EXPERIMENTALSTUDYOF19CVIAGAMMA-RAYLIFETIMEANDKNOCKOUTMEASUREMENTSByKennethAaronWhitmoreADISSERTATIONSubmittedtoMichiganStateUniversityinpartialful˝llmentoftherequirementsforthedegreeofPhysicsDoctorofPhilosophy2016ABSTRACTEXPERIMENTALSTUDYOF19CVIAGAMMA-RAYLIFETIMEANDKNOCKOUTMEASUREMENTSByKennethAaronWhitmoreThenuclearhaloisauniquephenomenonoccurringatthelimitofnuclearstability.Previousstudieshaveestablishedanenhancedlow-energyelectricdipolestrengthasachar-acteristicfeatureofhalonuclei.Despitesuchextensiveworkontheelectricresponse,thereisnoexperimentalevidenceonthemagneticresponseofhalos.Agamma-raylifetimemea-surementhasbeenperformedontheone-neutronhalonucleus19C,whichrepresentsthe˝rstmeasurementofamagnetictransitionbetweenboundstatesinahalonucleus.Thismeasurementalsoservesasameanstoconstrainthespin-paritiesofthestatesin19C.Thelifetimeofthe˝rstexcitedstatein19Chasbeenmeasuredusingboththeline-shapemethodandtheRecoilDistanceMethod.ThededucedB(M1;3=2+!1=2+)transitionstrengthrepresentsoneofthemosthinderedM1transitionsamonglightnuclei.Theresultiscom-paredtolarge-scaleshellmodelcalculations,whichpredictastronghindranceduetothedegeneracyofthe1s1=2and0d5=2neutronorbitals.TheresultestablishestheM1hindranceasanotherfeatureofhalonucleiwhicharedominatedbys-wavecon˝gurations.Theone-protonknockoutreactionof20Nisusedtostudythestructureoftheboundstatesin19Caswellasthegroundstatein20N.Eikonalreactionmodelcalculationsarecomparedtothemeasuredinclusivecrosssection.Thesmallinclusivecrosssectionindicatesthesigni˝cantdi˙erencebetweenthewavefunctionsofthelow-lyingstatesof20Nand19C.Theresultssupportthespin-parityassignmentoftheexcitedstatein19Cobtainedfromthelifetimemeasurement.Thecouplingoftheproton0p1=2orbitaltoa5=2+corewithin20Nissuggestedfromlarge-scaleshell-modelcalculationsperformedforthe2groundstatein20N,indicatingthatthedegeneracyofthe1s1=2and0d5=2orbitalsthatoccursin19Cdoesnotpersistin20N.Boththelifetimeandknockoutreactionanalysessupporttheexclusionofabound5=2+statein19C.Thecombinedresultspresentaconsistentpictureofthestructureof19Candprovideimportantdatatoestablishtrendofthe1s1=2and0d5=2singleparticleenergiesfortheN=13isotones.ACKNOWLEDGMENTSItisimpossibleformetofullyexpressmygratitudetoallofthepeoplewhohavecon-tributedtomysuccessatMSU.IwilltrytousethisspacetogiveasmuchthanksasIcan.First,Iwouldliketoexpressmydeepestgratitudetomyresearchadvisor,Prof.HironoriIwasaki.Hehasinspiredandchallengedme,andIknowIamabetterscientistbecauseofhim.Hisendlesspatienceandoptimismhavekeptmegoing,evenwhenIgetboggeddowninstressofwork.Iwouldalsoliketothankthemembersofmythesiscommittee,Profs.MortenHjorth-Jensen,ArtemisSpyrou,CarlSchmidt,andMeganDonahue.Theyhaveprovidedextremelyvaluableinsightovertheyears.ImustalsoacknowledgethecontributionsofmanyotherscientistsattheNSCL,includingthesta˙scientistswhokeptthisandotherexperimentsrunningsmoothly,aswellastheprofessorswhoseclassesI'vehadtheprivelegeoftaking.IalsogivecredittomyprofessorsatWilliamJewellCollege.Theyencouragedmetoexploremanyresearchopportunities,andultimatelyallowedmetodiscovermyenthusiasmfornuclearphysics.IamespeciallygratefultothevariousmembersoftheLifetimegroup:AntoineLemasson,DuaneSmalley,NobuyukiKobayashi,CharlesLoelius,RobElder,MaraGrinder,andofcourse,ChristopherMorse,whoservedasanexcellentleaderasthepioneerLifetimegraduatestudent.IamalsogratefultothemembersoftheGammagroupwhoprovidedmewithtechnicalexperienceandexpertise.IalsoamverygratefulforalloftheotherMSUgraduatestudentswhomadelivinginMichiganfun,andwithwhomIamveryluckytobefriends.Finally,Iwouldliketothankmyfamily,whohasbeennothingbutsupportivethroughallivofmyeducationalendeavors.Ourweeklychatshelpedtokeepmesaneandliftedmyspiritsduringthemostdi˚culttimes.Withouttheirsupport,noneofthiswouldbepossible.vTABLEOFCONTENTSLISTOFTABLES....................................viiiLISTOFFIGURES...................................ixChapter1Introduction...............................11.1TheNuclearLandscape..............................11.2NuclearShellModel...............................61.3HaloNuclei....................................111.4ElectromagneticTransitions...........................171.5NuclearReactions.................................20Chapter2ExperimentalTechniques.......................242.1DevelopmentofRadioactiveIonBeams.....................242.2Gamma-rayinteractionsinmatter........................252.3RelativisticDopplere˙ect............................302.4Line-shapeMethod................................342.5RecoilDistanceMethod.............................362.6Single-nucleonKnockout.............................41Chapter3ExperimentalDevices.........................463.1BeamProduction.................................463.2TRIPLEXdevice.................................503.3Gamma-raydetection...............................553.3.1Calibrations................................583.4Charged-particleDetection............................603.4.1S800Spectrograph............................603.4.1.1TimingScintillators......................623.4.1.2CathodeReadoutDriftChambers...............633.4.1.3IonizationChamber......................643.4.1.4TrajectoryReconstruction...................653.4.1.5Calibrations...........................663.5SimulationSoftware................................69Chapter4Gamma-rayLifetimeMeasurementof19C.............744.1Motivation.....................................744.2OverviewofExperiment.............................774.3Line-shapeAnalysis................................804.4RecoilDistanceAnalysis.............................834.5Results.......................................854.6Discussion.....................................87vi4.6.1ShellModelCalculations.........................89Chapter5One-protonKnockoutMeasurementof20N............975.1MotivationandOverview.............................975.2CrossSections...................................1015.2.1TheoreticalCalculations.........................1035.3MomentumDistributions.............................1065.4Discussion.....................................108Chapter6Conclusions................................115REFERENCES......................................118viiLISTOFTABLESTable4.1:Summaryofsystematicerrorsobservedforthelifetimemeasurement.Errorduetothebeamproperties,plungergeometry,andgamma-raybackgroundshapeweresimilarforboththeline-shapeandrecoil-distancemethods.Theerrorfromdegraderreactionsonlyappearsfortherecoil-distancemeasurement..................86Table5.1:Theoreticalcrosssectionscomparedtotheexperimentalvalues,as-suminga2groundstatein20N.Theoreticalcrosssectionsarecalculatedasdescribedinthetext.Thetotalcrosssectionis˝rstcalculatedassumingtheobserved3=2+stateistheonlyboundex-citedstatein19C.Thesecondcalculationalsoincludesthecrosssectionassumingthelow-lying5=2+stateisbound,calculatedwiththeexcitationenergyfromthepreviousgamma-raystudy[89]....102Table5.2:Theoreticalcrosssectionscomparedtotheexperimentalvalues,as-suminga0groundstatein20N.Inthiscase,the5=2+stateisnotaccessiblethroughremovalofan`=1proton.............105Table5.3:Summaryofsystematicerrorsobservedfortheknockoutcross-sectionmeasurement.Errorsfortheinclusivecrosssection(left)andexclu-sivecrosssectiontotheexcited3=2+state(right)arelistedsepa-rately...................................111viiiLISTOFFIGURESFigure1.1:Thechartofthenuclides,whichdisplaysallknownnuclei.Eachsquarerepresentsanisotopewithaspeci˝ccombinationofprotonsandneutrons.Blacksquaresshowthelocationsofstablenuclei,whileothercolorsrepresentthehalf-livesforunstablenuclei.Fig-ureadaptedfromRef.[1]........................2Figure1.2:Adiagramshowingthesplittingofenergylevelswithinthenuclearshellmodel.Theleftcolumnshowstherelativeenergylevelsofthemajorshellsusingaharmonicoscillator(H.O.)potential.Themid-dlecolumnshowshowthelevelschangewhenaWoods-Saxon(WS)potentialisused.Here,thelevelsaresplitaccordingtotheorbitalmomentumquantumnumber`.Therightcolumnshowsthechangeswhenaspin-orbitalterm(VSO)isincludedinthepotential,andshowsfurthersplittingaccordingtothetotalangularmomentumquantumnumberj=`1=2.Inallcases,alargegapbetweenlevelsindicatesashellclosure,butonlytherightcolumnreproducestheobservedmagicnumbers.FigurefromRef.[4].................8Figure1.3:Aillustrationofthenaïveshell-modelcon˝gurationsforthegroundstatesofseveralcarbonisotopes.Theorbitalsforthes,p,andsdshellsareshowninthestandardordering,similartoFigure1.2.Pro-tonsareshowninred,andneutronsinblue.Inthestablenucleus12C(a),allsixprotonsandsixneutrons˝llthe0s1=2and0p3=2orbitals.Inthenear-stablenuclei14C(b)and16C(c),theadditionalneu-trons˝rst˝llthe0p1=2orbital,andthenbegintooccupythe0d5=2orbital.In19C,however,thethirteenthneutronoccupiesthe1s1=2orbital.The1s1=2orbitalin19Cisimportantintheformationoftheground-statehalostructure.......................10Figure1.4:Aplotofthecalculateddensitiesfortheprotons(solidline)andneutrons(dottedline)in22C,whichisknowntobeatwo-neutronhalo.Theneutrondensityfallsofmuchmoreslowlythantheprotondensityatlargeradius,whichistheprimaryfeatureofhalonuclei.FigureadaptedfromRef.[12]......................13Figure1.5:Therootmeansquareradiiofseverallightnuclei,deducedfromtheinteractioncrosssection.Thereisalargeenhancementinradiusbetween9Liand11LiwhichdeviatesfromthenormalA1=3trend.Thismeasurementwasthe˝rstindicationofthehalostructurein11Li.FigurefromRef.[13].......................14ixFigure1.6:AplotshowinganexampleofthedistributionoftheE1andM1strengthsinheavynuclei.Thepygmydipoleresonanceislocatedatlowerenergythanthegiantdipoleresonance(GDR).Forlighterneutron-richnuclei,thelow-energyE1responseisduetoasoftdipoleexcitationwithoutaresonancecharacter.FigurefromRef.[24]...16Figure1.7:Transversemomentumdistributionsof(a)6Hefollowingtwo-neutronremovalof8Heand(b)9Lifollowingtwo-neutronremovalof11Li.Thenarrowcomponentin(b)isindicativeoftheextendedwavefunc-tionoftheremovedneutrons.FigurefromRef.[15].........22Figure1.8:Parallelmomentumdistributionof18Cfollowingone-neutronknock-outof19C.Datapointsinblackarecomparedtotheoreticalcalcula-tionsfor`=0(solidline)and`=2(dashedline)neutronremoval.FigurefromRef.[23]..........................23Figure2.1:Aschematicofthephotoelectrice˙ect.Aphotonisabsorbedbyanelectronboundwithinanatom.Theelectrongainsenergyequaltothedi˙erenceofthephotonenergyanditsbindingenergywithintheatom.Theremainingpositivelychargedionalsogainsasmallamountofrecoilmomentum......................25Figure2.2:AschematicoftheComptonscatteringprocess.Inthiscase,theelectronisconsideredtobeafreeparticle,andthephotonisscatteredelasticallyo˙oftheelectron.Theenergyofthephotonchangesdependingonthescatteringangle...................26Figure2.3:Aschematicofpairproduction.Here,theincomingphotoninteractswiththestrongelectric˝eldoftheatomicnucleusanddisappears,creatinganelectronandapositron.Theenergyofthephotonissplitbetweenthemassoftheelectron-positronpairandthekineticenergytransferredtotheelectronandpositron.Thenucleusalsogainsaslightrecoilmomentum.Afterbeingsloweddowninthesurroundingmaterial,thepositronannihilateswithasecondelectron,creatingtwophotonswithenergiesof511keV.Theseannihilationphotonsareemittedinoppositedirections......................28Figure2.4:Absorptioncrosssectionsinsolidgermanium,plottedasattenuation,forenergiesbetween1keVand100MeV.Theplotshowsthecrosssec-tionsforphotoelectricabsorption(red),Comptonscattering(blue),andpairproduction(magenta).Thesumofthethreeprocessesisshowninblack.DatafromtheXCOMdatabase[45].........30xFigure2.5:AgraphshowingtherelativecontributionstheenergyresolutionasdescribedinEquation2.10.Theredlineshowstheresolutionduetotheuncertaintyintheangleofemission,thebluelineshowsthee˙ectduetotheuncertaintyinvelocity,andthegreenlineshowstheintrinsicenergyresolution.Theblacklineshowsthetotalres-olutionwhenthethreee˙ectsareaddedinquadrature.Thecurvesarecalculatedwiththevalues=0:43,=0:008,=1.5°,andEintr=0:002MeVfora1MeVgammarayinthecenter-of-massframe...................................33Figure2.6:Anillustrationoftheprinciplesoftheline-shapetechniquefordeter-mininglifetimes.In(a),threedecaycurvesforlifetimesintherangeof100nsareshownforabeamexitingareactiontarget.Sim-ulatedDoppler-shiftcorrectedspectracorrespondingtoeachlifetimeareshownin(b),aswellasapeakforadecaywithnolifetimee˙ect(0ps).Forlongerlifetimes,thepeakinthespectrumisshiftedtolowerenergies,andabroadtailformsonthelow-energysideofthepeak...................................36Figure2.7:AdiagramillustratingtheprinciplesoftheRecoilDistanceMethod.Afterreactinginsidethetarget,thebeamtravelsthedistanceDwithatimeof˛ighttf.Onceenteringthestopper,thebeamslowsdownduringtimets,withts˝tf.Adetectordetectsgammaraysemittedfromananglerelativetothebeam.Gammarays0emittedduringthetimeof˛ighttfareshiftedinenergy,whilegammarays0emittedafterthebeamisstoppedareunshifted,creatingtwopeaksatdi˙erentenergiesinthegamma-rayspectrum.Thelifetimeisdeterminedfromtheyieldoftheunshiftedpeakrelativetothetotalgamma-rayyield.FigureadaptedfromRef.[49]...........38Figure2.8:AschematicdemonstratingtheRecoilDistanceMethodusedwithafastradioactivebeam.Theuseofadegradertoslowdownthebeamcreatesfast(blue)andslow(green)regionsalongthebeampath.Astheseparationofthetargetanddegraderincreases,moregammadecaysoccurinthefastregion.Theresultinggamma-rayspectraforeachofthesettingsareshownontheright..............40Figure3.1:TheCoupledCyclotronFacilityattheNSCL.Astablebeamiscre-atedatanionsourceandacceleratedintheK500andK1200cy-clotrons.Thebeamthenreactsattheproductiontargettocreatearadioactivesecondarybeam.ThesecondarybeamcontinuesthroughtheA1900andisthensenttotheexperimentalarea.FigureadaptedfromRef.[57]..............................47xiFigure3.2:Theproductionratesofnucleifollowingthefragmentationofanen-ergetic86KrbeamatvariouspointsintheA1900.Part(a)showstheproductionrateimmediatelyaftertheproductiontarget.Part(b)showstheratesafterthe˝rstselectionbasedonthemagneticrigidityofthebeam.ThestraightlineofacceptednucleirepresentstheconstantA=Zratiowhichisisolatedaftertravelingthroughthe˝rsttwodipolemagnetsoftheA1900.Part(c)showsthetransmittedbeamattheendoftheA1900,wherethereisaclearselectionofthemassandchargeofinterest.......................50Figure3.3:ApictureoftheTRIPLEXplungerattheNSCL,withadiagramonthebottom.Thediagramshowsthevariouscomponents:(A)theoutersupportframe,(B)oneofthemotors,(C)theoutertubewhichconnectstotheseconddegrader,(D)thecentraltubewhichconnectstothe˝rstdegrader,(E)theinnertubewhichconnectstothetarget,(F)thetargetcone,(G)the˝rstdegradercone,and(H)theseconddegradercone.Theradioactivebeamsenterstheplungerfromtheleftandencountersthefoilsonthefarright.Thepictureshowstheplungerwiththefoilframesremoved.FigurefromRef.[63].....51Figure3.4:TheTRIPLEXplungerlocatedinsidethededicatedvacuumchamber.Electricalfeedthroughsforcontrolandmonitoringofthedevicearevisiblein(a).Theclose-upviewin(b)showsthescrewswhichareusedtoaligntheplungertothebeampath.FigurefromRef.[63]..52Figure3.5:ThethreetubeswhichcomprisethebearingunitoftheTRIPLEXplunger.Thestationarymiddletubeisattachedtotheinnersupportringandconnectstothe˝rstdegraderfoil.Theinnerandoutertubesmovealongthebeamdirectionandconnecttothetargetfoilandseconddegraderfoil,respectively.FigurefromRef.[63].....53Figure3.6:ThedesignofthedetectorcrystalsinGRETINA.Part(a)showsthetwoshapesusedto˝tthedetectorfacesaroundacircle.Part(b)showshoweachcrystaliselectricallysegmentedintosixradialsectionsandsixlayersdeep,creating36individualsegmentsusedforpositionresolution.FigureadaptedfromRef.[40]..........56Figure3.7:ThebeampipeusedintheexperimentshowingtheGRETINAclus-tersaroundthedownstreamportion.Thedetectorsontopshowhowadjacentmodules˝ttogether.Theradioactivebeamcomesinfromthebottomright.............................57xiiFigure3.8:Resultsofthee˚ciencycalculationsusing(a)theline-shapecon˝g-urationwithasingleberylliumtarget,and(b)theRecoilDistanceMethodsetup,withaberylliumtargetandtwotantalumdegraders.Thestandardsourcesinclude152Eu(redsquares)and133Ba(bluetriangles)forgammaraysbelow500keV.Simulatede˚cienciesarecomparedtothedatainblack.Unscaledsimulatede˚cienciesareshownbysolidcircles,andin(b),theopencirclesshowthesimula-tionscaledby0.91.Onlythee˚ciencyfortheline-shapesetupisusedinthecalculationofpartialcrosssectionsintheknockoutreactionstudy...................................59Figure3.9:TheS800spectrograph.ThesecondarybeamarrivesfromtheA1900attheobjectplane,andissenttothetargetarea,wheretheex-perimentaltargetanddetectorsarelocated.Afterthetarget,˝nalproductsaresentthroughspectrographandanalyzedinthefocalplane.FigurefromRef.[65].......................61Figure3.10:ThefocalplanedetectorsoftheS800.TheCRDCsmeasurethexandypositionsofthebeamrelativetothecentralaxis.Theion-izationchambermeasuresenergyloss,andtheplasticE1scintillatormeasuresthetimeof˛ightofthebeamandisusedasatrigger.Thehodoscopebehindthescintillatorcanbeusedtotaglong-livedde-cays,butwasnotusedinthepresentwork.FigurefromRef.[67].......................................64Figure3.11:AnexampleofthecalibratedmaskrunsforCRDC1(left)andCRDC2(right).TheholesandlinescorrespondtoholesinaspeciallymadeplateinordertomaptherawCRDCsignalstotheknowncoordinatesoftheplateholes.............................67Figure3.12:Plotsshowingthee˙ectsofcorrectionstothetimingsignalsfromtheA1900extendedfocalplane(XFP)andS800objectplane(OBJ)scintillators.Theleftplotsshowthetimingspectrawithoutanycor-rections,andtherightplotsshowthesamespectrawithcorrectionsbasedonthedispersiveangle(afp)measuredattheS800focalplane.Thecorrectionsmakeparticleidenti˝cationpossible.........68Figure4.1:Thelevelschemefor19Catthetimeofthepresentexperiment.Twogamma-raytransitionshavebeenobservedamongtheboundstates[88,89],andtworesonancesabovetheneutronseparationen-ergyhavebeenobserved[90,91]....................76xiiiFigure4.2:Theparticleidenti˝cationspectrumforthesecondarybeam.Onthex-axisisthetimetakenfromtheS800objectplanescintillator(OBJ),andthey-axisshowsthetimefromtheA1900extendedfocalplanescintillator(XFP)............................78Figure4.3:Theparticleidenti˝cationspectrumforthe˝nalproductsintheS800spectrograph.Thex-axisshowsthecorrectedtimeof˛ightfromtheS800objectplane(OBJ),andthey-axisshowstheenergylossthroughtheionchamber........................79Figure4.4:TheDoppler-correctedspectrumusingtheline-shapemethod.Asin-glepeakat209keVwithawidetailatlowerenergiesisclearlyvisible.Theplotshowsthedatainblack,andthebest-˝tsimulationisshowninred,whichincludesabackgroundtakenfrom9Lishowninblue.82Figure4.5:TheDoppler-correctedspectrumusingtheRecoil-DistanceDoppler-ShiftMethod.Adoublepeakat209keVisagainvisible,withthesamelow-energytail.Thefastpeakcenteredat209keVismuchsmallerthantheslowpeakat190keV.Thedataareshowninblack,andthebest-˝tsimulationisshowninred,withtheassumedback-groundarisingfrom9Lishowninblue.Inthisplotthereareaddi-tionalx-raysaround50keVbecauseoftheenergylossofthebeaminsidethetantalumdegrader......................84Figure4.6:The˜2distributionsobtainedby˝ttingthesimulatedgamma-rayspectratotheexperimentalspectrumforvariouslifetimesofthe209-keVtransitionin19C.Part(a)showsthedistributionfortheline-shapespectrum,and(b)showsthedistributionfortherecoil-distancespectrum.Quadratic˝tstothecurvesgiveminimaat1.98nsand1.90ns,respectively...........................85Figure4.7:AplotshowingthedistributionsofallB(M1)transitionstrengthsamongnucleiofmassA<40.(a)plotsthevaluesforonlythosetransitionswhichinvolvea1=2+state,withthepresentlymeasuredB(M1;3=2+!1=2+g:s:)for19Chighlightedinred,andtheanalogousB(M1;1=2+!3=2+g:s:)inblue.(b)showsthedistributionforallM1transitions.Inbothcases,itisclearthattheB(M1)strengthfor19Cliesamongtheweakesttransitions.DatafromRef.