MEASUREMENTSOFCHARGERADIIOFNEUTRON-DEFICIENTCALCIUM USINGCOLLINEARLASERSPECTROSCOPYATBECOLA By AndrewJacobMiller ADISSERTATION Submittedto MichiganStateUniversity inpartialentoftherequirements forthedegreeof Physics-DoctorofPhilosophy 2019 ABSTRACT MEASUREMENTSOFCHARGERADIIOFNEUTRON-DEFICIENT CALCIUMUSINGCOLLINEARLASERSPECTROSCOPYATBECOLA By AndrewJacobMiller Calciumisauniqueelement,possessingtwostabledoubly-magicnuclei.Cachargeradii havebeenachallengefornucleartheoriestoreproduceduetotheirunusualbehavior.The chainofstableCaisotopesspansfrom 40 Cato 48 Ca,andalthoughtheseby8neutrons, thenuclearchargeradiiarenearlyidentical.Inadditiontothispeculiarity,thereisapro- nouncedodd-evenstaggeringinthechargeradii,andanunexpectedincreasemovingtoward theneutron-richisotopes.Thisworkrepresentstheinvestigationofchargeradiiofthese tcalciumisotopes.Collinearlaserspectroscopy(CLS)hasbeenshownto beavaluabletooltoinvestigatethefundamentalpropertiesofnuclei,suchasnuclearspin, chargeradii,magneticdipolemoments,andspectroscopicelectricquadrupolemoments.By probingtheshiftofthehypspectrabetweenisotopes,thenuclearchargeradiuscanbe extracted.Inthecaseofnon-zeronuclearandatomicspins,thenuclearmagneticdipoleand electricquadrupolemomentcanalsobeobtainedfromthehypsplitting.Inthepresent work,CLSwassuccessfullyperformedon 36 ; 37 ; 38 ; 39 CaattheBEamCOolerandLAserspec- troscopy(BECOLA)facilityattheNationalSuperconductingCyclotronLaboratory(NSCL) atMichiganStateUniversity.Improvementstotheionsourceandlasersystem,anda newphotondetectionsystemmadeitpossibletostudy 36 Cawithonly ˘ 50ions = sdespitethe challengeoftheshort102mshalf-life,themostsensitivemeasurementatBECOLAtodate. Firstmeasurementsweremadeofthechargeradiiof 36 38 Ca,andthechargeradiusof 39 Ca isinagreementwithpreviousresults,whilereducinguncertaintybyafactorof3.Thecharge radiioftheseisotopes,especiallytheweakly-bound 36 Cahaverequiredanadvancedmodel ofchargeradiitounderstandtheectsofloosely-boundprotonsonthepairinginteraction, inthevicinityoftheprotondrip-line( 35 Caisthelastboundcalciumisotope).Despitethe factthattheprotonsinthesetsystemsareweakly-bound,thechargeradiiof 36 38 Cawerefoundtobeverycompactandtlysmallerthanpredictedbyprevious theories.WithNuclearDensityFunctionalTheory(DFT)usinganovelFayansinteraction whichcontainsnovelnuclear-densitygradientdependentsurfaceandpairingterms,thecom- pletechainofchargeradiiof 36 52 Cahavenowbeenreproducedandunderstood,andthe DFTmodelbestdescribingthesenewmeasurementsontheneutrontsidealsohas improvedagreementwithdataontheneutron-richside. Copyrightby ANDREWJACOBMILLER 2019 Dedicatedtomywife,ElizabethClaireMiller v ACKNOWLEDGMENTS Thisdissertationcouldnotbecompletewithoutthankingthemanystudents,scientists, andfamilymemberswhohavehelpedmealongtheway. IwouldliketothankallthemembersoftheBECOLAgroup,especiallymyadvisorKei Minamisono.Thankyouforyourexampleinthelab,promotingsafety,invitingquestions, andbeingopentonewideasandimprovements,evenduringonlineexperiments.Imust thankDominicRossiandCalebRyderwhohelpedintroducemetotheBECOLAsystem. Eachofthemspentmanyhourshelpingmealignthelasersystemandlearntoperform thedataanalysis.OthermembersoftheBECOLAgroup,includingDavidGarand,Jeremy Lantis,SkyyPineda,andRobertPowel,thankyouallformakingourtimeworkingtogether bothenjoyableandproductive. Thankstotheexperimenterswhotraveledfromnearandfartohelp,especiallythe spokespersonofthisexperiment,AndrewKlose.Iamalsoparticularlygratefultothose whohelpeddesign,manufacture,andinstallthenewphotondetectionsystemusedinthis experiment:BernhardMaaˇ,Wilfriedauser,andJacobWatkins.Thisexperiment wouldnothavebeenpossiblewithoutthehelpofChandanaSumithrarachchiandtheothers inthegascellgroupworkingtoprovidethebestbeamspossibletoBECOLA.Otherswho helpedmakethisexperimentasuccessareYuanLiu,FelixSommer,andAndreaofer. Withouttheinputfromtheorists,thisworkwouldneverhavehadsuchanimpact,so IwouldliketothankWitekNazarewiczandPaul-GerhardReinhardfortheirinvaluable contributionstothediscussionandinterpretationofthechargeradiiresultsandAlexBrown forhiscontributionsregardingthenuclearmoments. Thankyoutothosewhospenttheirtimeandenergyservingonmycommittee:Kei vi Minamisono,MattComstock,PaulMantica,WitekNazarewicz,andJaideepSingh.Your commentsandquestionshelpedmeprogresssmoothlythroughmytimehere,andyour interestinmyresearchandmysuccessaregreatlyappreciated. Iwouldalsoliketothankmyfellowstudents,especiallythosewhospentlonghours workingthroughhomeworkwithme,playingvideogames,andthrowingbirthdaypartiesin ourLisaCarpenter,DennisForen,AliciaPalmisano,RoyReady,ZacTilocco,and manyotherswerealwayswillingtojoinmeinstudyingorrelaxing. ThankyoutothemanyprofessorsIhadtheprivilegeoflearningfrom,bothatMichigan StateUniversityandmy almamater ,AbileneChristianUniversity. Inadditiontomyacademiccomradesandmentors,Imustalsothankmyfamilyfor theirsupport.Thankyoutomyparentsfordedicatingsomuchoftheirtimeandto mybrothersandmethroughhomeschooling.Althoughmanypeoplereminisceabouttheir favoritegradeschoolteacher,IamblessedtosaythatIhadmyfavoriteteacherswithme everyyear,frompreschoolthroughhighschool,nottomentionthephenomenaltrips wewenton.YourhardworkhasallowedmetoreachwhereIamtodaybothacademically andspiritually,andIcannotthankyouenoughforallofyourcontinuedsacrThank youtomybrothersforyourencouragementandsupport,evenasyoueachcontinuewith yourownamazingachievementsinthearts. Mostimportantly,Ithankmywifeandson.Thankyouforfollowingmeacrossthe countryforschoolandwork,andfordemonstratinghowexcitinglearningnewthingscan be.Thankyouforgoingthroughlifebymyside,encouragingme,andkeepingmefocused onwhatreallymatters.Withoutyourloveandsupport,thiswouldnothavebeenpossible. Therearemanyothers,includingfamilymembers,fellowstudents,andteacherswhoI don'thavespacetothankherebyname.Thankyouallforyoursupport. vii TABLEOFCONTENTS LISTOFTABLES .................................... x LISTOFFIGURES ................................... xiii KEYTOABBREVIATIONS ............................. xxiv Chapter1Introduction ............................... 1 1.1Chargeradii....................................2 1.2Calcium......................................5 1.3CollinearLaserSpectroscopyatBECOLA...................8 1.4Dissertationoverview...............................9 Chapter2Experiment ................................ 11 2.1Basicprinciples..................................11 2.1.1Atomictransition.............................14 2.1.2Lasersystem...............................15 2.1.3Hypinteraction...........................17 2.1.4Bunchedbeambackgroundsuppression.................20 2.1.5Doppler-shift...............................23 2.2Isotopeproduction................................28 2.2.1Rareisotopes...............................28 2.3Systemupgrades.................................31 2.3.1Referenceisotopes............................32 2.3.1.1PIGsourceupgrades......................33 2.3.1.2Isotopiclimitations.......................34 2.3.2Photondetectionsystem.........................35 2.3.2.1Motivation...........................36 2.3.2.2PerformanceStudies......................37 Chapter3Analysis .................................. 43 3.1Calibration....................................43 3.2HypSpectra.................................45 3.2.1FittingFunction.............................45 3.2.2FittingProcedure.............................47 3.2.2.1Hypcouplingconstants.................47 3.2.3Results...................................48 3.3NuclearMoments.................................49 3.4Chargeradii....................................53 3.4.1Isotopeshift................................53 3.4.2Kingplot.................................55 viii 3.4.3Results...................................58 Chapter4Discussion ................................. 61 4.1Modelingchargeradii...............................61 4.2Empiricalextrapolation.............................68 Chapter5Conclusion ................................ 72 APPENDICES ...................................... 75 APPENDIXAHardware ............................ 76 APPENDIXBSoftwaretools ......................... 107 APPENDIXCPersonalcontributions .................... 112 REFERENCES ...................................... 116 ix LISTOFTABLES Table2.1:Thesetlaserfrequencyandscanningvoltageusedforeachmeasure- ment.ThesetlaserfrequencyisthewavenumberoftheTi:Sapphire laserpriortothefrequencydoubler,whilethescanningregionin- dicatestherangeofthescanningpotentialappliedtothedetection regionasthespectrawererecorded..................29 Table2.2:Thehalf-life( T 1 = 2 ),bunchingperiod( T b ),ratesatthecooler/buncher fortherareisotopes,andthetotalmeasurementtimeusedforeach isotope.Thebunchingrepetitionratesfor 36 ; 37 Cawerechosenbased ontheshorthalf-lives,whilethebunchingperiodsof 38 ; 39 Cawere bythehighionratesandthespace-chargelimitationsofthe trap.Becausethereferenceisotopes 40 ; 44 Cacanbeproducedwith highioncurrents,theyhavetlimitations,whicharediscussed inSection2.3.1.2.............................31 Table3.1:Theobtainedhypcouplingconstantsof 37 ; 39 Caforthe4 s 2 S 1 = 2 and4 s 2 P 3 = 2 states.Thestatisticalandsystematicuncertainties arereportedintheandsecondparentheses,respectively.The systematicuncertaintycomesfromthehighvoltagecalibration,and alsothedeviationofthe A hf -factorratiomeasuredfor 39 Cafrom theliteraturevaluefrom 43 Ca.Thisunexplaineddiscrepancyisthe dominantsourceofsystematicerror...................49 Table3.2:Thenuclearmomentsof 37 ; 39 Caobtainedinthiswork.Thetotal uncertaintyisreportedintheparentheses. Q ( 39 Ca)isinagree- mentwiththepreviousexperimentalvalueof3 : 6(7) e 2 fm 2 [56],al- thoughthesignof Q ( 39 Ca)isexperimentallydeterminedhereforthe time. ( 39 Ca)reportedheredoesnottakeintoaccountthehf anomaly,whichthepreviousandmoreprecise -NMRmeasurement, +1 : 0217(1) N [57],isinsensitiveto.Formorediscussionofthese results,seeref.[37]............................53 Table3.3:Isotopeshifts,deducedtialmean-squarechargeradii,andab- solutechargeradii.Thenumbersinparenthesesarethestatistical andsystematicuncertainties,respectively.Thetialmscharge radiusof 39 Carelativeto 40 Caisconsistentwiththepreviousvalue of 0 : 127(20)fm 2 [64].Rmschargeradiiwereobtainedusingthe chargeradiusof 40 Ca3 : 4776(19)fmfromref.