ISOSPINMIXINGANDTHE30P(p;)31SREACTIONRATEByMichaelBennettADISSERTATIONSubmittedtoMichiganStateUniversityinpartialfulllmentoftherequirementsforthedegreeofPhysics{DoctorofPhilosophy.2016ABSTRACTISOSPINMIXINGANDTHE30P(p;)31SREACTIONRATEByMichaelBennettThe30P(p;)31SreactionrateiscriticalformodelingthenalelementalandisotopicabundanceofONenovanucleosynthesis,thecalibrationofproposednovathermometers,andtheidenticationofpresolarnovacandidategrains.Unfortunately,therateisessen-tiallyunconstrainedexperimentally,despitenumerousstudiesusingavarietyofexperimentaltechniques,largelyduetouncertaintiesinthespinsandparitiesofthenarrow,isolated31Sprotoncaptureresonancestatesthatlikelygoverntheprotoncapturereactionrate.Thebetadecayof31Cl,whichpreferentiallypopulatesimportantl=030Pprotoncaptureresonancesin31S,isausefultoolforstudyingthepropertiesofthesestates.Usinganacceleratedbeamof31Cl,wehaveobservedthebeta-delayedgammadecayofanumberof31SstatesuptoexcitationenergyEx=7200,includingstateswithinthe30P(p;)31SGamowwindowforpeaknovatemperatures.Hereinwereporttheresultsofthisstudy,including:theproductionofa31Clbetadecayschemewithovertwiceasmany31Sgammatransitionsasthemostrecentliteraturescheme;theobservationofisospinmixingofaresonancewithisospinT=1=2atEx=6390:2(7)keVwiththenearbyT=3=231Sisobaricanalogstate(IAS),givingitanunambiguousspinandparityof3=2+;andtheclearidenticationofthesecondT=3=231Sstateandtheresultsofseveraltestsoftheisobaricmultipletmassequationforboththelowestandsecond-lowestA=31;T=3=2quartets.Copyrightby
MICHAELBENNETT
2016Ofmakingmanybooksthereisnoend,andmuchstudyisawearinessoftheesh.Eccl.12:12ThisdissertationisdedicatedtoH.MichaelSommermannandWarrenF.Rogers,therstpeopletoteachmethatbeingagoodphysicistismorethanthemakingofmanybooks.ivACKNOWLEDGMENTSOfcourse,noneofthiscouldhavebeenpossiblewithouttheworkofmyadvisor,Dr.ChrisWrede.Chrishasatremendousabilitytoseekoutimportantproblemsinnuclearastrophysicsandconceiveofnewwaystoanswerthosequestions,andIamverygratefulfortheopportunitytohavebeeninvolvedinmultipleimportantprojectsashisgraduatestudent.ChriswaspatientwithmeasImadebasicallyeverymistakeitispossibletomakeasagradstudent,andcontinuouslyencouragedmetobecomethebestresearcherIcould.IamtrulyindebtedtohimforhelpingmedevelopasmuchasIdidduringmytimeatMichiganState.Ialsohavetothankmyincrediblethesiscommittee:B.AlexBrown,LauraChomiuk,CarlSchmidt,HendrikSchatz,andUlrikeHager.Theirinsightsthroughouttheprocessofplanning,executing,andanalyzingtheresultsofmythesisexperimenthelpedtomakemeabetterandmorethoughtfulscientist.Ulrikeespeciallydeservesspecialthanksforexhibitingsuperhumanexibilityandsteppinginatthelastmomenttowitnessmydissertationdefensewhenacommitteememberwasunavailable.InadditiontomyadvisorChris,IamindebtedtotheothermembersoftheWredegroupfortheirsupport,especiallyDavidPerezLoureiro.WithoutDavid'scodingexpertise,theanalysisofmythesisexperimentwouldhavebeentremendouslyinecient.Davidnotonlyhelpedmewiththesortoftheexperimentdata,buthelpedmedevelopmyowninsightintohowtheanalysissoftwarecouldbeused.SarahSchwartz,BrentGlassman,andCathleenFryallprovidedassistanceandsupportbefore,during,andaftertheexperiment,andIamgratefultohavenotbeenanacademic\onlychild."IcanonlyhopeIprovidedthemasmuchsupportintheirowngrowth.vTheexperimentitselfwouldhavebeenaspectacularfailureifIhadbeenlefttomyowndevicestoprepareforandexecuteit.Ourwonderfulcollaboratorsalldeservethanksinthisarea,butIamgreatlyindebtedtoSeanLiddickandhisBetaGroup,especiallyChrisProkop,NickiLarson,andScottSuchyta,fortheirhelpwithsimulations,detectorsetupandmaintenance,calibration,dataacquisition,andmanyotherthings.OtherNSCLfacultyandstaplayedimportantrolesintheexperimentaswell,fromCraigSnowandthedesigngroup,toTomGinterandtheA1900group,toJimVincent,KurtKranz,andtheotherelectricalengineers,tothecyclotronoperatorswhoworkedtirelesslytosavetheexperimentwhentheacceleratorfailedwithintenminutesofthestartofdatataking.TheNSCLcommunitycontributedmanyseemingly-smallbutvitally-importantthingstotheexperiment,andIcannotthanktheworkershereenough.Graduateschoolisrarelyapleasantandsimpleendeavor,andatanumberoftimesthroughoutmytenureasastudentatMSU,Idespairedofsucceedingatthechallenge.Iamluckytohavehadsomanyincrediblefriends,frommyDungeons&Dragonsgroup(thanksforthreeyearsofincrediblestorytelling{butit'sspelled\T-H-I-N-N-U-S")tomydie-hardboardgamefriendswhodidn'tshyawayfromfourteen-hourspaceoperagames(\MAUOOOOB!")tomybrothersatHouseChaosDunk(therearetoomanyinsidejokestochoosejustonehere),tothosewillingtodropeverythingandmeettodiscussphilosophyatPeanutBarrelatoneinthemorning,toAlannaPawlak,whohasstoodbymethroughthickandthinforthepasttwoyears.Academiacanbealonely,nervousplace,andithasbeenalife-savertohavecalming,supportiverelationshipswithpeoplewhohavealwaysmademefeellikeIbelong.MyfamilydeservespraisefortheirconstantsupportandpatienceduringtheveyearsittookmetocompletemyPh.D.TheyneverwaveredinbelievingIcouldachievethismilestone,viandIcannotthankthemenoughfortheirlifelongsupportofmyacademicprogression.TABLEOFCONTENTSLISTOFTABLES....................................xiLISTOFFIGURES...................................xiiiChapter1Introduction...............................11.1OverviewofPertinentPhysics..........................11.1.1AnIntroductiontoNuclearPhysics...................11.1.2NuclearAstrophysicsandtheLivesofStars..............41.2ClassicalNovae..................................7Chapter2NuclearAstrophysics:MotivationforStudyof30P(p;)31S..112.1ElementalandIsotopicAbundances.......................112.1.1MeteoritesandTheirAnalysis......................132.1.2IsotopicAbundancesinClassicalNovae.................142.1.3MoreNucleosynthesis:NovaThermometers..............182.2PresolarGrains..................................202.2.1PresolarNovaGrains...........................22
2.2.2ComparisonwithNovaModels......................252.3The30P(p;)31SReactionRate:PurposeofThisWork............26Chapter3FormalismandTheoreticalConsiderations............283.1DerivationoftheThermonuclearReactionRate................293.2DirectandResonantCaptureReactions....................333.3ReactionRate:TheoreticalConsiderations...................363.3.1TheHauser-FeshbachStatisticalModel.................373.4TheNuclearShellModel.............................383.5MirrorNucleiandtheConceptofIsospin....................423.5.1IsospinandBetaDecay..........................453.5.2SelectionRules..............................46
3.5.3TheIsobaricMultipletMassEquation.................473.5.4IsospinMixing..............................48Chapter4ExperimentalConsiderationsandNSCLExperiment12028..524.1TechniquesforIndirectStudyof30P(p;)31S.................554.1.1Single-NucleonTransferReactions....................554.1.2In-BeamGamma-RaySpectroscopy...................564.1.3Charge-ExchangeReactions.......................564.1.4BetaDecay................................574.2PreviousStudiesof30P(p;)31S........................584.2.1CurrentStateof31SExperimentalUnderstanding...........614.3NSCLExperiment12028.............................62viii4.3.1BeamProductionattheCoupledCyclotronFacility..........634.3.2BeamPuricationUsingtheRFFS...................664.3.3ExperimentalSetup............................684.3.4BriefDiscussionofDetectorMechanisms................754.3.5TheNSCLDigitalDataAcquistionSystem...............78Chapter5E12028AnalysisProcedureandResults..............835.1DataProcessing..................................845.1.1OnlineAnalysis..............................865.2CalibrationProcedure..............................895.2.1EnergyCalibrations...........................895.2.2ThePeakShape..............................93
5.2.3EnergyCalibrationSystematics.....................955.2.4EciencyCalibration...........................965.2.5Eciency/IntensitySystematics.....................985.3DataReductionandFinalResultsSpectra...................1005.3.1GammaDecaySelectionRules......................1065.3.2PhotopeakIdentication{BuildingtheDecayScheme........1095.3.3SpinandParityAssignments:......................1155.3.4BetaFeedingsandGamma-rayAbsoluteIntensities..........1165.4AnalysisAnomalies................................1205.4.1ForbiddenTransitions:TheLevelsat3349keVand4970keV....1205.4.1.14970-keVLevel.........................122
5.4.1.23349-keVLevel.........................1285.4.2OverlappingPhotopeaks:the1368-keV,2100-keV,and6129-keVGammaRays....................................129
5.4.2.1The2100-keVPeak.......................130
5.4.2.2The1368-keVPeak.......................131
5.4.2.3The6129-keVPeak.......................134Chapter6DiscussionofE12028Results.....................1416.1IsospinMixing:ANewJˇ=3=2+Resonance.................1416.1.1AstrophysicalRelevanceandImplications................1476.2IsobaricMultipletMassEquationStudiesandtheSecond31ClT=3=2State1526.2.1LowestA=31;T=3=2Quartet.....................1526.2.2SecondA=31;T=3=2Quartet.....................1596.3DiscrepancieswiththeNuclearDataSheetsandComparisontoPreviousWork1646.3.1UnobservedPreviously-ReportedTransitions..............1656.3.2SpinandParityDiscrepancies......................166Chapter7Outlook:TheFutureof30P(p;)31S.................1697.1Conclusions....................................1697.2OutlookandFutureWork............................1707.3FinalThoughts..................................172ixAppendices........................................174APPENDIXA:TheoreticalTools...........................175APPENDIXB:DetectorDataSheets.........................209BIBLIOGRAPHY....................................213xLISTOFTABLESTable5.1:31SlevelexcitationenergiesEx,beta-decayintensitiesIandcorrespond-inglog(ft)values,andtransitionsfromeachlevelobservedtobepopulatedinthebetadecayof31Cl(thedesignationJˇndenotesthenthstateofagivenspinandparity).Alsoincludedforeachtransitionarethegamma-rayenergiesE,relativegamma-raybranchingratios(B.R.),andabsolutegamma-rayintensityper100betadecays,I.................123Table5.2:Allgamma-raysobservedinE12028assignedtothebetadecayof31Cl.Foreachgamma-ray,thetransitionenergyEtandgammaenergyEaregiveninkeV.Thetransitionandtheabsoluteintensityper100betadecaysIarealsolistedforeachtransition........................126Table6.1:CalculatedexcitationenergiesExandmixingmatrixelementsVofthetripletofisospin-mixedstates,includingtheT=3=2IAS,in31SforbothUSDBandUSDEinteractions.ThematrixelementslistedarebetweenthelistedT=1=2stateandtheT=3=2state.AllvaluesareinunitsofkeV.147Table6.2:Thermonuclear30P(p;)31SreactionrateNAh˙viinunitsofcm3mol1s1asafunctionoftemperature(reportedinGK,commonlynotatedT9,ashere),forcommonly-attainednovatemperatures.Here\RC"denotestheresonantcapturethroughthe3=2+stateat6390keV.NAistheAvogadronumber.TherateispresentedherewithoutuncertaintylimitsbecausetheonlyexperimentaluncertaintyusedinthecalculationwasthatfortheresonanceenergyEr;thisuncertaintyaectedtheresonancestrengthbylessthan0.2%..................................149Table6.3:Ground-statemassexcessandexcitationenergyExvaluesusedasinputfortheIMMEtsofthelowestA=31;T=3=2quartet.Exceptfortheobservedexcitationenergyofthe31SIAS,whichisfromRef.[89],allvaluesarethesameasinRef.[102]......................153Table6.4:OutputcoecientsforthequadraticandcubicIMMEtsforthelowestA=31;T=3=2quartetusinginputdatafromTable6.3.AllcoecientvaluesareinunitsofkeV.Thecubictdidnotcontainanydegreesoffreedom,sothe˜2=valueisundenedandhenceommitted.......153Table6.5:Ground-statemassexcessandexcitationenergyExvaluesusedasinputfortheIMMEtsofthelowestA=31;T=3=2quartet.Exceptfortheunperturbedexcitationenergyofthe31SIAS,whichisfromthepresentwork[89],allvaluesarethesameasin[102].................154xiTable6.6:OutputcoecientsforthequadraticandcubicIMMEtsforthelowestA=31;T=3=2quartetusinginputdatafromTable6.5.AllcoecientvaluesareinunitsofkeV............................154Table6.7:CalculatedexcitationenergiesExandmixingmatrixelementsVofthetripletofisospin-mixedstatesincludingthelowestT=3=2statein31PforbothUSDBandUSDEinteractions.ThematrixelementslistedarebetweenthelistedT=1=2stateandtheT=3=2state.AllvaluesareinunitsofkeV...................................158Table6.8:Ground-statemassexcessandexcitationenergyExvalues[79]usedasinputfortheIMMEtsofthesecond-lowestA=31;T=3=2quartet...161Table6.9:OutputcoecientsforthequadraticandcubicIMMEtsforthesecond-lowestA=31;T=3=2quartetusinginputdatafromTable6.8.AllcoecientvaluesareinunitsofkeV......................161Table6.10:CalculatedexcitationenergiesExandmixingmatrixelementsVofthetripletsofstatesinvolvedinmixingwiththesecond-lowestT=3=2statesinboth31Sand31P,forbothUSDBandUSDEinteractions.ThematrixelementslistedarebetweenthelistedT=1=2stateandtheT=3=2state.AllvaluesareinunitsofkeV..........................163xiiLISTOFFIGURESFigure1.1:Thechartofnuclides.TheverticalaxisrepresentsthenumberofprotonsZinagivennucleusandthehorizontalaxisrepresentsthenumberofneutronsNinthenucleus.AllisotopesofagivenZpossessidenticalchemicalproperties,butthenuclearphysicspropertiesofagivenelementaredierentfromoneisotopetothenext.Theblacksquaresalongthecenterofthedistributionarethestablenuclei;thegreenregionrepresentsunstablenucleithathavebeenobserved,whiletheyellowregionrepresentsnucleiexpectedtoexistbutasofyetunobserved..............2Figure1.2:Thepp-Ichain,whichproduces4Hefromfusionoffourprotons(1H).Thearrowsindicatetheorderofprocesses,whilethetimesnotedatthebottomofeachsection(demarcatedbythedashedgraylines)denotethetimeforthereactiontooccurinastellarenvironmentlikethatoftheSun.Theextremelylongtimescaleoftherstreaction,p+p!2H+e++,isduetotherepulsiveCoulombbarrierbetweenthetwoprotons.ThislongtimescaleisosetbythecolossalnumberofprotonsintheSun,˘1057.TheppchaindominatesenergyproductioninstarsasmassiveastheSun.5Figure1.3:TheCNO-Icycle,which,similartotheppchain(Fig.1.2),producesa4Henucleusfromthefusionoffourprotons.Unlikethepp-chain,however,theCNOcycleiscatalytic,consumingthefourprotonsnecesaryfor4Heproductionbutnotthe12Cnucleusthroughwhichthe4Heiscreated.TheCNOcycledominatesenergyproductioninstarsofgreaterthanabout1.5M.......................................6Figure1.4:Anartist'sdepictionofaclassicalnovainabinarysystem.Theaccretiondisk,consistingofhydrogen-richmaterialowingfromthemainsequencecompanionstar(right),isshownattheequatorofthewhitedwarf(left).ImagecopyrightDavidA.Hardy,www.astroart.org............9Figure2.1:Solarsystemelementalabundances4.56billionyearsagoasafunctionofelementalnumberZ,plottedfromdatainRef.[16].Theverticalaxishasbeennormalizedsothatthehydrogenabundancevalueissetat1012.Theotherelementalabundancesarerelativetothatnumber........12xiiiFigure2.2:Asolarphotosphereabsorptionspectrumtakeninthevisibleregionoftheelectromagneticspectrumbetween392nm(blue)and692nm(red).Thedarklinesatparticularwavelengthsthroughoutthespectrumareduetoatomicabsorptionofphotonsofthatwavelength,correspondingtoatransitiontoanexcitedatomicstateofthatatom.Sinceeachatomicelementpossessesitsownenergyscale,thepatternofabsorptionlinesinthesolarspectrummaybeusedtoinfertheelementalmakeupofthephotosphere..................................13Figure2.3:Iron(topleft),stony-iron(top-right),achondrite(bottom-left),andchon-drite(bottom-right)meteorites.Allofthemeteoriteswiththeexceptionofthechondritehaveundergonevaryingdegreesofchemicaldierentiationasaresultofheating.PhotoCredits:Ji-Elle,DougBowman,Captmondo,andH.Raab.................................15Figure2.4:Anexcerptofthechartofnuclidesshowingthe\hot"CNOcycle.Asillustratedhere,thehotCNOcyclebreaksoutfromthe\cold"cyclewhen13Ncapturesaprotoninsteadofundergoingbetadecay;thisisduetotheincreasedtemperatureinthestellarenvironment,makingtherelativetimeforprotoncaptureshorterthanthe13Nbeta-decayhalf-lifeof10minutes.16Figure2.5:Asectionofthechartofnuclidesshowingtheextentofclassicalnovanucleosynthesis.Thedarkblueboxesrepresentstableisotopes,andeacharrowrepresentseitheraprotoncapture,betadecay,or(p;)reaction.Theredcirclenearthetopoftheguredenotestheendpointofnovanucleosynthesis,aroundcalcium.Figurecredit:Ref.[20].........17Figure2.6:Theeightnovathermometersproposedasthemostusefulforconstrain-ingpeaknovatemperaturesinRef.[25].Eachlinerepresentstheratioofthenotatedelementalabundancesasafunctionoftemperature.Thefourmoststeeply-varyinglinesallinvolveeitherphosphorusorsulfurabun-dances,makingprecisedeterminationoftheseelementalabundancesahighpriority.Figurecredit:Ref.[25]....................19Figure2.7:ASEMimageofasilicon-carbide(SiC)grain,takenfromtheMurchi-sonmeteorite,acarbonaceouschondrite.Notethegrain'slarge(severalmicrons)size.Isotopicanalysishasrevealedthatthisgrainisindeedapresolargrain,withanoriginbeforethesolarsystem.Photocredit:Max-Planck-InstitutFurChemie.........................23xivFigure2.8:Aplotoftheratiosofcarbonandnitrogenisotopicabundancesasdeter-minedforanumberofSiCpresolargrainsandsomegraphitegrains.Thedottedcrossinthecenteroftheplotdenotessolarsystemabundances,andthevariouspopulationsofgrains,denotedwithdierentsymbols,areshowntobedistinctbasedonthecombinationofthetworatios.Asanexample,theso-callednovagrainsareshownheretobedecientbynearlyanorderofmagnitudeinboth12Crelativeto13Candin14Nrelativeto15N,whencomparedwithsolarabundances.Figurecredit:Ref.[35]...24Figure2.9:Aplotoftheratiosoftheabundancesofthestablesiliconisotopes.Thedottedcrosscenteredattheorigindenotessolarsystemabundances.Thenotationusedhereis\permil:"29;30Si=28Si=[(29;30Si=28Si)=(29;30Si=28Si)1]1000,suchthataplacementof100alongtheaxisrep-resenta10%higherratiothansolar,andaplacementof1100representsa110%higherratio.Figurecredit:Ref.[35]................25Figure3.1:AnabstractguredenotingtheGamowwindowandtheGamowpeakforprotoncaptureinastellarenvironment.ThebluelineshowstheMaxwell-Boltzmannenergydistributionoftheparticleintheenvironment;thatis,itshowsthedecreasingprobabilityofndingaparticleatagivenenergyastheenergyincreases.Theredlineshowstheenergydependenceofthepenetrability;thatis,itshowsthatthehigher-energytheprotonis,themorelikelyitistoapproachnearenoughtothenucleustointeract.Thepurplelineshowsthecombinationofthesetwofactorstocreatearegionofincreasedreactionprobability,theGamowwindow.Thepurplelinehasbeenmultipliedbyafactorof100inthisplottoillustratetheeect...32Figure3.2:Asimplegureshowing(a)directprotoncaptureintoanucleusand(b)resonantcaptureintothesamenucleus.Indirectcapture,theinitialnucleuscombineswithaproton,thenemitsagamma-rayintoalowerboundstateofthecompoundnucleus.Theenergyofthegamma-rayisonlyafunctionoftheinitialnucleusenergy,theprotonenergy,andthenalstateenergy.Inresonantcapture,theinitialnucleuscapturesaprotonintoahigher,unboundstateinthecompoundnucleus.Thatstatemayeitherre-emittheprotonoremitagamma-ray,deexcitingintoalowerstateofthecompoundnucleus.Inthiscasetheenergyofthegamma-rayistheenergydierencebetweentheresonancestateandthelowerstate...................................34Figure3.3:TheWoods-Saxonpotentialcommonlyusedtomodelthenuclearforce.Theformofthepotentialis:V(r)=Vo=[1+exp(rR=a)]whereVoisthepotentialdepth,R=1:25A1=3fm(Athemassnumber),andarepresentsthe\surfacethickness"ofthenucleus..............39xvFigure3.4:Comparisonoftheenergylevelsofthenucleususingthesimpleharmonicoscillatorpotential(left)andusingthefullshellmodel(right).Asshown,thespin-orbitinteractionsplitstheharmonicoscillatorlevels,andthelargegapsinbindingenergythatcausetheshellsdonotnecessarilycor-respondtothegapsbetweenharmonicoscillatorlevels.Figurecredit:Bakken(GPL).................................40Figure3.5:Comparisonoftherstveenergylevelsof13Nand13C.Theground-stateenergiesofthetwolevelshavebeenadjustedtobeequal,andtherelativeheightsofthestatesineachnucleusrepresenttherelativeexcita-tionenergiesofthosestates.Thespinsandparitiesofthestatesarealsoshown,andcolorisgiventodenotepositiveornegativeparitystates.Therelativespacingoflevelsinagivennucleusisnottoscaleasthisgure'spurposeissimplytoshowthemirrorsymmetrybetweenthesetwonuclei.43Figure3.6:AsimplegraphicaldepictionofT=1=2isobaricdoubletsin13Cand13NandT=3=2isobaricquartetsin13B,13C,13N,and13O.Heretheenergylevelshavebeenequalizedtodemonstratethemanifestationofperfectisospinsymmetrymoreclearly.Asshown,thegroundstatesandmultipleexcitedstatesin13Nand13CmakeuptheTz=+1=2andTz=1=2membersofthersttwoT=1=2doublets,andcertainexcitedstatesof13Nand13CcomprisetheTz=+1=2andTz=1=2membersoftheT=3=2quartetscompletedbyanalogousTz=+3=2andTz=3=2statesin13Oand13B,respectively.Thespacingbetweenlevelsisarbitraryandismeantonlytoshowthesymmetryoftheisobaricanalogstates..44Figure4.1:AschematicofthecoupledcyclotronfacilityandtheA1900fragmentsep-arator.TheionsourcereleasesstableionstotheK500,whichacceleratestheionstoˇ13km/s.BetweentheK500andK1200theionspassthroughagoldfoilstripperwhichremovesmoreelectrons,thentheK1200acceler-atestheparticlestoapproximatelyhalfthespeedoflightbeforeimpingingthemupontheproductiontarget.ThefourdipolemagnetsoftheA1900areillustratedinred,withtheseparatorwedgeinbetweendipoles2and3inyellow.ThefocalplaneoftheA1900housesascintillatorwhichisusedtocountthebeamcurrentafterpurication.Afterthefocalplane,thebeamcanbedirectedintooneofseveralexperimentalvaults.....65Figure4.2:AschematicandphotographoftheRFFS.Thebeamentersthechamberatleft,experiencesdeectionaccordingtotheelectriceldthroughoutitsightbetweenthetwoelectrodeplates,andexitstotheright.TheRFcouplerattopisusedtocoupletheoscillationfrequencyoftheRFFStothatofthecyclotron.ThisgureisreproducedfromtheoriginalRFFSpaper,Ref.[66]................................69xviFigure4.3:Aparticleidenticationplotfor31Clshowing31Clandstrongestlikelycontaminants.Asdescribedabove,thebeamconstituentsareseparatedbothbytimeofight,inthiscasebetweentheA1900focalplanescintilla-torandthePINdetectorattheexperimentalsetup,andbyenergylossinthePINdetector.Itiscustomarytoplottimeofightonthehorizontalaxisandenergylossontheverticalaxis.Thelargeblobneartherightedgeoftheplotisthe31Cl,whiletheadditionalblobsarebeamcontam-inants,whichhavebeenidentiedeitherbytheirpresenceinthegammaspectrumorinferredfromtheexpectedbeamcompositionattheA1900providedbytheNSCLbeamgroup.Thehorizontallyrepeatingstructureislikelyanartifactcausedbyreectionwithintheelectronics,creatingafalsesignalwiththeappropriateenergybutinatedtimeofight....71Figure4.4:Asimpleschematicshowingthecentralscintillatorandthegermaniumcloverdetectors.Theninecloverssurroundthescintillatorasshown.The31Clbeamimplantsinthescintillator,whichdetectsthebetadecays(blue).Thesubsequentgammadecayofthedaughter31Snucleuscreatesgamma-rays(green)whicharedetectedbythecloverarray........73Figure4.5:AfullCADdrawingoftheexperimentalsetup,includingthepneumaticdriveattachedtothePINdetectors(leftinsert),thecentralscintillatorwhichwasattachedtothecloverframebyametalarm(rightinsert),andthefullcloverframewithallninecloverdetectors(bottom).Theliquidnitrogendewarsforthecloversareshowninlightblue,andthecloverdetectorsthemselvescanbeseenasthelightgrayextensionsintothecenterofthearray............................74Figure4.6:AsimpliedelectronicsdiagramofE12028.Datasources(bothdetectorsandNSCLapparati)aremarkedasgreenboxes.IntermediatemodulesandtheNSCLDDASaremarkedaswhiteboxes.Destinationsfordata(boththedataacquisitioncomputerandthepatchpanelstotheData-U)aremarkedasorangeboxes,andconnectionstoexteriorpatchesaremarkedasorangearrows.EnergyinformationwaspassedfromthesourcesandintotheDDASaccordingtotheblackarrows,andtiminginformationfortime-of-ightmeasurementswaspassedfromthesourcesandintotheDDASaccordingtothebluearrows.Briefdescriptionsofthevariouscomponentsaregiveninthetext......................82Figure5.1:AsimplieddiagramshowingtheprocessingthattheReadoutprogramdoestoallowcoincidencesortingforincomingdata.Eventsfrom(inthecaseofE12028)thecloverdetectors(blackarrow)andthescintillatordetector(bluearrows)arebueredintotheReadoutprogram,whichseg-regatesthemaccordingtotheeventwindow(greyboxes)andpushesthemintoaneventleforanalysis.........................85xviiFigure5.2:AscreenshotoftheSpecTclanalysisprogramspectrumwindowshowinggamma-rayspectraforeachcrystalintherstfourcloverdetectors(thatis,thedetectorsintheringupstreamofthescintillator).Thesespec-trahavehadapreliminaryenergycalibration(seeSection5.2.1)appliedtothem,andtheydemonstratetheslightdierencesthateachdetectorcrystalseesasitrecordsinformation(forexample,comparetherelativeheightsofthepeakat2234keV).......................88Figure5.3:AcomparisonbetweenthepeakshapefunctionusedintheanalysisofE12028(redsolidline)andthestandardGaussianpeakshape(bluedashedline).Thetwofunctionshavebeennormalizedtohavethesamemaximumvalue,centroid,andstandarddeviationparameter,˙.Thelargelow-energytailistheresultoftheconvolutionwiththeexponential.....93Figure5.4:Upperpanel:theeciencycurvegeneratedfromttingtherelativee-cienciesofthe152Euand32Cldata.Thepolynomialrequiredsixterms,listedintheboxattop-rightalongwiththereducedchi-squaredvalueandthep-valueforthet.Lowerpanel:theuncertaintyenvelopeadoptedacrosstheenergyregion,basedontheenvelopeinRef.[73]andtheotherconsiderationsdetailedinthetext......................99Figure5.5:Ahistogramshowingtheuncalibratedscintillatorenergyspectrum.Be-causethecentralscintillatorwasonlyusedtocountthenumberofim-plantsanddecaysandallowforgatingofcloverevents,acalibrationwasnotneeded,althoughsincetheQ-valueofthe31Clbetadecayisapproxi-mately12MeV,itislikelythattheuncalibratedspectrum,whichappearstoshowadistributionofeventssimilartothatexpectedfora12-MeVbetaendpointenergy,isclosetoanyactualcalibrationthatwouldbeapplied.101Figure5.6:Ahistogrampopulatedbycalculatingthetimedierencebetweenaneventinthescintillatorandaneventinanyclovercrystal.Mostofthe31Cleventsoccurinthelargepeaknearthecenteroftheplateauwhiletherestoftheplateaueventsarelikelycausedbyaccidentalcoincidences.ThetroughtotheleftofthecentralpeakandthehigherplateaubackgroundtotheleftofthecentralpeakarelikelycausedbyreectionsinthedataconnectionsbetweenthedetectorandtheDDAS..............102Figure5.7:Ahistogramshowinganenergy-scalecalibratedcloverspectrum,includingall31cloversusedforanalysis.Thisspectrumrepresents,byovertwomagnitudes,thehighest-statistics31Clbeta-delayedgammarayspectrumtodate.....................................104xviiiFigure5.8:Maingure:Acomparisonofthescintillator-gated(blue)andtiming-gated(green)cloverspectra.Asillustratedhere,thebackgroundisre-ducedconsiderably,especiallyatlowerenergies,whilethephotopeakin-tegralfor31Cleventsisonlyslightlyreducedaccordingtotheeciencyofthescintillatortodetectbetaparticles.Inset:Azoomed-inregionbe-tween1100keVandˇ1600keV,demonstratingtheeectsofthescintillator-timinggatecomparedtothescintillator-onlygate.Asillustratedhere,severalphotopeakscausedbyroombackground,includingtheverypromi-nent1460-keVpeak,arealmostcompletelyeliminated..........105Figure5.9:Forphotopeaksofseveralenergiesbetween1MeVand7MeV,theratioofthemeasuredphotopeakintensityinthetimingspectrumtothemeasuredintensityintheungatedspectrum,themeanofthemeasurements(blackdashedline,80.6%)andtheone-sigmaenvelopedenotingthestandarddeviationofthepointsaboutthatmean(reddashedlines,0.7%).Thisgureillustratesthatthescintillatoreciencywasessentiallyconstantoveritsentireenergyrange,regardlessofthebetaendpointenergyofanyparticular31Cltransition...........................111Figure5.10:Acomprehensivespectrumshowingtheassignmentsforthephotopeaksusedinanalysisaswellasthoseofidentiedbeamcontaminants.Eachphotopeakislabeledbytheemittingnucleusanditsenergy.Peakslabeledwithoneortwoasteriskscorrespondtosingleanddoubleescapepeaks,respectively.Peaksmarkedwithasingledaggeraresumpeaksandthesummationisnoted,andpeaksmarkedwithadoubledaggerhavemultiplecontributionsandarediscussedindetailinSection5.4..........112Figure5.11:Thetiming-gatedspectrum(green)andthetiming-gatedspectrumaddi-tionallygatedontransitionsfromtherstexcited31Sstate,atEx=1248keV,tothegroundstate.Theoverallstatisticsofthespectrumarere-ducedbyseveralordersofmagnitude,butseveralfeaturesarenonethelessvisible,includingenhancedpeaksatenergiescorrespondingtotransitionstothe1248-keVstate,suchasthepeaksat985keV(2234!1248),2035(3283!1248),and5031(6279!1248).Manyotherpeakscanbeseentobeenhancedaswell;theseenhancementswereusedtohelpconrmtheplacementofseveralofthetransitionsshowninFig.5.12andreportedinTable5.1...................................114xixFigure5.12:Thenal,comprehensive31CldecayschemeproducedfromtheanalysisofE12028.Foreachlevel,thelevel'senergyroundedtothenearestkeVisreportedontheleftwingofthelevelanditsspinandparityJˇarereportedontherightwing.ThepreciseexcitationenergiesExofeachlevelarereportedinTable.5.1.EachbetadecaytransitionisdepictedbyaredarrowontherightsideandincludesitsbetafeedingI,alsoreportedinTable5.1.Bluecoloringforalevelindicatesthatthelevelhasneverbeenobservedin31Clbetadecaybefore.Gamma-raytransitionsbetween31Slevelsarealsodenotedinthetablebytheverticalarrows.Eachtransitionincludesthegamma-rayenergyEandbranchingratio(B.R.),whicharebothreportedinTable5.1.Aswiththepopulatedlevels,gammatransitionswhichhaveneverbeenobservedin31Clbetadecaybeforearecoloredblue.Theschemealsoreportstheadoptedbranchesforbeta-protonandbeta-alphaandunobservedgamma-rays..........121Figure5.13:Aportionofthe31Clgammaspectraintheregionaround1368keV.Thebluehistogramistheungatedgamma-rayspectrumwhilethegreenspectrumisthetiming-gatedspectrum.Asshown,theroombackgroundlinesareeliminatedwhilethedecay-relatedpeaksat1368keV,1412keV,and1433keVremain.Notehoweverthatthe1368-keVpeakisreducedmoresubstantiallythantheothertwopeaks,whichareknowntooriginatefrom31Clbetadecay.Thisimpliesthatthe1368-keVpeakdoesnotoriginatesolelyfrom31Clbetadecay....................132Figure5.14:Threespectrashowingthe2779-keVtransitionfromthe6129-keVstatetothe3349-keVstate.Blueline:thetiming-gatedspectrumshowingthe2779-keVphotopeakwithoutanyothercoincidencegatingapplied.Greenline:thecoincidencespectrumproducedbygatingonthe1248-keVtransitionfromtherstexcitedstatetothegroundstate.Redline:thecoincidencespectrumproducedbygatingonthe2100-keVtransitionfromthe3349-keVstatetothe1248-keVrstexcitedstate.The2779-keVphotopeakisenhancedinbothcoincidencespectra,indicatingthatthethreegammasformacascadeaddingupto6129keV.........136Figure5.15:Aportionofseveralgamma-rayspectraillustratingtheeectsofthevar-iousscintillatorgatesonbotha31Clphotopeak(thepeakat2565keVistherstescapepeakofthe3076-keVgamma)andaroombackgroundpeak(thepeakat2614keVisfrom208Tl).Blue:Ungatedgammaspec-trum.Red:Scintillator-gatedgammaspectrum,showingslightreductionofthe2614-keVpeak.Green:Timing-gatedgammaspectrum,showingalmostcompleteeliminationofthe2614-keVpeakandslightenhancementofthe2565-keVpeak.Purple:Scintillator-gatedspectrumgatedONLYonhigh-energyscintillatorevents,showingtheenhancementofthe2614-keVpeakandreductionofthe2565-keVpeak.Turquoise:Timing-gatedspectrumgatedONLYonhigh-energyscintillatorevents.........139xxFigure6.1:Asimplied31Cldecayschemefocusingonthe31Slevelsat6279(IAS)and6390keV.Theblueverticalarrowsindicatepreviouslyunobservedtransitions.EnergiesandintensitiesforthesetransitionsarelistedinTable5.1...................................143Figure6.2:Selectedportionsofthe-coincident-rayspectrum(blue)showingtran-sitionsfromthe6279-and6390-keV31Sstatestothegroundstateandrsttwoexcitedstates(Jˇ=1=2+;3=2+;5=2+,respectively).Thebottomtwopanelsalsoshow--spectra(green)withadditionalcoincidencecondi-tionsonthe1248-and2234-keVrays,respectively.Otherphotopeaksobservedfromthedecayof31Claremarkedwithblackcircles.Doubleescapepeaksaremarkedwithdoubleasterisks...............144Figure6.3:Ratiosofthe30P(p;)31Sthermonuclearreactionratescalculatedforboththenew3=2+stateat6390-keV[solidblueline]andthe6280-keVIAS[dashedgreenline]totheoverallHauser-Feshbachrate[98]........150Figure6.4:ResidualsforthequadraticIMMEtofthelowestA=31;T=3=2quartet(Tables6.5and6.6)afteraccountingfortheobservedisospinmixingin31S..................................155Figure6.5:Isospinmixingmatrixelement,including1˙condenceband,ofahypo-theticalstateengagedinisospinmixingwiththe31PIASat6381keVasafunctionoftheobservedexcitationenergyofthesecondstate.ThebandisderivedundertheassumptionthattheIMMEprovidesagoodtofthedataafteraccountingforisospinmixing.Thedotted(left)anddot-dashed(right)linesshowthe1˙boundsobtainedusingthispredictionwhentheUSDmixingmatrixelementand6461-keVstateenergy,respectively,areusedasinputs.................................157Figure6.6:ResidualsforthequadraticIMMEtofthesecond-lowestA=31;T=3=2quartet(Tables6.8and6.9)..........................162FigureA.1:AscreenshotofLise++,showingthespectrometersetupwindowinthecenteroftheimageandanassembledsetupcorrespondingtotheA1900fragmentseparatorandtheRFFSontheleft.Each\block"ofmaterialisinsertedintothebeamlineusingthesetupwindowandhasitsownsetofoptions,dependinguponthetypeofblockused.............177FigureA.2:AseriesofgraphsfromLise++,showingthebeam'ssimulatedcharacter-isticsattheRFFS.Includedcalculationsaretheyieldofeachisotope,thedispersionangle,andpositionforbothhorizontalandverticaltransversedirections,therigidityofthebeamconstituents,andtheirenergy.....178xxiFigureA.3:TheresultofaLise++calculationtondtheoptimalthicknessoftheberylliumtargetforproductionof31Cl.AlthoughLise++givesanes-timateofthebestthickness,experimentersareasalwayslimitedbytheavailabilityoftargetsofvaryingthicknesses................178FigureA.4:TheresultofaLise++calculationshowingtheimplantationdepthofseveralbeamconsituentsinsideatargetblock.Thecalculationshowstheyieldofeachspeciesanditsrangeinthematerial..............179FigureA.5:TheinputcardforPace.Asshown,thecalculationrequirestheAandZofbothtargetandprojectileaswellasthelaboratoryenergyofthebeam.180FigureB.1:AschematicofthesiliconPINdetectorsusedforE12028.........210FigureB.2:TheHammamatsuEJ200PlasticScintillatorDataSheet.........211FigureB.3:Aschematicoftheplasticscintillatorusedintheexperiment.......212xxiiChapter1
Introduction
1.1OverviewofPertinentPhysics
1.1.1AnIntroductiontoNuclearPhysics
TheperiodictableofelementsisorganizedaccordingtotheincreasingnumberZofprotonsinanatomofeachelement,butforeachelement,multipleisotopesexistwithidenticalZbutdieringnumbersNofneutrons.Theseisotopesarecommonlyorganizedaccordingtotheso-calledchartofnuclides(Fig.1.1),withthenumberofprotonsinthenucleusincreasingalongtheverticalaxisandthenumberofneutronsinthenucleusincreasingalongthehorizontalaxis.AsshowninFig.1.1,isotopesofanelementarearrangedinhorizontalrowsand,becausetheysharetheprotonnumberZ,haveidenticalchemistrywhilepossessingdierentnuclearproperties.NucleiwithidenticalneutronnumberNarecalledisotones,andnucleiwiththesametotalnumberofnucleons(so-called\massnumber"A=Z+N)arecalledisobars.Onlyasmallfractionofallknownnucleiarestable;thesearerepresentedinthechartofnuclides(Fig.1.1)bytheblacksquares.Theremainingnucleiareradioactiveandundergosomeformofnucleardecay,transformingsequentiallyintostablenuclei.Thetwoprimaryformsofdecayforlightnucleiareparticleemissionandbetadecay(commonlyabbreviatedwiththeGreeksymbol).Forexample,neutron-decientnucleitotheleftofstabilityon1Figure1.1:Thechartofnuclides.TheverticalaxisrepresentsthenumberofprotonsZinagivennucleusandthehorizontalaxisrepresentsthenumberofneutronsNinthenucleus.AllisotopesofagivenZpossessidenticalchemicalproperties,butthenuclearphysicspropertiesofagivenelementaredierentfromoneisotopetothenext.Theblacksquaresalongthecenterofthedistributionarethestablenuclei;thegreenregionrepresentsunstablenucleithathavebeenobserved,whiletheyellowregionrepresentsnucleiexpectedtoexistbutasofyetunobserved.2theNaxisofthechartofnuclidesdecayeitherviaprotonoralpha-particleemission,orbyundergoingbeta-plusdecay.Beta-plusdecay,inwhichthenucleustransformsaprotonintoaneutronandemitsapositron(abeta-plusparticle,+)andelectronneutrino,occursviatheweakinteraction.Anucleus,likeanatom,mayexistinoneofanumberofquantizedenergystates;particleemissionsuchasprotondecayrequirestheparentnucleus(thatis,thenucleusthatisundergoingdecay)tobeinastatewithmassgreaterthanthatofthedaughternucleus(thatis,thenucleusintowhichtheparentdecays)plustheparticle.Thisistrueforallnucleiincludingthosebeyondtheedgeofthechartofnuclides,the\dripline,"investigationofwhichisanactiveeldinnuclearscienceandisbeyondthescopeofthiswork.Iftheenergyofthisstateishigherthanthegroundstate,itmaybeattained,forexample,bytheabsorptionofagamma-raythataddsitsenergytothatoftheparentnucleus.Forprotonemissionof,forexample,16O,thisprocesswouldbenotated:16O(;p)15N.Inprinciple,inanenvironmentwithanabundanceofprotons,thereversealsohappens:suchprotoncaptureisnotatedinaverysimilarfashion:15N(p;)16Orepresentsthecombi-nationofa15Nnucleusandaprotontoform16O;theresultingphotoncarriesawayenergygivenointhecapturereaction(moredetailsinSection3.2).Ontheneutron-richsideofthechartofnuclides,neutronemissionandbeta-minusdecayoccurtomirrorthedecayofunstablenucleiontheneutron-decientside.And,insomecircumstances,anucleusmayemitanalphaparticle(),a4Henucleus.Ineachoftheabovecasesofdecay,thenucleuschangesinZ,N,and/orA.Aheaviernucleus(A>56)mayalsossion,breakingintoconstituentdaughterproductssuchthatthebindingenergypernucleonineachdaughternucleusishigher.Thesevariousdecayprocessesallmovethenucleustowardstability.Sincebeta-decaypreservesmassnumberA,alighternucleuscouldconceivablybetransformedtoaheavier3oneviaaseriesofparticlecapturesandbetadecays.However,sincemostunstablenucleiareextremelyshort-lived,ithashistoricallybeendiculttostudytheirproperties.Theeldofnuclearastrophysicsoriginatedthroughattemptstoanswerthisandotherlarge-scalequestions:Whatistheoriginoftheelements?Howdostarsgeneratetheirenergy?Andhowdotheenergeticstellareventsobservedthroughoutthegalaxysuchassupernovae,neutronstarmergers,andothercataclysmiceventscontributetotheobserveddistributionofisotopesthroughoutthegalaxy?
1.1.2NuclearAstrophysicsandtheLivesofStars
Themoderneldofnuclearastrophysicswasbornfromworkculminatinginthelate1950swiththepublicationofthetreatiseSynthesisoftheElementsinStars[1]andtheindependentformationofthelectureseriesStellarEvolution,NuclearAstrophysics,andNucleogenesis[2].Intheseworks,boththeoriginoftheelementsandtheenergygenerationofstarswereproposedastheconsequencesofnucleosynthesis:thebuildingupofprotonsandneutronsintolightelements,andthesubsequentbuildingofheavyelementsfromthoselighterpieces.Variousgalacticsourceswereproposedassitesfornucleosynthesis.TheBigBangproducedhydrogen(protons),neutrons,and,throughnucleosynthesisofthesetwobuildingblocks,theadditionallightnuclei3He,4He,and7Li[3].Withfewexceptionsoriginatingfromprocesseslikecosmicrayinteractions,allelementsbesideshydrogen,helium,andlithiumareproducedinstars.Starsarecategorizedaccordingtotheircomposition.\PopulationI"starslikeourSunhavecompositionsaectedbythematerialsejectedinthedeaththroesofolder\PopulationII"stars.Buteventheseancientstarsexhibitsomemetallicity{infact,everystarthathaseverbeenobservedhassomemetalsinit.Aproposedrstgenerationofstars,the4p+ p:2H + e++ vep+ p:2H + e++ ve2H + p:3He + 2H + p:3He + 3He + 3:4He + 2p1 billion years1 second1 million yearsFigure1.2:Thepp-Ichain,whichproduces4Hefromfusionoffourprotons(1H).Thearrowsindicatetheorderofprocesses,whilethetimesnotedatthebottomofeachsection(demarcatedbythedashedgraylines)denotethetimeforthereactiontooccurinastellarenvironmentlikethatoftheSun.Theextremelylongtimescaleoftherstreaction,p+p!2H+e++,isduetotherepulsiveCoulombbarrierbetweenthetwoprotons.ThislongtimescaleisosetbythecolossalnumberofprotonsintheSun,˘1057.TheppchaindominatesenergyproductioninstarsasmassiveastheSun.