[99].......88xivFigure4.8:Theexperimentalandshellmodelpredictionsforthelow-lyingstatesin19C.The1=2+groundstateisshowninblack,the3=2+isred,andthe5=2+stateisblue.Theexperimentallyobservedlevelsshownontheleftincludesthetwosuggestedlocationsofthe˝rstexcited5=2+state[89,90].Foralltheoreticalmodels,the1=2+groundstateiscorrectlyreproduced,whiletheorderofthe3=2+and5=2+statesarereversed...............................90Figure4.9:TheexperimentalandshellmodelpredictionsfortheB(M1;3=2+!1=2+)transitionstrengthin19C.Thecurrentexperimentalvalueisshownontheleft,andtheresultsforeachshellmodelcalculationareshownontheright.Thelightgreybarsindicatethevaluescalculatedwithoutanycorrections.ThedarkgreybarsfortheSFO-tlsandYuaninteractionsindicatecalculationswiththeloosely-bounde˙ects[103].Thestripedbarsadditionallyincludethemodi˝cationoftheM1operatorasdescribedinthetext.Inallcases,boththelooselybounde˙ectsandmodi˝edM1operatorimprovethepredictionsrelativetotheobservedvalue............................93Figure4.10:Theprimarycon˝gurationsforthegroundstate(a)andexcitedstate(b)in19C.The1=2+groundstateisprimarilyformedbythe1s1=2neutroncoupledtoa0+core.The3=2+excitedstateisdividedbetweentwomaincon˝gurations.Ontheleft,the1s1=2neutroniscoupledtoa2+core,andontheright,twoneutronsinthe1s1=2orbitalcoupleto0+,andthethreeneutronsinthe0d5=2orbitalcoupleto3=2+.Thecalculatedamplitudesofeachcon˝gurationaregiveninthetext.............................94Figure4.11:DecompositionofthecalculatedB(M1)strengthsin19C,23Na,and23Mgintothespin,orbital,andtensorcomponentsforbothprotonsandneutrons.AllcalculationsaremadeusingtheWBPinterac-tion[102].Itisclearthattheindividualmatrixelementsaresmallerfor19C,whilethesmallB(M1)valuesin23Naand23Mgareduetocancellationbetweenthecomponents.................96Figure5.1:Comparisonofthelow-lyinglevelsobservedin20Nwithshell-modelcalculationsusingtheWBPinteraction.Thespinandparitiesoftheexperimentallevelsarebasedonsimilarshellmodelcalculations[110],andno˝rmassignmentshavebeenmadeforanystates.......98xvFigure5.2:Threeplotswhichshowtheevolutionoftheneutrons1=2andd5=2or-bitalsinlightneutron-richisotopes.In(a),theenergiesoftheorbitalsareplottedrelativetotheneutronseparationenergyforN=7iso-tones.For13C(Z=6),the1s1=2orbital(red)isclearlylocatedbelowthe0d5=2orbital(blue),andfor14N,theorbitalsarenearlydegener-ate.Plot(b)showsthesametrendfortheN=9isotones.Here,for15C,the1s1=2orbital(red)isbelowthe0d5=2orbital(black),whiletheyoverlapin16N.Plot(c)showstheenergyofthe1s1=2orbital(red)relativetothe0d5=2orbital(blue)fortheN=11isotones.Inthiscase,theorbitalsarenearlydegeneratein17C,whilein18N,the1s1=2orbitalislocatedwellabovethe0d5=2orbital.FiguresfromRefs..............................100Figure5.3:Theprimarycon˝gurationsforthelowest2(left)and0statesin20N,calculatedwiththeWBPinteraction.Protonsareshowninred,andneutronsinblue.The2con˝gurationisformedbythecouplingofthevalence0p1=2protontothe(0d5=2)5J=5=2neutrons.Theshowncon˝gurationaccountsfor61%ofthetotalstrengthofthe2state.The0con˝gurationisformedbythecouplingofthesameprotontothe(0d5=2)4J=01s1=2neutrons.Thisaccountsfor74%ofthetotalstrengthinthe0state.........................106Figure5.4:Themomentumdistributionalongthebeamaxispkforthe209-keVstatein19Cfollowingtheone-protonknockoutof20N.TheblackpointsarethedatameasuredintheS800.Eikonalcalculationsareshownfors-wave(bluesolidline),p-wave(reddashedline),andd-wave(greendot-dashedline)protonremoval.ThecalculationshavebeennormalizedtothedataintheregionbetweenMeV/c,wheretheparticleswerefullyacceptedbytheS800..........107Figure5.5:Themomentumdistributionalongthebeamaxispkforthegroundstatein19Cfollowingtheone-protonknockoutof20N.TheblackpointsarethedatameasuredintheS800.Eikonalcalculationsareshownfors-wave(bluesolidline),p-wave(reddashedline),andd-wave(greendot-dashedline)protonremoval.ThecalculationshavebeennormalizedtothedataintheregionbetweenMeV/c,wheretheparticleswerefullyacceptedbytheS800..........109xviFigure5.6:Aplotshowingthesystematicdi˙erencebetweenexperimentalandtheoreticalone-nucleoncrosssections.Theplotincludesproton(blue)andneutron(red)knockoutusingfastbeamsaswellasprotonknock-outfromelectronscattering(black).ThereductionfactorRsisplot-tedasafunctionofS,whichisameasureofthedi˙erenceintheFermienergiesoftheprotonsandneutrons.FigurefromRef.[123].112xviiChapter1Introduction1.1TheNuclearLandscapeAlloftheobservablematterintheuniverseiscomposedofprotons,neutrons,andelectrons.Inthecenterofanatom,positivelychargedprotonsandunchargedneutronsarelocatedinsidethenucleus,whilenegativelychargedelectronsorbitaroundthenucleus.Allofthepositivechargeinsideofanatomiscontainedinsidethenucleus,whilethenegativechargeisspreadoutbythemotionofelectrons.Theidentityofanatom,whichisitselement,isdeterminedbythenumberofprotonsinsidethenucleus.Thisisthebasisofthechemicalperiodictableoftheelements,whicharrangesalltheelementsinorderofthenumberofprotons.Unliketheatom,thenucleusisde˝nedbyboththenumberofprotonsandneutronspresent.Thisformsthebasisforthechartofthenuclides,showninFigure1.1.Inthischart,eachsquarerepresentsanucleuswithagivennumberofprotonsandneutrons.Theprotonnumber,Z,isshownonthey-axis,andtheneutronnumber,N,isshownonthex-axis.Themassnumber,A,isde˝nedasthesumA=Z+Noftheprotonandneutronnumbers,becauseprotonsandneutronshaveapproximatelythesamemass(1:6731027kgforprotonsand1:6751027kgforneutrons),whilethemassofelectronsismuchsmaller(9:1091031kg).Thusthemassofanatomisnearlyequaltothesumofthenumberofprotonsandneutrons,whicharejointlycallednucleons.Agivennucleusisfullyspeci˝edbyfourterms:theelement(X),protonnumber(Z),neutronnumber(N),andmassnumber1Figure1.1:Thechartofthenuclides,whichdisplaysallknownnuclei.Eachsquarerepre-sentsanisotopewithaspeci˝ccombinationofprotonsandneutrons.Blacksquaresshowthelocationsofstablenuclei,whileothercolorsrepresentthehalf-livesforunstablenuclei.FigureadaptedfromRef.[1](A).TheseareusedtonotatethenucleusasAZXNwhichidenti˝esallthesetermsatonce.However,thisfullnotationisnotalwaysnecessary,becausetheelementisdeterminedbytheprotonnumber,andtheneutronnumbercanbesolvedfromthemassnumberasN=AZ.ThusthetypicalnotationusedinnuclearphysicsisAX,whichfullydescribesthenucleus.Althoughalmostallofthemassofanatomiscontainedwithinthenucleus,thenuclearsizeismuchsmallerthantheatomicsize.Thesizeofthenucleusdependsonthetotalnumberofprotonsandneutronspresent,andistypicallyontheorderofseveralfemtometers,or1015meters.Incontrast,theradiusoftheatomisdeterminedbytheorbitsofelectronsaroundthenucleusandisontheorderof0.1nanometers,or1010meters.Thereareseveraltermswhichareusefulindescribingnuclideswithspeci˝ccombinations2ofprotonandneutronnumbers.Nucleiwhichhavethesamenumberofprotonsbutdi˙erentnumbersofneutronsarecalledisotopes.TheseappearashorizontallinesinFigure1.1.Nucleiwiththesamenumberofneutronsbutdi˙erentnumbersofprotonsarecalledisotones,andappearasverticallines.Finally,nucleiwiththesamemassnumber,butdi˙erentnumbersofprotonsandneutronsarecalledisobars.IsobarsformdiagonallinesinFigure1.1,movingfromtheupperlefttothebottomright.OnefeaturethatisshownprominentlyinFigure1.1isthestabilityofindividualnuclei.Stablenucleiareshownbyblacksquares,whileunstable,orradioactive,nucleiareshownbyothercolorsbasedontheirhalf-lives[1].Itisclearfromthechartthatthereisapatterninthepositionsofstablenuclei.Thelocationofthestablenucleiacrosstheentirechartiscalledthevalleyofstability.ForlightnucleiwithsmallZandN,stabilityoccurswhentheprotonandneutronnumbersarenearlyequal,andN=Zˇ1.Heaviernuclei,however,experienceastrongerrepulsiveCoulombforceduetothelargenumberofprotons,andadditionalneutronsarenecessaryforthenucleitoremainstable.TheheaviestnucleicanreachratiosofN=Zˇ1:5.Inaddition,afurtherdistinctionismadeinthechart.TherearecombinationsofZandNattheedgesofthechartwhichdonotformnuclei;thesearecalledunboundbecausetheextremenumbersofprotonsorneutronspreventanuclearcompoundfromformingforany˝niteamountoftime.Theboundariesbetweentheboundandunboundnucleiarecalledthedriplines.Theprotondriplineislocatedtotheleftofthevalleyofstability,andtheneutrondriplineislocatedtotheright.AsshowninFigure1.1,therearemorethan3000unstablenuclei,whichformthemajorityofallknownnuclei.Thesenucleidecayintomorestablenuclei,andcandosoinawidevarietyofways.Themostcommontypesofradioactivedecayarealphadecay,betadecay,gammadecay,and˝ssion.3The˝rsttypeofradioactivedecay,alphadecay,alsocalledalphaemission,occursonlyfortheheaviestnuclei,typicallywithmassnumbersaboveA=200.Inthisprocess,anucleusejectstwoprotonsandtwoneutronstogetherasa42Henucleus,whichisalsocalledanalphaparticle.Becauseofthelossoftwoprotonsfromthenucleus,theelementischangedduringthealphaemission.Throughthisprocess,heavyunstablenucleilocatedinthetop-rightcornerofthechartofnuclidesmovetowardthestablenucleibymovingdownandtotheleftonthechart.ThealphaparticleistypicallyemittedwithenergyintherangeMeV.Duetoconservationofmomentum,theremaininglargenucleusobtainsasmallkickduringthedecayandsharessomeofthedecayenergyaskineticenergy.Thesecondtypeofradioactivity,betadecay,occursinthemajorityofnucleiwhicharenotstable.Thereareactuallytwotypesofbetadecay,denoted+decayanddecay.Indecay,aneutronistransformedintoaprotonthroughtheemissionofanenergeticelectronaswellasanantineutrino.Theoppositeoccursin+decay,inwhichaprotontransformsintoaneutron,emittingapositron(thepositivelychargedantiparticleofanelectron)andaneutrino.Analternativeto+decayiselectroncapture,inwhichanatomicelectronisabsorbedbythenucleuswhileaprotonischangedintoaneutron.Inallthreedecays,theprotonnumberofthenucleusischanged,whilethemassnumberisunchanged,becausethetotalnumberofnucleons,andthereforethemassnumber,remainsunchanged.Thesedecaysallownucleitomovetowardsthestablenuclei.Fordecay,nucleiwithexcessneutronsapproachstabilitybymovingupandtotheleftinthechartofnuclides.In+decayandelectroncapture,proton-richnucleiaremoveddownandtotherightinthechartofnuclides.Thethirdtypeofdecay,gammadecay,occursthroughtheemissionofaphoton.Inthegamma-decayprocess,allnucleonsremaininthenucleus.Thegammarayonlycarriesaway4energyasthenucleusde-excitesintoalowerenergystate.Typically,gammadecayoccursafteranotherformofradioactivedecaywhichleavesthenucleusinanexcitedstate.MostgammaraysareemittedwithenergiesintherangeMeV.The˝ssionprocessoccurswhenaheavynucleussplitsapartintotwosmallernuclei.Duringthe˝ssionprocess,alargeamountofenergyisreleased,about200MeV,whichismostlygiventothedaughternucleiaskineticenergy.Unliketheprevioustypesofdecay,the˝nalproductsarenot˝xed,butaredistributedamongthemedium-massnuclei.Theseproductstendtobeneutron-richbecauseoftherelativeneutronexcessforheavynuclei.Afterbeingcreated,thedaughternucleimaythenbetadecaytowardsstability.Inadditiontotheaforementioneddecayprocesses,othertypesofradiationarepossible,althoughmuchlesscommon.Protonradioactivityresultsintheemissionofaprotonfromthenucleus,andoccursforproton-richnuclei.ThiscanhappenwhenanucleusisinanexcitedstatewithanenergylargerthantheprotonseparationenergySp.Thisisde˝nedasthebindingenergyoftheleast-boundprotonandcanbecalculatedfromSp=[m(Z;N)+m(Z1;N)+m1H]c2(1.1)wherem(Z;N)andm(Z1;N)aretheatomicmassesofnucleiwithprotonnumbersZand(Z1),andneutronnumberN.TheatomicmassofHydrogenm1Histhesumoftheprotonmassmp=938:3MeV=c2andelectronmassme=0:511MeV=c2.Emissionofoneortwoprotonsalsooccursfromthegroundstateofnucleibeyondtheprotondripline,wheretheprotonseparationenergyislessthanzero.Similarly,theneutronseparationenergySnisde˝nedasSn=[m(Z;N)+m(Z;N1)+mn]c2(1.2)5wheremn=939:6MeV=c2istheneutronmass.1.2NuclearShellModelOnefeaturewhichishighlightedinFigure1.1istheexistenceofmagicnumbers.Theseareshowninthehorizontalandverticalboxesforprotonandneutronnumbersof8,20,28,50,82,and126.ThoughnotshowninFigure1.1,2isalsoconsideredasoneofthemagicnumbers.Thesevaluesarecalledmagicbecausenucleiwhichhavethismanyprotonsorneutronsexhibitincreasedstability.Thisstabilityisevidentintheformofhigherbindingenergies,higherseparationenergies,andhigher-lyingexcitedstatescomparedtoneighboringstablenuclei.Theobservationofmagicnumbershasledtothedevelopmentofthenuclearshellmodel[2].Withintheshellmodel,nucleons˝llupthenucleusby˝rstoccupyingsingle-particlestateswiththelowestenergies.Thepresenceofalargegapinenergybetweenorbitalscreatesashell,andthenumberofnucleonswithineachshellcorrespondstothemagicnumbers.Thesimplestformofthenuclearpotentialistheharmonicoscillator(H.O.)potential[3]:VHO(r)=12m!2r2:(1.3)whichgivesthepotentialVHOasafunctionofradiusrwiththefrequencyparameter!.Inthreedimensions,thispotentialleadstosingle-particlestatescharacterizedbyquantumnumbersnand`.Here,nisthenumberofradialnodesinthewavefunction,and`istheorbitalangularmomentumofthewavefunction.Theenergyofanucleonwithquantum6numbersnand`isgivenby:E(n;`)=(2n+`+3=2)~!:(1.4)Herethereisadegeneracyintheenergyfordi˙erentcombinationsofnand`.Thus,itiscustomarytode˝nethemajoroscillatorquantumnumberN=2n+`whichdeterminestheenergyofalldegeneratewavefunctions.ThisisshownontheleftsideofFigure1.2,wheretheenergylevelsforeachNareplotted.Each`contains2(2`+1)degeneratewavefunctions,sothatthetotaldegeneracyforeachoscillatornumberN=0,1,2...is2,8,20,40,70,112...[3].Thisreproducesthesmallermagicnumbers(2,8,and20),butfailstoaccountforthelargermagicnumbers.Theharmonicoscillatorpotentialpresentsinherentproblemsindescribingthenucleusbecausethestrengthdivergesforlargeradius,creatinganin˝nite-rangeforce.Becausethenuclearforceisknowntohavea˝niterange,amorerealisticformofthenuclearpotentialwhichtakesintoaccountthe˝niterangeisnecessarytobetterexplainnuclearproperties.OneexampleofsuchapotentialcomesfromtheWoods-Saxonform[3]:VWS(r)=V01+exp((rR0)=a):(1.5)Here,R0=1:25A1=3fmistheradiusofthepotential,anda=0:67fmisthedi˙usenessofthepotential.Theprimaryadvantageovertheharmonicoscillatorpotentialisthatthispotentialdoesnotdivergeatin˝nity.Inthiscase,energylevelsareagaindeterminedbynand`,andformmajoroscillatorshells.However,di˙erent`valuesarenotdegenerate,andhigher-`orbitalshavelowerenergiesthanlower-`orbitalswithinthesameshell.This7Figure1.2:Adiagramshowingthesplittingofenergylevelswithinthenuclearshellmodel.Theleftcolumnshowstherelativeenergylevelsofthemajorshellsusingaharmonicoscillator(H.O.)potential.ThemiddlecolumnshowshowthelevelschangewhenaWoods-Saxon(WS)potentialisused.Here,thelevelsaresplitaccordingtotheorbitalmomentumquantumnumber`.Therightcolumnshowsthechangeswhenaspin-orbitalterm(VSO)isincludedinthepotential,andshowsfurthersplittingaccordingtothetotalangularmomentumquantumnumberj=`1=2.Inallcases,alargegapbetweenlevelsindicatesashellclosure,butonlytherightcolumnreproducestheobservedmagicnumbers.FigurefromRef.[4]8isshownbythecenterofFigure1.2,wherethesplittingoftheorbitalangularmomentumrelativetotheharmonicoscillatorisevident.Theimportantterminthepotentialwhichallowsforthecorrectpredictionofmagicnumbersisthespin-orbitterm.Thiswas˝rstintroducedbyHansJensen[5]andMariaGoeppert-Mayer[2],whosharedtheNobelPrizeinphysicsfordiscoveryofthisterm.Thespin-orbitcouplingterminthepotentialhastheformVSO(r)=V`sddr11+exp((rR0)=a)~`~s:(1.6)ThistermisaddedtotheradialtermofEquation1.5[3],andtheresultsareshownontherightsideofFigure1.2.Withthisterm,theenergylevelsforeach`aresplitbythequantumnumberj,thetotalangularmomentum.Thetotalangularmomentumisformedbythevectoradditionoftheorbitalangularmomentum~`andthespin~sofeverynucleon.Becausetheintrinsicspinsis1/2forprotonsandneutrons,j=`1=2.Thestrengthofthespin-orbittermV`sisnegative,whichmeansthattheenergyislowerforj=`+1=2states,when~`and~sareparallel.Thesplittingoftheenergyincreaseswithincreasingj,sothatforlargej,theorbitalscanmoveacrossthemajorshellstocreatenewenergygapsandthereforenewmagicnumbers.AsshowninFigure1.2,theadditionofthespin-orbittermcorrectlyproducesthemagicnumbers28and50.Themagicnumber28occurswhenthe`=3;j=7=2orbital(denoted0f7=2)isloweredinenergybelowtheN=3shell.Similarly,thegapat50isformedwhenthe`=4;j=9=2orbital(0g9=2)movesbelowtheN=4shell.Theinclusionofthespin-orbittermalsopredictsthemagicnumbers82and126,whichcompleteallknownmagicnumbersasshowninFigure1.1.Thenuclearshellmodelhasbeenverysuccessfulinpredictingpropertiesofmanynuclei.9Figure1.3:Aillustrationofthenaïveshell-modelcon˝gurationsforthegroundstatesofseveralcarbonisotopes.Theorbitalsforthes,p,andsdshellsareshowninthestandardordering,similartoFigure1.2.Protonsareshowninred,andneutronsinblue.Inthestablenucleus12C(a),allsixprotonsandsixneutrons˝llthe0s1=2and0p3=2orbitals.Inthenear-stablenuclei14C(b)and16C(c),theadditionalneutrons˝rst˝llthe0p1=2orbital,andthenbegintooccupythe0d5=2orbital.In19C,however,thethirteenthneutronoccupiesthe1s1=2orbital.The1s1=2orbitalin19Cisimportantintheformationoftheground-statehalostructure.10However,thereareseveralchallengesassociatedwiththeshellmodel.Onesuchexampleisheavynucleiwithnon-sphericalshapes.Calculationofpropertiesofthesenucleiarebetterservedbyothermodels,suchastheNilssonmodeloracollectivemodel[6].Fornucleiawayfromstability,theenergiesofsingle-particlestatescanchangerapidly,causingtheorderingoforbitalstodeviatefromthestandardpictureshowninFigure1.2.AnexampleofthisdeviationisshowninFigure1.3,whichshowstheprimarycon˝gurationsofprotonsandneutronsinseveralcarbonisotopes.Forthenuclei12C,14C,and16C,theneutrons˝llorbitalsasexpectedbythestandardpicture,butforthedrip-linenucleus19C,the0d5=2and1s1=2orbitalsareoccupiedsimultaneously.Thepresentworkprovidesinsighttotheshell-modelcon˝gurationsin19Cbymeasuringthegamma-raytransitionstrengthbetweentheboundstates.Thechangesinsingle-particleenergiesalsocreatesenergygapsbetweendi˙erentorbitals.Thiscancausenewmagicnumberstoappear,suchasN=16[7]andN=32[8],whileotherconventionalmagicnumbersmaydisappear,suchasN=8[9,10],N=20[11].1.3HaloNucleiOneofthechallengestotheshellmodeldescriptionofnucleiistheappearanceofexoticstructuresinradioactivenuclei.Nearthelimitsofstability,astheenergylevelsofvalencenucleonsapproachtheparticledecaythreshold,anewstructurecalledahalocanemerge.Ahaloconsistsofweaklyboundneutronorprotonthatislargelydecoupledfromthenuclearcoreandhasawavefunctionwithanenhancedradius.Formationofthehaloisfavoredforneutronsinlowangularmomentumorbitals,wherethereislittleornocentrifugalorCoulombbarrierarisinginthepotential.Thisallowstheneutrontotunneloutsideofthenuclearcore11andformalow-densityaroundthecore.Forexample,thedensityoftheprotonsandneutronsforthetwo-neutronhalo22C[12]isshowninFigure1.4.Theexistenceofaneutroncloudisapparentfromtheneutrondensity,whichdecreasesatlargeradiusmuchmoreslowlythantheprotondensity.Experimentalidenti˝cationofhalonucleihasbeenmadethroughthreetypesofmeasurements:interactioncrosssection[13,14],momentumdistributionfollowingnucleonremoval[15,16],andCoulombbreakupreactionsEachofthesemeasurementsisassociatedwithadi˙erentfeatureofhalonuclei,andtogethertheyestablishthehaloasauniquephenomenonofweaklyboundnuclei.Theearliesthalonucleustobeidenti˝edwas11Li,whichexhibitsatwo-neutronhaloarounda9Licore.The˝rstindicationofanunusualstructurecamefromtheobservationofanenhancedinteractioncrosssection[13].ThisisdemonstratedinFigure1.5,whichplotsthermsradiusdeducedfrominteractioncrosssectionsforseverallightnuclei.Thelargeenhancementfor11Licomparedtotheotherlithiumisotopesisapparent.Subsequentmeasurementsofthemomentumdistributionof9Lirecoilsfollowingtwo-neutronremoval[15],andtheelectromagneticdissociationof11Li[20]˝rmlyestablished11Liasahalonucleus.Sincethe˝rstdiscoveryofthehalostructure,severalotherhalonucleihavebeenidenti˝ed,uptothemedium-massnucleus37Mg[21].Thepresentworkdescribesameasurementofthegamma-raytransitionin19C,whichexhibitsahalostructureinitsgroundstate.The1s1=2orbital,whichisresponsibleforthehaloformation,isshowntobeanimportantfactorintheobservedtransitionrate.Theprimaryfeaturepresentinhalonucleiisanenhancedradiuscomparedtoneighboringnuclei.Measurementoftheradiushasprimarilycomefrommeasurementsoftheinteractioncrosssection.Theinteractioncrosssection˙Iisthecrosssectiontochangeeithertheprotonorneutronnumberwhenaprojectilenucleusisincidentonatarget.Therelationbetween12Figure1.4:Aplotofthecalculateddensitiesfortheprotons(solidline)andneutrons(dottedline)in22C,whichisknowntobeatwo-neutronhalo.Theneutrondensityfallsofmuchmoreslowlythantheprotondensityatlargeradius,whichistheprimaryfeatureofhalonuclei.FigureadaptedfromRef.[12].˙IandthenuclearsizecanbemostsimplyunderstoodintermsoftheinteractionradiusRI[13]:˙I=ˇ[RI(p)+RI(t)]2(1.7)whereRI(p)andRI(t)aretheprojectileandtargetradii,respectively.Althoughnotdirectlyrelatedtotheinteractionradius,therootmeansquareRrmsisamoreusefulmeasureofthenuclearsize.Thisisobtainedfrom˙Iby˝ttingthecrosssectionobtainedfromaGlaubermodelcalculation[13,22].Forstablenuclei,theradiifollowsasimpletrend[4],withR=r0A1=3(1.8)usingatypicalradiusparameterr0=1:25fm.Forhalonuclei,however,theradiuscanbemuchlarger.ThisisdemonstratedinFigure1.5,whichshowstheRrmsforseverallight13Figure1.5:Therootmeansquareradiiofseverallightnuclei,deducedfromtheinteractioncrosssection.Thereisalargeenhancementinradiusbetween9Liand11LiwhichdeviatesfromthenormalA1=3trend.Thismeasurementwasthe˝rstindicationofthehalostructurein11Li.FigurefromRef.[13].nucleideterminedfromtheinteractioncrosssections[13].Thereisaclearenhancementintheradiusfor11Liwhichdeviatesfromthetrendshownbyseveralothernuclei.Anothermeanstoidentifyhalonucleiismeasurementofthemomentumdistributionoffragmentsfollowingremovalofthehalonucleon.Thesemeasurementsaremadebybom-bardingthenucleusonalighttargetathighincidentbeamenergy,andthemomentumoftherecoilingnuclearfragmentcanbemeasuredeitherparallelortransversetothedirectionofthebeam[23].Themomentumdistributionofthefragmentscanberelatedtothemo-mentumoftheremovednucleon,whereanarrowdistributionofthefragmentindicatesalowmomentumofthenucleonwithinthehalonucleus.Therelationbetweenalargeradius14andanarrowmomentumdistributionfollowingnucleonremovalhasbeeninterpretedwithinthecontextoftheuncertaintyprinciple,whereawavefunctionwithalargerspatialdistri-butionhasalowermomentum[15].Thecombinedresultspresentaconsistentpictureofthestructureofthehalowithaweaklyboundnucleonwithanextended,low-momentumwavefunction.ThethirdtypeofmeasurementwhichisusefulforstudyinghalonucleiisCoulombbreakupreactions.Inthismeasurement,thehalonucleusisincidentonaheavy,high-Ztargetatahighbeamenergy.TheabsorptionofavirtualphotonfromthestrongCoulomb˝eldofthetargetcausesthehalonucleontobereleasedintothecontinuum.Theenergyandmomentumofboththenuclearfragmentandtheemittednucleonaremeasured,andtherelativeenergyspectrumofthebreakupisreconstructedfromtheinvariantmassofthefragmentandnucleon.Theenergydistributionofthebreakupcanberelatedtothewavefunctionofthehalo[18]asd˙CDdErel=16ˇ39~cNE1(Ex)hqjZeArY1mj~r)i2(1.9)where˙CDistheCoulombdissociationcrosssectionasafunctionoftherelativebreakupenergyErel,NE1(Ex)isthenumberofvirtualphotonsforexcitation,hqjisthewavefunctionofthenucleoninthecontinuum,Y1misan`=1sphericalharmonic,andj~r)iisthewavefunctionofthehalonucleusbeforebreakup.Becauseofthelargeradialwavefunction,themostimportantcomponentofCoulombbreakupofhalonucleiistheelectricdipole,orE1breakup[18].TheE1operatorhastheformrY1,andtheradialtermprovidesasensitiveprobeofthewavefunctionforahalo.Severalmeasurementshaverevealedlargebreakupcrosssectionsatlowexcitationenergies,whichhavebeenobservedasafeature15Figure1.6:AplotshowinganexampleofthedistributionoftheE1andM1strengthsinheavynuclei.Thepygmydipoleresonanceislocatedatlowerenergythanthegiantdipoleresonance(GDR).Forlighterneutron-richnuclei,thelow-energyE1responseisduetoasoftdipoleexcitationwithoutaresonancecharacter.FigurefromRef.