( x ; y ) positionsofthechargedfragmentsaremadetoensurethedetectoraccep- tancesarereproduced.Thesimulatedneutronsarerequiredtohavephysicaltimes,and thesametwo-neutroncutsusedontheexperimentaldata(Section4.2.4)arealsoappliedtothe simulationoutput. 4.4.6DecayModel Therelativespeed v rel j ~ v n jj ~ v f j istheobservableofinterestforthisexperiment,andthis quantityisbytheamountofenergyavailableinthedecay.Alargerdecayenergy willproduceabroaderdistributionof v rel whilealowerdecayenergywillproduceanarrower distribution.Furthermore,sincetothedecayenergyspectrumwillnotbeusedtoextractany informationabout 26 O,adecaymodelwasusedtoreproducetheexperimentalspectra. 107 Figure4.41:Fragmentandneutronpositionspectrafor 24 Oeventsincoincidencewithtwohits inMoNA / LISA.Simulatedpositionspectra(bluecurves)areoverlaidonthecorrespondingex- perimentalspectra(blackpoints).ThetopleftandrightpanelsplottheCRDC2 x and y spectra respectively.Themiddleleftandrightpanelsshowthe x and y distributionsforthetime- sortedhitinMoNA / LISA.Thebottomleftandrightpanelsshowthe x and y distributionsforthe secondhitinMoNA / LISA. 108 4.4.6.1Oneneutrondecays Thedecayenergyforthesingleneutronemission 23 O ! 22 O + n wasmodeledasa -functionat E decay = 50keVbasedonpreviousmeasurementsofthisdecay(107;108;109).Thismeansthat foreverysimulatedevent,theamountofenergysharedbetweenthe 22 Odaughterfragmentandthe neutronwas50keV.Thisenergywaspartitionedbetweenthetwoparticlesaccordingtoeq.2.4. Thecenter-of-massmomentumvectorsareequalandopposite.Theorientationofthedecayaxis, whichiscollinearwiththemomentumvectors,israndomlychosensothatthedecayisisotropic inthecenterofmassframe.Finally,theneutronandfragmentmomentaareboostedintothelab frame. 4.4.6.2Twoneutrondecays Two 26 Odecaychannelsweresimulatedinthiswork.First,thedecayofthe 26 Ogroundstate wasmodeledasasimultaneousemissionoftwouncorrelatedneutronswherethedecayenergy distributionwasa -functionat50keV.Forthepurposesofthiswork,thisprocesswillbereferred toasaphase-spacedecayandituniformlysamplestheinvariantmassofthefragment-neutron andneutron-neutronpairsusingthe TGenPhaseSpace classasit'simplementedinROOT(110). Second,thedecayofthe 26 O1.28MeVexcitedstatewasmodeledasatwoneutronsequential decaythroughthe750keVstatein 25 O.Thealloweddecayenergiesfortheone-neutronemission 26 O ! 25 O + n wereuniformlydistributedbetween600and800keVandthedecayenergies forthesecondone-neutronemission 25 O ! 24 O + n wereuniformlydistributedbetween50and 450keV.Thismodelforthe 26 Oand 25 Olevelstructureisclearlyunphysicalbutitservestofold anybackgroundprocesses(suchasthesequentialdecayofthe1.28MeV 26 Oexcitedstate)intoa singlesimulation. 4.5Extracting T 1 = 2 Aftereventscorrespondingtothe 27 F ( 1 p ) ! 26 O ! 24 O + 2 n processwereextracted fromtheexperimentaldataset,therelativespeed v rel wascalculatedforeachoftheseevents 109 andtheresultingdistributionwascomparedtosimulationusinganunbinnedmaximumlikelihood technique.Thissectiondescribesthereasonforusingthistechniqueandthentheprocedureby whichitwasimplemented. Thesurvivaltimeofeach 26 Onucleusisnotdirectlymeasured,sothehalf-lifecannotbe extractedbyanexponentialfunctiontoadistributionofmeasuredtimes.Insteadthee ect ofa ˘ 1pshalf-lifeontheneutron-fragmentrelativespeedprovidesanindirectmeasurementof thehalf-life(seeSection3.10).Theproblemofextractingavalueforthehalf-life, T 1 = 2 ,becomes oneofparameterestimationwherethetruedistributionofrelativespeedsisdistortednotonly bydetectoracceptancesandresolutionsbutalsobythekinematicsofthedecayprocess.The underlyingprobabilitydensityfunction(p.d.f.)canbethoughtofastherelativespeeddistribution producedbymodelingtheincomingbeam,the1 p removalreaction,the 26 O ! 24 O + 2 n decayand thedetectorresponses.Modelparameters,exceptforthehalf-lifeofthedecay,areconstrainedby theexperimentalsetup.Thedecayhalf-lifeisdeterminedbythevalue ‹ T 1 = 2 thatmaximizes thelog-likelihood,ln L ( T 1 = 2 ) ,orequivalently,minimizesthenegativelog-likelihood.Thelatter signconventionwillbeadoptedfortheremainderofthisdocument. Thep.d.f.fromthepreviousparagraphcannotbedescribedbyananalyticfunctionsothelog- likelihoodwasconstructedfromaMonteCarlosimulationfollowingtheprescriptiondescribedin Ref.(111).Theprocedureisoutlinedasfollows: 1. Generateasimulateddatasetwithavalueforthehalf-life(e.g. T 1 = 2 = 1 : 5ps) 2. Specifyanarrowrange r inrelativespeedaroundeachvalue s i fromtheexperimentaldata set 3. Calculatethetotalnumberofsimulatedeventsthatfallwithintherange s i r foreach s i 4. Dividethesumfromstep3bythetotalsizeoftherange,2 r ,andthetotalnumberofsimu- latedeventsfornormalization 110 5. Takethenaturallogarithmofthequotientfromstep4andsumoverallexperimentaldata points s i Thisprocedurecanbesummarizedbythefollowingequation ln L = X i ln " P j 1 if ( e s j 2 ( s i r )) 2 rN # where e s j arethesimulatedrelativespeedsand N isthetotalnumberofeventsinthesimulation. Theprocedureoutlinedabovecanintroducesystematicerrorsfromtwosources:(1)nonlinear- ityinthesimulatedrelativespeeddistributionoverthesmallrangeand(2)statistical