\PopulationIII"stars,wasformedoutofhydrogenandheliumwithinabout109yearsaftertheBigBang[4].Stellarnucleosynthesisistheprocessoffusingthenucleithatmakeuptheinitialcom-positionofastarintoprogressivelyheaviernuclei.Starsspendthemajorityoftheirlifeburninghydrogenintoheliumbecausethisfusionyieldsmoreenergythansubsequentreac-tions.Throughnucleosyntheticprocessessuchastheppchain(Fig.1.2)andtheCNOcycle(Fig.1.3),1Hisfusedinto4He,proceedingthroughthenuclei2Hand3Heintheprocess.Beyondtheseelements,starsproduceheavierelementsvianucleosyntheticprocessessuchasthetriple-process,whichcombinesthree4Henucleiinto12C,andthe12C(;)reaction,whichproduces16O.Thesereactionsreleaseenergy,causingthestartoexpandandcool,andkeepingitinso-calledhydrostaticequilibriumagainstgravitationalcollapse.Inprinciple,astarmaygenerateenergybyfusingsuccessivelyheavierelementsinits512C13N14O14N15O15N(p,)(p,)(p,)(+)(+)(p,.)16O14C13CZNFigure1.3:TheCNO-Icycle,which,similartotheppchain(Fig.1.2),producesa4Henucleusfromthefusionoffourprotons.Unlikethepp-chain,however,theCNOcycleiscatalytic,consumingthefourprotonsnecesaryfor4Heproductionbutnotthe12Cnucleusthroughwhichthe4Heiscreated.TheCNOcycledominatesenergyproductioninstarsofgreaterthanabout1.5M.coreuntilitproduces56Feand56Ni,afterwhichtheenergyreleasedfromnuclearfusionislessthanthereleasefromssionofheaviernucleiintolighterconstituentpieces.Astar'snucleosyntheticendpointisdeterminedprimarilybyitsmass;lessmassivestars,ontheorderofthemassofthesun(M),donotreachsucientinternaltemperaturesforfusionbeyondoxygen,whilestarslessthanabout0.3Mwillnotevenfusetocarbon.Starsofmassabout8-10Mwillproduceneon.Despitethevarianceinnucleosyntheticendpoint,however,thesestarsallhavethesamefate:theywillrunoutoffuel,shedtheirouterlayers,andcollapseintoacompactstarknownasawhitedwarf.Starswithmassgreaterthanabout10M,however,produceelementsuptoA=56viafusionofneon,oxygen,andsiliconandwilleventuallyendtheirlivesasaneutronstarorblackhole.ElementsheavierthanA=56areproducedeitherviathe\slowneutron-captureprocess"whichisbelievedtooccurmostlyingiantstars,orincataclysmiceventssuchasmergers6ofneutronstarsorthedeathofstarsmuchmoremassivethantheSun.Inthelattercase,thecoreofthemassivestarcannolongerproduceadditionalthermalenergythroughnucleosynthesisandisconsequentlyunabletoghtogravity.Itthereforecollapses,releasinggravitationalpotentialenergythatpowersanexplosionknownasasupernova,duringwhichnucleosynthesisofheavyelementscanoccurintheejectedmaterial.Thecoresimultaneouslycollapsesintoaneutronstarorblackhole,dependinguponthemassoftheprogenitorstar.Theejectedmaterial,includingamixofheavierelements,isjettisonedointospaceinacore-collapsesupernova;thematerialmaytheneventuallybecomepartofagascloudwherenewstarsmayform.Inthisway,stellarmaterialisrecycledforuseasfuelinfuturegenerationsofthesenuclearfurnaces.
1.2ClassicalNovae
Asmentionedabove,starswithmassesontheorderofonesolarmassMdonotfuseelementsheavierthanneon,orpossiblymagnesium[5].Assuchastarnishesthemainsequenceofitslifeandrunsoutofhydrogeninitscore,itexpandstobecomearedgiant.Duringthistime,thestar'scorecontractsuntilitbecomeshotenoughtofuseheliumintocarbonandoxygen,whilehydrogenburningcontinuesinashellsurroundingthecore.Whenthestarrunsoutofheliuminitscore,thecore,whichdependinguponthemassofthestarisnowcomposedpredominantlyofeithercarbonandoxygen(forlessmassivestars)orofoxygenandneon(formoremassivestars),contractsyetagain.Atthispoint,thestarjettisonsitsouterlayersintoacloudknownasaplanetarynebula.Thehotcoreofthestarbecomesawhitedwarf,asmallstarofmassupto1.4Mwithacompositiondependentuponthecompositionofthecoreoftheprogenitor,againeithercarbon-oxygen7(a\"CO"whitedwarf)oroxygen-neon(an\ONe"whitedwarf).LighterstarsthantheSunwillproducewhitedwarfscomposedofhelium,whilestarswithmass8-10Mwillproducewhitedwarfscomposedofoxygenandneon.Nofusionoccursinthewhitedwarf;instead,furthercollapseispreventedbyelectrondegeneracypressure.Whitedwarfsradiatetheirstoredthermalenergy,becominglesshotandlessvisibleovertime.Forisolatedstars,thisrepresentstheendofnucleosynthesis.However,ithasbeenes-timatedthatuptoathirdofstarsexistasbinaries{systemswheretwostarsorbitagravitationalcenter[6].Inthesecases,thestarsmaygravitationallyinteractandcanhaveprofoundeectsoneachother'sevolution.Inasystemwhereawhitedwarfstarco-orbitsaless-massivemainsequencestar,thehydrogen-richstarmayoverowitsownRochelobe,theregionofspace,boundbyagravitationalequipotentialsurface,withinwhichthestar'sgravityattractsnearbymaterial.Thehydrogen-richmaterialmaythenowintotheRochelobeofthewhitedwarfviaaprocessknownasaccretion,slowlyspiralingontothesurfaceofthewhitedwarfinadiskorbitingthestar.Thisaccretionslowlybuildsupanenvelopeofhydrogen-richmaterialonthesurfaceofthewhitedwarf[7].Becausethewhitedwarfsurfaceissomuchmoredensethanthehydrogen,theincomingmaterialdoesnotfullymixwiththematerialthatformsthewhitedwarf.Whilesomemixingdoesoccurinthebottom-mostlayerofthehydrogenenvelope[8],themajorityoftheenveloperemainshydrogen-richandbecomesincreasinglyhotterasitiscompressed.Thiscontinuesuntilthehydrogen-richmaterialbecomeselectrondegenerateitself.Atsomepointafterthis,thematerialbecomeshotenoughthathydrogenburningoccurs,butbecausethematerialisdegenerate,itcannotexpandtocoolitselfoandmaintainequilibrium[9].Consequently,theenvelopebecomeshotterandhotterinaso-calledthermonuclearrunaway,whichcontinuesunabateduntilthedegeneracyislifted.Intheprocess,thematerialaccreted8Figure1.4:Anartist'sdepictionofaclassicalnovainabinarysystem.Theaccretiondisk,consistingofhydrogen-richmaterialowingfromthemainsequencecompanionstar(right),isshownattheequatorofthewhitedwarf(left).ImagecopyrightDavidA.Hardy,www.astroart.org
ontothewhitedwarfblowsoinanexplosionknownasaclassicalnova[10].Novaearecategorizedaccordingtothetypeofwhitedwarfonwhichtheyoccur:COnovaeoccuronthelessmassivewhitedwarfscomposedprimarilyofcarbonandoxygen,whileONenovaeoccurontheheaviestwhitedwarves,composedofoxygenandneon.Anartist'sdepictionofanovainabinarysystemisshowninFig.1.4.Novaearepowerfulexplosions,releasing˘1045ergsofenergy[11]andcausingthewhitedwarftoincreaseinbrightnessbyfactorsanywherefrom1,600to107,withpeakluminositiesbetween104and105L[12].Unlikethemorepowerfulsupernovae,however,classicalnovaedonotdestroythestaronwhichtheyoccur,meaningthattheycanberecurrent,withaperiodequaltotheamountoftimebeforeaccretedmatterontothewhitedwarfexplodes9again.Theyoccurfarmorefrequentlythansupernovaeinourgalaxy,withanexpectedoccurrencerateof3010peryear[13],comparedwithanexpectedsupernovarateofoneevery4010years[14].Approximately10novaeareobservedinourgalaxyperyear(thedierencebetweenexpectedandobservednumberisduetointerstellardustinthegalacticplane,whichobscuresthelightemittedfromthenova,aswellasthelackofsystematicsurveytechniques),allowingforobservationalstudyusingbothgroundandspace-basedtelescopes.Lightcurves,whichmeasurethebrightnessofanovaasafunctionoftime,havebeenrecordedfornovaeinradio,infrared,optical,ultraviolet,X-ray,andgamma-rayregionsoftheelectromagneticspectrum.Thesedierentmeasurementsofnovaehavehelpedtodeterminevariouspropertiesofthesestellarexplosions:therateanddurationoftheenergyoutput,thenova'sdistancefromEarth,andthedensityandtemperatureoftheejectedmaterialthroughouttheexplosion.Spectroscopyhasalsobeenusedtopartiallydeterminetheelementalcompositionofthenovaejecta.Similartosupernovae,materialfromtheexplosionisjettisonedoutintospace;novaehoweverarelessenergeticthansupernovaeandreleaseasmalleramountofmatter,between107and103M[15,10],comparedtobetween1and10Mforsupernovae.Asthismaterialexpandsoutwardfromthewhitedwarf,itcools,formingdustgrainsthatpreservethespeciccompositionofthenovamaterialatthetimeitcondenses.Thesegrains,carryingarecordofthenovaconditions,travelthroughoutthegalaxyandmayintimebecaughtupintheformationofanewstarorplanetarysystem.Thus,thesecataclysmicexplosionsmayleaveadistinct{andmeasurable{imprintonnascentstarsandtheirattendantsystems.10Chapter2
NuclearAstrophysics:Motivationfor
Studyof30P(p;)31S2.1ElementalandIsotopicAbundances
AsdiscussedinChapter1,thedistributionofelementsistheresultofnucleosynthesisatvarioussitesthroughouttheUniverse'slifetime.Itispossibletoquantifythisdistributionbothobservationallyandtheoretically.Forexample,temperatureanddensitymodelsoftheearlyUniversehavebeenusedtoexplainthecurrently-observeddistributionofroughly75%hydrogenand25%heliumbymassasaresultoftherelativenumbersofprotonsandneutronsproducedintheBigBang.[4].Thedistributionoftheremainingelementsismainlyduetostellarnucleosynthesis;however,becausestarsofdierentagescontaindierentamountsofheavyelementstobeginwith,therelativedistributionofelementsmaydierbetweenanytwogivenstarsorplanetarysystemsinthegalaxy,orevenbetweentwogalaxies[16].Animportantconceptinnuclearastrophysicsistheideaofabundance,therelativemea-sureoftheamountofagivenelementfoundinaspeciclocation.Theabundancesofelementsinagivensystemplayanimportantroleinitsconstruction.BecauseourownsolarsystemformedaroundanNth-generationstar,forexample,itcontainedheavyelementssuchasiron,andwasabletoformterrestrialplanets.Figure2.1depictstheelementalabundances11Z020406080100Abundance, log N(H) = 12024681012Figure2.1:Solarsystemelementalabundances4.56billionyearsagoasafunctionofele-mentalnumberZ,plottedfromdatainRef.[16].Theverticalaxishasbeennormalizedsothatthehydrogenabundancevalueissetat1012.Theotherelementalabundancesarerelativetothatnumber.
presentinoursolarsystematthetimeofitsformation.Ofinteresttonuclearastrophysicsistherelatedconceptofisotopicabundance;measuringtherelativeamountsofcarbon-12tocarbon-14,forexample,allowsforradioactivecarbondatingofancientorganiccompoundsonEarth'ssurface.Inprinciple,solarsystemabundancesmaybedeterminedinseveralways.Solarabsorp-tionandemissionspectra(Fig.2.2)maybeusedtodeterminetheelementalmakeupoftheSun,whichispresumedtohaveundergoneverylittleelementalchangefromtheformationoftheSunoutofthepresolarnebula.Unfortunately,thesespectraarebasedonatomictransitionsanddonotyieldinformationaboutisotopicsolarsystemabundances.Terrestrialmaterials,ontheotherhand,arereadilyaccessibleandmaybestudiedindepth,allowingforthedirectdeterminationofpreciseisotopicabundances.However,becauseofthehightemperaturesandpressurespresentduringtheformationofEarth,chemicalfractionationofterrestrialmaterialsmeansthatabundancesdeterminedfromterrestrialsourcesdonot12Figure2.2:Asolarphotosphereabsorptionspectrumtakeninthevisibleregionoftheelectromagneticspectrumbetween392nm(blue)and692nm(red).Thedarklinesatparticularwavelengthsthroughoutthespectrumareduetoatomicabsorptionofphotonsofthatwavelength,correspondingtoatransitiontoanexcitedatomicstateofthatatom.Sinceeachatomicelementpossessesitsownenergyscale,thepatternofabsorptionlinesinthesolarspectrummaybeusedtoinfertheelementalmakeupofthephotosphere.typicallymatchthecompositionofthesolarsystematlarge.Apromisingmethodtodeter-mineisotopicabundancesisthusdirectstudyofmaterialformedinthepresolarnebulathatwasunexposedtothehightemperaturesandpressuresthatresultedintheformationoftheplanets.Suchmaterialexistsintheformofancientmeteorites,comets,andasteroids.2.1.1MeteoritesandTheirAnalysis
Over50thousandmeteoriteshavebeendiscoveredonEarth.Ofthese,thevastmajorityoriginatedinprimordialasteroidsformedinthenascentsolarsystem.Thus,meteoritescompriseapromisingpathtodataonthecompositionofthesolarsystemasitwas4.6billionyearsago,aswellastheprocessesthatoccurredasitformed.Meteoritesareclassiedintothreecategoriesbasedontheircompositions:iron,stony-13iron,andstony.Boththeironandstony-ironmeteorites,whichtogethercompriseonlyapproximately6%ofmeteoritefalls,haveundergonesignicantmelting,makingthemun-suitableforabundancedetermination.Stonymeteorites,whichcontainthehighestamountoforganiccompounds,arethemselvessplitintotwobroadcategories,chondritesandachon-drites,basedonthepresenceorabsenceofsmall,roundgrainscomprisedmostlyofsilicatematerialcalledchondrules.Achondrites,likestony-ironandironmeteorites,havealsobeensubjecttoigneousprocesses;however,thesecompriseonly8%ofmeteoritefalls.There-maining86%ofmeteoritesarechondrites[17].ExamplesofeachofthesemeteoritetypesareshowninFig.2.3.Chondritesarethoughttohaveoriginatedinprimordialasteroidswhichformedintheprotostellardiskbutnevergrewlargeenoughtoheatupandundergothechemicaldier-entiationpresentinplanetarybodies.Assuch,thechondrulespresentinthemarethoughttohaveremainedlargelyunchangedfromtheformationofthesolarsystemaswell.Chon-dritesthemselvesareclassiedintothreegroups;ofthese,thecarbonaceouschondritesareofgreaterimportancetostudiesofabundancesthantheenstatiteandordinarychondrites[16]becauseasubsetofthegroup,theIvuna-typecarbonaceouschondrites(CIchondrites),wereneverheatedabove323K[18].CIchondritesexhibittheclosestagreementoverallwithsolarabundances,withtheexceptionofH,C,N,O[19],andthenoblegases[16].Thesemeteoritescanthusbeusedtoobtainnotonlyprimordialelementalabundancesbutisotopicabundancesaswell,givingawindowontothecompositionofthesolarsysteminitsinfancy.2.1.2IsotopicAbundancesinClassicalNovae
LikestellarnucleosynthesisoftheSunandmostotherstars,thethermonuclearrunawayofaclassicalnovaispoweredprimarilybyhydrogenburning.Theinitialburningisspurredby14Figure2.3:Iron(topleft),stony-iron(top-right),achondrite(bottom-left),andchondrite(bottom-right)meteorites.Allofthemeteoriteswiththeexceptionofthechondritehaveundergonevaryingdegreesofchemicaldierentiationasaresultofheating.PhotoCredits:Ji-Elle,DougBowman,Captmondo,andH.Raab1512C13N14O14N15O15N(p,v)(p,v)(p,v)(t+)(t+)(p,r)16O14C13CZNFigure2.4:Anexcerptofthechartofnuclidesshowingthe\hot"CNOcycle.Asillustratedhere,thehotCNOcyclebreaksoutfromthe\cold"cyclewhen13Ncapturesaprotoninsteadofundergoingbetadecay;thisisduetotheincreasedtemperatureinthestellarenvironment,makingtherelativetimeforprotoncaptureshorterthanthe13Nbeta-decayhalf-lifeof10minutes.
theppchain,buttheexplosiveburningthatoccursasthenovareachesitspeaktemperaturesispoweredbythehotCNO-cycle(Fig.2.4).Inthisenvironment,nucleosynthesison\seednuclei"suchas17Ocanoccur,fusingheavierelementsuptoAˇ40viaaseriesof(p;)and(p;)reactionsand+decays.Thisprocessisknownasnovanucleosynthesis(Fig.2.5).Aswithanynuclearprocess,novanucleosynthesisyieldsadistributionofisotopespro-ducedbytheburning.Thesenalabundancescharacterizethenovaandarethemselvesdependentuponenvironmentalfactorsinthenova:forinstance,maximumtemperatureachieved.AswiththeSun,thepresenceofspeciclinesinnovaspectraarecluestothenalelementalabundances[20].However,thesameproblemexistsaswithsolarspectra:onlyelementalabundancesmaybeinferred.Fornovae,acomprehensiveinvestigationofisotopicabundancesisevenmoredicult,sinceitisnotpossibletosamplethesysteminwhichthenovaoccursfromacrossthegalaxybeyondspectrallines(carbonandoxygenisotopicratios16Figure2.5:Asectionofthechartofnuclidesshowingtheextentofclassicalnovanucle-osynthesis.Thedarkblueboxesrepresentstableisotopes,andeacharrowrepresentseitheraprotoncapture,betadecay,or(p;)reaction.Theredcirclenearthetopoftheguredenotestheendpointofnovanucleosynthesis,aroundcalcium.Figurecredit:Ref.[20].17canbeinferredfrommolecularlinesfromCOnovae).Typically,theoreticalmodelsareusedbyastrophysiciststoestimatenovaisotopicabun-dances.Suchmodelsmay,forexample,useaone-dimensionalhydrodynamiccodetosimulatetheevolutionofthenovaenvironmentintegratedwithanuclearreactionnetworktosimu-latethenucleosynthesis[21].Thesenuclearreactionnetworkstakeinputabundancesofseednucleiand,giventabulatedratesasafunctionoftemperatureofeachreactioninvolvedinnucleosynthesis,producetheoreticalisotopicabundancesforthenova.Becauseonlyafewhundrednuclearreactionsareinvolved,andthosereactionslieclosetothelineofstability,itispossibletousemostlyexperimentally-determinedreactionratesinthemodels[10].Thankstoexperimentalworktomeasureandcharacterizethevariousreactionsinvolvedinnucle-osynthesis,mostreactionsofimportancetonovayieldshavebeenexperimentallymeasuredandcharacterizedsuciently.Untilrecently,onlythreereactionsstoodoutaschallengestoconstrainingnovaabundances:25Al(p;)26Si,18F(p;)15O,and30P(p;)31S,anda2013studyprovidedanexperimentallydetermined25Al(p;)26Sireactionrate[22,23].However,experimentallyconstrainingatleastthe30P(p;)31Sreactionratehasbeenchallenging.Infact,arecentevaluationofreactionrateuncertainties[24]concludedthatattemptingtocal-culateameaningful30P(p;)31Sreactionratefromtheavailableexperimentalinformationwasfutilebecausethenuclearphysicswassopoorlyunderstood.2.1.3MoreNucleosynthesis:NovaThermometers
Inadditiontoaectingthenalisotopicabundancesofnovanucleosynthesisoftheisotopesinthemassrange30A40,the30P(p;)31Sreactionratealsoaectsthecalibrationofso-callednovathermometers.Arecentstudy[25]proposedtheuseofrelationshipsbetweensimulatedabundanceratiosofvariouselementsandpeaknovatemperaturesasameansto18Figure2.6:TheeightnovathermometersproposedasthemostusefulforconstrainingpeaknovatemperaturesinRef.[25].Eachlinerepresentstheratioofthenotatedelementalabundancesasafunctionoftemperature.Thefourmoststeeply-varyinglinesallinvolveeitherphosphorusorsulfurabundances,makingprecisedeterminationoftheseelementalabundancesahighpriority.Figurecredit:Ref.[25].
constrainthehighesttemperatureanovacouldachieve.Thisstudyusedanuclearreactionnetworkwithanupdatedlibraryofreactionrates[26]andhydrodynamicnovamodelsoverarangeofprogenitorwhitedwarfmassesfrom1.151.35MtosimulateONenovaexplo-sionsandsubsequentnucleosynthesis.Itrecordedthenalelementalabundancesandpeaktemperaturesreachedbythenovaeduringthesimulationsandplottedtherelationships(Fig.2.6).Thestudyfoundthat,oftheeightproposedelementalabundanceratioswhichvariedmoststronglywithpeaktemperature,thetwomoststronglytemperature-dependentratioswereO/SandS/Al,withtheratiosO/PandP/Alfollowingcloselybehind.However,thestudyconcludedthattheapplicabilityofthesethermometerswaslimited,partlydueto19limitationsintheobservationandspectralanalysisofONenovaejecta,butalsodueheavilytothelackofavailableexperimentalinformationonthe30P(p;)31Sreactionrate.Infact,asinRef.[24],thestudywasnotevenabletouseexperimentaldatatoproduceareactionrate,insteadoptingforthepurelytheoreticalHauser-Feshbachstatisticalmodel[27](seeSection3.3.1).ThestudycalledfornewlaboratorymeasurementsoftherateinordertoascertainthevalidityofthenovathermometersO/S,S/Al,O/P,andP/Al.Inprinciple,isotopicabundancesarenotonlyusefulforatheoreticalunderstandingofnovanucleosynthesisoracomputationalstudyoftheirpeaktemperatures.InthesituationdescribedinChapter1,agrainofdustmaycondensefromanovaoutowandtraveltoayoungplanetarysystemintheprocessofformation.Ifthematerialintowhichthisgrainembedsitselfneverdrawscloseenoughtoitsparenttocausechemicalchangesthroughheating,thegrainmayretainadistinctrecordoftheisotopicabundancesofitsparentnova.Intheeventthatsuchagrain,havingenteredoursolarsystemduringitsformation,thenmakesitswaytoEarth,analysismaybecarriedoutinthesamemannerasforasolarsystemgrainandthenovagrain'sisotopiccompositionmaybedeterminedexperimentallyandcomparedtonovamodels.Suchagrain,createdbeforethebirthofthesolarsystem,canthusallowforin-laboratorystudyofanastrophysicalprocessthatoccurredsomewhereinthegalaxyover4.5billionyearsago.
2.2PresolarGrains
Despitetheknowledgethatanumberofdiversenuclearprocessescreatedthemixtureofelementsthatformedthesolarnebula[1],itwasoriginallythoughtthatthetheformationprocesshadhomogenizedthepresolarmaterialswhichwouldformthesolarsystem,resulting20inuniformsolarsystemisotopicabundances.Therstindicationsofanomalousisotopiccompositioninmeteoriticgrainsoccurredin1954and1964withtheobservationofdivergentamountsofhydrogen[28]andxenon[29].Furtherevidencein1973intheformofameteoriticexcessof16Oopenedthedoortotheideathatnotallmaterialpresentattheformationofthesolarsystemhadbeenhomogenized,andthesearchforsignaturesofpresolarmaterialcontinuedwithobservationofisotopicdivergencesinanumberofotherelements[30].Isotopicallyanomalousmeteoriticgrainsmayhaveoneofanumberoforigins[30].Theymaybetheresultofdecayprocessesoflong-livedradioactiveisotopessuchas26Al,whichisformedthroughoutthegalaxyingiantstarsandatothersites,withahalf-lifeof˘1106years;thisradioactive26Aliscaughtupintheformationofthesolarsystemandlendsitssignaturetothesesamplesintheformofitsbeta-decaydaughternucleus,26Mg.Theymaybeso-calledcalcium-aluminum-richinclusions,whichmaybeformedfromcrystallizationormeltingofcondensedliquidintheearlysolarsystem[31].Theymayalsosimplybetheresultoflocalinhomogeneitiesinthesolarnebula.Ineachofthesecases,thesamplemaybetheresultofincompletemixinginthesolarnebula,butwhileitretainssomeofthecharacteristicsofitssiteoforigin,itdoesnotexhibitisotopicratiosdivergentenoughtoprecludecreationintheyoungsolarsystem.Thosemeteoriticgrainswhoseisotopicabundancesaredierentenoughfromsolarsystemabundancestoprecludeasolaroriginareknownaspresolargrains.Presolargrainsmayoriginatefromanystellarprocesswhichproducesdustinuencedbynucleosynthesis.Eachgraincarriesauniquerecordoftheprocessthatspawnedit,andasaresultpresolargrainsfromdierentsourcesexhibitisotopicabundancesthataredistinctfromoneanother,evenastheyaredistinctfromsolarabundances.Thesediversegrainsalsoexhibitdiversecompositions:forexample,amajorityofsupernovagrainsaremadeofcarbonintheformofdiamond,whileamajorityofgrainsformedinasymptoticgiantbranch21(AGB)starsaresiliconcarbide(SiC)grains.SupernovaorotherprocessesmayalsoproduceSiCgrainswiththeirownuniqueisotopicabundances;thus,evengrainsofsimilarlarge-scalecompositionmayrevealdiverseoriginsuponexamination.
2.2.1PresolarNovaGrains
SiCgrainsarethemostextensivelystudiedofthepresolargrainsbecausetheyarebothcomparativelylargerthanothertypesofgrainsandbecausetheyaremorenumerousandeasilyfoundinmeteoritessuchasCIchondrites.Becausethesegrainsarelargeenoughtostudyindividuallyasopposedtoinbulk,techniquessuchassecondaryionmassspectroscopy(SIMS)[32]mayevenallowformeasurementsofelementswithsmallchemicalpresencesinthegrain.Laserablationandresonanceionizationmassspectroscopy(RIMS)[33]havealsoprovenusefulformeasurementsofheavierelementssuchasstrontium,zirconium,andmolybdenum[34].SiCgrainshaveanomalousisotopicabundancesofnotonlysiliconandcarbon,butofasizeablelistofotherelements:N,Mg,Ca,Ti,noblegases,andrefractoryelementssuchasSr,Zr,Mo,Ba,Nd,andSm.[30].AscanningelectronmicroscopeimageofaSiCgrainisshowninFig.2.7.Inquantifyingthepreciseisotopicabundancesofkeyspeciesinparticulargrains,sev-eraldierentpopulationsofSiCgrainshavebeenidentied.Inordertoeasilydierentiatebetweenthesepopulationsandtheirprospectiveoriginprocesses,itispossibletoplotchar-acteristicisotopicabundanceratiosasshowninFigs.2.8and2.9.Ineachofthesecases,dierentpopulationsofgrainsareshowntohavedistinctratiosofisotopicspeciesofkeyel-ements.MainstreamgrainsandtypesAandBgrainsarethoughttobeproducedinvarioustypesofcarbonstarsinwhichtheCNOcycle,heliumburning,andtheslowneutroncap-tureprocess(s-process)occur.TypesYandZgrainsaremostlikelyproducedinlow-mass22Figure2.7:ASEMimageofasilicon-carbide(SiC)grain,takenfromtheMurchisonme-teorite,acarbonaceouschondrite.Notethegrain'slarge(severalmicrons)size.Isotopicanalysishasrevealedthatthisgrainisindeedapresolargrain,withanoriginbeforethesolarsystem.Photocredit:Max-Planck-InstitutFurChemie23Figure2.8:AplotoftheratiosofcarbonandnitrogenisotopicabundancesasdeterminedforanumberofSiCpresolargrainsandsomegraphitegrains.Thedottedcrossinthecenteroftheplotdenotessolarsystemabundances,andthevariouspopulationsofgrains,denotedwithdierentsymbols,areshowntobedistinctbasedonthecombinationofthetworatios.Asanexample,theso-callednovagrainsareshownheretobedecientbynearlyanorderofmagnitudeinboth12Crelativeto13Candin14Nrelativeto15N,whencomparedwithsolarabundances.Figurecredit:Ref.[35].
AGBstarswithrelativelysmallamountsofmetalinwhichmixingbetweenlayersoccurred.X-typegrainshavebeenproposedtooriginateinsupernovae.Forathoroughdiscussionofthenaturesofdierentpresolargrainpopulations,seeRef.[30].Theso-called\novagrains"exhibitcharacteristicallyhigh30Si/28Siratiosalongwithcharacteristicallylow12C/13Cand14N/15Nratios.Novamodelsbroadlypredictthatasthemassofthewhitedwarfonwhichthenovaoccursincreases,therelativeamountofSiincreasesaswell,asthetemperaturesreachedduringnucleosynthesisincreaseenoughtoproducesiliconisotopes.Thus,oxygen-neon(ONe)novae,whichoccuronthemostmassivewhitedwarfs,areapossibleoriginprocessforthesecandidatenovagrains[36].24Figure2.9:Aplotoftheratiosoftheabundancesofthestablesiliconisotopes.Thedottedcrosscenteredattheorigindenotessolarsystemabundances.Thenotationusedhereis\permil:"29;30Si=28Si=[(29;30Si=28Si)=(29;30Si=28Si)1]1000,suchthataplacementof100alongtheaxisrepresenta10%higherratiothansolar,andaplacementof1100representsa110%higherratio.Figurecredit:Ref.[35].
2.2.2ComparisonwithNovaModels
AsmentionedinSection2.1.2,theoutputisotopicabundancesofnovamodelsmaybecom-paredwiththeobservedisotopicabundanceratiosofcandidatenovagrains.Inprinciple,becausetherearesofewnuclearreactionswhoserateshavenotbeenconstrainedbyexperi-ment,itshouldbestraightforwardtoassesswhetherornotcandidatenovagrainstrulydooriginateinclassicalnovae.Indeed,thecomparisonofisotopicratiosofcarbonandnitro-genisotopesinnovamodelstothoseinanalyzedgrainsrevealsagenerallygoodagreement[35]:boththemodelsandthegrainsthemselvesexhibitdecitsofboth12Cand14Nwhencomparedto13Cand15Nrespectively.However,theseagreementsareovershadowedbytheuncertaintiesinthemodelabundancesofthesiliconisotopes,whicharesolargeastocompletelyprecludeadeniteassessment.Theseuncertaintiesaretheresultofuncertainty25inonesinglereaction:30P(p;)31Sprotoncapture.30Siisproducedviathebetadecayof30P,30P(+)30Si.Thealternatepathforde-structionof30Pinnovaeisviaprotoncapture,30P(p;)31S.Theseprocessescompetetodeterminethetotalamountof30Siproducedinnovae:ifthe30P(p;)31Srateiscompara-tivelyfast,more30Pisdestroyedviaprotoncaptureandless30Siisproduced.Conversely,aslowrateyieldsmore30Si.Thehalf-lifefor30Pbetadecayiswell-knowntobe2.498(4)min,buttheprotoncapturereactionis,comparatively,essentiallyunconstrained[24].Infact,ifthelowerandupperlimitsoftherateareadoptedinsteadofthecentralrate,theamountof30Sirelativeto28Siproducedinthenovamodelsbecomesanexcessofafactorof˘6oradecit,respectively[35].Betterconstraintsforthe30P(p;)31Sreactionratearethuscriticaltoansweringthequestionofwhethercandidatenovagrainstrulydooriginateinclassicalnovae.
2.3The30P(p;)31SReactionRate:PurposeofThisWorkAsdiscussedabove,therateofthe30P(p;)31Sreactionplaysanintegralpartinanswer-inganumberofimportantquestionsforclassicalnovastudiesduetoitsinuenceonnovaobservables.Whatisthenalisotopicabundancedistributionofnovaeinthemassregionabove30P?Howhighatemperaturecannovaeactuallyreach?Anddopresolarnovagrainstrulycomefromnovae?Thersttwoquestionsareintertwined,asthemaximumtem-peraturesofnovaearerelevanttothequestionofCNOcyclebreakout,aprocessbywhichnovaecouldproducedramaticallydierentisotopicabundances.Theanswerstoallofthesequestions,however,involveconstrainingthe30P(p;)31Srate.Unfortunately,aswillbedis-26cussedinmoredetailinChapter4,studying30P(p;)31Sisnotrivialtask,andtheratestillremainslargelyunconstrainedexperimentally,despitenumerousstudies.Understandingthe30P(p;)31Sreactionrateinvolvesaddressinganumberofconsiderations:theenvironmentinwhichthereactiontakesplace,theenergiesoftheparticlesinvolvedinthereaction,andeventhespecicnatureofthecapturereactionitself.Allofthesearefactoredintothederivationofthethermonuclearreactionrate,therateofthereactionintheastrophysicalenvironment.Thepresentworkconstitutesanexperimentalstudyusingthebetadecayof31Cltopop-ulateanumberof31Senergystatesimportantto30P(p;)31Sandmeasuringtheirpropertiesviatheirgamma-decay.TheexperimentitselfwillbediscussedinChapter4,whileresultsandconclusionswillbediscussedinChapters5and6.Beforediscussingtheexperiment,however,itisusefultoengageinabriefdiscussionofthenuclearstructureconceptsandformalismusedtoderiveandinterpretresultsintheexperiment{thisisthefocusofthefollowingChapter3.27Chapter3
FormalismandTheoretical
Considerations
Theterm\reactionrate"inreferenceto30P(p;)31Sissomewhatloaded.Thereareinprincipleanumberofwaystoquantifythecombinationofaprotonanda30Pnucleusinto31S:thecrosssection,afactorwithgeometricunitsdescribingthelikelihoodofcombinationofthetwobodies;thereactionrate,whichfactorsintherelativespeedsoftheparticles;thethermonuclearreactionrateperparticlepair,whichnormalizesoverthedistributionsofenergiesoftheparticlesinanastrophysicalenvironment.Inpractice,whendiscussingortabulatingthe\reactionrate"ofagivenreaction,astrophysicistsareconcernedwiththislastquantity.Thetermthermonuclearreactionratereectsthefactthattheenergiesofparticlesinastrophysicalenvironmentslikenovaeareduetotheirthermalmotion.Inthepresentsection,webeginwithashortderivationofthethermonuclearreactionrateequationandadiscussionofthenatureofparticlecaptureinastrophysicalenvironments,includingdirectandresonantcapturereactions.Followingthat,wediscusswaystocalculatethereactionratewhennoteveryparameterinvolvedinthecalculationcanbeconstrainedexperimentally,includingtheHauser-Feshbachstatisticalmodel.Lastly,althoughtheyarenotstrictlyrelatedtoastrophysics,wediscusstwoconceptsimportanttonuclearstructureandtothederivationandinterpretationofresultsinthepresentwork,thenuclearshell28modelandtheisospinmodel.3.1DerivationoftheThermonuclearReactionRate
Ingeneral,tocalculateaparticlecapturereactionratebetweenaprojectile(say,apro-ton)andatarget(say,anucleusinastellarenvironment),severalfactorsmustbetakenintoaccount.Theseinclude:ageometricalfactorˇ2,where=2ˇ~p2mEisthedeBrogliewavelengthoftheprojectile;theinteractionbetweentheprojectileandtarget,whichcanberepresentedbyamatrixelementjMj2;andafactorcalledthepenetrability,essentiallytheprobabilitythattheprojectilewillapproachnearenoughtothetargettointeract.ThispenetrabilityfactorPl(E)accounts,forexample,fortherelativeangularmomentumoftheprojectilewithrespecttothetarget{radialmotionoftheprojectilewithrespecttothecenterofthetargetresultsinaneectiveenergybarrierwhichtheprojectilemustovercometointeract.Combiningthesefactorsyieldsthefollowingproportionalityforthecrosssectionforcapture:˙/1EjMj2Pl(E)(3.1)whereEistheenergyoftheprojectile.Chargedparticlereactionssuchasprotoncapturerequireconsiderationofadditionalfactors:becauseboththeprojectile(proton)andtarget(nucleus)ofthereactionarecharged,thepenetrabilitymustaccountfortheCoulombbarrier,therepulsivepotentialbetweenthenucleusandtheproton.ThepenetrabilityPl(E)factorofthes-wave(thatis,l=0)protoncapturecanbeshownusingtheexpansionofthetransmissioncoecient^Ttobedependent29upontheSommerfeldparameter:Pl(E)/e2ˇ;r2EZ1Z2e2~(3.2)whereZ1andZ2aretheprotonnumberoftheprojectileandtarget,respectively,isthereducedmassoftheprojectile-targetsystem(=M1M2M1+M2),eisthefundamentalcharge,andEisagaintheprojectileenergy.Asexpected,thispenetrabilityincreasesasthecenter-of-massenergyofthereactantsincreases.Fromanastrophysicalstandpoint,theprotoncapturerateisalsodependentuponanumberofenvironmentalfactorsincludingtherelativedensitiesofprojectileprotonsandtargetnucleiandtemperature,thelatterofwhichaectstheenergyoftheprojectiles.Itispossibletoderiveanexpressionfortherateofanuclearreactioninawaythatillustratesthisfact.ForthereactionX+Y!Z+W,whereXandY,are,forexample,aprotonandanucleus,theratemaybewritten:rXY=NXNYv˙(v)(3.3)whererXYisthereactionrateperunitvolumeandtime,NXandNYare,forXandYrespectively,thenumberperunitvolume,vistherelativevelocityoftheprojectile-targetsystem,and˙(v)isthereactioncrosssectionintermsofthatvelocity.Sincethevelocitiesofprojectilesandtargetsintheastrophysicalenvironmentarenotconstant,itispossibletouseageneralizeddistributionfortherangeofpossiblevelocitiesandwrite:rXY=NXNYZ10vf(v)˙(v)dv(3.4)30wherewecanadditionallydeneh˙viXYR10vf(v)˙(v)dvasthereactionrateperparticlepair.Asmentionedabove,itistypicallythisvaluethatistabulatedinplaceofthereactionraterXY,normalizedbytheAvogadronumberNA:NAh˙viXY.Thisthermonu-clearreactionrateisinunitsofcm3mol1s1,andreects,asmentioned,thefactthatthekineticenergyofthenonrelativisticprojectileandtargetistheirthermalmotion,dependentuponthetemperatureofthestellarplasma[3].Infact,becausetheparticlesinvolvedinthereactionarenonrelativisticandnondegen-erateandareinthermalequilibriumwithoneanother,therelativevelocitydistributionoftheprojectile-targetsystemcanbedescribedusingaMaxwell-Boltzmanndistribution,f(v):f(v)=2ˇkT3=2ev2=(2kT)4ˇv2dv(3.5)whereandvareagainrespectivelythereducedmassandcenter-of-massvelocityoftheprojectile-targetsystem,Tisthetemperatureofthestellarenvironment,andkistheBoltzmannconstant.Usingthisasthedistributionintheequationforthereactionrateperparticlepair,andconvertingfromavelocitydistributiontoanenergydistributionusingv=p2E=anddE=dv=v,thefollowingresultisreached:h˙viAB=Z10vf(v)˙(v)dv=Z10vf(E)˙(E)dE=r8ˇ1(kT)3=2Z10E˙(E)eE=kTdE(3.6)wherethereactionratedependsuponthetemperatureT,center-of-massenergyE,re-ducedmass,andthecrosssection˙(E),whichitselfdiersbetweendierentreactionsandisdependentuponthefactorsabove.Forprotoncaptureinastellarenvironment,thereactionrateasnotedheredepends31Energy (arb. units)Cross Section (arb. units)Figure3.1:AnabstractguredenotingtheGamowwindowandtheGamowpeakforpro-toncaptureinastellarenvironment.ThebluelineshowstheMaxwell-Boltzmannenergydistributionoftheparticleintheenvironment;thatis,itshowsthedecreasingprobabilityofndingaparticleatagivenenergyastheenergyincreases.Theredlineshowstheenergydependenceofthepenetrability;thatis,itshowsthatthehigher-energytheprotonis,themorelikelyitistoapproachnearenoughtothenucleustointeract.Thepurplelineshowsthecombinationofthesetwofactorstocreatearegionofincreasedreactionprobability,theGamowwindow.Thepurplelinehasbeenmultipliedbyafactorof100inthisplottoillustratetheeect.
notonlyupontheMaxwell-BoltzmannexponentialfactoreE=kTbut,asabove,theprotoncapturepenetrabilityfactor(Pl(E)/e1=pE,Equation3.2).Thesetwofactorscombinedyieldadistributionwithawell-denedmaximum(Fig.3.1);thismaximumisnamedtheGamowpeak.ThenarrowareaofincreasedreactionprobabilitysurroundingtheGamowpeakisknownastheGamowwindow,andphysicallyreectsboththefactthatanincreas-inglyenergeticprotonisincreasinglymorelikelytopenetratetheCoulombbarrierandtherapidlydiminishingprobabilityofthestellarenvironmentproducingsuchaprotonasenergyincreases.Thus,forexperimenterswishingtomeasurethereactionratedirectly,Equation3.6sug-32geststhatitisnecessarytomeasurethecross-section˙(E)atenergiesrangingthroughouttheGamowwindowinordertodeterminethethermonuclearrate.Thiscanbedicultexperimentally,butforcertainreactionsthereisanotherconsideration:thepresenceofresonances.3.2DirectandResonantCaptureReactions
Equation3.6isageneralizedreactionrateforparticlecaptureasdescribedinChapter1,eectivefordeterminingthedirectcapturereactionrateovertherelevantenergyregion.Forchargedparticlecaptures,thisregionistheGamowwindow.Physically,adirectcaptureresultswhenanucleusAZcapturesaprotonintoaboundstate,simultaneouslyemittingaphotonwhichcarriesawaytheenergydierencebetweentheinitialstateofthetargetnucleusAZplustheprotonandtheboundstateoftheproductnucleusA+1(Z+1)(Fig.3.2).However,intheeventthattheproductnucleushasanexcitedstateintheGamowwindow,therewillbeaso-calledresonantcapturecontributiontothereactionrateaswellasthedirectcapturerate.Inthiscase,thetargetnucleusandprojectileprotonhaveacenter-of-massenergyErsuchthatwhentheycombine,theyformtheproductnucleusinitsexcitedstate(Fig.3.2).Ifthisresonancestatethenundergoesgammadecay,theenergyoftheresultingphotonreectsonlythedierencesinenergiesbetweentheresonancestateandthestateintheproductnucleustowhichtheresonancedecays.Becausethisstateisunbound(i.e.itisabovetheprotonemissionthreshold,thebindingenergyforthenucleusplustheproton),itmayalsore-emittheprotonintothegroundstateofthetargetnucleusoroneofitsexcitedstates.Therelativeprobabilitiesfordecaythroughvariouschannels33EAZA+1(Z+1)SppEAZA+1(Z+1)Spp(a)(b)Figure3.2:Asimplegureshowing(a)directprotoncaptureintoanucleusand(b)resonantcaptureintothesamenucleus.Indirectcapture,theinitialnucleuscombineswithaproton,thenemitsagamma-rayintoalowerboundstateofthecompoundnucleus.Theenergyofthegamma-rayisonlyafunctionoftheinitialnucleusenergy,theprotonenergy,andthenalstateenergy.Inresonantcapture,theinitialnucleuscapturesaprotonintoahigher,unboundstateinthecompoundnucleus.Thatstatemayeitherre-emittheprotonoremitagamma-ray,deexcitingintoalowerstateofthecompoundnucleus.Inthiscasetheenergyofthegamma-rayistheenergydierencebetweentheresonancestateandthelowerstate.suchasprotonemissionorgammadecayaregivenbythepartialwidthsp,,or,forthegeneraldecaychanneli,i.Resonantcaptureischaracterizedbyanextremelyenhancedcross-sectioninaregionpeakingaroundtheresonanceenergyEr(theenergyabovetheprotonthreshold)oftheresonancestate.Thispeakisdescribedinpartbyitsfullwidthathalf-maximum,,thetotalwidthoftheresonance.Thistotalwidthisequaltothesumofpartialwidthsofalldecaychannels:=ii.Dependinguponthevalueof,aresonancemaybecategorizedasbroadornarrow.Thepresentdiscussionislimited,forthepurposesofthiswork,tonarrowresonances,whereismuchlessthanfewkeVandtheresonancewidthissuchthatthedecaypartialwidthsidonotvarywithenergy.Becauseofthesharppeakincross-sectionaroundtheresonanceenergy,theformulafordirectcapturedoesnotcompletelydescribetheenergydependenceofthecross-section.The34resonant-capturecross-sectionforanisolatedresonanceinvolvingtwodecaychannels(e.g.protonemissionandgammadecay)isdescribedbytheone-levelBreit-Wignerformula:˙(E)=4ˇ(2J+1)(2JA+1)(2JB+1)12(ErE)2+2=4(3.7)whereJandErarethetotalspinandresonanceenergyoftheresonancestate,JAandJBarethespinsoftheprojectileprotonandtargetnucleus,and1,2,andarethepartialwidthsandtotalwidth,respectively.DerivingthisBreit-Wignerdescriptionofnarrowresonancesisbeyondthescopeofthiswork,butanexcellentreviewmaybefoundinRef.[3].Usingthiscrosssectioninthereactionratecalculation(Equation3.6),thefollowingresultisobtained:NAh˙vi=NAp2ˇ~2(kT)3=2!Z1012(ErE)2+2=4eE=kTdE(3.8)where!(2J+1)=[(2JA+1)(2JB+1)].Asabove,foranarrowresonancethepartialwidthsiareconstantoverthetotalwidthoftheresonance.Inaddition,theMaxwell-BoltzmannfactoreE=kTcanbeapproximatedasaconstant,evaluatedatEr,allowingallthreetobepulledoutoftheintegral.Theintegralistheneasilyanalyticallycalculated:NAh˙vi=NAp2ˇ~2(kT)3=2!12=2eEr=kTZ10=2(ErE)2+2=4dE=NAp2ˇ~2(kT)3=2!12=2eEr=kTˇ=NA(2ˇkT)3=2~2!eEr=kT(3.9)where12=.Thequantity!isknownastheresonancestrength,andinpracticethereactionrateforanarrowresonancedependsonlyonthisquantityandontheresonance35energy.Thismeansthat,unlikeforthedirectcapturecasewherethecrosssectionmustbemeasuredatanumberofenergiesinordertodetermineitsenergydependence,thereactionratethroughagivenresonancemaybecalculatedfromthemeasurementofonlyafewparameters:thespinsoftheprojectile,targetnucleus,andresonancestateofthecompoundnucleus,andthepartialwidthsofdecayforeachchannelfromtheresonancestate.Inthecasewhereoneofthepartialwidthsismuchsmallerthantheother,i˝j,=iji+jˇi.Whenanumberofnarrow,isolatedresonancescontributetothereactioncross-section,theircontributionsaresimplysummed:NAh˙vi=NA(2ˇkT)3=2~2Xi(!)ieEri=kT(3.10)Thus,forcapturereactionswhereexcitedstatesoftheproductnucleusliewithintheGamowwindow,theresonantcapturereactionratedominatesoverthedirectcapturerateanddeterminationofthevariousnuclearparametersinvolvedintheresonancestatesbecomesparamounttoaccuratelydescribingtherate.Becausetheseparametersareintrinsictothestatesandnotafunctionoftemperatureorexternalenergy,theymaybemeasuredthroughanumberofdierenttechniques.Infact,because30P(p;)31Sisthoughttobedominatedbynarrow,isolatedresonances,studyofthesestatesoersapotentiallyfruitfulmeansofconstrainingtherate.Someofthetechniquesthathavebeenusedtostudythe30P(p;)31SreactionwillbediscussedinChapter4.