[24].uniquetohalonuclei[25].AnexampleofthedistributionoftheE1strengthinnucleiisinFigure1.6,whichshowsthetypicaldistributionoftheelectricdipole(E1)andmagneticdipole(M1)responses.Athighexcitationenergies,thereisalargecomponentfromtheGiantDipoleResonance(GDR).However,atlowexcitationenergies,thereareadditionallysmallerresonances.Forheavynuclei,thesearecalledpygmyresonances,butforlighthalonuclei,thelow-energystrengthisintheformofsoft-dipolemodes,whichwasoriginallyinterpretedasamotionoftheouterneutronsagainstthenuclearcore[26].Thissoft-dipolemodehasbeenwell-studiedforhalonuclei.However,themagneticresponseofhalonucleiisnotaswellstudied,hamperingdetailedcharacterizationofhalosystems.Inadditiontotheirinterestfromastructureperspective,halonucleihaveattractedinterestfortheiruseinnuclearreactions.Forexample,theneutron-capturereactionhasbeenstudiedfor14Cleadingtotheground-statehaloin15CInthiscase,theextended16wavefunctionofthehaloneutronin15Cleadstothedominanceofp-wavecapture,whichisdistinctfrommostastrophysicalcapturereactions,wherethes-wavecapturereactionisusuallydominant[29].Inaddition,severallightneutron-halonucleihavebeeninvestigatedinfusionreactionstoobserveapossibleenhancementofthefusioncrosssection.Theuseofneutron-richnucleiisregardedasusefulinthesynthesisofneutron-richheavyelementswithZ>114[30].However,theresultsofthesestudiesaremixed,withsomereportingenhancementofcrosssectionsbelowtheCoulombbarrier[31,32],whileothersshownoenhancement[33,34].ThesuppressionofthecrosssectionabovetheCoulombbarrierhasbeenexplainedasastrongpreferencefortheneutrontransferreaction,aphenomenonuniquetohalonuclei[35].1.4ElectromagneticTransitionsOneofthemostusefulwaystostudyandcharacterizenuclearstructureisbymeasuringgamma-raytransitionsbetweenboundstatesinnuclei.Measurementofgammaraysgivesdirectaccesstotheelectromagneticpropertiesofnuclei.Theprobabilitytoemitagammaray,whichisdirectlyrelatedtothelifetimeofastate,canberelatedtothewavefunctionsoftheinitial( i)and˝nal( f)statesofthenucleusas(˙L)=8ˇ(L+1)~L[(2L+1)!!]2E~c2L+1h fjjM(˙L)jj ii22Ji+1:(1.10)Here,(˙L)istherateofthetransitionbetweeninitialand˝nalstates iand f.Thedecaycanbeeitherelectric(˙=E)ormagnetic(˙=M)innature,withamultipolarityL.Themostcommonmultipolaritiesaredipole(L=1)andquadrupole(L=2).Eisthe17energyofthedecay.M(˙L)istheelectromagneticoperatorwhichgovernsthetransition.ThelastterminEquation1.10isthereducedtransitionprobabilityB(˙L;i!f):B(˙L;i!f)=h fjjM(˙L)jj ii22Ji+1:(1.11)Thistermcontainsallofthestructureinformationabouttheinitialand˝nalstates.Theelectricandmagneticoperatorshavedistinctformsanddependonthemultipolarityofthetransition:M(EL)=Xk=p;nek(rk)LYLm(k;˚k)(1.12)M(ML)=Xk=p;ngs;k~sk+2L+1g`;k~`krkh(rk)LYLm(k;˚k)iN:(1.13)Bothoperatorsaresummedovereachoftheknucleons.Thechargesofthenucleonsarerepresentedbyek,withep=+eanden=0.TheYLmarethesphericalharmonics.Thespinandorbitalg-factorsofthenucleonsarerepresentedbygs;kandg`;k,respectively,andN=0.105efmisthenuclearmagneton.Inordertodeterminethelifetime˝ofastate,thetotalratemustbecalculatedfromalltransitionsfromtheinitialstatetoallpossible˝nalstates:1˝==X˙L(˙L):(1.14)Thus,measurementofthelifetimeofagamma-raydecaycanprovidedetailedinformationaboutthestructureofanucleus.Itisimportanttoconsiderwhattypesofgammadecaycanbepresentforagiveninitialexcitedstate.Theavailabledecaysbetweentheinitialand˝nalstatesareconstrainedby18theangularmomentumJandparityˇofboththeinitialand˝nalstates.Mostimportantly,theangularmomentaoftheinitialand˝nalstatescannotdi˙erbymorethantheangularmomentumofthegammaray:jJiJfjLJi+Jf:(1.15)Thisiscalledthetrianglerulebecauseitisaconstraintthatarisesfromtheconservationofangularmomentum,sothatthevectors~Ji,~Jf,and~Lmustformaclosedtriangle.Inaddition,considerationmustbemadefortheparityˇofthestates,whichcaneitherbepositiveornegative.Iftheparityofthe˝nalstateisthesameastheinitialstate,thentheparityoftheelectromagneticoperatormustbeˇ=+1.Ifthestateshaveoppositeparity,thentheparityoftheoperatormustbeˇ=1.Theparityoftheelectricandmagneticoperatorsaregivenbyˇ(EL)=(1)L(1.16)ˇ(ML)=(1)L+1:(1.17)Theseconditionslimitthemultipolarities˙Lforallowedtransitions,whicharesummedinEquation1.14.Becausethecoe˚cientinthefrontofEquation1.10dependsstronglyonL,thetransitionratedecreasesdrasticallyforincreasingL.Thismeansthat,foragivenpairofinitialand˝nalstates,usuallyonlythetransitionwiththelowestmultipolaritycontributessigni˝cantlytothelifetime.Becauseofthe2L+1dependenceinEquation1.10,theabsolutetransitionratesdependstronglyonthetransitionenergyE.Inordertounderstandtherelativestrengthofatran-19sition,theWeisskopfunitisintroduced,whichprovidesanestimateofthetransitionrateforagivenmultipolarity[36].Theestimationismadebyassumingthatthetransitioniscausedbyasingleprotonmovingbetweenpuresingle-particleshell-modelstates.Additionally,thecalculationassumesthattheradialwavefunctionsoftheinitialand˝nalstatesareconstantwithinthenucleus,withaspherewhoseradiusisgivenbyEquation1.8.Weisskopfestimatesfortheelectricandmagneticstrengthsarede˝nedasB(EL)=14ˇ3L+32(1:2A1=3)2Le2fm2L(1.18)B(ML)=10ˇ3L+32(1:2A1=3)2L22Nfm2L2:(1.19)Theseestimationsnormalizethetransitionstrengthsfordi˙erentnucleiandprovideanestimationofthenumberofnucleonsparticipatinginthetransition.Forexample,ifatransitionhasastrengthontheorderof1W.u.,thenthatisanindicationthatthetransitionislikelyapuresingle-particletransition.IfthestrengthismuchgreaterthantheWeisskopfestimate,thenitislikelythatmorethanonenucleonisinvolved,andthestatesareconsideredtobecollectivestates.1.5NuclearReactionsOneofthemostimportantmethodsforobservinghalonucleiisthemeasurementofthemomentumdistributionsofnucleifollowingbreakupreactions.Earlymeasurementswerebasedonfragmentationreactions,inwhichastablebeamisacceleratedtohighenergy,typicallyabove100MeV/nucleon,andreactsonastationarytarget.Fragments,whichhaveanynumberofnucleonsremovedfromtheprojectilenucleus,arethendetectedafteremerging20fromthetarget.Themomentumofthefragmentsismeasured,eitheralongthedirectionoftheincomingbeam(parallelmomentumpk)orinadirectionperpendiculartothebeam(transversemomentump?).TheinterpretationofthemomentumdistributionswasbasedontheGoldhaberdescription[37],inwhichthedistributionoffragmentswasdescribedbyaGaussianformwithamomentumwidthparameter˙givenby˙2=˙20AF(AAF)A1(1.20)where˙istheobservedwidth,AandAFarethemassnumbersofthebeamandfragment,respectively,and˙0isaconstantwhichisusuallyintherangeMeV/c.ThiscanbeseeninFigure1.7(a),whichshowsthetransversemomentumof6Hefragmentsfollowingfragmentationofa8Hebeamat790MeV/nucleon[15].Oneoftheearliestindicationsofahalonucleuscamefromthemomentumdistributionof9Lifollowingtheremovaloftwoneutronsfrom11Li[15].Here,thedistributionincludedacomponentwhichwasmuchnar-rowerthanthatexpectedfromEquation1.20.ThiscanbeseeninFigure1.7(b),wherethenarrowdistributionwith˙0=23MeV=nucleonisseenontopofadistributionwiththemoretypical˙0=71MeV=nucleon.Thenarrowdistributionwasdiscussedasresultingfromtheweakbindingenergyofthetwovalenceneutronsin11Li.Alongwiththisresultcamenewinsightsintothereactiondynamics.First,itwasnotedthatthemomentumdistributionoftheprojectile-likefragmentmatchesthedistributionoftheremovednucleons[15].Sec-ond,themomentumdistributioncouldbequalitativelyunderstoodintermsofHeisenberg'suncertaintyprinciple.Largerspatialdistributionsfortheneutronwavefunctionsinhalosystemsleadtonarrowmomentumdistributionsfollowingtheirremoval[15,25].Thus,theobservationofthenarrowmomentumdistributionindicatedbothasmallmomentumofthe21Figure1.7:Transversemomentumdistributionsof(a)6Hefollowingtwo-neutronremovalof8Heand(b)9Lifollowingtwo-neutronremovalof11Li.Thenarrowcomponentin(b)isindicativeoftheextendedwavefunctionoftheremovedneutrons.FigurefromRef.[15].valenceneutronsandalargeneutronwavefunction,providingaconsistentpictureforthehalostructurein11Li.Morerecentdevelopmentsintheoryhaveallowedforadirectconnectionbetweenmo-mentumdistributionsandthesingle-particlestructureofnucleonsbyusingsingle-nucleonknockoutreactions[23].Theoreticalmomentumdistributionscanbecalculatedforagivenwavefunctionforanucleonwithintheprojectile,andtheshapeofthesedistributionsaresensitivetotheangularmomentumofthewavefunction.ThiscanbeseeninFigure1.8,whichshowstheparallelmomentumpkof18Cnucleifollowingtheone-neutronknockoutreactionof19Cat57MeV/nucleon[23].Thedataarecomparedtotheoreticalcurvesfor22Figure1.8:Parallelmomentumdistributionof18Cfollowingone-neutronknockoutof19C.Datapointsinblackarecomparedtotheoreticalcalculationsfor`=0(solidline)and`=2(dashedline)neutronremoval.FigurefromRef.[23].`=0(s-wave)and`=2(d-wave)neutronremoval.Thedi˙erenceinthetwocurvespro-videsaclearwaytoextracttheangularmomentumofthevalenceneutron,andcon˝rmsthelowangularmomentumofthehaloneutroninthegroundstateof19C.Bytaggingongamma-rayswhicharecoincidentwithoutgoingfragments,thismethodcanbeusedtoextractexclusivecrosssectionsleadingtobothgroundandexcitedstatesofthereactionproducts.Thus,measurementofmomentumdistributionsfollowingsingle-nucleonknockoutreactionshasbecomeapowerfultoolfornuclearstructuredetermination.23Chapter2ExperimentalTechniquesInthepresentstudy,theM1transitionrateforthe3=2+!1=2+g:s:transitionin19Cwasmeasuredusingtheline-shapetechniqueandRecoilDistanceMethod.Bothtechniquesarebasedongamma-rayspectroscopyusingafastbeamofradioactivenuclei.The19Cnucleiwereproducedwithaone-protonknockoutreactionfromabeamof20N,andthereactiondatawasanalyzedtocon˝rmthespin-parityassignmentsofthestatesin19Caswellasthegroundstatein20N.Thefollowingsectionsgiveanoverviewoftheuseofgamma-rayspectroscopywithradioactiveion(RI)beams,introducethemethodsusedtoextractthelifetimeofnuclearexcitedstatesfromthegamma-rayspectra,anddiscussthetheoreticalbasisforknockoutreactionswithfastRIbeams.2.1DevelopmentofRadioactiveIonBeamsTheabilitytoproducebeamsofradioactivenucleiouttotheprotonandneutrondriplineshasbeenanimportantstepintheprogressofnuclearstructurestudies[38].Beamsofradioactivenucleiareproducedfromhigh-energyreactionsofstablebeams.Becauseofthehighenergyatwhichtheradioactivesecondarybeamsareproduced,theRIbeamscanbeusedinsecondaryreactionstostudythestructureoftheradioactivenuclei.Theuseofinversekinematics,whichusesaheavybeamandlighttarget,incombinationwiththehighRIbeamenergy,meansthatsecondaryreactionproductsareforward-focused.Thisallows24Figure2.1:Aschematicofthephotoelectrice˙ect.Aphotonisabsorbedbyanelectronboundwithinanatom.Theelectrongainsenergyequaltothedi˙erenceofthephotonenergyanditsbindingenergywithintheatom.Theremainingpositivelychargedionalsogainsasmallamountofrecoilmomentum.forevent-by-eventtrackingofthereactionproducts[23].OnechallengetoovercomewiththeuseofRIbeamsislowerintensitiesrelativetostablebeams.Thisproblemiso˙setbytheabilitytousethicksecondaryreactiontargetsontheorderofg=cm2,sincethetotalreactionyieldisgivenbytheproductofthecrosssectionandtargetthickness.Forexample,Coulombexcitationreactionswithradioactivebeamscanbeperformedwithratesofonlyafewparticlespersecond[38].Besidesthelowintensities,anotherchallengetousingRIbeamsistheDopplerbroadeningofgammaraysemittedin˛ight.Thishasmotivatedthedevelopmentofposition-sensitivegermaniumdetectors[39,40],allowingfordetailedgammaspectroscopyofradioactivenuclei.Theseadvancementshavemadegamma-rayspectroscopywithfastRIbeamsapowerfultooltostudyexoticnuclei,andin-˛ighttechniquesforgamma-rayspectroscopyhavebeensuccessfullyappliedtoseveraltypesofexperiments,suchasknockoutreactions[41],transferreactions[42],andlifetimemeasurements[43].2.2Gamma-rayinteractionsinmatterGamma-raydetectorsmeasuretheenergydepositedinmatterbyincidentgammarays.Incontrasttochargedparticles,whichloseenergycontinuouslywhenmovingthroughmatter,25Figure2.2:AschematicoftheComptonscatteringprocess.Inthiscase,theelectronisconsideredtobeafreeparticle,andthephotonisscatteredelasticallyo˙oftheelectron.Theenergyofthephotonchangesdependingonthescatteringangle.photonsdepositlargeramountsofenergyindiscreteinteractions.Intheenergyrangeofthegammaspectrum,photonsinteractprimarilyinthreeways:photoelectricabsorption,Comptonscattering,andpairproduction.Inallthreeinteractions,photonstransfermostoftheirenergytoelectrons,whichareultimatelysloweddowninsideofthedetectorastheylosekineticenergythroughcontinuouscollisionswiththesurroundingatoms.The˝rstwayaphotoncandepositenergyintoamaterialisthroughthephotoelectrice˙ect.TheprocessisillustratedinFigure2.1.Inthisprocess,aphotonisincidentuponanelectronwhichisboundwithinanatom.Theelectronisejectedfromtheatom,andthephotondisappears.TheenergyEeoftheejectedelectronisgivenbyEe=EEb(2.1)whereEistheenergyoftheincidentphoton,andEbisthebindingenergyoftheelectronwithintheatom.Inordertoconservemomentum,therestoftheatomalsoreceivessomerecoilmomentum.However,becausetheatomismuchheavierthantheelectron,therecoilenergyoftheatomcanbeneglected.Atgamma-rayenergiesaround1MeV,thebindingenergyoftheouterelectronsismuchsmallerthanthephotonenergy,sothatalmostallofthephotonenergyistransferredtothekineticenergyoftheelectron.26ThesecondprocessisComptonscattering.TheprocessisillustratedinFigure2.2.Inthisprocess,aphotoniselasticallyscatteredo˙ofafreeelectron.Inreality,electronsaretypicallyboundwithinanatom,butthebindingenergyforouterelectronswherethescatteringusuallyoccursismuchsmallerthantheenergytransferredtotheelectron,sothisisneglected.Duringthescatteringprocess,thephotonisde˛ectedatsomeanglerelativetoitsinitialmotionandtransfersenergyandmomentumtotheelectron.Duetotheconservationofenergyandmomentum,theenergylossofthephotondependsonthescatteringangle.The˝nalenergyE0ofaphotonwithincidentenergyEisgivenbyE0=E1+(E=mec2)(1cos)(2.2)wheremec2=511keVistherest-massenergyoftheelectronandisthescatteringangleofthephoton.Theenergyoftherecoilingelectronisthedi˙erencebetweentheincidentand˝nalenergiesofthephoton:Ee=(E)2(1cos)mec2+E(1cos):(2.3)Twoextremesoftheelectron'senergycanbenoted.Forasmallscatteringangleˇ0,thephotonretainsalmostallofitsenergy,andEeˇ0.Atthemaximumscatteringangle,=ˇ,thephotontransfersthemaximumenergy,andEe=2(E)2mec2+2E:(2.4)Scatteringcanhappenatanyanglebetweenthesetwoextremes,andthedi˙erentialcross27Figure2.3:Aschematicofpairproduction.Here,theincomingphotoninteractswiththestrongelectric˝eldoftheatomicnucleusanddisappears,creatinganelectronandapositron.Theenergyofthephotonissplitbetweenthemassoftheelectron-positronpairandthekineticenergytransferredtotheelectronandpositron.Thenucleusalsogainsaslightrecoilmomentum.Afterbeingsloweddowninthesurroundingmaterial,thepositronannihilateswithasecondelectron,creatingtwophotonswithenergiesof511keV.Theseannihilationphotonsareemittedinoppositedirections.sectionforComptonscatteringisgivenbytheKlein-Nishinaformula[44]:d˙d=Zr2e11+(1cos)21+cos221+2(1cos)2(1+cos2)[1+(1cos)](2.5)whereZistheatomicnumberofthematerial,re=2:821015mistheclassicalelectronradius,and=E=mec2istheratioofthephotonenergyandelectronrest-massenergy.Afterscattering,thenewlower-energyphotoncanthengoontoscatteragainorbeabsorbed.The˝nalprocessispairproductionandisillustratedinFigure2.3.Here,aphotoninteractswiththestrongelectric˝eldwithinanucleusanddisappearswhilecreatinganelectron-positronpair.Apositronistheanti-particleoftheelectron,meaningithasthesamemass,size,andintrinsicspinastheelectron,butispositivelycharged.Thetotalkineticenergyoftheelectron-positronpairisequaltothedi˙erenceofthephotonenergyandtwicetheelectronrest-massenergy:KEe+KEe+=E1:022MeV:(2.6)28Becauseoftheenergythatgoesintocreatingtheelectron-positronpair,thisprocesscannothappenbelowphotonenergiesof1.022MeV,andtheprocessdoesnotbecomeimportantuntilseveralMeV.Asinthecaseofphotoelectricabsorption,thenucleusgainssomeoftheincidentmomentum,butitsenergycanbeignored.Theelectronandpositronthenloseenergyastheymovethroughthematerial.Whenthepositronisatrest,itthenencountersasecondelectron,andthetwoparticlesannihilateeachotherandcreatetwophotons.Thetwophotonsarebothcreatedwiththeenergyequaltotheelectronrest-massenergy,511keV.Additionally,becauseboththeelectronandpositronwereessentiallyatrestbeforeannihilation,theyhavezerototalmomentum,whichmeansthetwophotonsmustalsohavezerototalmomentum.Thus,thetwophotonsareemittedinoppositedirectionswithequalmomentum,andtheyeachhaveanenergyof511keV.TheseannihilationphotonsmaygoontointeractinthesurroundingmaterialviaphotoelectricabsorptionorComptonscattering,ormayescapefromthematerial.Themainconsequenceofexitingthematerialisthattheenergyoftheincidentphotonisonlypartiallydepositedinthesurroundingmaterial.AcomparisonofthecrosssectionsforthesethreeprocessesisshowninFigure2.4.Theattenuation(incm2/g)ingermaniumisshownasafunctionofenergyintherange1keVto100MeV.Itisclearthatatenergiesbelowabout100keV,thephotoelectricabsorptiondominates.Betweenabout200keVand1MeV,Comptonscatteringisthemostcommonprocess.AboveseveralMeV,pairproductionbecomesdominant.Mostnucleartransitionshaveenergiesbetweenabout50keVand5MeV,soallthreeprocessesarerelevantforgamma-rayspectroscopymeasurementsofatomicnuclei.29Figure2.4:Absorptioncrosssectionsinsolidgermanium,plottedasattenuation,forenergiesbetween1keVand100MeV.Theplotshowsthecrosssectionsforphotoelectricabsorption(red),Comptonscattering(blue),andpairproduction(magenta).Thesumofthethreeprocessesisshowninblack.DatafromtheXCOMdatabase[45].2.3RelativisticDopplere˙ectLifetimemeasurementsofexcitednuclearstatesaredonebymeasuringgammaraysemittedfromtheseexcitedstates.Thepresentexperimentinvolvesmeasuringgammaraysemittedfromaradioactivebeammovingatasigni˝cantfractionofthespeedoflight.Forsuchrelativisticsources,theemittedgammaraysaresigni˝cantlyDoppler-shiftedintheframe30ofthelaboratory.TheenergyofgammaraysobservedinalaboratoryframeisgivenbytheDoppler-shiftequation:Eobs=Ecm(1cos):(2.7)Here,Eobsistheenergyofthephotonobservedinthelaboratoryframe,Ecmisthetransitionenergyinthecenter-of-massframeofthemovingsource,=v=cisthespeedofthemovingsourcerelativetothespeedoflight,andistheangleofthephotonemissionrelativetothemotionofthesource,measuredinthelaboratoryframe.TheLorentzfactorisgivenby=1p12:(2.8)Theobservedenergyofagammaraydependsonboththedirectionitisemittedandthespeedofthemovingsource.Forsmallanglesofgammaemission(inthesamedirectionasthemotionofthesource),theenergyobservedinthelaboratoryframeisincreasedrelativetothecenter-of-massenergy.Atlargeanglesofemission(intheoppositedirectionofthemovingsource),thelaboratory-frameenergyisdecreasedrelativetothecenter-of-mass.Forlargerspeeds,thedeviationfromthecenter-of-massenergyincreasesatallangles,whileforsmallerspeeds(ˇ0),theobservedenergyapproachesthecenter-of-massenergy.Inordertodeducethetruedecayenergyofagammaray,aDoppler-shiftcorrectionisperformed,whichistheinverseofEquation2.7:EDC=Eobs(1cos):(2.9)Here,EDCistheDoppler-correctedenergy.Thiscorrectionismadeusingexperimentallydeterminedquantities.Theangleofemissionisdeterminedbytheinteractionpointwithina31detectorrelativetothebeamtrajectoryatthereactiontarget,andthespeedofbeamnucleicanbemeasuredafterthereaction.Whendealingwithagamma-raydetectorusedwithfastbeams,therearethreesourcesofuncertaintytothe˝nalenergyresolutionofDoppler-correctedgamma-rays.Thesee˙ectsincludetheuncertaintyinthevelocity()ofthebeam,theuncertaintyintheemissionangle()ofthegammaray,andtheintrinsicresolution(Eintr)ofthedetector[39].Thesethreee˙ectsareaddedinquadraturetogivethetotalenergyresolution:EE2=sin1cos2)2+cos(12)(1cos)2)2+EintrE2:(2.10)TheseareplottedinFigure2.5,wheretherelativecontributionsofeachtermareshownasafunctionoftheangleofgamma-rayemissionmeasuredinthelaboratory.ThevaluesusedinFigure2.5arebasedonthoseusedinRef.