[4]...........59 x Table3.4:Sourcesofuncertaintyinthentialmschargeradii, h r 2 i 40 ;A . Statisticaluncertaintycomesfromtheofthespectra.Thecalibra- tionuncertaintyisfromtheintheextractedcalibration voltage,describedinSection3.1.Atomicfactoruncertaintyisdueto thetotaluncertaintyfromtheatomicfactors, k and F asnotedin Equations3.13and3.14.........................59 TableA.1:ThedesignofthenewPDSwascomparedtotheexistingsystemus- ingsimulationswithinFRED,aray-tracingtool.Signalphotonswere simulatedusingraysemittedinalldirectionsfromrandompoints withinthevolumeofacylinder,approximatingtheoverlappinglaser andionbeam.Backgroundphotonswereapproximatedbyraysorig- inatingfromrandompointswithinacircle,andemanatinginaran- domdirectioninthehemispherefacingthePDS.Fortheoldsystem, thisbackgroundsourcewassimulatedintwotpositions(see SectionA.1.1)leadingtotherangesofbackgroundacceptanceand SNR.BothsystemsweresimulatedwithperfectThefrac- tionofsignalandbackgroundreachingthedetectorwerecompared toseveralactualmeasurementsusingreferenceionsandtheexisting system,inordertodeterminetheappropriateratioofincomingsig- nalandbackgroundphotons.TheSNRinthetableareextrapolated fromthesimulationsandthesereferencemeasurements,assigningthe measurementfromtheexistingsystemanSNRof1.Duetothefact thatthe 56 Femeasurementrequiresthecharge-exchangecell,there isanadditionalbackgroundsourcecomingfromtheionbeamitself, whichiscollectedinthesamewayasthesignal.Becauseofthis,the strategyemployedbythenewsystem,tominimizenoiseevenatthe costofsomesignal,isnotase,duetothefactthatthenoise fromthecharge-exchangeprocesscannotbereducedthisway....86 xi TableA.2:ThesechannelsareusedtocontroltheDVMs.Identicalchannelsexist for MTER N0002 . V RD isthechannelwherethevoltageisactually displayed. POLL TXT istheSCPImessagesenttopollthedevice. Usingacommandotherthantheoneshownheremayinadvertently resetthemeasurementsettingsonthedevice. V RD.SCAN isusedto determinehowoftenthevoltageisread. RST CMD mustbesetto 1 eachtimetheDVMispowercycled,inordertorunthe RST TXT commands.The SCPI channelscanbeusedtosendcommandsand queriestotheDVM,ifaqueryissenttheresponseisshowninthe replychannel.Thecommandsinthe RST TXT channelsherereset theDVM,thenthevoltagerangeandprecision,and enablethehighimpedancemode.Notethatahigherprecisionand thusslowerpollingrateisusedfortheFuG,whilealowerprecision andfasterpollingrateisusedfortheMatsusada.Formoredetails ontheSCPIcommandswhicharerecognizedbytheDVMs,seethe SCPIprogrammingguidefromAgilent.................102 xii LISTOFFIGURES Figure1.1:Figuremofrom[16]highlightingnuclideswhichhavebeenmea- suredusingopticalspectroscopy.Thenuclidesarepositionedwith neutronnumberincreasingtowardtheright,andprotonnumberin- creasingvertically.Stablenucleiarehighlightedinblack,whileblue squaresindicaterareisotopeswhichhavemeasurementsfromoptical spectroscopy.Boxesinmagentaindicatetherareisotopeswhichhave beenmeasuredatBECOLA.Theabilityofspectroscopytogainnu- clearinformationfromacrossallmassrangesandreachshortlived isotopesfarfromstabilityisevident.Forinterpretationoftherefer- encestocolorinthisandallotherthereaderisreferredto theelectronicversionofthisthesis...................3 Figure1.2:Single-particlelevelsofnucleonscalculatedfromthenuclearshell model.Ontheleftsidearetheenergystates,andtowardstheright, thesplittingfromthespin-orbitinteractionisincluded.Theoccu- pancyoftheselevelsisnotedaboveeach.Atshellclosureswhere largegapsoccurbetweenonelevelandthenext,magicnumbersare observed.Whenmovingawayfromstability,theselevelsshift,lead- ingtothedevelopmentofnewmagicnumbers,anddisappearanceof others...................................4 Figure1.3:ExamplesofchargeradiifromNetoKrisotopes.Experimentaldata isfrom[4].Neutronmagicnumbersareindicatedbyverticaldashed lines.Alocalminimumcanbeobservedinmanychainsfor N =28 and N =50,howeverthissignatureisnotobservedacross N =20.6 Figure1.4:Experimentalcalciumchargeradiipriortothiswork.Stableandlong- livedisotopesareplottedinblack,whileunstableisotopesareplotted inred.Datatakenfrom[4,26].Theparabolicshapebetween A =40 and A =48,oddevenstaggering,andthedramaticrisebeyond A = 48havebeenfortheoriestoreproduce[23,24,25,21,26,27].7 Figure2.1:TherareisotopeproductionattheNSCL.Forthisexperiment,stable 40 Caweregeneratedandthenacceleratedto140MeV = nucleonusing thecoupledcyclotrons,K500andK1200.Followingthisacceleration, thebeamwasimpingedonaberylliumtarget.Awiderangeofiso- topeswasproduced,andtheseisotopeswerethenpassedthroughthe A1900fragmentseparator,inordertoselectthoseofinterest.Fol- lowingthisschematic,thefastbeams(0 : 5c)werepassedtothegas cell,wherethebeamisthermalized..................12 xiii Figure2.2:ThegasstoppinglayoutattheNSCL.Fastbeamsofrareisotopes arebroughtinfromtheleftandareintroducedtothegascell,fol- lowingasoliddegrader.Theionsarethenthermalizedinthehelium gas.Onextractionfromthegascell,thedesiredcharge/mass ratiocanbeselectedusingadipolemagnet.Thelowenergybeam isthentransportedtotheBECOLAfacility,wherecollinearlaser spectroscopyisperformed........................12 Figure2.3:ThecollinearlaserspectroscopysetupatBECOLA.Aboveisacar- toonshowingthebeampathofthelaserandtheionbeams.The radio-frequencyquadrupole(RFQ)cooler/buncherisabletoaccept thebeamfromeitherthePIGsource,ortheradioactive\on- line"beamcomingfromthegascell.Asshown,thevoltageatthe detectionregionisscannedinordertoscanthetransitionenergybe- ingprobed,andthetwophotondetectionsystemscollecttheresonant photons.Below,amoretruetolifediagramispictured,inorderto giveanaccuratesenseofscale.....................13 Figure2.4:Atomicstates,andhypsplitting.Ontheleft,theelectronic levelsforCa ii aredrawn[41].Ontheright,theselectedtransition isdrawn,alongwiththehfsplittingfor I =3 = 2,asin 37 ; 39 Ca[42]. Theallowedhftransitionsaremarkedinred..............16 Figure2.5:Thelasersetupusedinthisexperiment.TheSirahMatisseTS Ti:SapphirelaserwasfrequencylockedusingthestabilizedHe-Ne andthewavelengthmeter,bysampling1%ofthelightasshown. Themajorityofthislightwasfrequencydoubled,andtransportedto theexperimentalareausingaeropticcable.Here,afterpassing throughapolarizingcube,10%ofthelightwassampledbyapower meterinordertocontroltheLPC,whichmodulatedthepoweren- teringthefrequencydoubler.Thisallowedpowerintro- ducedbythetransmissionthroughtheertobeeliminated,and ensureastable300 µ WbeamforCLS..................17 Figure2.6:Therecordedtimespectraforthe 36 Caand 44 Cabunchedbeams. 36 Cahadthefewestionscontainedineachbunch,thusgivingthe narrowesttime-width,while 44 Cahadthemostionsperbunch,due tocontaminationfromthemoreabundant 40 Ca(seeSection2.3.1.2), leadingtoaslightlywiderpeakintime.The 36 Catime-spectrum istakenfromthefull33hrunningtimewiththatisotope,whilethe 44 Catime-spectrumistakenfromasinglereferencemeasurementof < 1h.Otherisotopesmeasuredinthisexperimentfallwithinthese twoextremes,withatime-widthofapproximately1 µ s........21 xiv Figure2.7:Aboveisaschematicofthecooler/buncherusedatBECOLA.A uniqueaspectofthisdesignistheseparatecoolingandbunchingsec- tions,whichareoperatedwithtbugaspressures.Thedif- ferentialpumpingschemeusedtomaintainthecentralhigh-pressure regioncanbeseeninFigure2.8.Theionsarecontainedlaterallyby thepseudo-potentialofaradio-frequencyquadrupole(RFQ),whilea dragpotentialismaintainedinthelongitudinaldirectionusingseg- mentedelectrodes.Thegraphbelowtheschematicshowsthedrag andtrappingpotentialalongthebeamaxis,thedashedlineonthe rightindicatesthepotentialusedtoreleasetheionbunch.Figure fromref.[44]...............................23 Figure2.8:Illustrationofthetialpumpingchannel.Thehighandlow pressureregionsareconnectedviaanarrowchannelwithrelatively lowconductance(marked\C").Theheliumisintroducedintothe highpressureregion,andpumpedfromthelowpressureregionusing aturbomolecularpump(marked\S TMP ").Figurefromref.[44]..24 Figure2.9:Thebunchingasafunctionofcoolingtime.Asshown bytheblacksquares,withsmallquantitiesofionsthereisminimal reductioninforholdingtimesuptoonesecond.These measurementsweremadeusingpotassiumions.Figurefromref.[45].25 Figure2.10:Aplotshowingthethatthebunchingperiod, T b ,hasonthe signalandnoisecollectedwhenusingunstableisotopes,assuming thattheiencyofthecoolerisbythebunchingperiod, andthatthemeasurementtimeforeachbunchis1 µ sregardlessof theholdingperiod.Theverticalaxisrepresentsthenoiseandsignal countscollectedafteragiventime(seeEquations2.8and2.9),and theirratio(Equation2.11).Nounitsareprovidedonthevertical axis,astheabsolutequantitiesalsodependonotherfactors,such asthehalf-life,incomingbeamrates,andthemeasurementtimeof thebunch.Thesignalcountsareshowninblue,noisecountsinred, andSNR(scaledarbitrarily)isshowninblack.Astheperiodis increased,theionsheldinthetrap,andthusthesignalperbunch, willeventuallyplateau,however,theincreasedtimebetweenbunches leadstoacontinuedreductioninsignalcollected.Thenoisecollected fromthelaserisalsoreducedduetothedecreaseintotalbackground causedbymeasuringthedatainfewerbunches.Althoughbothsignal andnoisearemonotonicallydecreasing,thedrasticreductioninnoise causestheSNRtobemaximizedwhen T b ˇ 1 : 8 T 1 = 2 .......26 xv Figure2.11:Thisplothighlightsthebofantmeasurementwithre- gardtoDoppler-broadening.Foragiventemperatureofions,there willbeasmallspreadinkineticenergy,leadingtoaspreadinve- locity(blackbands).Whentheionsareacceleratedoveralarge potential,althoughtheenergyspreadremainsthesame(andissmall comparedtotheenergyfollowingacceleration),thecorrespondingve- locityspreadisreduced(redbands).Becausethevelocitydetermines theefrequencyofthelaserlightthattheionsareinteracting with,thisvelocitybunchingresultsinnarrowerresonancepeaks...27 Figure2.12:Experimentaldatafromthegascellgroupshowingthemass-to-charge spectrumof 39 Caextractionfromthegascell.Thehorizontalaxis showstheselectedmass/chargeratio(amu/ e ),whiletheverticalaxis indicatesthedecayrateseenintheextractedbeam.Thepeakat39is attributedtosingly-charged 39 Caions,whilethepeakat19 : 5contains thedoubly-chargedionsofthisspecies.At m=q =57,apeakcan alsobeseenduetosingly-charged 39 Cawithasinglewatermolecule (18amu)attached.Thisspectrumwastakenusing400Vforthe dissociationpotential.Asshown,thisresultsinatportion ofsingly-ionized 39 CawhichwasthentransportedtoBECOLAto performspectroscopy..........................30 Figure2.13:Renderingsofboththeoldandnewsystems.ThePMToftheold systemisplacedatthefocalpointoftheellipsoidalThenew systemimagehastwocompletephotondetectionregions,eachwith theirownellipticalaperture,CPCandPMT.Toenablea betterviewofthedesign,thedetectionregionclosesttotheviewer hashadtwopiecesremovedfromboththeellipticaland theCPC(green).Thedetectionregionfurtherfromtheviewerhas hadtheCPCandPMTremovedtoallowaviewoftheadjustable aperture(darkblue)whichsitsjustoutsidethewindow.Fourrods supportthetworegions,andareheldinsidethebeampipeusingfour adjustableinsulatingfeetoneachend(cyan),inordertoallowthe scanningpotentialtobeappliedtothePDS..............37 Figure2.14:SNRresultsusingvariousopeningwidthsatthefocalplane.The verticalaxisistheSNRofthedetectionregionofthenewsys- tem,normalizedusingtheseconddetectionregionofthenewsystem withtheadjustableplatesopenaswideaspossible.Itwasexpected thatthesignalwouldbeconcentratedatthecenterofthefocalplane, meaningthattherewouldbeapointwheretheSNRbeginstode- crease,asanywideropeningonlyservestoallowmorestrayphotons intothesystem.Asshown,thiswasnotthecase.Thewiderthe opening,thebettertheSNRbecomes.................39 xvi Figure2.15:ResultsexaminingtheSNRatvariouspointsalongtheY-axisof thefocalplane.Theplatesinonedetectionsystemweresettoa 3mmwideslit,andacalciumresonancewasrecordedwiththisslit atvariouspositionsalongtheY-axisofthefocalplane.Thevertical axisistheSNRofthesystembeingscanned,dividedbytheSNRof thecontrolsystem.RedcirclesshowthenormalizedSNR(compared totheotherdetectionsystemwhichwasusedasacontrol),whilethe graydottedlineindicatestheshapeexpectedfromthesimulation. Ratherthanobservingthesignalconcentratedintoasinglepeak(see FigureA.3),thehighestSNRwasobservedintwopeaksoneither sideofthecenter.Furthersimulationstoexplorethisbehaviorcan befoundinSectionA.1.3........................40 Figure2.16:SNRresultsusingvariousmaskstoselectthetwopeaksseenatthe focalplane.TheverticalaxisistheSNRofthenewsystem,divided bytheSNRrecordedwiththeoldsystem.Themaskusedforeach measurementisshownunderneaththeplot,grayareasarewhere thefocalplanewasblocked,whilewhiteareasshowtheregionof thefocalplanewherelightwasallowedtopassthroughthewindow. Whiletheadjustableplateswereusedtoblockthetopandbottom, a5mmor15mmwidestripofanodizedfoilwasusedtoblockthe centralportion.ThefocalplaneislabeledalongtheXandYaxes withunitsofmm.Selectingthetworegionsofpeakintensityseenin Figure2.15doesprovideabetterSNR,butinallcases,evenwiththe focalplanefullyopen,theSNRwasmorethanthatobtainedusing theoldsystem..............................41 Figure3.1:Thespectrummeasuredfor 36 Ca,alongwiththeresidualsfromthe usingapseudo-VoigtfunctionasdescribedinSection3.2.....50 Figure3.2:Thehypspectrummeasuredfor 37 Ca,alongwiththeresiduals fromtheusingsixpseudo-VoigtfunctionsasdescribedinSec- tion3.2..................................50 Figure3.3:Thespectrummeasuredfor 38 Ca,alongwiththeresidualsfromthe usingapseudo-VoigtfunctionasdescribedinSection3.2.....51 Figure3.4:Thehypspectrummeasuredfor 39 Ca,alongwiththeresiduals fromtheusingsixpseudo-VoigtfunctionsasdescribedinSection 3.2....................................51 Figure3.5:Atypicalspectrummeasuredforthereference 40 Ca,alongwiththe residualsfromthetusingapseudo-Voigtfunctionasdescribedin Section3.2................................52 xvii Figure3.6:Atypicalspectrummeasuredforthecalibration 44 Ca,alongwiththe residualsfromthetusingapseudo-Voigtfunctionasdescribedin Section3.2................................52 Figure3.7:Diagramfromref.