duetothenumberofeventsgeneratedintheMonteCarlosimulation.Inordertodetermineanap- propriaterange,thenegativeloglikelihoodwascalculatedfortwosimulateddatasetswithknown half-lives( T 1 = 2 = 2 ; 4ps).Thesepseudodatasetsconsistedof100simulatedeventsinorderto reproducethestatisticsoftheexperimentaldataset.ThenegativeLnLcurvesextractedfromthe pseudodatasetsareplottedinFigure4.42.Foreachdataset,fourrangevalueswereusedtocarry outtheLnLcalculation: r = 0 : 005cm / ns,0 : 050cm / ns,0 : 100cm / nsand0 : 500cm / ns.Whenthe rangevalueistoolarge( r = 0 : 5cm / ns,graycrosses),sensitivitytothehalf-lifeisreduced.When thevaluefor r istoosmall( r = 0 : 005cm / ns,blackdots),thesuminStep3becomessensitive tostatisticalinthenumberofsimulateddatapoints.Inprinciple,therangecouldbe madevanishinglysmallinthelimitofanlargenumberofsimulatedevents,however, thiswouldnotchangetheextractedhalf-lifeorthestatisticaluncertainty.Therefore,avalueof r = 0 : 100cm / nswaschosensinceitresultedinextractingtheexpected T 1 = 2 forbothpseudodata setswhileutilizingareasonablesimulateddatasetsize.EachpointinthenegativeLnLcurves correspondstoasimulateddatasetcontainingsixmillionevents. 111 Figure4.42:NegativeLnLcurvesextractedfromapseudodatasetwith T 1 = 2 = 2ps(left)anda secondpseudodatasetwith T 1 = 2 = 4ps(right).Inbothpanels,rangevaluesof0.005cm / ns,0.050 cm / ns,0.100cm / ns,and0.500cm / nscorrespondtotheblackdots,redtriangles,bluediamonds, andgraycrossesrespectively. 112 CHAPTER5 RESULTSANDDISCUSSION Inthischapter,theresultsofthelifetimemeasurementarepresentedfollowedbyadiscussion abouttheperformanceofthesegmentedtarget.Theperformanceisevaluatedthroughcomparing twoversionsoftheanalysis:(1)wheretheaddbackforthefragmentkineticenergyisestimated assumingasinglethickberylliumtargetand(2)wheretheprocedurediscussedinSection4.1.1.2 isusedtomakeanevent-by-eventoftheberylliumsegmentinwhichthe 26 Owas produced. 5.1Half-lifeMeasurement Thesignatureofameasurable( T 1 = 2 ˘ 1ps)half-lifeisashiftintherelativespeeddistribution. DetailsofthistechniquearepresentedinSection3.10.Tosummarize,ifthedecayof 26 O doesnotoccurinstantaneously,thenthenucleuswillslowdownasittravelsthroughthetarget material.Asaresult,theneutronsareemittedwithaslowerspeedthanifthedecayhappens instantaneously.Thisshiftsthecentroidoftherelativespeeddistributionbelowzero. Figure5.1:Half-lifeextractedfrom 23 O ! 22 O + n data.Theleftpanelshowstherelativespeed distributionfordata(blackpoints)andthreesimulationswith T 1 = 2 = 0ps,4ps,7ps.Theright panelshowsthenegativeloglikelihoodasafunctionofhalf-life.Theredlineindicatestheupper limitof1.7pscorrespondingtoa95%level. 113 5.1.1Results Forconsistencywiththepreviousanalysis(1)anunbinnedmaximumlikelihood(LnL)technique (111)wasusedtoextractthestatisticaluncertainties,seeSection4.5.Thesystematicuncertainty wasdeterminedbyexaminingthedecayofthe 23 Ounboundexcitedstate: 23 O ! 22 O + n ,re- portedinRef.(107).Inthecurrentexperiment,thisdecaywasmeasuredatthesametimeandwith anidenticalsetuptothe 26 Omeasurement:segmentedtarget,magnetsetting,MoNA tion,Sweeperdetectorsettingsandcalibrations.Sincethe 23 O decayhasahalf-life T 1 = 2 . 10 20 s,itprovidesameanstoquantifythesystematicuncertaintyassociatedwithextractingalifetime fromashiftintherelativespeeddistribution.Thedistributionofrelativespeedsbetween 22 Ofrag- mentsandtheneutrondetectedinMoNA / LISAisplottedintheleftpanelofFigure5.1.In additiontotherequiredfragment-neutroncoincidence,thelightdepositedisrequiredtobegreater than1MeVeeandthetwo-bodydecayenergyisrestrictedtobelessthan300keVinordertosup- pressbackgroundevents.Theupperlimitonthe 23 O half-lifewasfoundtobe1.7pswitha95% level,seeFigure5.1.Thisvaluerepresentsthesystematicuncertaintyassociatedwith extractingahalf-lifefromtherelativespeeddistributionmeasuredwiththeexperimentalsetup describedinChapter3. Turningtothe 26 Ohalf-lifemeasurement,therelativespeedsfor 24 Ofragmentsandneu- tronhitsareplottedintheleftpanelsofFigure5.2.Thetopleftpanelshowsthedistribution ofrelativespeedsbetween 24 OandtheneutronhitforeventswhereMoNA / LISAregistered atleasttwohits,bothofwhichdepositatleast1MeVeeoflight.Thethree-bodydecayenergy isrequiredtobelessthan300keVinordertosuppressbackgroundevents.Causalitycuts,see Section4.2.4.1,wereappliedtosuppresseventswherethetwohitsresultedfromtwoseparate detectionsofthesameneutron.Thespatialseparationbetweentwohitswasrequiredtobegreater than25cmandthevelocitydi erence, j ~ v 01 j = j ~ v 1 ~ v 0 j wasrequiredtobebetween7cm / nsand30 cm / ns.Inthebottomleftpanel,therelativespeeddistributionisplottedbutthethree-bodydecay energygateisrelaxedto350keVandthedecisionforestisusedtoselecttrue2 n events, seeSection4.2.4.2,insteadofthecausalitycuts.TherightpanelsinFigure5.2plotthenegative 114 Figure5.2:TherelativespeeddistributionsandLnLcurvesfortwosetsofanalysisgatesusedto isolateeventscorrespondingtothe 26 O ! 