3.3ReactionRate:TheoreticalConsiderations
Intheeventthatoneormoreoftheparametersinvolvedinaresonantcapturereactionareunknownexperimentally,theoreticalestimationsmaybeusedtosupplementexisting36information.Forprotoncapturereactions,maybeestimatedusing,forexample,thenuclearshellmodel,andpmaybeestimatedusingthefollowingformula:p=2~Rn2Er1=2Pl(Er;Rn)C2S2s.p.(3.11)whereErisagaintheresonanceenergy,Rn=1:25(A1=3A+A1=3B)fmistheinteractionra-dius,Pl(Er;Rn)isthepenetrabilitywhichitselfcanbecalculatednumericallybycomputingtheregularandirregularCoulombwavefunctionsFlandGlrespectively(Pl=1=[F2l+G2
l])[37],CisaClebsh-Gordancoecient,Sisthesingle-particlespectroscopicfactor,and2s.p.isthesingle-particlereducedwidth.Thefactor2s.p.containsinformationregardingthenu-clearstructureofthecompoundlevelanddependsupontheinteractionradiusRn,orbitalangularmomentuml,andnumberofnodesninthesingle-particleradialwavefunction˚l(Rn):2s.p.=Rn2˚2
l(Rn)(3.12)Thisestimationofpmaybeusedinplaceofanexperimentally-measuredpartialwidth,particularlyatenergiesclosetotheprotonemissionthresholdwheretheenergyofthein-comingprotonislowandtheprotonpartialwidthisthusmuchlowerthanthegamma-decaypartialwidth,p˝.Inthiscasethetotalwidthmayalsobeapproximatedˇp,asabove.
3.3.1TheHauser-FeshbachStatisticalModel
Unsurprisingly,onlyasmallsubsetofthevastnumberofnuclearreactionsrelevanttothenumerousstellarburningprocesseshavebeenmeasuredexperimentally.Incaseswhereno37experimentalparametershavebeenmeasured,itispossibletoestimatethereactionrateusingtheHauser-Feshbachstatisticalmodel[27].Derivationofthemodelformalismisbeyondthescopeofthiswork,butbriey,themodelassumesthereisahighleveldensityinthecompoundnucleusandthatconsequentlythereisalargenumberofresonancestatesthroughwhichthereactioncanproceed.Itthencalculatesacross-sectionthrougheachoftheseresonancesusingtheleveldensityandotherinputparameterssuchasthetransmissioncoeecientandaveragesovertheenergyregion,providingastatisticalestimateofthereactioncrosssectionintheregionofinterest.Hauser-Feshbachcalculationsforprotoncapturereactionsaretypicallyreliablewithinafactorofˇ23{butonlyiftheleveldensityofthecompoundnucleusislargeenoughintheregionofinterest[3].Incaseswheretheleveldensityisrelativelylow,Hauser-Feshbachmayprovideanunreliableestimateofthecross-sectionandthusthereactionrate.30P(p;)31SisexpectedtotobeattheedgeofapplicabilityfortheHauser-FeshbachratebecausethedensityofstatesintheGamowwindowisrelativelylow.
3.4TheNuclearShellModel
Analogoustothemodelforelectronsinatomicorbitals,itispossibletomodeltheenergystatesofagivennucleusbytakingintoaccountthefactthatprotonsandneutronsarebothfermionswhichobeythePauliexclusionprinciple.Theshellmodeltreatsthenuclearpo-tentialaccordingtotheWoods-Saxonpotential(Figure3.3)withanadditionalinteractioncouplingthenucleonspintoitsorbitalangularmomentum.Protonsandneutronsinde-pendentlyllorbitals,withcertain\magicnumbers"ofnucleonscorrespondingtoincreasedstabilityaccordingtotheclosureof\shells"ofnuclearorbitalswithquantumnumbersn(the38Figure3.3:TheWoods-Saxonpotentialcommonlyusedtomodelthenuclearforce.Theformofthepotentialis:V(r)=Vo=[1+exp(rR=a)]whereVoisthepotentialdepth,R=1:25A1=3fm(Athemassnumber),andarepresentsthe\surfacethickness"ofthenucleus.
numberofnodesinthewavefunction),l(theorbitalangularmomentum),andj(thetotalangularmomentum),typicallyusingthenotationnljtodenotethecombinationofquantumnumbersthatdenesthestate.The1sorbitalhasl=0,forexample,andcanthusholdonlytwonucleons(spin-upandspin-down;thatis,totalj=1=2andjz=1=2),the1porbitalwithl=1canholdsixnucleons,twointhe1p1=2orbital(j=1=2;jz=1=2)andfourin1p3=2(j=3=2;jz=3=2;1=2).The\magicnumbers"canthusbederivedfromthetotalnumberofnucleonscontainedattheshellclosures:2(1s,closingthes-shell),8(1s+1p,closingthep-shell),20(1s+1p+1d+2sclosingthesd-shell),etc.Unliketheatomicshellmodel,thenuclearshellsdonotstrictlycorrespondtoaparticularquantumnumberandinsteadrepresentonlyplaceswherethebindingenergyofthenucleushaslargegapsbetweenorbitals,henceexamplessuchasthesd-shell,whichincludesthe1d5=2;2s;and1d3=2orbitals.AgraphicalrepresentationoftheshellmodelanditscomparisontotheharmonicoscillatorisshowninFigure3.4.Byaccountingfortheseconsiderations,theshellmodelcanbeusedtopredictthespinandparityofthegroundstateofagivennucleus.Forexample,30P,with15protonsand15neutrons,llsthe1s,1p,and1dorbitalsforbothprotonsandneutrons,withoneproton39Figure3.4:Comparisonoftheenergylevelsofthenucleususingthesimpleharmonicoscilla-torpotential(left)andusingthefullshellmodel(right).Asshown,thespin-orbitinteractionsplitstheharmonicoscillatorlevels,andthelargegapsinbindingenergythatcausetheshellsdonotnecessarilycorrespondtothegapsbetweenharmonicoscillatorlevels.Figurecredit:Bakken(GPL).40andoneneutroneachintheirrespective2sorbital.Thenucleuscanherebethoughtofasaninertcoreof14protonsand14neutrons(thatis,aninert28Sicore)withtotalangularmomentumj=0,becauseitcompletesthe1d5=2orbital,andtwoextravalencenucleons,eachwithj=1=2.Thetwoextranucleonssumtoatotalj=1.Becausethetwonucleonsareinanorbitalwithl=0,theparityispositiveandtheshellmodelcorrectlypredictsaspinandparityofthe30PgroundstateofJˇ=1+.Inprinciple,theshellmodelcanalsobeusedtopredicttheenergies,spins,andparitiesofexcitedstatesinthenucleus.However,althoughthegroundstatespinandparitycanbeinferredfromthespinsandparitiesofindividualnucleons,thisisnotnecessarilytrueforexcitedstates.Thisisduetothecollectivityofthenucleus,orthetendencyofmanynucleonstobesimultaneouslyinvolvedinnuclearexcitations.Aswiththecalculationforthe30Pgroundstate,itiscustomaryforshellmodelcalculationstotreatagivennucleusasaninertcorewithoneormorevalencenucleons,limitingthevalencespaceofthecalculationtoafeworbitalsabovetheclosedcoretoreducecalculationcosts.Becausethecalculationforthenucleus'sstructureisarbitrarilytruncatedatsomepoint,thecalculationmustuseaso-calledectiveinteraction,whichaccountsforthetruncationintheHamiltonianusedforthecalculation.Oneverycommonly-usedHamiltonianisthe\Universalsd-shell"(USD)interaction[38].TheUSDmodelsUSDAandUSDBproducetwo-bodyeectiveHamiltonianmatrixelementsmeanttotreattheinteractionbetweennucleonsinthesd-shellspace.Thesematrixelementsareinpartderivedfromatofexperimentally-knownenergylevelsonnucleiinthemassregionfrom17Oto39K,andcanbeusedincalculationstonotonlypredicttheenergies,spins,andparitiesofnuclearstatesinagivensd-shellnucleus,butprobabilitiesforelectromagnetictransitionsbetweenstates(e.g.B(En),wherenistheorderoftheelectrictransition)as41wellastransitionprobabilitiesforbetadecay(amorethoroughdiscussionaboutbeta-decaystrengthsfollowsinSection3.5.1).Forthecalculationofastrophysicalreactionrates,inthecasewhenenergylevelsforagivennucleus,gamma-decaybranchesforthoselevels,andbeta-decayprobabilitiestothoselevelsfromaparentnucleuscanbecalculated,measurementsmaybecomparedtotheoryinordertoconstrainthespinandparityofobservedstates.Incaseswhereparameterssuchaspareneededtocalculatearesonancestrengthofthelevel,quasi-theoreticalpvaluesbasedonobservedenergiesandcalculatednuclearstructurecomponentsmaybeusedtogiveameaningfulestimateoftheresonancestrengthand,hence,thereactionrate,withreducedtheoreticaluncertaintieswhencomparedwiththeHauser-Feshbachstatisticalmodel.3.5MirrorNucleiandtheConceptofIsospin
Anothertoolusefulfordeterminingthespinandparityofnuclearstatesistheexistenceofmirrornuclei,nucleiwithidenticalmassnumberAbutconjugateprotonnumberZandneutronnumberN.Inprinciple,thenuclearstructureofmirrornucleishouldmatchclosely:asshowninFigure3.5,forexample,theenergiesofexcitedstatesin13Cand13Nareverysimilar,andthespinsandparitiesofthestatesareidentical.Thus,ifparameterssuchasspinandparityareknowpreciselyinagivennucleus,thatinformationcanbeusedtohelpconstrainmatchingstatesinthemirrornucleus{assumingthatthestates'mirrorassignmentcanbemadeaccurately.Theusefulnessofmirrornucleiisaconsequenceoftheisospinmodel,aconceptintroducedbyWernerHeisenbergin1932[39].Becausethenuclearforceischarge-independentandthemassesoftheprotonandneutronarenearlythesame,itispossibletotreatthemnotas4213N1/2-1/2+3/2-5/2+5/2+13C1/2-1/2+3/2-5/2+5/2+EFigure3.5:Comparisonoftherstveenergylevelsof13Nand13C.Theground-stateenergiesofthetwolevelshavebeenadjustedtobeequal,andtherelativeheightsofthestatesineachnucleusrepresenttherelativeexcitationenergiesofthosestates.Thespinsandparitiesofthestatesarealsoshown,andcolorisgiventodenotepositiveornegativeparitystates.Therelativespacingoflevelsinagivennucleusisnottoscaleasthisgure'spurposeissimplytoshowthemirrorsymmetrybetweenthesetwonuclei.individualparticles,butasprojectionsinaso-calledisospindoubletofasingleparticle,thenucleon.Thisparticlehasbothspinj=1=2andisospinT=1=2,withprojectionsTz=+1=2forneutronsandTz=1=2forprotons.Inthisway,agivennucleuscanbeseentohavetotalisospinprojectionTz=(NZ)=21.However,eachstateofagivennucleus,althoughsharingthesameprojection,doesnotnecessarilycarrythesametotalisospinT.Themathematicalformalismforisospinisverysimilartothatforstandardangularmomentum.Forexample,astatewithagiventotalisospinTcanhaveanyofthe2T+1projectionsT<Tz<T.Inthecaseof13Cand13NasshowninFigure3.5,thesimilaritiesbetweenthelevelschemescanthusbeinterpretedastheobservationofisospinstateswiththesameisospinacrossnucleibutwithdierentisospinprojections.Inthemostbasicexampleofthiscase,thegroundstatesofthetwonucleicanbeinterpretedastheTz=+1=2(13C)1Thisistheconventionfornuclearphysics.TheconventiondiersfromtheoriginalconceptionoftheprotonasaTz=+1=2particleandthemodernconventioninparticlephysics.Thisconventionwaschosensothatthemorenumerousneutron-richnucleihavepositiveisospinprojection4313O13N13BT = 1/2T = 1/2T = 3/213CT = 1/2T = 3/2Tz= +3/2ETz= +1/2Tz= -1/2Tz= -3/2Figure3.6:AsimplegraphicaldepictionofT=1=2isobaricdoubletsin13Cand13NandT=3=2isobaricquartetsin13B,13C,13N,and13O.Heretheenergylevelshavebeenequalizedtodemonstratethemanifestationofperfectisospinsymmetrymoreclearly.Asshown,thegroundstatesandmultipleexcitedstatesin13Nand13CmakeuptheTz=+1=2andTz=1=2membersofthersttwoT=1=2doublets,andcertainexcitedstatesof13Nand13CcomprisetheTz=+1=2andTz=1=2membersoftheT=3=2quartetscompletedbyanalogousTz=+3=2andTz=3=2statesin13Oand13B,respectively.Thespacingbetweenlevelsisarbitraryandismeantonlytoshowthesymmetryoftheisobaricanalogstates.
andTz=1=2(13N)membersofaT=1=2doublet.Ingeneral,statesinthesenucleiwillsharetheisospinofthegroundstate.However,certainhigherenergystates,suchasthoseanalogoustothegroundstatesof13B(Tz=+3=2)and13O(Tz=3=2),willhaveisospincorrespondingtotheirownmultiplet(e.g.,membersofaT=3=2quartet).Thesestatescanbeseenasmembersofanisospinmultiplet,asshowninFigure3.6.Generally,stateswithsimilarwavefunctionsthatshareanisospinvalueacrossnucleiareknownasisobaricanalogstates(IAS)ofoneanother;thesestatessharethesamespinandparityaswell,allowingforthemirrorsymmetrydescribedabove.443.5.1IsospinandBetaDecay
Sinceboththeprotonandneutroncarrytheirisospinanduniqueisospinprojection,anyprocessthataectsthenumberofnucleonsinanucleus,forexamplebetadecay,changesattheveryleasttheprojectionTz,andusuallytheisospinTaswell.Inthecaseofbetadecay,thenucleonnumbersNandZchangebutthetotalmassAremainsconstant.ThischangestheisospinprojectionTz=1;thatis,anucleonipsitsisospinprojection.Betadecayproceedsthroughtwocouplingsoftheweakinteraction,thevector(V)andaxial-vector(A)components,withdecayratefrominitialnuclearstatetonalnuclearstategivenbytheequation:Wi;f=(f=Ko)[g2VBi;f(F)+g2ABi;f(GT)](3.13)wherefisadimensionlessconstantthatdependsontheQ-valueofthedecay,Ko=1:88441094erg2cm6s,gVandgAarethecouplingconstantsforthevectorandaxial-vectordecays,andB(F)andB(GT)arethetransitionstrengthsforthevectorchannel(\FermiDecay")andaxial-vectorchannel(\Gamow-TellerDecay"),withdenoting+anddecay.TheoperatorsoftheFermiandGamow-TellertransitionsareO(F)=itiandO(GT)=i˙iti,whereiisthesummingindexoverallnucleonsinthenucleus,tistheraising/loweringladderoperatorthatchangestheisospinprojectionTzbyone,and˙isthenucleonspinoperator.InthecaseoftheFermioperatoritcanbeseenthattheeigenvalueoftheoperatoractingonastateisgivenby:tjT;Tzi=pT(T+1)Tz(Tz+1)jT;Tz1i;(3.14)45theFermistrengthB(F)issimplygivenby:B(F)=jhTf;TzfjtjTi;Tziij2=(T(T+1)Tzf(Tzf+1))TfTiTzfTzi1;(3.15)allowingforquickcalculationoftheexpectedFermistrengthofagivenbetadecaytran-sitionbasedontheisospinandisospinprojectionoftheinitialstate.3.5.2SelectionRules
FromtheseoperatorsitcanbeseenthattheFermitransitionmovesonlybetweenspinprojectionsTzofisospinmultipletmemberswithoutchangingthetotalisospinT.Thisresultsinabetadecayselectionrule:Fermitransitionsarethoseexclusivelybetweenagroundstateanditsisobaricanalogstate:Tz=1;T=0;J=0.Becauseoftheadditionofthespinoperator,theselectionrulesforGamow-Tellertransitionsarelessstrin-gent:Tz=1;T=0;1;J=0;1(J=09J=0).TheFermitransitioncanbethoughtofasproceedingbetweentwostateswithinthesameisospinmultiplet,whiletheGamow-Tellertransitioncanbethoughtofasproceedingbetweentwostatesfromdierentmultiplets(whichmayormaynotsharethesameisospin).BecausethestrengthoftheFermitransitionisconcentratedintoasinglestate(theIAS),FermitransitionsgenerallyproceedatanincreasedratecomparedtoGamow-Tellertransi-tions.TheGamow-Tellertransitionstrength,bycomparison,isusuallyfragmentedbetweenmanynalstates.FermiandGamow-Tellertransitionsarelabeled\allowed"transitions(J=0!J=0decayswithT=0arepureFermitransitionsandarelabeled\super-allowed").Transitionsnotallowedbytheseselectionrules(e.g.,J=2)arelabeledas\forbidden"{thatis,thematrixelementofthetransitioniszeroandtheyaretheoretically46prohibitedfromhappening.Inrealitythesetransitionsdoproceed,butatagreatlyreducedrate.Asshownhere,bothspinandisospinselectionrulesplayapartinconstrainingwhichnalstatesarepopulatedinanuclearprocess.Forexample,inprotoncaptureonto30P,theT=1=2protoncouplestotheT=030Pgroundstate.Thismeansthatonly31SstateswithT=1=2arepopulated.Conversely,forthebetadecayoftheT=3=231Clgroundstateto31S,thereisastrongtransitiontotheT=3=2IAS.Unfortunately,eventhoughthisstateispopulatedviatheFermitransition,makingitsspinandparitypreciselyknown(Jˇ=3=2+)itisnotexpectedtobepopulatedatallinthe30P(p;)31Sprotoncapture,ifisospinisagoodquantumnumber.
3.5.3TheIsobaricMultipletMassEquation
Sofar,wehavediscussedisospinasthoughitwereaperfectsymmetry,andthenucleonsasthoughtheirchargeswereirrelevant.Ifthisweretrulythecase,isobaricanalogstatesbetweennucleiwouldallsharethesamemassexcess2,asshowninFig.3.6.Thefactthattheydonot(asshowninFigure3.5,forexample)isduetothechargeoftheproton.BecausetheCoulombforceactsonlyonprotons,theenergyvalueofmultipletmemberstatesisperturbedsystematicallyacrossthemultipletaccordingtotheprotonnumberZ.Thisshiftcanbetreatedwithrst-orderperturbationtheory:theisobaricmultipletmassequation(IMME),rstproposedbyWigner[40,41],predictsthatisobaricanalogstateswithinamultipletofgivenThavemassexcessesrelatedtooneanotheraccordingtotheir2Massexcessisthedierencebetweenthemassofanucleusandthemassofitsconstituentnucleons,M(Z;A)Amu.Foranexcitedstate,themassexcessisthis(negative)valueplusthe(positive)excitationenergyExofthestate.47isospinprojection:(Tz)=a+bTz+cT2z(3.16)wherea,b,andcarecoecientsthatcanbecalculatedfromthetheoryorobtainedbyttingthemeasuredmassexcessesofthemultipletmemberstateswithaquadraticfunction.Ifthemassexcessesofenoughmultipletmembersareknown(forexample,forthreemembersofanisospinquartet),theIMMEcanbeusedtopredictthemassexcessesoftheunobservedmemberstates.Ifthegroundstatemassexcessofthenucleusofsuchastateisknownprecisely,apredictedexcitationenergyExcanbecalculatedaswell.TheIMMEhashistoricallybeenverysuccessful,onlyfailinginafewknowncases[42,43,44,45].ReasonsforthebreakdownoftheIMMEcouldinclude:thepresenceofmany-bodycharge-dependentforces[46];afailureonthepartoftheperturbationtheory,indicatinganeedforhigher-orderterms;inaccuratemeasurements;orisospinmixingofthemultipletmemberwithanearbystateofadierentisospin[47].
3.5.4IsospinMixing
TheperturbativeeectsoftheCoulombforcecanbethoughtofastheresultoftwofactors:therelativerangedierencebetweenthenuclearforceandtheCoulombforce(i.e.,theformeronlyactsbetweencloselyneighboringnucleonswhilethelatteractsoverthewholenucleus)andtherelativestrengthsoftheinteractionsasmassincreases(i.e.,theformerincreases,torstorder,linearlywithnucleonnumber,whilethelatterincreases˘Z2)[48].Thus,forincreasinglyheavynuclei,theCoulombinteractionmayhaveagreatereectontheviabilityofisospinasagoodquantumnumber.48Alongwiththeperturbationofisobaricanalogstatemassexcessesdescribedabove,theCoulombinteractioncanresulttosomedegreeinthemixingofstateswithdieringisospins.Inthiscase,theHamiltonianmatrixfortheinteractionsinvolvingthesestatescontainso-diagonalelementscorrespondingtoanisospininteractionbetweenthem.Forasimpletwo-levelmodelwithstatesj1i=jJ1T1iandj2i=jJ2T2i,H=0
B
@H11H12H21H221
C
AwhereH11andH22includethecontributionstotheHamiltonianfromnuclearinterac-tionsandtheo-diagonalelementsH12andH21containtheisospin-mixingterms.BecausetheCoulombforceisrotationallyandtime-reversalinvariant,thematrixisrealandsym-metric,withH12=H21=VJ1J2,whereVisthemixingstrengthandtheJ1J2denotesthefactthatonlyisospin-mixingtermsaecttheo-diagonalelementsduetotherotationalinvariance.Now,insteadofjJ1T1iandjJ2T2ibeingisospineigenstatesoftheHamiltonian,theyformthebasisforneweigenstates(withJ1=J2=J):j 1i=cosjJT1i+sinjJT2i(3.17)j 2i=sinjJT1i+cosjJT2i(3.18)wherethemixingangleisgivenbytan2=2VD,whereDistheenergydierencebetweentheoriginalstates.Fromthisexpression,itcanbeseenthatonlystateswhoseenergiesareneareachother(thatis,Dissmall)willmixnon-negligibly.Thistwo-statemixingsystemfollowsthestandardtwo-statemixingformalismforquantummechanics:the49levelsareperturbedinenergyaccordingtotheirunperturbedspacingDandthemixingmatrixelementVaccordingtotheequationsE=pD2+4V2(3.19)E=D+2(3.20)whereEistheobserved(perturbed)energydierencebetweenthestates,Disagaintheunperturbedenergy,Visthemixingmatrixelement,andistheperturbation.InthecasewhenoneofthemixedstatesisanisobaricanalogstatepopulatedbyaFermitransition,themixinganglecanbecalculatedexperimentallyfromthedeterminationoftheobservedFermistrengthstothetwomixedstates:R=tan=sB2B1(3.21)whereB2andB1arethestrengthsofthetransitionstothesecondandrststatelistedinEquations3.17and3.18,respectively.TheunperturbedmixingmatrixelementVandunperturbedenergyspacingDmaythenbecalculatedpurelyfromquantitiesbasedonobservables:D=E1R21+R2(3.22)V=ER1+R2:(3.23)50Inthiscase(mixingbetweenanIASandanearbystate),theFermitransitionconstrainsthespinandparityoftheIAS,whichinturnconstrainsthespinandparityofthemixedstate.Thus,observationofisospinmixingbetweenanIASandanearbynuclearstateinanenergyregionwherethespinsandparitiesofstatesarepoorlyconstrainedcouldprovideaconstraintonthespinandparity{propertiesofthatstaterelevantforcalculatingtheresonancestrengthofareaction.51Chapter4
ExperimentalConsiderationsand
NSCLExperiment12028
AsmentionedinSection2.1.2,the30P(p;)31Sreactioniscriticaltoansweringanumberofastrophysicalquestions,butitsrateissopoorlyexperimentallyconstrainedthatuntil2007theadoptedratewasbasedonthestatisticalHauser-Feshbachcalculation.However,theHauser-Feshbachmodelcalculatestheratebasedonanassumedhighdensityofstates,whichistypicallyonlyvalidatmassnumbersand/orenergieshigherthantheGamowwindowforthisreaction.Inaddition,themodeldoesnottakeintoaccountindividuallow-energyresonances,whichaspreviouslydiscussedarelikelytodominatetherateatnovatemperatures.Thus,experimentalmeasurementoftheresonancestatesintheenergyregiondirectlyabovetheprotonthresholdisofkeyimportanceformovingtowardanaccuratecalculationoftherate.Ideally,thepropertiesoftheresonancesinvolvedinthisreactionwouldbemeasureddirectly.Inprinciple,protoncapturereactionsmaybemeasuredinnuclearphysicsfacilitieseitherbyacceleratingahydrogenbeamandimpingingituponaheavytarget(\regularkinematics")orthereverse,acceleratingabeamofthetargetnucleusandimpingingituponahydrogentarget(\inversekinematics").Since30Phasahalf-lifeofjust2.5minutes,maintainingatargetlongenoughtoperformadirectmeasurementusingahydrogenbeam52withsucientstatisticsisunfeasible,leavingthereverse-kinematicsmethodtheonlyfeasiblemeansofstudyingthereactiondirectly.Therearetwomajorcurrently-usedmethodsofproducingaradioactivebeamsfordirectstudyofthe30P(p;)31Sreactionbyinversekinematics.Fragmentationinvolvesproducingabeamofareadily-availablestableisotope,acceleratingittohighenergies(theexactenergydependsupontheacceleratorusedandthecharge-to-massratioofthenucleusinquestion{forexample,thecoupledcyclotronfacilityattheNationalSuperconductingCyclotronlaboratorytypicallyacceleratesstableionstobetween80MeV/u1and170MeV/u),andimpingingituponathin,lighttarget.Thisso-calledprimarybeamisfragmentedtoproduceanumberoflighterconstituentpieces,includingthedesiredshort-livednucleus.Theresult-ing\cocktail"beamofvariousisotopespeciescanthenbelteredusingmagneticeldstoproduceapurersecondarybeamofthedesiredisotopeforstudy.Thefragmentationtech-niquecanallowforrelativelyhighratesofproduction,butbecausethebeamisacceleratedtoveryhighenergiesbeforeproductionofthedesiredisotope,itisdiculttostudylow-lyingresonancesdirectlywithoutslowingthebeamdowntostellarenergies.The\IsotopeSeparationOn-Line"(ISOL)techniqueinvolvestheaccelerationofabeamoflight,stablenuclei,whichbombardsatargettoproduceradioactiveisotopes.Thechoiceofbothstablebeamandproductiontargetaectswhichtypeofisotopesareproduced,andthedesiredisotopesmustalsobeextractedfromthetarget;theselow-energyisotopesarethenionizedatasourceandseparatedbymasstoproduceapurebeamoflowenergyions.Theseionsareacceleratedanddeliveredtotheexperimentalsetup.Unlikefragmentation,theseionscanbeacceleratedtoavariabledesiredenergyandatamuchbetterpurity,allowingformorestraightforwarddirectstudyofcapturereactionssuchas30P(p;)31S.The1thatis,MeVpernucleon53ISOLtechniqueisnotableinthatitcanproducerelativelypurebeamswhencomparedtofragmentation,sincethemassseparationoccursbeforeacceleration.However,theintensityofthebeamisdependentuponanumberofchemistry-relatedfactorsthatmaymakeitdicultforcertainspeciesofinteresttoleavethetarget,meaningthatnoteveryisotopethatisdesiredforstudyispracticaltoproduceviathistechnique.Unfortunately,bothofthesemethodsareinecientatproducing30Pfordirectstudyof30P(p;)31S.ISOLdoesnotproduceasucientlyintensebeambecausephosphorusisa\refractory"element,andevenusingtechniquessuchasgasstoppingofbeamsandreaccel-eration,fragmentationfacilitiesarecurrentlyunabletoproduceabeamwithhighenoughintensityandlowenoughenergytopopulatetherelevantresonancestates.Becauseofthis,anumberofindirectmethodshavebeenemployedtopopulatetheseimportantstatesandmeasuretheirproperties.Becausetheproperties(spin,parity,resonanceenergy,totalandpartialwidths)ofthenuclearstatesinvolvedinthereactionareindependentofthereac-tionmechanism,andbecauseforresonantcapturethecross-sectionisentirelydependentupontheseparameters,populatingthemindirectlyandmeasuringtheirdecay2canprovideconstraintsontheparametersthatconstrainthereactionrate.Indeed,anumberofindirectmethodshaveprovidedaconsiderableamountofinforma-tiononthestatespotentiallyinvolvedinthe30P(p;)31Sreaction.Boththediverseandingenuitivetechniquesusedtostudy31Saswellastheexperimentsperformedthemselvesareworthdiscussingbriey.2Thismethodofmeasuringthesestatesiscontingentupontheso-calledBohrindependencehypothesis,oramnesiaassumption,theassumptionthatthemodeofdecayofacompoundnucleusisindependentofitsmodeofformation.Thishypothesisisbasedonthenotionthatthelifetimeofthecompoundsystemismuchlongerthanthetimescaleforformationanddecay(thatis,atleastseveraltimesthetimeittakesforanucleontotraversethenucleus),andthatthecompoundnucleustherefore\forgets"all\memories"exceptthosepertainingtoconservationlaws.Forhigherenergies,whereresonancesoverlap,thisassumptionhasbeenshowntobeinvalid.SeeRefs.[49,50,51,52]fordiscussionandrelevantexperiments.544.1TechniquesforIndirectStudyof30P(p;)31S4.1.1Single-NucleonTransferReactions
Single-nucleontransferstudiesof30P(p;)31Sinvolvepopulatingexcitedstatesof31Sbyimpingingafast,lightprojectilesuchasp,dor3Heontoaheavytransmissiontargetofastablenucleuswithamassnumberonelessthanoronegreaterthan31.Thelightprojectilegainsonenucleonfromorlosesonenucleontothetarget,gainingorlosingenergyaswellintheprocess.Thetransformedprojectile(ejectile)thencontinuesonitswayandcanbeanalyzedeitherusingsilicondetectorsthatrecorditsenergyorbyusingaspectrograph,whereitskineticenergyismeasuredintermsofitsrigidityinamagneticeld.Theresultingspectrumwillshowpeaksatvariousenergies,whichcorrelatetotheenergytransferredbetweenthereactantstocreatethecompoundnucleusand,hence,theexcitationenergiesofthelevelspopulated.Inprinciple,thismeasurementmaybemadeatanumberofazimuthalanglesandtherelativeintensitiesofthepeaksofanyexcitedstatemaybecomparedtodeterminetheorbitalangularmomentumloftheprojectileandconsequentlythepopulatedstate,butinordertocondentlydeterminethetotalspinandparityJˇofthestate,itisusuallynecessarytocalculateatheoreticalangulardistributionfortheejectile,assumingtheallowedJˇvaluesforthestatebasedonthetriangleruleforangularmomentum,andcomparetoexperiment.Becausetheinitialspinofthetargetandprojectilearebothknown,itisalsopossibletocompareresultstoshellmodelcalculationsofexcitationenergiesandJˇvaluesandinferthespinsandparitiesoftheresonancestates.554.1.2In-BeamGamma-RaySpectroscopy
Bydenition,\gamma-rayspectroscopy"referstoanyexperimentalprocedurethatinvolvestheaccumulationandanalysisofanenergyhistogramofgamma-rays.In-beamgamma-rayspectroscopyexperimentalstudiesof31Sinvolveproductionofthe31Snucleusinanexcitedstate,whichthenemitsagamma-rayinde-excitation.Theproductiontargetissurroundedbyoneormoregamma-raydetectors,whichmeasurethede-excitationofthe31Sandallowforthecreationofagammaspectrum,fromwhichresonanceenergiescanbedetermined.Thesemaybeusedtocreateagamma-decayschemewithgammabranchesforeachexcitedstate{theschemecanbecomparedwith,forexample,thenuclearshellmodel,facilitatingtheinferenceofparameterslikethespinsandparitiesofstatespotentiallyrelevanttothe30P(p;)31Sreaction.4.1.3Charge-ExchangeReactions
Theexecutionofcharge-exchangereactionexperimentsissomewhatsimilartothatforsingle-nulceontransfer:impingealight,fastprojectileontoastabletarget,producethedesiredcompoundnucleusandmeasuretheenergyoftheejectiletolearnabouttheenergyand/orspinoftheresonancestatepopulatedintheexchange.However,theinteractionmechanismisdierent:theprojectilenucleusapproachescloseenoughtothetargetnucleussuchthatitfallswithinthepotentialwellofthenuclearforce.Thetwonucleiinteract,resultinginasinglenucleonineachnucleusippingitsisospinprojectionTz:theresultisthataprotonintheprojectiletransformsintoaneutronwhileaneutroninthetargettransformsintoaproton,orviceversa.ThemassnumberAofbothnucleiremainsthesamewhiletheirprotonnumbersZ56increaseordecreasebyonesuchthatthetotalZofbothnucleiremainsconstant.ThechangeinTzmayalsobeaccompaniedbyachangeinisosopinT=0;1,aswellasanuclearspinchangeS=0;1,andanytransferoforbitalangularmomentuml(inthiscasetheparityofthestatemayalsochangeaccordingtotherule:ˇ=(1)l).Inthecasewherenoorbitalangularmomentumistransfered,totalangularmomentumJchangesonlyby0or1;charge-exchangereactionscanherebeseentoobeytheselectionrulesmentionedinSection3.5.1forFermi(T=0)andGamow-Teller(T=1)transitions.Alongwiththeseselectionrules,theangulardistributionoftheejectilemaybeusedtoconstrainthespinandpossiblyparityoftheresonancestatespopulatedintheexchange.4.1.4BetaDecay
Asmentionedin1,betadecayinvolvesthetransformationofasinglenucleonwithinthenucleus:eitheraneutrontoaproton(),orvice-versa(+).Similartochargeexchange,betadecaychangesthemassandprotonnumbers(A;Z)ofanucleusto(A;Z1).Unlikecharge-exchangeinteractions,however,onlyasinglenucleusisinvolvedandthetransitionismediatedbytheweakinteraction.Theelectron/positronemissionconservesthechargeofthedecay,whilethesimultaneousemissionofanelectronantineutrino/neutrinoconservestheleptonnumber.AsmentionedinSection3.5.1,betadecayalsofollowstheselectionrulesforFermiandGamow-Tellertransitions.Theemissionofthetwospin-1/2particlesmeansthat,aswithcharge-exchangereactions,spinmaychangebetweentheparentanddaughternucleus,S=0;1;however,transitionswithgreatertotalmomentumchangeJthanthisarestronglysuppressed(\forbidden,"atermdenotingthattheyarenotallowedwithintheusualmathematicalformalismusedtodescribethetransition{althoughtheyareobviouslynotforbiddeninreality)sincethere57isnoincomingnucleusthatmaytransferorbitalangularmomentum.Thismeansthattheparityofthenalstateis,for\allowed"transitions,identicaltothatoftheinitialstate.Thus,beta-decayexperimentsareusefulincaseswherestrongconstraintsonthespinsandparitiesaredesired.
4.2PreviousStudiesof30P(p;)31SItshouldbenotedthatnosinglemethodmentionedabovehasbeenexclusivelyusedtoobtaininformationaboutthe30P(p;)31Sreaction.Inmanycases,duetothenaturesoftheexperimentalmethods,constraintsontheresonancestatesarenotperfectlyunambiguous.TheGamowwindowfor30P(p;)31Satnovatemperaturesextendsabout600keVabovethe31Sprotonthresholdat6130keV,sothenumberofresonancestatesinthatregion,aswellastheirproperties,isofkeyimportanceforconstrainingthe30P(p;)31Srate.Presentedhereisabriefoverviewofthehistoryof30P(p;)31Sstudieswhichhavebeenusedtopiecetogetherthecurrentunderstandingofthe31Sresonancesintheenergyregionofinterest.In1999,Vernotteetal.[53]publishedworkontheiruseofthesingle-protonandsingle-neutrontransferreactionson32S,32S(3He,)31Sand32S(d,3He)31P,topopulatelevelswithexcitationenergyEx8MeVinboth31Sanditsmirrornucleus31P.3131Slevelsand4131Plevelswereobserved.Themeasuredexcitationenergiesandlvalueswerecomparednotonlywithshellmodelcalculationsbutwitheachother,withthegoaltoproduceanidenticationofpositive-paritystatesin31Susingboththemirrornucleusandthesdshellmodel.Whilethisexperimentwasnotmotivatedbyastrophysics,itprovidedamodelfortwofuturetransferreactionsintendedtoprobethe30P(p;)31Sreactionfornovastudies[54,55].58ItwasnotuntilsevenyearsafterthepublicationofVernotteetal.thatworkwaspublishedfollowingupontheastrophysicalmotivationsforstudyof31S.Kankainenetal.publishedtheresultsofabeta-decaystudyof31Cl[56],whichpopulates31Sresonancesinitsdecay.Theexperimentmeasuredboththebeta-delayedprotonemissionandbeta-delayedgammadecayofstatesabovetheprotonthreshold.Multipleusefulresultswereobtained,includingthediscoveryofastatethenevaluatedat6921(15)keVthroughmeasurementofprotonemissions,gammarayscorrespondingtode-excitationofthersttwoexcited31Sstates,andtherstdenitiveidenticationofthe31Sisobaricanalogstatethroughmeasurementofitsgammadecay.Jenkinsetal.performedameasurementofboththe12C(20Ne,n)31Sand12C(20Ne,p)31Preactions,motivatedbothbyconsiderationsofthestructuraldierencesbetweenthetwomirrornuclei[57]andbyconsiderationsofthe30P(p;)31Srate[58].Theauthorsob-servedseveralproton-unboundstatesin31Sand,throughanalysisofgamma-rayangulardistributionsandcomparisonwithmirrorstatesin31P,inferredtheirspinsandparities.Althoughthismethodpreferentiallypopulatedhigh-spinstates,whicharelessimportantforthe30P(p;)31Sreactionduetotheheightenedcentrifugalbarrier,thestudyusedthisfacttoconstrainwhichotherresonanceswereactuallyimportantforthereactionrateandconcludedthattwospecicresonancesatEx=6257keVand6350keVdominatethe30P(p;)31SrateattemperaturesbelowT=0:2GKandthatanadditionaltworesonancesatEx=6543and6593keVdominatetherateatpeaknovatemperaturesabove0.2GK{despitenotevenobservingthesestates.Maetal.publishedin2007resultsofameasurementofthe32S(p;d)31Stransferreaction[54],whichobtainedangularandenergydistributionsofdeuteronsandusedcomparisonstotheorytodeducespectroscopicinformationfor31Slevelswithin1MeVoftheprotonemission59threshold.Becausetheinitialnucleus32ShasaspinandparityJˇ=0+,theauthorswereabletousel=0transferstounambiguouslyconrmtheexistenceofthe31SJˇ=1=2+statethenevaluatedat6263keV(6257keVinRef.[57],whichtheyconcludeddominatedthe30P(p;)31Srateatlowertemperatures,T<0:2GK.Aratewascalculatedusingtheexperimentalparametersobtainedinthisstudyandearlierstudies,buttheauthorsfoundthattheexperimentalrateonlyagreedwiththeHauser-Feshbachrateabove0.3GK;belowthattemperature,theratediereduptoseveralordersofmagnitude.Twocharge-exchangeinteractionstudieshavebeenperformed.TherstwasastudybyWredeetal.publishedin2007[59]whichmeasuredthemomentaofoutgoingtritonsfrom31P(3He,t)31S.Multipleresonanceswereeitherconrmedfromtentativeassignmentsinearlierworksordiscoveredforthersttime.Includedinthelattergroupwasastateat6400keV,forwhichevidencewasfound.Parikhetal.alsoperformedameasurementof31P(3He,t)31S,againobservingevidenceforthestateat6402keVandassigningitaspinandparityof7=2+.Thelatteralsousedtheangulardistributionsoftheoutgoingtritonstohelpconstrainspinsandparitiesofanumberofstates.Wredeetal.alsomeasured32S(d;t)31S,ndingmoreevidenceforthepeakat6400keV.Multiplein-beamgamma-rayexperimentsbuiltupontheworkbyJenkinsetal.In2007Della-Vedovaetal.publishedresultsofastudyof24Mg(16O,n)31Swheretheauthorsobservedseveraltransitionsfromhigh-spinstates[60,61],conrmingtheobservationsofJenkinsetal..Dohertyetal.usedthe28S(;n)31Stopopulateresonances,includinglow-spinstates,intheregionofinterestandused-and--coincidencestodeterminetheenergiesofresonancestatesintheregionofinterestwithveryhighprecision[62,63].Usingcomparisonstothemirrornucleus31P,theauthorsdeterminedthespinsandparitiesoftheresonances,includinganewly-discoveredresonanceat6393keV,whichtheylabeledasa60Jˇ=5=2+state.Theauthorsclaimedthattheircomparisonsto31Phadpositivelyidentiedallrelevantstatesintheenergyregion,butimpliedthatthe6402-keVstateobservedinRefs.[59,64]wasidenticaltothestateat6393keVandnotdistinct.Mostrecently,Irvineetal.publishedin2013astudyofthe32S(d;t)31Sreaction,measur-ingoutgoingtritonswithexcitationenergies6:3.Ex.7:1MeV.Themajorresultofthestudywasadditionalevidenceforthelevelat6402keVseenbyParikhetal.,contributingtothedebateonwhetherthelevelwasdistinctfromthe6393-keVlevelmentionedinRef.[62].Thestudyalsoproducednewmirrorassignmentsforthe31S-31Pmirrorpairandusedtheassignmentstoconcludethatthe6402-keVstate'sspinandparitywere7=2+.4.2.1CurrentStateof31SExperimentalUnderstandingAlthoughthesenumerousstudieshavehelpedbuildlarge,complementaryunderstandingsoftheenergyregionabovethe31Sprotonemissionthreshold,andinsomecaseshavealsopro-videdtightconstraintsonthepropertiesofresonancesintheregion,therearestillambiguitiespreventingacomplete,preciseexperimentalunderstandingofthesestates.Asmentionedexplicitlyabove,thereiscurrentlynoconsensusonthenumberofstatesintheenergyregionaroundEx=6400keV.TheresultsfromDohertyetal.[62,63]implythatthereareonlytwostatesintheregion,at6393and6394keV.TheresultsfromRefs.[59,64,55]takentogether,however,implytheexistenceofadistinctstateat6402keV,thespinandparityofwhichisnotidenticaltoeitherofthestatesmentionedinRefs.[62,63].Bothinterpreta-tionshavecomplications,andanaccuratereactionratedeterminationusingindirectstudyrequiresunambiguousenergies,spins,andparitiesoftheresonancestatesinvolvedinthereaction.Infact,severaladditionalstatesintheGamowwindowalsohaveambiguousspinsand61paritiesatpresent.Dependingonthemethodsusedtostudythestates,theconstraintsontheirparameterscanincasesbeatodds.Thus,experimentsthatstronglyconstrainspinsandparities,suchasstudyviabeta-decay,canbeveryusefultowardhelpingtosolvetheconundrum.