[22]describingthemassshiftandshiftprinci- ple.Theleftpanelshowsthemassshift.Atthetop,isarepresenta- tionofthenormalmassshift(NMS)whicharisesfromthebalanceof momentumbetweenthenucleusandeachelectron,whilethemiddle andlowerdiagramsdepictthespmassshift(SMS)whichvaries dependingonthemomentumcorrelationbetweentheelectronsin theatomicsystem.Thecenterpaneldescribestheshift.Apure Coulombpotentialforapoint-likenucleusisshownbythedottedline. Variationsinthechargevolumeofthenucleustheenergylevels oftheelectronsduetotheprobabilityoftheelectronsbeingwithin thenucleus.Thehorizontaldottedlinerepresentstheenergyofan s -electronforapoint-likenucleus,whiletheredandbluelevelson theleftandrightshowtheenergylevelsshiftedduetothevolumeof thenucleus A 0 and A respectively.Therightsectionprovidesanidea oftherelativecontributiontotheisotopeshiftofeachversus theatomicnumber.Astheshiftisthecomponentsensitiveto thecharge-radius,thechargeradiusbecomestoextractfor lighterisotopes,wherethemassshiftdominates...........56 Figure3.8:Kingplotfromref.[59]comparingthemodisotopeshiftofthe D1andD2transitionsinCa.Previousdatapointsareshownin red,whilethebluecirclesarethosemeasuredinthatwork.Both theverticalandhorizontalaxesareplottingthe\modisotope shift,whichisthelefthandsideofEquation3.12.Here represents theexpressioninvolvingthemasses.Thehighdegreeofprecisionand theexcellentlinearityofthepointsgivesgoodcausetoneglecthigher orderwhichmaycause F tovarybetweenisotopes......57 Figure4.1:Thechargeradiimeasuredinthiswork(redsquares)andpreviousex- perimentalvalues(blacksquares)arecomparedtoDFTpredictionsof SV-min(HFB), r ,BCS), r ,HFB),and r ,HFB)mod- els.Thevaluesof r ,HFB)for A> 37areverycloseto r ,HFB) results;hence,theyarenotshown.Therms.chargeradiiwereob- tainedusingtheknownchargeradiusof 40 Ca(ref.[4]),anditser- rorhasbeenincorporatedintothesystematicuncertainty(thegray band).ThesevaluesareshowninTable3.3..............63 xviii Figure4.2:Graphsshowingtheenergylevels,radii,andoccupationsofthesingle protonstatesintheCachain.Theuppergraph,a),highlightsthe weakly-boundnatureofthevalenceprotons.Asshownbythered triangles,thesingleprotonenergyofthe0 f 7 = 2 orbitalrisesabove theCoulombbarrierandbecomesunboundinthecaseof 36 38 Ca. Graphsb)andc)displaythesingle-protonrmsradiiandoccupa- tions,respectively.Inthecaseofthe r ,BCS)approach,the single-protonradiusincreasesinanon-physicalwayasthestatesrise abovetheFermienergyandcontinuumcomeintoplay(reddi- amonds).Witharealisticpairinginteraction,theseunboundstates donotgrowsodramaticallyinsize,asshownbythe r ,HFB) and r ,HFB)points(greencirclesandbluetriangles)......65 Figure4.3:Nuclearchargeradiiofcadmiumisotopes.Figuretakenfromrefer- ence[73].Experimentalvaluesandseveralmodelsareshown.The graybandrepresentsthesystematicuncertaintyoftheexperimental values,arisingfromtheuncertaintyoftheconstant.The insetalsoshowsthecorrespondingone-neutronseparationenergies. TheagreementoftheFayansfunctionalisquitegood,althoughsome overestimationoftheodd-evenstaggeringispresenthereaswell..66 Figure4.4:Nuclearchargeradiioftinisotopes.Figuretakenfromreference[74]. Experimentalvaluesandseveralmodelsareshown.Theinsetalso showsthebehaviorofthe r ,HFB)modelinthevicinityof 208 Pb. TheabilityofthisFayansfunctionaltoreproducethesekink-structures inheavyisotopeswithoutadditionaltuningisquiteremarkable...67 Figure4.5:Figurefromreference[75].Thisshowstherelativelocationsof thenuclidesusedinthefourempiricalchargeradiirelationships.In allfourcases, R in jp = R (1)+ R (2) R (3) R (4)=0,allowingthe predictionofachargeradiusfromtheotherthree.Thisrelationshipis mostrobustfor(a),wherethereisonlyastepof1protonorneutron betweenthefournuclidesbeingcompared,butholdsreasonablywell fortheotherthreerelationshipsaswell................69 xix Figure4.6:Empiricalpredictionsofchargeradiishownhereinpurplediamonds aretakenfromreference[75].Thesepredictionsusesomecombina- tionsofthefourrelationsshowninFigure4.5.Experimentalcharge radiiareshownusingtheredandblacksquares.Largedeviations fromtheempiricalpredictionscansuggestsuddenvariationsinshape ordensityofthenucleus.Itisinterestingtonotethatwhileprevious theoriesdramaticallyoverpredictedthesizeoftheproton-richnuclei inthischain,theseempiricalpredictionsarereasonablyaccurate,and infactunderpredictnucleibelow A =40,whichistobeexpected consideringtheoftheweakly-boundprotonsintheseisotopes. Thishighlightsthefactthatthereisnosuddenchangeofbehavior ofthechargeradiiinthisregion,andstrengthensthebeliefthata modelwhichperformswellinthischainmaybeapplicableacrossthe nuclearchart...............................71 FigureA.1:Schematicshowingthethreesimulationsetups.Raytracingsimula- tionswereperformedinFREDusingperfectlyesurfaces.The newsystemwassimulatedwithacylindricalvolumeemittingraysin randomdirections,whilethebackgroundwassimulatedasraysemit- tedinrandomdirectionsfromwithintheareaofacircleplacedatthe entrancetotheellipticalregion.Theoldsystemwassimulatedtwice, oncewiththesourcespositionedidenticallytothenewsystem,shown ontheright,andagainwiththebackgroundsourcemovedcloserto bewiththesystem,asshownontheleft.Thesetwosimulations oftheoldsystemwereusedtoevaluatethestabilityofthesimulations andobtainarangeofexpectedvalues.................78 FigureA.2:Simulationshowingthegeometricdistributionofphotonsarrivingat thefocalplaneusingtheexistingPDS.Thebluedashedcircleindi- catesthegeometricacceptanceofthePMT,collectingthemajority ofthesignal(redpoints),andexcludingaportionofthebackground (yellowpoints).Inthissimulation,44 : 8%ofthesignalphotonsreach thefocalplane,while25{40%ofthebackgroundphotonssimulated arriveatthefocalplane.IntheareacoveredbythePMT,13 : 4%of thesignal,and1 : 5{2 : 6%ofthebackgroundwerecollected.Therange ofvaluesforthebackgroundareduetoavariationinthepositionof thesimulatedbackgroundsourceasdescribedinSectionA.1.1...79 xx FigureA.3:Simulationshowingthegeometricdistributionofphotonsarriving atthefocalplane(left)andthePMT(right)withthenewPDS. Thebluedashedlineindicatesthegeometricoftheaperture atthefocalplane,collectingthemajorityofthesignal(redpoints), andexcludingaportionofthebackground(yellowpoints).Inthis simulation,42 : 9%ofthesignalphotonspassthroughtheapertureof thefocalplane,comparedtoonly15 : 3%ofthebackgroundphotons. FollowingtheCPC,thePMTaccepts8 : 3%ofthesignal,andonly 0 : 21%ofthebackgroundphotons...................80 FigureA.4:Simulationresultsdescribingtheangulardistributionofphotonsar- rivingatthefocalplaneofthenewPDS.Signalisshowninred, andbackgroundinyellow.Thepolarangleof0 ° correspondstothe perimeterofthehemisphere,whileanazimuthalangleof0 ° or 180 ° correspondstothebeamaxis.Theangulardistributionisalsoshown projectedontoahemisphereaboveeachgraph,thecenterofthehemi- spherefacesthePMT.Thebluedashedlineindicatestheangular chosenfortheCPC.Photonsarrivingwithapolarangleless than70 ° arerejectedexcludingalargepartofthebackground...81 FigureA.5:Simulationresultsdescribingtheangulardistributionofphotonsar- rivingatthefocalplaneoftheoldPDS.Signalisshowninred, andbackgroundinyellow.Thepolarangleof0 ° correspondstothe perimeterofthehemisphere,whileanazimuthalangleof0 ° or 180 ° correspondstothebeamaxis.Theangulardistributionisalsoshown projectedontoahemisphereaboveeachgraph,thecenterofthehemi- spherefacesthePMT.Asshown,theprimaryintensityofboththe signalandbackgroundlieatthesamepolarangle,preventingany improvementfromaddingaCPCtotheexistingsystem.......82 FigureA.6:Thissimulationhighlightsthedinsignalcollectionofthetwo systems.Whiletheoldsystemcollectsalargefractionofthesignal atthecenterofthedetectionregion(thefocusoftheellipsoid),the newsystemcollectssignalmoreconsistentlyacrosstheentireregion. Thetotalacceptanceofthenewsystem(areaunderthiscurve)is stilllessthantheoldsystem,however,theincreaseinnoiserejection allowsittooutperformtheoldsystemfortheCa ii measurement..83 FigureA.7:Aschematicofthebeampipeusedtoenclosethetwonewdetection regions.The8inonthetopandbottom,whicharefrom oneanother,arethelocationofthetwonewdetectionregions.A smalloppositeeachallowsahighvoltagefeedthroughinorder toapplythescanningpotentialtothedetectionregion,asshownin FigureA.12................................87 xxi FigureA.8:dataforthealuminumsheetingusedfortheesur- facesofthenewPDS.TheMIRO4300UPwasused,inordertoallow forgoodythroughouttheUVrange(250{450nm).....88 FigureA.9:Lookingfromtheendofthebeamcross,theinternalsupportsof thePDSareshown.Fourwhiteinsulatingfeetsupportitwithinthe beampipe,andthefourmainsupportrodsarepointingawayfrom theviewer,fromtheendsofthealuminum inthecenterofthebeam pipe.Thesmalltubeatthecenterofthe iswherethelaserand ionbeamstravel.Alignmentofthismainframewasaccomplishedby rotatingthedouble-threadedbrassconnectorsnexttotheinsulators.89 FigureA.10:PhotographsshowingtheassemblyofanellipticalOnthe top,thepartsareshownpriortoassembly.Theleftimageshows theassemblyminusonesideplate,andtheendplate.Whenthe endplateisattached,itpushestheesheetinshwiththe endoftheellipticalplates,causingittobowfurtheroutwardand layagainsttheellipticalplates.Therightphotoshowsthe assembly,lookinginthroughthewindowoftheendplate.Notshown hereisthemetalmeshwhichcoversthiswindowinordertoensure thattheentiredetectionregionisenclosedbythedesiredpotential whenscanning..............................90 FigureA.11:Photographsshowinghowtheellipticalrsareattachedtothe supportrods.Onthelefttheinteriorofthebeampipeisshown. Asmall\J"shapedhookholdstheellipticalassemblyup againstthesupportrods.Itislockedinplacewithasetscrewonthe sidefacingoutthesideandsoitcanbeaccessedfromoutside withcarefuluseoftweezersandalonghexwrench,asshowninthe photoontheright............................91 FigureA.12:Photographsshowingtheinstallationofthehighvoltagefeedthrough usedforthescanningpotential.Asmallspringallowscontactwith thebackofoneoftheellipticalTheleftphotoshowsthe highvoltageconductorincontactwiththesystem,whiletheright showsthefeedthroughremovedfromthesystem...........91 FigureA.13:AphotographoftheCPCassembly.TwohalvesofoneCPCare shownintheforeground,whileintheback,afullyassembledCPC canbeseen.Thewidesideattachestothesidewindowofthebeam pipe,andthePMTisattachedtothenarrowend..........92 xxii FigureA.14:Aphotographshowingtheassemblyoftheexteriorcomponentsof thePDS.Anadjustableapertureofaluminumplatesisattachedto thewindow,withtheCPCfollowing.AttherightendoftheCPC, themountinghardwareforaPMTisshown.ThefaceofthePMT