24 O + 2 n decay.Thetoprowshowstheresultsforevents extractedusingthecausalitycutsandthebottomrowshowstheresultsusingthedecisionforest. Intherelativespeedplots,theblackpointsrepresentdataandthecurvesarefromsimulationswith T 1 = 2 = 0ps,4ps,and8ps.TheverticalredlinesontopoftheLnLcurvesdenotethe1 ˙ statistical uncertainties. 115 Figure5.3:Thegrayverticallinesindicatetheupperandlowerlimitsonthehalf-lifeobtained fromadding,inquadrature,thestatisticalandsystematicuncertaintiesquotedinRef.(1).Thered verticallinesdenotetheupperandlowerlimitsandtheblackpointsshowthenegativeloglikelihood curvefromthecurrentanalysisusingthecausalitycuts. loglikelihoodsforthehalf-livesextractedfromthedataunderthesetwosetsofgates.Usingthe causalitycuts, T 1 = 2 = 5 : 0 + 1 : 7 2 : 2 (1 ˙ statistical) 1 : 7(systematic)ps.Usingthedecisionforest2 n cuts T 1 = 2 = 5 : 0 + 2 : 0 1 : 6 (stat) 1 : 7(syst)ps. Thecurrentlyadoptedvalueof4 : 5 + 3 : 2 3 : 4 ps(112)forthehalf-lifeofthe 26 Ogroundstatecomes fromthemeasurementreportedinRef.(1).The + 3 : 2psand 3 : 4psuncertaintiescomefrom addingthestatisticalandsystematicerrorsfromRef.(1)inquadrature.Applyingthesameproce- duretotheresultsstatedattheendofthelastparagraphgives T 1 = 2 = 5 : 0 + 2 : 4 2 : 8 psforthecausality cutsanalysis.Theupper(lower)errorbarisreducedby0.8ps(0.6ps)comparedtothecurrently adoptedvalue.Figure5.3illustratesthereductionintheupperandlowerlimitsfortheanalysisus- ingthecausalitycutssincethisisthesamemethodthatwasusedinRef.(1)toselecttwo-neutron events.Forthedecisionforestanalysis, T 1 = 2 = 5 : 0 + 2 : 6 2 : 3 ps,andtheupper(lower)errorbaris reducedby0.6ps(1.1ps). Focusingonthedecisionforestscenariobecauseithasslightlyhigherstatistics,98counts 116 comparedto80counts,onecanconcludethat 26 Ohasanon-zerohalf-lifewitha95% level.Thisisdeterminedbythe95%intervaltobe1 : 9ps < T 1 = 2 < 9 : 9ps. Whenthelowererrorbarat1 : 9psiscombinedwiththe1.7pssystematicerrorbar,theresulting lowerlimit T 1 = 2 = 0 : 2psexcludeszero.Inasimilarfashion,theresultfromthecausalitycuts suggestsanon-zerolifetimeata90%level,wherethecorrespondingintervalis1 : 9 ps < T 1 = 2 < 8 : 1pssothelower1 : 6 ˙ errorbarislocatedat1.9ps.Theseresultsthe observationof2 n radioactivityfromthegroundstateof 26 O,reportedinRef.(1).Ultimately, ahigherlevel,e.g. > 5 ˙ (113;114),isneededtoestablishaveobservationof anewformofradioactivity.Analysisiscurrentlyunderwaytoextractthe 26 Ohalf-lifefroma measurementcarriedoutattheRIKENNishinaCenter'sRadioactiveIsotopeBeamFactory. 5.1.2Implications TheexperimentalconstraintsshowninFigure2.2havebeenadjustedtothe T 1 = 2 > 0 : 2ps resultfromthisanalysisandtheupdatedplotisshowninFigure5.4.Themeasurementfrom RIKENestablishesthe 26 Odecayenergy E decay = 18 7keV(48)asindicatedbytheredvertical linesinFigure5.4.ThealsoshowsthatonlyoneofthecalculationsfromRef.(58)is consistentwithboththehalf-lifeanddecayenergymeasurements.Thereddottedlinecorresponds tocalculationswherethe n n potentialhasbeenreducedbyafactoroffour.Settingthis n n interactiontozero,seethebluedashedlineinFigure5.4,over-predictsthehalf-life.Calculations withthefullpotential,seethepurpleshort-dashedlineinFigure5.4,predicta T 1 = 2 thatisshorter thanthevaluereportedhere. 5.2SegmentedTargetEvaluation AsdiscussedinSection2.4,theuncertaintyintroducedbytheunknownreaction / decayposition isakeycomponentthatdirectlythedecayenergyresolution.Theanalysisprocedure (2)fromaboveresultsinanimproveddecayenergyresolutioncomparedtoprocedure(1)because itenablesabetterestimateoftheenergyaddback.Thisresolutionimprovementisdemonstrated 117 Figure5.4:Decaywidth / half-lifeasafunctionofdecayenergyfor2 n emissionfrom 26 O.The graylineassumesapureorbital[ d 2 ]coupledtothetotalangularmomentum L = 0. Thesolidblackcurveshowstheresultswhennostateinteraction(FSI)andan 24 O massissimulated.ThebluedashedlineplotstheresultswhennoFSIandthecorrect 24 Omassis simulated.Thereddottedlineshowsthecalculationresultswhenthe n n FSIisscaledby0.25, andthepurpleshort-dashedcurveincludesthefull n n FSI.Theverticalredlinesroughlyindicate theexperimentalresultsfrom(48),andthegreenshadedareaindicateshalf-lifelimitsobtainedin thiswork.Imageadaptedfrom(58). throughthesimulationanddataanalysisofthe 26 Ohalf-lifemeasurement. 5.2.1TargetThickness-Simulation Toillustratethee ectthatthetargetthicknesshasonthedecayenergyresolution,thereaction 27 F ( 1 p ) ! 26 O ! 