4.3NSCLExperiment12028
Experiment12028(E12028)wascarriedoutattheNationalSuperconductingCyclotronLaboratory(NSCL)usingtheCoupledCyclotronFacility(CCF).Itranfrom24Februaryto4Mar,2014andincludedcollaboratorsfromMichiganStateUniversity,NotreDameUniversity,OakRidgeNationalLaboratory,theUniversityofTennessee,andtheUniversityofSouthernIndiana.Broadly,theexperimentproducedtwofastbeamsfortwopurposes:(1)abeamof32Clionsforcalibrationpurposesand(2)abeamof31Clionsfortheexperimentalmeasurementitself.Bothofthesebeamsweredeliveredtoanexperimentalsetupwheretheywereimplantedintoadetectorusedtomeasurethebetadecayto31;32S.Surroundingthecentraldetectorwasanarrayofgamma-raydetectorsusedtomeasurethegammade-excitationofthe31;32Sexcitedstatespopulatedinthedecay.The32Clbeamwasproducedrst,inordertocalibratethesedetectors.ThepurposeofE12028wastomeasurethesegamma-raysanddetermineboththerelativegammabranchingsfortheexcitedstatesandthebetafeedingsofthestatesthemselves,usingthisinformationtoconstructadecayschemeforthebetadecay.Thisbetadecayscheme,includingpreciseinformationonthebetafeedingsandgammabranchingsofthe1=2+;3=2+;and5=2+statesintheregion,couldthenbecomparedwithatheoreticaldecayschemeproducedviashellmodelcalculations.Bycomparingtheexperimentally-determined62parameters(suchasexcitationenergy)ofthestatestothoseofthetheoreticalstates,itwouldbepossibletoassociateobservedstateswithshellmodelstates,whichincludespinandparityassignments.Then,theoretically-calculatedparametersforthestates,suchasp,couldbecombinedwithexperimentally-derivedparameterstoproduceresonancestrengthsand,subsequently,acalculationofthereactionratewhoseuncertaintywastiedtotheuncertaintyofthetheoreticalparametersusedinthecalculation{aboutafactoroftwo.4.3.1BeamProductionattheCoupledCyclotronFacility
TheprinciplebehindtheCoupledCyclotronFacilityatNSCLissimple:stableionsfromanECRionsource3areinjectedintoatransversemagneticeld,causingcircularmotionaccordingtotheLorentzforce.Whiletheionsarecirclingthecyclotron,theyareaectedbyanoscillatingelectriceldinthedirectionofmotion,causingtheionstoaccelerateastheymove.Whentheionshavebeenacceleratedtotheproperspeedandenergy,theyexitthecyclotronandareimpingeduponafragmentationtarget,producingacocktailbeamwhichincludesthedesiredradioactiveisotopeforstudy.TheCCFusestwocyclotrons,theK500andtheK1200,toacceleratestableionsuptoanenergyof50-180MeV/u,atanintensityofanywherefrom0.1pnA4(foruraniumbeams)to175pnA(foroxygen).InthecaseofE12028,astable\primarybeam"of36Arwasproducedandaccelerateduptoanenergyof150MeV/uintheCCF,atabeamcurrentof75pnA.The36Arwasthenimpingedupona9Betargetofthickness1627mg/cm25andfragmentedtoproducea3ECRstandsfor\electroncyclotronresonance."TheECRtechniqueinvolvesusingelectromagneticradiationinconjunctionwithamagneticeldtomovefreeelectronsinagasaccordingtotheelectroncyclotronresonancefrequency.Thefreeelectronscollidewithgasmoleculesinthesource,ionizingthemforextractionandacceleration.4pnA,\particlenano-Ampere,"isaunitofbeamcurrent.Thetransformationfromchargecurrent(e.g.nA)tobeamcurrentrequiresdividingbythechargestateoftheion.5ˇ0.88cm.Theunitmg/cm2isamoreeasily-understoodmeasureofthetargetmass(likelihoodofinteraction)perunitarea(of,e.g.,thebeam).Toconvertbetweenthetwo,simplydividebythedensityof63cocktailbeamthatincludedboth32Cland32Clasconstituents.Thethicknessofthetargetwaschosentomaximizeproductionofbothoftheseisotopeswithouthavingtoswitchoutthetarget.Inordertoisolatethedesiredisotope,inthiscaseeither32Cl(forthecalibration)or31Cl(fortheexperimentitself),itispossibletoseparatethebeamconstituentsbytheirrigidity,theratioofmomentumtocharge,againusingtheLorentzforce:mv=q=Br.Thecocktailbeamissentthoughoneormoretransversemagneticeldsandthevariousconstituentsexperiencevaryingmagnitudesofforce,causingcircularmotionwithradiusdependentupontherigidityoftheparticularisotope.Specieswithhigherrigidityhaveahigherradiusofcurvatureinaconstantmagneticeld,whilethosewithlowerrigiditybendmoresharply.Thus,dierencesinrigiditybetweenthebeamconstituentsresultinaspatialdistributionofspecies.Collimatingslitsmaythenbeusedtoselectisotopesaccordingtospatialposition.Inordertopurifythebeamfurther,anenergydegrading\wedge"maybeused:ionswithasharedrigidityBrbutdierentatomicnumberslosedierentamountsofmomentuminthewedge,furtherdispersingthemandallowingforadditionalpuricationinmagneticelds.TheNSCLusestheA1900fragmentseparator[65]topurifythefragmentedbeam.TheA1900fragmentseparationisaccomplishedthroughtheuseoffourdipolemagnets,twobeforeandtwoafterthedispersivewedge.Themagneticeldsofthedipolesmaybetunedtoaspecicrigidity,suchthat,afterfragmentation,thedesiredisotopeiscenteredonthebeamline,whilelighterandheavierbeamconstituentsarebenttoo-centerpathsandblocked.TheCCFandtheA1900areshownschematicallyinFigure4.1.ForE12028,thedipolesoftheA1900wererigidity-tunedtocenter32Cland31Clse-quentiallyonthefocalplaneattheendofthefragmentseparator.Themagneticseparationthetargetmaterial(Be:ˆ=1848mg/cm3)64K500cyclotronK1200cyclotronA1900ion sources20 ft10 mRT-ECRST-ECRstripperfoilwedgefocalplaneproductiontargetimage 1image 2image 3Figure4.1:AschematicofthecoupledcyclotronfacilityandtheA1900fragmentseparator.TheionsourcereleasesstableionstotheK500,whichacceleratestheionstoˇ13km/s.BetweentheK500andK1200theionspassthroughagoldfoilstripperwhichremovesmoreelectrons,thentheK1200acceleratestheparticlestoapproximatelyhalfthespeedoflightbeforeimpingingthemupontheproductiontarget.ThefourdipolemagnetsoftheA1900areillustratedinred,withtheseparatorwedgeinbetweendipoles2and3inyellow.ThefocalplaneoftheA1900housesascintillatorwhichisusedtocountthebeamcurrentafterpurication.Afterthefocalplane,thebeamcanbedirectedintooneofseveralexperimentalvaults.65wasassistedbytheinclusionofawedgeof145mg/cm2Al.Themagnetsrequiredretuningbetweentheproductionofthe32Clbeamandthe31Clbeam;thistuningwasontheorderof5%rigiditydierence.AtthefocalplaneoftheA1900,thebeamcurrentandpurityofthe32Clbeamwereroughly635,000particlespersecond(pps)and11%,respectively,withexpectedcontaminants26Mg(stable),27Al(stable),29Si(stable),30P((T1=2=2:5m),31Si(T1=2=157:3m),and33Ar(T1=2=173:0ms).Forthe31Clbeam,thebeamcurrentandpuritywereroughly13,000ppsand3%,respectively,withcontaminants24Na(T1=2=15h),25Mg(stable),26Al6,28Si(stable),29P(T1=2=4:1s),30S(T1=2=1:2s),and32Ar(T1=2=100:5ms).4.3.2BeamPuricationUsingtheRFFS
TheexperimentwasconductedintheS2vaultoftheNSCLexperimentalarea.ThislocationallowedforfurtherpuricationofthebeamusingtheRadio-FrequencyFragmentSeparator(RFFS)[66].TheconceptbehindtheRFFSallowsforseparationofisotopesaccordingtothevaryingspeedsatwhichtheyexitthetarget:produceatransversetime-oscillatingRFelectriceld,whosefrequencyissynchronizedwiththeacceleratorRF.Becausethebeamconstituentshavedieringspeedscomingoutofthetarget,theyreachtheRFFSchamberatphasedierences˚withrespecttotheoscillatingeld.Astheparticletraversestheeld,itexperiencesadeectioncommensuratewiththefractionoftheoscillationperioditspendswhiletheeldispointedinonedirection.AsnotedinRef.[66],thedeectionangle6Ground-state26Alhasabeta-decayhalf-lifeof7.2e5y,makingiteectivelystable.However,thefragmentationprocessmayproduceisotopesinexcitedstates;inthecaseof26Al,therstexcitedstateisanisomerwithbeta-decayhalf-lifeof6.3s,meaningthatsomeofthe26Alproducedwasradioactive.Noneoftheproducedradioactive26Alresultedingamma-raydetection,duetopuricationproceduresdiscussedinSection4.3.2.66dierencebeforeandafterapplicationoftheRFeldisb=1BrVg!(cos(˚)cos(!T+˚))(4.1)whereVistheRFvoltage,!istheRFfrequency,whichitselfiscoupledtotheoperatingfrequencyoftheCCF(ˇ25MHz),˚isthephasedierencebetweentheparticleandthecavity,gisthegapacrosstheelectrodecreatingtheeld,andTisthetimetakenbytheparticletotravelacrosstheelectrode.Fromthisequation,itcanbeseenthatmaximumdeectionoccurswhentheparticleenterstheseparatorchamberinphasewiththeeld(thatis,˚=0),sothattheelddeectionincreasesmonotonicallyduringtheparticle'straversalofthechamber.Conversely,minimumdeectionoccurswhentheeldvariessuchthatitdeectstheparticleequallyinbothtransversedirections,leavingitinthecenterofthebeamlinewhenitexitsthechamber.AswiththeA1900,collimatingslitsmaybeusedtoselectbeamconstituentsofacertaindeectionangle.Toselectthedeectionforthedesiredisotope,thephaseoftheoscillatingeldintheRFFSmaybesettoadesiredvalue.ForE12028,theeldproductionvoltageoftheRFFSwassetto100kV.InordertoguardagainstthepossibilityofRFFSfailure,whichwouldresultintheentiretyofthebeampassingthroughundeected,thephaseoftheRFFSwassetsuchthatthedesiredisotope(32Cland31Cl,sequentially)wasdeectedtothemaximumangle,andtheotherconstituentsweredeectedlesssharply.Thecollimatingslitswerethenusedtoblockthetransitofthecontaminantions,allowingthedesiredisotopethrough.Thenominalphaseanglewasˇ60degrees;thepurityofthebeamwasmonitoredduringperiodicbeamdiagnos-ticchecksthroughouttheexperiment(seeSection4.3.3),andtheanglewasadjustedslightlytomaximizethepurityandtransmissionofthedesiredisotope.Thepurityofthebeamafter67lteringintheRFFSwasassessedattheexperimentalsetup.AnimageandaschematicoftheRFFSareshowninFigure4.2.
4.3.3ExperimentalSetup
AfterpassingthroughtheRFFS,thebeamcontinuedtotheexperimentalsetup,whichincludedascintillatorintowhichthebeamwasimplanted,asuiteofhigh-puritygermaniumdetectorstomeasurethebeta-delayedgamma-raysfromthedecayof31Cl,andtheNSCLdigitaldataacquisitionsystem(DDAS).Aboutonemeterupstreamoftheexperimentalarea,apairofsiliconPINdetectorswereusedtomeasurethepurityandcompositionofthebeam.PIN(positive-intrinsic-negative)detectorsaresemiconductingdetectorswithanundopedintrinsicsemiconductorsandwichedbetweenasectionofp-typesemiconductor(thatis,asemiconductordopedwithanimpurityresultinginfewerelectronsandagreaterconcen-trationofholes)andasectionofn-typesemiconductor(dopedwithanimpurityresultinginahigherconcentrationofelectrons).Ifavoltageisappliedinthen-to-pdirection(thatis,electronsowinthep-to-ndirection,aso-calledreversebias),anelectriceldextendsintotheintrinsicsemiconductorregion,creatingaregionwhereholes7andelectronsarequicklyswepttooppositesidesofthedetectorandmakingthedetectorusefulformeasuringionizationintheregion,suchasthatcausedbyanincomingbeamparticle.Theamountofchargedetected,andthustheamplitudeofthevoltagepulsetheampliersendstothedataacquisitionsystem,isdependentupontheenergydepositedinthedetector.TomeasurethepurityandcompositionofthebeamexitingtheRFFS,thesiliconPINdetectorswereinsertedintothebeamline.Thebeampassedthroughacollimatortomake7Thatis,anelectronhole,apositioninthevalencebandwhereanelectronisexpectedtobebutisabsent.68Figure4.2:AschematicandphotographoftheRFFS.Thebeamentersthechamberatleft,experiencesdeectionaccordingtotheelectriceldthroughoutitsightbetweenthetwoelectrodeplates,andexitstotheright.TheRFcouplerattopisusedtocoupletheoscillationfrequencyoftheRFFStothatofthecyclotron.ThisgureisreproducedfromtheoriginalRFFSpaper,Ref.[66].69sureitonlyimpingedupontheactivedetectorareaofthedetectors,thendepositedenergyinthedetectorsasitpassedthrough,withdierentbeamconstituentsdepositingvaryingamountsofenergyaccordingtotheirmass.BothPINdetectorsmeasuredenergydeposition,butoneofthedetectorswasalsousedtomeasurethetimeofight(TOF)oftheionsbetweenthePINsandboththecyclotronandascintillatoratthefocalplaneattheendoftheA1900.ThedetectorwasconnectedtoaTC241sfasttimingamplierandaCanberra454constantfractiondiscriminator8,thensplitintotwoseparatesignalsandconnectedasthe\start"signalfortwoseparatetime-to-amplitudeconverters(TACs),whichusedtheCCFRFsignalandthesignalfromascintillatorattheA1900focalplane,respectively,as\stop"signals9.WiththeseenergyandTOFmeasurements,atwo-dimensionalspectrumwascreatedwhichseparatedincomingbeamionsbytheirmassandcharge(Figure4.3).Fromthis2Dspectrum,itwasasimplemattertointegratevariousclustersofincomingionstodeterminethepurityofthebeamaswellaswhichcontaminantswerepresent.The32Clbeamwasfoundtobe99%pure32Cl,withonlytraceamountsofothercontaminants.The31Clbeamwasfoundtobeˇ85%pure;however,thestrongestcontaminantwasthestableisotope28Si,sotheradio-purityofthebeamwasfoundtobe95%,with24Na(˘2%)and29P(˘1.5%)thestrongestradioactivecontaminants.Inprinciple,thePINscouldbeinsertedintothebeamindenitely,sothatparticleidentication(PID)andpuritydiagnosticscouldbeperformedcontinuouslythroughouttheexperiment.However,boththe32Cland31Clbeamshada8Aconstantfractiondiscriminatormeasuresthetimingofavoltagepulsefromadetectorbysplittingtheincomingsignal,invertingpartofit,anddelayingtheinvertedsignalsuchthatthesumofthetwosplitsignalsalwayscrossesthezerovoltagemarkatthesamefractionoftheoriginalpulse'srisetime.Itallowsforprecisetimingmeasurementsofpulseswithnonzerorisetimes.9Althoughthismethodoftime-of-ightmeasurementmayappearbackward,itisimportanttousethedownstreamdetectorasthe\start"signal,sincenoteveryparticlecreatingasignalattheA1900upstreamwillreachtheexperimentalsetup.ThismethodpreventsthemeasurementofTOF\starts"withouttheiraccording\stops."Inordertoreceivethesignalsintheproperorder,acabledelayboxwasusedwithlengthsofwiredesignedsimplytodelaythesignalfromtheA1900reachingtheTAC.70Figure4.3:Aparticleidenticationplotfor31Clshowing31Clandstrongestlikelycontam-inants.Asdescribedabove,thebeamconstituentsareseparatedbothbytimeofight,inthiscasebetweentheA1900focalplanescintillatorandthePINdetectorattheexperimentalsetup,andbyenergylossinthePINdetector.Itiscustomarytoplottimeofightonthehorizontalaxisandenergylossontheverticalaxis.Thelargeblobneartherightedgeoftheplotisthe31Cl,whiletheadditionalblobsarebeamcontaminants,whichhavebeenidentiedeitherbytheirpresenceinthegammaspectrumorinferredfromtheexpectedbeamcompositionattheA1900providedbytheNSCLbeamgroup.Thehorizontallyre-peatingstructureislikelyanartifactcausedbyreectionwithintheelectronics,creatingafalsesignalwiththeappropriateenergybutinatedtimeofight.veryhighintensity,evenafterpuricationintheRFFS.TopreventthedestructionofthePINdetectorsviaradiationdamage,theywereattachedtoahydraulicdrivewhichcouldberaisedandloweredintothebeamline.Atvariouspointsthroughouttheexperiment,thebeamintensitywasreducedbyafactoroftenatthecyclotron,atwhichpointthePINswereinsertedintothebeamfordiagnosticspurposes.Inpractice,thisprocedurewasperformedonceper8-hourexperimentershift.Theexperimentalsetup,asmentioned,consistedofacentralscintillatorcomposedofBC408plasticsurroundedbyhigh-puritygermaniumdetectors.Thescintillator(51517125mm)wasmountedinalumninumsquarehousingwithanentrancewindowofaluminizedmylaratamuchlowerdensityandthicknesstopreventinteractionsbeforeimplantation.Itwasattachedona5121mmsidetoaHamamatsuR1924Aphotomultipliertube(PMT)10whichwasconnecteddirectlytotheNSCLDDAS.ThePMTneededtobebiasedinordertoproducetheelectroncascadeandsubsequentvoltagepulse,andwaskeptatanoperatingvoltageof1kVthroughouttheexperiment.Incomingionswereimplantedintothescintillator,depositedenergy,anddecayedafteratime.Boththeimplantsanddecaysproducedenergyinthescintillator,whichwasnotabletodierentiatebetweenthetwoexceptforbroadenergydistributionshapeinspectralanalysis.Thescintillator'smainpurposeintheexperimentwassimplytorecordwheneachimplantordecayhappened,sothatcoincidencesortingcouldbeperformedduringanalysis(seediscussioninChapter5).Notiminginformationfromthescintillatorbeyondthearrivaltimeofthepulseswasrequired,soitwassucienttoconnectthePMToutputtotheDDASinput.Surroundingthecentralscintillatorwasanarrayof9high-puritygermanium\clover"detectors{theYaleClovershareArray.Cloverdetectorsconsistoffoursemiconductinggermaniumcrystals,packedinasquareclover-likeformationwithinthedetector.SimilartothesiliconPINdetectors,thegermaniumcrystalsinthecloverhaveapositive-intrinsic-negativestructurewhichallowsforrapiddetectionofionizingradiationsuchasgamma-rays.Germaniumisusedforthegammadetectorsinsteadofsiliconbecausethedepletedintrinsicsemiconductorregioncanbemanufacturedtobemorethanafewmillimetersthick,allowingforasignicantprobabilityofcompleteabsorptionofincomingphotons.LikethePINdetectors,thecloverdetectorsneededtobebiased:theoperatingvoltageof10APMToperatesaccordingtothephoto-electriceect:incomingphotonsreleaseelectronsonapho-tocathode.Theseelectronsthencascadethroughaseriesofdynodes,multiplyingthetotalcharge.Theelectronsarethendetectedatananode,withtheresultingpulseamplitudebeingproportionaltotheinitialphotoelectronuxgeneratedatthephotocathode.72Figure4.4:Asimpleschematicshowingthecentralscintillatorandthegermaniumcloverdetectors.Theninecloverssurroundthescintillatorasshown.The31Clbeamimplantsinthescintillator,whichdetectsthebetadecays(blue).Thesubsequentgammadecayofthedaughter31Snucleuscreatesgamma-rays(green)whicharedetectedbythecloverarray.thedetectorswasaround5kV.Becausethebandgapofgermaniumislow,thermalgenerationofchargecarriersinthedetectorscancause\leakagecurrent,"whichcompromisestheenergyresolutionofthedetectors(manifestingasabroadeningoftheenergypeakinaspectrumandadecreaseintheprecisionofmeasurementofthepeak'scentroid)andcanevendestroythedetector.Topreventthis,thecloversarekeptinavacuumchamberwhichisattachedtoadewarofliquidnitrogen.Theliquidnitrogencoolsthedetectorstoˇ75K,coldenoughtopreventthermally-creatednoisefrombeingaproblem.Thedewarsforthecloverdetectorswerelledtwiceadaythroughouttheentireexperimenttoreplenishevaporatednitrogen.The9cloverdetectorswerearrangedintwo\rings"offourdetectorseach,withoneringupstreamofthescintillatorandoneringdownstreamofthescintillatorsuchthatthegermaniumcrystalswerecenteredonthescintillatorandasclosetoitaspossible.Theninthcloverwascenteredonthebeamaxisbehindthescintillator,insertedintothe\ring"madebythebackfourgermaniumcrystals.Thecloversandthescintillatorwereallheldin73Figure4.5:AfullCADdrawingoftheexperimentalsetup,includingthepneumaticdriveattachedtothePINdetectors(leftinsert),thecentralscintillatorwhichwasattachedtothecloverframebyametalarm(rightinsert),andthefullcloverframewithallninecloverdetectors(bottom).Theliquidnitrogendewarsforthecloversareshowninlightblue,andthecloverdetectorsthemselvescanbeseenasthelightgrayextensionsintothecenterofthearray.74placebyanaluminumframedesignedandfabricatedattheNSCL.SchematicsofthecloversandscintillatorareshowninFigures4.4and4.5.Notiminginformationfromthecloverswasneededotherthanthearrivaltimeofthepulsesgeneratedbyenergydepositionofthegamma-rays,soitwassucienttoconnecttheenergyoutputforall36crystalstotheDDASwithoutfurthermodication.
4.3.4BriefDiscussionofDetectorMechanisms
ThedetectorsmentionedinSection4.3.3utilizeanumberofdierenttechniquestoproduceasignalforprocessing.Itisusefulforinterpretingthespectraproducedbythedetectorstoengageinabriefdiscussiondetailingtheinteractionmechanismsoccurringwithinthedetectorthatultimatelyresultinthesignaloutputfromthedetectorandsentforprocessing.Forhigh-puritygermaniumdetectorssuchastheclovers,anincominggamma-raymayinteractinmultipleways:itmayundergophotoelectricabsorption,wherethephotonisabsorbedbytheatomandtheelectronisfreed;itmayComptonscatter,transferingsomeofitsenergytoanatomicelectroninthegermanium;or,itmayundergopairproductioninthepresenceofthestrongelectriceldoftheprotonsinthenucleioftheatomsthatmakeupthedetector[67].Intherstcase,theincidentphotondisappearsinaninteractionwithanatominthedetector.Anelectronisfreed,viathephotoelectriceect,fromoneoftheatomicshells;thiselectronhastotalkineticenergyequaltothephoton'senergyminusthebindingenergyoftheelectroninitsshell.Tollinthegapleftbytheelectron,theatomrearrangesitselectronconguration,releasingthebindingenergyofthevacatedshellintheformofanX-rayorAugerelectron.TheX-rayquicklyundergoesitsownphotoelectricabsorptionandisre-absorbedbytheatomwiththeresultoftheemissionofelectronsuchthatthesumoftheenergiesofallelectronsproducedthiswayisequaltotheoriginalphoton75energy.Theelectronsarecollectedattheedgeofthesemiconductorandavoltagepulsecorrespondingtothefullphotopeakenergyisproduced.Thecorrespondingobservableinthespectrumisasharppeakattheoriginalgamma-rayenergy.Thisisofcoursetheidealsituationforgamma-rayspectroscopy.Inthesecondcase,thephotondepositsonlysomeofitsenergy,causingtheelectrontorecoilandleavingatsomeoutgoinganglewithnewenergyaccordingtothestandardComptonformula:E0=E1+E=mec2(1cos)(4.2)wheremeissimplytheelectronmass,511keV/c2.InthiscasetherecoilingelectronthereforehasenergyEe=E0E.ThetwoextremecasesofComptonscatteringareaminimallygrazingcollision,whereˇ0andtheelectronleaveswithverylittleenergy,and=ˇ,ahead-oncollisionwheretheelectronleaveswithmaximalenergyEemax=E=(1+2E=mec2).ThedistributionofrecoilingelectronenergiesinthedetectorproducesaComptonspectrum,demarcatedbytheComptonedge,correspondingtothemaximumelectronenergyEemax.ThespecicshapeoftheComptonspectrumissomewhatdependentonthebindingenergyoftheelectronpriortothescatteringprocess.Inpairproduction,agammaofenergyE>2mec2(mec2=511keV)may,asaresultofinteractionwiththeintenseelectriceldnearprotonsinthenucleiofthedetectormaterial,disappearcompletelyandproduceinplaceofthegamma-rayanelectronpositronpairwithtotalsharedkineticenergyequaltoE2mec2.Boththeelectronandpositronwilltypicallytravellessthanafewmillimetersbeforedepositingtheentiretyoftheirkineticenergy(theoriginalphotopeakenergyminus1022keV)intothedetector.Thepositronwill76thenannihilatewithanelectronalreadyinthedetector,producingtwogammaraysof511keV,whichmayormaynotdeposittheirenergyinthedetector.Ifboth511-keVgammasdeposittheirfullenergy,thefullphotopeakenergyisrecoveredinthedetector.Ifoneorbothofthegammasescapefromthedetectorwithoutdepositinganyoftheirenergy,arstescapepeakorsecondescapepeak,respectively,isproducedinstead,withpeakenergieslowerthanthefullphotopeakenergyby511keVand1022keV,respectively.Theseescapepeaksareaveryconsistentfeatureofgamma-rayspectraandinsomecasescanbeusedtoinfertheexistenceofaphotopeakifthedetector'senergyrangedoesnotextendtohighenoughphotopeakenergies.Forplasticscintillatordetectors,theabovegamma-rayeectsmayoccurastheydoforgermaniumdetectors.However,sincetheatomsmakinguptheplastichavemuchlowerZvaluesthangermanium,thephotoelectriccrosssectionismuchlower.AvastmajorityofgammainteractionsinplasticscintillatorsaresimpleComptonscatteringevents,makingproductionofagamma-rayphotopeakalmostimpossible.OrganicscintillatorssuchastheplasticscintillatorusedinE12028are,ontheotherhand,oftenfavoredforbeta-decayspectroscopyduetotheirhydrogencontent[67].Lightproductioninscintillatorsbeginswhentheincomingparticles(betaparticles,nu-clei,etc)excitevalenceelectronsinthemoleculescomposingthescintillatormaterialfroma\groundstate"singletstate(S0)toanexcitedsingletstate(Sn).TheexcitedsingletstateimmediatelydecayswithoutemittingaphotontotheS1singletstate,whichcanthenquicklyemitaphotonwithenergyequaltothedierencebetweentheelectronsingletstatesS1andS011,producingpromptradiationknownasuorescence.However,theexcited11Inprincipletheexcitationorde-excitationcanoccurtoanyofthevibrationalstatesassociatedwiththestate;however,theenergyspacingbetweenthesevibrationalstatesissmallcomparedtothespacingbetweenelectronstates,sotheydonotfactorheavilyintothelightproduction.77singletstatescanalsodecaytoexcitedtripletstatesTn,whichwilldecay,similarlytotheSnstates,tothelowesttripletstateT1.Thisstatedecays,withalonglifetime,viaphotonemissiontothegroundstateS0.Theprolongedde-excitationoftheT1stateisknownasphosphorescence,orafter-glow.Theamountoflightproducedviauroescenceisthusseentobeafunctionofthebetaorchargedparticleenergy,ashigher-energyparticlescanexcitegreaternumbersofmolecules.However,forhigh-energyparticles,aneectknownasquenchingcanoccur,resultinginareducedlightoutputasthehigh-energyparticlesheavilyionizethescintillatormaterialinacompactregion.Thelightoutputforaprotoninaplasticscintillatoristypicallyˇ1/10thatforelectrons[67].Thismeansthat,forE12028,theeective\gain"intheplasticscintillatorfortheincomingˇ50MeV/u31Clbeamparticleswaslowerthanthatofthe.12MeVelectrons.Whilethechargedparticlesstopquicklyinthescintillator,however,betaparticlescanundergoaprocessknownasbackscattering.SimilartothesituationwithComptonscatteringforgammarays,thisscatteringchangesthedirectionofthetravelingelectronsandcreatesthepossibilitythattheymayescapethedetectorwithoutdepositingtheirentireenergy.Thisconsequentlyleadstoabetadetectioneciencythatislessthanunity.
4.3.5TheNSCLDigitalDataAcquistionSystem
Thissimplesetup{centralscintillatorsurroundedbygermaniumgamma-raydetectors{wasnonethelesssucientformeasuringalloftheobservablesweintendedtostudy.How-ever,performingthemeasurementisnotassimpleastakingvoltagepulsesfromdetectors.Traditionally,detectorshavebeenconnectedtoaseriesofanalognuclearinstrumentationmodules(NIM)designedtoproperlyshapetheincomingvoltagepulse,alignintimeany78signalswhicharemeanttobedetectedincoincidence(thatis,accountforthetimingdif-ferencescreatedbytheelectronicsdierencesbetweendetectorsthatmightreceivesignalssimultaneously),andthendigitizetheanalogvoltagepulsesothatitcanbereadintoadataacquisitioncomputer.TheNSCLDigitalDataAcquisitionSystemisacomputerizedsystemdesignedtore-placemanyofthehardwaremodulesusedinbetweenthedetectorandthedatacollectioncomputer.Oneormore16-channelPixie-16signalprocessingPXIcardsareconnectedtoanoperatingcomputercrate[68].EachchannelfromthePixie-16cardcanacceptsignalsfromavarietyofdetectors.Thesignalisdigitizedinthesystem,andpulseshapingandtimingcorrectionisaccomplishedbyusingthemodulesoftware.Thesystemusesthesoft-wareparameterstoobtainthepulsearea,andsendstheprocessedpulseinformationtothedataacquisitioncomputer.AnumberofparameterscanbesetintheDDAS,includingthetriggerthreshold(thevoltagelevelbelowwhichaneventinthedetectorwillnotregisterintheelectronics)andtheriseanddecaytimeofthepulse.AthoroughdiscussionoftheDDASimplementationattheNSCLisbeyondthescopeofthiswork,butthereareanumberofexcellenttreatmentsofthetopic[69,70,68].Inthecaseofthecloverdetectors,themostcriticalpulse-shapingparametertosetintheDDASisthedecayconstant˝ofthepulseshape.Sincethesignalscomingfromthesedetectorshaveaveryfastrisetimeandanexponentialdecaytime,itisimportanttosetthedecayconstant˝properlyforeachdetectorinordertoaccuratelyintegratetheoutputenergysignal.FortheNSCLDDAS,thisisaccomplishedbyrecordingthewaveformofanincomingpulseusingadigitizedspectrumgeneratorattachedtothePixiecrate.Thewaveformtypicallyhasaveryfastrisetimefollowedbyanexponentialdecay,sothedecayconstant˝maybesetsimplybyttingthewaveformforagivendetectorwithanexponential79oftheformet=˝.InthecaseofE12028,preciseenergydeterminationwasnotstrictlynecessaryforthescintillator,asthescintillator'smainpurposewassimplytothetimingofimplantsanddecays;however,preciseenergydeterminationforthecloverdetectorswascriticaltoobtaininganarrowenergyresolutionoftheincominggamma-rays.Thus,itwasnecessarytoindividuallyset˝foreachofthe36cloverdetectorsinDDAS.The˝valueforeachcrystalwassettoavaluebetween41.5sand57.4s,withmostofthecrystalshavinga˝valueofaround44s.TheelectronicsusedinE12028arerelativelysimple;abriefdescriptionofeachcomponentisgivenhere.TC241sAmp:apre-amplierusedtoshapethesignalfromthePINdetectorfortiming.ThesignalwaspatchedouttotheuserareaattheNSCL,theData-U.QCFD:aconstantfractiondiscriminatorwithfourinputs,usedforprecisetiming.Operatesbysplittingtheincomingsignalintotwoidenticalpulses,invertingandde-layingoneofthetwo,thensummingthetwopulsesbacktogether.Thezero-pointcrossoverofthemixedpulseisthepulsetime.TAC:time-to-amplitudeconverter.Recordsastartandstopsignal,thenproducesavoltagepulsewithheightproportionaltothetimedierencebetweenthetwo.2mm/LEMOConverterConvertsbetweentheDDAScablestandard(LEMO)andthestandardforthepatchpanelelectronics(2mmcable).FastTrigger:AlogicmodulethatreswhenitreceivesasignalindicatingthattheDDAShasreceivedapulsefromadetectorthatsurpassesthethresholdformeasure-ment.80FIFO:Fan-in,fan-out.Takesdatafromanumberofsources,combinesitintoasinglechannel,andcanoutputtoseveralsources.Delay:Alengthofwiremeasuredtoimpedethedeliveryofanelectronicsignalbythedesiredtime.Typicallycomesinadelayboxwithseverallengthsofwire.TheenergysignalsfromallofthedetectorsaswellastheTOFsignalsfromthePINdetectorsetupwerefedintoDDAS,whichprocessedthesignalsandsentthemtothedataacquisitioncomputer,whichwasrunningtheNSCLReadoutsoftware.Furtherdescriptionoftheuser-sideanalysisprocedureisthefocusofChapter5.Inordertoquicklyassesstheoperationofeachofthedetectors,theprocesseddetectorsignalsfromDDASwereexportedthroughanumberoflogicmodulesdesignedtoisolatespeciccomponentsofthesignalsorcombinesignalsfrommultiplesources.TherelevantsignalswerethenpatchedoutoftheS2vaultintheNSCLandintoapatchpanelintheNSCLData-Ucountingroomarea.Anelectronicsdiagram,includingthediagnosticpatches,isshowninFigure4.6.81Figure4.6:AsimpliedelectronicsdiagramofE12028.Datasources(bothdetectorsandNSCLapparati)aremarkedasgreenboxes.IntermediatemodulesandtheNSCLDDASaremarkedaswhiteboxes.Destinationsfordata(boththedataacquisitioncomputerandthepatchpanelstotheData-U)aremarkedasorangeboxes,andconnectionstoexteriorpatchesaremarkedasorangearrows.EnergyinformationwaspassedfromthesourcesandintotheDDASaccordingtotheblackarrows,andtiminginformationfortime-of-ightmeasurementswaspassedfromthesourcesandintotheDDASaccordingtothebluearrows.Briefdescriptionsofthevariouscomponentsaregiveninthetext.82Chapter5
E12028AnalysisProcedureand
Results
Althoughtheelectronicequipmentusedtoacquiredataforanuclearexperimentisacriticalcomponentinansweringtheresearchquestion,theequipmentbyitselfisworthlesswithoutananalysissuitetorecord,sort,andmakesenseofthedata.Forexample,theNSCLDigitalDataAcquisitionSystem(DDAS)receivesasinputthevoltagepulsesfromvariousdetectorsthatsignifytheenergydepositedinthedetectorfromvariousphysicalsources:incomingbeamparticles,implantations,betadecays,gamma-rays,etc.Theseincomingsignals(referredfromhereonasevents)mustbetime-stampedandsentfromtheDDAStoacomputerusedforprocessing,monitoring,andstoringthedatainameaningfulwayforfutureoineanalysis.Inpractice,notwoexperimentswilluseexactlythesamedataprocessingprocedure,eveniftheyhavesimilargoalsandobservables.Thisisduetoseveralfactorsincludingdierencesininstrumentationneedsandanalysisgoals.InthischapterwewilldiscussthespecicanalysisproceduresusedinE12028,includingsoftwareparameters,datareductionprocedures,particleidentication,datacalibration,andproceduresforthecalculationsre-quiredtoproducethe31Clbetadecayscheme.Wewillalsopresenttheresultsofanalysis,includingthedecayschemeitself,thedatausedtobuildit,andthegamma-rayspectraused83foranalysisandresults.
5.1DataProcessing
EacheventreceivedbytheDDASistime-stampedandsenttoadatabuerthatisattachedtotheNSCLReadoutprogram.Readoutreceivesthetime-stampedeventfromDDASandgroupsthemwithinaneventwindowaccordingtoatimingparametersetbytheexperi-menter,thuscreatingsmallgroupsofdatawiththesametop-leveleventwindowtime-stampinadditiontothesecond-leveltime-stampsoftheindividualevents(seeFig.5.1).Inthisway,voltagepulsesthatoccurwithinaseteventwindowarelabeledbythesystemasbeingcoincidentwithoneanother,allowingformorespecicgatingprocedureslateroninanaly-sis.Theoptimaleventwindowfortheexperimentisdependentupontherateofincomingexperimentaldataandthespecicprocessesthatproduceit.Forexample,gamma-gammacoincidencegatesaredesignedtolteroutofaspectrumallgamma-raysthatarenotpartofacascadewiththegatedgamma,buttheyonlydosowithintheeventwindow.Makingtheeventwindowtoolargethusincreasestheriskofaccidentalcoincidencesfromotherbeta-delayedgamma-rays,whichincreasethebackgroundinthecoincidencespectrum.InthecaseofE12028,ourtotalrateofbeamimplantationwashigh,upto9000particlespersecond.However,forevery31Climplantationanddecayto31Srecordedinthescintilla-tor,thereisasubsequentadditionaldecayofthe31Snucleusto31P,whichproducesitsownbeta-delayedgammaraysthatcanbeincoincidencewithgammasfromthebetadecaysofdierentnucleiincluding31Cl.Thisalsodoesnottakeintoaccounttheriskofaccidentalcoincidencesfromradioactivebeamcontaminantsorroombackgroundradiation.Tolimittheprobabilityofaccidentalcoincidencesatthishighrate,wesettheReadouteventwindow84CLOVERSTO ANALYSISSCINTILLATORREADOUTFigure5.1:AsimplieddiagramshowingtheprocessingthattheReadoutprogramdoestoallowcoincidencesortingforincomingdata.Eventsfrom(inthecaseofE12028)thecloverdetectors(blackarrow)andthescintillatordetector(bluearrows)arebueredintotheReadoutprogram,whichsegregatesthemaccordingtotheeventwindow(greyboxes)andpushesthemintoaneventleforanalysis.85tobe1swide.Datafromtheringbuerwasthussegmentedinto1-seventwindowsandstoredinaneventle,aleformatthatstoresdatain\words"denotingthetimeofanin-comingevent,thechannelonwhichtheeventarrived,andthesizeoftheevent(thatis,thepulseheight),allorganizedbyeventwindows.TheReadoutprogramiscapableofcreatingmultipleeventlesasneededaccordingtotheamountofincomingdata,butitiscommonpracticetoorganizedatainto\runs"ofafeweventleseachsothat,ifthebuereddatabecomescorrupted,datalossiskepttoaminimum.ForE12028ourprocedurewastostopandrestartdataacquisitionroughlyeveryhour,typicallyresultingina\run"ofthreetoveeventlesdependingupontheimplantationratethroughoutthehour.5.1.1OnlineAnalysis
Becausebeamtimeisatapremium,itisimportanttoanalyzetheincomingdataassoonasithasbeenprocessedintoaneventle.Thisallowsexperimenterstodeterminewhethertheexperimentislikelytomeetitsresearchgoalsatthecurrentexperimentalsettings,orwhetherchangesmustbemade.Inaddition,itallowsexperimenterstodiagnosewhetherchangesintheexperimentalelectronicsareneeded,e.g.whetherthedelaycablesusedinparticleidentication(PID)time-of-ightmeasurementsareanappropriatelengthtoleavethe\stop"signalfromtheupstreamA1900scintillatorlaterthanthe\start"signalfromthedownstreamPINdetectorsattheexperimentalsetup.Typically,experimentersattheNSCLwilluseaprogramcalledSpecTclfortheironlineanalysis.SpecTclcanbeattachedtoeithertheincomingonlinedataortostoredevtles,sotheprocedureforE12028wastoruntwoinstancesofSpecTclondierentcomputers,oneattachedtoonlinedatathatallowedforquickanalysischanges(newgates,calibrations,etc.)andoneattachedsequentiallytostoreddatatoproduceacomprehensiveaccumulationof86dataforthepurposeofassessingtheexperiment'sstatus.Fig.5.2showsatypicalSpecTcldisplay.8788Figure5.2:AscreenshotoftheSpecTclanalysisprogramspectrumwindowshowinggamma-rayspectraforeachcrystalintherstfourcloverdetectors(thatis,thedetectorsintheringupstreamofthescintillator).Thesespectrahavehadapreliminaryenergycalibration(seeSection5.2.1)appliedtothem,andtheydemonstratetheslightdierencesthateachdetectorcrystalseesasitrecordsinformation(forexample,comparetherelativeheightsofthepeakat2234keV).885.2CalibrationProcedure
Inordertomeaningfullyunderstandandinterprettheincomingdata,itmustbecalibrated{theDDASonlyforwardstoReadoutthetime-stampedvoltagepulsesfromthedetectors,eachofwhichhasbeenampliedaccordingtoitsuniqueelectronicgain.Thismeansthatattheanalog-to-digitalconversion(ADC)stepintheDDAS,gammaraysofthesameenergydetectedindierentdetectorswilllikelyresultinvoltagepulsesofdierentheightsandconsequentlybesortedintodierentdigitized\ADCchannel"histogrambins.Thus,plottingtheuncalibratedgamma-rayspectrumcombiningmultipledetectorsresultsinanonsensicalhistogram.Inordertoevenmakesenseofthedataatall,therstcalibrationappliedtothedatamustbeanenergyscalecalibration.Further,oncethecalibratedspectrahavebeenproduced,anyanalysisprocedurewhosegoalistoderiveintensitiesofgamma-raysmusttakeintoaccountthefactthatnodetectorisperfectlyecientfordetection.Notonlyisitpossiblethatagamma-rayofagivenenergywillnotdeposititsfullenergyintothedetector,orevenfailtointeractatall,thedetectorssimplydonotcoverthefull4ˇsolidanglesurroundingthegamma-raysource.Toproduceaccurateintensitiesofgamma-rays,aneciencycalibrationmustalsobeperformed.HerewewilldiscussbothoftheseproceduresforE12028.