pressesupagainstarubberO-ringontheCPC,whilethebackend isheldtightlyusingaplateheldbythreadedrodsprotrudingfrom theCPC.................................92 FigureA.16:PhotographoftheMFCandsurroundingcomponents.Theopen 9-pinconnectorseenonthe\T"connectorcanbeattachedtoacom- puterviaanRS232serialcablewhenthesystemisnotonhighvolt- age.Theinsulatorisnecessarytopreventcurrentleakagewhenthe PIGsourceisraisedonthehighvoltage.Originally,thebentalu- minumbracketwasthoughttobet,howevertherewasan excessiveamountofleakagecurrentwhentheMFCwasconnected theredirectly.WhennotoperatingthePIGsource,theshvalve canbeclosedtoallowabettervacuuminthePIGsource......97 FigureA.17:Schematicshowingthewiringofthecustom\T"connectorbeing usedfortheMFC.Thediagramistakenfromthemanualprovided byBronkhorst,andcolorlabelsareaddedtoindicatetheactualwires usedwithintheconnector.Theleftsideofthisconnectstothe MFC,andtheconnectiononthelowerrightallowsforpowerand analogcontrolstobeattached.Theupperrightconnectorisusedfor serialcommunicationwithaPC,andisnotusuallyconnected....98 FigureA.18:Datatakenusingdoubly-charged 40 Caions.Thecooler/buncherwas settoallowdoubly-charged 40 Caionstopassthrough,andsingly- chargedionswererejected(veusingspectroscopy).AstheCEC temperaturewasincreased,ionbeamcurrentsweremeasuredafter theCECwithandwithouttheionkicker.Bymeasuringthetotal currentfromthedoubly-chargedionbeampriortotheCEC,assum- ingatotaltransmissionthroughtheCECof98%,andmeasuring theneutralandchargedbeamcurrent,thefractionofatomsineach chargestatecanbedetermined.Whilethismethodisnotverypre- cise,itdoesshowthatthecrosssectionforthedoubleandsingle chargeexchangeprocessincreaseswithtemperature.........101 FigureA.19:PhotographoftheRaspberryPi3B+anditsconnectionsusedto interfacewiththewaveformgenerator.Thisisintherackinside thehighvoltagecage.Ratherthanconnectthewaveformgenerator directlytothePi,aUSBhubmustbeplacedinbetween,otherwise thePidoesnotrecognizethedevice..................105 xxiii KEYTOABBREVIATIONS ‹ BCS-Bardeen-Cooper-Schr ‹ BECOLA-BEamCOolerandLAserspectroscopy ‹ CLS-collinearlaserspectroscopy ‹ DVM-digitalvolt-meter ‹ FC-Faradaycup ‹ FS-shift ‹ FRIB-FacilityforRareIsotopeBeams ‹ FWHM-full-widthatthehalf-maximum ‹ hf-hyp ‹ HF-Hartree-Fock ‹ HFB-Hartree-Fock-Bogolyubov ‹ IS-isotopeshift ‹ KE-kineticenergy ‹ LPC-laserpowercontroller ‹ ms-mean-square ‹ MS-massshift ‹ NSCL-NationalSuperconductingCyclotronLaboratory(EastLansing,MI,USA) ‹ PDS-photondetectionsystem ‹ PIG-Penningionizationgauge ‹ PMT-photo-multipliertube ‹ RFQ-radiofrequencyquadrupole ‹ rms-rootmean-square ‹ SNR-signal-to-noiseratio ‹ UV-ultra-violet ‹ -NMR- -raydetectingnuclearmagneticresonance xxiv Chapter1 Introduction Sincethediscoveryofthenucleusoveracenturyago[1],experimentshavebeeninvestigating thesizeandshapeofallkindsofnuclei[2,3,4].Measurementsofglobalpropertiesofnuclei, suchasthechargeradius,magneticdipolemoment,andelectricquadrupolemomenthave broughtnewinsightstothebehaviorofthenuclearforceandthecomplexinteractionsof protonsandneutronswithinthenucleus[5,6].Theseexperimentalresultsareespecially usefulwhentheyextendourknowledgetounstablenuclei,towardstheprotonorneutron dripline,becausetheycanrevealevolutionsinnuclearshellstructurewhichmaynotexist intheregionofstablenuclides. OnemethodofstudyingrareisotopesisCollinearLaserSpectroscopy(CLS).Thistech- niqueenablesaveryprecisemeasurementoftheatomichyp(hf)structure,whichcanbe byvariationsinthenucleus'sizeandshape.Amajoradvantageofthistechniqueis theeaseofapplicationtoexoticisotopes,duetotmeasurementusingafastbeam[7]. Whilecollinearlaserspectroscopyhasbeeninuseforoverfortyyears[8],thetechniquehas seenmajoradvancementsinthe21stcenturyduetothepioneeringofthebunched-beam technique(atime-resolvedmeasurement)[9].Thishasdramaticallyenhanced thesensitivity,andinrecentyears,bunched-beamCLShasenabledmeasurementsofcharge radiiandnuclearmomentsofanexpandinggroupofrareisotopesasproductioncapabilities haveincreased[10,11]. Fromaprecisemeasurementofanatomichftransition,thistechniqueallowstheextrac- 1 tionofchargeradiiandnuclearmomentsofrareisotopes,andhasbeenusedwithgreat successfornucleiacrosstheentirenuclearchart,asshowninFigure1.1[12,13,14,15,16]. Theseobservablescangiveindicationsofthenuclearstructure,includinghighlightingthe socalled\magicnumbers"wherenucleiareobservedtobeparticularlywellbound,due togapsinthenucleonsingle-particleenergylevels,orshells,showninFigure1.2.Moving awayfromstability,variationsinthenuclearsingle-particleenergylevelscangiveriseto newmagicnumbers[17],andseveralunstablecalciumisotopeshavebeenpredictedtoshow characteristicsofthesenewshellclosures[18]. 1.1Chargeradii Oneofthemostimportantpropertiesofthenucleusisitssize.Indeed,theremarkably smallchargeradiusofthenucleuscomparedtotheatomicradiuswastheexperimental observationofthenucleus,beforeevenunderstandingitsconstituents[1].Tothisday, measurementsofnuclearchargeradiicontinuetobeanimportanttooltoinformand nucleartheories[19]. Fromscatteringexperiments,itisfoundthat,ingeneral,thechargedensityofthenucleus isapproximatelyconstant.Thisleadstoasimplerelationbetweentheradius, r ,andthe numberofnucleons, A : r = r 0 A 1 = 3 (1.1) where r 0 hasbeenfoundempiricallytobeapproximately1 : 2fm[20].Whilethiscanserve asaroughestimationforthechargeradiiofheaviernuclei,thedeviationsfromthisgeneral trendprovideinsightsintotheunderlyingstructureofthenucleus. Whenlookingatthevariationsinchargeradiiacrossachainofisotopes,magicnumbers 2 Figure1.1:Figuremofrom[16]highlightingnuclideswhichhavebeenmeasuredusing opticalspectroscopy.Thenuclidesarepositionedwithneutronnumberincreasingtoward theright,andprotonnumberincreasingvertically.Stablenucleiarehighlightedinblack, whilebluesquaresindicaterareisotopeswhichhavemeasurementsfromopticalspectroscopy. BoxesinmagentaindicatetherareisotopeswhichhavebeenmeasuredatBECOLA.The abilityofspectroscopytogainnuclearinformationfromacrossallmassrangesandreach shortlivedisotopesfarfromstabilityisevident.Forinterpretationofthereferencestocolor inthisandallotherthereaderisreferredtotheelectronicversionofthisthesis. 3 Figure1.2:Single-particlelevelsofnucleonscalculatedfromthenuclearshellmodel.On theleftsidearetheenergystates,andtowardstheright,thesplittingfromthespin-orbit interactionisincluded.Theoccupancyoftheselevelsisnotedaboveeach.Atshellclosures wherelargegapsoccurbetweenonelevelandthenext,magicnumbersareobserved.When movingawayfromstability,theselevelsshift,leadingtothedevelopmentofnewmagic numbers,anddisappearanceofothers. 4 canbeseenasalocalminimum,whereasharpkinkispresent[4,21].Thisfeatureiscaused bythenucleustakingonasphericalshapewhenanucleonshellclosureisreached,but becomingdeformedandincreasingtherootmean-square(rms)chargeradiuswhennucleons areaddedorremovedfromthesystem[22].Thissignatureispresentinmanychainsof radii,butatneutronnumber N =20theexpectedkinkstructurenearlyvanishes,asshown inFigure1.3[23]. UsingCLStoexamineaparticularhfspectrumallowsthetransitionenergytobeprecisely measuredandcomparedtothatofareferenceisotope.Thisinenergyisknown astheIsotopeShift(IS)andfromit,thetialchargeradiuscanbeextracted[12]. MeasurementsusingCLScanthereforecomparethechargeradiiofrareisotopestothoseof known,stableisotopes,providinginsightsintothestructureofnucleifarfromstability. 1.2Calcium DuetothevarietyofstableisotopesinCa(protonnumber Z =20),spanningbetweentwo doubly-magicnuclei, 40 ; 48 Ca( N =20 ; 28),Cahasproventobeauniqueelement.Despite theadditionof8neutrons,themean-squarechargeradiusof 48 Caisnearlythesamesize as 40 Ca,whiletheisotopesinbetweenriseandfallinaparabolicmannerandpossessa pronouncedodd-evenstaggering,asshowninFigure1.4.Nucleartheorieshavestruggled toreproducethisthelocalmaximumat 44 Ca[23],theodd-evenstaggering[24,25],and thedramaticminimumat 48 Ca[21,26,27].Becauseofthis,thechainofCaisotopeshas becomeaprovinggroundforchargeradiimodels. Beyondthecomplexpatternseeninthechargeradii,experimentalevidenceofdoubly- magicfeatureshasbeenseeninneutron-richCaat N =32and N =34throughprecision 5 Figure1.3:ExamplesofchargeradiifromNetoKrisotopes.Experimentaldataisfrom[4]. Neutronmagicnumbersareindicatedbyverticaldashedlines.Alocalminimumcanbe observedinmanychainsfor N =28and N =50,howeverthissignatureisnotobserved across N =20. 6 Figure1.4:Experimentalcalciumchargeradiipriortothiswork.Stableandlong-lived isotopesareplottedinblack,whileunstableisotopesareplottedinred.Datatakenfrom[4, 26].Theparabolicshapebetween A =40and A =48,oddevenstaggering,andthedramatic risebeyond A =48havebeenfortheoriestoreproduce[23,24,25,21,26,27]. 7 massmeasurementsand2 + excitationenergiesrespectively[28,29],andevidenceofdoubly- magicnucleiisexpectedtobeseenintheientregionaswell[18].Recently,an investigationofneutron-richCaisotopeswascarriedoutatCERN,andthechargeradiiof 49 ; 51 ; 52 Caweredeterminedforthetime[26].Thechargeradiioftheseisotopeswere surprisinglylargecomparedwiththeoreticalpredictions,contrastingwiththeexpectationof aspherical,doubly-magicnucleusat N =32. ThisinterestinthechainofCaisotopeshascontinued,andtheuniqueproductiontech- niqueusedattheNSCLhasenabledthestudyofisotopesontheneutsideof thechain,reachingallthewaytotheshort-lived( T 1 = 2 =102ms[30]) 36 Ca,andallowing measurementsofthechargeradiiandnuclearmoments. 1.3CollinearLaserSpectroscopyatBECOLA AttheBEamCOolingandLAserspectroscopy(BECOLA)facilityattheNationalSupercon- ductingCyclotronLaboratory(NSCL),aCLSbeamlinewasinstalledandcommissionedin 2014[31,32].Duetothetproductionmethod(projectile-fragmentreaction)mostly employedattheNSCL,BECOLAhasauniqueopportunitytostudyrareisotopeswhich aretoproduceintquantitiesatotherCLSfacilities,wheretheIsotopeSep- arationOnLine(ISOL)techniqueistypicallyused.Theprojectilefragmentationreaction resultsinreactionproductsbeingemittedinanarrowforwardangleatnearlytheprimary- beamvelocity.Becauseofthis,fragmentscanbeselectedtlyandquicklyusinga magneticspectrometer[33],andthenthermalizedinagascell[34]. AtotherCLSfacilities,suchastheCOLLAPScollaborationatISOLDE/CERN,the ISOLtechniqueisusedtoproducerareisotopes.Inthismethod,lightionsbombardathick 8 targetandstationaryreactionproductsareproducedwithinthetarget.Thesereaction productsthenneedtobeextracted,andlongreleasetimesfromthetargetcanleadto largedecaylosses.ThislimitationhasbeenpartiallyovercomeusingtheIonGuideIsotope SeparatorOn-Line(IGISOL)approach,whichcanallowaveryfastextractionforsome reactionproducts,howeveritisstilltoachievehighratesforneutron-dt isotopes[35]. AsshowninFigure1.1,laserspectroscopyhasbeenutilizedtostudyanumberofisotopes totheleftofthevalleyofstabilityintheheaviermassregion,howeverforlighterisotopes, theshortlifetimes,andthusyofproductionandtransport,makethisregion tostudy.AtBECOLA,severalsuccessfulmeasurementsoftheseneutront isotopeshavebeenperformed,includingthechargeradiiof 36 ; 37 K[23]and 52 ; 53 Fe[21]. 