24 O + 2 n wassimulatedusingthesoftwarepackagedescribedinSection4.4. Thesimulationoutputhasthesameformatasthecalibrateddata,soitisprocessedthroughthe samedecayenergycalculationthatisusedfortheexperimentaldata.Theresultingspectrumhas thedetectoracceptances,e cienciesandresolutionsfoldedinsoitisdirectlycomparabletothe experimentaldata. TherightpanelinFigure5.5comparestwodecayenergyspectrareconstructedfromsimula- tionsof(1)asinglethick(11.1mm)berylliumtarget(grayline)and(2)asegmentedtarget(red 118 Figure5.5:Resultsfromasimulationrunwithasingle11.1mmtarget(graycurves)andwith asegmentedtarget(redcurves).The11.1mmtargetisthesumofthethicknessesofthethree berylliumtargetsusedinthesegmentedtarget.Forthesegmentedtargetsimulation,thedetector andtargetthicknessesinthesimulationarethesameasthoselistedinTable3.1.Theleftpanel plotsthedi erenceinspeed j ~ v n jj ~ v f j betweenthedetectedneutronandthechargedfragment andtherightpanelplotsthethreebodydecayenergyreconstructedfromthesimulation.Deviation fromthe50keVinputdecayenergyisduetotheenergyaddback.Neutronresolutionsareturned o . line)consistingofthreeseparatepiecesofberyllium4.1mm,3.7mmand3.3mmthick.The inputdecayenergyforeveryeventinbothsimulationsis50keV.Forboththeredandthegray lines,thereconstructeddecayenergiesarebroadenedrelativetotheinputdistributionbecauseof theuncertaintyinthereaction / decaylocationwithinthetarget.Resolutione ectsfortheneutron measurementsareturnedo . TheleftpanelinFigure5.5comparestheneutron-fragmentrelativespeeddistributionsforthe thick(graycurve)andsegmented(redcurve)targetsimulations.Therelativespeediscalculated as j ~ v n jj ~ v f j where v f = v 0 + v addback and v addback / p E addback representstheadjustment (duetothekineticenergyaddback)tothemeasuredvelocity v 0 .Forthelowdecayenergycase simulatedhere,thelabframeneutronandfragmentvelocitiesaresimilartothebeamvelocityat thedecayvertex.Afterthedecaythefragmenttravelsthroughtheremainderofthetargetmaterial andlosesenergy.Theenergyaddbackservestoaccountforthislossandrecoverthekineticenergy atthevertex.Sincetheaveragereaction / decaypositionisthecenterofthetarget,theenergy addbackisestimatedastheenergylostbyafragmentproducedatthecenterofthetarget.The 119 deviationfromzerooftherelativespeeddistribution'scentroidhowwellthisestimate reproducestheevent-by-eventactualenergyloss.Thewidthoftherelativespeeddistributionis relatedtothethicknessofthetargetandthedecayenergyoftheunboundresonance.Ated decayenergy,athickertargetimpliesalargerspreadofreaction / decaypointsaroundthetarget centerwhichtranslatestoalargerspreadofactualenergylossesaroundthechosenaddbackvalue. Thisintroducesaspreadintherelativespeedproportionaltothetargetthicknessasillustratedby thedi erencebetweenthegrayandredcurvesintheleftpanelofFigure5.5. 5.2.2Improveddecayenergyresolution Figure5.6showstherelativespeedandthree-bodydecay-energyspectrareconstructedfromdata (points)andsimulation(curves).InthetoprowofFigure5.6,thereconstructionusedenergyloss informationfromthesilicondetectorstoidentifyevent-by-eventthesegmentinwhichtheproton knockoutoccurred.Basedonthistheappropriateenergyaddback(seeTable4.7)was thenusedtoreconstructthefragmentmomentum.InthebottomrowofFigure5.6,theaddbackto thecenterofthemiddleberylliumsegmentwasusedinthereconstructionforallevents.Justasin thesimulatedcase(Figure5.5)thereisasubstantialimprovementinthedecayenergyresolution ofthesegmentedtargetspectrumcomparedtothesinglethicktargetreconstruction. Asmentionedabove,thespreadoftherelativespeeddistributionisrelatedtothetargetthick- nessandtothedecayenergywhichispartitionedamongthefragmentandneutrons.Whenmore energyisavailableinthedecaythedi erencebetweenthemomentaofthedaughterproductswill belarger.Thisiswhytheupperandlowertails(e.g.theupperleftpanelinFigure5.6)ofthe relativespeeddistributioncorrespondtohigherdecayenergies(seealsoEq.3.3).Forlowdecay energies,likethe 26 Ogroundstate,alargemismatchbetweentheactualfragmentenergylossand theestimatedvalueresultsinhighdecayenergies;notethemuchlargertailonthelow energypeakinthebottomrightpanelcomparedtotheupperrightpanelinFigure5.6. 120 Figure5.6:Measuredrelativespeedandthreebodydecayenergyspectrafor 27 F ( 1 p ) ! 26 O ! 24 O + 2 n aredrawnasblacktrianglesandblackcircles,respectively.Thesolidcurvesarespectra reconstructedfromsimulation.Twodi erentdecaychannelsweresimulated(seeinsetoftopright panel):(1)directpopulationofthe 26 Ogroundstatefollowedbytwo-neutronemission(redsolid line)and(2)populationofthe 26 Oexcitedstatefollowedbyasequentialneutronemission throughthe 25 Ogroundstate(greendashedline).Inthetoprow,theenergyaddbackischosen event-by-eventbasedonwhichberylliumsegmentwasasthereactiontargetusingthe methoddescribedinSection4.1.1.2.Inthebottomrow,theaddbackforthemiddleberyllium segmentisappliedtoallevents. 5.2.3ResolutionImprovementsandtheHalf-lifeMeasurement Figure5.7plotstherelativespeeddistributionsandthenegativeloglikelihood(LnL)curvesex- tractedusingthetargettoinformtheenergyaddback(toppanels)andassuminga singlethicktarget(bottompanels).ThecomparisoninFigure5.7showsthattherelativespeed distributionforasinglethicktargetwouldnothaveprovidedasensitivemeasureofthehalf-life. Incaseswhereoneneedstodiscriminatebetweentwodecaychannelsliketheonesshownin Figure5.6,theimprovedresolutiono eredbythesegmentedtargetallowsforacleanergateonthe threebodydecayenergywhichprovidesbetterstatisticsforstudiesofasingleunboundresonance. 121 Figure5.7:Thetopleftplotshowstherelativespeedreconstructedusingthetarget (Section4.1.1.2)toinformtheenergyaddback.Theblackpointsaredataandthecurvesrepresent relativespeeddistributionssimulatedassumingvarious T 1 = 2 for 26 O-reddashedis0ps,black solidis4psandbluedottedcorrespondsto8ps.Inthebottomleftasinglevaluefortheaddback (correspondingtotheenergylossthroughhalfofasinglethicktarget)isusedforallmeasured andsimulatedevents.Therightpanelsshowtheextractednegativeloglikelihoodcurvesforthe segmented(top)andsinglethick(bottom)targetreconstructions. 