5.2.1EnergyCalibrations
ThegeneraltechniqueusedinanenergyscalecalibrationistoxtheADCscaleatseveralplaceswheretheenergyiswell-known.Thisistypicallydonebyselectingseveralprecisely-measuredgamma-rayphotopeaksintheuncalibratedgammaspectrum,plottingthemasaseriesoforderedpairsofADCchannelandknownenergy,andttingthepointswithan89analyticalfunctionthatcantranslateADCchannelintoenergyatallpointsalongtheaxis.Then,thedatacanbesortedagainusingthisanalyticfunctiontoproduceacalibratedspectrumwhosehorizontalaxisisanaccurateenergyscale.Thisprocedurecanbeperformedinanumberofwayswhenmultipledetectorsarein-volved.Eachindividualclovercrystalcanbeindividuallycalibrated,orthecrystalsasagroupcanbegainmatchedsothattheirADCscalesalignwithoneanother,andtheaggre-gatespectrumthencalibratedasone.Further,thechoiceofpeakstouseforcalibrationisatthediscretionoftheexperimenterandtheneedsoftheanalysis.InthecaseofE12028,severaloptionsexisted:wecouldforexampleusephotopeaksfrom31Sbeta-delayedgammaraysthemselves,orusepeaksfromcontaminantsorroombackgroundradiation.Eachoptionpresentsitsownsetofconcernsforproducinganaccuratecalibrationfunc-tion.Intherstcase,thenecessaryassumptionisthatpreviousstudiesof31Sgamma-rayshaveproducedaccuratevaluesforthephotopeakenergies.Thisassumptionmaynotbewarrantedhowever,giventhatthein-beamgamma-rayspectroscopyexperimentsmentionedinChapter4haveproducedexcitationenergiesforvarious31Sstatesthataresystematicallyhigherthanthe31Clbeta-decayexperimentsdoneto-datebyafewkeV[59,71,56,72].Inbothcases,theissueofenergyscaleextrapolationisalsoaconcern,sinceifthecurveisxedonlyatlowerenergiesandextrapolatedtohigherenergies,theerroronthecalibrationin-creasesmonotonicallywithenergy.Thisexperimentaimedtopreciselymeasuregamma-raysuptoenergiesof7100keV,andthehighest-energyknownphotopeakin31Sthatisnotintheenergyregionwewishedtostudyisthedeexcitationtothegroundstatefromthe6280keVIAS{analmost1MeVextrapolation.Acalibrationbasedonroombackgroundpeakswouldhaveintroducedevenmoreofaproblem,sincethehighest-energycommonly-usedbackgroundpeakisatonly2614keV.90AsmentionedinChapter4,weproduceda32Clbeamforthepurposeofcalibration,andoptedtouseitforbothenergyandeciencycalibration.Adetailedgamma-rayspectroscopystudyof32Clwaspublishedin2012[73]whichproduced32Sgamma-rayenergiesupto7189keV.Wetthe32Clpeaksupto7189keV,obtainedADCchannelmaxima(weusedthepeakmaximuminsteadofthecentroid,asdiscussedmorefullyinSection5.2.2),andttheobtainedpointstoobtainacalibrationcurveforeachclovercrystalforthe32Cldata.Weusedaquadraticfunctionforthecalibrationtoaccountforsmallnonlinearitiesinthegainofthedetectors,whichmayhappenoversuchalargeenergyrangeasours.Typicalquadraticcoecientsforthe32Clenergyscalecalibrationwereontheorderof109keV,indicatingthatthenonlinearityinthedetectorwasverysmall.Thecalibrationcurvesweproducedfortheclovercrystalswereusedtocalibratethe32Cldata,whichwastakenoveronlyafewhours.However,becausethegainsindetectorscandriftovertime,itisnotfeasibletoapplyasinglecalibrationtoadetectoroveranentireexperiment,whichmaylastforhundredsofhours.Thisgaindrifthastheeectofspreadingoutthephotopeak,creatingapeakwithaloweramplitudeandawiderpeakshape.Thiscanresultinlessprecisemeasurementsofthepeak,andcanevenpreventpeaksfromlow-intensitytransitionsfrombeingvisibleatall.Wefoundthatveofthe36clovercrystalsexhibitedimpracticallylargegaindriftovertimeaswellaschangesinresolution,whichproducedoddpeakshapes;datafromthesecrystalswasdiscardedentirely,astheywouldnothavecontributedproductivelytoproducinganaccurateandprecisecalibration.Inordertocounterthepotentialeectsofgaindriftinthe31Cldata,whichwastakenoverthecourseofmanydays,wegroupedrunsintosmallsetsthatwereobtainedclosetooneanotheranddidnotexhibitappreciablegaindrift.Wethenappliedthe32Clcalibrationtothesetofrunsthatwastakenclosestintimetothe32Clcalibrationruns(onlyabout91fourhoursdierence)toproduceacalibrated31Clspectrumbasedontheindependent32Clcalibration.Wetseveral31Sphotopeaksinthiscalibrated31Clspectrumtoproduceourownsetof31Clcalibrationvalues,towhichtheremainderofthe31Cldatawasgain-matched,producinga31Clcalibrationthatdidnotrequireextrapolationorrelianceonexternal31Clvalues.Inordertocheckthiscalibration,weperformedasecond,completelyseparatecalibrationontherstsegmentof31Cldatatobecalibratedwiththe32Clcalibrationcurveusingthecascadecrossovermethod.Thismethodutilizeswell-knownlow-energyroombackgroundpeakstoperformapreciselinearcalibrationinaknownenergyregion(e.g.,thewell-known40Kand208Tllinesat1460.8and2614.5keV,respectively).Withinthatregion,itisthenassumedthatphotopeakenergiesarepreciselydetermined.Intheeventthattwogamma-rayswithinthatregionarepartofacascade,theirtransitionenergies(whicharerelatedtothegamma-rayenergiesbyasimplereference-frametransformation)maybeaddedtogetthetransitionenergyofthegamma-raythatde-excitestheleveltowhichthetwocascadinggammassum.Thisnewgamma-ray,whichostensiblyliesoutsideofthecalibration,maythenbeusedtoextendthecalibration,extendingtheregioninwhichthecascadecrossovermethodmaybeapplied.Thistechniqueistime-consumingandarduous,butitdoesproduceacalibrationthat,exceptforthetwopeaksusedtobeginthecalibration,reliesonlyoninternally-obtainedenergies.Weusedthismethodtoverifytheaccuracyofthe32Clcalibrationmethod,butthetwomethodswerefoundtogiveconsistentresults,sothe32Clmethodwasusedontherestofthe31Cldataexclusively.92Energy [Arbitrary Units]490492494496498500502504506508Function Value [Arbitrary Units]0102030405060708090100gausFigure5.3:AcomparisonbetweenthepeakshapefunctionusedintheanalysisofE12028(redsolidline)andthestandardGaussianpeakshape(bluedashedline).Thetwofunctionshavebeennormalizedtohavethesamemaximumvalue,centroid,andstandarddeviationparameter,˙.Thelargelow-energytailistheresultoftheconvolutionwiththeexponential.5.2.2ThePeakShape
Ideally,gamma-raysofagivenenergywoulddeposittheentiretyoftheirenergyintotheGedetectorandproduceapeakthatfollowstheGaussiandistribution,withacentroidandmaximumattheenergyofthegamma-rayandastandarddeviationbasedontheresolutionofthedetector.However,becauseofanumberofphysicaleectsinthedetector,theactualphotopeakshapeisskewedasshowninFigure5.3.Themostprominentfeatureofthisshapeisthelow-energytail.Thisisduetoincompletechargecollectioninthedetectorofelectron-holepairs[74].Inprinciple,pile-upeectsintheADCcouldalsochangetheshapeofthepeak;however,thecharge-collectiontailisamuchstrongereectandthelattercanusuallybesafelyignored.Forttingphotopeaksinthegamma-raydata,weusedthefollowingfunction,created93throughtheconvolutionofaGaussianwithanexponential,thecomplementaryerrorfunc-tion:N2˝exph˙22˝2+x˝i1erfh˙2+˝xp2˙˝(5.1)whereNisthepeakintegral(notsimplytheamplitude;thepeakshapefunctionrelatestheintegralNtotheamplitudeAbythefactor2˝),isthecentroid,˙istheGaussianstandarddeviation,and˝isthe\decayconstant"oftheexponential.Notethatthis˝isnotrelatedtothelifetime˝ofthenuclearstatewhosedeexcitationmaycreatethephoto-peakbeingt,northewaveformdecayconstant˝usedinsettingtheDDASpulse-shapingparmeters.ThefunctionshowninEquation5.1representsasinglepeakwithnobackground;throughoutanalysiswecombinedmultiplepeak-shapefunctionsasneededtomodeldoubletsorseriesofpeaks,andwefoundasimplelinearbackgroundA+Bxtobesucientforourbackgroundmodeling.ThismodiedGaussianpeakshapeisalsoappropriatetomodelthesystematicwideningofthephotopeaksduetotheenergyabsorptioneectsinthedetectorasthegamma-rayenergyincreases.Atlowenergies,thepeakshapegenerallylooksmoreGaussian,whereasathigherenergiesthedeviationbecomesmorepronounced.Inpractice,both˙and˝aectthewidthofthephotopeak;thus,least-squaresminimizationsofthetwillproduceunreliabletswithcorrelateduncertaintiesiftheseparametersarenotconstrainedsomehow.Toincreasethereliabilityofthepeak-shape,wetanumberofstrong,isolatedpeaksfroma152Eucalibrationsourceatanumberoflowerenergies.Whenwewereassuredthatthetwasgood(thatis,a˜2=valuecloseto1andatresultwheretheminimizationfunctiondidnothavecorrelatederrors)foreachofthesepeaks,weplottedthe˙and˝valuesasfunctionsof94energyandtthemwithparametrizationfunction:˙=pA+BE),˝=A+BE+CE2.Inthisway,wewereabletosystematicallyvaryboth˙and˝acrosstheentireenergyspectrum,allowingthepeakshapetowidenmonotonicallywithincreasingenergyandelim-inatingthepotentialforcorrelatederrorsinvolving˙and˝.Wealsooptedtousefortherecordedenergyvalueofthephotopeakthemaximumofthepeakshape1insteadofthecentroidasdescribedabove.Wechosetousethemaximuminsteadofthefunctioncentroidbecause,formultipletsofthesamepeakwithslightlydierenttboundaries,themaxi-mumwasfoundtovarylessthanthecentroid.Weusedthecentroidparameter'sstatisticaluncertaintyfromthetfortheuncertaintyinthemaximum.
5.2.3EnergyCalibrationSystematics
Forourenergycalibration,weobtainedboththeenergiesandthestatisticaluncertaintiesoftheenergiesfromthepeakshapeusedtotthephotopeaks.However,acalibrationcannotbemoreprecisethanthedatausedtoperformit,soweneededtotakeintoconsiderationthesystematicerrorasaresultofthecalibration.Toassesstheaccuracyofthecalibrationandassignsystematicuncertaintiesatlowenergies,wemeasuredthepeak-shapemaximumofanumberofroombackgroundpeaksknowntoaveryhighprecision,andassigneda0.2-keVblanketsystematicuncertaintyupto2.7MeVbasedonthevarianceofthepeakcentroidsaroundtheirliteraturevalues[75].Above2.7MeV,wemeasuredtheexcitationenergyofvariouslevelsusingthecascade-crossovermethodwithmultiplecascadesofthesamelevelandcomparedthespreadintheexcitationenergyderivedfromthecascades.Wewereable,usingthismethod,toproduceawideninguncertaintyenvelopetohighenergies:0.2keVfor1Tondtheenergycorrespondingtothemaximumofthetfunctionpeak,weusedacomputerfunctionthatiterativelystepsthroughthefunctionoverthetrangeandsearchesforthemaximumvalueofthefunction.Itthenndstheenergyvalueatthatmaximumfunctionvalue.95E<2:7MeV,0.3keVfor2:7MeV<E<4:8MeV,and0.6keVforE>4:8MeV.5.2.4EciencyCalibration
Asmentionedabove,thedetectorarraywasnotcapableofmeasuringeverysinglegamma-rayemittedbythe31Cldecay.Inordertoproperlydeduceaccurateabsoluteintensitiesforthegammasemittedasaresultof31Clbetadecay,itwasnecessarytoperformaneciencycalibration,providingascalefactorfortheintegralofthepeakasafunctionofenergy:N(Epeak)=(Epeak)I(Epeak)whereNisthenumberofgamma-raysdetected(thepeakintegral),Iistheactualintensityofthepeak(eectively,thenumberofgamma-raysemittedatthesource),andtheeciencyissimplytheratiobetweenthetwo{alloftheseareevaluatedateachpeakenergyEpeak.Inpractice,whenexperimentersdiscuss\eciency,"theytypicallyrefertotheabsoluteeciency,whichistheeciencyasdenedhere:simplytheratioofthenumberofgammasmeasuredbythedetectorarrayduringthemeasurementtimeintervaltothenumbergammasemittedfromthesourceduringthattime.However,theabsoluteeciencyofadetectorhasitselftwofactors,thegeometricaleciencyandtheintrinsiceciencyofthedetector:A(E)=GI(E).ThegeometricaleciencyEGisessentiallyameasureofhowthedetectorsarearrayed,howmuchofthesolidanglesurroundingthesourceiscoveredbymaterialthatmaydetectthegamma-ray.TheintrinsiceciencyEI(E)isameasureofhowlikelythedetectormaterialistoabsorbtheincominggamma-ray,whichdependsupontheenergyofthephoton.Whilethesefactorsmayinprinciplebedeterminedindependently,itisusuallyeasiertomeasurethecombinedabsoluteeciencydirectly.Typically,theproceduretodetermineabsoluteeciencyistousean\absolutelycali-brated"source,whoseactivityataspecicpointintimewaspreciselyknown,anddetermine96itscurrentactivityoveraspanoftimebyapplyingtheknownhalflife.Then,itispossibletocountthenumberofgamma-raysinagivenphotopeakdetectedduringtheintervalandcompareittotheexpectedyieldofgamma-raysforthatphotopeakusingtheradiationlaw:Npeak(t)=IpeakRoeln(2)t=T1=2,whereRoistheactivitymeasuredatthepointatwhichthesourcewasabsolutelycalibrated,tisthetimesincethatpoint,T1=2isthehalf-lifeofthesource,Ipeakisthebranchingintensityofthephotopeakbeingmeasured(thatis,theabsoluteintensityofthatgamma-rayperdecayofthesource),andNpeakisthustheex-pectednumberofcountsinthatpeak.TheabsoluteeciencythenissimplydeterminedasNcounted=Nexpected.Thisprocesscanbeperformedforanumberofpeaksandanana-lyticeciencycurvemaybeproducedfromatofthedata,allowingfordeterminationofabsoluteeciencyanywhereonthecurve.Incaseswhereanabsolutelycalibratedsourceisnotavailable,itisnotpossibletoproduceanabsoluteeciencycurve.However,therelativeeciencyofthedetectorarrayasafunctionofenergymaystillbedetermined,sincetheratiosofecienciesatanytwoenergiesisconstant.Inthiscase,theexperimentercansimplymeasurethephotopeakintegralatananchor-pointenergyandusetheabsoluteintensityofthatgamma-raytoproduceaoating\anchor"fortherestofthecurve.Forexample,theeciencycouldbesettounity;thentherelativeeciencyoflower-energypeakswouldbecomparablyhigherandtherelativeeciencyofhigher-energypeakswouldbelower,withthecurveoverallretainingthesameshapeasanabsoluteeciencycurve.Aswiththeenergycalibrationcase,standardcalibrationsourcestypicallydonotproducegamma-raysofsucientenergyforprecisiondeterminationofaneciencycalibrationcurveoveralargerangeofenergies.InthecaseofE12028,wehad32Cldataupto7189keVandthe152Eucalibrationsourcedatabelow1400keV.Weproducedarelativeeciencycurve97forourdatausingthesetwodatasets.The152Euwasusedtoproduceacurveupto1400keV.Thiscurvewasthenextrapolatedbyextendingthefunctionupto1547keV,wheretherst32Clphotopeakwasanchoredtothe152Eucurve.Therelativeintensitiesofthe32Clphotopeaks[73,76]werethenusedtoextendthecurvefrom1547keVto7189keV,producingarelativeeciencycurveovertheentireenergyrangeofourdata.Thespecicshapeofthiscurvewasanexponentialoftheform(E)=exp[iAiln(E)i],where(E)istheeciencyatagivenenergyEandtheexponentialcontainsapolynomialoforderiwithargumentln(E),thenaturallogoftheenergy.TheenergycurveanduncertaintyenvelopeisshowninFig.5.4.
5.2.5Eciency/IntensitySystematics
Althoughthe152Eusourcewasabsolutelycalibrated,ourprocedurefordeterminingtheabsoluteintensitiesofthegamma-raysonlyrequiredrelativeeciencies,sowedidnotprop-agatetheerrorduetothecalibratedactivityofthesource.However,ourderivedintensitiesstillreliedondatafromRef.[73]andourowncalibrationprocedure,soitwasnecessarytopropagateuncertaintiesintheeciencythroughtheintensitycalculation.Todothis,weincludedaateciencyuncertaintyof0.7%acrossallenergiesbasedonvariationsinthephotopeakintegralasaresultofvariationsinthepeakttingprocedure,anuncertaintyof0.2%forE<1547keVfromspreadbetweenthedatapointsusedforcalibrationandthecalibrationcurveitselfinthe152Eudata,aat1.4%uncertaintyforE>1400keVfromtheextrapolationofthe152Eudatato1547keV,andtheenergy-dependentuncertaintyenvelopevaluesaboveE=1547keVtakenfromRef.[73]:0.4%for1:5MeV<E<3:5MeV,1%for3:5MeV<E<5MeV,and5%forE>5MeV.TheenvelopefromthesecombineduncertaintiesisshownalongwiththeeciencycurveinFig.5.4.98Energy [keV]01000200030004000500060007000Eff. [%]024681012 / ndf 2c 4.562 / 9Prob   0.8707
p0        0.01096± 8.399 p1        0.001724±p2        0.0002485±p3       ±p4       ± 0.001498 p5       ± 0.0001753 p6       ± / ndf 2c 4.562 / 9Prob   0.8707
p0        0.01096± 8.399 p1        0.001724±p2        0.0002485±p3       ±p4       ± 0.001498 p5       ± 0.0001753 p6       ±Exp. Source + Norm. 32Cl Efficiencies01000200030004000500060007000Residual [rel %]0510 / ndf 2c 4.563 / 15Prob   0.9952p0        0.1206± / ndf 2c 4.563 / 15Prob   0.9952p0        0.1206±GraphFigure5.4:Upperpanel:theeciencycurvegeneratedfromttingtherelativeecienciesofthe152Euand32Cldata.Thepolynomialrequiredsixterms,listedintheboxattop-rightalongwiththereducedchi-squaredvalueandthep-valueforthet.Lowerpanel:theuncertaintyenvelopeadoptedacrosstheenergyregion,basedontheenvelopeinRef.[73]andtheotherconsiderationsdetailedinthetext.99Inadditiontothesystematicuncertaintyderivedfromoureciencycalibrationproce-dures,wealsoincludedauniform1%uncertaintyintheecienciesatallenergiesduetosumming.Intheeventthatmultiplegamma-raysenteradetectorsimultaneously,theymaybedetectedsimultaneously,resultinginarecordedenergythatishigherthantheactualenergyofeithergammaray.Evenifoneofthetwogammaraysonlydepositssomeofitsenergy,scatteringoutofthedetectorafterward,itwillcausetheapparentphotopeakintensityoftheothergammatodecrease,asaneventoftheparticularphotopeakenergywillnolongerberecorded.Toarriveatthe1%uncertainty,wesimulatedtheinteractionofgamma-raysinthecloverdetectoroverarangeofenergiesandrecordedthetotaleciencyofthegammainthedetector(thatis,weintegratedtheentiresimulatedspectrum,notjustthephotopeak).This1%uncertaintywasfactoredintothephotopeakintensitiesusedtodeterminethebranchingratiosandabsoluteecienciesforeachtransition.5.3DataReductionandFinalResultsSpectra
Weobtainedcumulativespectraforboththecentralscintillatorandthesurrounding36clovercrystals.ThecumulativescintillatorspectrumisshownanddescribedinFigure5.5.Becausethepurposeofthescintillatorwasonlytogateoutroombackground,wedidnotneedtocalibrateitsenergyscale.Thesoftwaregate,asdescribedabove,loopsoveralleventswithintheeventwindowanddetermineswhethertheymeetthegatecriteria.Ifaneventmeetsthegatecriteria(forexample,\theeventoccurswithinacertaindetector"or\theeventoccurswithinacertainamountoftimeafterorbeforeanotherevent"),thesoftwarecontinuestoprocesstheothereventsintheeventwindowaccordingtotherulesofthegate.Typically,gatessuchasthisareusedtoeliminateunwantedeventswhilekeepinggoodevents100Energy [uncalibrated]50001000015000200002500030000Counts / 10 bins02004006008001000120014001600310´ScintEnergyFigure5.5:Ahistogramshowingtheuncalibratedscintillatorenergyspectrum.Becausethecentralscintillatorwasonlyusedtocountthenumberofimplantsanddecaysandallowforgatingofcloverevents,acalibrationwasnotneeded,althoughsincetheQ-valueofthe31Clbetadecayisapproximately12MeV,itislikelythattheuncalibratedspectrum,whichappearstoshowadistributionofeventssimilartothatexpectedfora12-MeVbetaendpointenergy,isclosetoanyactualcalibrationthatwouldbeapplied.forprocessing{thisprocedureisknownasdatareduction.Inthecaseofthescintillatorgate,weproducedaspectrumplottingthetime-stampdierencebetweeneventsinthesamewindowrecordedinthescintillatorandinanyofthecloverdetectors,showninFig.5.6.Thevastmajorityofclovereventsrelatedto31Cldecayoccurveryquicklyafterthedecayitself,whereasclovereventsoriginatingfromothersources,suchasroombackground,occurrandomlythroughouttheeventwindow.AsshowninFig.5.6,thismanifestsasabroadplateauofaccidentalcoincidencetimingvalues,withacentraltruecoincidencetimingvaluepeak.Wegatedonthecentralpeak,awindowofapproximately1sascomparedtothebroad10-splateau.10102004006008001000Counts / 10 ns510610710Figure5.6:Ahistogrampopulatedbycalculatingthetimedierencebetweenaneventinthescintillatorandaneventinanyclovercrystal.Mostofthe31Cleventsoccurinthelargepeaknearthecenteroftheplateauwhiletherestoftheplateaueventsarelikelycausedbyaccidentalcoincidences.ThetroughtotheleftofthecentralpeakandthehigherplateaubackgroundtotheleftofthecentralpeakarelikelycausedbyreectionsinthedataconnectionsbetweenthedetectorandtheDDAS.102Asmentionedabove,veclovercrystalswerefoundtobeinappropriateforanalysisandwerenotincludedinthecalibrationprocedure.Theremainingclovercrystalswerecali-bratedandgain-matchedtoproducethecumulativespectrumshowninFig.5.7.Thisspectrumrepresentsthehighest-statisticsgamma-rayspectrumresultingfrom31Clspec-trumproducedto-datebyovertwoordersofmagnitude.Ithasnotbeenprocessedbeyondtheenergyscalecalibration(thatis,thereisnocoincidencegateapplied).Gatingonthescintillator/clovertimingpeakproducedagammaspectrumwithsignicantlyreducedroombackgroundpeaksandonlyveryslightlyreduced31Speaks.Acomparisonoftheungatedandscintillator/timinggatedspectrum(referredfromthispointforwardasthe\timing-gatedspectrum,"forthetimedierence),includinganinsertofaregionwithgreatly-reducedback-groundpeaks,isshowninFig.5.8.Thedeterminationofgamma-rayenergiesandintensitiesforE12028wascarriedoutonthisgatedspectrum.103104Energy [keV]1000200030004000500060007000Counts / keV210310410510610710Ungated Clover EventsFigure5.7:Ahistogramshowinganenergy-scalecalibratedcloverspectrum,includingall31cloversusedforanalysis.Thisspectrumrepresents,byovertwomagnitudes,thehighest-statistics31Clbeta-delayedgammarayspectrumtodate.105Figure5.8:Maingure:Acomparisonofthescintillator-gated(blue)andtiming-gated(green)cloverspectra.Asillustratedhere,thebackgroundisreducedconsiderably,especiallyatlowerenergies,whilethephotopeakintegralfor31Cleventsisonlyslightlyreducedaccordingtotheeciencyofthescintillatortodetectbetaparticles.Inset:Azoomed-inregionbetween1100keVandˇ1600keV,demonstratingtheeectsofthescintillator-timinggatecomparedtothescintillator-onlygate.Asillustratedhere,severalphotopeakscausedbyroombackground,includingtheveryprominent1460-keVpeak,arealmostcompletelyeliminated.5.3.1GammaDecaySelectionRules
WhilethebetadecayselectionrulesdiscussedinSection3.5.2narrowtherangeofstateswhicharelikelytobepopulatedin31Clbetadecay,theydonotinprincipleaecthowthosestatesundergogammadecay.Theelectromagnetictransitiondoesnotchangethemassnumberofnucleoniccompositionofthenucleus,soitdoesnotrequirespecicisospinvaluesbetweenlevels.However,thereareafewgammadecayselectionruleswhichaecttheobservedgammabranchingofexcitedstates;thesewillbediscussedbrieyhere.AswiththeGamow-Tellerbetadecaytransitions,theelectromagnetictransitionoperatorcouplestonuclearspinJandprohibitsJ=0!J=0transitions.Electromagnetic\allowed"transitionssimilarlyhaveJ=0;1.However,unlikeallowedbetadecays,allowedgammadecaysrequireaparitychange,ˇf=ˇi.Todeterminewherethisadditionalrulecomesfrom,wecanwriteoutthetransitionprobabilityforgammadecay[77]:T(˙)fi=20~+1[(2+1)!!]2E~c2+1jh˘fJfmfjM˙j˘iJimiij2(5.2)whereEisthegamma-rayenergyandM˙isthematrixelementassociatedwiththetransition.Theindices˙,,andarethemultipoleradiationeldnumbers:˙delineatestheelectriceld(˙=E)orthemagneticeld(˙=M),andandarethelandmmultipolenumbersoftheelectromagneticeldsphericalharmonics.ItispossibletorewriteMintermsoftheseparateelectricandmagneticcomponents:ME=Q;MM=M(5.3)106whereQ=EAXj=1e(j)rjY(j);M=N~cMAXj=1h2+1g(j)ll(j)+g(j)ss(j)irj[fjY(j)];(5.4)e(j)theelectriccharge,rjistheradiusofthenucleonj,Y(j)isthesphericalharmonicoforder,Nisthenuclearmagnetone~2mp,l(j)ands(j)aretheorbitalandspinangularmomentaofnucleonj,glandgsaretheorbitalandspingyromagneticratios,gl=1forprotonsand0forneutrons,gs=5:586forprotonsand3:826forneutrons.EandMarephasefactors,E=iandM=i1[77].Fromtheaboveequationsitcanbeseenthattheelectricandmagneticoperatorsfortheelectromagneticrequiredierentparityconstraints:theparityofYis(1),andtheparityofris+1,sotheparityofrjYandhencetheelectrictransitionoforder,E,isˇ=(1),whereistheorderofthetransition.Similarly,sincetheparityofris1andtheparityofrjYis(1),theparityofthemagnetictransitionMis(1)1.Theseconstraintsaretheoriginoftheparityselectionruleandalsohelptoinfertheprobabilityoftransitionsbetweendierentstates.Transitionprobabilitydecreasesbyaboutapproximatelyoneorderofmagnitudeperincreasingorderofmultipolarity{thus,themostlikelytransitionsareelectricandmagneticdipoletransitions,E1andM1.LesslikelythantheirrespectivedipoletransitionsbyanorderofmagnitudearethequadrupoletransitionsE2andM2{however,thestrengthsofmagnetictransitionsaretypicallyanorderofmagnitudeweakerthanthecomparableelectrictransitions,meaningthatE2transitionscompetenotwithM2transitionsbutwithM1.The\allowed"electromagnetictransitionsreferencedabovecanthereforebeseentocorrespond107toE1transitions,whileMandE(2)transitionscanbethoughtofasthevariousordersof\forbidden"transitions.Inthisway,experimenterscanestimatetheprobabilityoftransitionstoagivenlowerenergystateusingtherule-of-thumbselectionrules.Inaddition,theQ-valueofthetransition(inthiscase,thedierenceinenergybetweenthetwonuclearstates)aectstheprobabilityofdecay:ahigherenergydierencebetweenthetwostatesincreasestheprobabilitythatthephotonwillbecreatedwiththerequisiteangularmomentum,andtransitionsbetweenstateswithalargeJbutlowQ-valueproceedmoreslowlythantransitionsbetweenstateswiththesamedierenceinangularmomentumbutagreaterdierenceinenergy2Infact,asetofrule-of-thumbequationsknownas\Weisskopfestimates"[78]canbeusedtogiveaballparknumberfortheexpectednumberofgammadecayspersecondforelectricandmagnetictransitionsoforderl:El=CElA2l=3Q2l+1Ml=CMlA(2l2)=3Q2l+1(5.5)withcoecients[78]:l12345CEl:1:010147:31073:41021:11052:41012CMl:3:110132:21071:01013:31061:01013.Theseestimatesincludeanumberofsimplifyingassumptions,includingthatonlyonesingleprotonchangesstatesinthetransitionandthatthenalstateofthatprotonhasangularmomentum1/2.Themethodsusedforestimatingthetransitionstrengthmatrix2Anotableexampleistheisomericstateoftantalum-180,180mTa.Thegroundstateof180TahasspinandparityJˇ=1+,whiletheisomericsecondexcitedstatehasJˇ=9butanexcitationenergyofonly77.2(12)keV.WithonlythegroundstateandtheJˇ=2+rstexcitedstatetowhichtodecay,180mTaisexpectedtohaveahalflifenolessthan1015years.108elementMinsophisticatedcomputationaltoolssuchastheUSDshell-modelcalculationsperformedfortheanalysisinthepresentworkaremuchmorecomplex,buthavethesamebasicgoal:provideprobabilitiesforelectromagnetictransitionsbetweenstatesinagivennucleus,whichcanbenormalizedintheformofgammadecaybranchingratios,relativeprobabilitiesforagivenstatetodecaytoagivenlower-energystateinanucleus.Thesetheoreticalbranchingratiosmaythenbecomparedtothosecalculatedfromobservedgammatransitionsinanexperiment.
5.3.2PhotopeakIdentication{BuildingtheDecayScheme
Asmentioned,E12028producedabeta-delayedgamma-rayspectrumwithstatisticsovertwoordersofmagnitudehigherthanthetwo31Clbeta-decaystudiesto-date[56,72].Thewealthofdataprovedinvaluabletotheconstructionofanew,morecomprehensiveandprecisebeta-decayscheme.However,theprocessofbuildingthedecayschemewasnotwithoutitschallenges.Eventhoughthe31Clbeamwasverypure(95%),thehighintensitymeantthateventhesmallcontaminantactivitycouldproducedetectablepeaksinthegamma-rayspectrum.Evenpeaksthatcouldnotbepositivelyidentiedasaproductofcontaminantprocesseswerenotassumedtocomefrom31Cldecay.Andevenamongthosepeaksthatcouldbeshowntobe31Cl()peaks,itwasnotknownaprioriwhereinthedecayschemetoplacethem.Weutilizedtwomethodsoflteringoutcontaminantpeaksandidentifyingandplacingpeakscorrespondingtotransitionsfrom31Clbetadecay.First,wefoundthat,whenweappliedthetiminggateandcomparedtheintegralsofthegatedpeakstotheungatedpeaks,thetiminggatereducedknown31Clpeaksbyaconstantfactor,yieldingaratiooftiming-gatedintensitytoungatedintensityof0.806(7)%,anumberostensiblyrelatedtothebeta109detectioneciencyofthescintillator.Thisratiowasobservedforanumberof31ClpeaksovertheentireenergyrangeandisshowninFig.5.9.Incomparison,peaksknowntocomefrombeamcontaminantswerefoundtohaveratiosthatweredierentfromthisvalue,typicallylowerbyˇ0.2%,ostensiblyduetoimplantationdepth(simulationsperformedpriortotheexperimentpredictedthatthe31Clwouldimplantthefarthestintothescintillatorandthatthecontaminants,havinglowerenergy,wouldimplantatshallowerdepths).Inthisway,wewereabletousethetiminggatetoinferthatanygivenphotopeakdidordidnotinfactcomefrom31Cldecay,withafewnotableexceptionsthatwillbediscussedinSection5.4.Acomprehensivebeta-delayedgamma-rayspectrumwithpeaklabelsisshowninFig.5.10.Second,becauseofboththehighstatisticsandthelargenumberofdetectors,wewereabletomakegreatuseofgamma-gammacoincidences.Similartothebeta-gammacoincidences,itispossibletogateonarangeofspecicgamma-rayenergies(e.g.,onaspecicpeak)and,whenoneclovercrystaleventwithintheeventwindowfallsinsidethatenergygate,processonlythoseeventsthatoccurindierentcrystalsofthedetector,orevenindierentwholedetectors.Thisgatecanthenbecombinedinthesoftwarewithothergates,forexamplethescintillatortiminggate,toreducebackgroundevenfurther.Thebenetofgamma-gammacoincidencegatesisintheidenticationofcascades:sincecascadinggamma-raysareemittedessentiallysimultaneouslycomparedtotheimplantationandbeta-decayrate,gatingononegamma-rayinacascadewillallowfortheproductionofaspectrumwithgreatlyreducedbackground,butwithastrongenhancementofothergammasinthecascade.Intheeventthatmultiplecascadesincludede-excitationfromaparticularlevel,agamma-gammacoincidencespectrumproducedfromgatingonthatde-excitationcanyieldmuchinformationabouttransitionsfromhigherstates.Thiskindofgateisparticularlyeectivewhenapplied110Energy of Photopeak1000200030004000500060007000Ratio (ScintGated)/Ungated0.720.740.760.780.80.820.840.860.880.9 / ndf 2c 19.27 / 16Prob   0.2548
p0        0.001192± 0.8106 p1       ±GraphFigure5.9:Forphotopeaksofseveralenergiesbetween1MeVand7MeV,theratioofthemeasuredphotopeakintensityinthetimingspectrumtothemeasuredintensityintheungatedspectrum,themeanofthemeasurements(blackdashedline,80.6%)andtheone-sigmaenvelopedenotingthestandarddeviationofthepointsaboutthatmean(reddashedlines,0.7%).Thisgureillustratesthatthescintillatoreciencywasessentiallyconstantoveritsentireenergyrange,regardlessofthebetaendpointenergyofanyparticular31Cltransition.11131S(985)31S*31S(1049)31S**31P(1266)31S(1248)29Si(1273)31S(1283)Á(1368)31S(1413)31S(1433)40K(1461)31S*31S(1561)31S(1673)31S(1738)31S(1759)31S(1827)31Si(1852)31S**31S**31S(2035)31S**31S(2071)Á34Clm(2127)31S(2183)31S(2186)31S(2234)31S**31S**31S*31S**31S*31S(2484)31S*31S**‡Áˆ˘31S*31S(2780)31S(2788)31S*31S(2838)31S(2844)31S*31S(2959)31S(2971)31S(2995)31S**31S(3076)31S(3106)31S*31S**31S(3202)31S(3283)34Clm(3304)31S(3313)31S(3435)31S(3469)31S*31S(3618)31S(3541)31S*31S(3656)31S*31S(3773)31S*31S(3908)31S*31S(4020)31S(4045)31S(4085)31S**31S**31S(4156)31S(4187)31S(4207)31S*31S(4519)‡˘208Tl(2614)31S*31S*31S(4717)31S(4866)31S**31S(4970)31S(5030)31S(5141)31S(5156)31S**31S**31S**31S*31S*31S(5401)31S(5435)31S*31S*31S*31S(5890)31S(5901)31S**31S(6129)31S(6254)31S(6278)31S(6390)31S*‡ˆ˝)31S(7049)214Bi(1120)30P(2342)29Si(2426)31S**31S*24Mg(7070)Figure5.10:Acomprehensivespectrumshowingtheassignmentsforthephotopeaksusedinanalysisaswellasthoseofidentiedbeamcontaminants.Eachphotopeakislabeledbytheemittingnucleusanditsenergy.Peakslabeledwithoneortwoasteriskscorrespondtosingleanddoubleescapepeaks,respectively.Peaksmarkedwithasingledaggeraresumpeaksandthesummationisnoted,andpeaksmarkedwithadoubledaggerhavemultiplecontributionsandarediscussedindetailinSection5.4.112tolowerexcitedstates{thespectrumandaspectrumgatedonthetransitionfromtherstexcitedstateat1248keVisshowninFig.5.11.113114Energy [ keV]010002000300040005000600070008000Counts / keV110210310410510610710810different crystal coincidences, 1248 keVFigure5.11:Thetiming-gatedspectrum(green)andthetiming-gatedspectrumadditionallygatedontransitionsfromtherstexcited31Sstate,atEx=1248keV,tothegroundstate.Theoverallstatisticsofthespectrumarereducedbyseveralordersofmagnitude,butseveralfeaturesarenonethelessvisible,includingenhancedpeaksatenergiescorrespondingtotransitionstothe1248-keVstate,suchasthepeaksat985keV(2234!1248),2035(3283!1248),and5031(6279!1248).Manyotherpeakscanbeseentobeenhancedaswell;theseenhancementswereusedtohelpconrmtheplacementofseveralofthetransitionsshowninFig.5.12andreportedinTable5.1.5.3.3SpinandParityAssignments:
AsmentionedinChapter4,31ClbetadecayselectivelypopulatesstateswithJˇ=1=2+;3=2+;5=2+.Thisreducesthenumberofoptionsforassigningspinandparity,butdoesnotprovideim-mediatelyunambiguousconstraints.Typically,experimenterswillusetheresultsofcom-plementaryexperimentstopopulatenuclearstatesandmakespinandparityconstraints.However,asdiscussedinChapter4,despitetherelativewealthofexperimentaldataontheresonancesinvolvedinthe30P(p;)31S,unambiguousconstraintsfortheresonancesarestillelusive.Inpartthisisduetoconictingspinandparityconstraintsfromdierentexperi-ments,soinordertoapproachtheexperimentalstudyofthesestatesfreefromtheconcernsofpreviousexperiments,weutilizedapartiallytheoreticalmethodofassigningspinandparity.PriortotheanalysisofE12028,weperformedshell-modelcalculationsusingtheUSDBinteraction[38]topredictboththebetafeedingsofthevarious31Sstatespopulatedinallowed31Clbeta-decaytransitionsandthegammabranchingsforanumberofpositive-parity31Sstates,includingthoseaddressedinthebeta-decaycalculation.TheresultsofthesecalculationsarereportedinAppendixA.Duringanalysis,wethenassignedspinandparitybasedoncomparisonbetweenobservedbetafeedingsandgammabranchingsforeachstate(calculatedaccordingtotheprocedureinthefollowingsection)andthosepredictedbyboththeUSDBcalculationsandbynewUSDcalculationsperformedfollowingtheexperiment(seediscussioninSection6.1).Manyofthe31Sstatesbelowtheprotonthresholdthatarepopulatedby31CldecayalreadyhaveunambiguousJˇvalues,sotheywereusedtoconrmthattheUSDcalculationswerecorrect.Then,forthestateswithuncertainspinandparity,includingstatesneverseen115beforein31Clbetadecay,weappliedtheUSDresultstomakethespinandparityassignment.InsomecasesthisallowedforamoredeniteassignmentthanisreportedintheA=31NuclearDataSheets[79](forexample,thestateat3283keV),andinonecase,thestateat4867keV,ourassignmentbasedontheshell-modelcalculationsdisagreedwiththespinandparityreportedinNDS(Jˇ=1=2+inouranalysiscomparedto(1=2;3=2)inNDS).Formoststates,theobservedquantitiesandpredictedquantitieswereinverygoodagreement;wheretheagreementwasnotverygood,wehaveindicatedatentativeassignmentintheconventionalway:(Jˇ)inparentheses.Again:nosinglereactionordecayexperimentcanabsolutelydeterminethespinsandparitiesoftheimportantstates,butrelyingonourshell-modelcalculationsratherthanthepreviousexperimentalworkattheleastallowedusthefreedomtomakecomplementaryassignmentsindependentlyofanyprecedentssetbythoseexperiments{whichmayormaynotbeaccurate.
5.3.4BetaFeedingsandGamma-rayAbsoluteIntensities
Constructingabetadecayschemerequirestwothings:betafeedingsforeachlevelandgammaintensitiesforeachobservedtransition.Thelatterisoftenpresentedbothintermsofrelativegammabranchingsforeachlevel,denotingtherelativeintensityofallgamma-raysthatde-excitealevel,andintermsofabsolutegamma-rayintensityperunitbetadecay,denotingthenumberoftimespersingledecay(or100,forexample),thatagamma-rayofthatenergywillbedetected.Thetypicalprocedurefordeterminingabsoluteintensityofagivengamma-rayperbetadecayissimplytocountthenumberofgamma-raysobserved,usethedetectoreciencytodetermineanabsolutephotopeakintensity,andthendividebythenumberbetadecaysobserved.Onecommonmethodfordeterminingthenumberofimplantsandthusthenumber116of31Cldecaysissimplytocountthenumberof31ClionsthatpassthroughthePINdetectorinthePIDspectrum,givingthenumberofimplanted31Clions.Then,thebeta-detectioneciencyofthecentraldetector(inthiscasethescintillator)canbeusedtoscalethisnumbertoarriveatthenumberofdecaysthatresultindetectedgamma-raysinthebeta-gammaspectrum.Determinationofbetafeedingsisaslightlymoreinvolvedprocessbecauseagivenenergylevelcanbepopulatedbothbybetafeedingandbygammade-excitationfromhigherlevels.Sinceitisnotknownaprioriwhichofthetwoprocessesade-excitinggammarayoriginatesfrom,theabsolutephotopeakintensitycannotbeusedbyitselftodeterminethebetafeedingofthelevel.Instead,theprocedureforagivenlevelistosumtheintensitiesofallgammasde-excitingthelevel,thensubtractthesumofallgammasfeedingthelevel.Theassumptionhereisthateverygammafeedingthelevelwillresultinagammade-excitingthelevelaswell,whichisavalidassumptionforlevelsbelowtheprotonthreshold,wheregammadecayistheonlyenergeticallyallowedchannel.Theresultingquantityisthenumberofgamma-raysde-excitingthelevelnotfromgammafeeding;eachofthesegammasmustthereforecomefromabetadecaywhichfeedsthelevel.So,thebetabranchforthestateistypicallycalculatedbydividingthisvalue,whichrepresentsthenumberofbetadecaysfeedingthelevel,bythetotalnumberofbetadecays.InthecaseofE12028,bothoftheseprocesseswerecomplicatedbythefactthatthePINdetectorshadtobeextractedfromthebeamlineforthemajorityoftheexperiment.Thus,anynormalizationtotheestimatednumberofimplanted31Clionswouldbeapproximateandpotentiallyinaccurate.Inaddition,sinceweonlyusedarelativeeciencycurve,wedidnothaveabsolutephotopeakecienciesfortheobservedgamma-raytransitions,mak-ingdeterminationoftheabsolutenumberofgammaraysinanygivenphotopeakdicult.117Instead,wecircumventedcalculationsusingthenumberofbetadecaysandperformedanor-malizationforthebetafeedingsandabsolutegamma-rayintensitiesperbetadecayentirelyfromthemeasuredgamma-rayphotopeakintensities.Tocalculatethebetafeedingforeachlevel,werstadoptedbranchesforthebetadecaytothegroundstate(whichcannotbedeterminedfrom31Sgammade-excitationsastherearenonetomeasure),beta-alphaandbeta-protonemission(sinceanumberofdecaysareabovetheemissionsthresholdforthesetwoparticles),andestimatedunseenbeta-gammabranches.FollowingRef.[56]weadopteda7(2)%betafeedingforthegroundstate;thisvalueisbasedonthe31Sibetadecaybranchtothe31Pgroundstate,themirrorprocessfor31Clbetafeedingofthe31Sgroundstate.Thisvaluewascorroboratedbya7.9%ground-statebetafeedingintheUSDshell-modelcalculationweperformedtohelpinterpretthedatawegenerated(Section5.3.3).Thebeta-alphaandbeta-protonbranchweadopted,1.4(6)%,wasbasedonimprovementstothevalueusedinRef.[56]byRef.[72]andourshellmodelcalculations.Weestimatedaconservative0.5(5)%branchforunseengammaraysbasedoncomparisonsbetweenthegammabranchesweexpectedourfromshellmodelcalculationsandwhatweactuallyobserved.Wesummedthesebranchestoatotal8.9(22)%forunobservedbetafeedingand,reasoningthattheremainingbetafeedingwassplitbetweenthelevelsfromwhichgammadecaywasobserved,usedtheremaining91.1(22)%ofthebetafeedingasthetotalfeedingforallobservedlevels.Then,wesubtractedtherelativeintensityofthegamma-raytransitionsfeedingthelevelfromtherelativeintensityofthegamma-raytransitionsde-excitingthelevel.Theseadjustedrelativeintensitiesforeachlevelwerethenallsummedtogethertogivethetotalrelativenumberofbetadecays,andthebetabranchingsweredeterminedbytakingtheratioofeachlevel'ssubtractedrelativegammafeedingstothetotalsumofall118subtractedrelativegammafeedings.Theabsolutegamma-rayintensitiesperdecaywerethendeterminedaccordingtotheformula:I;abs;i=I;rel;i100(Ip+I+I;g:s:+I;unsn)(nIn;allIm;all)+jI;rel;j(5.6)whereI;abs;iistheabsoluteintensityoftransitioniperunitbetadecay,I;rel;iistherelativeintensityofthetransition,determinedfromtheintegralandtheeciencycurve,IpandIarethebeta-protonandbeta-alphabranches,respectively,I;g:s:istheground-statebetabranch,I;unsnistheadoptedunseengammabranch,nIn;allisthesumofallad-justedrelativeintensitiesforeachleveln(thetotalrelativenumberofbetadecays),Im;allistheadjustedrelativeintensityofthelevelmwhichtransitionide-excites,andjI;rel;jisthesumofalltransitionsjde-excitinglevelm.Theuncertaintyonthisvaluefactoredintheuncertaintyintheadoptednon-observed-gammabranches(essentiallyapurelysys-tematicuncertainty),thestatisticalandsystematicuncertaintiesoftherelativegamma-rayintensities,andtheuncertaintiespropagatedthroughthederivationoftheadjustedrelativegammaintensitiesforeachlevel.Withthesequantities,andusingtheanalysistoolsmentionedabove,wewereabletoproduceabetadecayschemewith62totalgamma-raytransitions{overtwiceasmanyasthenumberoftransitionsreportedinthe2012A=31NuclearDataSheets[79],whichreported29transitions,andinthemostrecent31Clbeta-decaystudy[72],whichreported27.Inaddition,weobservedbeta-decaytransitionstotenlevelspreviouslyunobservedin31Clbetadecay.Afulldecayschemewithbetaintensitiesandgamma-rayenergiesandrelativebranchingratiosisshowninFig.5.12,andlistsofall31Sgamma-raytransitionsobserved119inthedecay,includingtheirenergiesandabsoluteintensities,arereportedinTables5.1and5.2.Theseresults,includingtheeectonthe30P(p;)31Sreactionrate,willbediscussedin-depthinChapter6.
5.4AnalysisAnomalies
Inanyanalysisprocedure,casesrequiringspecialattentionwilloftencropup.Inbuildingthe31Cldecayschemeanddeterminingtheabsolutegamma-rayintensitiesandbetafeedings,forexample,severalcasesoccurredwhichrequiredmodicationofthegeneralanalysistechniquedescribedabove.HerewediscusstheanalysisproceduresusedtodeterminethequantitiesreportedinFig.5.12andTable5.1.First,wewilldiscusstheobservationofso-called\forbiddentransitions,"gamma-raytransitionscorrespondingtobetadecayspopulatingstatesforbiddenbytheFermiandGamow-Tellerselectionrules(Section3.5.2).Wewillalsodiscusstheamendedanalysisproceduresusedwhengamma-rayphotopeaksfrommultiplesourcesoverlapintheenergyspectrum.ThesignicanceofthesendingsisdiscussedinChapter6.