1.4Dissertationoverview Thisdissertationwilldescribetherecentchargeradiimeasurementoft 36 ; 37 ; 38 ; 39 CaatBECOLA[36].Fromthehfspectra,thetialmean-square(ms)charge radiusofeachisotopewasextracted,aswellastheground-statenuclearmomentsof 37 ; 39 Ca. Whilethechargeradiusof 39 Caisknown,allothersaremeasurements.Thenu- clearmomentsof 37 Cawerealsodeterminedforthesttime[37],butwillnotbediscussed indetailinthisthesis.Theseresultsgivevaluableinsightintonuclearstructureandits evolutionmovingawayfromthevalleyofstability,hintingatthesub-shellclosurein 36 Ca, andspelucidatingtheimportanceofthepairinginteractionwhenweakly-bound statescomeintoplay. Chapter2presentstheexperimentalsetup.ItprovidesanintroductiontoCLSand 9 describestheisotopeproductiontechniquesusedinthisexperiment.Severalupgradesper- formedatBECOLAarealsodescribed,includingthenewphotondetectionsystemwhich wasbuiltandinstalledforuseinthisexperiment.Chapter3containstheanalysisprocedure usedtocalibrate,andextractthenuclearobservablesfromthedata.Thecalibrationus- ingtworeferenceisotopes,whichwasusedinthisexperimentforthetimeatBECOLA, isdescribed,andthehypspectraof 36 ; 37 ; 38 ; 39 Caarepresented.Theextractionofthe nuclearmomentsismentioned,andamoredetaileddescriptionofthechargeradii extractionispresented.Chapter4discussestheresultingmschargeradiiindetail.The ofweakly-boundprotonlevelsisdiscussed,andthereproductionoftheseresultsus- ingastate-of-the-artmodelofchargeradiiispresented.Theexperimentalresultsarealso comparedtoempiricalpredictionsofchargeradii.Chapter5summarizestheexperimental andtheoreticalresults,andgivesanoverviewoffuturework.AppendixAprovidesamore detailedlookatsomeexperimentaltoolsintheBECOLAbeamline,includingthephotonde- tectionsystemandtheionsource.AppendixBgivesinformationaboutsoftwaretools usedforthisexperiment,providingdetailsabouttheROOTdatastructuresandscriptsused fortheanalysis.AppendixCoutlinesmypersonalcontributionstotheworkatBECOLA, includingmentionoftalksandpresentationsgiven 10 Chapter2 Experiment Schematicsshowingtheexperimentalsetup,includingtherareisotopeproductionsystemat theNSCL,thegasstoppingsystem,andtheCLSsetupisshowninFigures2.1,2.2,and2.3, respectively. 2.1Basicprinciples CollinearLaserSpectroscopyhasbeenusedwithgreatsuccessinpastdecadestoexamine thesizeandshapeofnuclei[12,13].Earlystudieswerelimitedintheirusefulnessby resolutionandsensitivity,butrecenttechnologicaladvancementshavedramaticallyincreased thesensitivity,allowinghighprecisionmeasurementsofexoticisotopes[14,16,38,39].Laser spectroscopyhasservedasavaluabletoolforexploringchargeradiibeyondstability,and hasbeenusedtostudynucleifromlighttoheavyelementsacrossthewholenuclearchart. Thefundamentalstrategyistopreciselycomparetheshiftofthehypspectrabetween twoisotopes.Fromthisinenergy,theintheirmean-squarechargeradii canbeextracted.Foranisotopewithanon-zeroground-statenuclearspin,thehyp splittingofthetransitioncanalsorevealtheground-statenuclearmomentsofthatisotope. Toachievetherequiredprecisiontoextractthesenuclearpropertiesfromradioactivebeams withlowrates,thesignal-to-noiseratio(SNR)mustbecarefullyoptimizedandmeasurements ofstablereferenceisotopesmustalsobemadewithhighprecision. 11 Figure2.1:TherareisotopeproductionattheNSCL.Forthisexperiment,stable 40 Cawere generatedandthenacceleratedto140MeV = nucleonusingthecoupledcyclotrons,K500and K1200.Followingthisacceleration,thebeamwasimpingedonaberylliumtarget.Awide rangeofisotopeswasproduced,andtheseisotopeswerethenpassedthroughtheA1900 fragmentseparator,inordertoselectthoseofinterest.Followingthisschematic,thefast beams(0 : 5c)werepassedtothegascell,wherethebeamisthermalized. Figure2.2:ThegasstoppinglayoutattheNSCL.Fastbeamsofrareisotopesarebrought infromtheleftandareintroducedtothegascell,followingasoliddegrader.Theions arethenthermalizedintheheliumgas.Onextractionfromthegascell,thedesired charge/massratiocanbeselectedusingadipolemagnet.Thelowenergybeamisthen transportedtotheBECOLAfacility,wherecollinearlaserspectroscopyisperformed. 12 Figure2.3:ThecollinearlaserspectroscopysetupatBECOLA.Aboveisacartoonshow- ingthebeampathofthelaserandtheionbeams.Theradio-frequencyquadrupole(RFQ) cooler/buncherisabletoacceptthebeamfromeitherthePIGsource,ortheradioac- tive\online"beamcomingfromthegascell.Asshown,thevoltageatthedetectionregion isscannedinordertoscanthetransitionenergybeingprobed,andthetwophotondetection systemscollecttheresonantphotons.Below,amoretruetolifediagramispictured,inorder togiveanaccuratesenseofscale. 13 2.1.1Atomictransition AfavorableatomictransitionisnecessaryforCollinearLaserSpectroscopy,spone thatisaccessibletothelasersystemandproducesastrongsignal.Detectionofthespectral resonancerequiresabsorptionofthelaserlighttotheexcitedstate,andsubsequentspon- taneousemissioninwhichthephotonscanbedirectedtransversetothebeam.Becauseof this,thesignalratedependsontheabsorptionandspontaneousemissionratesofthetran- sition.TheseratesaredependentonthepopulationoftheatomicstatesandtheEinstein cots, A 21 and B 12 respectively.ThelargertheEinsteincotsofthetransition are,themorequicklytheionswillabsorbandemitphotonsthroughspontaneousemission. Itisalsoimportantthatthetransitionhasalargebranching-ratiofromtheupperstateback tothelowerstate,sothatthemajorityofdecaysarrivebackatthegroundstate,rather thandecayingtoastatewhichisnotbeingprobed,asthetransitionmaybeexcitedseveral timesastheionstravelthroughthedetectionregions. Inpreparationfortheexperiment,spectroscopyoftransitionsinCa i andCa ii wereper- formed.Usingthespectroscopicnotationforchargestates,Ca i referstoneutralcalcium atoms,whileCa ii ,Ca iii ,etc.refertothesingly-,doubly-,andhigherionizedstatesrespec- tively.Bothsingly-anddoubly-chargedionsofstable 40 Cawereproducedoasthe ratioofCa ii andCa iii tobedeliveredfromthegascellwasanunknownfactorleadingup totheonlineexperiment.IntheeventofanonlinebeamconsistingprimarilyofCa iii ,a charge-exchangeprocessmustbeutilized,asallground-stateatomictransitionsinCa iii are toohighinenergytobeaccessedbythecontinuouswavelasersystemsatBECOLA.Several testswereperformedtoverifythatcharge-exchangeandspectroscopycouldbeperformedat BECOLAusingCa iii (seeSectionA.2.2).Followingthechargeexchangeprocesshowever, 14 theCa iii ionbeamisfractionalizedintoCa i andCa ii ,andvariousatomicstateswithin each[40].Duetothis,theidealsituationwouldbetoobtainabeamconsistingentirelyof Ca ii ,avoidingthefractionalizationsubsequenttochargeexchangeandtheadditionalback- groundphotonsproducedasexcitedelectronicstatesdecayfollowingthechargeexchange process.Ultimately,theonlinebeamcomingfromthegascellwasapproximatelyevenly splitbetweensingly-anddoubly-chargedions(seeSection2.2.1),andsospectroscopywas performedonthe4 s 2 S 1 = 2 $ 4 p 2 P 3 = 2 (393 : 3663nm, A 21 =1 : 47 10 8 s 1 [41])transition insingly-chargedcalciumions.TheatomiclevelsforCa ii areshowninFigure2.4. 2.1.2Lasersystem ToexcitethistransitioninCa ii ,aSirahMatisseTSTi:Sapphireringlaserwasused.This laserfrequencywaslockedto787nmlightusingaHighFinesseWSU-30wavelengthmeter, whichwascalibratedbyafrequency-stabilizedHe-Nelaser.The787nmlightwasthen frequency-doubledusingaSpectraPhysicsWaveTrain,inordertoreachthenearUVlight (393nm)required.Afterbeingtransportedthroughaer-opticcable,300 µ Wofthis lightwasthencollimateddownthebeamline.Thelaserpowerwasstabilizedjustprior toenteringthebeamlineusingalaser-powercontroller(LPC)[43].Thecontrollingoptic oftheLPCwasplacedimmediatelyfollowingtheTi:Sapphirelaser,priortothefrequency doubler.Thetransmittanceofthisdevicewascontrolledusingvoltageinputfromapower metermonitoringa10%sampleofthebluelaserbeamafterexitingtheer-opticcableand passingthroughapolarizingcubetotheverticalpolarizationofthelight.Thiswas thetimethatthissetupwasusedinanonlineexperimentatBECOLA,andensuresthat anylongtermdriftsintransmittanceoftheerorrotationofthepolarizationexitingthe ercanbecounteractedbytheLPC,allowingastable300 µ WbeamtoentertheBECOLA 15 Figure2.4:Atomicstates,andhypsplitting.Ontheleft,theelectroniclevelsforCa ii aredrawn[41].Ontheright,theselectedtransitionisdrawn,alongwiththehfsplittingfor I =3 = 2,asin 37 ; 39 Ca[42].Theallowedhftransitionsaremarkedinred. 16 Figure2.5:Thelasersetupusedinthisexperiment.TheSirahMatisseTSTi:Sapphirelaser wasfrequencylockedusingthestabilizedHe-Neandthewavelengthmeter,bysampling1% ofthelightasshown.Themajorityofthislightwasfrequencydoubled,andtransportedto theexperimentalareausingaeropticcable.Here,afterpassingthroughapolarizingcube, 10%ofthelightwassampledbyapowermeterinordertocontroltheLPC,whichmodulated thepowerenteringthefrequencydoubler.Thisallowedpowertuationsintroducedby thetransmissionthroughtheertobeeliminated,andensureastable300 µ Wbeamfor CLS. beamlineforthespectroscopy.AschematicofthissetupcanbeseeninFigure2.5. 2.1.3Hypinteraction 36 ; 38 ; 40 ; 44 Caareeven-evennuclei,possessingaground-statenuclearspinof0.For 37 ; 39 Ca however,theground-statenuclearspinis3 = 2,duetoanunpairedneutroninthe0 d 3 = 2 shell. Withanon-zeronuclearspin,thealignmentbetweenthespinofthenucleus, I ,andthe atomicspin J createsahypsplittingofthelevelsintheatomictransition,leadingto sixallowedtransitionenergiesforthe 37 ; 39 Caspectra. 17 Theshiftofanelectroniclevelrelativetotheenergyisgivenbytheequation: E = K 2 A hf + 3 K ( K +1) 4 I ( I +1) J ( J +1) 8 I (2 I 1) J (2 J 1) B hf (2.1) where K = F ( F +1) I ( I +1) J ( J +1), F isthequantumnumberbythevector F = I + J ,and A hf and B hf arethehypercouplingconstantsoftheatomiclevel. TheCa ii transitionusedinthisworkconnectstheupperstate 2 P 3 = 2 ( J =3 = 2)andthe lowerstate 2 S 1 = 2 ( J =1 = 2),usingthenotation 2 S +1 L J .Fromthenuclearspin, I =3 = 2 presentin 37 ; 39 Ca[42],possiblevaluesof F intheupperstateareintegerstepsbetween (3 = 2+3 = 2)and(3 = 2 3 = 2),leadingtothevalues F upper =3 ; 2 ; 1 ; 0 : (2.2) Possiblevaluesof F inthelowerstatearethenintegerstepsbetween(3 = 2+1 = 2)and (3 = 2 1 = 2),giving F lower =2 ; 1 : (2.3) Thus,theupperlevelissplitintofourenergylevels,whilethelowerissplitintotwo.Due totheselectionrulesofthetransition, F =+1 ; 0 ; 1,sixtransitionsareallowed: ( F upper ;F lower )=(3 ; 2) ; (2 ; 2) ; (1 ; 2) ; (2 ; 1) ; (1 ; 1) ; (0 ; 1) : (2.4) Toobtaintheinenergyfromthecentroid,Equation2.1canbeusedto 18 E upper and E lower ,givingatransitionenergyof E = E 0 + E upper E lower (2.5) where E 0 isthecentroidenergyofthetransition.Aschematicdiagramofthishfsplitting, highlightingtheallowedtransitionsisshowninFigure2.4. Thehypcouplingconstants A hf and B hf areas, A hf = 0 = ( IJ ) ; (2.6) B hf = eQV zz : (2.7) Here and Q arethemagnetic-dipoleandspectroscopicelectric-quadrupolemomentofthe nucleus, e istheelementarycharge,and B 0 and V zz arethemagneticand gradientrespectively,whichareproducedatthenucleusbytheorbitalelectrons.Inthecase ofthelowerlevel( 2 S 1 = 2 )theelectricissphericallysymmetric,leadingtoagradientof0 atthenucleus,andthusa B hf of0.Thismeanstherearethreehypcouplingconstants foreachisotope. Whenahfspectrumismeasured,thepositionsofthepeakscanbettedusingthe energiesdescribedfromEquation2.5byallowingthesehypnecouplingconstantstovary. Fromthisthehypecouplingconstantscanbedetermined,andthusthenuclear momentscanbeextractedusingsomeknowledgeof B 0 and V zz ,whichwillbethesamefor eachisotopeassumingapoint-likenucleus[37](seeSections3.2.2.1and3.3). 