122 Thehalf-lifemeasurementrequiresselectingeventswherethedecayofthe 26 Ogroundstatewas observed.Eventswherethedecayofthe2 + statewasobservedrepresentbackgroundbecausethis stateisnotexpectedtobelong-lived.Event-by-eventdiscriminationbetweenthesetwotypesof eventsisenhancedbytheimproveddecayenergyresolutionprovidebythesegmentedtarget. 123 CHAPTER6 SUMMARYANDCONCLUSIONS 6.1Half-lifemeasurement Insummary,thelifetimeofthe 26 Ogroundstatewasextractedusingtwodi erentapproaches forsuppressingfalse2 n (background)events.Bothresultsareconsistentwithinthe1 ˙ statistical errors.Thesystematicuncertaintywasdeterminedbyextractinganupperlimitforthehalf-life oftheunboundexcitedstatein 23 O.Theresultsofthehalf-lifemeasurementareconsistent withthoseofRef.(1)andsuggestahalf-lifeforthe 26 O2 n decayontheorderofpicoseconds whichmeetsthecriteriaforradioactivity.Alowerlimit T 1 = 2 > 0 : 2pswasdeterminedata95% level.ThisresulttogetherwiththedecayenergymeasurementfromRef.(48)constrains thestrengthofthe n n stateinteractionusedinthecalculationsfromRef.(58). Alogicalextensionofthisworkistoconsiderwhether2 n radioactivityisuniqueto 26 Oorif othernucleimightexhibitthisexoticdecaymode.Apurelyqualitativediscussionofthisquestion canbebasedaroundthethreecriterialistedinSection2.2.3.Naively,onecouldexpectthatthe valenceneutronsincertainheaviersystemscouldoccupyorbitalswithhigherangularmomenta thusincreasingthepotentialbarrierandsubsequentlythehalf-lifefortwo-neutronemission.How- ever,theincreasingnuclearleveldensityneartheneutronthresholdforheaviersystems(115;116) mustbetakenintoaccount.Thistrendcorrespondstoadecreaseinthelikelihoodtohavenoin- termediatestatesinthetwo-neutronemissionprocess(seeFigure2.1),andthiswiththe requirementstatedinitem(1)fromthelistinSection2.2.3.Abalancebetweenthesetwogeneral trendscouldallowfor2 n radioactivityfromahandfullofnucleiheavierthan 26 O. Furthermore,thedevelopmentoftheoreticalframeworksliketheGamowshellmodel(117; 118;119)hashighlightedtheimportanceofincorporatingscatteringanddecaychannelsintoshell modelcalculations.Aninterestingconjecturebasedonthesetheoreticaldevelopmentsholdsthat couplingtoclusterdecaychannels(e.g.2 n emission)imprintsclustercorrelationsontheshell 124 modelwavefunctions(120;121).Fromthisperspective,theorganizationofnucleonsintoclus- tersinsidethenucleusisviewedasanear-thresholdphenomenonandnotasaconsequenceof propertiesoftheHamiltonianorsomesymmetryofthenuclearmany-bodyproblem.Thisin- terpretationsuggeststhatcertainsystemsnearthe2 n emissionthresholdcouldhaveenergylevel schemesthatmeetthesimultaneous(cluster)emissionrequirement(seeFigure2.1anditem(1) fromSection2.2.3),thusenhancingthepossibilitytomorecasesof2 n radioactivity. 6.2SegmentedTarget Anewdevicewasdevelopedforuseininvariantmassspectroscopyofneutronunboundstates attheNSCL.Itcurrentlyconsistsofthreeberylliumtargetsinterleavedbetweenfoursiliconde- tectors.Theenergylossmeasuredineachsiliconallowsthereactionthatproducestheunbound statetobelocalizedtoaparticularberylliumsegment.Thisimprovestheaccuracyoftheenergy addbackusedtoreconstructthefragmentmomentum.Theresultisathickerreactiontargetfor improvedreactionyieldwithoutdecayenergyresolution. Inordertoevaluatetheperformanceofthesegmentedtarget,twoversionsoftheanalysiswere conducted:(1)asingletargetequalinthicknesstothesumofthethreeberylliumsegmentswas assumedwhencalculatingtheenergyaddbackand(2)the dE measurementswereusedtoidentify thetargetsegmentcontainingthereactionandtheaddbackforthatparticularsegmentwasapplied. Method(2)resultsinabetterreconstructionofthefragmentmomentum.Consequently,therelative speedanddecayenergyresolutionsweremuchimprovedinmethod(2)comparedtomethod(1). TheincreasedreactionyieldwascriticalinlightoftheCRDC1electronicsboardfailuredis- cussedinSection4.1.2.2.Thismalfunctionreducedthee ciencyofthechargedparticleposition measurementbyafactoroftwo.Basedonsimulationswithtwicethestatistics,thetotalwidthof the1 ˙ intervalcouldbereducedbyapproximately20%relativetoits1 : 7 + 2 : 2 = 3 : 9 ps(2 : 0 + 1 : 6 = 3 : 6psforthedecisionforestanalysis)widthreportedinthepreviouschapter. 