5.4.1ForbiddenTransitions:TheLevelsat3349keVand4970keVAcursoryanalysisofthe31Clbetadecayscheme(Fig.5.12)showsthatwhilemostofthespinandparityassignmentsof31Sstatesobservedenablethemtobepopulatedbythebeta-decayselectionrulesforallowedtransitions,twostates,thestatesat3349keVand4970keV,wereassignedspinsandparitiesofJˇ=7=2+and3=2,respectively.Only1=2+;3=2+;and5=2+statesareallowedbytheselectionrulesofbetadecayfromthe3=2+31Clgroundstate;120Figure5.12:Thenal,comprehensive31CldecayschemeproducedfromtheanalysisofE12028.Foreachlevel,thelevel'senergyroundedtothenearestkeVisreportedontheleftwingofthelevelanditsspinandparityJˇarereportedontherightwing.ThepreciseexcitationenergiesExofeachlevelarereportedinTable.5.1.EachbetadecaytransitionisdepictedbyaredarrowontherightsideandincludesitsbetafeedingI,alsoreportedinTable5.1.Bluecoloringforalevelindicatesthatthelevelhasneverbeenobservedin31Clbetadecaybefore.121Fig.5.12(cont'd)Gamma-raytransitionsbetween31Slevelsarealsodenotedinthetablebytheverticalarrows.Eachtransitionincludesthegamma-rayenergyEandbranchingratio(B.R.),whicharebothreportedinTable5.1.Aswiththepopulatedlevels,gammatransitionswhichhaveneverbeenobservedin31Clbetadecaybeforearecoloredblue.Theschemealsoreportstheadoptedbranchesforbeta-protonandbeta-alphaandunobservedgamma-rays.
transitionstothesetwostatesareknownasforbiddentransitionsbecausetheyviolatetheselectionrulesdiscussedinSection3.5.2:inthecaseofthe3=2+state,theparityselectionruleisviolated,ˇ+!ˇ(arstforbiddentransition),andinthecaseofthe7=2+state,J=2insteadof0or1(asecondforbiddentransition).AsmentionedinSection3.5.2,however,theforbiddenappelationdoesnotmeanimpossible,simplyproceedingatagreatlyreducedrate.ThehighstatisticsofE12028thereforepotentiallyallowforobservationofthesetransitions.Wewilldiscussheretheanalysisprocedurefor,andinterpretationof,bothoftheselevels.
5.4.1.14970-keVLevel
WeobservedagammarayataphotopeakenergyofE=4970:2(9)keV(reportedinTable5.1).Itwasnotobservedtobeincoincidencewithanyothergammaray,andtheratioofitsgatedintensitytoitsungatedintensitywasfoundtobeconsistentwiththeaverageratioof0.806(7),indicatingalikelyoriginfrom31Clbetadecay.TheA=31NuclearDataSheets[79]reportastateatEx=4971(3)basedonanumberofnuclearreactionexperiments,someofwhichwerediscussedinChapter4[80,81,82,83,84,85]withaspinandparityassignmentJˇ=3=2;noneoftheseexperiments,however,measuredgamma-raytransitions.Wecalculatedabetafeedingforthislevelof0.037(7),whichyields,usingthesameprocedureasforthe6279-keVand6390-keVstates,alog(ft1=2)valueof6:61,whichisconsistentwiththeinterpretationthatthislevelwasfedviatherst-forbiddenbeta122Table5.1:31SlevelexcitationenergiesEx,beta-decayintensitiesIandcorrespondinglog(ft)values,andtransitionsfromeachlevelobservedtobepopulatedinthebetadecayof31Cl(thedesignationJˇndenotesthenthstateofagivenspinandparity).Alsoincludedforeachtransitionarethegamma-rayenergiesE,relativegamma-raybranchingratios(B.R.),andabsolutegamma-rayintensityper100betadecays,I.ExIlog(ft)TransitionEB.R.I1248.43(20)2.5(6)5.83=2+
1!1=2+
11248.40(20)10012.3(5)2234.06(20)47(4)4.35=2+
1!1=2+
12233.97(20)99.7(62)53.2(27)5=2+
1!3=2+
1985.62(23)0.35(2)0.187(9)3076.40(31)2.58(18)5.31=2+
2!1=2+
13076.24(20)93(6)2.82(14)1=2+
2!3=2+
11827.93(25)6.8(5)0.205(14)3283.76(31)4:64(32)5.05=2+
2!1=2+
13283.57(31)16.1(9)1.11(6)5=2+
2!3=2+
12035.24(20)63.6(35)4.38(22)5=2+
2!5=2+
11049.66(21)20.3(9)1.40(5)3349.30(32)<0:01>7.77=2+
1!3=2+
12100.79(25)1000.076(14)3434.90(33)0:64(5)5.83=2+
2!1=2+
13434.70(32)54.7(35)0.420(24)3=2+
2!3=2+
12186.33(33)45.3(30)0.348(21)4085.4(8)0:74(5)5.65=2+
3!1=2+
14085.2(8)2.3(10)0.019(8)5=2+
3!3=2+
12837.60(32)73(5)0.614(34)5=2+
3!5=2+
11852.19(25)25.0(14)0.211(14)4207.7(31)4:15(27)4.83=2+
3!1=2+
14207.43(31)63.8(21)3.12(18)3=2+
3!3=2+
12959.09(31)36.2(21)1.77(9)4519.63(32)1:13(9)5.33=2+
4!1=2+
14519.28(32)1001.20(7)123Table5.1(cont'd)4717.72(32)1:55(9)5.15=2+
4!1=2+
14717.34(32)37.5(24)0.618(37)5=2+
4!3=2+
13469.13(31)6.9(5)0.113(8)5=2+
4!5=2+
12483.60(22)28.7(17)0.472(26)5=2+
4!5=2+
21433.89(22)24.3(14)0.399(22)5=2+
4!7=2+
11368.34(29)1.10.0185=2+
4!3=2+
21283.32(37)2.6(4)0.043(7)4866.2(6)1:64(10)5.01=2+
3!1=2+
14865.8(6)41.2(27)0.71(4)1=2+
3!3=2+
13617.40(31)58.8(39)1.01(6)4970.7(9)0:037(7)6.63=2
1!1=2+
14970.2(9)1000.037(7)5021.9(5)0:273(21)5.75=2+
5!3=2+
13773.2(5)28.6(30)0.078(7)5=2+
5!5=2+
12787.7(8)6.4(15)0.0173(39)5=2+
5!5=2+
21738.52(36)23.3(28)0.063(7)5=2+
5!7=2+
11672.53(29)41.8(38)0.114(9)5156.1(6)0:93(10)5.23=2+
3!1=2+
15155.7(62)90(11)0.84(8)3=2+
3!3=2+
13907.3(4)9.8(23)0.091(8)5435.9(9)0:023(7)6.63=2+
5!1=2+
15435.4(9)86(38)0.020(7)3=2+
5!3=2+
14187.4(15)14(5)0.0034(7)5775.4(4)0:254(25)5.55=2+
6!5=2+
13541.10(27)1000.254(21)5890.3(8)0:269(21)5.43=2+
6!1=2+
15889.7(8)26.0(35)0.070(9)3=2+
6!5=2+
13656.01(37)63(6)0.170(12)3=2+
6!5=2+
22605.9(5)10.6(18)0.029(5)6129.3(10)0:0253(31)6.35=2+
7!1=2+
16128.7(10)4.470.00125=2+
7!7=2+
12779.5(6)1000.0253(18)124Table5.1(cont'd)6255.0(6)0:57(6)4.91=2+
5!1=2+
16254.3(6)80(10)0.46(4)1=2+
5!5=2+
14020.2(5)9.7(13)0.055(6)1=2+
5!5=2+
22970.9(4)10.1(14)0.058(6)6279.0(6)18:7(11)3.43=2+
7!1=2+
16278.4(6)16.9(17)3.15(30)3=2+
7!3=2+
15030.1(6)10.4(10)1.94(18)3=2+
7!5=2+
14044.7(30)60.6(37)11.3(6)3=2+
7!1=2+
23202.2(4)0.432(39)0.081(7)3=2+
7!5=2+
22995.04(31)6.16(37)1.15(6)3=2+
7!3=2+
22843.9(4)0.452(39)0.084(7)3=2+
7!5=2+
32192.63(28)0.59(5)0.110(9)3=2+
7!3=2+
32071.11(22)3.09(19)0.577(32)3=2+
7!3=2+
41759.05(34)0.39(5)0.072(8)3=2+
7!(5=2+
4)1561.01(29)0.56(5)0.104(8)3=2+
7!1=2+
31412.91(30)0.44(4)0.082(7)6390.2(7)3:38(18)4.13=2+
8!1=2+
16389.5(7)5.4(6)0.181(18)3=2+
8!3=2+
15141.3(6)10.8(11)0.368(36)3=2+
8!5=2+
14155.84(31)44.4(27)1.51(9)3=2+
8!1=2+
23313.56(33)11.8(7)0.401(22)3=2+
8!5=2+
23106.28(31)21.6(12)0.734(39)3=2+
8!3=2+
22182.52(25)6.0(5)0.210(16)7050.0(8)0:047(6)5.71=2+
6!1=2+
17049.2(8)1000.047(5)7149.8(9)0:059(8)5.55=2+
8!3=2+
15900.8(8)1000.059(7)125Table5.2:Allgamma-raysobservedinE12028assignedtothebetadecayof31Cl.Foreachgamma-ray,thetransitionenergyEtandgammaenergyEaregiveninkeV.Thetransitionandtheabsoluteintensityper100betadecaysIarealsolistedforeachtransition.EtransETransitionI985.63(23)985.62(23)5=2+
1!3=2+
10.187(9)1049.68(21)1049.66(21)5=2+
2!5=2+
11.40(5)1248.43(20)1248.40(20)3=2+
1!1=2+
112.3(5)1283.35(37)1283.32(37)5=2+
4!3=2+
20.043(7)1368.37(29)1368.34(29)5=2+
4!7=2+
10.0181412.94(30)1412.91(30)3=2+
7!1=2+
30.082(7)1433.93(22)1433.89(22)5=2+
4!5=2+
20.399(22)1561.05(29)1561.01(29)3=2+
7!(5=2+
4)0.104(8)1672.58(29)1672.53(29)5=2+
5!7=2+
10.114(9)1738.57(36)1738.52(36)5=2+
5!5=2+
20.063(7)1759.10(34)1759.05(34)3=2+
7!3=2+
40.072(8)1827.99(25)1827.93(25)1=2+
2!3=2+
10.205(14)1852.19(25)1852.19(25)5=2+
3!5=2+
10.211(14)2035.31(20)2035.24(20)5=2+
2!3=2+
14.38(22)2071.18(22)2071.11(22)3=2+
7!3=2+
30.577(32)2100.87(25)2100.79(25)7=2+
1!3=2+
10.076(14)2182.60(25)2182.52(25)3=2+
8!3=2+
20.210(16)2186.41(33)2186.33(33)3=2+
2!3=2+
10.348(21)2192.71(28)2192.63(28)3=2+
7!5=2+
30.110(9)2234.06(20)2233.97(20)5=2+
1!1=2+
153.2(27)2483.71(22)2483.60(22)5=2+
4!5=2+
10.472(26)2606.0(5)2605.9(5)3=2+
6!5=2+
20.029(5)2779.6(6)2779.5(6)5=2+
7!7=2+
10.0253(18)2787.8(8)2787.7(8)5=2+
5!5=2+
10.0173(39)2837.74(32)2837.60(32)5=2+
3!3=2+
10.614(34)2844.0(4)2843.9(4)3=2+
7!3=2+
20.084(7)2959.24(31)2959.09(31)3=2+
3!3=2+
11.77(9)2971.1(4)2970.9(4)1=2+
5!5=2+
20.058(6)2995.19(31)2995.04(31)3=2+
7!5=2+
21.15(6)3076.40(31)3076.24(20)1=2+
2!1=2+
12.82(14)3106.45(31)3106.28(31)3=2+
8!5=2+
20.734(39)3202.4(4)3202.2(4)3=2+
7!1=2+
20.081(7)126Table5.2(cont'd)3283.76(31)3283.57(31)5=2+
2!1=2+
11.11(6)3313.75(33)3313.56(33)3=2+
8!1=2+
20.401(22)3434.90(33)3434.70(32)3=2+
2!1=2+
10.420(24)3469.34(31)3469.13(31)5=2+
4!3=2+
10.113(8)3541.32(27)3541.10(27)5=2+
6!5=2+
10.254(21)3617.63(31)3617.40(31)1=2+
3!3=2+
11.01(6)3656.24(37)3656.01(37)3=2+
6!5=2+
10.170(12)3773.5(5)3773.2(5)5=2+
5!3=2+
10.078(7)3907.5(4)3907.3(4)3=2+
3!3=2+
10.091(8)4020.4(5)4020.2(5)1=2+
5!5=2+
10.055(6)4044.94(30)4044.7(30)3=2+
7!5=2+
111.3(6)4085.4(8)4085.2(8)5=2+
3!1=2+
10.019(8)4156.14(31)4155.84(31)3=2+
8!5=2+
11.51(9)4187.71(15)4187.4(15)3=2+
5!3=2+
10.0034(7)4207.7(31)4207.43(31)3=2+
3!1=2+
13.12(18)4519.63(32)4519.28(32)3=2+
4!1=2+
11.20(7)4717.72(32)4717.34(32)5=2+
4!1=2+
10.618(37)4866.2(6)4865.8(6)1=2+
3!1=2+
10.71(4)4970.7(9)4970.2(9)3=2
1!1=2+
10.037(7)5030.6(6)5030.1(6)3=2+
7!3=2+
11.94(18)5141.7(6)5141.3(6)3=2+
8!3=2+
10.368(36)5156.1(6)5155.7(62)3=2+
3!1=2+
10.84(8)5435.9(9)5435.4(9)3=2+
5!1=2+
10.020(7)5890.3(8)5889.7(8)3=2+
6!1=2+
10.070(9)6129.3(10)6128.7(10)5=2+
7!1=2+
10.00126255.0(6)6254.3(6)1=2+
5!1=2+
10.46(4)6279.0(6)6278.4(6)3=2+
7!1=2+
13.15(30)6390.2(7)6389.5(7)3=2+
8!1=2+
10.181(18)7050.0(8)7049.2(8)1=2+
6!1=2+
10.047(5)7149.8(9)5900.8(8)5=2+
8!3=2+
10.059(7)127decayfrom31Cl.WehavethereforelabeledthestateasJˇ=3=2inthedecayscheme;thede-excitinggammarayistherstmeasuredgammade-excitationofthestateandthecalculatedexcitationenergyof4970.7(9)keVismoreprecisethantheNDSreportedvalue.5.4.1.23349-keVLevel
Weobservedagamma-rayof3773keV,whichwasobservedtobeincoincidencewiththede-excitationoftherstexcitedstateat1248-keV.Initially,thesimplestinterpretationofthisgamma-rayseemedtobethatitwasthede-excitationofaknownlevelat5021keVtotherstexcitedstate.However,wealsoobservedtwogamma-rays,energies2100and1673keV,whichwereobservedtobeincoincidencewithoneanotherandtogethersumto3773keV.The2100-keVpeakwasalsoobservedtobeincoincidencewiththe1248-keVgammaray.Itwasnotimmediatelyclearwhetherthesepeakscorrespondedtoacascadeaddinguptosomenewlevelat3773keV,orwhethertheycorrespondedtoacascadede-excitationofthe5021-keVstate.Afterfurtheranalysis,includingconsiderationofourshell-modelcalculations,wefoundthatthe5021-keVstateisexpectedtohavefourprimarygammabranches:tothe1248-keVstate,tothe2234-keVstate,tothe3283-keVstate,andtoaJˇ=7=2+stateatatheoreticalenergyof3477keV.TheA=31NuclearDataSheets[79]reporta7=2+levelat3351.30(19)keV,whichisexpectedtodecayprimarilyviaa2102.4(2)-keVgammaray.GiventheagreementwiththeNDSandwiththeshell-modelcalculations(whichareonlyaccuratetowithinafewhundredkeV),weconsidereditlikelythatthe2100-keVgammaraywasthede-excitationofastateatanexcitationenergyEx=3349:30(32)keV,andthe1672-keVgammaraywasfeedingthisstateviathede-excitationofthe5021-keVstate.Wealsoobservedtwootherrelevantgammarayswhoseenergiescorrespondtode-excitationsofknown31Sstates:a1368-keV128gammarayde-excitingthestateat4717keV,anda2779-keVgammarayde-excitingastateweobservedtobeat6129keV.ThesetransitionsareallreportedinTable5.1.The31Clbeta-decaytransitiontothestateat3349keVwouldbearstforbiddentransition.Wecalculatedabetafeedingupperlimitforthisstateof<0:01.AsdescribedinSection5.3.4,theprocedureforcalculatingthebetafeedingsofthe31Sstatesinvolvedsubtractingthesumoftheintensitiesofthegammaraysfeedingthelevel(inthiscase,the1368,1672,and2779-keVgammas)fromthesumoftheintensitiesofthegammaraysde-excitingit(inthiscase,the2100-keVgamma).However,theprocedurefordeterminingthebetafeedingofthisstatedeviatedsignicantlyfromthestandardprocedure,asboththephotopeakofthe2100-keVgammade-excitingthisstateandthephotopeakofthe1368-keVgammafeedingitoverlappedwithphotopeaksfromotherprocesses.Thisatypicalprocedure,includingthecalculationofthebetafeedingupperlimit,isdetailedinthefollowingsection.5.4.2OverlappingPhotopeaks:the1368-keV,2100-keV,and6129-keVGammaRaysIdeally,theanalysisprocedureforhigh-resolutiongamma-rayspectroscopyissimple:foreachphotopeakinthespectrum,identifythepeak'soriginandmeasureitsintensity.Thisdatamaythenbeusedtoproduce,forexample,abetadecayschemeandabsolutegammaintensities.However,occasionallyitisthecasethatvariousfactorspreventthesimpleiden-ticationandanalysisofaphotopeak.AsdiscussedinChapter5,peakscorrespondingtobackgroundprocesses(e.g.,roombackgroundpeaks)areoftenpresentinthespectrum.Althoughroombackgroundpeakscanbegatedoutofthespectrum,photopeakscorrespond-ingtotransitionsfrombeamcontaminantswillalsoshowupinthespectrum,evenifitis129gatedonbetadecays.Statistically,somenumberofpeaksfromdisparateprocesseswillhaveenergiessimilarenoughthattheyoverlapintheenergyspectrumhistogram.Thismeansthatsimplyintegratingsuchaphotopeakwithoutcarefulconsiderationwillgiveaninaccu-rate,unphysicallyhighintensity;specialcaremustthereforebetakentodisentanglethetwocontributingsourcesofthephotopeak.Inouranalysisofthe31Clbeta-gammaspectrum,weoccasionallyencounteredphoto-peakswithcontributionsfrommultiplesources.Twoofthesehavealreadybeenmentioned:the1368-keVgammarayfeedingthe3349-keVlevel,andthe2100-keVgammarayde-excitingit.Wealsoobservedtwophotopeakscorrespondingtotransitionsde-excitingastateat6129keV:a2779-keVtransitiontothestateat3349keVandaphotopeakat6129keVweassignedtoatransitiontothegroundstate.The2779-keVgammawasgreatlyen-hancedinthe2100-keVcoincidencespectrum,lendingcredencetothehypothesisofastateat6129keV.However,6129keVisalsotheenergyofaknowngammaray(E=6129:89(4))correspondingtoatransitionfromthesecondexcitedstateof16O,allowingforthepossibil-ityofcontributionsfromtwosourcesintheintensityofthispeak.Thisandthetwocasessurroundingthe3349-keVstatewillbediscussedhere.
5.4.2.1The2100-keVPeak
The2100-keVgammarayde-excitingthe3349-keVstatewasmeasuredatapeakenergyofE=2100:79(25)keV.Thisenergyisunfortunatelyonly3keVlowerthantheenergyoftherstescapepeakofawell-knownstrongroombackgroundpeakatE=2614:511(10)keVfromthedecayof208Tl.Althoughthetiminggateonthegamma-rayspectrumgreatlyreducesroombackgroundcontributions,particularlyintensepeakssuchasthe2614-keVpeakmaynotbesupressedentirely,andalthoughthecontributionfromtheescapepeak130islikelyverysmallandrelegatedtotheuppertailofthepeak,itcouldstillresultinanerroneouslyhighintensityofthe2100-keVpeak.Toaccountforthepossiblecontributiontothepeakfromtheroombackgroundescapepeak,wetthe2614-keVpeakanditsrstescapepeakinaroombackgroundrun,arunwherenobeamwasimpingentuponthetarget.Fromthesets,weobtainedtheratiooftheescapepeaktothephotopeak.Wethent2614-keVpeakinthetiming-gatedspectrumtoobtainanintensityandusedthisratiotoobtainthelikelyintensityofitsrstescapepeakinthemixed2100-keVpeak.Wesubtractedthiscontributionfromthemeasuredphotopeakintensityofthe2100-keVpeaktoobtainthecontributionofthede-excitationofthestateat3349keVtothe2100-keVpeak.
5.4.2.2The1368-keVPeak
Ourshell-modelcalculationspredictasmallbutnot-insignicantgammaraybranchde-excitingthe4717-keVstateandfeedingthe3349-keVstate.Wedidinfactobserveagamma-rayatenergyE=1368:34(29)keV,whichwasenhancedincoincidencewiththe2100-keVgammade-excitingthe3349-keVstate.Inanalyzingtheintensityofthispeak,however,itbecameclearthatmultiplesourceswerecontributingtothepeak.Thegammabranchfromthe4717-keVstatewasmuchhigherthanexpected,andtheratioofthetiming-gatedpeaktotheungatedpeakwas0.61(5),lowerthanandinconsistentwiththeaverageratioof0.806(7).Azoomed-inviewofthe1368-keVpeakandnearbypeaksisshowninFig.5.13.24MgisastableisotopewitharstexcitedstateenergyofEx=1368:626(5)keV,soifourbeamcontaminantsproduced24Mgthroughdecay,gammasfromthedecayprocesscouldcontributetothephotopeakat1368keV.Unfortunately,boththebeta-plusdecayof24Alandthebeta-minusdecayof24Naproduce24Mg,andbothhaveahighprobabilityof131Energy [keV]132013401360138014001420Counts / keV5000060000700008000090000Clover gated on Scint + DT HitsFigure5.13:Aportionofthe31Clgammaspectraintheregionaround1368keV.Thebluehistogramistheungatedgamma-rayspectrumwhilethegreenspectrumisthetiming-gatedspectrum.Asshown,theroombackgroundlinesareeliminatedwhilethedecay-relatedpeaksat1368keV,1412keV,and1433keVremain.Notehoweverthatthe1368-keVpeakisreducedmoresubstantiallythantheothertwopeaks,whichareknowntooriginatefrom31Clbetadecay.Thisimpliesthatthe1368-keVpeakdoesnotoriginatesolelyfrom31Clbetadecay.132producinga1368-keVgammaray,andtheyieldofthevarious24Mgbeta-delayedgammasisdierentbetweenthesetwoprocesses;consequently,ifbothofthesenuclideswereproduced,itwouldnotevenbestraightforwardtouseother24Mgpeakstoseparatethecomponentsofthe1368-keVpeak.Ground-state24Nahasahalf-lifeof14.997(12)h,meaningthataroombackgroundruntakenshortlyafterremovalofbeamfromtheexperimentalsetupshouldshowphotopeakscorrespondingto24Nabeta-delayedgammaraysinatiming-gatedgamma-rayspectrum.Infact,weobservedboththe1368-keVphotopeakandthe2754-keVphotopeak,correspondingtothetransitionfromthe24Mgrstexcitedstatetothegroundstateandthetransitionfromthesecondexcitedstatetotherst,respectively.Wealsoobserved,inthetiming-gated31Sspectrum,aphotopeakwithenergyE=7070:9(16)(˙=3:4).Theonlygammawithwhichthisvalueisconsistentcorrespondstoatransitionfromthe24MglevelatEx=8439:36(4)keVtothe1368-keVstate.Consequently,itappearedasthoughboth24Na,whichwasobservedinthePIDspectrum,and24Al,whichwasnotclearlyidentiedinthePIDspectrumbutofwhichtraceamountscouldbeincludedinthebeamorproducedthroughinteractionsbetweenthebeamandthescintillator,contributedtothe1368-keVphotopeak.Inordertoaccuratelydeterminethecontributiontothepeakfrombeamcontaminants,itwasthereforenecessarytoseparatethe24Alcontributionfromthe24Nacontributionandseparatebothcontributionsfromthe1368-keVphotopeak.Ordinarily,experimenterstrytoreducethecontributiontothespectrumfrombeamcontaminantsasmuchaspossible;inthiscasehowever,ourpurebeamactuallymadeitmorediculttoinvestigatethenatureofthe24Aland24Nacontribution,sinceveryfewpeakswereavailabletodeterminetheratioofthetiming-gatedcontaminantphotopeakstotheungatedpeaks.Themostpronouncedpeakwefoundwasthepeakat2754keV,butitisproducedindierentamountsbetween24Aldecay133and24Nadecay,requiringfurtherdisentanglement.Thepeakat7070keV,however,hasanenergyhigherthantheQ-valueofthe24Nabeta-minusdecay(5513.6(6)keV),meaningthatonly24Alcontributedtothepeakintegralinourspectrum.Wemeasuredtheintegralofthispeakinbothgatedandungatedspectraandusedtherelativeyieldsperbetadecayofthe2754-keVand7070-keV24Albeta-delayedgammastodeterminetheexpectedcontributiontotheintegralofthephotopeakat2754keVfrom24Al(14(1)%).Theremainingintegralofthephotopeak(87(6)%)wasthentakenastheexpectedcontributionfrom24Na.Todisentanglethevarioussourcesofthe1368-keVgammaray,wethenusedtherelativeyieldsperbetadecayofthe1368-keVgammabranchesforboth24Aland24Nacomparedtothe7070-keVpeakand2754-keVpeak,respectively,todeterminetheexpectedcontributionofthosetwocontaminantstothe1368-keVphotopeak:25(2)%expectedcontributionfrom24Alanda67(5)%expectedcontributionfrom24Na.Subtractingthesevaluesfromthephotopeakintegralyieldedthe\leftover"contributionfrom31Clbetadecay{only6(4)%ofthetotalpeakintegral.Sincethisvalueisnearlycompatiblewithzero,wetreatedthecontributionfrom31Clasanupperlimitinthecalculationsforthebetafeedingandabsolutegammaintensities.
5.4.2.3The6129-keVPeak
Althoughourbeamdidnotcontainany16Oasacontaminant,andany16Oproducedviafragmentationwouldbeinitsgroundstatebythetimeitwasdeliveredtotheexperimentalsetup,excited16Ocouldbeproducedfromnuclearreactionsbetweentheincominghigh-energybeamparticlesandtheatomsintheplasticscintillator.Infact,16Omaybeproducedviathe13C(;n)16Oreactionrateusing13Cnucleipresentintheplasticscintillatorandalphaparticlesproducedviasecondaryreactionsinvolvingthe31Clandtheplasticinthe134target.16OproducedthiswayhasahighlikelihoodofyieldingagammarayofenergyE=6128:63(4)keV,correspondingtoade-excitationofthesecondexcited16Ostate[86,87,88].AsreportedinTable5.1,weobservedaphotopeakatE=6128:7(10)keV,whichweinterpretedasatransitiontothegroundstatefromalevelatEx=6129:3(10)keVandtowhichweassignedatentativespinandparityofJˇ=(5=2)+basedoncomparisonwithourshell-modelcalculations.Thebranchtothe3349-keVstatewasfoundtoagreewiththeshell-modelcalculations,andthe2779-keVgammacorrespondingtothede-excitationofthisstatewasobservedtobestrengthenedincoincidencespectragatedbothonthede-excitationofthe3349-keVstatetotherstexcitedstate(E=2100keV)andonthe1248-keVde-excitationoftherstexcitedstate(bothspectraarepresentedinFig.5.14).The6129-keVstateexcitationenergyderivedfromaddingthetransitionenergiesofthecascadinggammas(Ex=6128:9(5)keV)wasconsistentwiththeenergyderivedfromthe6129-keVphotopeak(Ex=6129:3(10)keV).However,theobservedgammabranchtothegroundstatewashigherthanexpected,especiallyconsideringthatwedidnotobservetwootherbranches,which,intheshell-modelprediction,werepredictedtobemoreintensethantheground-statebranch.Thus,itseemedprudenttoconsiderwhetherreactionsinthescintillatorwerecontributingtothe6129-keVline.UsingPace,afusion-evaporationreactioncalculatorincludedintheprogramLise++(seeAppendixA),wecalculatedtherateofproductionofalphaparticlesinthescintillatorforbeamenergiesupto50MeV/u,usingthestoppingpoweroftheBC408plastictomodelthelikelyenergythebeamattendierentimplantationdepths,uptothedepthatwhichthebeamstopped(approximately2.5mmaccordingtoLise++).Wefoundthat,forabeamofrate6000pps,uptoˇ1700alphaparticlespersecondcouldbeproducedthroughfusion-evaporationatthehighestinteractionprobability.Thesealphascouldtheninteract135Energy [keV]2700272027402760278028002820284028602880Counts / keV110210310410Clover Energies total gated on timingFigure5.14:Threespectrashowingthe2779-keVtransitionfromthe6129-keVstatetothe3349-keVstate.Blueline:thetiming-gatedspectrumshowingthe2779-keVphotopeakwithoutanyothercoincidencegatingapplied.Greenline:thecoincidencespectrumproducedbygatingonthe1248-keVtransitionfromtherstexcitedstatetothegroundstate.Redline:thecoincidencespectrumproducedbygatingonthe2100-keVtransitionfromthe3349-keVstatetothe1248-keVrstexcitedstate.The2779-keVphotopeakisenhancedinbothcoincidencespectra,indicatingthatthethreegammasformacascadeaddingupto6129keV.136withthe13Cthatmakesupˇ1%ofthecarbonintheBC408.Foralphaswithenergybetween5MeVand10MeV,the(;n)crosssectionforproduc-tionofthe6129-keVgammarayhasbeenmeasured[88].UsingtheaveragealphaenergycalculatedbyPACE,andthecrosssectionsinRef.[88],wecalculated,forthealphapro-ductionratesabove,6129-keVgammaproductionratesofupto3.9gammaspersecond.FactoringintheeciencyoftheCloverdetectorsat6129keV(0.63%),weestimatedathe-oreticalcountrateof0.0213C(;n)16O6129-keVgammarayspersecond.Thisvaluewascomparedtothetheoreticalyieldof6129-keVgammasfromourshell-modelcalculations:forabeamofthesamerategiventheshellmodelbetabranchtothe6129-keVstate(0.04%),thegammabranchtothegroundstate(7.94%),andtheeciencyofthedetectorsatthatenergy(0.63%),weestimatedadetectionrateofonly0.00131Clbeta-delayed6129-keVgammarayspersecond,approximately5%therateofdetectionforreaction-producedgammas.Thesetheoreticalcalculationsseemtoimplythatthe6129-keVphotopeakismostlytheresultof13C(;n)16Oreactions.Toassessthevalidityofthisprediction,wecheckedtheratiobetweenthetiming-gatedphotopeakintegralofthe6129-keVpeakandthatoftheungatedphotopeak(thestan-dardprocedurefordeterminingapeak'sorigins,detailedabove).Wefoundtheratiotobe0.78(16),consistentwiththeaveragevalueof0.806(7),butwithamuchhigheruncertainty.Itcouldbehoweverthattheslightlylowerratioisevidence,asinthecaseofthe1368-keVpeak,ofmultiplecontributingsources.Wealsolookedatanumberofspectragatedondierentconditionstotryanddeterminewhetherthe6129-keVpeak'soriginscouldbeex-perimentallydetermined.Werstlookedatagamma-rayspectrumtowhichthetiming-gatewasapplied,butonlyforhigh-energyscintillatorevents(E>15000\channels"),ostensiblygatingoneventsabovethe31Clbetadecayendpointenergy(Fig.5.15).Thisproduced137aspectrumwithgreatlyreduced31Clbeta-delayedgamma-rays,butenhancedroomback-groundpeaksatlowenergies.Thepeakat6129-keVwasalsoslightlyenhancedinthisspectrum,indicatingthatitlikelyhasalargecontributionfromapromptprocessthatisnot31Cldecay.Totryanddeterminethelikelycontributiontothe6129-keVpeakfromthenon-decayprocess(inthiscasethe13C(;n)16Oreaction),weobtained,foranumberofphotopeakscorrespondingtoroombackgroundtransitions,theratioofthephotopeakintegralinagammaspectrumgatedonlyonthehigh-energyscintillatoreventstotheintegralinthestandardscintillator-gatedspectrum(weusedthescintillator-onlygateinsteadofthetiminggatebecausealmostalloftheroombackgroundpeakswerereducedtoomuchtomeasureinthetiming-gatedspectrum).Thesevalueswerefoundtobemoreorlessconsistentwithoneanother.Todetermineareasonableestimatefortheratio,wetookaweightedaverageofthedatapointsandfound,withap-valueof0.48,aratioof0.2447(20).Thatis,theratioofthephotopeakofanon-decaygammaeventgatedonhigh-energy(non-decay)scintillatoreventstotheratioofanon-decaygammaeventgatedonanyenergyscintillatoreventwasapproximately25%.Wethencomparedtheratioofthe6129-keVpeakintegralinthehigh-energyscintillator-gatedspectrumtothepeakintegralinthestandardscintillator-gatedspectrum.Theratioforthatpeakwasfoundtobe0.27(16){slightlyhigher,butstillconsistentwiththeaveragevalueandwithalargeuncertainty.Ifthe6129-keVpeakweretheresultofboth31Clbetadecayandthe13C(;n)16Oreaction,thisvalueshouldactuallybelowerthan0.2447,sincethe31Clcomponentwouldbereducedmuchmorestronglythanthereactioncomponent.Therefore,itseemsunlikelythat31Clbetadecaycontributesstronglytothispeakinourdata.Tosetanupperlimitonthe31Clcontribution,weusedthe5%31Clcontributionfrom138Energy [keV]25902600261026202630Counts / keV310410510Ungated Clover EventshCloverScintEntries    6.179043e+10
Mean     6198
RMS      84.2hCloverDTHighEntries    2.770546e+07
Mean     1769
RMS     68.57Figure5.15:Aportionofseveralgamma-rayspectraillustratingtheeectsofthevariousscintillatorgatesonbotha31Clphotopeak(thepeakat2565keVistherstescapepeakofthe3076-keVgamma)andaroombackgroundpeak(thepeakat2614keVisfrom208Tl).Blue:Ungatedgammaspectrum.Red:Scintillator-gatedgammaspectrum,showingslightreductionofthe2614-keVpeak.Green:Timing-gatedgammaspectrum,showingalmostcompleteeliminationofthe2614-keVpeakandslightenhancementofthe2565-keVpeak.Purple:Scintillator-gatedspectrumgatedONLYonhigh-energyscintillatorevents,showingtheenhancementofthe2614-keVpeakandreductionofthe2565-keVpeak.Turquoise:Timing-gatedspectrumgatedONLYonhigh-energyscintillatorevents.139thecombinationofourshell-modelcalculationsandthePACEcalculationstoestimatethatnomorethan5%ofthephotopeakintegralisdueto31Clbetadecay.TheresultingupperlimitisreportedinTable5.1.140Chapter6
DiscussionofE12028Results
Asshowninthepreceedingchapter,E12028successfullyproducedanew31Clbetadecayschemewithtennewobservedbetatransitionsand40newobservedbeta-delayedgammatransitions.Infact,thenumberofnewgammatransitionsobservedisnearlydoublethenumber(22)ofpreviously-seentransitions,allofwhichwereobservedagaininthisexperi-ment.Thisexpansionofthedecayschemeisusefulingeneralfornuclearstructurestudies,butasstatedinChapter2,themotivationforE12028wasthepursuitofconstraintsonthespinsandparitiesofpotentially-important31Sstatesinthe30P(p;)31SGamowwindow,inordertoprovideexperimentalconstraintsonitsrate.Here,wepresentresultsfromE12028ofastrophysicalimportance,aswellasotherresultsfromtheexperiment.TheprimaryresultofE12028wasthediscoveryandunambiguousidenticationofanewJˇ=3=2+resonancestate(Section6.1),butanalysisalsoallowedfortestsoftheisobaricmultipletmassequa-tion(Section6.2)andcomparisonwithpreviousexperimentalspinandparityassignments(Section6.3).
6.1IsospinMixing:ANewJˇ=3=2+ResonanceNote:Thissectionisadaptedfromapaperpreviouslypublishedbytheauthorandcollaboratorsonthissubject,Ref.[89]141AsmentionedinSection3.5.4,IsospinmixingmayoccurbetweentwostatesofidenticalspinandparitybutdierentisospinT,causingeachobservedstatetobeasuperpositionofisospinstates,j 1i=cosjJT1i+sinjJT2iandj 2i=sinjJT1i+cosjJT2i(Eqs.3.17and3.18).Inthiscase,themixedstateswill,accordingtothestrengthsofthemixing,adoptcharacteristicssimilartooneanother{observingthesecharacteristics,then,canbeobservationalevidenceofisospinmixing.For31Clbetadecay,thetransitiontothe6279-keVisobaricanalogstate(IAS)isgreatlystrengthenedbecauseoftheFermitransitionfromtheT=3=231Clgroundstate.InouranalysiswefoundthebetafeedingofthestatetobeI=18:7(11)%.OnlytheIASshouldhavesuchahighbetafeedingatsuchhighexcitationenergyin31S.Gamow-TellertransitionstostateswithsimilarlyhighexcitationenergiesareshowninFig.5.12tohavebetafeedingsontheorderofhalfapercentorless,withonenotableexception:thestateat6390-keV.ThisstatewasfoundtohaveabetafeedingofI=3:38(18)%.ThebetafeedingofthisstateisabnormallyhighanddidnotmatchthepredictionfromourUSDBshell-modelcalculations.Further,asreportedinTable5.1,thegammabranchingofthisstatewasobservedtobequalitativelysimilartothatfortheIAS,especiallythehighlystrengthenedbranchtothe5=2+secondexcitedstateat2234keV.Intotal,weobservedsixtransitionsde-excitingthisstate.AsimpliedbetadecayschemeisdepictedinFig.6.1,andselectedportionsofthetiming-gatedspectrumshowingtransitionstotherstthree31Sstates,alongwithgamma-gammacoincidencespectrashowingtheenhancementsofthetransitionsduetothegates,areshowninFig.6.2.Similartothebetafeedingforthelevel,thegammabranchingofthisstatedidnotmatchthepredictionfromourUSDBshell-modelcalculations;basedonthesedivergencesfrompredictionandsimilartieswiththeIAS,weinvestigatedthepossibilityofisospinmixingbetweenthetwostates.142Figure6.1:Asimplied31Cldecayschemefocusingonthe31Slevelsat6279(IAS)and6390keV.Theblueverticalarrowsindicatepreviouslyunobservedtransitions.EnergiesandintensitiesforthesetransitionsarelistedinTable5.1.1436250630063506400Counts / keV310410·½½ G.S.®6279 G.S.®    63905000505051005150Counts / keV10210310410510·½ 1248®6279 1248®     6390½Energy [keV]4000405041004150Counts / keV210310410510610··½½**** 2234®6279 2234®     6390Figure6.2:Selectedportionsofthe-coincident-rayspectrum(blue)showingtransitionsfromthe6279-and6390-keV31Sstatestothegroundstateandrsttwoexcitedstates(Jˇ=1=2+;3=2+;5=2+,respectively).Thebottomtwopanelsalsoshow--spectra(green)withadditionalcoincidenceconditionsonthe1248-and2234-keVrays,respectively.Otherphotopeaksobservedfromthedecayof31Claremarkedwithblackcircles.Doubleescapepeaksaremarkedwithdoubleasterisks.144TheexpectedFermistrengthforthetransitiontotheIASiseasytocalculateaccordingtoEq.3.15:B(F)=(T(T+1)Tzf(Tzf+1))=3=2(3=2+1)1=2(1=2+1)=3.Usingtheobservedbetafeedingsofthetwostates,wewerealsoabletocalculateexperimentalFermistrengths.ByrewritingEq.3.13intermsofthehalf-life,andcombiningconstants,wecanproducethe\ft"value,ameasureofthetransitionrate:ft1=2=C[B(F)+(gA=gV)2B(GT)](6.1)wherehereC=ln2Ko=(gV)2=6170(4),avaluedeterminedfromseveralmeasurementsofpure0+!0+Fermitransitions[90],andt1=2isthepartialhalf-life,thetotalhalf-lifeofbetadecaydividedbythebetafeedingtothelevelinquestion.ThisfvalueistheresultofanumberofcorrectionstoananalyticresultderivedforanucleusofZ=0,fZ=0,whichcanbecalculatedusingtheQ-valueofthebetadecayandtheexcitationenergyofthelevel.Thesecorrectionsaccountfordistortionofthewavefunctionfromtheelectronandthediusenessofthenuclearchargedistribution.Theycanallbecalculatedcomputationally;aderivationispresentedinRef.[90]andcoecientsforthecalculationoftheelectrondistortioncorrectioncanbefoundinRef.[91].IfitisassumedthatB(F)ismuchgreaterthanB(GT)(validfortheIASandanystatemixingstronglywithit,sincetheGamow-Tellertransitionisfragmentedacrossallstates),E.6.1canbesolvedfortheFermistrength:B(F)=C=fVt1=2.HerewehaverewrittenfasfVtodenotetheinclusionoftheFermiphase-spacefactorcorrection,sincethetransitionisaFermidecay.ForboththeIASandthestateat6390keV,wecalculatedfVandt1=2andsubsequentlytheFermistrength:theresultswereastrengthofB(F)=2:4(1)fortheIASandB(F)=0:48(3)forthestateat6390keV,whichtogethersumto2.9(1).Theinated145Fermitransitionstrengthtothestateat6390keV,thereducedtransitionstrengthtotheIAS,theirsum,andthesimilartiesbetweentheirgammabranchesarestrongevidencethattheFermitransitionissplitviaisospinmixing,primarilybetweenthesetwostates.Todeterminethestrengthofthemixingweadoptedthetwo-levelmixingformalismdescribedinEqs.3.19,3.20,3.21,3.22,and3.23andcalculatedR=tanfromtheFermistrengths.UsingRalongwiththeobservedenergyspacingbetweenthelevelsE=6390:2(7)6279:0(6)=111:2(9)keV,wecalculatedanempiricalmixingmatrixelementV=41(1)keVandanunperturbedlevelspacingD=74(2)keV.Wealsousedthemix-ingangleitselftodeterminethewavefunctionsforthetwostates:j IASi=0:408jT=1=2i+0:913jT=3=2iandj 6390i=0:913jT=1=2i0:408jT=3=2i.Becausethebetafeedingandgammabranchingofthe6390-keVstatewasnotcorrectlypredictedbyourUSDBshell-modelcalculation,B.A.BrownrepeatedtheUSDtforthecalculationwiththeHamiltonianfromRef.[38],butthistimeusingonlyexcitationenergiesoflevels(thatis,excludingthebindingenergiesofthenuclei).ThisnewUSDt,termedbyB.A.Brownas\USDE,"givesasimilarroot-mean-squaredeviationfromtheexperimentally-observedenergylevelsintheregion(126keV,comparedto122keVRMSdeviationforUSDB),butgivesastrengthenedbetafeedingtothe6390-keVlevelandmorequantitativelysimilargammabranching.B.A.Brownperformedanisospin-mixingcalculationwithbothUSDBandUSDEandfoundthatbothinteractionsyieldedatripletof3=2+levelsinvolvedinisospinmixing.TheexcitationenergiesofthethreestatesandthecalculatedmixingmatrixelementsforthetwoT=1=2statesforboththeUSDBandUSDEcalculationsarereportedinTable6.1.Basedonthevaluesofexcitationenergyandmatrixelementspredictedbytheseshell-modelcalculations,aswellasthetheoreticaluncertaintiesimpliedbythedierencesbetween146Table6.1:CalculatedexcitationenergiesExandmixingmatrixelementsVofthetripletofisospin-mixedstates,includingtheT=3=2IAS,in31SforbothUSDBandUSDEinteractions.ThematrixelementslistedarebetweenthelistedT=1=2stateandtheT=3=2state.AllvaluesareinunitsofkeV.JˇUSDBExUSDBVUSDEExUSDEVE1(T=1=2)3=2+620535609530E2(T=3=2)3=2+65206184E3(T=1=2)3=2+638212637527theUSDBcalculationandtheUSDE,theoryisconsistentwiththepresentexperimentalresult.TheexperimentalresultsshowthattheisospinmixingoftheIASisdominatedbythenearby6390-keVstate.ThebestexperimentalcandidatefortheotherT=1=2;Jˇ=3=2+stateintheUSDBandUSDEtripletsisat5890keV[62,63]andhasanobservedbetafeedingofjust0.269(21)%.TherelativelylargeenergydierencebetweenthislevelandtheIASisconsistentwiththeshellmodelcalculations,andboththelargespacingandtherelativelysmallbetafeedingrendertheisospinmixingnegligiblewhencomparedtothemixingwiththe6390-keVstate{andthereforenegligibleforthepurposesofthepresentwork.6.1.1AstrophysicalRelevanceandImplications
ThepresentworkconstitutestherstclearobservationofisospinmixingbetweenaT=3=2stateandaT=1=2inthesdshell,withthepossibleexceptionofacontroversialcaseforA=23[92,93,94].AsdescribedinSection3.5.4,themixed6390-keVstatemusthavethesamespinandparityastheIAS,Jˇ=3=2+.Noexperimentalstudyhasidentiedthislevelbefore,makingthisresulttherstcompletelyunambiguousidenticationofthisstate.Furthermore,the3=2+spinandparityofthestatemakeitanimportantl=0resonancestateforprotoncapture,locateddirectlyintheheartoftheGamowwindowwhereitislikelytogreatlyimpactthe30P(p;)31Srate.Usingtheformalismdescribed147inSections3.2and3.3,weattemptedtocalculatethethermonuclearreactionrateusingEq.3.10withourexperimentally-determinedspinfor!,theexperimentalresonanceenergy(Er=ExSp;259:3(8)keV)andaprotonpartialwithpcalculatedaccordingtoEq.3.12.Fortheprotonpartialwidth,wecalculatedaspectroscopicfactorC2S(whichincludedtheClebsh-Gordancoecient)of0.0087usingtheUSDBinteraction.WescaledthisvaluebythesquareoftheT=1=2componentofthe6390-keVstate(0:9132=0:83)toaccountforthemixingwiththeIAS.UsingthevaluesforFlandGlgeneratedbythecodedescribedinRef.[95],wecalculatedapenetrationlengthPl=2:369109.Alongwithasingle-particlereducedwidth2s:p=0:553calculatedusingthetablesinRef.[96],wecalculatedavalueoftheprotonpartialwidthp=36eV.Thisvalue,whencombinedwiththespinoftheresonanceandagamma-raypartialwidthof0.97eVastakenfromTableIIIofRef.[97],leadstoa30P(p;)31Sresonancestrengthof!=24eV.Usingthisresonancestrengthandtheresonanceenergy,wecalculatedthethermonuclearreactionrateforthe3=2+stateat6390keV.ThisrateistabulatedinTable6.2forpeaknovatemperatures.Inordertodeterminetheeectofourcalculatedresonantcapturerateontheoveralldeterminationofthe30P(p;)31Srate,wecomparedourratetotheratecalculatedusingtheHauser-Feshbachstatisticalmodel(Section3.3.1).WechosetocomparetotheHauser-Feshbachratebecause,asdiscussedinSections4.2and4.2.1,theuncertaintysurroundingboththenumberofresonancesintheGamowwindowandtheirspinsandparitiesmakescalculatingtheratethroughthoseresonancesaspeciouspursuitatbest.PlottedinFig.6.3istheratioofthereactionratecalculatedthroughthe6390-keVresonancetotheHauser-Feshbachrate.AsdiscussedinSection3.5.2,onlyT=1=231Sstatesareallowedbyisospinselectionrulestobepopulatedviaprotoncaptureon30P.However,becausetheisospinmixingoftheIASandthe6390-keVstateresultsinasmallT=1=2componentforthe148Table6.2:Thermonuclear30P(p;)31SreactionrateNAh˙viinunitsofcm3mol1s1asafunctionoftemperature(reportedinGK,commonlynotatedT9,ashere),forcommonly-attainednovatemperatures.Here\RC"denotestheresonantcapturethroughthe3=2+stateat6390keV.NAistheAvogadronumber.TherateispresentedherewithoutuncertaintylimitsbecausetheonlyexperimentaluncertaintyusedinthecalculationwasthatfortheresonanceenergyEr;thisuncertaintyaectedtheresonancestrengthbylessthan0.2%.T93=2+RCT93=2+RC0.018.002E-1280.237.246E-050.0151.583E-840.241.173E-040.026.199E-630.251.821E-04
0.032.034E-410.262.728E-04
0.041.026E-300.273.957E-04
0.052.511E-240.285.579E-04
0.064.336E-200.297.667E-04
0.074.447E-170.31.030E-03
0.087.847E-150.311.355E-03
0.094.295E-130.321.750E-030.11.038E-110.332.222E-030.111.388E-100.342.778E-03
0.121.190E-090.353.425E-03
0.137.264E-090.364.169E-03
0.143.396E-080.375.015E-03
0.151.283E-070.385.969E-03
0.164.081E-070.397.032E-03
0.171.126E-060.48.211E-03
0.182.764E-060.421.092E-02
0.196.143E-060.441.410E-020.21.256E-050.461.776E-020.212.389E-050.482.188E-02
0.224.274E-050.52.645E-02149Temperature [GK]0.10.150.20.250.30.350.400.10.20.30.40.5+Rate(3/2Figure6.3:Ratiosofthe30P(p;)31Sthermonuclearreactionratescalculatedforboththenew3=2+stateat6390-keV[solidblueline]andthe6280-keVIAS[dashedgreenline]totheoverallHauser-Feshbachrate[98].