19 2.1.4Bunchedbeambackgroundsuppression Inrecentyears,backgroundsuppressionbyusingabunchedionbeamhasledtomajorad- vancementsinCLS[38,39].StrayphotonsreachingthePMTfromthelaserareat sourceofbackground.Thesecannoteasilybeoutduetothefactthatthe cencephotonsarethesamewavelengthasthelaserlight.Bybunchingtheionbeamand countingphotonsonlyinthewindowoftimewheretheionbunchcomespastthedetector,a largeportionoftheconstantbackgroundcausedbystraylightfromthelasercanbeignored. If,forexample,theionbeambunchisreleasedoncepersecond,andthebunchhasatime spreadof1 µ s,thiswillreducethemeasuredlaserbackgroundbyafactorof10 6 ,without reducingtheamountofsignalobserved(assumingperfectbunching ThistechniqueisemployedatBECOLAusingatwo-sectioncooler/buncher[44,45].The sectionisoperatedwithapressureofapproximately0 : 1mbar[45],to tlycooltheincomingions.Ionsarecontainedlaterallyinthetrapusingaradio- frequencyquadrupole(RFQ),whichissensitivetothecharge/massratiooftheions.As theionsmovethroughalongitudinaldragtheymoveintothebunchingsection,where thepressureisapproximately100timeslower[45],andaweakerlateralcontainmentis present.Thislowerpressuresectionallowsforbetteremittanceintheionbunch,andallows alowenergyspread,resultinginatimewidthof1 µ sforeachbunch(FWHM).Aschematic ofthecooler/buncherisshowninFigure2.7,andthetialpumpingdesigncanbe seeninFigure2.8. Thebunchestypicallyhaveatime-spreadof1 µ sFWHM,howeverdependingonhowfull thetrapis,thetimespectrumatthedetectionregioncanbeInFigure2.6,thetwo mostextremetimespectrafromthisexperimentcanbeseen; 36 Caand 44 Ca,whichwere 20 Figure2.6:Therecordedtimespectraforthe 36 Caand 44 Cabunchedbeams. 36 Cahad thefewestionscontainedineachbunch,thusgivingthenarrowesttime-width,while 44 Ca hadthemostionsperbunch,duetocontaminationfromthemoreabundant 40 Ca(see Section2.3.1.2),leadingtoaslightlywiderpeakintime.The 36 Catime-spectrumistaken fromthefull33hrunningtimewiththatisotope,whilethe 44 Catime-spectrumistaken fromasinglereferencemeasurementof < 1h.Otherisotopesmeasuredinthisexperiment fallwithinthesetwoextremes,withatime-widthofapproximately1 µ s. operatedwiththefewestandgreatestnumberofionsinthetrap,respectively. Thisbunchedbeamtechniqueiscrucialtoreducethelaserrelatedbackground,which isthesamewavelengthasthesignal,andisthepredominantsourceofbackgroundforthe Caisotopemeasurements.Severalfactorscanlimittheenessofthebunchedbeam techniquehowever.Toavoidlosingsignal,lossofionsduetotheholdingtimemustbetaken intoaccount.Atlowincomingratesofions,theBECOLAcooler/buncherisconsistently abletoholdionbunchesforupto1sbeforeseeingadecreaseinthetrapping,as showninFigure2.9.Whendealingwithunstableionshowever,thehalf-lifeoftheisotope canbecomealimitingfactor,andthereductioninbackgroundmustbebalancedagainst thelossofsignalcausedbytheincreaseofdecaystakingplaceinthetrap.Whenthe trapforanextendedtime,therateofincomingionswillbecomebythedecayofions currentlyinthetrap,andthesignalperbunchwillplateau.Regardlessofthetimebetween bunches,thetimespreadofeachbunchwillalwayspanacross1 µ s,resultinginaconsistent amountoflaserrelatedbackgroundrecordedperbunch.Becauseofthis,bunchingtimes tlylongerthanthehalf-lifeoftheisotopeinquestionwillleadtoaconsistentSNR 21 perbunch,however,becausethereisnogaininthesignalstrengthfromtheincreasedtime, thelongertimebetweenbunchesservesonlytoreducethefrequencyatwhichsignaland noisearebeinggathered. Todeterminetheoptimalbunchingtimefortheshort-livedCaisotopes,thisreduction insensitivitywastakenintoaccount.ExaminingSNRasafunctionofthebunchingperiod, T b ,apeakcanbefoundat ˘ 1 : 8 T 1 = 2 .Becauseaconsistentamountofbackgroundis recordedduringthetimewindowofeachbunch,thebackgroundrate(pertime,notper bunch)variesas B ( T b ) / 1 =T b : (2.8) Thesignalrecordedpertimedependsontheamountofionspresentatthetimethebunchis released,andtherateatwhichbunchesarereleased.Becauseofthedecayswithinthetrap, theionsheldinthetrapatthetimeofreleaseisproportionalto(1 )(1 e b ),where isthedecayconstantoftheisotope.Thesignalrateisthen S ( T b ) / (1 )(1 e b ) =T b (2.9) whentakingintoaccounthowfrequentlythesebunchesofsignalarrive.TheSNRcanthen bewritten SNR= S ( T b ) p B ( T b ) / (1 )(1 e b ) p T b (2.10) Expressingthisintermsofthehalf-life, T 1 = 2 =ln(2) ,wearriveatamoreusefulexpression, SNR / q T 1 = 2 (1 e T b =T 1 = 2 ) q T b =T 1 = 2 (2.11) Inthisform,theSNRcanplottedversus T b =T 1 = 2 asshowninFigure2.10.Fromthisitwas 22 Figure2.7:Aboveisaschematicofthecooler/buncherusedatBECOLA.Auniqueaspect ofthisdesignistheseparatecoolingandbunchingsections,whichareoperatedwitht gaspressures.Thetialpumpingschemeusedtomaintainthecentralhigh- pressureregioncanbeseeninFigure2.8.Theionsarecontainedlaterallybythepseudo- potentialofaradio-frequencyquadrupole(RFQ),whileadragpotentialismaintainedin thelongitudinaldirectionusingsegmentedelectrodes.Thegraphbelowtheschematicshows thedragandtrappingpotentialalongthebeamaxis,thedashedlineontherightindicates thepotentialusedtoreleasetheionbunch.Figurefromref.[44]. determinedthatthemaximumSNRisobtainedusingabunchingtime T b ˇ 1 : 8 T 1 = 2 .This wasusedtosetthebunchingperiodusedduringtheexperimentfor 36 ; 37 Ca.Duetothe longerhalf-livesandhigherrates,thebunchingperiodsselectedfor 38 ; 39 Cawereshorterthan theseidealcalculations,becauseofthespace-chargelimitationsofthetrap.Thesebunching periodsandratesareshowninTable2.2. 2.1.5Doppler-shift Theprecisionwhichcanbeachievedthroughcollinearlaserspectroscopyisduetouseof theDoppler-shift.Whilethelaserisstabilizedandheldataspcfrequency,thevelocity oftheionbeamcanbemobyscanningthesmallvoltageappliedatthedetection 23 Figure2.8:Illustrationofthetialpumpingchannel.Thehighandlowpressure regionsareconnectedviaanarrowchannelwithrelativelylowconductance(marked\C"). Theheliumisintroducedintothehighpressureregion,andpumpedfromthelowpressure regionusingaturbomolecularpump(marked\S TMP ").Figurefromref.[44]. region.Scanningacrossthetransitionisaccomplishedbyutilizingthefactthattheionsare movingalongtothelightpath,allowingaprecisevariationintheelaserfrequency. Inaddition,byperformingantmeasurementat30keV,thevelocityoftheionbeam reducesDoppler-broadeningduetothermalmotionofionswithinthebunch,asshownin Figure2.11,resultinginanarrowlinewidth[46,47].Becausetheareaofthepeakisnot changed,thepeakheightisincreasedbythesamefactorasthelinewidthisreduced,thus thiskinematicalcompressionincreasesbothsensitivityandresolution[12]. Whenacceleratedacrossthepotential U ,kineticenergy(KE)of qU isgiven toeachion.UsingtherelativisticexpressionforKE, KE= qU = 1 p 1 2 mc 2 mc 2 (2.12) where m istherestmassoftheion,and isthevelocityinunitsof c .Thisexpressioncan 24 Figure2.9:Thebunchingasafunctionofcoolingtime.Asshownbytheblack squares,withsmallquantitiesofionsthereisminimalreductioninforholding timesuptoonesecond.Thesemeasurementsweremadeusingpotassiumions.Figurefrom ref.[45]. 25 Figure2.10:Aplotshowingthethatthebunchingperiod, T b ,hasonthesignaland noisecollectedwhenusingunstableisotopes,assumingthattheofthecooleris bythebunchingperiod,andthatthemeasurementtimeforeachbunchis1 µ s regardlessoftheholdingperiod.Theverticalaxisrepresentsthenoiseandsignalcounts collectedafteragiventime(seeEquations2.8and2.9),andtheirratio(Equation2.11). Nounitsareprovidedontheverticalaxis,astheabsolutequantitiesalsodependonother factors,suchasthehalf-life,incomingbeamrates,andthemeasurementtimeofthebunch. Thesignalcountsareshowninblue,noisecountsinred,andSNR(scaledarbitrarily)is showninblack.Astheperiodisincreased,theionsheldinthetrap,andthusthesignal perbunch,willeventuallyplateau,however,theincreasedtimebetweenbunchesleadstoa continuedreductioninsignalcollected.Thenoisecollectedfromthelaserisalsoreduced duetothedecreaseintotalbackgroundcausedbymeasuringthedatainfewerbunches. Althoughbothsignalandnoisearemonotonicallydecreasing,thedrasticreductioninnoise causestheSNRtobemaximizedwhen T b ˇ 1 : 8 T 1 = 2 . 26 Figure2.11:Thisplothighlightsthebofantmeasurementwithregardto Doppler-broadening.Foragiventemperatureofions,therewillbeasmallspreadinkinetic energy,leadingtoaspreadinvelocity(blackbands).Whentheionsareacceleratedover alargepotential,althoughtheenergyspreadremainsthesame(andissmallcomparedto theenergyfollowingacceleration),thecorrespondingvelocityspreadisreduced(redbands). Becausethevelocitydeterminestheefrequencyofthelaserlightthattheionsare interactingwith,thisvelocitybunchingresultsinnarrowerresonancepeaks. 27 besolvedfor togive, = s 1 1 1+ q 2 ( U=m ) 2 +2 qU=m (2.13) where isthevelocityinunitsof c , q isthechargeoftheionsinunitsof e , U isthepotential inVolts,and m isthemassoftheionsinunitsofeV/ c 2 .Fromthisvelocity,the Doppler-shiftedfrequencyisexpressed, f obs = f s 1 1+ (2.14) where f obs istheDopplershiftedfrequencyobservedbythecollinearparticlesastheymove awayfromthesource,and f isthestablefrequencyofthelaser.Inthisway,eachvoltage stepattheinteractionregioncanbepreciselyconvertedtoafrequency. Forthisexperiment,thesingly-ionizedCaionswerereleasedfromthecooler/buncher withanenergyofapproximately29850V[36].Thisresultsinashiftoftheobservedlaser frequencyequivalenttoapproximately16MHzforeachvoltappliedtothedetectionregion. Thesetlaserfrequency,aswellastherangesofthescanningvoltageappliedtothedetection regionforeachisotopeareshowninTable2.1. 2.2Isotopeproduction 2.2.1Rareisotopes OneuniqueaspectoftheBECOLAfacilityistheaccesstotisotopesthat theNSCLcanprovide.Byusingfastightproductionandseparation,rareisotopeswith 28 Table2.1:Thesetlaserfrequencyandscanningvoltageusedforeachmeasurement.The setlaserfrequencyisthewavenumberoftheTi:Sapphirelaserpriortothefrequencydou- bler,whilethescanningregionindicatestherangeofthescanningpotentialappliedtothe detectionregionasthespectrawererecorded. SetLaserFreqScanningRegion(s) A (1/cm)(V) 3612724 : 149665{45 3712723 : 92541 80{30,45{105 3812723 : 7097 15{65 3912723 : 51262 105{ 45,70{120 4012723 : 30257 15{65 4412722 : 5675 15{65 shorthalf-livescanbeproducedintquantitiestoperformCLS.Afastprimarybeam of 40 Ca(140MeV = nucleon)isobtainedusingthecoupledsuperconductingcyclotrons,and impingedonaBetarget,withathicknessof658mg = cm 2 .Fromtheprojectile-fragmentation reactions,reactionproductsareproducedinht,andspisotopescanbeselectedusing theA1900fragmentseparator[33].ThistechniqueiscomplementarytotheISOLtechnique employedatotherCLSfacilities[35],inthatitallowsbeamsofrareisotopestobeproduced withoutregardstotheirchemicalproperties. Thefastbeamistheninjectedintoagascellstopper[34]wheretheions arethermalized.AfterthestoppingrangeselectioninthecellCa-ionsareextractedatan energyof30keVandtransportedtotheBECOLAfacility.Onextractionfromthegascell, apotentialof400Vwasusedforcollision-induceddissociationtodissociatewatermolecules attachedtothecalciumions.Thisvoltagewasselectedtogivealargeportionofsingly- chargedionswithnoattachedmolecules,withoutgreatlyreducingtheoveralland degradingtheemittanceofthebeam.Thedistributionofthesetstateswhenthis dissociationvoltagewasusedareshowninFigure2.12. AtBECOLA,therareisotopesareinjectedintothecooler/buncher,andextractedin 29 Figure2.12:Experimentaldatafromthegascellgroupshowingthemass-to-chargespectrum of 39 Caextractionfromthegascell.Thehorizontalaxisshowstheselectedmass/charge ratio(amu/ e ),whiletheverticalaxisindicatesthedecayrateseenintheextractedbeam. Thepeakat39isattributedtosingly-charged 39 Caions,whilethepeakat19 : 5contains thedoubly-chargedionsofthisspecies.At m=q =57,apeakcanalsobeseenduetosingly- charged 39 Cawithasinglewatermolecule(18amu)attached.Thisspectrumwastaken using400Vforthedissociationpotential.Asshown,thisresultsinacantportionof singly-ionized 39 CawhichwasthentransportedtoBECOLAtoperformspectroscopy. 