125 BIBLIOGRAPHY 126 BIBLIOGRAPHY [1] Z.Kohley etal. ,Phys.Rev.Lett., 110 ,152501,2013. [2] ThisMonthinPhysicsHistory,APSNews.March2008,Vol.17,No.3. [3] From NobelLectures , Physics 1901-1921.ElsevierPublishingCompany,Amsterdam,1967. [4] H.Becquerel,ComptesRendusAcadSci,Paris, 122 ,501Œ3,1896. [5] H.Becquerel,ComptesRendusAcadSci,Paris, 122 ,501,1896. [6] H.Becquerel,ComptesRendusAcadSci,Paris, 122 ,559,1896. [7] H.Becquerel,ComptesRendusAcadSci,Paris, 122 ,689Œ694,1896. [8] H.Becquerel,ComptesRendusAcadSci,Paris, 122 ,762,1896. [9] H.Becquerel,ComptesRendusAcadSci,Paris, 122 ,1086,1986. [10] H.Becquerel, Surdiversespropriétésdesrayonsuraniques (Gauthier-Villars1896). [11] ThisMonthinPhysicsHistory,APSNews.May2006,Vol.15,No.5. [12] ThisMonthinPhysicsHistory,APSNews.May2007,Vol.16,No.5. [13] From NobelLectures , Physics 1922-1941.ElsevierPublishingCompany,Amsterdam,1965. [14] G.Audi,F.G.Kondev,M.Wang,W.Huang,andS.Naimi,ChinesePhysicsC, 41 (3), 030001,2017. URL http://dx.doi.org/10.1088/1674-1137/41/3/030001 [15] M.G.Mayer,Phys.Rev., 74 ,235Œ239,1948. URL http://dx.doi.org/10.1103/PhysRev.74.235 [16] M.G.Mayer,Phys.Rev., 75 ,1969Œ1970,1949. URL http://dx.doi.org/10.1103/PhysRev.75.1969 [17] K.Krane, IntroductoryNuclearPhysics (Wiley,NewYork,NY1988). URL https://cds.cern.ch/record/359790 [18] M.G.Mayer,Phys.Rev., 78 ,16Œ21,1950. URL http://dx.doi.org/10.1103/PhysRev.78.16 [19] M.G.Mayer,Phys.Rev., 78 ,22Œ23,1950. URL http://dx.doi.org/10.1103/PhysRev.78.22 [20] O.Haxel,J.H.D.Jensen,andH.E.Suess,Phys.Rev., 75 ,1766Œ1766,1949. URL http://dx.doi.org/10.1103/PhysRev.75.1766.2 127 [21] A.Brown, LectureNotesinNuclearStructurePhysics ,nationalSuperconductingCyclotron LaboratoryandDepartmentofPhysicsandAstronomy,EastLansing,MI,USA,2011. [22] M.Thoennessen,ReportsonProgressinPhysics, 67 (7),1187Œ1232,2004. URL http://dx.doi.org/10.1088/0034-4885/67/7/r04 [23] M.Pfützner,M.Karny,L.V.Grigorenko,andK.Riisager,ReviewsofModernPhysics, 84 (2),567Œ619,2012. URL http://dx.doi.org/10.1103/RevModPhys.84.567 [24] M.Thoennessen, TheDiscoveryofIsotopes (Springer,Cham2016). URL https://doi-org.proxy1.cl.msu.edu/10.1007/978-3-319-31763-2 [25] J.Erler,N.Birge,M.Kortelainen,W.Nazarewicz,E.Olsen,A.M.Perhac,andM.Stoitsov, Nature, 486 (7404),509Œ512,2012. URL http://dx.doi.org/10.1038/nature11188 [26] G.Gamow,ZeitschriftfürPhysik, 51 (3),204Œ212,1928. URL http://dx.doi.org/10.1007/BF01343196 [27] R.W.GurneyandE.U.Condon,Nature,1928. URL http://dx.doi.org/10.1038/122439a0 [28] G.Knoll, RadiationDetectionandMeasurement (Wiley1989). URL https://books.google.com/books?id=UCtlQgAACAAJ [29] R.Dunlap, AnIntroductiontothePhysicsofNucleiandParticles (ThomsonBrooks / Cole 2004). URL https://books.google.com/books?id=UZN9QgAACAAJ [30] K.Jackson,C.Cardinal,H.Evans,N.Jelley,andJ.Cerny,PhysicsLettersB, 33 (4),281Œ 283,1970. URL http://dx.doi.org/https://doi.org/10.1016/0370-2693(70)90269-8 [31] S.Hofmann,W.Reisdorf,G.Münzenberg,F.P.Heßberger,J.R.H.Schneider,andP.Arm- bruster,ZeitschriftfürPhysikAAtomsandNuclei, 305 (2),111Œ123,1982. URL http://dx.doi.org/10.1007/BF01415018 [32] Y.B.Zel'dovich,J.Exptl.Theoret.Phys.(U.S.S.R.), 38 ,1123Œ1131,1960. [33] V.I.Goldansky,NuclearPhysics, 19 ,482Œ495,1960. URL http://dx.doi.org/10.1016/0029-5582(60)90258-3 [34] D.F.Geesaman,R.L.McGrath,P.M.S.Lesser,P.P.Urone,andB.VerWest,Phys.Rev.C, 15 ,1835Œ1838,1977. URL http://dx.doi.org/10.1103/PhysRevC.15.1835 [35] O.V.Bochkarev,A.A.Korsheninnikov,E.A.Kuz'min,I.G.Mukha,A.A.Ogloblin,L.V. Chulkov,andG.B.Yan'kov,ZhETFPismaRedaktsiiu, 40 ,204,1984. 128 [36] G.J.KeKelis,M.S.Zisman,D.K.Scott,R.Jahn,D.J.Vieira,J.Cerny,andF.Ajzenberg- Selove,Phys.Rev.C, 17 ,1929Œ1938,1978. URL http://dx.doi.org/10.1103/PhysRevC.17.1929 [37] R.A.Kryger etal. ,Phys.Rev.Lett., 74 ,860Œ863,1995. URL http://dx.doi.org/10.1103/PhysRevLett.74.860 [38] M.Pfützner etal. ,TheEuropeanPhysicalJournalA-HadronsandNuclei, 14 (3),279Œ285, 2002. URL http://dx.doi.org/10.1140/epja/i2002-10033-9 [39] H.Liu, ReactionMechanismDependenceofthePopulationandDecayofHe-10 ,Ph.D. thesis,MichiganStateUniversity,2019. [40] M.Meister etal. ,Phys.Rev.Lett., 91 ,162504,2003. URL http://dx.doi.org/10.1103/PhysRevLett.91.162504 [41] M.S.Golovkov etal. ,Phys.Rev.Lett., 93 ,262501,2004. URL http://dx.doi.org/10.1103/PhysRevLett.93.262501 [42] M.S.Golovkov etal. ,Phys.Rev.C, 72 ,064612,2005. URL http://dx.doi.org/10.1103/PhysRevC.72.064612 [43] Z.Kohley etal. ,Phys.Rev.C, 87 ,011304,2013. URL http://dx.doi.org/10.1103/PhysRevC.87.011304 [44] Y.Aksyutina etal. ,PhysicsLettersB, 666 (5),430Œ434,2008. URL http://dx.doi.org/https://doi.org/10.1016/j.physletb.2008.07.093 [45] A.Spyrou etal. ,Phys.Rev.Lett., 108 ,102501,2012. URL http://dx.doi.org/10.1103/PhysRevLett.108.102501 [46] E.Lunderberg etal. ,Phys.Rev.Lett., 108 ,142503,2012. [47] C.Caesar etal. ,Phys.Rev.C, 88 ,034313,2013. URL http://dx.doi.org/10.1103/PhysRevC.88.034313 [48] Y.Kondo etal. ,Phys.Rev.Lett., 116 ,102503,2016. URL http://dx.doi.org/10.1103/PhysRevLett.116.102503 [49] A.H.Wuosmaa etal. ,Phys.Rev.C, 95 ,014310,2017. URL http://dx.doi.org/10.1103/PhysRevC.95.014310 [50] L.V.Grigorenko,I.G.Mukha,C.Scheidenberger,andM.V.Zhukov,Phys.Rev.C, 84 , 021303,2011. URL http://dx.doi.org/10.1103/PhysRevC.84.021303 [51] D.Guillemaud-Mueller etal. ,Phys.Rev.C, 41 ,937Œ941,1990. URL http://dx.doi.org/10.1103/PhysRevC.41.937 129 [52] M.Fauerbach,D.J.Morrissey,W.Benenson,B.A.Brown,M.Hellström,J.H.Kelley, R.A.Kryger,R.Pfa ,C.F.Powell,andB.M.Sherrill,Phys.Rev.C, 53 ,647Œ651,1996. URL http://dx.doi.org/10.1103/PhysRevC.53.647 [53] S.M.LukyanovandY.E.Penionzhkevich,PhysicsofAtomicNuclei, 67 ,1627Œ1632,2004. URL http://dx.doi.org/10.1134/1.1802348 [54] A.Schiller,T.Baumann,J.Dietrich,S.Kaiser,W.Peters,andM.Thoennessen,Phys.Rev. C, 72 ,037601,2005. URL http://dx.doi.org/10.1103/PhysRevC.72.037601 [55] M.Thoennessen,G.Christian,Z.Kohley,T.Baumann,M.Jones,J.K.Smith,J.Snyder,and A.Spyrou,NuclearInstrumentsandMethodsinPhysicsResearch,SectionA:Accelerators, Spectrometers,DetectorsandAssociatedEquipment,2013. URL http://dx.doi.org/10.1016/j.nima.2013.07.035 [56] LookingatPhysicsHistory,CapitolHillQuaterly(APS).July2000,Vol.6,No.2. [57] W.vonSchlippe,relativisticKinematicsofParticleInteractions,March2002 http://www.physics.utah.edu/~jui/5110/hw/kin_rel.pdf . URL http://www.physics.utah.edu/~jui/5110/hw/kin_rel.pdf [58] L.V.Grigorenko,I.G.Mukha,andM.V.Zhukov,Phys.Rev.Lett., 111 ,042501,2013. URL http://dx.doi.org/10.1103/PhysRevLett.111.042501 [59] L.V.GrigorenkoandM.V.Zhukov,Phys.Rev.C, 91 ,064617,2015. URL http://dx.doi.org/10.1103/PhysRevC.91.064617 [60] K.HaginoandH.Sagawa,Phys.Rev.C, 93 ,034330,2016. URL http://dx.doi.org/10.1103/PhysRevC.93.034330 [61] L.V.Grigorenko,R.C.Johnson,I.G.Mukha,I.J.Thompson,andM.V.Zhukov,Phys. Rev.Lett., 85 ,22Œ25,2000. URL http://dx.doi.org/10.1103/PhysRevLett.85.22 [62] C.R.Ho man etal. ,Phys.Rev.Lett., 100 ,152502,2008. [63] G.Cowan, StatisticalDataAnalysis ,Oxfordsciencepublications(ClarendonPress1998). URL https://books.google.com/books?id=ff8ZyW0nlJAC [64] G.Christian etal. ,PhysicalReviewLetters, 108 (3),1Œ5,2012. URL http://dx.doi.org/10.1103/PhysRevLett.108.032501 [65] S.Leblond etal. ,Phys.Rev.Lett., 121 ,262502,2018. URL http://dx.doi.org/10.1103/PhysRevLett.121.262502 [66] T.Baumann,A.Spyrou,andM.Thoennessen,ReportsonProgressinPhysics, 75 (3), 036301,2012. URL http://dx.doi.org/10.1088/0034-4885/75/3/036301 130 [67] Y.Ayyad etal. ,NuclearInstrumentsandMethodsinPhysicsResearchSectionA:Acceler- ators,Spectrometers,DetectorsandAssociatedEquipment,2018. URL http://dx.doi.org/https://doi.org/10.1016/j.nima.2018.10.019 [68] T.Furuno etal. ,NuclearInstrumentsandMethodsinPhysicsResearchSectionA:Acceler- ators,Spectrometers,DetectorsandAssociatedEquipment, 908 ,215Œ224,2018. URL http://dx.doi.org/https://doi.org/10.1016/j.nima.2018.08.042 [69] A.Laird etal. ,NuclearInstrumentsandMethodsinPhysicsResearchSectionA:Acceler- ators,Spectrometers,DetectorsandAssociatedEquipment, 573 (1),306Œ309,2007,pro- ceedingsofthe7thInternationalConferenceonPosition-SensitiveDetectors. URL http://dx.doi.org/https://doi.org/10.1016/j.nima.2006.10.384 [70] S.Beceiro-Novo,T.Ahn,D.Bazin,andW.Mittig,ProgressinParticleandNuclearPhysics, 84 ,124Œ165,2015. URL http://dx.doi.org/https://doi.org/10.1016/j.ppnp.2015.06.003 [71] W.Rogers,A.Kuchera,J.Boone,N.Frank,S.Mosby,M.Thoennessen,andA.Wantz,Nu- clearInstrumentsandMethodsinPhysicsResearchSectionA:Accelerators,Spectrometers, DetectorsandAssociatedEquipment, 943 ,162436,2019. URL http://dx.doi.org/https://doi.org/10.1016/j.nima.2019.162436 [72] F.Marti,P.Miller,D.Poe,M.Steiner,J.Stetson,andX.Y.Wu,inF.Marti(editor), Cy- clotronsandTheirApplications2001 ,volume600of AmericanInstituteofPhysicsConfer- enceSeries ,64Œ68(2001). URL http://dx.doi.org/10.1063/1.1435199 [73] D.J.Morrissey,B.M.Sherrill,M.Steiner,A.Stolz,andI.Wiedenhoever,NuclearInstru- mentsandMethodsinPhysicsResearchB, 204 ,90Œ96,2003. URL http://dx.doi.org/10.1016/S0168-583X(02)01895-5 [74] I.Tanihata,NuclearPhysicsA, 553 ,361Œ372,1993. URL http://dx.doi.org/https://doi.org/10.1016/0375-9474(93)90636-C [75] M.S.Basunia,NuclearDataSheets, 112 (8),1875Œ1948,2011. URL http://dx.doi.org/https://doi.org/10.1016/j.nds.2011.08.001 [76] M.D.Jones, SpectroscopyofNeutronUnboundStatesinO-24andN-23 ,Ph.D.thesis, MichiganStateUniversity,2015. [77] M.D.Bird,S.J.Kenney,J.Toth,H.W.Weijers,J.C.DeKamp,M.Thoennessen,andA.F. Zeller,IEEETransactionsonAppliedSuperconductivity, 15 (2PARTII),1252Œ1254,2005. URL http://dx.doi.org/10.1109/TASC.2005.849553 [78] N.Frank, SpectroscopyofNeutronUnboundStatesinNeutronRichOxygenIsotopes ,Ph.D. thesis,MichiganStateUniversity,2006. 131 [79] B.Luther etal. ,NuclearInstrumentsandMethodsinPhysicsResearchSectionA:Acceler- ators,Spectrometers,DetectorsandAssociatedEquipment, 505 (1),33Œ35,2003,proceed- ingsofthetenthSymposiumonRadiationMeasurementsandApplications. URL http://dx.doi.org/https://doi.org/10.1016/S0168-9002(03)01014-3 [80] T.Baumann etal. ,NuclearInstrumentsandMethodsinPhysicsResearch,SectionA:Ac- celerators,Spectrometers,DetectorsandAssociatedEquipment, 543 (2-3),517Œ527,2005. URL http://dx.doi.org/10.1016/j.nima.2004.12.020 [81] K.Stiefel, MeasurementandModelingofFragmentsandNeutronsProducedfromProjectile FragmentationReactions ,Ph.D.thesis,MichiganStateUniversity,2018. [82] W.Peters, StudyofNeutronUnboundStatesUsingtheModularNeutronArray ,Ph.D.thesis, MichiganStateUniversity,2007. [83] G.A.Christian, SpectroscopyofNeutron-UnboundFluorine ,Ph.D.thesis,MichiganState University,2011. [84] J.Paul,NuclearInstrumentsandMethods, 96 (1),51Œ59,1971. URL http://dx.doi.org/https://doi.org/10.1016/0029-554X(71)90436-8 [85] O.B.TarasovandD.Bazin,NuclearInstrumentsandMethodsinPhysicsResearch,Section B:BeamInteractionswithMaterialsandAtoms, 266 (19-20),4657Œ4664,2008. URL http://dx.doi.org/10.1016/j.nimb.2008.05.110 [86] S.Matsuno etal. ,Phys.Rev.D, 29 ,1Œ23,1984. URL http://dx.doi.org/10.1103/PhysRevD.29.1 [87] M.D.Jones etal. ,Phys.Rev.C, 92 ,051306,2015. URL http://dx.doi.org/10.1103/PhysRevC.92.051306 [88] J.K.Smith etal. ,Phys.Rev.C, 90 ,024309,2014. URL http://dx.doi.org/10.1103/PhysRevC.90.024309 [89] Z.Kohley etal. ,Phys.Rev.C, 87 ,011304,2013. URL http://dx.doi.org/10.1103/PhysRevC.87.011304 [90] A.Spyrou etal. ,Phys.Rev.Lett., 108 ,102501,2012. URL http://dx.doi.org/10.1103/PhysRevLett.108.102501 [91] Z.Kohley etal. ,Phys.Rev.Lett., 109 ,232501,2012. URL http://dx.doi.org/10.1103/PhysRevLett.109.232501 [92] C.R.Ho man etal. ,Phys.Rev.C, 83 ,031303,2011. URL http://dx.doi.org/10.1103/PhysRevC.83.031303 [93] T.Nakamura etal. ,Phys.Rev.Lett., 96 ,252502,2006. URL http://dx.doi.org/10.1103/PhysRevLett.96.252502 132 [94] F.M.Marqués,M.Labiche,N.A.Orr,F.Sarazin,andJ.C.Angélique,NuclearInstruments andMethodsinPhysicsResearchA, 450 ,109Œ118,2000. URL http://dx.doi.org/10.1016/S0168-9002(00)00248-5 [95] A.Hoecker,P.Speckmayer,J.Stelzer,J.Therhaag,E.vonToerne,andH.Voss,PoS, ACAT , 040,2007. [96] R.BrunandF.Rademakers,Nucl.Instrum.Meth., A389 ,81Œ86,1997. URL http://dx.doi.org/10.1016/S0168-9002(97)00048-X [97] N.Frank,A.Schiller,D.Bazin,W.Peters,andM.Thoennessen,NuclearInstrumentsand MethodsinPhysicsResearchSectionA:Accelerators,Spectrometers,DetectorsandAsso- ciatedEquipment, 580 (3),1478Œ1484,2007. URL http://dx.doi.org/http://dx.doi.org/10.1016/j.nima.2007.07.008 [98] S.Agostinelli etal. ,NuclearInstrumentsandMethodsinPhysicsResearchSectionA:Ac- celerators,Spectrometers,DetectorsandAssociatedEquipment, 506 (3),250Œ303,2003. URL http://dx.doi.org/https://doi.org/10.1016/S0168-9002(03)01368-8 [99] J.Allison etal. ,IEEETransactionsonNuclearScience, 53 (1),270Œ278,2006. URL http://dx.doi.org/10.1109/TNS.2006.869826 [100] J.Allison etal. ,NuclearInstrumentsandMethodsinPhysicsResearchSectionA:Acceler- ators,Spectrometers,DetectorsandAssociatedEquipment, 835 ,186Œ225,2016. URL http://dx.doi.org/https://doi.org/10.1016/j.nima.2016.06.125 [101] B.Roeder,DevelopmentandvalidationofneutrondetectionsimulationsforEURISOL, EURISOLDesignStudy,ReportNo.10-25-2018-006,2008. URL http://ns.ph.liv.ac.uk/eurisol/M3_in-beam_validations.pdf [102] Z.Kohley etal. ,NuclearInstrumentsandMethodsinPhysicsResearchSectionA:Accel- erators,Spectrometers,DetectorsandAssociatedEquipment, 682 ,59Œ65,2012. URL http://dx.doi.org/https://doi.org/10.1016/j.nima.2012.04.060 [103] J.BiersackandL.Haggmark,NuclearInstrumentsandMethods, 174 (1),257Œ269,1980. URL http://dx.doi.org/https://doi.org/10.1016/0029-554X(80)90440-1 [104] J.F.ZieglerandJ.P.Biersack, TheStoppingandRangeofIonsinMatter ,93Œ129(Springer US,Boston,MA1985). URL http://dx.doi.org/10.1007/978-1-4615-8103-1_3 [105] A.Goldhaber,PhysicsLettersB, 53 (4),306Œ308,1974. URL http://dx.doi.org/https://doi.org/10.1016/0370-2693(74)90388-8 [106] K.VanBibber etal. ,Phys.Rev.Lett., 43 ,840Œ844,1979. [107] A.Schiller etal. ,Phys.Rev.Lett., 99 ,112501,2007. URL http://dx.doi.org/10.1103/PhysRevLett.99.112501 133 [108] N.Frank etal. ,NuclearPhysicsA, 813 (3),199Œ211,2008. URL http://dx.doi.org/https://doi.org/10.1016/j.nuclphysa.2008.09.009 [109] G.Christian etal. ,NuclearPhysicsA, 801 (3),101Œ113,2008. URL http://dx.doi.org/https://doi.org/10.1016/j.nuclphysa.2008.01.004 [110] F.James,MonteCarloPhaseSpace,CERN,1968. URL http://cds.cern.ch/record/275743/files/CERN-68-15.pdf?version=1 [111] D.M.Schmidt,R.J.Morrison,andM.S.Witherell,NuclearInst.andMethodsinPhysics Research,A, 328 (3),547Œ552,1993. URL http://dx.doi.org/10.1016/0168-9002(93)90674-7 [112] M.BasuniaandA.Hurst,NuclearDataSheets, 134 ,1Œ148,2016. URL http://dx.doi.org/https://doi.org/10.1016/j.nds.2016.04.001 [113] G.Aad etal. ,PhysicsLettersB, 716 (1),1Œ29,2012. URL http://dx.doi.org/https://doi.org/10.1016/j.physletb.2012.08.020 [114] S.Chatrchyan etal. ,PhysicsLettersB, 716 (1),30Œ61,2012. URL http://dx.doi.org/https://doi.org/10.1016/j.physletb.2012.08.021 [115] H.A.Bethe,Phys.Rev., 50 ,332Œ341,1936. URL http://dx.doi.org/10.1103/PhysRev.50.332 [116] P.LeboeufandJ.Roccia,Phys.Rev.Lett., 97 ,010401,2006. URL http://dx.doi.org/10.1103/PhysRevLett.97.010401 [117] N.Michel,W.Nazarewicz,M.andK.Bennaceur,Phys.Rev.Lett., 89 ,042502, 2002. URL http://dx.doi.org/10.1103/PhysRevLett.89.042502 [118] R.IdBetan,R.J.Liotta,N.Sandulescu,andT.Vertse,Phys.Rev.Lett., 89 ,042501,2002. URL http://dx.doi.org/10.1103/PhysRevLett.89.042501 [119] N.Michel,W.Nazarewicz,M.andT.Vertse,JournalofPhysicsG:Nuclear andParticlePhysics, 36 (1),013101,2008. URL http://dx.doi.org/10.1088/0954-3899/36/1/013101 [120] J.Okwicz,M.andW.Nazarewicz,ProgressofTheoreticalPhysicsSupple- ment, 196 ,230Œ243,2012. URL http://dx.doi.org/10.1143/PTPS.196.230 [121] M.andJ.Okwicz,arXive-prints,arXiv:1910.06526,2019. 134