T=3=2IAS,thereisasmallcontributiontothereactionratethroughthatlevelaswell,plottedasaratiototheHauser-Feshbachratealongsidethatofthe6390-keVstate.AsshowninFig.6.3,theresonantcapturereactionrateforthissingleresonanceap-proaches50%oftheHauser-Feshbachrate.Thisisasubstantialcontributionforasingleresonance,consideringthattheHauser-Feshbachrateismeanttoestimatethetotalreactionratethroughalllevels.DespitetheexperimentalprogressmadebythestudiesdiscussedinSection4.2,theHauser-FeshbachmayremainthebestestimateofthereactionratebecauseofthenumerousambiguitiessurroundingtheresonancesintheGamowwindow.Thefactthatthissingleresonanceapproacheshalfofthetheoreticalestimateatpeaknovatemper-aturesmeansthatitisthemostimportantresonancewithanunambiguousspinandparityassignmentand,hence,ameaningfulestimateofthereactionrate.150Giventhatthe30P(p;)31SrateforthissingleresonanceproducesanestimatethatisasizableportionoftheHauser-Feshbachrate,itispossiblethattheoverallreactionrateishigherthanpreviousstudieshaveestimated.Inadditiontothepositive-paritystatesobservedinthisstudyandmodeledusingtheUSDBandUSDEinteractions,the30P(p;)31Srateislikelyaectedbythepresenceofanumberofnegative-paritystatesinthe6-7MeVenergyregion.Arecentstudy[97,99]concludedthatthesenegative-paritystates,someofwhichhavenotyetbeenobservedexperimentally,aremostimportantfordeterminationofthe30P(p;)31Srate.Thepresentworkcalculatedaratethroughonlyonesinglepositiveparityresonanceandfoundacontributionofupto50%oftheHauser-Feshbachrate;iftheresonantcaptureratesthroughthenegativeparitystatesmentionedinRef.[97]areaslargeasorlargerthanthatforthe6390-keVstate,theoverall30P(p;)31SratecouldbeseveraltimeshigherthantheHauser-Feshbachrate.Ahigher30P(p;)31Sratecouldhelptoaddressthediscrepancybetweenobserved30Si/28Siratiosfrompresolarnovagrainsandpredictedratiosfromnovamodels.Iftheactual30P(p;)31Srateweremuchhigherthanthecurrentestimates,itwouldbethecasethatmuchless30Siwasproducedfromthebetadecayof30P,sincemuchmore30Pwouldbedestroyedviaprotoncapture.Thus,novanucleosynthesiswouldproduceamoremodestexcessof30Si,potentiallybringingtheoreticalpredictionsofSiisotopicabundancesmoreinlinewiththeobservedabundancesingrains.1516.2IsobaricMultipletMassEquationStudiesandtheSecond31ClT=3=2StateNote:Thissectionisadaptedfromapaperpreviouslypublishedbytheauthorandcollaboratorsonthissubject,Ref.[100]
6.2.1LowestA=31;T=3=2QuartetTheisospinmixingoftheIASmeansthattheobservedexcitationenergyofEx=6279:0(6)isperturbedfromthevalueoftheexcitationenergythatwouldbepredictedbytheisobaricmultiplemassequation(Section3.5.3).FittingtheIMMEwiththeobservedexcitationenergyofthisstateanditsT=3=2analogsin31Cl,31P,and31SicouldthereforeresultinabreakdownoftheIMME,likelyrequiringacubicterm.Ithashistoricallybeendicult,however,totesttheIMMEforthelowestA=31;T=3=2quartetbecauseofimprecisionintheexperimentalmassexcessvalueof31Cl.Untilrecently,thevalueusedinIMMEtestswasfroma1977experimentalmeasurementofthe36Ar(3He,8Li)31ClQ-value:=707050keV[101].AmuchmoreprecisePenningtrapmassmeasurementof31Cl,publishedin2016,obtainedavalueforthegroundstatemassexcessthatwas15timesmoreprecisethanthisvalue[102].ThisobservationdidinfactleadtoanobservedbreakdownoftheIMME,requiringalargecubictermofd=3:5(11)keV.Giventheobservationofstrongisospinmixingin31S,itisinterestingtoconsiderwhetherornotthismixingplaysanappreciableroleinthebreakdownoftheIMMEforthelowestA=31;T=3=2quartet.Ref.[102]usedthevalueofthe31Clground-statemassexcessobtainedinthatstudyalongwiththeliteraturevaluesfortheotherground-statemassexcessesofthequartetmembersandtheexcitationenergies.Toconrmtheresultsofthat152Table6.3:Ground-statemassexcessandexcitationenergyExvaluesusedasinputfortheIMMEtsofthelowestA=31;T=3=2quartet.Exceptfortheobservedexcitationenergyofthe31SIAS,whichisfromRef.[89],allvaluesarethesameasinRef.[102].NucleusTz[keV]Ex[keV]31Cl3=27034:7(34)031S1=219042:52(23)6279:0(6)31P+1=224440:5411(7)6380:8(17)31Si+3=222949:04(4)0Table6.4:OutputcoecientsforthequadraticandcubicIMMEtsforthelowestA=31;T=3=2quartetusinginputdatafromTable6.3.AllcoecientvaluesareinunitsofkeV.Thecubictdidnotcontainanydegreesoffreedom,sothe˜2=valueisundenedandhenceommitted.QuadraticCubica15466:3(9)15464:1(10)b5302:4(10)5295:2(20)c209:2(9)209:9(10)d4:3(11)˜2=16:0=1studyasastartingpoint,wehavereplicatedthattwiththeonlydierencebeingtheuseofourexperimentally-determinedexcitationvaluefortheIAS,Ex=6279:0(6).Weusedthisvalueratherthantheliteraturevalue,whichisbasedonatofgamma-rayenergiesfromaprevious31Clbeta-decaymeasurement[72]thatiscurrentlyunpublishedandhasgreateruncertaintyonthevalue(ˇ2keV;thetwoexcitationenergiesarethusconsistent).Theresultofusingourexcitationenergyvalueis,however,similartoRef.[102]:theIMMEfails,requiringanevenlargercubictermd=4:3(11)keV,andthereducedchi-squaredvalueofthepurelyquadratictincreasesfrom˜2==11:6inRef.[102]to16.0.TheinputsandoutputsforthisIMMEtarereportedinTables6.3and6.4,respectively.ItcouldbethecasethattheIMMEtwouldworkif,insteadofusingtheobservedexcitationenergyofthe31SIAS,thetwereperformedusingtheunperturbedexcitation153Table6.5:Ground-statemassexcessandexcitationenergyExvaluesusedasinputfortheIMMEtsofthelowestA=31;T=3=2quartet.Exceptfortheunperturbedexcitationenergyofthe31SIAS,whichisfromthepresentwork[89],allvaluesarethesameasin[102].NucleusTz[keV]Ex[keV]31Cl3=27034:7(34)031S1=219042:52(23)6297:6(13)31P+1=224440:5411(7)6380:8(17)31Si+3=222949:04(4)0Table6.6:OutputcoecientsforthequadraticandcubicIMMEtsforthelowestA=31;T=3=2quartetusinginputdatafromTable6.5.AllcoecientvaluesareinunitsofkeV.QuadraticCubica15453:0(12)15454:6(12)b5307:0(10)5316:1(24)c206:4(10)205:2(10)d5:0(12)˜2=17:0=1energyofthatstate.Totestwhetherisospinmixingin31ScouldaccountentirelyfortheIMMEbreakdown,weperformedanIMMEtincludingtheunperturbed31SIASenergy,keepingtheotherinputsconstantfromRef.[102].However,farfromsolvingtheIMMEbreakdown,usingtheunperturbed31SIASenergyactuallyexacerbatestheproblem:theresultisareducedchi-squaredvalueof˜2==17:0inthequadratict,increasedfromboththetinRef.[102]andourowntusingtheobservedIASenergy,andacubictermevenlargerinmagnitude,+5:0(12)keV.InputandoutputparametersforthistarereportedinTables6.5and6.6,respectively,andresidualsforthequadratictareshowninFig.6.4.Clearly,isospinmixingin31SalonecannotaccountfortheIMMEbreakdown.However,giventhemirrornatureof31Sand31P,itislikelythatthereissimilarisospinmixingpresentinthelatter'sisobaricanalogstate,whichhasnotyetbeendirectlyobserved.Weattempted154zT00.511.5IMME Fit Residual [keV]0246810 Figure6.4:ResidualsforthequadraticIMMEtofthelowestA=31;T=3=2quartet(Tables6.5and6.6)afteraccountingfortheobservedisospinmixingin31S.155touseourunperturbed31SIASenergyandtheresultsfromRef.[102]topredictthe\unperturbed"energyforthe31PIAS,thelowestT=3=2stateinthatnucleus.TheresultofthepredictionusingthequadraticIMMEisa31PstateatEx=6390:8(24)keV,only10keVabovethecurrentobservedexcitationenergyvalueof6380:8(17)keV.Intheeventthattheobservedexcitationenergyistheresultofaperturbationduetoisospinmixingwithanearbyhigher-energyT=3=2stateorfragment,itsactualenergycouldbehighenoughtorevalidatetheIMMEafteraccountingforsaidmixing.Acursoryglanceatthe2013A=31NuclearDataSheets[79],however,seemstoimplythatnosuchstateisknowntoexist.Nonearbystatesintheevaluationhavespinandparityrequiredformixingwiththe31P(Jˇ=3=2+)state.Itispossible,usingthetwo-levelmixingschemefromSection3.5.4toderivecombinationsofmixingmatrixelementVandunperturbedexcitationenergyDforsuchastate,suchthattheIMMEisrevalidated.Usingthetwo-levelmixingequationsandtheobservedandpredictedenergiesofthe31PIAS,thecurveshowninFig.6.5istheresultforEx>6401keV.Thesolutionat6401keVcorrespondstothelimitingcaseoftwodegeneratestatesatExˇ6391keV,bothpertrubedby10keV(thatis,D=0andE=2V).Usingthiscurve,itispossibletomakeanaiveempiricalpredictionoftheenergyofthe31Pstateinvolvedinmixingwiththe31PIAS,assumingthattheunperturbedenergyspacingisthesame(74(2)keV).ThisvalueyieldsasecondstateatEx=6454:8(35)keV,withanassociatedmixingmatrixelementof27.2(35)keV.Coincidentally,thispredictedstateisnearaknown31PstateatEx=6460:8(16)keV.TheA=31NuclearDataSheets[79]listthislevelashavingspinandparityJˇ=5=2+.Althoughthisassignmentisbasedonexperimentalworkevaluatedpreviouslyasleadingtoanunambiguousspinandparity[103],multipleexperimentalstudies[104,105,106,107,53],whilepotentiallyfavoringthe156Excitation Energy [keV]640064106420643064406450646064706480Mixing Matrix Element [keV]1015202530Figure6.5:Isospinmixingmatrixelement,including1˙condenceband,ofahypotheticalstateengagedinisospinmixingwiththe31PIASat6381keVasafunctionoftheobservedexcitationenergyofthesecondstate.ThebandisderivedundertheassumptionthattheIMMEprovidesagoodtofthedataafteraccountingforisospinmixing.Thedotted(left)anddot-dashed(right)linesshowthe1˙boundsobtainedusingthispredictionwhentheUSDmixingmatrixelementand6461-keVstateenergy,respectively,areusedasinputs.157Table6.7:CalculatedexcitationenergiesExandmixingmatrixelementsVofthetripletofisospin-mixedstatesincludingthelowestT=3=2statein31PforbothUSDBandUSDEinteractions.ThematrixelementslistedarebetweenthelistedT=1=2stateandtheT=3=2state.AllvaluesareinunitsofkeV.JˇUSDBExUSDBVUSDEExUSDEVE1(T=1=2)3=2+62588:361184:2E2(T=3=2)3=2+63646236E3(T=1=2)3=2+657910:9638312:75=2+assignment,havenotexcludedthe3=2+assignment.Furthermore,asnotedinRef.[79],onestudy[108]hasevenlabeledthestateasJˇ=1=2+,furthercomplicatingthematter.IfthestatedidinfacthavespinandparityJˇ=3=2+,itcouldmixwiththe31PIASat6381keV.Tocomplementthisrudimentaryempiricalapproachandfacilitatethesearchforthehypothetical31Pmixingwiththe31PIAS,B.A.Brownperformedshell-modelcalculationsusingboththeUSDBandUSDEinteractionstopredictmixingmatrixelementsandexci-tationenergiesforthe31Pandnearbystates.Aswiththe31ScasedescribedinSection6.1,theresultsofthecalculationsareatripletofJˇ=3=2+,includingtheIAS,allinvolvedinisospinmixing.TheresultsofthesecalculationsarereportedinTable6.7.Asshowninthetable,themixingmatrixelementsforthe31Pcasearemuchsmallerthanforthe31Scase(Table6.1).Whiletentativeexperimentalcandidatesforthelowermixedstateinthetripletexistat6233keVand6158keV[79],nohighercandidateisapparent,withtheexceptionofthestateat6461keV.Thisstate,however,requiresasignicantlyhighermixingmatrixele-mentthanthecalculationsusingUSDpredict;assuch,itshouldberegardedasatentativesolutionatbest.Theshell-modelmatrixelementsforthemixedstatesandthefunctionalforminFig.6.5canbeusedtoderivetheoreticalupperandlowerboundsfortheexcitationenergyofthemixed31PstateofEx=6406keVandEx=6402keV,respectively.158Experimentalsearchesarethereforeneededtouncoverpotential31PstatesintheenergyregionslightlyaboveEx=6400keVwhichcouldfullltheroleoftheJˇ=3=2+statemixingwiththeIAS,aswellastodeterminewithcertainythespinandparityofthe6461-keVstate.Asaninterestingastrophysicalaside,thespinandparityassignmentsofRefs.[62,63]usedthe5=2+assignmentforthisstate,additionallyusingthemirrorstateassignmenttoinferthesamespinandparityforanobserved31Slevelat6393keV.Ifthis6461-keVstateisinfacta3=2+state,itwouldmeanthatitisthemirrorofthe6390-keV31Sleveldiscoveredinthepresentwork,ratherthanthemirrorofthe6393-keV31SleveldescribedinRefs.[62,63].Ifanothernearby31PstateisdiscoveredtobeJˇ=3=2+stateinstead,itcouldinverselyimplythatthetwo31Sstatesat6390keVand6393keVaredistinct.6.2.2SecondA=31;T=3=2QuartetThelowestA=31;T=3=2quartetincludesthe31Cland31SigroundstatesandthelowestT=3=2excitedstatesin31Sand31P.However,asillustratedforA=13inFig.3.6,thereisalsoaT=3=2quartetforeachexcitedstatein31Cland31Si,whichalsoincludesT=3=2statesin31Sand31Pwithexcitationenergiesgreaterthantheexcitationenergiesoftheground-stateisobaricanalogs.Aswiththelowerquartet,ithasbeenhistoricallydiculttotesttheIMMEinthesecondquartet,largelyduetouncertaintiesassociatedwithboththeexcitationenergyoftherstexcitedstatein31ClandambiguityintheidentityofthesecondT=3=2statein31S.Infact,atentativemeasurementoftherst31Clexcitedstatevia31Ardecay[109]wastheonlyevidencefortheobservationofthatstate[83]untilarecentCoulomb-breakupexperimentwasperformedtoconrmtheexistenceofthestate[110].Inthisstudy[110],theexcitationenergywasfoundtobeEx=782(32)keV,leavingtheidentityofthesecond15931ST=3=2stateastheprimaryambiguityinthequartet.Varioussourceshavereportedexcitationenergiesforthe31SstaterangingfromadeniteT=3=2assignmentforastateatEx=7006(25)keVusingthe29Si(3He,n)31Sreaction[81]withsomewhatlowprecisiontoarelativelyprecise,buttentative,assignmentforastateatEx=7036(2)keV[111],withalternativecandidatesat6975(3)[97,111]and7053(2)keV[111].AlthoughthisJˇ=1=2+31Sstateisexpectedtobenearly1MeVabovetheprotonthreshold,theprotonemissionisisospinforbiddenand,therefore,itshouldhaveasubstantialgamma-decaybranchunliketheotherlow-spinlevelsintheregion[112].Preciseobservationofahighenergy-raytransitionfromalow-spinstateinthisregionwouldbeasignatureofthesecondT=3=2state,allowingforaprecisedeterminationofitsenergy.Theshellmodelpredictsthatthestatedecayspredominantlytothegroundstate,andshellmodelcalculationsusingbothUSDB[113]andtherecently-developedUSDE[89]modelspredicta31Sstate745(50)keVabovethe31SIASenergyofEx=6279keV.Intheshellmodel,thisstatehasa31Clfeedingof0.03(2)%andaground-state-decaybranchofg:s:
==0:95(4).AnalysisofE12028resultedintheidenticationofanumberofnew31Sstates.AsdiscussedinSection5.3.3,thespinsandparitiesforamajorityofnewstateswereinferredbycomparingtoshellmodelcalculations.Isospinmixingofthestateat6390keVandtheIASallowedforthedeductionofthatstate'sspinandparitywithprecision.WedidnotobserveanyphotopeaksbetweenEx=6400andEx=7000keV.Slightlyabovethisenergyregion,weobservedaphotopeakatE=7049:2(8)whichdoesnotappearincoincidencewithanyotherpeak,indicatingthatitislikelyatransitiontothegroundstate.Thisstate'sexcitationenergyofEx=7050:0(8)keV,approximately770keVabovethe6279-keV31SIAS,isalsoconsistentwithourshell-modelcalculations,whichpredictasecondT=3=2level770keVabovethetheoreticalIAS.Theobservedbetafeedingofthisstate,0.047(5)%,isalso160Table6.8:Ground-statemassexcessandexcitationenergyExvalues[79]usedasinputfortheIMMEtsofthesecond-lowestA=31;T=3=2quartet.NucleusTz[keV]Ex[keV]31Cl3=27034:7(34)782(32)31S1=219042:52(23)7050:0(8)31P+1=224440:5411(7)7141:1(18)31Si+3=222949:04(4)752:23(3)Table6.9:OutputcoecientsforthequadraticandcubicIMMEtsforthesecond-lowestA=31;T=3=2quartetusinginputdatafromTable6.8.AllcoecientvaluesareinunitsofkeV.QuadraticCubica14697:3(14)14698:6(22)b5307:2(19)5306:0(25)c205:0(18)211(8)d4(5)˜2=0:51=1consistentwithshellmodelpredictions,andnoothergamma-raytransitionsde-excitingthestatewereobserved.Basedontheobservedagreementwiththeshellmodelpredictions,thesingulargammabranchtothegroundstate,andasmall,previouslyobservedbeta-protonbranch[83],wehaveidentiedthisstateasthesecond31ST=3=2state,withJˇ=1=2+.Similarlytothelowestquartet,wetthesecondA=31;T=3=2quartet,includingthenew7050-keVstateexcitationenergyandtheexcitationenergiesoftheotherthreequartetmembers,usingtheIMME.TheinputmassexcessesandexcitationenergiesarereportedinTable6.8,andtheoutputparametersarereportedinTable6.9.Asshown,thequadratictyieldsareducedchi-squaredvalueof˜2=v=0:51=1andap-valueof0.48,indicatingagoodt.Thisfurtherconrmsthatthestateat7050keVisindeedthe31SmemberofthesecondT=3=2;A=31quartet.TheresidualsforthistareshowninFig.6.6.AlthoughthequadraticIMMEtisverygoodofthemeasuredmassexcessesandex-161zT00.511.5IMME Fit Residual [keV]010 Figure6.6:ResidualsforthequadraticIMMEtofthesecond-lowestA=31;T=3=2quartet(Tables6.8and6.9).162Table6.10:CalculatedexcitationenergiesExandmixingmatrixelementsVofthetripletsofstatesinvolvedinmixingwiththesecond-lowestT=3=2statesinboth31Sand31P,forbothUSDBandUSDEinteractions.ThematrixelementslistedarebetweenthelistedT=1=2stateandtheT=3=2state.AllvaluesareinunitsofkeV.31SJˇUSDBExUSDBVUSDEExUSDEVE1(T=1=2)1=2+72347:864216:8E2(T=3=2)1=2+72716944E3(T=1=2)1=2+78142271179:431PJˇUSDBExUSDBVUSDEExUSDEVE1(T=1=2)1=2+72516:564173:1E2(T=3=2)1=2+73106982E3(T=1=2)1=2+78616:171277:2citationenergiesofthequartetmembers,itispossiblethat,aswiththerstquartet,asmallamountofisospinmixingperturbstheexcitationenergiesofthe31Sor31Pmemberstates.PotentialcandidateT=1=2statesexistinbothnuclei,butnoexperimentalevidencewasobservedinE12028topositivelyidentifyanysuch31Sstate.Asmentionedpreviously,no31Sgamma-raytransitionswereobservedcorrespondingtostatesbetweenEx=6400andEx=7000.Toestimatethepotentialamountofmixing,B.A.BrownusedboththeUSDEandUSDBinteractionstopredictenergylevelsandmixingmatrixelementsforboth31Sand31P,aswasdoneforthelowestquartet(Tables6.1and6.7).Themodelsproducesmallmixingmatrixelements,consistentwiththelackofobservationalevidencefromourexperimentaldata.Theresultsoftheshell-modelcalculationsarereportedinTable6.10.Usingourhigh-precisionmeasurementoftheexcitationenergyoftheT=3=2statein31S,itispossibletotesttherecentmeasurement[110]ofthe31ClrstexcitedstateusingtheIMME.ByincludingthenewmeasurementofthesecondT=3=231Sstateexcitationenergywiththeexcitationenergiesandground-statemassexcessvaluesfor31Pand31Sitoproduce163theIMMEcurve,andaccountingfortheuncertaintyintroducedbythepossibilityofisospinmixingviathedcoecientinthecubict,wederiveanIMMEmassexcessforthe31Clrstexcitedstateof=6276(10)keV.Whencombinedwiththenew,preciseground-statemassexcessfromRef.[102]anditsuncertainty,wecalculateanexcitationenergyEx=759(11)keVforthestate,avalueconsistentwiththemeasuredvalueofEx=782(32)keV[110].Itisalsopossibletousetherecentvalue[102]ofthe31Clprotonseparationenergy,Sp=265(4)keVandthe30S+presonanceenergybasedonthebeta-protonmeasurement,Er=461(15)keV[83,109]:theresultofthecalculationisanexcitationenergyofEx=726(16)keV,whichisconsistentwithourpredictiontowithin1.8combinedstandarddeviationsandwiththevaluefromRef.[110]towithin1.6combinedstandarddeviations.Giventheslighttensionbetweenthevaluebasedonthebeta-protonmeasurement[109]andtheothertwovalues,anewmeasurementof31Arbetadecay[114,115]withhighsensitivitytolow-energyprotonswouldbeaninterestingstudy.
6.3DiscrepancieswiththeNuclearDataSheetsandComparisontoPreviousWorkE12028isoneofonlyafewbetadecayexperimentsperformedtodate,buthasneverthelessyieldedovertwoordersofmagnitudehigherstatisticsforanalysisthanthemostrecentexperiment[72],resultinginadecayschemewithdoublethenumberofobservedgamma-raytransitionsandninenewobservedbetatransitions.TheminimumnitebetafeedingobservedwasI=0:023(7)%,tothestateat5436keV.Thisimprovementinsensitivitytellsusmuchaboutthe31Snucleus,butitalsoallowsforinsightintotheresultsofprevious164experimentsaswell.
6.3.1UnobservedPreviously-ReportedTransitions
Inparticular,severalstatesreportedintheA=31NuclearDataSheetsbetweenEx=5MeVandEx=8MeVwerenotobservedinE12028.TheexcitationenergiesofthesestatesreportedinthebetadecayschemeoftheNDSare:5408.2keV(Jˇunconstrained),5786.2keV(Jˇ=(1=2;3=2;5=2)+),6420.7keV(Jˇ=(1=2;3=2;5=2)+),7280.0keV(Jˇunconstrained),7416.8keV,(Jˇunconstrained),7631.8keV(Jˇ=(1=2;3=2;5=2)+),and7644.5keV(Jˇ=(1=2;3=2;5=2)+).Thestatesat5786,6421,7280,7417,7632,and7645keVarefromtentativeassignmentsmadeintheprevious31Cldecaystudy[72],butourexperimentwasmuchmoresensitiveanddidnotobservethesepeaks.Therefore,weattributethemtocontaminantsintheexperimentofRef.[72]andhaveelectedtoomitthemfromournormalizationanddecayscheme.TheA=31NuclearDataSheetsalsoreportatransitionfromthestateat3076keVtothestateat2234keV,basedonthereportsofRefs.[72]and[116],butnoteintheirdecayschemethatthetransitionwasnotincludedintheleast-squarestthatresultedinthequantitiesreportedinthetable.TheNDSevaluatorsnoteaswellthatthedierenceinenergybetweenthetwostatesostensiblyinvolvedinthetransitionisonly842keV.Ref.[72]assignedthetransitionanabsolutegammaintensityper100betadecaysofI=1:1(1),implyingthatthepeakshouldbestrongerthanthenearby985-keVgammaraymarkingthetransitionbetweenthe2234and1248-keVstates,whichisassignedanabsoluteintensityofonly0.2(1).However,despiteobservingthe985-keVgammaray,wedidnotobserveaphotopeakat845keVinourbeta-delayedgamma-rayspectrumorinthe2234-keVcoincidencespectrum.Afterattemptingtotthisregion,weobtainedanupperlimitontheintensityofthistransition165ofI=0:018(4)per100betadecays,andhaveomitteditfromthenormalizationanddecayscheme.
6.3.2SpinandParityDiscrepancies
AsdiscussedinSection4.2,thespinandparitiesofseveralimportantresonancestatesinthe30P(p;)31SGamowwindowarenotknownunambiguously,duetodiscrepanciesinassignmentsbetweendierentexperiments[64,62,63].Since31ClpreferentiallypopulatesJˇ=1=2+;3=2+;and5=2+states,wecancompareourassignmentstoanyassignmentswiththosespinsandparitiesinpreviousstudies.Refs.[64]and[63]reportanumberofspinandparityassignmentsthatareatoddswithoneanother;thesensitivityofourexperimentallowsustocommentontheassignmentsofRefs.[64]and[63]andthelikelihoodoftheseassignmentsbeingcorrect.Ref.[64],forexample,reportstwostates:oneat6328.6(9)keVandoneat6356.1(9)keV,withJˇ=1=2+and3=2+,respectively.Ref.[63],however,assignsthesestatesspinsandparitiesof3=2and5=2,respectively.GiventhattheselevelswerenotobservedinE12028,webelieveitismorelikelythatthespinandparityassignmentsofRef.[63]arecorrectforthesestates.Ref.[64]alsoreportsanumberofotherstatesthatmighthavebeenvisibleintheE12028data,butareabsent:alevelat6719.9(9)keVwithJ=5=2,alevelat6749.0(9)keVwithJˇ=3=2+,alevelat6936.7(17)keVwithJˇ=(1=25=2)+,alevelat6959.6(16)keVwithJˇ=1=2+,andalevelat7033.5(13)keVwithJˇ=(1=25=2)+.Thesestates,whicharecomparativelyhigherabovetheprotonemissionthresholdthanthelow-lyingresonancesobservedinE12028,decayprimarilyviaprotonemission,sothelackofobservationinE12028isnotasignicantcauseforconcern.The7033-keVstateisinterpretedasthestatethatRef.[71]tentativelyidenties,andE12028hasconrmedas,thesecond166T=3=231Sstate.WealsoobserveafewdiscrepancieswithRef.[63].ThelevelreportedinRef.[63]atEx=4527:8(2)keVisgivenaJˇassignmentof3=2+.Wedidnotobserveanystateatthisenergy;theclosestcandidateisthestateatEx=4519:63(32)keV,whichwealsoassignedJˇ=3=2+.Ref.[63]alsoreportedastateatEx=4710:1(8)withJˇ=5=2+.Weagaindidnotobserveastateatthisenergy,butinsteadhaveidentieda5=2+stateatthenearbyenergyEx=4717:72(32)keV.Wedidnotobservethe5=2+stateRef.[63]reportsat5401.5(8)keVorthe5=2+stateat5518.3(3)keV,andthecloseststateweobservedtotheseenergies,at5435.9(9),wasassignedJˇ=3=2+.Wedidobserveastateat5775.4(4)keV,whichwehaveassignedaspinandparityof5=2+,butwhichRef.[63]didnotobserve.Ref.[63]makesacoupleoftentativeassignments,whichtheresultsofE12028couldhelpnarrowdown.Ref.[63]reportsastateat6138.3(21)keV,identiedbyboththetransitiontothe7=2+stateat3349keVandthetransitiontothe3=2+rstexcitedstateat1248keV,withspinandparityJˇ=(3=2;7=2)+.Thisstateisnearinenergytothestateweidentiedat6129keV,butweassignedthisstateaspinandparityof5=2+basedonacomparisontoourshellmodelcalculations.TheassignmentinRef.[63]ismadebasedonboththeangulardistributionsmeasuredforthetransitionsandthemirrornucleusassignmenttoa31Pstateat6233.4(15)keV.Interestingly,however,thetwogammabranchesobservedinRef.[63]arethetwostrongestbranchesreportedinourcalculations,andthe31Pstatedoesnothaveacertainparity,insteadreportedintheNDSasJˇ=(3=2;5=2;7=2)+.Ifthesetwostatesarethesame,andtheshellmodelcalculationsarecorrect,itcouldimplythattheassignmentinRef.[63]isincorrect.ThemostinterestingpotentialdiscrepancybetweenthepresentworkandRef.[63]isthestateat6390keV.Ref.[63]reportsastateatthenearbyenergyof6392.5(2)keV(thisstate167wasbrieymentionedattheendofSection6.2.1),butassignsitaspinandparityof5=2+,basedontheangulardistributionofthetransitionobservedandthemirrorassignmentwiththe6461-keV31Pstate.Wedidnotobservethisstateinouranalysisatallandtheclosest5=2+statetothe6390-keVstatepredictedbyourshell-modelcalculationsisˇ300keVhigher,implyingeitherthatitsbetafeedingisextremelylow(inwhichcaseitsphotopeakwouldlikelybeentirelyoverwhelmedbythenearbystrong6390-keVphotopeak),orthattheassignmentinRef.[63]isincorrect.However,preliminaryresultsfromindependentworkbytheauthorsofRef.[63]seemtoindicatethatthegammabranchesofthestateat6393keVareinconsistentwiththebranchesweobservedforthe6390-keVstate[117],implyingthatthetwostatesarenonethelessdistinct.168Chapter7
Outlook:TheFutureof30P(p;)31S7.1Conclusions
AsaresultofNSCLexperiment12028,anew30P(p;)31Sresonancestatewasdiscoveredandfoundunambiguously,viaisospinmixingwiththenearbyisobaricanalogstate,tohaveaspinandparityJˇ=3=2+[89].Thisresonancewasfoundtocontributestrongly{upto50%ofthetheoreticalHauser-Feshbachstatisticalmodelrate{tothe30P(p;)31Srate,makingitthemostimportantunambiguouslyidentiedresonanceandameaningfulmeansofestimatingtheoverall30P(p;)31Srate.Therevelationoftheimportanceofthe6390-keVstatetothe30P(p;)31Sreactionrateandtheimplicationthattheratemaybehigherthanpreviouslyestimatedmeanthatdiscrepanciesbetweentheslight30Siexcessobservedinnovagrainsandtherelativelylargeexcesspredictedbysomenovamodelsmaybelessatoddsthanpreviouslythought:ifthe30P(p;)31Sproceedsmorequicklythancurrentmodelssuggest,relativelyfewer30Snucleiwillultimatelybeproducedvia30Pbetadecay,reducingthe30Siyieldinmodels.Theobservationofisospinmixingitselfisamilestoneinitsownright:therstclearidenticationinthesdshellofmixingbetweenaT=3=2stateandaT=1=2state.ThenewUSDinteraction\USDE"wascreatedbyB.AlexBrownspecicallyforanalysisoftheresultsofisospinmixingin31Sandhasalreadybeenusedtostudypotentialisospin169mixinginanothersd-shellnucleus,31P.E12028allowedforthedeniteidenticationofthesecondT=3=2stateof31Sviathegammade-excitationofthe7050-keVstatetothegroundstate31S[100]and,alongsidetherecentprecisionmeasurementofthe31Clsecondexcitedstateenergy[110],hasallowedfortherstprecisiontestoftheIMMEforthesecondA=31;T=3=2quartet.TheIMMEbreakdownobservedaftertherecent31Clmassmeasurement[102]hasnotyetbeensolved,butweconcludeitislikelythatisospinmixingin31Ptomatchthemixingin31Slikelyplaysarole.Theresultsofthisworkwillprovideanimportantstartingpointforthesearchforthepotentialmixed31Pstate.TheanalysisofE12028hasalsoproducedthemostcompleteanddetailed31Clbetadecayschemetodate,with40newobservedgamma-raytransitionsandtennewobservedbetafeedings,includingnewobservationsofforbiddenbetadecay.Thisschemereectsovertwoordersofmagnitudehigherstatisticsacquiredcomparedtopreviousworkandhasalsoallowedforthepotentialexclusionofseverallevelstentativelyidentiedinpreviousworkasbelongingto31S.Theimprovedaccuracyoflevelidenticationin31Sshouldassistineortstoassignspinsandparitiesbasedon,forexample,comparisonsto31Pmirrorstates,whichitselfcouldhelpcorrectlydeducetheparametersoftheimportantresonancesthatcontributetothe30P(p;)31Sreaction.7.2OutlookandFutureWork
TheresultsofE12028areanimportantsteptowardansweringthequestionofwhetherpresolar\nova"grainsdoindeedoriginateinclassicalnovae.However,theexperimentbyitselfhasnotentirelysolvedthemysteryoftheseancientstellarmessengers.Theexpectedstrengthofthe6390-keVresonancemayindicatethattheoverallreactionrateishigherthan170thepreviouslyestimatedrate,butunfortunately,constraintsonimportant31Sresonancesotherthanthe6390-keVstateremainelusive,anditisstillnotknownwithcertaintyhowmany31SstatesexistbetweenEx=6390keVandEx=6410keV.Inaddition,theratecalculatedfortheresonantcapturethroughthe6390-keVstate,whilebasedonexperimentalndings,didrequiretheoreticalestimatesofboththeprotonpartialwidthpandthegammapartialwidth.Conveniently,theisospinmixingofthisstatewiththenearbyIASanditsconsequentlylargebetafeedingmakeitidealforstudyusing31Clbetadecay.Abeta-delayedprotondecaymeasurementofthisstatecouldproduceanexperimentalmeasurementoftheprotonbranchingratiowhich,alongsideameasurementofthelifetimeofthestate,couldyieldanentirelyexperimentalresonancestrengthandthersttrulyexperimentalcalculationofalowerlimitonthe30P(p;)31Srate.Withtherelativelylargeresonancestrengthexpectedforthe6390-keVstate,itisalsopossiblethattheresonancecouldbeacceseddirectlyviaprotoncaptureon30Patthenextgenerationofrareisotopebeamfacilities[118,119].Finally,itisprobable[97]thatanumberofnegative-parityresonancescontributestronglytothereactionrateatpeaknovatemperatures;studyoftheseresonanceswasasamatterofdesigneectivelyprohibitedinE12028,butfutureexperimentalworkwillalsolikelyneedtofocusontheeectsofthesenegative-paritystatesonthe30P(p;)31Srate.AlthoughtheIMMEbreakdownreportedinRef.[102]hasnotbeensolvedwiththeadditionofisospinmixingin31S,adiscoveryofisospinmixingin31PcouldhelprevalidatethequadraticIMMEinthelowestA=31;T=3=2quartet.The6461-keVstateisthebestcandidateformixingwiththe31PIAS,butitisnotconsistentwithUSDshell-modelpredicitionsandshouldbeinvestigatedfurther.Futurenuclearstructurestudiescouldfo-cusongammaspectroscopyofenergylevelsintheenergyregionaroundthe31PIASto171complementthenumerouschargedparticlemeasurementscarriedouttodateanddeterminewhetheralllevelsintheregionhavebeenobserved.InthesecondA=31;T=3=2quartet,threeoutoffourmembersnowhavemass-excessuncertanties<2keVandtheirmassesarewell-describedbythequadraticformoftheIMME,buttheextentofisospinmixing,althoughlikelysmall,isstillnotknownwithcertainty.Furtherstudiesofthismultipletcouldfocusonreducingtheuncertaintyinthe31Clrstexcitedstateexcitationenergy.Boththemeasurementof31Arbeta-delayedprotonsandmeasurementviain-beamgammarayspectroscopycouldprovideindependentcontributionstowardreducingthisuncertainty:theformercouldhelpaddresstheapparentdisagreementbetweenRef.[109]andRefs.[110,100]andthelattercouldpresentanovelapproachtomeasuringtheexcitationenergyofthisstate.