[39].Actualvaluesforeachtermdependontheexperimentalconditions,butgenerallythee˙ectsofeachtermareclearfromthe˝gure.Thecontributionfromtheangularuncertainty(E)islargestatcentralanglesofˇ40°°andvanishesatlargeandsmallangles.Theuncertaintyduetotheuncertaintyinvelocity(E)hastheoppositee˙ect:itislargestatlargeandsmallangles,andsmallestatcentralangles.Theintrinsicenergyresolutionofadetectordependsontheenergyofdetectedgamma-rays.BecausetheDoppler-shiftedenergydependsontheangle,thereisanangulardependenceoftheintrinsicresolution.Thisangulardependenceisminimal,however,andtheintrinsicresolutionissmallerthanthesumoftheothertwocomponentsatallangles.Ingeneral,allofthecontributionstotheuncertaintyarelargerathigherbeamvelocities,andforastationarysourceonlyaconstantintrinsicresolutionispresent.Thesensitivitytothedi˙erentcomponentsforfastbeamscanbeexploitedinwelldesignedgamma-ray32Figure2.5:AgraphshowingtherelativecontributionstheenergyresolutionasdescribedinEquation2.10.Theredlineshowstheresolutionduetotheuncertaintyintheangleofemission,thebluelineshowsthee˙ectduetotheuncertaintyinvelocity,andthegreenlineshowstheintrinsicenergyresolution.Theblacklineshowsthetotalresolutionwhenthethreee˙ectsareaddedinquadrature.Thecurvesarecalculatedwiththevalues=0:43,=0:008,=1.5°,andEintr=0:002MeVfora1MeVgammarayinthecenter-of-massframe.measurements,andthisformsthebasisforthelifetimemeasurementtechniquesdescribedinthefollowingsections.Inmanyexperimentalcases,theDoppler-correctedenergyisnotnecessarilyequaltothecenter-of-massenergyofthegammaray.Exploitingthedi˙erencebetweenthesetwovaluesiskeytothelifetimemeasurementtechniqueswhichhavebeendevelopedtostudyexcited-statelifetimesusingfastradioactivebeams.Thefollowingsectionsdiscusstwotechniquesusedinanexperimentwith19Cwhichwillbediscussedinthisthesis.Thetwomethods,theline-shapemethodandtheRecoilDistancemethod(RDM),relyonprecisemeasurements33ofDoppler-shiftedgammarays.Thetwomethodsarecomplementaryintheirsensitivitytolifetimes;theline-shapemethodisusefulforlifetimesontheorderof100ns,whiletheRDMisbettersuitedforshorterlifetimesontheorderof1ns.Inadditiontolifetimemeasurements,thetheoreticalframeworkfortheone-nucleonknockoutreactionispresented,whichwasusedtoproduce19Cinthesameexperiment.Whilethelifetimemeasurementswereusedexclusivelytostudythestructureoftheexcitedstatesin19C,theknockoutreactionwasusedtostudythegroundstateof20Naswellasalltheboundstatesin19C.2.4Line-shapeMethodThe˝rstpartoftheexperimentwasbasedontheso-calledline-shapemethod[46,47].Thistechniqueisbasedontheemission-pointdistributionofgammaraysemittedin-˛ightafterareactiontarget.Forabeamwithvelocitycandanexcited-statelifetime˝,gamma-raydecaysaredistributedexponentiallyalongthebeamline,andtheaveragedecaypositionzalongthebeamlineisgivenbyz=˝c:(2.11)Atavelocityofˇ0:3,thebeammovesabout1mmin10ps.Ifthelifetimeoftheexcited-statedecayismorethan100ps,thendecaysoccur,onaverage,severalcentimetersdownstreamofthetarget.Thisshiftinzresultsinachangeintheemissionangleofdetectedgammarays.ThechangeintheobservedenergydEresultingfromachangeintheemissionangleisgivenbydE=sin1cosE:(2.12)34IntheDopplercorrectionofexperimentallyobservedgammarays,theemissionangleiscalculatedusingthemeasuredinteractionpointofthegammaraywithinthedetectorandassumingthegammaraysareemittedfromthetargetposition.AnychangeinthezpositioncannotbetakenintoaccountintheDopplercorrection,sodecaysthatoccurdownstreamfromthetargetwillappearintheDoppler-correctedspectrumatenergiesbelowthetruedecayenergy.Becausethedecaysarespreadoutcontinuouslyacrossthepathofthebeam,theresultisashiftinthecentroidoftheDoppler-correctedpeakaswellasatailextendingtolowenergies.Bothofthesee˙ectsareincreasedforlongerlifetimes.Theline-shapee˙ectisdemonstratedinFigure2.6.InFigure2.6(a),threedi˙erentdecaycurvesforlifetimesbetween100psand1nsareshownforabeamexitingatarget.Figure2.6(b)showssimulatedDoppler-shiftcorrectedspectraforeachlifetime.Areferencespectrumforadecaywithnolifetimeisalsoplotted.Thepeakwithoutanylifetimee˙ectsisnarrowandsymmetric,whilepeaksarisingfromthelonglifetimesareshiftedtolowerenergiesandexhibitasymmetryduetolongtailsatlowenergy.Asthelifetimeincreases,thetailbecomeswiderand˛atter.Thesensitivityofthepeakshapestothelifetimedependsontheexperimentalconditions.Foradetectorwhichislocatedontheorderof10cmfromthebeamline,thee˙ectbecomesnoticeablewhentheaveragedecaypositionzisatleastseveralmillimetersbehindthetarget,whichcorrespondstoalifetimearound100psforˇ0:3.Theupperlimitofsensitivityisdeterminedbythesizeofdetectors,sothatiftheaveragedecaypositionislargerthanthedetectorsize,mostdecayscannotbedetected.Inthiscase,thepeakshapebecomes˛atandlosesallsensitivitytothelifetime.Foradetectorsizeofseveraltensofcentimeters,thepracticalupperlimitofthelifetimeforthismethodisaround10ns.35Figure2.6:Anillustrationoftheprinciplesoftheline-shapetechniquefordetermininglifetimes.In(a),threedecaycurvesforlifetimesintherangeof100nsareshownforabeamexitingareactiontarget.SimulatedDoppler-shiftcorrectedspectracorrespondingtoeachlifetimeareshownin(b),aswellasapeakforadecaywithnolifetimee˙ect(0ps).Forlongerlifetimes,thepeakinthespectrumisshiftedtolowerenergies,andabroadtailformsonthelow-energysideofthepeak.2.5RecoilDistanceMethodTheRecoilDistanceMethod[48]hasbeendevelopedtomeasurethelifetimeofnuclearstates.Themethodisbestsuitedforlifetimesbetween1psand1ns(1012109s)[49].Indeed,experimentsusingthetechniquehavespannedthisentirerange[50,51].ThissectiondescribestheprinciplesoftheRecoilDistanceMethod,andhowithasbeenadaptedforcurrentexperimentswithradioactivebeams.TheearlyapplicationoftheRecoilDistanceMethodtostudynuclearlifetimeswithlow-energyexperimentswithstablebeamsisdescribedinRef.[52]andillustratedinFigure2.7.36Themethodusesadevicecalledaplunger,whichholdsathintargetmaterialandathickerstoppermaterial,whichareseparatedalongthebeamline.Thedistancecanbeadjustedbymovingthestopperrelativetothe˝xedtarget.Anuclearreactionoccurswithinthetarget,creatinganucleusinanexcitedstate.Thenucleusthenexitsthetarget,movingtowardthestopperwhichisseparatedfromthetargetbyawellde˝neddistanceD.Thereactiononthetargetisde˝nedtohappenattimet=0,andthetimeof˛ighttf=D=vofthebeamacrossthetarget-stopperseparationdependsonDandthevelocityvofthebeamafterthetarget.Iftheaveragelifetime˝ofthenuclearexcitedstateissimilartothetimeof˛ight,thensomenucleiwilldecaywhilein˛ightinfrontofthestopper,andtheremainingnucleiwilldecayafterbeingstopped.Ifadecayoccurswhilethebeamisin˛ight,thegammarayenergyinthelaboratoryframewillbeDoppler-shiftedaccordingtoEquation2.7.Ifthedecayoccursafterthebeamhasstoppedinthetarget,therewillbenoDopplershiftintheenergy.ThustwopeaksappearintheDoppler-correctedgamma-rayspectrum:ashiftedandanunshiftedpeak.Theintensitiesofthetwopeaks(IsandIu,respectively)dependontherelativemagnitudesof˝andD=v:Is=N(1eD=v˝)(2.13)Iu=NeD=v˝(2.14)whereNisthetotalnumberofgammaraysemitted.Measurementofthepeakintensitiescanbemadeatseveraldistances,andtheratioR=IuIs+Iu=eD=v˝(2.15)37Figure2.7:AdiagramillustratingtheprinciplesoftheRecoilDistanceMethod.Afterreactinginsidethetarget,thebeamtravelsthedistanceDwithatimeof˛ighttf.Onceenteringthestopper,thebeamslowsdownduringtimets,withts˝tf.Adetectordetectsgammaraysemittedfromananglerelativetothebeam.Gammarays0emittedduringthetimeof˛ighttfareshiftedinenergy,whilegammarays0emittedafterthebeamisstoppedareunshifted,creatingtwopeaksatdi˙erentenergiesinthegamma-rayspectrum.Thelifetimeisdeterminedfromtheyieldoftheunshiftedpeakrelativetothetotalgamma-rayyield.FigureadaptedfromRef.[49].canbedeterminedasafunctionofD.Theslopeoftheexponentialcurvegivesthevalue1=v˝,sothat˝canbedeterminedwhenvisknown.Inastopped-beamexperiment,visdeducedfromtheenergyoftheDoppler-shiftedpeak[52].Withtheavailabilityoffastradioactive-beamfacilities,theRecoilDistanceMethodhasbeenadaptedtoimproveitscapabilitiesandsensitivities[49].Withbeamspeedstypicallyabovev=c˘0:3,thebeamcannolongerbestoppedwithintheplunger.Instead,thesecondfoilisusedasadegraderwhichlowersthevelocityofthebeam.Theprocessisillustrated38inFigure2.8.Inthiscase,theprinciplesofthemethodarethesimilartothelow-energyexperiments.Thedegraderdividesthebeamintotworegions:afastregioninfrontofthedegrader,andaslowregionbehindit.Thevelocityofthebeamineachregionisdistinct,creatingtwodi˙erentDoppler-shiftedenergies.Intheresultinggamma-rayspectrum,theseareseenasafastpeakandaslowpeak.Fortheoriginalmethod,noDopplercorrectionoftheobservedgamma-rayspectrumisnecessary,becausetheunshiftedportionofthespectrumappearsatthetruedecayenergy.Forexperimentswithafastbeam,however,boththefastandslowpeaksinthespectrumaresigni˝cantlyDoppler-shifted.Thus,aDoppler-shiftcorrectionisrequiredtorecoverthetruedecayenergy.Typically,individualgamma-rayenergiesarecorrectedusingEquation2.9byassumingthegammaraysareemittedatthefastvelocityimmediatelyafterthetarget.Withtheseassumptions,thefastpeakislocatedatthetruedecayenergy,andtheslowpeakbecomesshiftedtolowerenergies,becauseitsDopplercorrectionisperformedincorrectly.OneoftheadvantagesofusingtheRDMwithfastbeamsisthecapabilitytodetecttheoutgoingparticlesincoincidencewithgammarays.Theparticlescanbeidenti˝edonanevent-by-eventbasisusingtime-of-˛ightandenergy-lossmeasurements.Particleidenti-˝cationafterthereactionallowsforareductionofbackgroundduetotheeliminationofbeamcontaminantsinthe˝nalspectrum[49].Inaddition,thickertargetsmaybeusedwithfastbeams,whichallowsforfeasiblereactionratesevenforlowbeamintensities.Thereare,however,somecomplicationintheanalysisandinterpretationofthedatawiththeuseoffastbeams.Becauseofthetargetthickness,somedecaysmayoccurwithinthetargetorthedegrader,reducingtheintensityofthefastandslowpeaksinthegamma-rayspec-trum.Athighenergies,reactionsofthesecondarybeammayalsooccurwithinthedegrader,whichincreasestheyieldsfortheslowpeakonly.Thus,thelifetimecannotbeextractedin39Figure2.8:AschematicdemonstratingtheRecoilDistanceMethodusedwithafastra-dioactivebeam.Theuseofadegradertoslowdownthebeamcreatesfast(blue)andslow(green)regionsalongthebeampath.Astheseparationofthetargetanddegraderincreases,moregammadecaysoccurinthefastregion.Theresultinggamma-rayspectraforeachofthesettingsareshownontheright.40themannerdescribedabove.Instead,thelifetimeisdeterminedbycomparingthedatatosimulationsoftheexperiment.2.6Single-nucleonKnockoutSingle-nucleonknockoutreactionshavebecomeapowerfultoolinthestudyofthesingle-particlestructureofnuclei[23].Thesearedescribedasdirectreactions,inwhichaprotonorneutronisremovedinasinglestep[53].Inaknockoutreaction,aprojectilenucleusmovingathighenergy,typicallyabove˘50MeVpernucleon,encountersastationarytargetnucleus.Inthereaction,asinglenucleonisremovedfromtheprojectile.Inthesimplestapproximationofthisreaction,theprojectileisdescribedasasinglevalencenucleonandtheremainingnucleonsintheprojectile,wherethelattercomponentsarecalledacore.Duringthereaction,thevalencenucleonisremovedinstantaneouslyfromtheprojectilewhilethecoreremainsintact.Withintheframeworkofadirectreaction,thecoredoesnotinteractatallwiththetarget,excepttobeelasticallyscattered,sothattheinternalcon˝gurationofthecoreisnotchanged.Theknockoutreactionisdescribedtheoreticallyintheeikonal,orGlaubermodel[53,54].Withinthismodel,theprojectileisassumedtotravelinastraightlineasitpassesthetarget,anditswavefunction (~r)canbeseparatedintoaplanewaveandacylindricalterm:~r)=eikz (b;z)(2.16)wheretheprojectilemovesinthezdirectionwithmomentumk,andtheimpactparameterbisthedistancebetweentheprojectile'strajectoryandthetarget.TheHamiltonianincludesatermV(b;z),whichisthepotentialbetweenthetargetandprojectile.Applyingthe41Hamiltonianandignoringsecond-orderderivativesof givestheeikonalequation:@ @z=i~vV(b;z) :(2.17)Thishasthesolution (b;z)=ei˜(b;z)(2.18)wheretheeikonalphase˜(b;z)isde˝nedas˜(b;z)=i~vZzV(b;z0)dz0:(2.19)Fromtheeikonalphase,theelasticS-matrixSisde˝ned:S=ei˜(b;+1):(2.20)Theprobabilitytoscattero˙ofthetargetelasticallyisgivenbyjSj2,andtheparticlecanbeabsorbedbythetargetifjSj2<1,whichoccursforcomplexpotentials.Tocalculateknockoutcrosssections,theS-matricesScandSvarecalculatedseparatelyforthecore(c)andvalence(v)nucleon,respectively.ForcalculationoftheS-matrices,thepotentialV(b;z)isobtainedbyfoldingthetargetthetargetdensity,corenuclearorvalencenucleondensity,andanappropriatenucleon-nucleonpotential.ThetargetandcoredensitiesareusuallytakentobeaWoods-Saxonform,andthevalencenucleonisassumedtobeapointparticle.TherelativewavefunctionofthevalencenucleonwithinthecorenucleusiscalculatedinaWoods-Saxonpotentialwhichisadjustedtomatchtheexperimentalseparationenergyofthenucleonwithintheprojectile.42Therearemechanismsthroughwhichtheprojectilecanreactwiththetarget:strippinganddi˙ractivedissociation.Thestrippingreactionoccurswhenthevalencenucleonisab-sorbedbythetargetnucleus,whilethecoreisscatteredwiththeremainingnucleonsinthesamestateasbeforethereaction.Thisisaninelasticreactioninwhichthetargetdoesnotremaininitsgroundstate.Thecrosssectionforstrippingisgivenby˙str=12j+1ZXmh jmj(1jSvj2)jScj2j jmidb:(2.21)Withintheintegral,theterms(1jSvj2)andjScj2aretherespectiveprobabilitiesforthevalencenucleontobeabsorbedandthecorenucleustoscatterelastically.The jmisthewavefunctionofthenucleonrelativetothecorenucleus,andtheintegralistakenovertheimpactparameterbofthecenterofmassofthenucleon-coresystemrelativetothetarget.Di˙ractivedissociationoccurswhenboththevalencenucleonandtheremainingcorearescatteredelastically.Becausetheentirereactioniselastic,thetargetnucleusisleftintactinitsgroundstate.Thedi˙ractivedissociationcrosssectionisgivenby˙diff=12j+1ZXm;m0hh jmjj1ScSvj2j jmim;m0jh jm0j(1ScSv)j jmij2idb(2.22)Thisequationarisesfromthecompletenessoftheboundandunboundwavefunctionsofthescatteredparticlesafterthereaction,andavoidsintegrationofthestatesinthecontinuum.Thetotalsingle-particlecrosssection˙spisthesumofthestrippinganddi˙ractivedissociationcrosssections,additionallyincludingaCoulombbreakupterm˙C:˙sp=˙str+˙diff+˙C:(2.23)43TheCoulombbreakuptermisduetoastrongCoulombforcefromhigh-Ztargetsandisnotimportantforlighttargets,asisthecaseforthecurrentwork.Thesingle-particlecrosssectionisapplicableonlyforpuresingle-particlestates.Actualcrosssectionsmustbecorrectedtoaccountforthemixtureofdi˙erentcon˝gurationsinthewavefunction.Thesedi˙erentcon˝gurationsinboththeinitialand˝nalwavefunctionsmeanthatdi˙erentnucleonsmayberemovedtogivethe˝nalstate.Therelativecontribu-tionsforremovalofthevariousnucleonsisgivenbyaspectroscopicfactor.ThespectroscopicfactoriscalculatedfromtheoverlapofthewavefunctionsoftheinitialprojectilewithAnucleonsAiandthecorewavefunctionwithA1nucleonsA1f:hA1fjAii=Xn`jcfi(n`j) n`j(2.24)where n`jisthenormalizedwavefunctionforthevalencenucleonwithquantumnumbersn`j,andeachofthecfi(n`j)isacoe˚cientoffractionalparentagewhichdescribestherelativecontributionofeachsingle-particlewavefunctiontoconnecttheinitialand˝nalnuclearstates.ThespectroscopicfactorSfi(n`j)isthengivenbySfi(n`j)=cfi(n`j)2:(2.25)ThespectroscopicfactorcanbedescribedastheprobabilitythattheprojectileinaninitialstateAiwillformacoreina˝nalstateA1fbyremovinganucleonwithquantumnumbersn`j.Forremovalofanucleonoccupyingapuresingle-particlestate,Sfi(n`j)=1,andforremovalofanucleonfromafully˝lledorbital,Sfi(n`j)=2j+1,whichre˛ectsthetotalnumberofavailablenucleonsintheorbital.Ifthespectroscopicfactoriscalculatedinthe44isospinformalism,itmustbereplacedbyC2S(n`j;T),usuallysimplywrittenasC2S(n`j),whereCisanisospinClebsch-Gordancoe˚cient.Finally,ifthespectroscopicfactoriscalculatedinaharmonicoscillatorbasis,thereisanadditionalcenter-of-masscorrection[A=(A1)]Nwhichmustbeapplied[55],whereN=2n+`isthemajoroscillatorshelloftheremovednucleon.Inanactualknockoutexperiment,theinitialstateoftheprojectileis˝xed,anditissu˚cienttodenotethespectroscopicfactortothe˝nalstateofthecorewithS(Jˇ;n`j).Thetotalcrosssection˙thtoagiven˝nalstateisgivenby˙th=Xn`jAA1NC2S(Jˇ;n`j)˙sp(n`j;Seffn):(2.26)Thiscrosssectionisanincoherentsumofallsingle-particlecrosssectionsforthenucleonwavefunctionswhichcanconnecttheinitialprojectileand˝nalcorestates,normalizedbyeachofthespectroscopicfactors.Ingeneral,boththestrippinganddi˙ractivedissociationcrosssectionsarecalculatedbasedonremovinganucleonwithane˙ectiveseparationenergySeffn:Seffn=Sn+Ex(Jˇ):(2.27)Thee˙ectiveseparationenergyisbasedontheseparationenergySnoftheremovednucleonfromthegroundstateoftheprojectilenucleus,andleadsastateJˇinthe˝nalnucleuswithexcitationenergyEx(Jˇ).45Chapter3ExperimentalDevicesThepresentexperimentalworkwasperformedattheNationalSuperconductingCyclotronLaboratory(NSCL)atMichiganStateUniversity.InexperimentsattheNSCL,aprimarybeamofstablenucleiisacceleratedbytheCoupledCyclotronFacility(CCF)[56]inordertoproducearadioactivesecondarybeam.Thesecondarybeamisanalyzedandpuri˝edbytheA1900fragmentseparator[57]anddeliveredtotheexperimentalarea.Secondaryreactionswiththeradioactivebeamoccuronatargetintheexperimentalarea,and˝nalreactionproductsareseparatedandidenti˝edbytheS800spectrograph[58].Thedetailsofeachoftheexperimentalstepsaredescribedinthefollowingsections.3.1BeamProductionAttheNSCL,radioactiveionbeamsareproducedasreactionproductsofstablebeamsonastabletarget.Theaccelerationofthestablebeambeginswiththeionizationofstableisotopes.TheNSCLhastwomachineswhichperformthisfunction,theSuperconductingSourceforIons(SuSI)[59,60]andtheAdvancedRoomTemperatureIonSource(ARTEMIS-B)[61].Inbothcases,thebeamisextractedfromastablesourceandinjectedintoaplasmaforionization.Gaseousspeciesareextracteddirectlyfromagassource,whilesolidsourcesareheatedinanovensothattheatomsevaporateintotheair.Withintheplasma,thestableatomsareionizedthroughmultiplecollisionswithenergeticelectronswhichare46Figure3.1:TheCoupledCyclotronFacilityattheNSCL.AstablebeamiscreatedatanionsourceandacceleratedintheK500andK1200cyclotrons.Thebeamthenreactsattheproductiontargettocreatearadioactivesecondarybeam.ThesecondarybeamcontinuesthroughtheA1900andisthensenttotheexperimentalarea.FigureadaptedfromRef.[57].acceleratedusingelectroncyclotronresonance.Inthismethod,theelectronsintheplasmaareacceleratedbyapplyingmicrowaveradiation,andtheelectronsarecon˝nedtoacircularorbitbyaconstantmagnetic˝eld.TheprocesscanbeunderstoodbyequatingtheLorentzforceonamovingelectroninamagnetic˝eldandthecentripetalforceoftheresultingcircularmotion:qvB=mev2ˆ:(3.1)Here,qisthechargeoftheelectron,visthevelocityoftheelectronperpendiculartothemagnetic˝eld,Bisthemagnetic˝eldstrength,mistheelectron'smass,andˆistheradiusofrotation.Rearrangingthisequationyieldsthecyclotronfrequency!c:!c=qBme(3.2)which,intheclassicallimit,isindependentoftheenergyoftheelectronandtheradiusofmotion,allowingtheplasmaelectronstobeexcitedmanytimesatthesameelectromagnetic47frequency.Afterbeingionized,thestableatomsaresenttotheCoupledCyclotronFacilityforacceleration.TheCoupledCyclotronFacility[56]comprisestwocyclotrons:theK500andK1200.ThesecanbeseenontheleftsideofFigure3.1.Thepartiallyionizedatomsare˝rstinjectedintotheK500,wheretheyareacceleratedbasedonthecyclotronmotionoftheions.Theionsarecon˝nedwithinthecyclotronbyamagnetic˝eld,andacceleratedbetweenthreesetsofelectrodeswithradio-frequencyalternatingelectric˝elds.Astheionsgainenergy,theyremainincircularmotionduetothemagnetic˝eld,buttheradiusˆoftheirorbitincreaseswithincreasingenergy:ˆ=mvqB:(3.3)Oncethebeamreachesthemaximumenergyofabout10MeV/nucleon,orthevelocityv=cˇ0:15,itexitstheK500throughaportontheouteredgeofthecyclotron.ThebeamistransportedthroughacouplingbeamlinetotheentranceoftheK1200cyclotron,whereathincarbonfoilremovesallormostoftheremainingelectrons.AfterenteringtheK1200,theaccelerationprocessbeginsagain,andthebeamisaccelerateduptoa˝nalenergyofMeV/nucleon,orv=cˇ0:5.Aftertheionbeamisfullyacceleratedbythecyclotrons,itisejectedfromtheK1200andimpingedonaproductiontargettoproduceseveralradioactivespecies.Radioactiveionsareproducedinthetargetprimarilythroughprojectilefragmentationofthestablebeamnuclei.Duringtheprojectilefragmentationprocess,aprefragmentis˝rstproducedinanexcitedstate.Afterthereaction,nucleonsarestatisticallyemittedfromtheprefragment,resultingina˝nalfragmentwithanynumberofprotonsorneutronsfewerthantheincidentnucleus.Afteremergingfromthetarget,thebeamiscomposedofmanydi˙erentnuclearspecies,48includingstableandradioactivespecies.ThisisshowninFigure3.2(a),whichshowsanexampleofallthenucleiproducedbyfragmentationofastablebeamof86Kr,drawnonthechartofnuclides.Toproducethesecondarybeamthatissuitableforexperiments,thedesiredisotopemustbe˝lteredfromalltheremainingfragmentationproducts.Thisisdonethroughisotopicseparationofthebeam[62].ThebeamisdirectedintotheA1900fragmentseparator[57],whichconsistsoffoursuperconductingdipolemagnetsaswellasanaluminumwedgewhichtogetherachieveisotopicseparation.Inthe˝rststepofseparation,thesecondarybeampassesthroughthe˝rsttwomagnets,whichachieveseparationbasedonthemagneticrigidityBˆofthebeam.BasedonEquation3.3,themagneticrigidityofthebeamisBˆ=mvq(3.4)whichdependsonthemomentummvandchargeqofthenucleus.Becausethevelocityoftheparticlesremainsnearlyconstantduringfragmentationreactions,themagneticrigidityselectionisessentiallyaselectioninthemass-to-chargeratioA=Zofthebeam[62].TheresultofthiscanbeseeninFigure3.2(b),whichshowsthenucleialongalineofconstantA=Zwhichpassthroughthe˝rsttwodipolemagnetsoftheA1900.ThenextstageinseparationoccursthroughtheuseofanaluminumdegraderlocatedbetweenthesecondandthirdmagnetsoftheA1900.Thedegraderisintheshapeofawedgetoprovideavariablethicknesswhichisadjustedforeachexperiment.Thisisusedtoselectivelydecreasethevelocityoftheproducts.TheenergylossofaheavychargedparticlethroughamaterialisgivenbytheBethe-Blochformula[44]:dEdx=4ˇe4q2mv2NZln2mv2Iln1v2c2v2c2:(3.5)49Figure3.2:Theproductionratesofnucleifollowingthefragmentationofanenergetic86KrbeamatvariouspointsintheA1900.Part(a)showstheproductionrateimmediatelyaftertheproductiontarget.Part(b)showstheratesafterthe˝rstselectionbasedonthemagneticrigidityofthebeam.ThestraightlineofacceptednucleirepresentstheconstantA=Zratiowhichisisolatedaftertravelingthroughthe˝rsttwodipolemagnetsoftheA1900.Part(c)showsthetransmittedbeamattheendoftheA1900,wherethereisaclearselectionofthemassandchargeofinterest.Here,q,m,andvarethecharge,mass,andvelocityoftheheavyparticle.Zistheatomicnumberofthesurroundingmaterial,andIisanempiricallydeterminedparameterwhichrep-resentstheaverageionizationpotentialofthematerial.Afterpassingthroughthe˝naltwomagnets,thebeamisagainseparatedbymagneticrigidity.Inthiscase,theparticlesdonothavethesamevelocity,andthebeamisselectedbasedonaconstantratioofA2:5=Z1:5[62].ThisisseeninFigure3.2(c),wherethe˝nalacceptanceislimitedtoasmallrangeofnuclei.Foragivenexperiment,themagnetic˝eldsanddegraderthicknessareadjustedtoproduceasecondarybeamwiththeoptimalbalanceofenergy,intensity,andpurity.3.2TRIPLEXdeviceAtthemainexperimentalarea,theTRIplePLungerforEXoticbeams(TRIPLEX)[63]wasusedtoprovideatargetanddegrader.ThisdevicewasrecentlydevelopedforlifetimemeasurementsattheNSCL.TheTRIPLEXisamodi˝edversionoftheso-calledsingleplungerandcanholduptothreemetalfoils.Theadditionofaseconddegraderallowsfornewtypesoflifetimemeasurementsandextendsthesensitivitytolifetimes[63,64].Aphoto50Figure3.3:ApictureoftheTRIPLEXplungerattheNSCL,withadiagramonthebottom.Thediagramshowsthevariouscomponents:(A)theoutersupportframe,(B)oneofthemotors,(C)theoutertubewhichconnectstotheseconddegrader,(D)thecentraltubewhichconnectstothe˝rstdegrader,(E)theinnertubewhichconnectstothetarget,(F)thetargetcone,(G)the˝rstdegradercone,and(H)theseconddegradercone.Theradioactivebeamsenterstheplungerfromtheleftandencountersthefoilsonthefarright.Thepictureshowstheplungerwiththefoilframesremoved.FigurefromRef.