30 bunchesperiodically.AsdiscussedinSection2.1.4,therepetitionrateofthebuncheswas selectedforeachisotopetooptimizethesignal-to-noiseratioofthesystem.For 36 ; 37 Ca,the bunchingratesweredeterminedbasedonthehalf-life,howeverinthecaseof 38 ; 39 Caand thereferenceisotopes 40 ; 44 Ca,therepetitionratewaslimitedbythenumberofionsable tobecontainedinthetrapwithoutdisturbingthebeamemittance.Theapproximateion beamintensityatBECOLAandtherepetitionratesusedforthespectroscopyarepresented inTable2.2. Table2.2:Thehalf-life( T 1 = 2 ),bunchingperiod( T b ),ratesatthecooler/buncherfortherare isotopes,andthetotalmeasurementtimeusedforeachisotope.Thebunchingrepetition ratesfor 36 ; 37 Cawerechosenbasedontheshorthalf-lives,whilethebunchingperiodsof 38 ; 39 Cawerebythehighionratesandthespace-chargelimitationsofthetrap. Becausethereferenceisotopes 40 ; 44 Cacanbeproducedwithhighioncurrents,theyhave tlimitations,whicharediscussedinSection2.3.1.2. T 1 = 2 T b RateMsmtTime A (ms)(ms)(1/s)(h) 361021805033 : 3 3718133096042 : 3 38440220135009 : 8 39859 : 630600008 : 5 40|30See2.3.1.2 < 1 44|30See2.3.1.2 < 1 2.3Systemupgrades Inordertoachievethesensitivityrequiredtomeasureallfourisotopesduringthisonline run,severalupgradesandoptimizationswerecarriedoutattheBECOLAbeamline.The mostdramaticchangeforthisexperimentwastheinstallationofanewly-designedphoton detectionsystem(PDS),butotherexperimentalstrategieswerealsoemployedforthe timeinthisonlineexperiment,suchastheuseoftworeferenceisotopesforcalibration. 31 2.3.1Referenceisotopes Toobtainaprecisemeasurementoftheisotopeshift,thespectrumoftworeferenceiso- topesweretaken.Bymeasuringanisotopewithaknowntransitionfrequencyperiodically throughouttheexperiment,notonlydoesitserveasareferencefortheisotopeshift,butit alsoallowsanunderstandingofthesmallthatmaytakeplaceduringthehours ordaystakentomeasuretheunknownisotope.Theknownisotopeshiftbetweenthetwo referenceisotopescanalsobeusedtounderstandthereleasevoltageofthecooler/buncher (seeSection3.1).Thesemeasurementsrequireanionsourcewithatrate andastablebeamcurrent.Usinganion-sourcewithahighrateallowsforthesecalibration measurementstobetakenquickly,inordertominimizetheinterruptionoftherareisotope beam.Becauseoftherelativelyshorttimeframeforeachofthesemeasurements( < 1h), thestabilityofthesourceiscrucial,topreventinthebaselineofthespectrum duringthemeasurement.Ifasourcerequiresalongperiodoftimetostabilize,thisagain takestimeawayfromthemeasurementoftheunknownisotope. AtBECOLA,aPenningIonizationGauge(PIG)sourcewasusedtoservethispurpose withgreatsuccess.Thissourcerequiresrelativelylittlewarmup,andprovidesaconsistent ioncurrent.Itcanalsobeusedtoproduceawiderangeofelementsbyusinganelectrode orgaswhichcontainsthedesiredelement(s).Ionsfromthesourceareproduced fromboththecathodematerialandthegas.Byutilizingagaswhichhasa tlytmassthantheisotopeofinterest,theRFQinthecooler/bunchercan beusedasamasstoselectonlytheisotopeofinterest.Althoughthissourcewas designedandinstalledin2015,severalupgradeswereperformedpriortothisexperimentin ordertoimprovethestabilityandwarm-uptime. 32 2.3.1.1PIGsourceupgrades ThedesignofthePIGsourceusedatBECOLAprovidesaconvenientwaytoproducelarge currentsofavarietyofstableions(typicallyontheorderof10nAofbeamcurrentfollowing thecooler/buncher).ThewofthegasintothePIGsourceisthemost aspectofthesourcetostabilizeandcontrol.WhenoperatingthePIGsource, inthegaspressurehavebeenobservedtodirectlytheioncurrentexitingthesource, includingthecurrentafterthecooler/buncher.Inpreviousexperiments,thispressurewas controlledbyadjustingahandvalve,whichrequiredloweringthehighvoltageinorderto physicallyadjusttheknob,aswellasaperiodoftimetostabilizeaftermakingadjustments. Toimprovetheperformanceandreducethetimerequiredtostartthesource,aw controller(MFC)wasobtainedfromBronkhorst,Inc.withawrangeof0 : 014{0 : 7ml n = min. Whilethisupgradegreatlythestart-upprocedureforthesource,italsohasthe bofenablingeasycontrolofthegaspressure,allowingamorethoroughandrepeatable explorationofthesourceparameters.TheMFCallowsverystablepressurestobeachieved quickly,andcanbecontrolledremotelywithoutloweringthehighvoltage.Becauseofthis easeofcontrol,thePIGsourcewasabletobeoperatedsuccessfullyusingaHegas, whichwasnotpossiblewhenusingthehandvalveduetothenarrowrangeofgaspressures whichenabletheplasmageneration.Formoredetailsregardingthesetupandoperationof theMFC,seeAppendixA. InordertousethissourcetoproducethetworeferenceCaisotopes,cathodeswere orderedfromACIAlloys,Inc.consistingofacustomalloywithanatomicratioof2parts Snto1partCa,withgreaterthan99%purity.ThemassbetweenSnandCa isttoouttheSninthecooler/buncher.DuetobeingmixedwithSn,the 33 cathodesareexpectedtobelessreactivethanpureCa,however,carewastakenduring shippingandinstallationtominimizecontactwithairtoavoidoxidizationofthesurface. TheprimaryexposuretoairoccurredduringtheinstallationofthecathodesintothePIG source,aprocesstakingapproximately30min,andnofromoxidizationwereobserved intheoperationofthePIGsource. 2.3.1.2Isotopiclimitations ThePIGsourceproducesmuchlargerquantitiesofionsthanareneeded,howevertheprimary limitationisduetothefactthatthenaturally-occurringisotopesareproducedinratio withtheirabundancewithinthecathodematerial.Becausetheyaresocloseinmass, thecooler/buncherisunabletobetweenthereferenceisotopes,andthespace-charge limitationsofthetrapthendictateamaximumamountofsignalperbunchforagiven isotope. Theprimaryreference, 40 Ca,hasanaturalabundanceof96 : 94%andservesasthe referencefortheisotopeshiftoftherareisotopes,inordertoextractthechargeradii.A secondisotope, 44 Ca,wasalsomeasuredthroughouttheexperiment,inordertocalibrate thespectraasdescribedinSection3.1. 44 Caisthesecondmostabundantisotopeatonly 2 : 09%,makingitmoretomeasuretly. Whenmeasuringthetransitionin 44 Ca,theiontrapwasslightlyovinorder toachieveanacceptablereferencemeasurementinunder1h.Thisresultsinaslightly widertimespreadoftheionbunchatthedetectionregion(2 µ sFWHM),howeverseveral measurementsof 44 Caweretakenusingthenarrowertimewidthaswellinordertoverify thatthissmallchangedidnotthepeakpositionintheenergyspectrum.Thetime spectraof 36 Caand 44 CaareshowninFigure2.6. 34 2.3.2Photondetectionsystem Whilethebunchedbeamtechniqueallowsfordramaticimprovementstothelaserback- ground,thesensitivityofthesystemisstilldependentontheabilityofthephotondetection system(PDS)tocollectresonantphotonsfromthelaser-excitedionbeam,andexcludestray lightfromthelaser.Signalcollectionisperformedusingaphoto-multipliertube(PMT) whichisalignedradiallytothebeamaxis.Thegoalistoallowphotonsfromthelaserto simplypassbythedetector,whilethosewhichareabsorbedandspontaneouslyemittedfrom theionbunchmaybeemittedtransversetothesharedbeampath,andcanthenbecollected usingaegeometrytodirectasmanyphotonsaspossiblefromtheinteractionregion towardsthePMT. ThePDSoriginallyinstalledatBECOLAin2013doeswellatcollectingalargeamount ofthesignalproduced,andwillfromherebereferredtoasthe\oldsystem".Anupgraded systemwasdesignedbycollaboratorsatTUDarmstadt,withastrongemphasisonthe rejectionofthenoisefromstraylaserlight,whichwillhereafterbereferredtoasthe\new system".Inpreparationforthisexperiment,simulationsofthenewsystemwereperformed tocompareittotheoldsystem.Thesimulationsindicatedthatthisnewsystemwould provideanincreasedsensitivityfortheCa ii measurement(seeTableA.1fordetailedresults ofthesimulations).ThecomponentsweremanufacturedatTUDarmstadtandinstalled atBECOLAinthesummerof2017,andanidenticalsystemwasalsoinstalledatTU Darmstadt.Afterinstallationofthenewsystem,performancestudieswerecarriedoutto evaluatethereal-lifeperformanceofthetwosystems,andthenewsystemwasutilizedfor theonlineexperiment. 35 2.3.2.1Motivation TheoldsystematBECOLAinvolvesanellipsoidalasshowninFigure2.13.The laserandionbeamspassinandoutoftheellipsoidandcrossafocalpointoftheellipsoid. Lightwhichisemittedatthefocalpointisfocusedtothesecondfocalpointoftheellipsoid, wherethePMTispositioned.Thissystemdoesanexcellentjobatcollectingtheresonant photonsproducedatthefocalpoint,butthesensitivitycouldbeimprovedthroughbetter noiserejection.ThestrategyoftheoldsystemistoincreasetheSNRbymaximizingthe signal.AnalternativemethodofachievingahighSNRistominimizethenoisereachingthe detector. Thenewsystemutilizesanellipticalsection,whichspatiallyseparatesthesignalfrom thenoise.Unliketheoldsystemwhichprimarilycollectedphotonsemittedfromthefocal pointoftheellipsoidaltheellipticalofthenewsystemallowscollection ofphotonsfromtheentiresectionofbeampassingthroughthedetector,asthefocalregion oftheellipticalreisalineinsteadofapoint(seeFigureA.6).Thesecondfocal regionofthisellipticalislocatedjustoutsideofthevacuum-enclosedbeamline,just througha5inwindow.Apairofmovableplatesoutsidethiswindowcreateahorizontal aperturewhichallowsthefocusedsignaltopassthrough,whileblockingamajorityofthe background.Followingthisgeometriccut,acompoundparabolicconcentrator(CPC)is usedtorejectphotonswhichareenteringwithanangleofmorethan20 ° fromthenormal. ThisangularcutfurtherreducesthenoisereachingthePMT,althoughitdoescomeatthe costofsomesignal.Arenderingshowingthecomponentsofthenewsystemisshownin Figure2.13.Detailsabouttheopticalsimulationsusedtodecidespcharacteristicsof thenewsystem,includingexpectedperformancecomparedtotheoldsystem,canbefound 36 Figure2.13:Renderingsofboththeoldandnewsystems.ThePMToftheoldsystemis placedatthefocalpointoftheellipsoidalctor.Thenewsystemimagehastwocomplete photondetectionregions,eachwiththeirownellipticalraperture,CPCandPMT. Toenableabetterviewofthedesign,thedetectionregionclosesttotheviewerhashad twopiecesremovedfromboththeellipticalandtheCPC(green).Thedetection regionfurtherfromtheviewerhashadtheCPCandPMTremovedtoallowaviewofthe adjustableaperture(darkblue)whichsitsjustoutsidethewindow.Fourrodssupportthe tworegions,andareheldinsidethebeampipeusingfouradjustableinsulatingfeetoneach end(cyan),inordertoallowthescanningpotentialtobeappliedtothePDS. inSectionA.1.1. 2.3.2.2PerformanceStudies Thefabrication,installation,andalignmentprocedureofthenewsystemiscoveredinSec- tionA.1.2.Withboththepreviousandupgradedsystemsinplaceinthebeamline,sys- tematictestswereperformedusingstablebeamsatBECOLA.Optimizationofthelaser backgroundinallthreedetectionregionswasperformedbycarefullyaligningthelaser. Unfortunately,thelaserbackgroundinthethreeregionswereeachminimizedatdt 37 alignments,makingcomparisonThisislikelyduetoarotationalmisalignmentof theoldsystem,whichcannotcurrentlybeadjusted.Forcomparisonbetweenthesystems,an alignmentwasusedwhichallowedalldetectionregionstobeneartotheirminimum.Inthe actualexperiment,thepreviousdetectionsystemwasremoved,allowingfurtheralignment tolowerthebackgroundratesinthetwonewdetectionregions. Theprimaryvariableallowingustogaugeperformanceofthenewsystemwasthegeo- metriccut(mask)followingtheellipticalAnadjustableplateoneachsideallowsa slitofvariablewidthtobeformed.Itwasexpectedthattheellipticalchamberwouldfocus thelighttoasinglestrip,sothattheadjustableaperture(showninFigure2.13)couldbe tomaximizetheSNR(seeFigureA.3forthesimulated Whileoneofthenewdetectionregionshadtheslitatthefocalplaneopenedaswide aspossible,thewidthoftheotherwasvaried,andateachpointtheSNRbetweenthetwo detectionsystemswascompared.Surprisingly,theSNRsimplyincreasedastheslitswere openedwider,asshowninFigure2.14. Duetothisunexpectedresult,theintensityofsignalacrossthefocalplanewasexamined. Toevaluatetheintensityatthefocalplane,theplateswereadjustedtoa3mmslit whichwasthenmovedalongtheY-axisofthefocalplane.Thisslitwasscannedacross thefocalplaneandateachposition,theresonanceof 40 Cawasmeasured.Becauseeach measurementpositionrequiresatlyhigh-qualityscanofthecalciumresonanceline, asinglescanoftheslitacrossthefocalplanetakesatamountoftime.Tominimize theofanydriftintheionsourceintensityduringthistime,theslitpositionwasadjusted ononeofthenewdetectionsystems,whiletheotherwasheldconstantwiththeslitwide open.TheSNRwasthennormalizedbycomparingtothewideopendetectionsystem.The resultsareshowninFigure2.15andanapproximateexpectedfromsimulationsis 38 Figure2.14:SNRresultsusingvariousopeningwidthsatthefocalplane.Thevertical axisistheSNRofthedetectionregionofthenewsystem,normalizedusingthesecond detectionregionofthenewsystemwiththeadjustableplatesopenaswideaspossible.It wasexpectedthatthesignalwouldbeconcentratedatthecenterofthefocalplane,meaning thattherewouldbeapointwheretheSNRbeginstodecrease,asanywideropeningonly servestoallowmorestrayphotonsintothesystem.Asshown,thiswasnotthecase.The widertheopening,thebettertheSNRbecomes. 