7.3FinalThoughts
Ultimately,nooneexperimentcancurrentlyaddressconclusivelythenumerousquestionssurroundingpresolarnovagrainsandthe30P(p;)31Srate.E12028,likethemanyexperi-mentalstudiesthatprecededit,representsonlyoneimportantsteponthejourneytowardansweringwhetherornottheseancientmicroscopicgrainsdoindeedcomefromclassicalno-vae.E12028hasalsoplayedanimportantroleinhelpingtomotivatefuturenuclearphysicsexperimentsbyposingseveralquestionsthatcanbeansweredusingtechniquesrelatedtothebeta-delayedgamma-rayspectroscopyemployedforthisstudy.Complementarybeta-delayedprotonmeasurementsrepresentapromisingpathwaytowardunderstandingofthenuclearstructureof31Sandotherimportantnuclei.Asithappens,acompactbeta-delayedprotondetectorisalreadyunderdevelopmentattheNationalSuperconductingCyclotron172Laboratory,withdesignstoanswertheseandmanyotherinterestingquestions.Withtheadventnewdetectorslikethis,andtheabilitytoputthemtoexcellentusewithnext-generationfacilitiesliketheFacilityforRareIsotopeBeamsandthehigh-intensityexoticbeamssuchfacilitiespromise,thefutureofexperimentalstudyofboththenuclearaspectandtheastrophysicsaspectofclassicalnovaelooksincrediblybright.173APPENDICES174APPENDIXA
TheoreticalTools
Asdescribedinanumberofpreviouschapters,wereliedonanumberoftheoreticalandpredictivetoolsbothtohelpplantheexperimentandtohelpinterprettheresultsofanalysis.Tojustifytheexperimentandreceivebeamtime,experimentersmustbeabletopredicthowmanyhoursofbeamtheywillneed,andwhatlevelofstatisticsarerequiredtoachievetheexperimentalgoals.Inaddition,experimentersmustbeabletoassessthefeasibilityofproducingthenuclideforstudyintherstplace.Forexample,asmentionedinChapter4,itisnotfeasibletoproduce30Pinlargequantitiesfordirectstudyof30P(p;)31S.Itwouldbecostlytolearnthisfactonlyaftersecuringdozensofhoursofbeamtime,settingupanexperimentalsetup,andstartinganexperiment;consequentlytoolsforsimulatingtheproductionofrareisotopesarerequired.InthisAppendix,wediscusstwosimulationtools,Lise++andGeant4,whichweusedbothtoassessthefeasibilityofE12028beforeitsexecutionandtointerpretandcheckexperimentalresultsduringitsanalysis.WewillalsodiscussbrieytheUSDshellmodelcalculationsweperformedduringanalysisandwillreporttheresultsofthosecalculations.Lise++:ExoticBeamProductionwithFragmentSepa-
rators
Lise++isafreely-availableprogramdeveloped\tocalculatethetransmissionandyieldsoffragmentsproducedandcollectedinaspectrometer."[120]Lise++allowstheexperimenter175toselectastableprojectileandanenergyandintensitypost-acceleration,simulatethefragmentationoftheprojectileonatarget,andtunethewidthofthetargetsoastoproducethemaximumsimulatedamountofadesiredexoticnuclide\fragment."Theexperimentercanthenbuildasimulatedbeamline,includingfragmentseparatorssuchastheA1900andtheRFFSaswellasdetectorssuchassiliconPINdetectorsortheplasticscintillatorusedinE12028(Fig.A.1).Lise++containsalargelistofmaterialswithwhichexperimenterscanbuild\blocks"inthesimulation;Lise++alsocalculatesthetime-of-ight,energyloss,position,angulardistribution,andotherparametersforeachoftheseblocks(Fig.A.2).InthiswaytheexperimentercanuseLise++tosimulatethingslikebeampuricationandproductionrateinordertodeterminethebestexperimentalcongurationtomeettheexperimentalneeds.ForE12028,weperformedaLise++simulationthatsimulated31ClfragmentproductionattheNSCLforthe36Arbeamthatwasultimatelyusedfortheexperiment.WewereabletousethesimulationtodetermineabaselinefortheoptimaltuningparametersfortheA1900aswellasthethicknessofthe9Betargetusedforproduction(Lise++isabletocalculateanoptimaltargetthickness(Fig.A.3)tomaximizeproductionofthedesiredfragment)andtheoptimalsettings(voltage,phaseangle)oftheRFFSupstreamoftheexperimentalsetup.Wewerealsoabletosimulatetheimplantationdepthofvariousbeamconstituents(Fig.A.4),aquantitysubsequentlyusedinourGeant4calculationstofacilitatepredictionsofgamma-rayeciencyforthecloverdetectors.Inadditiontothebasicbeamlinesimulationcomponents,Lise++containsanumberofusefulextrasincludingacalculatortoquicklydetermineenergylossthroughblocksandtoolstosimulatetheinteractionofthebeamparticleswiththoseblocks.Oneofthesetools,mentionedinChapter5,isthefusion-evaporationcalculatorPACE.PACEisaMonteCarlo176FigureA.1:AscreenshotofLise++,showingthespectrometersetupwindowinthecenteroftheimageandanassembledsetupcorrespondingtotheA1900fragmentseparatorandtheRFFSontheleft.Each\block"ofmaterialisinsertedintothebeamlineusingthesetupwindowandhasitsownsetofoptions,dependinguponthetypeofblockused.177FigureA.2:AseriesofgraphsfromLise++,showingthebeam'ssimulatedcharacteris-ticsattheRFFS.Includedcalculationsaretheyieldofeachisotope,thedispersionangle,andpositionforbothhorizontalandverticaltransversedirections,therigidityofthebeamconstituents,andtheirenergy.FigureA.3:TheresultofaLise++calculationtondtheoptimalthicknessoftheberylliumtargetforproductionof31Cl.AlthoughLise++givesanestimateofthebestthickness,experimentersareasalwayslimitedbytheavailabilityoftargetsofvaryingthicknesses.178FigureA.4:TheresultofaLise++calculationshowingtheimplantationdepthofseveralbeamconsituentsinsideatargetblock.Thecalculationshowstheyieldofeachspeciesanditsrangeinthematerial.
simulationtoolthattakesasinputthemassesoftwonucleiandthebeamenergy(Fig.A.5)andproducesnotonlyalistoflikelyfusion-evaporationproductsbutangularandenergydistributionsfortheseproductsaswellasforprotons,neutrons,andalphaparticlesproducedthroughtheprocess.PACEproducesresultsinatextle;anoutputexampleisproducedbelowinSectionA.179FigureA.5:TheinputcardforPace.Asshown,thecalculationrequirestheAandZofbothtargetandprojectileaswellasthelaboratoryenergyofthebeam.180181PACEOutput
ThefollowingcodeexampleistheoutputforatypicalPACEfusion-evaporationcalculation.Theinputnucleiinthiscaseare31Cland12Candthebeamenergyissimulatedat50MeV/u(1550MeVtotalbeamenergy).\texttt{v.Version4.2013:4630-06-16PACE4modifiedJULIAN***********projectionangular-momentumcoupledevaporationMonteCarlocode**********************angulardistributionsobtainedusingM-statesofangularmomentum***********MODE=1************FusionxsectiontakenfromBassmodel
BassfusionxsectionforE=1550.0MeVis897mb
Fusionradius=5.30fm.Barrierheightis15.92MeV182Transmissionprobabilityforaone-dimens.barrier:Classical---------------------------------------------------------------------------StartingconditionsZNASpinProjectile1714311.5
Target66120.0
Compoundnucleus232043Bombardingenergy(MeV)1550.00
Centerofmassenergy(MeV)432.56
Compoundnucleusexcitationenergy(MeV)443.508
Q-valueofreaction(MeV)10.950
Compoundnucleusrecoilenergy(MeV)1117.442
Compoundnucleusrecoilvelocity(cm/ns)7.086e+00
Compoundnucleusvelocity/c2.362e-01
Beamvelocity(cm/ns)9.829e+00
Beamvelocity/c3.276e-01***InputtransmissioncoefficientsdeterminedbyinputvalueofTLdiffuseness.***diffuseness=2.00183***Opticalmodelinputcalculationbypasses.*********Experimentalfusioncrosssection(mb)897
FusionL-grazing70.907FusionL-difusseness2.000
Yrastspinatmaximumexcitationenergy82
Compoundnucleusformationcrosssection(mb)897---------------------------------------------------------------------------Partialcrosssections(mb)---------------------------------------------------------------------------JSIG(J)|JSIG(J)|JSIG(J)|JSIG(J)|JSIG(J)|---------------------------------------------------------------------------0.50.18|17.56.3|34.512|51.518|68.518|
1.50.7|18.56.7|35.513|52.519|69.516|
2.51.1|19.57|36.513|53.519|70.514|
3.51.4|20.57.4|37.513|54.519|71.511|
4.51.8|21.57.7|38.514|55.520|72.58.3|
5.52.1|22.58.1|39.514|56.520|73.56|
6.52.5|23.58.4|40.514|57.520|74.54.1|1847.52.8|24.58.8|41.515|58.521|75.52.7|
8.53.2|25.59.1|42.515|59.521|76.51.8|
9.53.5|26.59.5|43.515|60.521|77.51.1|10.53.9|27.59.8|44.516|61.522|78.50.7|
11.54.2|28.510|45.516|62.522|79.50.44|
12.54.6|29.511|46.516|63.522|80.50.27|
13.54.9|30.511|47.517|64.522|81.50.17|
14.55.3|31.511|48.517|65.522|82.50.1|
15.55.6|32.512|49.518|66.521|83.50.057|
16.56|33.512|50.518|67.520|84.50.028|---------------------------------------------------------------------------***Sphericalnucleusleveldensity
***SierkbarriernotfoundforA=43Z=23
***Inputfissionbarrier=39.70MeVatL=0takenfromSierk
***G.S.littleAmultipliedbyfactor1.000obtainsaddleleveldensity***Nofissioncalculationforbarrierabove30.0MeV
***Little-A=MASS/10.0185Energyrangeforneutronprotonalphagamma
minimal0.010.420.850.00
minimal40.0029.2651.7720.00***Internalprobabilitydiscriminatorofprogramsetto0.0020Numberofcascadesis1000OpticalmodelparametersforlightemittedparticlesV*E*E**2R0RARDR0CW0*E*E**2R01AIDRMCHDNPDIMAGIRAD47.010-0.267-0.0021.2960.6600.0009.520-0.0530.0001.2570.4800.000250SURF157.468-0.5500.0001.2500.6501.25013.5000.0000.0001.2500.4700.000250SURF150.0000.0000.0005.8680.5764.0987.5150.0000.0005.8680.5760.000250VOL0E.M.TransitionstrengthsinWeisskopfunitsE1=0.000080M1=0.025000E2=4.800000M2=0.019500***Gilbert-Cameronspincutoffparameterused
------Outputresultsforcompoundnucleusdecay-----1861.YieldsofresidualnucleiZNAeventspercentx-section(mb)
121426Mg10.1\%0.897
121224Mg20.2\%1.79
111324Na10.1\%0.897
121123Mg10.1\%0.897
111223Na90.9\%8.07
101323Ne20.2\%1.79
13922Al10.1\%0.897
111122Na40.4\%3.59
101222Ne121.2\%10.8
111021Na70.7\%6.28
101121Ne171.7\%15.291221F20.2\%1.79101020Ne343.4\%30.591120F80.8\%7.1810919Ne30.3\%2.6991019F121.2\%10.818781119O50.5\%4.48
9918F20.2\%1.79
81018O131.3\%11.7
9817F80.8\%7.18
8917O595.9\%52.9
71017N30.3\%2.69
9716F10.1\%0.897
8816O10310.3\%92.4
8715O282.8\%25.1
7815N11211.2\%100
6915C30.3\%2.69
8614O10.1\%0.897
7714N171.7\%15.2
6814C323.2\%28.7
7613N101\%8.97
6713C13713.7\%123
5813B30.3\%2.69
6612C10510.5\%94.2
6511C131.3\%11.7
5611B757.5\%67.3
4711Be10.1\%0.8971885510B10.1\%0.897
4610Be10.1\%0.897
549B141.4\%12.6
459Be353.5\%31.4
369Li10.1\%0.897
448Be141.4\%12.6
437Be70.7\%6.28
347Li313.1\%27.8
325Li60.6\%5.38
235He50.5\%4.48
213He10.1\%0.897
123H10.1\%0.897Totalfission363.6\%32.3
TOTAL1000100\%897Mode=ALPHTotalnumber=113
15-301..............................12.00.0
30-458..............................82.00.018945-6064............................104.02.4
60-7581............................92.61.6
75-9012............................35.32.4
90-10542............................63.72.4105-12015............................66.21.9
120-13532............................54.02.4
135-15021............................33.72.4
150-16522............................44.52.5
165-180..1............................17.00.0
180-19511............................24.52.5
210-225..1............................17.00.0
225-240..2............................27.00.0
240-2552..1..........................35.34.7
255-270..1............................17.00.0
270-285..2............................27.00.0
285-300..2............................27.00.0
300-315..1............................17.00.0
315-330....3..........................312.00.0
330-345..1............................17.00.0
345-360....11........................214.52.5
360-375....12........................315.32.4190375-390....1..3......................419.54.3
390-405..12511....................1016.55.2
405-420........2......................222.00.0
420-435....21..1....................417.06.1
435-450....1..53131..............1427.78.0Ex/J-4-9-14-19-24-29-34-39-44-49-54-59-64-69-74-79sumavrgstdvSum3932129115131..............----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Geant4:SimulationofCloverEciencies
Inordertoassessthefeasibilityofanexperiment,theamountofstatisticsrequiredtoobservemeaningfulresultsmustbeestimatedaccurately.Experimentersmustthereforetakespecialcaretodeterminenotonlyifthebeamproductionfacilitiesareabletoproducethenuclidetheywant,theymustalsodeterminewhethertheelectronicsanddetectorsavailabletothemarecapableofrecordingenoughdatatomakeanalysisworthwhile.Forgamma-raydetectorssuchasthecloverarray,thismeansestimatingtheeciencyofthedetectorsatanumberofdierentenergies;suchanestimationallowstheexperimentertomakepredictionsabouttheprobabilityofobservinganygivensignalforagivennumberoftotalevents.Thisinturnallowsexperimenterstoknowwithgreatercertaintyhowmuchbeamtimetoaskfor.Toaddressthisissue,weusedasuiteofcodescalledGeant4.Geant4isafreely-availablesuiteofcodes\forthesimulationofthepassageofparticlesthroughmatter."[121].Atitsmostbasic,Geant4createsparticlessuchasprotons,electrons,compoundparticlessuchasnuclei,photons,orfundamentalparticlesand,usingparametersspeciedbytheuserandphysicsinteractionroutines,simulatestheinteractionoftheparticlewithageometricalbodymadeofauser-speciedmaterial.ThismaterialcanbeofanymakeupandusersmaydenetheirownmaterialwithinGeant4'ssourcecode.WeusedGeant4priortoE12028inordertoassessthefeasibilityofusingtheYaleCloversharearrayinsteadofusingNSCL'sownSegmentedGermaniumArray(SeGA,acompactgermaniumarraycomposedofsixteenthirty-two-foldsegmentedgermaniumcrys-tals[122]).PreliminarysimulationssuggestedthattheCloversharearraywouldhaveroughlydoubletheeciencyofSeGAatallenergies,soweperformedanumberofsimulationsusingasimulatedCloversharearraytodeterminetheneededstatisticsforE12028.191TodeneageometricalbodyinGeant4,theusermustdenethematerialoutofwhichthebodyiscreated,thenplaceitaroundanoriginpointthecode'ssimulationspace.Weusedasetofdenitionsforthecloverarrayprovidedbyacollaborator,tweakingthepositionoftheninecloverdetectorstosuitourspecicexperimentalneeds(e.g.theuseofasmallscintillatorforimplantationasopposedtoalargergermaniumimplantationdetector).Wealsodenedanumberofcentralimplantationdetectors(plasticscintillator,NaIscintillator,LaBrdetector)and,usingtheestimatesoftheimplantationdepthfromLise++,performedsimulationsforanumberofdierentgamma-rayenergiestodetermineboththeeciencyofthecloverdetectorsandtheattenuationofthegammasinthecentraldetector.Thesesimulationshelpedfuelboththedecisiontouseaplasticscintillatorinthecenterofthearray,andthethicknessofthedetector.WealsoutilizedGeant4duringanalysistohelpassesstheaccuracyofourrelativee-ciencycalibration.Weperformed,foreach152Euand32Clpointweusedinthecalibration,aGeant4simulationoftheeciencyofthecloverdetectorswherethesourcewasplacedinthesimulationattheestimatedpositionofthecalibrationsource(152Eu)ortheestimatedbeamimplantationdepth(32Cl).Thesesimulationswereusedtoproduceatheoreticalrela-tiveeciencycurveasdescribedinChapter5,whichwasthencomparedtotheexperimentaleciencycurve.Weobservedthetwocurvestobeverysimilar,dieringlessthan10%atallenergiesandfrequentlybylessthan4%.Insituationswhereexperimentally-determinedecienciesareunavailable,theoretically-generatedcurvessuchasthecurveweproducedus-ingGeant4maybeusedinstead,butaswedidnotrequireextrapolationtohighergammaenergiesthan7MeVorextremelylowenergies,weoptedtouseonlytheexperimentalcurveinouranalysis.192Geant4InputMacroExample
Beforebeingabletorun,Geant4requirestheusertoinputaseriesofcommandsdeningtheparametersofthesimulation:whichdenedgeometricalbodiesshouldbe\turnedon"forthesimulation,whichparticlesshouldthecodeproduceandatwhatenergies,howmanyiterationsofthesimulationshouldbedone,etc.WeutilizedaGeant4\macro"lewiththesecommandspre-loaded.Then,atruntime,wewereablesimplytoinvokethemacroleandinputalltheparametersforthesimulationatonce.Belowisanexampleofasimplemacroledesignedtoproduceasimulationof1e61-MeVgamma-raysusingtheplasticscintillatorandthecloverdetectors.Itisworthnotingthattherearemanycommandswhichcanbeusedthatdonotappearinthissimpleexample;thereaderisdirectedtotheGeant4documentationforfurtheredication[123].###########################################################################THISTESTGENERATESGAMMARAYSATTHECENTREOFTHETARGET#
#WHICHISTHEORIGIN.#
###########################################################################/p26/analysis/dirname
/p26/analysis/filename1MeVTest
/p26/phys/SelectPhysicsLowEnergy_EM
###usetheLaBrcrystal?
#/p26/det/UseLaBrDetectorfalse
###usetheplasticscintillator?
/p26/det/UsePlasticScinttrue193/p26/det/UseSegafalse
/p26/det/UseClovertrue
/run/initialize
/run/verbose0
/event/verbose0
/tracking/verbose0
/grdm/verbose0
#/grdm/noVolumes
#/grdm/selectVolumeDetector
/grdm/allVolumes
###SOURCEINFO###
###Gammas###
/gps/source/intensity1
/gps/particlegamma
/gps/ang/typeiso
/gps/ene/mono1.000MeV
############
#NodaughterdecaysinanalogueMCmode
#/grdm/analogueMC1
/run/setSeed344
/run/beamOn1e6
#/vis/enable
#/vis/reviewKeptEvents
################################################################################################################################################194ShellModelCalculations:USDEResults
AsmentionedinChapters5and6,onesetoftheoreticaltoolsthatwasinvaluabletotheanalysisofE12028wereanumberofshell-modelcalculationsperformedbyB.A.BrownusingtheUSDinteraction,rstmentionedinChapter3.Sincethenatureofshellmodelcalculationshasalreadybeenmentionedandtheresultsofmanyofthecalculationshavealreadybeendiscussed,wepresentheresimplytheresultsofourUSDEcalculationsforboththebetafeedingsandgammabranchingsofthetheoreticalstates.ThesecalculationresultsarereprintedwithpermissionfromB.A.Brown,andaretruncatedafterthehighestenergyweusedinourexperimentalanalysis(7.455MeVinthecalculations,whichweinterpretedasthestateweobservedat7149keV).
BetaFeedingCalculations
cl310s_310sdpnf92epninputq-value=11.972MeVcalculatedt1/2=0.2154E+00sec
experimentalt1/2=0.0000E+00(+/-)0.0000E+00sec
ji,ti=1.51.5
nongamow-tellerbr=0.00%
ft=6177/[qf*(1.260**2.)b(gt)+b(f)]whereqf=0.60forthequenchingfactorsumb(gt)b(f)=2.758352.92792
centroid=7.24777
sumb(gt)/3|n-z|=0.30648forcl312y
th[br*b(gt)/3|n-z|]=0.01496
exp[br*b(gt)/3|n-z|]=0.00000sumsumjftfnfex(MeV)br(%)br(%)log(ft)qf*b(gt)qf*b(gt)log(fa)b(f)1/21/210.0007.964592.0365.5690.01050.01055.1370.00001953/21/211.2124.952687.0835.5320.01140.02194.8930.00003/21/211.2120.000287.08310.0070.00000.02194.8930.00005/21/212.27941.474145.6094.3690.16640.18844.6530.00001/21/223.2301.798143.8115.4930.01250.20094.4140.00005/21/223.3046.587937.2234.9090.04800.24894.3940.00003/21/223.6000.188937.0346.3700.00170.25054.3130.00003/21/223.6000.016037.0187.4420.00000.25054.3130.00025/21/234.2300.774036.2445.5750.01040.26094.1300.00003/21/234.3433.515832.7284.8830.05100.31194.0960.00003/21/234.3430.000832.7278.5420.00000.31194.0960.00003/21/244.6070.177632.5506.0960.00310.31504.0120.00003/21/244.6070.045732.5046.6860.00000.31504.0120.00135/21/244.8311.418831.0855.1210.02950.34443.9400.00001/21/234.9113.570527.5154.6930.07880.42333.9130.00005/21/255.1240.288427.2265.7130.00750.43083.8400.00001/21/245.3840.034127.1926.5490.00110.43193.7480.00003/21/255.7100.056627.1366.2060.00240.43433.6260.00003/21/255.7100.086327.0496.0230.00000.43433.6260.00595/21/265.7410.165926.8835.7270.00730.44163.6140.00003/21/266.1020.017026.8666.5710.00100.44273.4700.00003/21/266.1020.581226.2855.0390.00000.44273.4700.05653/21/276.3170.467125.8185.0430.03530.47793.3790.00003/21/276.31719.30036.5183.4270.00000.47793.3792.31305/21/276.3450.04356.4746.0610.00340.48133.3670.0000
3/21/286.3830.17816.2965.4330.01440.49573.3500.0000
3/21/286.3834.27202.0244.0530.00000.49573.3500.5469
1/21/256.4210.42391.6005.0400.03550.53123.3340.0000
1/21/267.0880.01621.5386.1400.00280.54073.0170.0000
5/21/2107.4550.48810.4754.4650.13330.78122.8200.0000------------------------------------------------------------------------------------------------------------------------------------------------------------------196GammaBranchingCalculations
--------------------------------------------------------------------------------gammadecayfors_310y.deoBRgreaterthan0.000
!modelspace=sdpn
!interaction=f9e2pn
!e_p=1.360e_n=0.450E2
!g_sp=5.000g_sn=-3.440M1spin
!g_lp=1.174g_ln=-0.110M1orbital
!g_pp=0.240g_pn=-0.160M1tensorEiJinitauT_(1/2)M1momentQmomentwidth(MeV)(psec)(psec)(u_N)(e^2fm^2)(eV)
0.0001/2+10.0000000.000000-0.3900.00-----0.0000E+00EfJfnfBRdelB(1)B(2)A_pA_n1.2123/2+13.0707482.1284891.177-8.44-----0.2143E-03EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+1100.00000.790.6376E-020.3925E+02-7.093-6.4022.2795/2+10.2959450.2051340.4732.66-----0.2223E-02EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+199.7069999.000.0000E+000.4472E+02-9.040-9.0821.2123/2+10.2931-0.210.4434E-030.2485E+000.930-0.0983.2301/2+20.0123900.0085880.1500.00-----0.5311E-01EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+190.34790.000.1229E+000.0000E+000.0000.000
1.2123/2+19.6339-0.310.4911E-010.1634E+023.5611.941
2.2795/2+10.0182999.000.0000E+000.1542E+023.1742.7501973.3045/2+20.1033410.0716310.9801.28-----0.6367E-02EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+115.3164999.000.0000E+000.3072E+01-3.4410.8571.2123/2+165.8987-0.470.3231E-010.2380E+026.6346.4982.2795/2+118.7849-0.220.9146E-010.6036E+02-10.797-9.6513.2301/2+20.0000999.000.0000E+000.7225E+001.0661.4043.4777/2+10.1938600.1343742.375-1.82-----0.3394E-02EfJfnfBRdelB(1)B(2)A_pA_n
1.2123/2+198.5153999.000.0000E+000.6956E+0213.28912.2522.2795/2+11.4744-0.970.1298E-020.1216E+025.4705.385
3.3045/2+20.0103-0.060.5827E-020.1150E+02-5.001-6.2003.6003/2+20.0125340.0086880.4436.35-----0.5250E-01EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+166.3579-0.510.5096E-010.1495E+02-4.154-4.6331.2123/2+133.57730.390.9728E-010.3642E+026.4807.236
2.2795/2+10.0425-2.350.1277E-030.5823E+01-2.219-4.0173.2301/2+20.01770.030.1581E-010.1463E+01-1.708-0.2133.3045/2+20.0047-0.070.8100E-020.5688E+01-2.342-3.5213.4777/2+10.0000999.000.0000E+000.7974E+000.9870.9854.2305/2+30.0077610.0053800.8860.70-----0.8478E-01EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+10.0137999.000.0000E+000.1061E-011.010-2.4921.2123/2+172.51070.260.1806E+000.1962E+025.3867.843
2.2795/2+127.3727-0.060.2688E+000.3565E+012.4372.9113.2301/2+20.0003999.000.0000E+000.2718E+00-0.513-1.2883.3045/2+20.00701.120.2865E-030.6064E+013.5442.693
3.4777/2+10.02020.740.2231E-020.3119E+02-7.439-7.9253.6003/2+20.0754-0.280.2049E-010.5698E+02-10.311-9.9271984.3433/2+30.0222380.0154141.0778.03-----0.2959E-01EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+157.4115-0.380.1565E-010.1715E+01-1.191-2.2221.2123/2+141.3628-0.180.3331E-010.1633E+011.1712.1422.2795/2+10.5977-1.210.7044E-030.3480E+01-2.262-1.4533.2301/2+20.59290.540.8473E-020.2911E+026.4184.586
3.3045/2+20.0096-5.910.6052E-050.2816E+01-2.257-0.6373.4777/2+10.0003999.000.0000E+000.2007E+000.830-0.5183.6003/2+20.0253-0.240.1492E-020.2210E+011.8870.902
4.2305/2+30.0000-1.590.2095E-050.5934E+01-2.740-2.5474.6073/2+40.0237710.0164770.802-7.92-----0.2768E-01EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+114.41992.010.6980E-030.1913E+012.184-0.4531.2123/2+165.4809-0.220.3806E-010.2395E+012.442-0.5032.2795/2+116.38370.080.3082E-010.5603E+00-0.925-0.5323.2301/2+20.46170.420.3582E-020.4873E+01-2.415-2.5123.3045/2+22.96370.140.3140E-010.5171E+01-2.505-2.5353.4777/2+10.0129999.000.0000E+000.2402E+01-1.951-0.9943.6003/2+20.23670.130.5446E-020.1308E+011.5250.473
4.2305/2+30.03940.060.1750E-010.6574E+012.7623.047
4.3433/2+30.00120.130.1498E-020.4827E+01-2.675-1.6814.8315/2+40.0186400.0129200.12814.52-----0.3530E-01EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+148.2236999.000.0000E+000.8023E+01-3.863-3.7421.2123/2+12.5966-5.290.5754E-040.1768E+01-2.026-1.1152.2795/2+122.70860.370.3674E-010.1080E+025.0642.583
3.2301/2+20.0282999.000.0000E+000.1172E+011.5981.0653.3045/2+222.9195-0.080.1948E+000.8566E+013.2995.9613.4777/2+11.1901-0.060.1455E-010.4385E+00-0.528-2.0073.6003/2+22.11620.170.3359E-010.9335E+01-4.686-2.4691994.2305/2+30.1762-0.130.2435E-010.1536E+02-5.408-4.9914.3433/2+30.03930.080.1023E-010.3498E+01-2.565-2.4284.6073/2+40.0001-0.050.2398E-030.1557E+00-0.653-0.1754.7387/2+20.0016-0.010.6236E-010.4783E+01-2.045-5.7244.9111/2+30.0017890.001240-0.1840.00-----0.3679E+00EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+158.04740.000.1556E+000.0000E+000.0000.000
1.2123/2+141.00090.200.2477E+000.9986E+013.0030.856
2.2795/2+10.0280999.000.0000E+000.1010E+011.0120.1013.2301/2+20.88250.000.5900E-010.0000E+000.0000.000
3.3045/2+20.0001999.000.0000E+000.5317E-010.1710.2073.6003/2+20.0195-0.350.2442E-020.2536E+011.3141.033
4.2305/2+30.0000999.000.0000E+000.4816E+000.6100.3394.3433/2+30.00370.090.6407E-020.2296E+01-1.461-0.3464.6073/2+40.0179-0.020.2026E+000.7231E+01-1.952-2.5514.8315/2+40.0000999.000.0000E+000.4802E+000.6160.3155.1245/2+50.0468100.0324461.2940.20-----0.1406E-01EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+10.3076999.000.0000E+000.1518E-01-0.0270.7541.2123/2+123.3452-0.060.4715E-020.1546E-01-0.0780.9112.2795/2+122.7511-0.240.1132E-010.1189E+010.8583.3433.2301/2+20.0890999.000.0000E+000.6364E+00-0.864-1.7333.3045/2+233.41920.090.6676E-010.2191E+012.1481.567
3.4777/2+118.7246-0.140.4990E-010.5042E+01-3.831-0.6453.6003/2+20.3078-0.430.8874E-030.1037E+01-1.9850.4584.2305/2+30.3298-0.010.5600E-020.4046E-020.270-1.1624.3433/2+30.05570.680.9685E-030.1062E+02-4.298-4.7474.6073/2+40.0419-0.590.2722E-020.5163E+02-10.269-8.0864.7387/2+20.62770.010.1324E+000.4879E+00-0.725-1.6134.8315/2+40.0005-0.550.1690E-030.8518E+01-4.034-3.6954.9111/2+30.0000999.000.0000E+000.2072E+01-2.7160.3732005.3841/2+40.0023960.001661-0.4270.00-----0.2746E+00EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+193.73990.000.1424E+000.0000E+000.0000.000
1.2123/2+13.930024.880.2069E-040.1057E+02-2.412-2.9282.2795/2+10.1346999.000.0000E+000.1588E+01-0.728-1.7593.2301/2+20.65970.000.1565E-010.0000E+000.0000.000
3.3045/2+20.0425999.000.0000E+000.3721E+011.7680.7183.6003/2+21.2192-0.090.5047E-010.1944E+01-0.901-1.6604.2305/2+30.0011999.000.0000E+000.1878E+01-1.360-0.1964.3433/2+30.17600.110.3653E-010.5872E+012.2230.899
4.6073/2+40.0701-0.080.3517E-010.5872E+01-1.512-3.0464.8315/2+40.0000999.000.0000E+000.1636E+01-1.002-0.9944.9111/2+30.02680.000.6007E-010.0000E+000.0000.000
5.1245/2+50.0000999.000.0000E+000.3937E+011.3282.2225.7103/2+50.0036600.0025371.2466.76-----0.1798E+00EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+177.8778-0.010.6490E-010.6053E-02-0.114-0.0021.2123/2+115.2811-0.020.2605E-010.4489E-02-0.4361.0202.2795/2+10.03330.670.8864E-040.4814E-01-0.1981.575
3.2301/2+24.6813-0.170.4633E-010.3015E+012.1481.226
3.3045/2+20.05335.020.2268E-040.1416E+01-2.1111.092
3.4777/2+10.0205999.000.0000E+000.8236E+000.4452.6873.6003/2+20.59330.040.9781E-020.6228E-010.524-0.473
4.2305/2+30.0329-0.750.1007E-020.3717E+01-2.082-2.2764.3433/2+30.2281-0.020.1385E-010.3561E-010.1300.445
4.6073/2+40.90220.050.1040E+000.3312E+01-1.474-3.6364.7387/2+20.0003999.000.0000E+000.6716E+00-0.516-2.0834.8315/2+40.1082-0.050.2465E-010.1350E+01-1.169-1.6324.9111/2+30.0625-0.120.1874E-010.6010E+01-2.511-3.3065.1245/2+50.11780.030.9078E-010.3122E+011.9451.975
5.3841/2+40.0075-0.070.3340E-010.2166E+02-4.901-5.8725.4817/2+30.0000999.000.0000E+000.7320E+012.8993.2632015.7415/2+60.0165480.0114700.94011.29-----0.3976E-01EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+15.8503999.000.0000E+000.4626E+00-1.7341.5371.2123/2+112.7606-0.550.3621E-020.7661E+001.5740.0082.2795/2+149.21950.300.3727E-010.4128E+013.1101.661
3.2301/2+24.3081999.000.0000E+000.2128E+025.9397.1603.3045/2+20.6739-5.510.5093E-040.3743E+01-2.982-1.5203.4777/2+17.8246-0.020.2314E-010.1727E-010.578-1.0313.6003/2+28.55760.100.2961E-010.1005E+01-1.577-0.6894.2305/2+31.5854-0.050.1573E-010.2908E+00-0.842-0.3924.3433/2+35.8696-0.150.7218E-010.1145E+02-4.942-3.4794.6073/2+40.00153.850.2255E-050.3725E+000.9880.335
4.7387/2+20.8737-0.100.2944E-010.3974E+012.5423.168
4.8315/2+42.4278-0.020.1105E+000.7877E+00-0.955-1.9434.9111/2+30.0022999.000.0000E+000.2785E+012.1852.4805.1245/2+50.0383-0.070.5569E-020.1087E+01-1.331-1.6535.3239/2+10.0000999.000.0000E+000.1476E+01-1.883-0.9215.3841/2+40.0001999.000.0000E+000.5171E+013.2892.4365.4817/2+30.0065-0.050.1267E-010.5671E+01-3.073-3.6755.7103/2+50.00000.000.3862E-010.5515E-020.241-0.3236.1023/2+60.0038800.0026891.049-5.25-----0.1696E+00EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+134.39500.370.1949E-010.1031E+011.1201.129
1.2123/2+10.8614-0.200.1037E-020.2457E-01-0.7711.6342.2795/2+142.0900-0.220.1052E+000.5018E+01-2.804-1.4833.2301/2+20.64730.230.3795E-020.3582E+000.2551.888
3.3045/2+214.14170.120.9321E-010.2385E+011.5992.032
3.4777/2+10.5454999.000.0000E+000.9202E+013.5202.8423.6003/2+25.6506-0.290.4877E-010.9269E+013.2513.706
4.2305/2+30.04040.810.5457E-030.1456E+011.4500.980
4.3433/2+30.7172-0.140.1891E-010.1782E+01-1.610-1.0694.6073/2+40.02950.080.1284E-020.5818E-01-0.3590.013
4.7387/2+20.0063999.000.0000E+000.2802E+011.9621.5122024.8315/2+40.01760.320.1137E-020.1051E+011.3630.436
4.9111/2+30.6646-0.070.5726E-010.3253E+011.7682.672
5.1245/2+50.12550.020.1963E-010.1043E+00-0.0391.553
5.3841/2+40.0334-0.260.1239E-010.2271E+02-5.276-5.2375.4817/2+30.0000999.000.0000E+000.1313E+00-0.5470.0435.7103/2+50.03200.060.7748E-010.2254E+025.2985.089
5.7415/2+60.00230.000.7048E-020.7448E-02-0.4591.002
5.7857/2+40.0000999.000.0000E+000.1892E+01-1.366-1.9845.9837/2+50.0000999.000.0000E+000.2404E+000.4960.6816.3173/2+70.0012190.0008451.029-8.66-----0.5398E+00EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+111.1342-0.450.1711E-010.1250E+01-2.3302.0721.2123/2+115.0990-0.170.5144E-010.7948E+00-1.8001.4802.2795/2+159.7927-0.060.4219E+000.1132E+011.2570.9303.2301/2+21.20880.020.1914E-010.7006E-02-0.4160.884
3.3045/2+28.53430.080.1444E+000.1549E+01-1.344-1.4693.4777/2+10.0035999.000.0000E+000.1258E+00-0.286-0.7123.6003/2+20.0868-0.220.1923E-020.1802E+00-0.496-0.3874.2305/2+30.1625-0.040.8317E-020.4435E-010.184-1.4924.3433/2+32.67420.030.1618E+000.5307E+00-0.695-1.1374.6073/2+40.00880.520.6431E-030.8519E+00-1.5020.437
4.7387/2+20.0010999.000.0000E+000.6488E+00-0.158-3.1024.8315/2+40.2920-0.040.4139E-010.4796E+00-0.558-1.3914.9111/2+30.84780.020.1421E+000.5446E+000.8170.812
5.1245/2+50.11930.010.3272E-010.1480E-01-0.024-0.4675.3841/2+40.0136-0.040.7806E-020.2179E+000.737-0.1545.4817/2+30.0001999.000.0000E+000.1160E+011.817-0.7065.7103/2+50.00980.030.2045E-010.7208E+001.542-0.887
5.7415/2+60.01130.000.2767E-010.6223E-030.212-0.753
5.7857/2+40.0000999.000.0000E+000.2006E+00-0.8310.5225.9837/2+50.0000999.000.0000E+000.3347E-020.199-0.8606.1023/2+60.0002-0.020.8510E-020.9254E+001.438-0.0706.3455/2+70.0044770.0031030.799-2.79-----0.1470E+00203EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+17.9359999.000.0000E+000.1407E+01-1.848-0.8701.2123/2+123.08140.090.2148E-010.9398E-010.633-0.2442.2795/2+118.0718-0.120.3363E-010.4155E+00-1.029-0.3993.2301/2+20.0575999.000.0000E+000.3572E+00-0.162-2.7643.3045/2+20.95380.640.3060E-020.1931E+01-2.178-0.9833.4777/2+141.65680.090.2223E+000.2960E+012.8710.689
3.6003/2+20.2610-0.200.1539E-020.1179E+00-0.8010.5524.2305/2+32.0329-0.110.2693E-010.1051E+011.4281.265
4.3433/2+32.0244-0.120.3157E-010.1583E+012.584-0.9614.6073/2+40.94770.050.2284E-010.2882E+000.9020.197
4.7387/2+22.54380.030.7768E-010.4862E+00-1.088-0.5074.8315/2+40.01590.540.4527E-030.8125E+001.4300.585
4.9111/2+30.1020999.000.0000E+000.3065E+027.3277.9945.1245/2+50.05180.700.2416E-020.1153E+02-4.413-5.1485.3239/2+10.0002999.000.0000E+000.3427E+00-1.1820.3855.3841/2+40.0001999.000.0000E+000.2162E+000.8130.0755.4817/2+30.0169-0.080.3295E-020.3841E+000.9300.563
5.7103/2+50.06890.010.3416E-010.2031E+000.5710.726
5.7415/2+60.0494-0.020.2844E-010.6481E+001.0771.129
5.7857/2+40.11780.020.8511E-010.1378E+011.8880.684
5.8229/2+20.0000999.000.0000E+000.4423E+001.215-0.0525.9837/2+50.00980.010.2632E-010.5551E+001.377-0.106
6.0339/2+30.0000999.000.0000E+000.1961E-010.0690.5536.1023/2+60.0001-0.030.7623E-030.1554E+000.2331.443
6.3173/2+70.00000.000.7526E-020.3309E-010.2970.0926.3833/2+80.0018790.0013030.722-7.91-----0.3501E+00EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+115.9169-0.250.1738E-010.3931E+001.342-1.2681.2123/2+16.2754-0.100.1357E-010.7829E-010.901-1.4792.2795/2+150.29960.090.2182E+000.1504E+011.1451.991
3.2301/2+27.50020.050.7217E-010.2159E+000.3231.090
3.3045/2+213.5578-0.180.1362E+000.6350E+01-3.270-1.3192043.4777/2+10.0212999.000.0000E+000.4436E+001.609-1.9043.6003/2+20.4143-0.230.5513E-020.5498E+001.504-1.2504.2305/2+30.46270.120.1382E-010.6076E+00-0.968-0.5384.3433/2+33.3777-0.090.1192E+000.3656E+01-1.899-2.7574.6073/2+40.3841-0.090.2056E-010.7285E+00-0.840-1.2534.7387/2+20.0324999.000.0000E+000.1167E+02-4.786-0.7164.8315/2+40.32000.140.2541E-010.2776E+01-2.008-1.3354.9111/2+31.3862-0.080.1305E+000.5691E+012.4583.172
5.1245/2+50.0146-0.160.2153E-020.5256E+00-0.910-0.4725.3841/2+40.0127-0.130.3788E-020.9063E+00-1.6430.7345.4817/2+30.0000999.000.0000E+000.1172E-01-0.5622.1795.7103/2+50.02330.030.2307E-010.5768E+000.4472.025
5.7415/2+60.0005-0.130.5525E-030.3469E+00-0.661-0.6205.7857/2+40.0000999.000.0000E+000.8658E-020.340-1.4415.9837/2+50.0000999.000.0000E+000.2836E+00-0.558-0.6816.1023/2+60.00050.040.7379E-020.2104E+011.3312.425
6.3173/2+70.00000.000.7683E-020.4167E+001.075-0.380
6.3455/2+70.00000.000.2678E-010.5134E-02-0.8862.3606.4211/2+50.0040320.002795-0.9930.00-----0.1632E+00EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+187.62920.000.4663E-010.0000E+000.0000.000
1.2123/2+10.5384-0.860.3078E-030.1212E+00-0.9271.7072.2795/2+15.1445999.000.0000E+000.8541E+012.7031.0163.2301/2+21.87750.000.8141E-020.0000E+000.0000.000
3.3045/2+22.3516999.000.0000E+000.1618E+02-3.183-3.0223.6003/2+21.8544-0.230.1106E-010.1050E+010.7181.051
4.2305/2+30.1542999.000.0000E+000.6181E+011.5833.0304.3433/2+30.1744-0.380.2387E-020.1167E+010.8070.958
4.6073/2+40.1086-0.100.2537E-020.1135E+00-0.175-0.5314.8315/2+40.0332999.000.0000E+000.6610E+01-2.460-0.6464.9111/2+30.00000.000.1356E-050.0000E+000.0000.000
5.1245/2+50.0039999.000.0000E+000.2167E+011.0411.4815.3841/2+40.03130.000.3949E-020.0000E+000.0000.000
5.7103/2+50.0788-0.130.3041E-010.1417E+02-2.726-3.5942055.7415/2+60.0002999.000.0000E+000.2782E+011.3131.2746.1023/2+60.0193-0.030.8352E-010.8765E+012.2242.582
6.3173/2+70.00050.010.5756E-010.7924E+01-1.678-3.7746.3455/2+70.0000999.000.0000E+000.5638E-030.172-0.5956.3833/2+80.00010.000.2476E+000.3479E+02-4.595-4.6497.0881/2+60.0003070.0002120.9790.00-----0.2147E+01EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+196.66280.000.5030E+000.0000E+000.0000.000
1.2123/2+10.21460.750.1256E-020.2929E+00-0.7770.649
2.2795/2+10.2448999.000.0000E+000.2533E+012.147-1.4863.2301/2+20.51210.000.1653E-010.0000E+000.0000.000
3.3045/2+20.0036999.000.0000E+000.1231E+000.867-1.5193.6003/2+20.9135-0.150.3900E-010.1048E+011.143-0.2354.2305/2+30.0017999.000.0000E+000.2317E+00-0.6430.4304.3433/2+30.12800.250.1079E-010.1291E+010.7571.284
4.6073/2+40.1666-0.180.1958E-010.1477E+01-1.5670.9184.8315/2+40.0030999.000.0000E+000.1384E+010.6551.7174.9111/2+30.78350.000.1407E+000.0000E+000.0000.000
5.1245/2+50.0000999.000.0000E+000.3903E-01-0.078-0.3875.3841/2+40.03060.000.1148E-010.0000E+000.0000.000
5.7103/2+50.24880.020.1762E+000.3128E+000.736-0.466
5.7415/2+60.0006999.000.0000E+000.3548E+011.8620.2926.1023/2+60.03320.020.6408E-010.2864E+000.1241.306
6.3173/2+70.02660.040.1075E+000.3798E+011.4051.877
6.3455/2+70.0000999.000.0000E+000.1738E+000.560-0.3826.3833/2+80.0145-0.050.7644E-010.4873E+01-2.4630.5056.4211/2+50.01140.000.7110E-010.0000E+000.0000.000
6.7025/2+80.0000999.000.0000E+000.1762E-010.0620.2297.0105/2+90.0000999.000.0000E+000.1157E+010.8850.7047.4555/2+100.0047280.0032771.533-2.61-----0.1392E+00EfJfnfBRdelB(1)B(2)A_pA_n
0.0001/2+10.4127999.000.0000E+000.3093E-010.515-0.5992061.2123/2+133.10500.170.1586E-010.1778E+000.792-0.0962.2795/2+10.5511-0.310.4344E-030.2310E-010.2170.172
3.2301/2+20.3433999.000.0000E+000.4401E+000.9350.7863.3045/2+212.2493-0.130.2021E-010.3056E+00-0.768-0.6883.4777/2+10.31023.400.4704E-040.4948E+00-1.7531.469
3.6003/2+21.2961-0.810.1644E-020.1038E+01-1.692-0.4334.2305/2+36.1183-0.070.2180E-010.1499E+000.3970.909
4.3433/2+30.0528-0.050.2099E-030.6801E-03-0.0910.1334.6073/2+429.1054-0.200.1453E+000.1068E+02-3.974-5.7784.7387/2+20.5800-1.160.1485E-020.3870E+01-2.870-2.0354.8315/2+42.2139-0.020.1471E-010.1889E-01-0.0010.7524.9111/2+30.2467999.000.0000E+000.3995E+013.0461.6755.1245/2+50.23873.660.1574E-030.5570E+013.1743.255
5.3239/2+10.0002999.000.0000E+000.9584E-020.0270.4525.3841/2+40.0048999.000.0000E+000.2162E+000.2611.7445.4817/2+35.95690.100.9213E-010.3320E+012.0423.749
5.7103/2+51.8912-0.120.4212E-010.3020E+01-2.477-1.9755.7415/2+60.2458-0.150.5741E-020.6010E+001.675-0.8415.7857/2+40.02090.610.3929E-030.7590E+00-0.995-1.7365.8229/2+20.0029999.000.0000E+000.4283E+001.0320.4455.9837/2+51.0836-0.040.4074E-010.4982E+001.2250.141
6.0339/2+30.0095999.000.0000E+000.2821E+012.4241.8176.1023/2+60.11410.690.3760E-020.1394E+024.9745.289
6.3173/2+73.1906-0.020.2600E+000.8348E+00-1.485-0.4866.3455/2+70.27380.210.2305E-010.1162E+024.5494.805
6.3833/2+80.2339-0.030.2278E-010.3146E+00-0.694-0.9576.4211/2+50.0088999.000.0000E+000.1287E+024.7225.2626.4877/2+60.0543-0.050.7176E-020.2360E+00-0.558-0.9596.7025/2+80.05550.030.1562E-010.2489E+00-0.689-0.6337.0105/2+90.0009-0.020.1267E-020.4298E-010.3380.106
7.0459/2+40.0000999.000.0000E+000.1170E+01-1.487-1.3947.0881/2+60.0000999.000.0000E+000.9744E-010.4230.4197.1201/2+70.0000999.000.0000E+000.4755E+001.0320.6347.1573/2+90.02570.020.1166E+000.1078E+02-4.061-5.5957.1667/2+70.00000.480.1354E-040.5257E+00-0.817-1.4797.3193/2+100.00310.010.1474E+000.1191E+024.3055.778207------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------208APPENDIXB
DetectorDataSheets
Inordertofacilitteuseofthisthesisasareferencetool,wehaveincludedherethedatasheetsforanumberofdetectorsspecictoourexperimentalsetup.WhilewereferthereadertotheappropriateliteraturefordevicesliketheA1900[65]andtheRFFS[66],wehavehereincludeddatasheetsforthesiliconPINdetectorsusedintheexperiment,theplasticscintillator,anditsattachedphotomultipliertube.209210FigureB.1:AschematicofthesiliconPINdetectorsusedforE12028.210    ELJEN TECHNOLOGY  Tel: (325) 235-4276 or (888) 800-8771  PO Box 870, 300 Crane Street Fax: (325) 235-0701  Sweetwater TX 79556 USA  Website: www.eljentechnology.com  EJ-200 EMISSION SPECTRUM0.00.20.40.60.81.0380400420440460480500WAVELENGTH  (nm)AMPLITUDE EJ-200 PLASTIC SCINTILLATOR  
This plastic scintillator combines the two important properties of long optical attenuation length and fast 
timing and is therefore particularly useful for time-of-flight systems using scintillators greater than one 
meter long.  Typical measurements of 4 meter optical attenuation length are achieved in strips of cast 
sheet in which a representative size is 2 cm x 20 cm x 300 cm. 
 
The combination of long attenuation length, high light output and an emission spectrum well matched to 
the common photomultipliers recommends EJ-200 as the detector of choice for many industrial 
applications such as gauging and environmental protection where high sensitivity of signal uniformity are 
critical operating requirements. 
 Physical and Scintillation Constants: Light Output, % Anthracene ..................................... 64 Scintillation Efficiency, photons/1 MeV e- ................. 10,000 Wavelength of Max. Emission, nm .......................... 425 Rise Time, ns ........................................................... 0.9 Decay Time, ns ........................................................ 2.1 Pulse Width, FWHM, ns ........................................... ~2.5 No. of H Atoms per cm3, x 1022 ................................ 5.17 No. of C Atoms per cm3, x 1022 ................................ 4.69 No. of Electrons per cm3, x 1023 ............................... 3.33 Density, g/cc: ........................................................... 1.023  
Polymer Base: ––––. Polyvinyltoluene Light Output vs. Temperature: Refractive Index: –––.1.58 At +60oC, L.O. = 95% of that at +20oC Vapor Pressure: –––.. Is vacuum-compatible No change from +20oC to -60oC Coefficient of Linear  Expansion: –––––– 7.8 x 10-5 below +67qC   
Chemical Compatibility:  Is attacked by aromatic solvents, chlorinated solvents, ketones, solvent bonding cements, etc.  It is stable in water, dilute acids and alkalis, lower alcohols and silicone greases.  
It is safe to use most epoxies and fisuper gluesfl with EJ-200.  FigureB.2:TheHammamatsuEJ200PlasticScintillatorDataSheet.211212P.O. Box 1433980 CC BunnikThe NetherlandsTel.  31 (0)30 657 0312Fax.  31 (0)30 656 7563Radiation Detectors & CrystalsSCIONIX HOLLAND BVSCIONIX
˘˜˛ 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