[63].oftheTRIPLEXdeviceandacorrespondinglabeleddiagramareshowninFigure3.3.TheTRIPLEXconsistsofthreemainparts:asupportstructure,bearingunit,andfoilunit.Thissectiondescribeseachpartoftheplungeraswellasitsexperimentaloperation.ThesupportstructureconsistsofanouterframewhichisattachedtoaninnersupportringaswellasadedicatedvacuumpipewhichhousestheTRIPLEXplungerduringexperi-ments.Thesupportringisthefoundationforalloftheimmobilecomponentsintheplunger.Theseincludethecentralplungertubewhichholdsthe˝rstdegraderfoilandthemotorswhichmovetheinnerandoutertubes.Duringexperiments,theplungerisplacedwithinthevacuumpipeasshowninFigure3.4.Sixscrewsholdtheouterframeinplaceandcanbeadjustedtoaligntheplungerfoilstothebeampath.Thebeamchamberalsocontains51Figure3.4:TheTRIPLEXplungerlocatedinsidethededicatedvacuumchamber.Electricalfeedthroughsforcontrolandmonitoringofthedevicearevisiblein(a).Theclose-upviewin(b)showsthescrewswhichareusedtoaligntheplungertothebeampath.FigurefromRef.[63].twosetsofelectricalfeedthroughs,whichallowsthecablingforcontrolandmonitoringoftheplungertobeconnectedthroughthebeampipe.Thebearingunitoftheplungercomprisesthreeconcentrictubeswhichattachtothethreeplungerfoils.ThetubescanbeseeninFigure3.5.Themiddletubeisattachedtotheinnersupportringandis˝xedinplace.Theinnerandoutertubesareheldinplacebyfourslidingbearingswhichareattachedtothemiddletube.Theinnerandoutertubescanbemovedalongthecentralaxistoadjustthefoildistances,andareattachedtotwoindependentmotors.Eachmotorisconnectedtothetubeviaasmallwireattachedtoaringclampedaroundeachmovabletube.Thisallowsthetubestobemovedwithout52Figure3.5:ThethreetubeswhichcomprisethebearingunitoftheTRIPLEXplunger.Thestationarymiddletubeisattachedtotheinnersupportringandconnectstothe˝rstdegraderfoil.Theinnerandoutertubesmovealongthebeamdirectionandconnecttothetargetfoilandseconddegraderfoil,respectively.FigurefromRef.[63].introducinganyradialforcewhichcoulddeformthetubes.Themovablerangeofeachmotorisapproximately2.5cm.ThefoilunitislocatedatthedownstreamendoftheTRIPLEXandcanbeseenontherightsideofFigure3.3.Attheendofeachtubeisabrassringwhichholdsaframetomountthetargetanddegraderfoils.Eachfoilismountedonaspeciallyshapedcone.Theconesareattachedbysmallscrewsandareheldinplacebysprings.Thecompressionofeachspringisadjustedtoensurealignmentofthefoilfaces.AsseeninFigure3.3,thesecond-degraderconehasalonger,narrowshapewhich˝tsinsidethe˝rst-degraderconesothatthesecond-degraderfoilcancomeintocontactwiththe˝rst-degraderfoil.Thisrequiresthatthesecond-degraderfoilbecutintoacircleto˝t.However,thereisnosuch˝ttingrequirementforthetargetand˝rst-degraderfoils,sotheseareusuallysquare-shaped.Theinnerdiameteroftheseconddegraderisabout4.6cm,whichislargeenoughforalloftheradioactivebeamtopassthrough.Accuratepositioningoftheplungerfoilsisvitalforrecoildistanceexperiments,especiallyforshort-lifetimemeasurements.Thepositionsofthetargetandseconddegraderfoilsaredeterminedfromthreeindependentmeasurements.First,themotorswhichcontrolthe53motionoftheinnerandoutertubesprovidethedisplacementinformationofthetubeswithsub-mresolution.Themaximumrangeofeachmotorisapproximately3cm;however,dependingonthecon˝gurationofthefoils,theachievablefoilseparationisusuallyreducedto2.5cm.Additionally,twomicrometerprobeshavebeeninstalledintheplungersupportwhichmeasurethedisplacementoftheinnerandoutertuberingsoverasmalldistance.Theseprobeshavearangeofonlyabout2mm,muchsmallerthanthetotalrangeofthemotors,buthaveasimilarmeasurementprecision.However,thesecanbeusedtoverifythemotordisplacementreadingsatsmalldistances,wheretheaccuratemeasurementismostimportant.Ithasbeenfoundthattheprobesshowsmalldeviationsinthedisplacementontheorderof5mattheedgeoftheirranges.Therefore,theseareonlyusedinalimitedcontextfordistanceveri˝cation.The˝nalmethodtodeterminefoilseparationisbasedoncapacitancemeasurements.Individualfoilsareelectricallyisolatedfromeachotherandfromtherestoftheplunger.Therefore,twofoilsplacedatsmalldistancesfromeachotheractasaparallelplatecapacitor.Thepositionofthefoilswhereelectricalcontactoccursde˝nesthesettingforzerodistanceofthefoils.Anyo˙setfromthetruezerodistanceisdeterminedthroughbymeasuringthecapacitanceatseveraldistancesnearthezero-distancesetting.Avoltagepulseissentintothe˝rstdegrader,andtheinducedvoltageinthetargetorseconddegraderisreadout.Foraparallelplatecapacitor,thecapacitance,andthereforetheinducedvoltage,isinverselyproportionaltothefoilseparation.A˝ttothecalibrationdeterminesthepointatwhichtheinducedvoltagediverges,andthedi˙erencebetweenthispointandthezero-distancesettingobtainedfromtheelectricalcontactdeterminestheo˙setforthetruezerodistance.Thiscalibrationmustbedoneforeachexperimentandeachsetoftargetanddegraderfoils.Theo˙setforatypicalexperimentisontheorderof10m.Thecapacitancemeasurementalsoservesasawaytoalignthefoils.Byadjustingthe54compressionofthesprings,thecapacitancebetweentwoadjacentfoilscanbemaximized,andthemaximumcapacitancecorrespondstothebestparallelalignment.3.3Gamma-raydetectionGammaraysemittedfromtherecoilingnucleiweredetectedbytheGamma-RayEnergyTrackingIn-beamNuclearArray(GRETINA)[40].GRETINAconsistsof7detectormod-ules,eachwith4high-puritygermaniumcrystals.Eachcrystalhasadeformedhexagonalface,andthefacesaredesignedto˝taroundaspherewitharadiusof18cm.Thedetectorcrystalsareeachabout7cmwideand9cmdeep,andadjacentcrystalswithineachmodulehavelessthana2.7mmgapbetweenthem.ThearrangementofthefourcrystalswithineachmodulecanbeseeninFigure3.6(a),whichshowsthetwotypesofhexagonsused.The28crystalscoveredasolidangleof1ˇ,orabout25%ofthesphere.ApictureoftheGRETINAcrystalssurroundingthebeampipeisfoundinFig.3.7.Thepictureshowshowthehexagonalcrystals˝ttogetheraroundthesphere.GRETINArepresentsalargeadvancefrompreviousdetectorsbecauseofitsexcellentpo-sitionandenergyresolution.Becauseofthestrongdependenceoftheobservedenergyontheemissionangle(Equation2.7),determinationofthegammay-rayinteractionpositionwithinthecrystalisvitalindeterminingtherest-frameenergyofthegammaray.ForGRETINA,thetypicalpositionresolutionachievedisabout2mm,correspondingtoanangularresolu-tionofabout10mrad.Thisresolutionisachievedthroughtheuseofelectricalsegmentationofthedetectorcrystals.Eachdetectorhas36individualelectricalcontactsaroundtheout-sideofthecrystal.Thesearearrangedinsixrowsalongthelengthofeachcrystal,andsixsectionsaroundtheedges.ThesegmentationpatternisshowninFigure3.6(b).When55Figure3.6:ThedesignofthedetectorcrystalsinGRETINA.Part(a)showsthetwoshapesusedto˝tthedetectorfacesaroundacircle.Part(b)showshoweachcrystaliselectricallysegmentedintosixradialsectionsandsixlayersdeep,creating36individualsegmentsusedforpositionresolution.FigureadaptedfromRef.[40].aninteractionoccursinonesection,thereleasedelectronsareallcollectedatasinglean-ode.Thetotalchargecollectedisthenreado˙fromthatanode.Additionally,anodesinadjacentsegmentswillrecordimagechargesiftheelectronsarecollectednearthatsegment.Therelativesizeoftheimagechargesonadjacentsegmentsdependsonthelocationofthegamma-rayinteraction.AlgorithmsusedbytheGRETINAsoftwarecanreconstructtheoriginalinteractionpositionwithinthesegmentbasedontheimagechargesofadjacentseg-ments.Acylindricalborethroughthecenterofeachcrystalcontainsthecentralcontactinthemiddleofeachcrystalwhichrecordsthetotalenergydeposited.GRETINAwasdesignedspeciallyforgamma-raytracking,whichisthereconstructionofthepathofthegammarayaftermultipleComptonscatteringevents.Thetrackingprocessisdoneinseveralsteps.First,individualinteractionpointsaregroupedaccordingtothelikelihoodthattheycamefromthesamephotonoriginatingfromthetarget.Withinthe56Figure3.7:ThebeampipeusedintheexperimentshowingtheGRETINAclustersaroundthedownstreamportion.Thedetectorsontopshowhowadjacentmodules˝ttogether.Theradioactivebeamcomesinfromthebottomright.group,theorderingoftheinteractionsisdeterminedbycomparingeverypossiblesequenceofinteractionsusingtheComptonequation(Equation2.2).Inthecaseof19C,thereisonlyasinglegammaraynear200keV.AtthecurrentbeamenergywiththepositioningofGRETINAandtheTRIPLEX,theresultinginlab-frameenergieswerebetween100and300keV.Attheseenergies,mostgammaraysdepositalloftheirenergyinasinglephotoelectricevent,andveryfewgammaraysareComptonscattered.Therefore,trackingwasnotimplementedintheanalysisof19Cdata,sincetrackingisdesignedtoreconstructCompton-scatteredgammarays.Instead,thedatawerecutonagamma-rayhitmultiplicityofone,meaningonlythoseeventswithasinglehitinGRETINAwereused.Thishadthee˙ectofreducingthegamma-raybackgroundwithoutsigni˝cantlyreducingcountsinthepeak.573.3.1CalibrationsCalibrationofGRETINAservestwopurposes.First,theenergycalibrationensuresthatthesignalsreadoutfromthedetectorscorrespondtothetrueenergydepositedbygammaraysinthedetector.Thisisdonebyusingstandardsources:smallsamplesofradioactiveisotopeswhichemitgammaraysatwellknownenergies.Afterasourcehasbeenmeasuredwiththedetectors,thesignalresponsescanbe˝ttedwithapolynomialtomapeachdetectedpeaktothecorrectenergy.ThisisdoneforeachcrystalofGRETINA,sincethecentralcontactofeachcrystalreadsoutthetotalenergyforthatcrystal.Forthecurrentexperiment,nocorrectionsneededtobemadetotheread-outenergies,becausetheexperimentwasruninthemiddleofalongcampaignwithGRETINA,sothedetectorsweremaintainedinacalibratedstate.Thesecondpurposeofthecalibrationsistoperformameasurementofthee˚ciencyofthedetectorswhichprovidesanessentialcalculationoftheprobabilityofdetectinggammaraysasafunctionoftheirenergy.Thee˚ciencycalibrationisvitaltotheknockoutreactionmeasurement,becausecalculationofexclusivecrosssectionstoexcitedstatesdependsonpreciseknowledgeofthedetectione˚ciency.Forpropere˚ciencycalibration,twopiecesofinformationarenecessary:thetotalactivityofthestandardsourceandtherelativeintensityofeachofthegammaraysoftheisotope.Inthisexperiment,twostandardsources,133Baand152Eu,wereusedforthee˚ciencycalibration,whichexhibitseveralgamma-raypeaksintheregionbelow500keV.Themeasurede˚ciencyfortwodi˙erentTRIPLEXcon˝gurationsareshownbytheredandbluepointsinFigure3.8.Additionally,simulatede˚cienciesareshownbytheblackpoints.Theprocessforsimulatingthee˚cienciesisdescribedinSection3.5.Figure3.8(a)showsthee˚ciencymeasuredintheline-shapecon˝gurationwith58Figure3.8:Resultsofthee˚ciencycalculationsusing(a)theline-shapecon˝gurationwithasingleberylliumtarget,and(b)theRecoilDistanceMethodsetup,withaberylliumtargetandtwotantalumdegraders.Thestandardsourcesinclude152Eu(redsquares)and133Ba(bluetriangles)forgammaraysbelow500keV.Simulatede˚cienciesarecomparedtothedatainblack.Unscaledsimulatede˚cienciesareshownbysolidcircles,andin(b),theopencirclesshowthesimulationscaledby0.91.Onlythee˚ciencyfortheline-shapesetupisusedinthecalculationofpartialcrosssectionsintheknockoutreactionstudy.onlyaberylliumfoilplacedinthetargetpositionoftheTRIPLEX.Figure3.8(b)showsthee˚ciencymeasuredwithaberylliumtarget,tantalum˝rstdegrader,andtantalumseconddegraderattachedtotheTRIPLEX.Thissecondcon˝gurationwasnotusedtotakedatainthecurrentexperiment,butmatchedthesettingsusedforasimilarexperimentwith17C[51].Inbothcases,thecalibrationistakenwiththesourceattachedtothedownstreamedgeofthetarget.InFigure3.8(b),thereisasharpdropine˚ciencyatenergiesbelow120keVduetothestrongattenuationofgammaraysthroughthethickdegraderswhicharenotpresent59intheprevioussetting.3.4Charged-particleDetectionAfterthesecondarybeamreactsintheTRIPLEXtarget,recoilingproductsaredetectedintheS800spectrograph[58].TheprimaryfunctionoftheS800istoidentifyparticlesthroughposition,momentum,andtimingmeasurements.TheS800achievesthisthroughitssuiteofparticledetectors.Inthepresentwork,themomentumoftherecoiling19Cparticleswasalsousedtoquantifytheknockoutreactionfromthe20Nsecondarybeam.ThissectiondescribesthedetailsoftheparticledetectorsintheS800andthecalibrationsusedtomakemeasurements.3.4.1S800SpectrographThelayoutoftheS800canbeseeninFigure3.9.TheS800consistsoftwoparts:ananalysislineandthespectrographitself.TheradioactivesecondarybeamenterstheanalysislinefromtheA1900andissenttotheexperimentalarea.Finalproductsarecollectedinthefocalplaneattheendofthespectrograph.TheS800canoperateintwodistinctmodes.Inthefocusedmode,thebeamisfocusedatthetargetposition,whileatthefocalplanethebeamisdispersedbasedontheenergyspreadofthereactionproducts.Inthismode,thereisalargeacceptance(2.5%)oftheincomingmomentum,butthetotalenergyresolutionislimitedbythemomentumwidth.Indispersionmatchedmode,thebeamisdispersedatthetargetposition.ThislowersthemomentumacceptanceoftheS800to0.25%,butsigni˝cantlyincreasestheenergyresolution.TheS800achievesparticleidenti˝cationmeasurementsthroughalargearrayofparticle60Figure3.9:TheS800spectrograph.ThesecondarybeamarrivesfromtheA1900attheobjectplane,andissenttothetargetarea,wheretheexperimentaltargetanddetectorsarelocated.Afterthetarget,˝nalproductsaresentthroughspectrographandanalyzedinthefocalplane.FigurefromRef.[65].detectors[66].AttheobjectplaneoftheS800(seeFigure3.9)isaplastictimingscintillatorwhichrecordsthetimingofeachincomingsecondarybeamparticle.Afterthetarget,therearetwodipolemagnetswhichseparatethe˝nalreactionproductsfromunreactedsecondarybeamnucleibytheirmagneticrigidity.ThefocalplaneoftheS800islocatedafterthemag-nets,andseveraldetectorsarelocatedtherewhichprovideidenti˝cationof˝nalproductsafterreactionofthesecondarybeam.AdiagramofthefocalplanedetectorscanbeseeninFigure3.10.ACathodeReadoutDriftChamber(CRDC)islocatedatthefocalplanepositionandmeasuresthepositionofparticlesatthefocalplane.AsecondCRDClocatedonemeterdownstreamofthefocalplaneisusedinconjunctionwiththe˝rstCRDCtoprovidethefulltrajectoryofbeamparticlesafterthetarget.AftertheCRDCsisanioniza-tionchamberwhichrecordstheenergylossofthebeam.ImmediatelyaftertheionizationchamberisanE1scintillator,whichprovidesatime-of-˛ightmeasurementfromtheobject61planescintillatorandisusedasatriggerfortheS800readout.BehindtheE1scintillator,ahodoscopecomposedofof32CsI(Na)crystalscanbeusedforcharge-stateidenti˝cationofthestoppedbeam.Allofthesedetectorsprovideevent-by-eventparticleidenti˝cation,meaningthateachparticlethatreachesthefocalplaneoftheS800isindividuallyrecorded.3.4.1.1TimingScintillatorsWithintheS800,beamparticlesencountertwoplasticscintillatordetectors.Whenachargedparticlemovesthroughascintillator,moleculeswithintheplasticareexcitedaselectronsareknockedoutbytheenergeticparticles.Themoleculesthendecaybackintothegroundstate,emittingphotons.Thislightiscollectedattheendsofthedetectorbyaphotomultipliertube,whichrecordsthetotallightoutput.Whilethesedetectorshavepoorenergyresolution,theyhaveverygoodtimingresolution.Thesearethereforeusedtomeasurethetimeof˛ightofthebeamacrosstheS800.Thetimeof˛ightofthereactionproductsmeasuredbetweentheobjectscintillatorandtheE1scintillatorcanbeunderstoodfromthemagneticrigidityoftheS800spectrograph:Bˆ=mvqˇAvZ=AZdT:(3.6)Becauseeachnucleustravelsthesamedistancedbetweenthescintillators,foragivenmag-neticrigidity,thetimeTtakenbyeachparticleisproportionaltoA=Z.Therefore,thetime-of-˛ightmeasurementisusedwithZ-identi˝cationtodeterminethe˝nalreactionproducts.TheA1900hassimilartimingscintillatorswhichareusedtoidentifytheincomingsecondarybeam.623.4.1.2CathodeReadoutDriftChambersAtthefocalplanearetwoCRDCs,whicheachmeasurethex(dispersive)andy(non-dispersive)positionsofthereactionproducts.ThesedetectorsareshowninFigure3.10.Eachdetectorhasanactiveareaof3059cm2andadepthalongthecentralbeamaxisof1.5cm,andthetwodetectorsareseparatedbyadistanceof1malongthebeamline.TheCRDCsare˝lledwithagasmixtureof80%CF4and20%C4H10.Asabeamparticlepassesthroughthegas,itknocksoutelectrons,ionizingthegasmolecules.Theelectronsthendriftacrossanappliedelectric˝eldandarecollectedbyananodewireorientedalongthexdirection.Theanodewireisarrangedalongaseriesof224cathodepads,andthexpositionisdeterminedbyreadingtheinducedchargeonthecathodescausedbytheaccumulatingelectrons.Achargeisinducedonseveraladjacentpads,andtherearedi˙erentwaystodeterminethepositionwherethebeamparticlepassedthrough.Inthiswork,thepositioniscalculatedbycalculatingtheweightedaverageoftheinducedchargelocation.Intheanalysis,thepadwiththehighestinducedchargeisfound,andtheaveragepositionoftheinducedchargeiscalculatedusingthe˝vepadsoneithersideofthecentralpad,witheachpadweightedbytheirinducedcharge.Thedistributionofchargescanalsobe˝tusingaGaussiandistribution,withthecentroidofthe˝tusedasthexposition.Theypositionisdeterminedfromthedrifttimeoftheelectronsmovingtowardtheanode.ThisismeasuredrelativetothetimingsignalfromtheE1scintillator.OncethexandypositionsofthebeamparticlesareknownatbothCRDClocations,theangleofthebeampathalongeachdirectioncanbecalculated.ThepositionandangularinformationfromtheCRDCsisusedtocalculatethebeampathatthetargetlocation.ThisisdescribedinSection3.4.1.4.63Figure3.10:ThefocalplanedetectorsoftheS800.TheCRDCsmeasurethexandypositionsofthebeamrelativetothecentralaxis.Theionizationchambermeasuresenergyloss,andtheplasticE1scintillatormeasuresthetimeof˛ightofthebeamandisusedasatrigger.Thehodoscopebehindthescintillatorcanbeusedtotaglong-liveddecays,butwasnotusedinthepresentwork.FigurefromRef.[67].3.4.1.3IonizationChamberAnionizationchambertomeasuretheenergylossofbeamparticlesislocatedbehindthesecondCRDC.Thechamberisagas-˝lleddetectorabouttentimesasthickalongthebeampathastheCRDCs.Itis˝lledwithP10gas,whichconsistsof90%Arand10%CH4.JustlikeintheCRDCs,beamparticlesloseenergyacrossthedetectorthroughcollisionswithgasmolecules,creatingionsbyknockingoutelectrons.Thegasissurroundedby16anode-cathodepairswhichcollecttheelectronsandgasions,andthetotalamountofchargecollectedisproportionaltotheenergylossofthebeam.Forafullyionizedparticle,theenergylossofbeamparticlesisproportionaltothesquareoftheatomicnumberofthe64particle(Equation3.5),andtheenergylossservesasaZ-identi˝cationof˝nalproducts.Thisisusedincombinationwiththetimeof˛ightmeasuredbythescintillatorstouniquelyidentifythemassandchargeofallparticlesintheS800.3.4.1.4TrajectoryReconstructionDeterminationofthebeampositionandangleatthelocationofthereactiontargetisnec-essaryforproperDopplercorrectionofthegamma-rayspectrum.Thelab-frameenergyofemittedgammaraysdependsontheangleofemissionrelativetothemotionofthemovingsourcenucleusaswellasthespeedofthesource.Therefore,Dopplercorrectionofobservedgammaraysdependsonboththelocationofthegammarayinteractionwithinthedetectorandthedirectionofthemovingnucleus.DeterminationofthebeamtrajectoryatthetargetlocationisdoneusingthecodeCOSYIn˝nity[68]andisbasedonthepositionmeasurementsfromtheCRDCsinthefocalplaneoftheS800.Thequantitieswhichareusedinthiscalcu-lationarethedispersiveposition(xfp),dispersiveangle(afp),non-dispersiveposition(yfp),andnon-dispersiveangle(bfp),measuredatthefocalplane.Thexfpandyfparesimplythepositionsmeasuredinthe˝rstCRDC.TheanglesatthedispersiveplanearecalculatedfromthexandypositionsofbothCRDCsandthe˝xedgap(1m)betweenthem:afp=tan1x2x11m(3.7)bfp=tan1y2y11m:(3.8)Here,x1andy1arethepositionmeasuredinthe˝rstCRDC,andx2andy2arethepositionmeasuredinthesecondCRDC.Fromthesequantities,theparticletrajectoryatthetarget65areaiscalculatedthroughaninversemappingS1:0BBBBBBBBB@ataytabtadta1CCCCCCCCCA=S10BBBBBBBBB@xfpafpyfpbfp1CCCCCCCCCA(3.9)whereata,yta,bta,anddtaarethedispersiveangle,non-dispersiveposition,non-dispersiveangle,anddeviationfromthecentralenergyatthetargetposition,respectively.Thein-versemapS1transformsthefocal-planeparameterstothetargetparameters.Thismapisnon-linearandextendsto˝fthorderinthefocal-planeparameters.Infocusedmode,thedispersivepositionatthefocalplaneismuchsmallerthanthedispersionduetothemomen-tumspreadoftheincomingsecondarybeam;thereforethedispersivepositionatthetarget(xta)cannotbecalculatedandisinsteadassumedtobezero.3.4.1.5CalibrationsInordertoproperlyusethedatafromtheS800detectors,thesevaluesmustbecalibratedbeforetheanalysisofexperimentaldata.ThissectiondetailsthestepstakentocalibratetheS800detectors.AlthoughtheCRDCsaredesignedtomeasurethexandypositionsofthebeamatthefocalplane,therawsignalsareapositionsignal(x)andatimesignal(y).Calibrationofthesesignalsarenecessarytoconverttherawsignalsintothetruexandypositions.ThecalibrationsaredonebyinsertingametalplateinfrontofeachCRDC.Theplatehasaspeci˝cpatternofholesdrilledintoitwhosepositionsarepreciselyknown.Thus,theeachCRDConlydetectsparticleswhichpassthroughtheholesandintothedetector.Thepattern66Figure3.11:AnexampleofthecalibratedmaskrunsforCRDC1(left)andCRDC2(right).TheholesandlinescorrespondtoholesinaspeciallymadeplateinordertomaptherawCRDCsignalstotheknowncoordinatesoftheplateholes.leftintherawCRDCspectrumcanbematchedtotheknownpattern,andthemeasuredvaluesattheholesareadjustedthroughalinearscalingtothetruepositionvalues.Thecalibratedpositionsaredesignedsothattheorigin(0,0)correspondstothecentralaxisofthebeamlineatthefocalplane.ThecalibrationisperformedseparatelyforeachCRDCs.Thisprocessisdoneseveraltimesduringanexperiment,becausethetimingsignalduetoionsdriftinginthegasdependsonseveralfactors,includingthepressureandtemperatureofthegas.AnexampleofthecalibrationsisshowninFigure3.11,whichshowsthematchingholepatterninbothCRDCs.ThetimingsignalsfromtheA1900andS800scintillatorsarevitalintheidenti˝cationofsecondarybeamfragmentsand˝nalparticles.Thetimeof˛ightofagivenisotopeisnotunique,however.Ifaparticle'smotiondeviatesfromthecentralaxisofthebeamline,thenthelengthofitstrajectorythroughtheS800,andthereforetimeof˛ight,willbechanged.ThisdeviationcanbecorrectedforusingthetrajectoryoftheparticlemeasuredintheS800focalplane.ThiscorrectionismadeforthesignalsofthetimingdetectorsattheA190067Figure3.12:Plotsshowingthee˙ectsofcorrectionstothetimingsignalsfromtheA1900extendedfocalplane(XFP)andS800objectplane(OBJ)scintillators.Theleftplotsshowthetimingspectrawithoutanycorrections,andtherightplotsshowthesamespectrawithcorrectionsbasedonthedispersiveangle(afp)measuredattheS800focalplane.Thecorrectionsmakeparticleidenti˝cationpossible.extendedfocalplane(XFP)andS800objectplane(OBJ).Fortheobjectscintillator,forexample,thecorrectionismadeaccordingtotheequationTOBJ;corrected=TOBJ+caafp+cxxfp(3.10)68whereTOBJ;correctedandTOBJarethecorrectedanduncorrectedtimingsignalsfromtheS800objectscintillator,afpandxfparethedispersiveangleandpositionatthefocalplane,andcaandcxarecorrectionfactors.AsimilarequationisusedtocorrectTXFP,thetimefromtheA1900extendedfocalplanescintillator.Thee˙ectofthecorrectionsisshowninFigure3.12.3.5SimulationSoftwareMostoftheanalysisofthelifetimeisdonebycomparinggamma-raydatatooutputfromasimulationpackage.ThesimulationsarebasedonGeant4[69].Geant4incorporatesparticletrackingthroughphysicalmaterials,anddealswithparticleandphotoninteractions.Thecurrentpackage[70]wasdevelopedspeci˝callyforlifetimemeasurementsattheNSCL.ThecodehaspreviouslybeenupdatedtoincludethreefoilsoftheTRIPLEXplungerandincludesnewgeometriesfortheGRETINAdetectors.Thissectiondescribesdetailsofthesimulationpackageandhowitisusedinlifetimeanalysis.Theprimaryinputstothesimulationarethepositionsoftheplungerfoilsandthedetectors,andthepropertiesoftheincomingsecondarybeam.Thesimulationcanincludeone,two,orthreefoils,correspondingtosingletarget,two-foilplunger,orthree-foilplungercon˝gurations,respectively.Eachfoilisassignedathicknessbasedontheweightofthefoilmeasuredbeforebeinginstalledintheplunger.Thedensityofthefoilscanbescaledtomatchtheenergylossofthesecondarybeamthrougheachfoilmeasuredduringtheexperiment.Thefoilsarepositionedbasedonfoilseparationsusedintheexperiment,andtheentireplungercanbetranslatedrelativetoothermaterialstomatchtheexperimentalconditions.EachGRETINAmodule,whichconsistsoffoursegmentedgermaniumcrystals,69canbeturnedonoro˙inthesimulationbasedontheexperimentalcon˝gurationofthedetectors.Eachdetectorisbasedonasetoffourcrystalshapeswhicharerotatedandtranslatedtotheappropriatepositionwithinthesimulatedvolume.Thesimulationdoesnotincludesegmentinformationwhenconsideringgamma-rayinteractionpositions.Instead,thepositionresolutionisreplicatedbyshiftingthephotoninteractionpositionaccordingtoaGaussiandistribution.Thewidthofthisdistributionis˝xedasaninputtomatchtheobservedpositionresolution.Thesimulationalsogivesacompletedescriptionoftheradioactivebeam.Protonandneutronnumbersaswellasthecharge-statedistributionofthesecondarybeamnucleusarespeci˝ed.Thebeamisalsogivenaspatial,angular,andmomentumspreadtomatchexperimentalconditions.Thereactiontocreatethe˝nalnucleusinsimulatedinasinglestepbyinstantaneouslychangingthenumberofprotonsandneutronsappropriately.Reactionkinematicsareincludedbyparameterswhichdescribetheaveragemomentumlossduringthereactionandthespreadofmomentumaroundthisaverage.Thelevelschemeofthe˝nalnucleusisconstructed,withtherelativepopulationandbranchingratiosofeachstatespeci˝ed.Duringeveryeventinthesimulation,andsinglenucleusiscreatedwhichissenttowardtheplungertarget.Withinthetargetanddegraderfoils,thebeamissloweddowncontinuously.Thereactionpositionisrandomlychosensomewhereinsidethetargetanddegraderfoils.Therelativenumberofreactionsoneachofthefoilsis˝xedbytwoparameters.Asthe˝nalnucleusispropagatedinthesimulation,itemitsphotonsaccordingtothelevelscheme.Thelocationofgamma-rayemissionisdeterminedbysamplinganexponentialdecayfunctionbasedontheinputlifetime.Withinthesimulation,photonsareemittedisotropicallyintherestframeofthenucleus,andthedistributionisLorentz-boostedintothelaboratory70frame.TheDoppler-shiftedphotonsthenmovethroughthevolumeofthesimulationandcanbeabsorbedorscatteredbyanyofthesurroundingmaterials.Photoninteractionswithinthedetectorsarerecorded,includingtheenergylossandpositionforeachscatteringorabsorptionevent.Inordertocomparewithexperimentaldata,thesimulationisrunmanytimes,withMonteCarlotrialstosimulaterandomprocesses.Positionandenergyinformationforthenucleialongtheirtrackissavedatseveralpoints,includingthereactionlocationandphotonemissionpoints.Photonhitinformationwithinthedetector,includingmultiplicity,isalsosaved.ThedataisstoredastreeswithintheROOTframework[71].DataareoutputtoaROOT˝le,andareeithersavedasrawROOTtreesorhistograms.Thesehistogramscanbecomparedtoexperimentaldataduringanalysis.Theanalysisofexperimentaldataconsistsofcomparingexperimentalandsimulatedgamma-rayspectra.Onceallphysicalandgeometricalparametersare˝xedinthesimulation,severalsimulationsareperformedbyvaryingthelifetimeofthestatesofinterest.Foreachlifetime,thesimulatedhistogramsare˝tusingPearson's˜2test[72].ForahistogramwithNbins,datapointsni,whereiistheindexofabin,andsimulatedpointsi,the˜2valueiscalculatedas:˜2=NXi=1(nii)2˙2i(3.11)where˙iistheuncertaintyinthecountsinbiniofthedata.BecausethecountsineachbinarePoisson-distributed,˙2i=ni.Thistestprovidesaoforeachsimulation.Ifthedataarewelldescribedbythesimulatedspectrum,thenthe˜2valuewillbesmaller,becausethedi˙erenceniiisdecreased.Ontheotherhand,ifthedataandsimulationareverydi˙erent,thenthe˜2willincrease.Onceathe˜2valueiscalculatedforseveraldi˙erent71lifetimes,adistributioncanbefound.Thelifetimewiththeminimumofthedistributionisthenthelifetime.Forsmallchangesaroundthebest-˝t,thedistributionisquadratic,andthelifetimevalueswhichareonestandarddeviationawayaregivenby˜2=˜2min+1:(3.12)Inordertovalidatethesimulationandevaluatesystematicerrors,thesimulationsoftwareisalsousedtosimulatesourcee˚ciencymeasurements.Inthiscase,noparticlesarecreated,butgammaraysareemittedisotropicallyfromasinglepoint,whichisde˝nedtomatchthepositionofasourceusedintheexperimentalcalibration.Theenergiesandbranchingratiosofallgammaraysusedinthecalibrationaresettomatchknownvaluesintheliterature.Aftersimulatingalargenumberofgammarays,thedetectorresponseisrecorded,andthee˚ciencyofeachpeakiscalculatedinthesamemannerastheexperimentaldata,takingintoaccountthetotalnumberofeventsinthesimulation.Examplesofthesimulatede˚ciencyareshowninFigure3.8,whichcomparesthesimulationto152Euand133Basourcemeasurementsintheenergyrangebelow500keV.InFigure3.8thee˚ciencyisshownforthecon˝gurationwithonlyaberylliumtarget,andthesimulationiswellmatchedtothedatawithoutanyscaling.InFigure3.8(b),thee˚ciencyiscomparedforthecon˝gurationwithaberylliumtargetandtwotantalumdegraders.Inthiscase,thesimulationoverestimatesthee˚ciency,whichisduetoincorrectdescriptionofgamma-rayabsorptionwithinthedegradermaterials.Tomatchthemeasurede˚ciency,thesimulationmustbescaledbyanempiricalcorrectionfactor.InFigure3.8(b),thesimulationisscaledby0.91,whichreproducesthee˚ciencybetween200keVand500keVwithin2%,whilethee˚ciencybelow150keVcannotbematchedbyscaling.However,thiscon˝gurationwasnotusedinthepresentexperiment,72andthediscrepancyisexpectedtobesmallerbecauseonlyasingledegraderwasusedfortherecoil-distancesetting.Infact,tocalculateknockoutcrosssectionstotheexcitedstatein19C,onlythee˚ciencyusingtheline-shapesettingisnecessary,wheretheshapeofthesimulatede˚ciencycurveisconsistentwiththedata.73Chapter4Gamma-rayLifetimeMeasurementof19C4.1MotivationTheprimarymotivationofthisexperimentwastomeasurethemagneticdipole(M1)re-sponseofahalonucleustoprovidenewinsightintoitsstructure.AsdiscussedinSection1.3,awellknowncharacteristicofhalonucleiisthelargeenhancementoftheelectricdipole(E1)responseatlowenergies,alsocalledthesoftE1excitation[73].ThissoftE1excitationhasbeenwellcharacterizedexperimentallyforseveralhalonuclei[18,20,74,75].However,therearenosimilarmeasurementsoftheM1propertiesofhalonuclei.Somestaticmag-neticpropertieshavebeenmeasuredforhalonuclei,suchasthemagneticmoment[76]andmagnetizationradius[77]oftheone-neutronhalo11Be.However,thesearestaticpropertiesonlymeasuredforasinglestate.AnM1transitionhasbeenobservedbetweenthe1=2+and3=2+statesin17C[51,78].Thistransition,withalifetimeofover500ps,isastronglyhinderedtransition,andthishindrancehasbeenascribedtothepossiblepresenceofahalostructureintheexcited1=2+state[78].However,thehalonatureofthisstatehasnotbeencon˝rmedexperimentally.Thus,thereisinterestinstudyingtheM1transitionrateinacon˝rmedhalonucleus.ThesearchforanM1transitionbetweenboundstatesisexpectedtobedi˚cult.The74greatestobstacleisthesmallnumberoftransitionsavailable.Becauseofthelowparticleseparationenergyrequiredforhaloformation,therearetypicallyveryfewboundexcitedstateswhichcandecayviagammaemission.Inaddition,thereisastructuree˙ectwhichtendstoblocktheoccurrenceofM1transitions.Inasimplisticmodelofthehalo,ans1=2neutroniscoupledtoaninert0+core.OneconsequenceoftheselectionrulesfortheM1transitionisthatitcanonlyoccurbetweenthetwomagneticsubstatesofasingle`orbital,resultinginaspin-˛ip[79].Forapure`=0state,theM1responsevanishesduetotheabsenceofaspin-˛ippartnerforthes1=2orbital.However,therealisticpicturecanbemorecomplex;forexample,non-negligiblecore-excitationcomponentshavebeensuggestedbyaninclusiveone-neutronremovalstudyfromthehalonucleus19C[80].Therefore,ameasure-mentofthemagneticresponsecanprobethepurityofthes-waveandcorecon˝gurationin19C.Thepresentworkaimstoquantifythemagnetictransitionstrengthin19Cinordertoidentifypossiblehindrance,andinvestigatetheroleofshellmodelcon˝gurationsresponsibleforsuchatransition.Thenucleus19Cpresentsanidealcasetoinvestigatemagneticresponsesofhalonuclei.The˝rstsuggestionofahalostructureinthegroundstateof19Ccamefromameasurementofthelongitudinalmomentumfollowingone-neutronremoval[81].Thegroundstatehasbeenfurtherstudiedthroughinteractioncrosssections[82],momentumdistributions85],Coulombdissociation[18],andknockoutreactioncrosssections[41,80,86].Theseresultshave˝rmlyestablishedtheground-stateone-neutronhalostructurewithspinandparityJˇof1/2+andone-neutronseparationenergySnof580(90)keV[87].Theground-statehaloappearsbecausethe˝nalneutronin19Coccupiesthe1s1=2orbital,asshowninFigure1.3.Inadditiontoground-statestudies,anexcitedstateatˇ200keVhasbeenestablishedbyin-beam-raystudies[88,89]whichproposeatentativeJˇof3/2+.Asecondpossible-ray75Figure4.1:Thelevelschemefor19Catthetimeofthepresentexperiment.Twogamma-raytransitionshavebeenobservedamongtheboundstates[88,89],andtworesonancesabovetheneutronseparationenergyhavebeenobserved[90,91].transitionatˇ70keVwasalsoobservedinoneexperiment[89],suggestingaJˇ=5/2+stateatˇ270keV.However,theexistenceofthisstatehasbeencalledintoquestionbecausea5/2+statewasobservedjustabovetheneutronseparationenergy[90],andone-neutronknockoutcrosssectionsexcludeabound5/2+state[80,86].Basedontheproposedlevelscheme,themultipolaritiesofboththe3=2+!1=2+g:s:andpossible5=2+!3=2+transitionsareexpectedtobeM1.Aprevioussearchforisomericstatesin19Cdidnot˝ndanystatewithalifetimeontheorderof50nsorlonger,indicatingarelativelyprompttransition[92].Aprecisemeasurementofthelifetimeoftheboundexcitedstatesin19Ccanhelptodeducethestructureofthesestates.764.2OverviewofExperimentTheexperimentwasperformedattheNationalSuperconductingCyclotronLaboratoryatMichiganStateUniversitytoinvestigatethelifetimeofboundexcitedstatesin19C[93].Sta-ble22NeionswereproducedbytheSuSIionsourceanddeliveredtotheCoupledCyclotronFacility(CCF).IntheK500cyclotron,the22Ne4+ionswereacceleratedtoanenergyof10.9MeV/nucleon.OncetheionsexitedtheK500,theyweresentthroughastripperwhichremovedallremainingelectrons,andthe22Ne10+ionswereacceleratedbytheK1200cy-clotronupto120MeV/nucleon.Theprimarybeamwasproducedatarateof150particlenanoamperes,orabout91011particlespersecond.The22Nebeamwasimpingedona5.8mmberylliumtarget.AcocktailofreactionproductswassentthroughtheA1900fragmentseparator,whichwastunedtoallow20Nfragmentstopassthrough.AlthoughtheCCFusuallyproducesnucleithroughfragmentationreactions,20Nionswereproducedinadi˙erentreaction.Fragmentationonlyinvolveslossofnucleons,butthecurrentreactionrequiredlossofthreeprotonsaswellasthepickupofoneneutronbythe22Nenuclei.Thisreactionsettingwaschosenbecauseitprovideda20Nbeamatasuitableenergyandrateaswellasahighbeampurity.A2.8mmaluminumwedgedegraderwasplacedinthemiddleoftheA1900toseparateproductsbasedontheirenergyloss.The20NionsexitedtheA1900andarrivedattheexperimentalareaatanaverageenergyof74MeV/nucleon.Thesecondarybeamarrivedattheexperimentalstationatarateof1104ionspersecond,andatotalmomentumwidthof2%aroundtheaveragemomentum.Thepurityofthesecondarybeamattheexperimentalstationwas91%.Identi˝cationofthesecondarybeamwasmadebasedontimingmeasurementsattheA1900extendedfocalplaneandtheS800objectplane.Theparticleidenti˝cationspectrumforthesecondarybeamis77Figure4.2:Theparticleidenti˝cationspectrumforthesecondarybeam.Onthex-axisisthetimetakenfromtheS800objectplanescintillator(OBJ),andthey-axisshowsthetimefromtheA1900extendedfocalplanescintillator(XFP).showninFigure4.2with20Nidenti˝edastheprimarycomponent.TheTRIPLEXplungerwaslocatedatthetargetlocationoftheS800.SevenGRETINAmoduleswereplacedaroundthebeamline.Fourmoduleswerelocatedinaforwardringincentereddownstreamoftheplunger,andthethreeremainingmoduleswerelocatedinacentralringcenteredimmediatelyinfrontoftheplunger.Theplungerwasshiftedsothatthetargetwaslocatedabout13cmupstreamfromthecenteroftheGRETINAsphere.Thiswasdoneinordertoallowforsu˚cientgamma-raydetectione˚ciencywhilemaintainingsensitivitytothevaryingdegreesofDopplershiftsduetothedi˙erentrecoilvelocitiesintheRecoilDistanceMethod(RDM)measurement.Withthiscon˝guration,theforwarddetectorscoveredanglesof25°°,andthecentraldetectorscoveredanglesof50°°.78Figure4.3:Theparticleidenti˝cationspectrumforthe˝nalproductsintheS800spectro-graph.Thex-axisshowsthecorrectedtimeof˛ightfromtheS800objectplane(OBJ),andthey-axisshowstheenergylossthroughtheionchamber.Line-shapedatawere˝rsttakenwithasingle2.0mmberylliumtarget.FortheRecoilDistanceMethod,a0.92mmtantalumdegraderwasaddedtotheplunger,withatarget-degraderseparationof5.0cm.Becauseofthelowratesoftheexperiment,RDMdatawereonlytakenatasingledistancesetting.AfterreactingintheTRIPLEXfoils,recoiling19Cnucleiweredetectedandidenti˝edintheS800spectrograph.Theparticleidenti˝cationspectrumobtainedduringtheline-shapeportionoftheexperimentisshowninFigure4.3.Thisspectrumwasproducedbygatingonthe20NsecondarybeamintheA1900particleidenti˝cation(Figure4.2).Alleventsoriginatingfroma20NparticleintheA1900areplotted.Cleanseparationof˝nalnuclearproductsisachievedbasedontheionchamberenergylossandtheS800focalplanetiming.79Thesetwomeasurementsallowuniqueidenti˝cationofreactionproducts.AccordingtoEquation3.5,theionsloseenergybasedonZ2,orthesquareofthechargeoftheion.Thus,theverticalaxisofservesasanidenti˝cationoftheatomicnumberofeachisotope.Thehorizontalaxisshowsthecorrectedtimeof˛ightTOBJfromtheS800objectplane.Foragivenmagneticrigidityofthespectrograph,thetimeof˛ightisrelatedtothemass-to-chargeratioofeachisotopeaccordingtoEquation3.6.Thus,thespectruminFigure4.3isusedtoidentifyeachproductformedfromthesecondarybeamreactionswithintheTRIPLEX.InFigure4.3,thelocationof19Cnucleiisindicated,aswellas9Li.Bymakingsoftwaregatesontheparticleidenti˝cationspectrum,eachnucleuscanbestudiedindividually.4.3Line-shapeAnalysisDatawere˝rsttakenwiththeline-shapemethod.Inthiscase,theTRIPLEXwasusedwithonlya2.0mmberylliumfoilplacedatthetargetposition.Thetargetwaslocated13cmupstreamofthecenterofGRETINA.Thespectrumwascreatedusingadetectormultiplicityofone,meaningthatonlythoseeventsinwhichasingleGRETINAcrystalrecordsahitareshown.Forthecurrenttransitionenergyof209keV,thelabframeenergiesofthegammaraysmeasuredinthedetectorsarebelow300keV.Attheseenergies,photoninteractionsingermaniumaredominatedbyphotoabsorption(seeFigure2.4);therefore,thiscutonmultiplicityresultedinlittlereductionofcountsinthegamma-raypeak,whilereducingcontributionsfromthebackground.TheenergyofeverydetectedgammaraywascorrectedaccordingtoEquation2.7.Thevelocity=0:362wasbasedonthemagneticrigidityoftheS800.TheemissionanglewascalculatedfromtheinteractionpositionwithinGRETINA,assumingthegammaraysoriginatedfromthecenterofthetarget.Corrections80totheemissionangleandvelocityweremadebasedontheparameterscalculatedfromtheS800inversemap(Equation3.9).TheDoppler-correctedgamma-rayspectrumisshownbytheblackhistograminFig-ure4.4.Inthespectrum,asinglegamma-raypeakisseeninthespectrumat209(2)keV.Theasymmetricpeakhasawidetailextendingtolowenergies.Themoderateslopeinthespectrumindicatesthedecaysoccurwhile19CrecoilsaremovingalongthebeampathsurroundedbyGRETINA.Thismeansthedecaylifetimeisonthesameorderasthe˛ighttimeforthebeamtopassthroughGRETINA.Thedetectorcoverageextendsabout30cmdownstreamofthetargetposition,whichcorrespondstoa˛ighttimeofabout3nsattheaverage19Crecoilvelocity.Atransitionfromasecondexcitedstatenear270keVisnotsigni˝cantlyobservedinthepresentdata.Agamma-gammacoincidenceanalysisgatedonthe209-keVpeakregionabove100keVplacesanupperlimitof10%onpossiblefeeding.Inordertoextractthelifetimeofthe209-keVexcitedstate,thedatawerecomparedtosimulatedspectrabasedonthelifetimeofthestate.Withinthesimulation,thespatialandangulardistributionsofthebeamwas˝rstmatchedtothemeasureddistributionsoftheincoming20Nbeamandoutgoing19Cbeam.Gammaraysinthesimulationwhichin-teractedwiththedetectorswereanalyzedinthesamemannerastheexperimentaldata,andaDoppler-correctedspectrumwasproduced.Thesimulatedspectrumdoesnotincludeabackground,sotocomparetotheexperimentalspectrum,thesimulationmustbeaddedtoanappropriatebackgroundshape.Typically,anexponentialshapeisusedfortheback-groundinsimulations,whichis˝tintheregionsaroundgamma-raypeaks[64,94].However,inthecurrentspectrum,itisdi˚culttoseparatethelow-energytailofthepeakfrom19Cwiththebackgroundatlowenergy.Therefore,thebackgroundwastakenfromgammaraysincoincidencewith9Lirecoils,whichalsoappearintheparticleidenti˝cationspectrumin81Figure4.4:TheDoppler-correctedspectrumusingtheline-shapemethod.Asinglepeakat209keVwithawidetailatlowerenergiesisclearlyvisible.Theplotshowsthedatainblack,andthebest-˝tsimulationisshowninred,whichincludesabackgroundtakenfrom9Lishowninblue.thesamesettingas19C(seeFigure4.2).Thenucleus9Liwaschosenbecauseithasasinglegamma-raytransitionat2.7MeV,sothelow-energyregionisexpectedtobedominatedbybackground.Thebackgroundwasscaledtomatchtheregionabovethepeak,andthesimulatedspectrumwasaddedandscaledto˝tthepeakregionabove100keV.Thesimula-tionwas˝tforseveralassumptionsofthebackgroundshape,itwasfoundthatthebest˝tlifetimedidnotsigni˝cantlydependonthechoiceofbackgroundparametrization.Severalspectrawereproducedintheabovemannerbyadjustingthelifetimeofthe209keVtransition,andthebest˝tlifetimewasdeterminedusingthe˜2-minimizationpro-ceduredescribedinSection3.5.The˜2distributionasafunctionofthesimulatedlifetimeisshowninFigure4.6(a).Aquadratic˝ttodistributionaroundtheminimumgivesabest-˝tlifetimeof1.98ns.Thesimulationusingthislifetimeisshownbytheredhistogramin82Fig.4.4.Basedonthe˜2˝tting,thestatisticaluncertaintyforthismeasurementis0.10ns.4.4RecoilDistanceAnalysisAfterdatawastakenwithonlyatarget,a0.92-mmtantalumdegraderwasaddedinordertomakeuseoftheRecoilDistanceMethod.Thetargetanddegraderwereseparatedby5.0cm,withthetargetlocated15cmupstreamfromthecenterofGRETINA.Thegamma-rayspectrumwascreatedagainbygatingonthedetectorcrystalmultiplicityofone.Aswasthecasefortheline-shapeanalysis,thespectrumwasobtainedbyDopplercorrectionoftheobservedgammaraysassumingthedecaysoriginatedfromthecenterofthetargetandwereemittedwithaspeedof=0:362.CorrectionswerealsomadetothevelocityandanglebasedonS800measurements.Inthiscase,gammaraysemittedbetweenthetargetanddegraderareemittedatthesamevelocityastheDopplercorrection,andappearinthespectrumatthetruedecayenergy.Nucleithatdecayafterthedegraderhavealowervelocity(=0:322)thanthosewhichdecaybeforethedegrader.Thisdi˙erenceinvelocitymeansthattheDoppler-correctedenergyisbelowthetruedecayenergy,andthesegammaraysappearasapeakatlowerenergy.TheDoppler-correctedspectrumisshownbytheblackpointsinFig.4.5.Adouble-peakedstructureisseenwithasimilarlow-energytailasseenintheline-shapespectrum.Thefastandslowpeaksarelabeledinthe˝gure.Thefastpeakiscenteredat209keV,whiletheslowcomponenthasapeakat190keV.Thelargeheightoftheslowpeakrelativetothefastpeakindicatesthatmostdecaysoccurbeyondthedegrader.Thusthelifetimemustbelongerthanthe˛ighttimeof500psacrossthe5cmdistancebetweenthetargetanddegrader.Similartotheline-shapemeasurement,theslowpeakhasabroadtailextending83Figure4.5:TheDoppler-correctedspectrumusingtheRecoil-DistanceDoppler-ShiftMethod.Adoublepeakat209keVisagainvisible,withthesamelow-energytail.Thefastpeakcenteredat209keVismuchsmallerthantheslowpeakat190keV.Thedataareshowninblack,andthebest-˝tsimulationisshowninred,withtheassumedbackgroundarisingfrom9Lishowninblue.Inthisplotthereareadditionalx-raysaround50keVbecauseoftheenergylossofthebeaminsidethetantalumdegrader.tolowenergies.Thisagainisanindicationofthenanosecond-orderlifetime.Inadditiontotherecoil-distancepeak,thereisapeakaround50keV.ThisisduetoX-raysoriginatingfromtantalumatomswhichareexcitedasthebeampassesthroughthedegrader.Theseareemittedinthelaboratoryframearound70keV,butappearatlowerenergiesafterDopplercorrection.SimulationswereperformedinthesamemannerdescribedinSection4.3,withaback-groundspectrumfrom9Lirecoilswasagainusedtomatchthebackgroundinthe19Cspec-trum.The˜2distributiongeneratedasafunctionoflifetimeisshowninFigure4.6(b).Thesameenergyrangewasusedforthe˝tprocedureinboththeline-shapeandrecoildistancemeasurements,andthisregionexcludesthetantalumX-raypeakbelow100keV.Inthis84Figure4.6:The˜2distributionsobtainedby˝ttingthesimulatedgamma-rayspectratotheexperimentalspectrumforvariouslifetimesofthe209-keVtransitionin19C.Part(a)showsthedistributionfortheline-shapespectrum,and(b)showsthedistributionfortherecoil-distancespectrum.Quadratic˝tstothecurvesgiveminimaat1.98nsand1.90ns,respectively.case,thebest-˝tlifetimeis1.90ns,withastatisticalerrorof0.12ns.Thesimulationusingalifetimeof1.90nsiscomparedtothedatabytheredhistograminFig.4.5.Thelifetimeobtainedfortherecoil-distancemeasurementisslightlylongerthanthevalueobtainedwiththeline-shapetechnique,althoughtheyarebothconsistentwithintheiruncertainty.4.5ResultsTheadoptedvalueofthelifetimeinthisworkwasdeterminedbytakingintoaccountsystem-aticerrorsforeachanalysis.TheerrorsaresummarizedinTable4.1.Forbothmeasurements,thelargestuncertaintyinthelifetimeisthestatisticaluncertaintyfromthe˝ttingprocedure.Sourcesofsystematicuncertaintycommontobothmeasurementsincludetheshapeofthebeampro˝leusedinthesimulation,thegeometryoftheTRIPLEX,andtheshapeofthegamma-raybackground.Thelargestsystematicuncertaintyof0.05ns(3%)camefromthekinematicpro˝leofthebeam.Theuncertaintyintheshapeofthebackgroundadded0.02ns85Table4.1:Summaryofsystematicerrorsobservedforthelifetimemeasurement.Errorduetothebeamproperties,plungergeometry,andgamma-raybackgroundshapeweresimilarforboththeline-shapeandrecoil-distancemethods.Theerrorfromdegraderreactionsonlyappearsfortherecoil-distancemeasurement.ComponentError(%)Di˙erenceofmeasurements4Beamproperties3Plungergeometry<1Backgrounddetermination1Degraderreactions2(3%),anduncertaintyfromthepositioningoftheTRIPLEXplungercontributedlessthan0.02ns(<1%).Fortherecoil-distancedata,anadditionalambiguityarisesinthespectrumfromreactionsproducing19Cinthedegrader,whichintroducebackgroundcontributionsinthelifetimemeasurement.Suchreactionsonlypopulatetheslowpeakandincreasetheapparentlifetimeofthetransition.Toaccountforsecondaryreactionsinthedegrader,aratioRisintroducedinthesimulationwhichde˝nestheratiooftheyieldoftarget/degraderreactions.Thisallowsthesimulatedspectratobeproperly˝ttotheslowpeak.Typically,Risdeterminedsimultaneouslywiththelifetime˝inatwo-dimensional˜2˝tting.Forthecurrentsetup,however,itwasfoundthatthereisalmostnosensitivitytochangesinthereactionratio.TheratioRassumedinthepresentsimulationis4.6(14),whichisestimatedfromratiosdeducedinpreviousexperimentsutilizinganalogousone-protonknockoutreac-tionsfromnitrogenprojectiles[95,96].ThelargeRinthismeasurementresultsinasmalladditionalerrorof2%.Byaddingthestatisticalandsystematicerrorsinquadrature,theresultsarededucedtobe1.98(12)nsand1.90(13)nsforthetwomeasurements.Becausethetworesultsareconsistent,theadoptedvalueisdeterminedtobe1.94(15)nsbytakingtheaverage.Inadditiontotheerrorsdiscussedabove,the˝nalvaluealsoincludestheerrorduetothedi˙erence(4%)betweenthetworesults.BasedonEquation1.10,the86B(M1)transitionstrengthiscalculatedasB(M1)=56:8E3˝2NMeV3fs:(4.1)Assumingthatthe209-keVtransitionin19CisapureM1transition,theB(M1;3=2+!1=2+)strengthisdeterminedtobe3.21(25)1032N.ComparingtotheWeisskopfestimateforanM1transition(Equation1.19),thiscorrespondstoastrengthof1.79(14)103Weis-skopfunits(W.u.).IfthereisanyadmixturefromtheE2multipolarityinthisdecay,theB(M1)strengthisreducedaccordingly.However,thee˙ectisexpectedtobenegligibleinthiscaseduetothe1/E2L+1dependenceofthepartiallifetimes.InthemassregionA<44,thelargestE2transitionstrengthsconnectingtogroundstatesareabout20W.u.[97].Withthisstrengthassumed,theB(M1)isreducedbyonly6%.Infact,theE2strengthsforthe2+!0+transitionsinneighboringevencarbonisotopesareonlyW.u.[95,96,98],sotheE2contributionin19Cmaybesafelyignored.Apossiblespinandparityassignmentof5=2+forthe209-keVstatewouldrequireapureE2transitionforthedecaytothe1=2+groundstate.ThiswouldresultinaB(E2)of350W.u.,farbeyondtherecommendedupperlimitof100W.u.[97].Thusthepresentlymeasuredlifetimesupportsthe3/2+assignmentpreviouslyproposedforthe˝rstexcitedstatein19C.4.6DiscussionToinvestigatethedegreeoftheM1hindrancein19C,thepresentresultiscomparedtoexistingdataforM1decaystrengthsinthemassregionA<40inFigure4.7[99].InFig-ure4.7(a),thestrengthsofalltransitionswhichinvolvea1=2+stateareplotted.Asisclear87Figure4.7:AplotshowingthedistributionsofallB(M1)transitionstrengthsamongnucleiofmassA<40.(a)plotsthevaluesforonlythosetransitionswhichinvolvea1=2+state,withthepresentlymeasuredB(M1;3=2+!1=2+g:s:)for19Chighlightedinred,andtheanalogousB(M1;1=2+!3=2+g:s:)inblue.(b)showsthedistributionforallM1transitions.Inbothcases,itisclearthattheB(M1)strengthfor19Cliesamongtheweakesttransitions.DatafromRef.[99]fromthe˝gure,theM1transitionin19C,highlightedinred,isamongthesmalleststrengthsobservedinthismassregion.Thestrengthisevenbelowthatmeasuredfortheanalogoustransitionin17C(5:7103W:u:)[51],shownbythebluepoint.InFigure4.7(a),twoadditionalpointsatA=23arevisible,denotingthe1=2+!3=2+g:s:transitionsinthemirrornuclei23Naand23Mg.TheseareconsideredtobeinterbandtransitionsbetweenNilssonorbits[2111/2]and[2113/2]ofwelldeformednuclei[100].Inashellmodelpicture,thehindranceisduetoalargecancellationbetweentheorbitalandspincontributionstotheM1strength[101].WhencomparedtoalltransitionsinthismassregionasshowninFigure4.7(b),theM1hindrancein19Cremainsevident,indicativeoftheunusualstructureof19C.884.6.1ShellModelCalculationsShellmodelcalculationswereperformedtoinvestigatetheoriginoftheM1hindranceaswellastheremainingstrength.Threeinteractions,WBP[102],SFO-tls[103],andYuan[104],wereusedwithinthepsdmodelspace,allowing~!,where~!representstheexcitationofasinglenucleontothenext-highestoscillatorshell.TheWBPinteractionwasconstructedbasedon˝tstoenergylevelsfornucleiofmass[102].Themodi˝edSFOinteraction,denotedSFO-tls,isbasedontheSFOinteraction[105],whichisamodi˝cationofthePSDMK2interaction[106].Furthermodi˝cationstotheSFOinteractionhavebeenmadebasedonRef.[103],includingadjustmentstothetensor(t)andspin-orbit(ls)componentsoftheinteraction.Theseadjustmentsweremadeinordertoreproducetheobservedmagneticpropertiesin17C[103].TheinteractionofYuanincorporatesamonopole-baseduniversalinteractionwhichincludesthebareˇ+ˆtensorforce[107].Thisisdesignedtoreproducematrixelementsinvolvingnucleonswithinthepandsdshells(hpsdjVjpsdi)andcross-shellmatrixelementsbetweenthepandsdshells(hppjVjsdsdi)whichhavenotbeenwellstudiedinotherinteractions[104].BoththeSFO-tlsandYuaninteractionshavefurthercorrectionswhichincludeso-calledlooselybounde˙ects.Thesecorrectionstakeintoaccountthelargesizeofthelooselybound1s1=2orbitalbyreducingthestrengthofmatrixelementswhichinvolvethisorbital[103].Theinteractionsincorporatingthelooselybounde˙ectsaredenotedSFO-tls+lbeandYuan+lbe.Forall˝veinteractions(WBP,SFO-tls,SFO-tls+lbe,Yuan,andYuan+lbe),thelevelschemeandB(M1;3=2+!1=2+)transitionprobabilityin19Carecalculatedandcomparedtopresentdata.Thecalculatedenergiesforthelow-lyingstatesin19CcanbeseeninFigure4.8.The˝gurealsodisplaystheexperimentallevelscheme,includingthelocationsofboththepro-89Figure4.8:Theexperimentalandshellmodelpredictionsforthelow-lyingstatesin19C.The1=2+groundstateisshowninblack,the3=2+isred,andthe5=2+stateisblue.Theexperimentallyobservedlevelsshownontheleftincludesthetwosuggestedlocationsofthe˝rstexcited5=2+state[89,90].Foralltheoreticalmodels,the1=2+groundstateiscorrectlyreproduced,whiletheorderofthe3=2+and5=2+statesarereversed.posedbound[89]andunbound[90]˝rst5=2+state.Allcalculationscorrectlypredictthe1=2+groundstate.However,inallcasesthe5=2+stateislocatedbelowthe3=2+,incontrasttotheproposedlevelscheme.Atestcalculationwasabletoreproducethelevelorderingifthesingle-particleenergyofthe1s1=2orbitalwasreducedby0.5MeVwithintheYuan+lbeinteraction.Althoughthelevelorderingwasconsistentwithexperiment,therewasnofundamentalchangeinthecomponentsofthewavefunctionsorinthecalculatedB(M1)transitionstrength.ThepurposeofthepresentworkistodeterminethemicroscopiccauseofthehinderedB(M1)strength.Becausethisunderstandingisnota˙ectedbythespeci˝corderingoflevels,thediscrepancyinthelevelschemeisnotimportantforthecurrentresults.CalculationsfortheB(M1;3=2+!1=2+)transitionstrengthfor19Ccanbeseenin90Figure4.9.Generally,theM1transitionstrengthiscalculatedbetweenaninitialstate ianda˝nalstate faccordingtoB(M1)=h fjjM(M1)jj ii22Ji+1(4.2)whereM(M1)istheM1operatorwhichallowstransitionsbetweenstatesforsinglenucleons,andJiisthespinoftheinitialstate.ThefullM1operatorisgiveby[79,101]M(M1)=r34ˇXi;˝zˆgs˝z~si;˝z+g`˝z~`i;˝z+gt˝zp8ˇhY2(^ri;˝z)~si;˝zi(1)˙N:(4.3)Thethreetermsinthesummationrepresentthespin,orbital,andtensorcomponentsoftheoperator.Theindexirepresentsthesumoverallnucleons,with˝z=prepresentingprotonsand˝z=nrepresentingneutrons.Theg-factorsgs,g`,andgtrepresenttherelativestrengththespin,orbital,andtensorterms,respectively.The˝rsttwotermsrepresentthestandardM1operatordescribedinEquation1.13,andimposetheM1selectionrule`=0;1.Thetensortermisamodi˝cationwhichallowsaforbiddentransitionwith`=2throughtheinclusionoftheY2sphericalharmonic.Thecalculationswere˝rstperformedusingthee˙ectiveg-factorsgsp=5:307,gsn=3:635,g`p=1:15,g`n=0:15,andgtp;gtn=0.Thesevalueswerepreviouslyusedtoexplainmagneticpropertiesin17C[103].TheresultsareshownbythesolidlightgreybarsinFigure4.9.Inallcases,thecalculationsunder-predicttheobservedvalue,byuptotwoordersofmagnitude.Severalcorrectionsarerequiredtomatchtheexperimentalvalue.Theadditionoflooselybounde˙ectstotheSFO-tlsandYuaninteractionsisshownbythesoliddarkgreybars.Inbothcases,thisadditionimprovesthepredictedvalues,butisstillinsu˚cientto91accountfortheobservedvalue.Afurthercorrectionisneededtomatchtotheexperiment,andthisoccursthroughthemodi˝cationofthee˙ectiveg-factorsintheM1operator.Themodi˝edvaluesaregsp=5,gsn=3:5,g`p=1:175,g`n=0:106,gtp=0:26,andgtn=0:17.Thesearebasedona˝ttoM1datausingsixparameterswiththeUSDAinteraction[101].Thenon-zerogttermsactivatethetensortermintheM1operator,whichisacouplingofthepositionandspinvectorsofthenucleons.ThepresenceoftheY2(~r)termallowsthe`=2transition,whichisotherwiseforbiddenbyselectionrules.Inthecaseof19C,thisallowsatransitionbetweenthes1=2andd3=2orbitalsforthevalenceneutron.Theinclusionofthemodi˝edM1operatorisshownbythehatchedbarsinFigure4.9.Inallcases,thisprovidesfurtherimprovementtothepredictedvalues.Basedontheshell-modelcalculations,theobservedM1hindranceisascribedtothelow-eringofthe1s1=2orbitalandresultingproximitytothe0d5=2orbitalcharacteristicofweaklyboundnuclei.Thedegeneracyofthesetwoorbitalsissupportedbyallthecalculations,asdemonstratedbythecompressedlevelschemesshowninFigure4.8.Theprimarycon˝gu-rationsofthevalenceneutronsinthegroundand˝rstexcitedstatesin19Caresimilarforallcalculations,andaregivenasfollows:j19C(1=2+)i=j(d5=2)4J=0+(s1=2)i+:::(4.4)j19C(3=2+)i=j(d5=2)4J=2+(s1=2)i+j(d5=2)3J=3=2+(s1=2)2i+::::(4.5)Thecon˝gurationslistedherearebuiltontopofa14Ccore,witha0+con˝gurationshownbyFigure1.3(b).Thesecon˝gurationsareshownschematicallyinFigure4.10.FortheWBP92Figure4.9:TheexperimentalandshellmodelpredictionsfortheB(M1;3=2+!1=2+)transitionstrengthin19C.Thecurrentexperimentalvalueisshownontheleft,andtheresultsforeachshellmodelcalculationareshownontheright.Thelightgreybarsindicatethevaluescalculatedwithoutanycorrections.ThedarkgreybarsfortheSFO-tlsandYuaninteractionsindicatecalculationswiththeloosely-bounde˙ects[103].Thestripedbarsadditionallyincludethemodi˝cationoftheM1operatorasdescribedinthetext.Inallcases,boththelooselybounde˙ectsandmodi˝edM1operatorimprovethepredictionsrelativetotheobservedvalue.andYuan(+lbe)interactions,therelativestrengthsofthecon˝gurationsarenearlyidentical,with2ˇ0.48,2ˇ0.29,and2ˇ0.26,whiletheSFO-tls(+lbe)gives2ˇ0.40,2ˇ0.26,and2ˇ0.23.Theagreementbetweenthesecalculationsclearlyshowstheprevalenceofthe1s1=2and0d5=2componentsinboththegroundandexcitedstates.For˝vevalenceneutronswithinthe(s1=2d5=2)5spaceabovea14C(0+)core,theonlypossiblecon˝gurationsforthe1=2+and3=2+statesarethoselistedinEquations4.4and4.5.TheM1transitionstrengthbetweenthesecon˝gurationsisexactlyzero.Moregenerally,allpossiblecon˝gurationsforthe(s1=2d5=2)5spaceare(a)(d5=2)5J=5=2+;(b)(d5=2)4J=0+(s1=2)J=1=2+;(c)(d5=2)4J=2+(s1=2)J=3=2+,5=2+;(d)(d5=2)4J=4+(s1=2)J=7=2+,93Figure4.10:Theprimarycon˝gurationsforthegroundstate(a)andexcitedstate(b)in19C.The1=2+groundstateisprimarilyformedbythe1s1=2neutroncoupledtoa0+core.The3=2+excitedstateisdividedbetweentwomaincon˝gurations.Ontheleft,the1s1=2neutroniscoupledtoa2+core,andontheright,twoneutronsinthe1s1=2orbitalcoupleto0+,andthethreeneutronsinthe0d5=2orbitalcoupleto3=2+.Thecalculatedamplitudesofeachcon˝gurationaregiveninthetext.9=2+;and(e)(d5=2)3(s1=2)2J=3=2+,5=2+,9=2+.TheB(M1)strengthiszerobetweenallpairsofthesecon˝gurationsexceptforthespin-˛iptransitionsbetweenthe3=2+and5=2+statesin(c),andthe7=2+and9=2+statesin(d).ItisalsoimportanttonotethattheM1strengthbetweenthe3=2+and5=2+statesin(e)isalsozerobecausethesearewithin94theidentical(d5=2)3space,andthistransitionisnotaspin-˛ip.Othertransitionsrequirea`=2,j=2transitionbetweenthe1s1=2and0d5=2orbitals.ThisistheprimarymechanismresponsibleforthesuppressedB(M1)valuein19C.AdditionalcomponentstothewavefunctionsbeyondthoselistedinEquations4.4and4.5couldallow˝nitetransitionstrengths,butthetotalB(M1)remainsdiminishedduetothesmalleramplitudesofthosecon˝gurations.AdecompositionoftheM1strengthintotheindividualmatrixelementsfortheprotonandneutronorbital,spin,andtensorcomponentsusingtheWBPinteractionisshowninFigure4.11.The˝gurealsoshowssimilarcalculationsforthe1=2+!3=2+transitionsin23Naand23Mg,whichalsohavesmallB(M1)valuesonthesameorderas19C.For23Naand23Mg,thereisalargecancellationbetweenindividualcomponentswhichresultsinthesmallvaluesforthetransitionstrengths.For19Chowever,thereisnotasstrongofacancellatione˙ect,butinsteadtheindividualmatrixelementsthemselvesaremuchsmaller.Whatisuniquefor19Cisthattheorbitalandspincomponentsareassmallasthetensorcomponents.Thus,theprominenceofthes1=2andd5=2orbitalsinboththegroundandexcitedstatesreducestheM1strengthdowntothelevelof1032N,wherecontributionsfromthe`-forbiddentransitionbetweenthes1=2andd3=2orbitalsbecomeimportant.95Figure4.11:DecompositionofthecalculatedB(M1)strengthsin19C,23Na,and23Mgintothespin,orbital,andtensorcomponentsforbothprotonsandneutrons.AllcalculationsaremadeusingtheWBPinteraction[102].Itisclearthattheindividualmatrixelementsaresmallerfor19C,whilethesmallB(M1)valuesin23Naand23Mgareduetocancellationbetweenthecomponents.96Chapter5One-protonKnockoutMeasurementof20N5.1MotivationandOverviewResultsfromthepresentexperimentwerealsoanalyzedtostudytheone-protonknockoutreactionfrom20N.Thisstudycanbeconsideredasanextensionofthelifetimemeasurementbecauseitisusedtoexaminethespinsoftheexcitedstatein19C.Thereactionisalsosensitivetothegroundwavefunctionin20N.Measurementoftheexclusivecrosssectionstostatesin19Cgivestheoverlapofthosestateswiththe20Ngroundstateandcanbeusedtoconstrainthegroundstatespinin20N.Additionally,measurementofthemomentumdistributionofthe19Cfragmentsafterthereactioncanbeusedtocon˝rmthecon˝gurationofthevalenceprotonin20N.Inastandardshellmodelpicturefor20N,theseventhprotonoccupiesap1=2orbital,andthethirteenthneutronoccupiesa0d5=2orbital.Couplingofthesetwoorbitalsisexpectedtoproducea2or3forthegroundstate.However,asshowninFigure1.3,thestandardpicturedoesnotexplainthegroundstateintheneighboringnucleus19C.Ifthesituationisthesamefor20N,thenanothergroundstatemaybepossible.CalculationswiththeWBPinteraction[102]predicta2groundstate.However,thereisverylittleexperimentaldataonthelevelsin20N.Abeta-decaystudyof20Nhasfounddecayratesconsistentwitha297Figure5.1:Comparisonofthelow-lyinglevelsobservedin20Nwithshell-modelcalculationsusingtheWBPinteraction.Thespinandparitiesoftheexperimentallevelsarebasedonsimilarshellmodelcalculations[110],andno˝rmassignmentshavebeenmadeforanystates.groundstate[109].Amorerecentgamma-raystudyobservedseveraltransitionsin20Nandmadetentativelevelassignmentsbasedonshellmodelcalculations,butdidnotmakeany˝rmspinassignments[110].Thelevelschemefor20NresultingfromthatexperimentisshowninFigure5.1,whichalsocomparesthelevelsobtainedwiththeWBPinteraction.Becauseveryfewstatesareassignedfromtheexperiment,itisclearmoreevidenceisnecessarytocharacterizethespininthegroundstateof20N.Thereisgreatinterestinidentifyingthelevelsin20N,becausesuchidenti˝cationcanhelptolocatethesingle-particleenergiesoftheneutron1s1=2and0d5=2orbitals.Itisknownthatthesetwoorbitalsmoverapidlyamongneutron-richnuclei.In22O,theloweringofthe0d5=2orbitalbelowthe1s1=2orbitalcausestheappearanceoftheN=14magicnumber[111],whilethegapisexpectedtodisappearinthenearbynucleus21C[112].Therelativeenergies98ofthe1s1=2and0d5=2orbitalsforseveralisotonicchainsareshowninFigure5.2115].FortheN=7andN=9chains,theorbitalsarenearlydegeneratein14Nand16N,andin13Cand15C,thelevelsareinverted,withthe1s1=2orbitalatalowerenergythan0d5=2.InthecaseofN=11,thelevelsareinvertedfor17C,butreturntonormalorderingin18N,withthe1s1=2abovethe0d5=2.InthepresentcaseofN=13,thetwoorbitalsaredegeneratein19C[93],asshownbythesimultaneous˝llingoftheseorbitalsinFigure1.3(d).Ifthenormalorderingreturnsin20N,thenthe˝vevalenceneutronsshouldoccupythe0d5=2orbital.Couplingofthese0d5=2neutronstothe0p1=2valenceprotoncreatesa2or3groundstatein20N,anddeterminationofthegroundstatewouldhelptocon˝rmtheexpectedtrendforN=13.Althoughthecurrentexperimentcanprovideevidenceforthegroundstatein20N,properassignmentofalllow-lyingstatesisnecessarytofullydeterminethelocationsofthe1s1=2and0d5=2single-particlestates,whichisbeyondthescopeofthecurrentmeasurement.Theknockoutreactionisalsointerestinginviewofthepossiblesimilarityoftheneutrondistributionsin20Nand19C.Previousmeasurementsoftheinteractionandcharge-changingcrosssectionsin20Nhavesuggestedthepresenceofaneutronskin[116,117].Aneutronskinisdescribedasanexcessofneutronsaroundthenuclearsurface,althoughwithouttheextendedtailthatispresentforhalos.Thecurrentmeasurementisthe˝rststudyofaone-protonknockoutreactionleadingtoahalonucleus,andcouldbeusefultoinvestigatetheconnectionbetweenskinsandhalos.99Figure5.2:Threeplotswhichshowtheevolutionoftheneutrons1=2andd5=2orbitalsinlightneutron-richisotopes.In(a),theenergiesoftheorbitalsareplottedrelativetotheneutronseparationenergyforN=7isotones.For13C(Z=6),the1s1=2orbital(red)isclearlylocatedbelowthe0d5=2orbital(blue),andfor14N,theorbitalsarenearlydegenerate.Plot(b)showsthesametrendfortheN=9isotones.Here,for15C,the1s1=2orbital(red)isbelowthe0d5=2orbital(black),whiletheyoverlapin16N.Plot(c)showstheenergyofthe1s1=2orbital(red)relativetothe0d5=2orbital(blue)fortheN=11isotones.Inthiscase,theorbitalsarenearlydegeneratein17C,whilein18N,the1s1=2orbitalislocatedwellabovethe0d5=2orbital.FiguresfromRefs.1005.2CrossSectionsTheexperimentwasperformedasdescribedinSection4.2.Thecrosssectionwascalculatedfromtheline-shapesetupwhichemploysonlya2mmberylliumtargetwithnodegrader.Theinclusivecrosssection˙1pfortheone-protonknockoutfrom20Nto19Ciscalculatedas˙1p=N19CAt20NˆtdtNA(5.1)whereN19Cisthetotalnumberof19Cnucleidetected,N20Nisthenumberof20Nnucleiincidentonthereactiontarget,At,ˆtanddtarethetargetmolarmass,massdensity,andthickness,respectively,NA=6:021023mol1isAvogadro'snumber,andisthee˚ciencytodetectthe19Cnuclei.Thenumberof19CnucleiwasobtainedbyintegratingtheparticleIDspectrum(seeFigure4.3).TheincomingbeamratewasgivenbytherateoftheS800objectscintillator.TransmissionofthesecondarybeamfromtheobjectscintillatortotheexperimentalareawasdeterminedbymeasuringthesecondarybeamratethroughS800withoutatargetinstalled.Becausethe2%totalmomentumspreadofthesecondarybeamissmallerthanthe5%momentumacceptanceoftheS800spectrograph,allofthebeamincidentatthetargetlocationisdetectedatthefocalplane,sothatthetransmissionmeasuredatthefocalplaneisidenticaltothetransmissionatthetarget.Thetotalone-protonknockoutcrosssection˙1pmeasuredforthecurrentexperimentis0.76(10)mb.Crosssectionstoexcitedstatesareobtainedindependentlyfromtheinclusivecrosssectionbyconsideringgamma-rayyieldsinGRETINA.Inthepresentexperiment,onlyasinglegamma-raytransitionwasobservedat209keV.Themeasuredlifetimesupportsthe3=2+assignmentforthisstate.Noevidenceforabound5=2+statewasfound.Thetotalyieldswerecountedfromthegamma-rayspectruminFigure4.4.Allcountsinthepeak,evendown101Table5.1:Theoreticalcrosssectionscomparedtotheexperimentalvalues,assuminga2groundstatein20N.Theoreticalcrosssectionsarecalculatedasdescribedinthetext.Thetotalcrosssectionis˝rstcalculatedassumingtheobserved3=2+stateistheonlyboundexcitedstatein19C.Thesecondcalculationalsoincludesthecrosssectionassumingthelow-lying5=2+stateisbound,calculatedwiththeexcitationenergyfromthepreviousgamma-raystudy[89].Ex(keV)Jˇ`,j˙sp(mb)C2S˙th(mb)˙exp(mb)01=2+1,3/213.10.0380.522093=2+1,1/212.00.0141,3/213.00.0861.362695=2+1,1/211.90.811,3/212.90.06811.1Inclusive(1=2+;3=2+)1.880.76(10)Inclusive(1=2+;3=2+;5=2+)13.0tothelow-energyregionbelow100keV,wereassumedtocomefromeitherthe3=2+!1=2+transitionorfromthebackground.Thegamma-rayyieldswerethenscaledbasedonthee˚ciencyofgamma-raydetection.Usingthismethod,thecalculatedcrosssectiontothe209-keVstate˙1p(209keV)is0.62(9)mb.Ingeneral,thecrosssectiontothegroundstate˙1p(g:s:)issimplycalculatedasthedi˙erencebetweentheinclusivecrosssectionandthecrosssectiontoallexcitedstates:˙1p(g:s:)=˙1pXEx˙1p(Ex):(5.2)Withonlyoneobservedexcitedstate,thecrosssectiontothegroundstatein19Cisfoundtobe˙1p(g:s:)=0:14(13)mb.1025.2.1TheoreticalCalculationsCalculationoftheoreticalcrosssectionscomprisestwoparts:spectroscopicfactorsC2S(Jˇ;n`j)forremovalofaprotonwithquantumnumbersn`jleadingtoa˝nalstatewithspinJˇ,andsingleparticlecrosssections˙sp(n`j;Seffp)basedonane˙ectiveprotonseparationenergySeffp.Spectroscopicfactorswereobtainedfromshell-modelcalculationswiththeWBPin-teraction[102].Statesin19Cand20Nwerecalculatedwithinthepsdmodelspaceallowing~!for19Cand~!for20N.Spectroscopicfactorswerecalculatedforeachprotonorbitalwhichconnectsthegroundstatein20Nwiththelow-lyingstatesin19C.Thecalcu-latedspinandparityofthegroundstatein20Nis2(seeFigure5.1).However,becausethegroundstateisnotexperimentallywelldetermined,the0stateat395keVwasalsoconsideredasaground-statecandidate.Therefore,twosetsofcalculationsweremadeforbothpossiblegroundstates.Single-particlecrosssectionswerecalculatedfromelasticS-matricesforthetarget-coreandtarget-protonsystems.TheS-matriceswerecalculatedbyfoldingthetargetandcore/protondensitieswiththee˙ectivenucleon-nucleoninteractions.Densitiesforthe9Betargetand19CcorewerecalculatedintheWoodsSaxonform:ˆ(r)=ˆ01+exp[(rc)=a]:(5.3)103TheparameterswereobtainedfromRef.[118]:a=0:54(5.4)c=(0:978+0:0206A1=3)A1=3(5.5)ˆ0=3A4ˇc3(1+ˇ2a2=c2):(5.6)Theprotonwastreatedasapointparticle.Thee˙ectivenucleon-nucleonwastakentobeazero-rangedeltafunctionwhosestrengthwasbasedonnucleon-nucleusscatteringathighenergy[119].Theprotonpotentialwithinthe19CcorewasaWoods-Saxonform,withtypicalparametersr0=1:25fmanda=0:7fm.Thepotentialdepthwasadjustedtomatchtheprotonseparationenergyof17.9MeV.Forthecrosssectiontotheexcitedstatein19C,thee˙ectiveseparationenergySeffpiscalculatedasthesumoftheground-stateseparationenergyandtheexcitationenergyof209keV.Theresultsofthecrosssectioncalculationsareshownfora2groundstateinTa-ble5.1,andfora0groundstateinTable5.2.Thetableslistthespectroscopicfactorsandsingle-particlecrosssectionsforthe1=2+groundstateand3=2+excitedstatein19C.Thetheoreticalcrosssectionforeachstateistheproductofthesefactorsscaledbyacorrectionfactorof[A=(A1)]N=[20=19]1whichisnecessarytocorrectforthecenter-of-massmotionforspectroscopicfactorscalculatedintheharmonicoscillatorbasis[55].Forthe2case,itispossibletoremovean`=1protonandproduce1=2+,3=2+,and5=2+statesin19C.Therefore,theinclusivecrosssectionwascalculated˝rstforgroundstateandobserved3=2+excitedstate,andadditionallyincludinga5=2+state.Thee˙ectiveseparationenergyofthisstatewasbasedontheexcitationenergy(269keV)fromthegamma-raystudywhich˝rstsuggestedabound5=2+state[89].Becausetheexcitationenergiesaremuchsmaller104Table5.2:Theoreticalcrosssectionscomparedtotheexperimentalvalues,assuminga0groundstatein20N.Inthiscase,the5=2+stateisnotaccessiblethroughremovalofan`=1proton.Ex(keV)Jˇ`,j˙sp(mb)C2S˙th(mb)˙exp(mb)01=2+1,1/212.10.8510.82093=2+1,3/213.00.0590.81Inclusive11.60.76(10)thantheprotonseparationenergy,thesingle-particlecrosssectionsdidnotdependlargelyonthechoiceofenergyofthisstate.Fora0groundstatein20N,itisimpossibletoaccessa5=2+statefrom`=1removal,sothiscalculationonlycompareswiththepresentlyobservedstates.Inallcasesofgroundstatespinsin20N,thecalculationsexceedtheexperimentalvalue.Witha2groundstate,thecalculatedcrosssectiontothe1=2+and3=2+statesin19Cis1.88mb,whichisonthesameorderastheexperiment.Whenincludingthe5=2+state,thecrosssectionincreasesbyanorderofmagnitudeto13mb.Forthecaseofa0groundstate,thecrosssectionisalsomuchlargerthantheexperiment,at11.6mb.Thelargedi˙erenceincrosssectionsbetweenthecalculationsisduetothedi˙erentspectroscopicfactorsobtainedforthetwopossiblegroundstatesin20N.Thedominantcon˝gurationsforthe2and0statesareillustratedinFigure5.3.IntheWBPinteraction,the2statein20Nisprimarilyformedfromtheˇp1=2(d5=2)5couplingofthevalenceprotonandvalenceneutrons,withastrengthof61%.Withthiscon˝guration,removalofanyp-waveprotonleavestheneutronsinthed5=2orbital.AsdiscussedinSection4.6.1,thesecon˝gurationsformsonlyasmallpartofthe1=2+and3=2+statesin19C,bothofwhicharedominatedbythe(d5=2)4(s1=2)1and(d5=2)3(s1=2)2con˝gurations.105Figure5.3:Theprimarycon˝gurationsforthelowest2(left)and0statesin20N,calculatedwiththeWBPinteraction.Protonsareshowninred,andneutronsinblue.The2con˝gurationisformedbythecouplingofthevalence0p1=2protontothe(0d5=2)5J=5=2neutrons.Theshowncon˝gurationaccountsfor61%ofthetotalstrengthofthe2state.The0con˝gurationisformedbythecouplingofthesameprotontothe(0d5=2)4J=01s1=2neutrons.Thisaccountsfor74%ofthetotalstrengthinthe0state.5.3MomentumDistributionsTheparallelmomentumpkoftherecoiling19CfragmentswasmeasuredattheS800focalplane.TheinclusivemomentumdistributionwasseparatedintocomponentsforthegroundandexcitedstatebygatingongammaraysdetectedbyGRETINA.Amomentumspectrumwas˝rstobtainedbyselectingeventswithagammaraybelow300keV,sincethisisthehighestobservableenergyforforward-angledetectors.Thisspectrumwascomposedofamixofbothtruedecaysfromthe209-keVstateandlow-energybackground.Aseparatespectrumwasproducedbyselectinggammaraysabove300keV,whereonlybackgroundeventsoccur.Thebackgroundmomentumspectrumwasscaledtotheexpectedbackgroundrateinthemixedspectrumandsubtractedfromthemixedspectrumtoproducethedistributionforthe209-keVstate.Thisensuredthattheshapeofthedistributionbestcorrespondedtotrue106Figure5.4:Themomentumdistributionalongthebeamaxispkforthe209-keVstatein19Cfollowingtheone-protonknockoutof20N.TheblackpointsarethedatameasuredintheS800.Eikonalcalculationsareshownfors-wave(bluesolidline),p-wave(reddashedline),andd-wave(greendot-dashedline)protonremoval.Thecalculationshavebeennormalizedtothedataintheregionbetween0MeV/c,wheretheparticleswerefullyacceptedbytheS800.decayevents.Thisspectrumwasthenscaledbasedontheaveragee˚ciencytodetectgammaraysfromthe19Cbeam.Becauseofthelonglifetimeoftheexcitedstate,mostdecaysdonotoccuratthetarget,sothee˚ciencywasestimatedbasedontheoveralle˚ciencyofthesimulation.TheresultisshownbytheblackpointsinFigure5.4.Finally,themomentumdistributionfordirectknockouttothegroundstatein19Cwasobtainedbysubtractingtheexcited-statedistributionfromtheinclusivedistribution.ThisisshownbytheblackpointsinFigure5.5.Theoreticalmomentumdistributionswerecalculatedusingtheeikonalprescription[23].Theoutputofthecalculationsisasymmetricdistributioninthecenter-of-massframeoftheincomingsecondarybeamandtargetnuclei.Tocomparewithdata,severalcorrections107weremade.Eachdistributionwas˝rstwidenedtothelab-framewidthbytheLorentzfactor=1:076,andtransformedtothemeasuredcentralmomentumofthe19Cproducts.Thelab-framedistributionwasthenfoldedwiththeobservedmomentumoftheincoming20Nbeam.Finally,thedistributionwaswidenedbyanadditional5%toaccountforthedi˙erenceinenergylossbetweenthehigh-andlow-energyportionsofthebeamthroughtheberylliumtarget.TheresultsareshowninFigure5.4forthe209-keVstateandinFigure5.5forthegroundstate.Calculationsweremadeforprotonremovalfroms-orbitals(`=0),p-orbitals(`=1),andd-orbitals(`=2).Thedistributionshavebeennormalizedtotheexperimentaldata.InFigure5.4,itisclearthatthedatamatcheitherthe`=0or`=1curves.However,thetotalmomentumspreadofthenucleiexceedsthe5%acceptanceoftheS800,sotheshapeofthedistributioniscuto˙athighandlowmomentum.Inthecentralregion,wherenearlyallofthebeamcanreachtheS800,the`=1distributionisaslightlybetter˝t.Thismatchestheresultsfromthecrosssection-analysis,wherethebestagreementwiththeorycomesfrom`=1protonremoval.InFigure5.5,thedi˙erentdistributionscannotbedistinguishedbecauseofthelargeerrorbars,buta2groundstatein20Nwouldagainrequire`=1removaltoreachthe1=2+groundstatein19C.5.4DiscussionSystematicerrorsoftheinclusiveandexclusivecrosssectionsareconsideredseparately.TheerrorsaresummarizedinTable5.3.Thedominantsourceoferrorinthetotalcrosssectionisduetotheuncertaintyinthee˚ciencytodetect19Cnuclei.Theuncertaintycanbebrokendownintotwofactors:themomentumacceptanceoftheS800,andtheoverallshapeofthemomentumdistribution.UncertaintyduetotheS800acceptanceoccursbecausetotal108Figure5.5:Themomentumdistributionalongthebeamaxispkforthegroundstatein19Cfollowingtheone-protonknockoutof20N.TheblackpointsarethedatameasuredintheS800.Eikonalcalculationsareshownfors-wave(bluesolidline),p-wave(reddashedline),andd-wave(greendot-dashedline)protonremoval.Thecalculationshavebeennormalizedtothedataintheregionbetween0MeV/c,wheretheparticleswerefullyacceptedbytheS800.109momentumspreadofthe19Cfragmentsisgreaterthanthe5%acceptanceoftheS800.Be-causethe19CbeamwascenteredintheS800,thisresultedinthebeambeingcuto˙forboththehighandlowenergysidesofthedistribution.Thetotalcrosssectionwascalculatedbymatchingthe`=1theoreticalmomentumdistributiontotheexperimentaldistributioninthecentralregionwheretheacceptanceisnearunity.Thisregionwasde˝nedbythemagnetsettingsoftheS800,andcorrespondedto67MeV/c.Theremainingcrosssectionfromtheparticleswhicharecuto˙wasobtainedbyscalingthetotalcountsinthecentralregionbythetheoreticalfractionofcountsinthisregion.Thisresultedinanestimated10%ofparticleswhichdidnotappearinthespectrum,andthiswastakenastheerrorontheacceptance.Theshapeofthemomentumdistributionalsointroduceserrorinthecrosssectiondetermination.Intheeikonalmodel,thedistributionissymmetric;however,severalpreviousknockoutexperimentshaveobservedasymmetriesinthefragmentmomentumdis-tributions,appearingastailsatlowenergyDi˙erenttheoreticalapproacheshavebeenusedtoexplainthesetails.Forremovalofweaklyboundneutronsfromhalonuclei,coupleddiscretizedcontinuumchannel(CDCC)calculationshavebeensuccessfullydescribedthedistributions[120],whilethetailformedfromremovalofstronglyboundnucleihasbeenexplainedasanenergydissipationmechanismduringthestrippingreaction[121].Becausethelow-energyportionofthedistributionisnotobserved,thepossibilityofatailaddsanadditionalestimated10%uncertainty.Othercontributionstotheerrorinthetotalcrosssection,forexampleduetothegateofselecting19Cintheparticleidenti˝cationspectrum,aremuchsmallerandcanbeconsiderednegligible.Inthecalculationofexclusivecrosssections,thelargesterrorcomesfromdeterminationofthein-beame˚ciencyofthegamma-raydetectors.Twoconsiderationsmustbemade:thee˚ciencyasafunctionofgamma-rayenergyandasafunctionofpositionofthegamma110Table5.3:Summaryofsystematicerrorsobservedfortheknockoutcross-sectionmeasure-ment.Errorsfortheinclusivecrosssection(left)andexclusivecrosssectiontotheexcited3=2+state(right)arelistedseparately.InclusiveExclusiveComponentError(%)ComponentError(%)S800acceptance10Simulatedabsolutee˚ciency9Momentumdistributionasymmetry10Simulatedpositione˚ciency10Particleidenti˝cation<1Backgrounddetermination2decay.Thelonglifetimeofthe3=2+statecausesdecaystooccur,onaverage,about20cmdownstreamofthetarget.BecausesourcecalibrationsforGRETINAwereonlymadeat˝xedpositionsofthetarget,itwasnecessarytorelyonsimulationstodeterminethee˚ciency.AsshowninFigure3.8(a),theenergydependenceiswellreproducedfortheline-shapecon˝guration.However,thescalingfactorrequiredtoreproducethee˚ciencyintheplungercon˝gurationmeansthattheerrorinthesimulatede˚ciencycanbe10%.Thereisalsoerrorinthesimulationduetotheabsorptionofgammaraysinthesurroundingmaterial.Becausetheshapeofthee˚ciencycurveiswellmatchedfortheline-shapecon˝gurationwherethereisnoheavydegrader,thisadditionalerrorisminimal.Forthepositiondependenceofthee˚ciency,itisnotpossibletocomparetoanyrealdata,sothereisanadditionaluncertaintyof10%.Anotherpossiblesourceoferroristhedeterminationofabsolutecountsinthegamma-rayspectrum,whichdependsonthechoiceofbackgroundassumed.Inthepresentcase,theerrorwasestimatedbycomparingthelow-energybackgroundincoincidencewith9Liand18Crecoils.Thedi˙erencebetweenthesetwocasesmadeasmall(1%)contributiontotheerror.Asdiscussedabove,alltheoreticalcrosssectionsfortheinclusivecrosssectionexceedtheexperimentalvalueof0.76mb.Asimilardi˙erencebetweenexperimentandtheoryhasbeen111Figure5.6:Aplotshowingthesystematicdi˙erencebetweenexperimentalandtheoreticalone-nucleoncrosssections.Theplotincludesproton(blue)andneutron(red)knockoutusingfastbeamsaswellasprotonknockoutfromelectronscattering(black).ThereductionfactorRsisplottedasafunctionofS,whichisameasureofthedi˙erenceintheFermienergiesoftheprotonsandneutrons.FigurefromRef.[123].112foundforseveralknockoutreactionsatenergiesnearorabove100MeV/nucleon[123].Atrendhasbeenobservedinwhichremovalofmorestronglyboundnucleonsresultsinalargerdi˙erencebetweentheexperimentandtheory.ThistrendisshowninFigure5.6.Here,areductionfactorRs=˙exp=˙thisde˝nedastheratiobetweentheexperimental(˙exp)andtheoretical(˙th)crosssections.ThisisplottedagainstanasymmetryparameterSfortheprojectilenucleus.Thisparameterisde˝nedasthedi˙erencebetweentheprotonandneutronseparationenergies,andservesasmeasureoftherelativeFermisurfacesofprotonsandneutrons.Thede˝nitionofSisdi˙erentforprotonorneutronknockout,andde˝nestherelativebindingoftheremovednucleon.Therefore,pointsontheleftarefromremovalofweaklyboundnucleons,wherethereductionfactorisnearunity,andpointsontherightareduetoremovalofstronglyboundnucleons,wherethereductionfactorismuchsmaller.For20N,theseparationenergiesareSn=2:16MeVandSp=17:94MeV[87],sothatS=15:78MeV.FromthetheoreticalcalculationsinSection5.2.1,theclosestvalueof1.88mbresultsfroma2groundstatein20Nandnobound5=2+statein19C.Forthisvalue,thereductionfactorRsis0.40.LookingatFigure5.6,thisvaluefallsinlinewiththeobservedtrend.Othertheoreticalcalculationswhichassumeadi˙erentgroundstatein20Norabound5=2+statein19Cgivecrosssectionsnearlyanorderofmagnitudelarger,resultinginreductionfactorsbelow0.07,muchlowerthananyotherpreviousvalue.Thus,thetrendintheknockoutcrosssectioncalculationssupportsthe2groundstatein20N,aswellastheunboundnatureofthe5=2+state.OnepointtonoteinFigure5.6istheneutron-knockoutreactionfrom20C[80],whichappearsasaredpointatS=26MeVandRsˇ1.Thisreactionalsoproduces19C,sothetheoreticalcalculationsaresimilartothepresentwork.Thetworesultsatoppositeendsofthe˝gurehighlighttheobservedtrendaswellastheneedforimprovedtheoreticalframeworktoexplainthetrend.113Inadditiontothegamma-raymeasurement[93],thecurrentmeasurementprovidesnosupportforabound5=2+statein19C.Thisisinlinewithneutronknockoutreactionsfrom20C[80,86],includingthemostrecentmeasurement[124].Thus,thereisnowaconsistentpictureof19Cwhichexcludesabound5=2+state.114Chapter6ConclusionsThisworkpresentedtwoexperimentalstudieswhichweredesignedtounderstandthestruc-tureoftheone-neutronhalonucleus19C.Alifetimemeasurementofthe˝rstexcitedstateprovidedthe˝rstmeasurementofanM1transitionbetweenboundstatesinahalonucleus.ThemeasuredB(M1)transitionstrengthconstrainedthespin-parityoftheexcitedstate.Datafromtheone-protonknockoutreactionof20Nusedinthesameexperimentalsosup-portedthespin-parityassignmentofthisstate.Theknockoutreactioncrosssectionwasalsousedtoexaminethespinofthegroundstatein20N.Thelifetimeofthe˝rstexcitedstateat209keVin19Cwasmeasuredusingtwocom-plementaryDoppler-shifttechniques,theRecoilDistanceMethodandline-shapetechnique.Forbothmethods,thelifetimewasdeterminedby˝ttingsimulatedgamma-rayspectratotheexperimentalspectra.Theresultsfromthetwotechniqueswereconsistent,andthelifetimeofthe209-keVexcitedstatewasmeasuredtobe1.94(15)ns.Thislifetimeex-cludeda5=2+!1=2+transitionandprovidedfurthersupportforthesuggested3=2+assignmentoftheexcitedstate.ThecalculatedB(M1;3=2+!1=2+)transitionstrengthof3:21(25)1032N,or1:79(14)103W:u:,indicatedastronglyhinderedtransition,andwasshowntobeoneoftheweakestM1transitionsamonglightnuclei.Shellmodelcalculationswereperformedinordertounderstandtheoriginofthehin-dranceofthe3=2+!1=2+transitionin19C.Theresultsofthecalculationsindicatedthatthehindrancewasduetothedominanceofs-wavecontributionstobothground-state115andexcited-statecon˝gurations.Thedominanceofthesecon˝gurationswerecausedbytheneardegeneracyoftheenergiesofthe1s1=2and0d5=2orbitalsin19C,whichisageneraltrendforlightneutron-richnuclei.Toaccountforthe˝nitestrengthofthetransition,amodi˝edM1operatorincorporatingatensorcomponentwasfoundtoplayasigni˝cantroleinthetransitionbyallowingthe`=2transitionbetweentheneutron1s1=2and0d3=2orbitals.Theimportanceofthetensortermwasshowntobeauniquefeatureof19C,whilesmallB(M1)valuesobservedinothersd-shellnucleiwereduetocancellationofindividualmatrixelements.Thisworkwasthe˝rstmeasurementofamagnetictransitionbetweenboundstatesinahalonucleus,andtheresultsestablishthehinderedM1transitionasacharacteristicofs-wavehalonuclei.Thestructureof19Cwasalsostudiedviaaone-protonknockoutreactionfromafastbeamof20N.Themeasuredinclusivecrosssectionwascomparedtotheoreticalcalculationsbasedontheeikonalreactionmodel.Twosetsofcalculationswereperformedbasedondi˙erentassumptionsfortheground-statespinof20N.Thecomparisonwasmadetodistinguishthepossiblespin-parityassignmentofthegroundstate.Thetwocalculationsdi˙eredbyanorderofmagnitude,o˙eringacleardistinctionbetweentheresults.Thebestagreementbetweenexperimentalandtheoreticalcrosssectionswasobtainedwitha2groundstatein20N.Thisstateisformedfromap1=2protoncoupledtoa5=2+con˝gurationinthe19Ccore.Thiscon˝gurationinthegroundstateof20Nindicatesthatthe1s1=2and0d5=2orbitalsarenolongerdegeneratein20N.Themeasuredinclusivecrosssectionof0.76(10)mbwassmallerthantheoreticalpredictions,andareductionfactorwasdeterminedfromthedi˙erencebetweentheexperimentalandtheoreticalvalues.Thereductionfactorof0.40obtainedforthe2groundstatein20Nwasconsistentwithageneraltrendestablishedbyseveralknockoutreactionstudiesspanningawiderangeofbindingenergies.Thedataobtainedin116thisexperimentmakeavaluableadditiontothistrend,andcanaidinunderstandingtheoriginofthereductionfactor.Inboththelifetimeandknockoutanalyses,noevidencewasfoundfortheexistenceofabound5=2+statein19C.Nogamma-raytransitionfromasecondstatein19Cwasobservedinthelifetimemeasurement,andthelargeincreaseintheknockoutcrosssectionfrom20Npredictedfrominclusionofthebound5=2+stateruledoutthepossibilityofsuchastate.Theresultsoftheseanalysespresentaconsistentpictureof19C,withasinglebound3=2+stateat209keV.Thisdescriptionisalsoinagreementwithneutron-knockoutreactionsfrom20C,whichsimilarlyexcludesuchabound5=2+state.Thepresentlifetimemeasurementcanserveasabenchmarkforfuturemeasurementsofthemagneticresponseinhalonuclei.Heaviersystemswithdeformedp-wavehalosmayreveallargerM1strengthsbecauseofthepresenceofaspin-˛ippartner,andthepresentmeasurementestablishescriteriatodeterminetheoccurrenceofsuchatransition.Thepresentknockout-reactionmeasurementdemonstratesthatthedegeneracyofthe1s1=2and0d5=2orbitalsobservedin19Cdoesnotpersistin20N.ToestablishthistrendamongtheN=13isotonesmorequantitatively,furtherstudiesontheexcitedstatesin20Narenecessary.Suchresultswillbeimportantinthedeterminationofbroadtrendsinthenuclearstructureofneutron-richnuclei.117REFERENCES118REFERENCES[1]ofhttp://www.nndc.bnl.gov/chart.[2]M.G.Mayer,Phys.Rev.75,1969(1949).[3]K.Heyde,BasicIdeasandConceptsinNuclearPhysics,2nded.(InstituteofPhysicsPublishing,London,1999).[4]J.Suhonen,FromNucleonstoNucleus(Springer-Verlag,Berlin,2007).[5]O.Haxel,J.H.D.Jensen,andH.E.Seuss,Phys.Rev.75,1766(1949).[6]R.F.Casten,NuclearStructurefromaSimplePerspective(OxfordUniversityPress,NewYork,1990).[7]A.Ozawa,T.Kobayashi,T.Suzuki,K.Yoshida,andI.Tanihata,Phys.Rev.Lett.84,5493(2000).[8]S.N.Liddick,P.F.Mantica,R.Broda,B.A.Brown,M.P.Carpenter,A.D.Davies,B.Fornal,T.Glasmacher,D.E.Groh,M.Honma,M.Horoi,R.V.F.Janssens,T.Mizusaki,D.J.Morrissey,A.C.Morton,W.F.Mueller,T.Otsuka,J.Pavan,H.Schatz,A.Stolz,S.L.Tabor,B.E.Tomlin,andM.Wiedeking,Phys.Rev.C70,064303(2004).[9]T.SuzukiandT.Otsuka,Phys.Rev.C56,847(1997).[10]H.Iwasaki,T.Motobayashi,H.Akiyoshi,Y.Ando,N.Fukuda,H.Fujiwara,Zs.Fülöp,K.I.Hahn,Y.Higurashi,M.Hirai,I.Hisanaga,N.Iwasa,T.Kijima,A.Mengoni,T.Minemura,T.Nakamura,M.Notani,S.Ozawa,H.Sagawa,H.Sakurai,S.Shimoura,S.Takeuchi,T.Teranishi,Y.Yanagisawa,andM.Ishihara,Eur.Phys.J.A13,55(2002).[11]C.Thibault,R.Klapisch,C.Rigaud,A.M.Poskanzer,R.Prieels,L.Lessard,andW.Reisdorf,Phys.Rev.C12,644(1975).[12]K.Tanaka,T.Yamaguchi,T.Suzuki,T.Ohtsubo,M.Fukuda,D.Nishimura,M.Takechi,K.Ogata,A.Ozawa,T.Izumikawa,T.Aiba,N.Aoi,H.Baba,Y.Hashizume,K.Inafuku,N.Iwasa,K.Kobayashi,M.Komuro,Y.Kondo,T.Kubo,M.Kurokawa,T.Matsuyama,S.Michimasa,T.Motobayashi,T.Nakabayashi,S.Nakajima,T.Nakamura,H.Sakurai,1R.Shinoda,M.Shinohara,H.Suzuki,E.Takeshita,S.Takeuchi,Y.Togano,K.Yamada,T.Yasuno,andM.Yoshitake,Phys.Rev.Lett.104,062701(2010).119[13]I.Tanihata,H.Hamagaki,O.Hashimoto,Y.Shida,N.Yoshikawa,K.Sugimoto,O.Ya-makawa,T.Kobayashi,andN.Takahashi,Phys.Rev.Lett.55,2676(1985).[14]I.Tanihata,H.Hamagaki,O.Hashimoto,S.Nagamiya,Y.Shida,N.Yoshikawa,O.Ya-makwa,K.Sugimoto,T.Kobayashi,D.E.Greiner,N.Takahashi,andY.Nojiri,Phys.Lett.B160,380(1985).[15]T.Kobayashi,O.Yamakawa,K.Omata,K.Sugimoto,T.Shimoda,N.Takahashi,andI.Tanihata,Phys.Rev.Lett.60,2599(1988).[16]N.A.Orr,N.Anantaraman,S.M.Austin,C.A.Bertulani,K.Hanold,J.H.Kelley,D.J.Morrissey,B.M.Sherrill,G.A.Souliotis,M.Thoennessen,J.S.Win˝eld,andJ.A.Winger,Phys.Rev.Lett.69,2050(1992).[17]T.Aumann,D.Aleksandrov,L.Axelsson,T.Baumann,M.J.G.Borge,L.V.Chulkov,J.Cub,W.Dostal,B.Eberlein,Th.W.Elze,H.Emling,H.Geissel,V.Z.Goldberg,M.Golovkov,A.Grünschloÿ,M.Hellström,K.Hencken,J.Holeczek,R.Holzmann,B.Jonson,A.A.Korshenninikov,J.V.Kratz,G.Kraus,R.Kulessa,Y.Leifels,A.Leis-tenschneider,T.Leth,I.Mukha,G.Múnzenberg,F.Nickel,T.Nilsson,G.Nyman,B.Petersen,M.Pfützner,A.Richter,K.Riisager,C.Scheidenberger,G.Schrieder,W.Schwab,H.Simon,M.H.Smedberg,M.Steiner,J.Stroth,A.Surowiec,T.Suzuki,O.Tengblad,andM.V.Zhukov,Phys.Rev.C59,1252(1999).[18]T.Nakamura,N.Fukuda,T.Kobayashi,N.Aoi,H.Iwasaki,T.Kubo,A.Mengoni,M.Notani,H.Otsu,H.Sakurai,S.Shimoura,T.Teranishi,Y.X.Watanabe,K.Yoneda,andM.Ishihara,Phys.Rev.Lett.83,1112(1999).[19]R.Palit,P.Adrich,T.Aumann,K.Boretzky,B.V.Carlson,D.Cortina,U.DattaPramanik,Th.W.Elze,H.Emling,H.Geissel,M.Hellström,K.L.Jones,J.V.Kratz,R.Kulessa,Y.Leifels,A.Leistenschneider,G.Münzenberg,C.Nociforo,P.Reiter,H.Simon,K.Sümmerer,andW.Walus(LAND/FRSCollaboration),Phys.Rev.C68,034318(2003).[20]T.Kobayashi,S.Shimoura,I.Tanihata,K.Katori,K.Matsuta,T.Minamisono,K.Sugimoto,W.Müller,D.L.Olson,T.J.M.Symons,andH.Wieman,Phys.Lett.B232,51(1989).[21]N.Kobayashi,T.Nakamura,Y.Kondo,J.A.Tostevin,Y.Utsuno,N.Aoi,H.Baba,R.Barthelemy,M.A.Famiano,N.Fukuda,N.Inabe,M.Ishihara,R.Kanungo,S.Kim,T.Kubo,G.S.Lee,H.S.Lee,M.Matsushita,T.Motobayashi,T.Ohnishi,N.A.Orr,H.Otsu,T.Otsuka,T.Sako,H.Sakurai,Y.Satou,T.Sumikama,H.Takeda,S.Takeuchi,R.Tanaka,Y.Togano,andK.Yoneda,Ph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