39 Figure2.15:ResultsexaminingtheSNRatvariouspointsalongtheY-axisofthefocalplane. Theplatesinonedetectionsystemweresettoa3mmwideslit,andacalciumresonancewas recordedwiththisslitatvariouspositionsalongtheY-axisofthefocalplane.Thevertical axisistheSNRofthesystembeingscanned,dividedbytheSNRofthecontrolsystem.Red circlesshowthenormalizedSNR(comparedtotheotherdetectionsystemwhichwasused asacontrol),whilethegraydottedlineindicatestheshapeexpectedfromthesimulation. Ratherthanobservingthesignalconcentratedintoasinglepeak(seeFigureA.3),thehighest SNRwasobservedintwopeaksoneithersideofthecenter.Furthersimulationstoexplore thisbehaviorcanbefoundinSectionA.1.3. shownaswell.Adoublepeakwasobserved,indicatinganunexpectedstructureinthesignal arrivingatthefocalplane.Inaddition,thepeakswerealsoabitwiderthanwhatwas expectedofthesinglepeak. TofurtheroptimizetheSNRofthenewsystem,severalothermaskswereusedtotryand takeadvantageofthestructureseeninFigure2.15.Inadditiontovaryingtheslitwidth,a stripoffoilwasplacedatthecenter,elycreatingtwoslits,allowingformorenoise 40 Figure2.16:SNRresultsusingvariousmaskstoselectthetwopeaksseenatthefocal plane.TheverticalaxisistheSNRofthenewsystem,dividedbytheSNRrecordedwith theoldsystem.Themaskusedforeachmeasurementisshownunderneaththeplot,gray areasarewherethefocalplanewasblocked,whilewhiteareasshowtheregionofthefocal planewherelightwasallowedtopassthroughthewindow.Whiletheadjustableplateswere usedtoblockthetopandbottom,a5mmor15mmwidestripofanodizedfoilwasusedto blockthecentralportion.ThefocalplaneislabeledalongtheXandYaxeswithunitsof mm.SelectingthetworegionsofpeakintensityseeninFigure2.15doesprovideabetter SNR,butinallcases,evenwiththefocalplanefullyopen,theSNRwasmorethanthat obtainedusingtheoldsystem. 41 rejectionatthefocalplane.Thistestwasdonewithvaryingwidthsblockingthecenter, andvariouspositionsofthetwoadjustableplates.Theresultsusingthesemasksareshown inFigure2.16.AlthoughtheSNRdoesincrease,itwasuncertainwhetherthedouble-peak structureseenatthefocalplanewouldbestabletoslightchangesinalignmentofthelaser orionbeam.Becausethisthoroughmeasurementoftheintensityacrossthefocalplane isnotfeasibleduringtheonlineexperimentduetothetimeconstraints,itwasdecidedto runtheexperimentwithoutthefoilblockingthecenter,andwiththeslitsopenaswideas possible,theleftmostconditioninFigure2.16.Bysimplyusingtheapertureatthefocal planefullyopenedhowever,thereislesspossibilityforaslightchangeofalignmentduringthe onlineexperimenttodrasticallyreducetheSNR.ThisdoesstillmarginallyimprovetheSNR overtheprevioussystem,butnotquitetotheextentexpectedfromthesimulations.This stillallowsforausablemeasurementof 36 Ca,especiallyconsideringthefact thattherearetwoidenticaldetectionsystemsinthenewinstallation.Furtherinvestigation ofthesebehaviorswasperformedusingsimulations,andcanbefoundinSectionA.1.3. Regardlessoftheseunexpectedbehaviorsatthefocalplane,thenewPDSwasable toperformbetterthantheoldsystemforspectroscopyonCa ii inthetests(see Figure2.16).Withaproperbehavioratthefocalplane,itshouldbepossibletoincreasethe performanceevenfurther,howeverthismayrequireareplacementofthefoilused fortheellipticalorabetteralignmentofthesystemtotheionandlaserbeam.In caseswherethechargeexchangecellisrequiredforthemeasurement,itisexpectedthatthe newsystemwillperformworsethantheoldsystem(seeTableA.1),sohavingbothsystems availableisstillanecessity. 42 Chapter3 Analysis RareisotopebeamsweresenttoBECOLAfromOctober7th,throughOctober15th,2017. Thedataobtainedwerecalibrated,andthehypspectrawereusedtoextractthenuclear momentsandchargeradii. 3.1Calibration ForpastexperimentsatBECOLA,calibrationhasbeenperformedusingasingle,stable isotope.AsthefrequencyisscannedusingaDopplershift,itiscrucialtodetermineboth thelaserfrequency,aswellastheenergyoftheionbeamwithhighprecision.Whenreleased fromthecooler/buncher,theionsareextractedwithapotentialthatis10Vlowerthanthe minimumpotentialofthetrap.Astheionsleavethehelium-trap,someresidualcooling canoccur,allowingtheabsoluteenergyofthebeamtobefromthenominal betweenthepotentialattheinteractionregionandthepotentialofthetrap.Inthepast, thisbehaviorwasexaminedbyrecordingthepotentialofthetrap,andusingtheknown absolutetransitionfrequencyofastableisotopetodeterminetheinvoltagewhich occursduringthereleaseofthebeam. DuetothevarietyofstableCaisotopesavailable,thedecisionwasmadetoinsteaduse themeasuredcentroidsoftwostableisotopes, 40 ; 44 Caforcalibrationduringtheexperiment. Byusingtheknownisotopeshiftbetween 40 ; 44 Ca,thereleasepotentialcanbeextracted 43 withoutrelyingonthemeasurementusingahighvoltagedivider.Thistechniquehasthe addedadvantagethat,asthesemeasurementsaremadethroughouttheexperiment,the actualvariationintheisotopeshiftmeasurementisobtained,asopposedtosimplyrecording theofanabsolutemeasurement.Thisallowsforabetterunderstandingofthe systematictakingplaceoverthedurationoftheexperiment. Everytwoorthreehoursduringtheexperiment,ameasurementof 40 Caand 44 Cawas takenusingthePIGionsource.Asanionbunchleavesthetrap,someresidualcoolingmay occurastheionsareacceleratedoutofthetrap,resultinginanuncertaintyofapproximately 10V.Byutilizingtworeferenceisotopeshowever,thisprimarysourceofuncertaintywas reducedbyafactoroftwo,andwaslimitedinsteadbytuationsofthehighvoltagepower supplythroughoutthecourseoftheexperiment. Thecalibrationofthedatawasperformedasfollows: Thescanningvoltagewasrecordedwithahighprecisiondigitalvoltmeter(DVM), KeysightTechnologies34465A,ateachscanningpositionduringtherun.Becauseofthis, measurementstakenatthesamevoltagestepmayeachhaveaslightlytvoltage recordedinthedata.Measurementsfromeachrunweresortedintovoltagebinswithawidth of0 : 99896419V,andthebincenterswereselectedforeachruntoensurethatthesumof alldeviations(betweentheDVMvoltageandthecenterofthebinwherethatmeasurement wasplaced)waszero.Thedeviationofthebinwidthfrom1Vwasduetothefactthat whenrequestingascanningvoltageof 1000Vfromthesystem, ˘ 999Varemeasured. Foreachreferencemeasurementtaken,thebeamenergywasconvertedintofrequency, andthetheknownshiftbetweentheseisotopes( 40 ; 44 =850 : 231(65)MHz[48]),wasused todeterminethepreciseenergyoftheionbeam.Thiscalibrationwasperformedusingthe spectrarecordedbytheupstreamdetector(channel1),astherewasmorelaserbackground 44 presentinthedownstreamdetector(channel2).Fortherareisotopemeasurements,the weightedaverageoftheneighboringreferencemeasurementswereusedtoshiftthebeam energy(voltage)ofeachrun.Theedgesofeachbinwereconvertedintofrequencyusing theDopplershiftandsetlaserfrequency.Thissamevoltageshiftwasthenappliedto theneighboringreferencemeasurementsof 40 Ca,andtheweightedaverageofthecentroid frequencieswasusedtothefrequencyofeachbinintherareisotopedata. Finally,nowthattherareisotopedatahasbeencalibratedandconvertedtoarelative frequency(usingtheprecedingandfollowingcalibrationmeasurements)thesehistograms werecombinedintoasinglehistogramforeachrareisotopeanddetector.Anuncertainty foreachvoltagecalibrationwasobtainedbasedonthestatisticalprecisionofthereference measurementsusedtodeducetheshift,aswellastheuctuationinvoltageshiftsbetween eachreferencemeasurement.Thissystematicuncertaintyaccountsforinthe cooler/buncherpotential,aswellasinthelaserfrequency,asitdependsdirectly onthevariationsinthereferencemeasurementsthroughouttheexperiment. Thedatafromthetwodetectionregionswerenotcombined,rather,theyweresepa- ratelyandtheparameterswerethenaveraged,withweightingbasedontheuncertainty. 3.2HypSpectra 3.2.1FittingFunction Afterconvertingtheresonancespectrafromvoltagetofrequency,thepeakswereusing anasymmetricpseudo-Voigtfunction.AtrueVoigtfunctionisaconvolutionofaGaussian andLorentzianlineshape,duetothecombinationofDopplerandcollisionbroadening,and thenaturallineshape,respectively[49].Thepseudo-Voigtfunctionemploysalinearcombi- 45 nationofaGaussianandLorentzian,inordertoserveasaclosed-formapproximationto theVoigtfunction.Forthisexperiment,thepseudo-Voigtusedwasoftheform I ( )= L ( )+(1 ) G ( )(3.1) where I istheintensityoftheresonanceasafunctionoffrequency , (0 << 1)describes thecombinationoftheLorentzianandGaussiancomponents, L and G ,whoselineshapes dependonthetotalareaunderthepeak A 0 ,thecentroidfrequency 0 ,andthetotalFWHM 0 .theLorentzianandGaussianfunctionsare G ( )= 2 A 0 0 r ln(2) ˇ e 4ln(2) 0 0 2 (3.2) L ( )= 2 A 0 ˇ 0 1 1+4 0 0 2 (3.3) ToapproximateatrueVoigtfunction, and 0 haveacomplicatedrelationshiptothe FWHMoftheLorentzianandGaussianfunctionswhichareconvoluted[50].Becausewe arefocusedonextractingthecentroidsofeachpeak,thetruewidthsofthedeconvoluted GaussianandLorentzianarenotofinterestsothisanalysissimplyusesthevaluesof and 0 intheprocess. Becauseofcollisionswhenexitingthecooler/buncher,thereisapossibilityofsome asymmetrypresentinthespectrum.Energycanbelostduringtheexit,andthusalow energytailmaybepresent.Toaccountforasymmetryusingtheaboveofthe pseudo-Voigt,afunctionofthefrequencyisusedinplaceof 0 ,resultinginonesidehaving 46 awiderlineshapethantheother[51]. ( )= 2 0 1+ e a ( 0 ) (3.4) Thisaddsanadditionalparameter a ,whichasymmetricallytheshapeofthespectrum. 3.2.2FittingProcedure Thelineshapeofthisasymmetricpseudo-Voigtfunctionisdeterminedbythe a and ,and inthecaseof 36 ; 38 Ca,theseparameterswereconstrainedtomatchthoseobtainedfromthe 40 Cacalibrationruns( = : 66, a =0 : 0007).Duetothelongerrunningtimes, inthehighvoltageofthecooler/bunchercanleadtoawiderresonancethanthereference measurements,whicharetakeninlessthan1h,andsothewidths( 0 )of 36 ; 38 Cawereeach freeparameters.Whilethereferencemeasurementshadatypicallinewidthof75MHz,the linewidthof 36 Cawas80MHz. 3.2.2.1Hypcouplingconstants Inthecaseof 37 ; 39 Ca,theunpairedneutronresultsinanon-zeroground-statenuclear spin.Asdescribedin2.1.3,theinteractionbetweentheelectronandnuclearspinleadstoa hypsplittingofthe4 s 2 S 1 = 2 $ 4 p 2 P 3 = 2 transition.ThesecanbeseeninFigures3.2 and3.4.Allsixpeaksofthesehypspectrawereusingacommonlineshapeand width,withtheircentroidpositionsgivenfromthethreehypcouplingconstantsas describedinSection2.1.3.Forthesetransitionsbetweenhyplevels,thetransition probability(andthustherelativeintensityofeachspectralline)canbedeterminedfrom theClebsch-Gordancotslinkingthelowerandupperstates.Thiscalculationdoes 47 requireaconsiderationofthehelicityofthelaserlightusedtoexcitethetransition,but because 37 ; 39 Caweremeasuredunderthesamelaserconditions(powerandpolarization)the lineshapeandrelativeintensitiesof 39 Cashouldmatchthoseof 37 Ca.Anotherconstraint, whichcanbeusedwhencomparingisotopes,istheratioofthe A hf couplingconstantsof theupperandlowerstate.Inthisratio,thenuclearfactorsfromthemagneticmomentand spincancelout,andtheonlyremainingcontributiontothisratiocomesfrom B 0 ofeach state,whichwillnotvarybetweenisotopes.Theratioofthe A hf constantsandtherelative intensityofeachpeakwereusedasconstraintsintheof 37 Ca,becauseoftheexcellent statisticsofthe 39 Cameasurement,. Noconstraintswereplacedonthehypcouplingconstantsof 39 Cawhenthe spectrum.TheobtainedvaluesareshowninTable3.1.Interestingly,the A hf ( 2 S 1 = 2 )and A hf ( 2 P 3 = 2 )hadaratioof26 : 24(4),whichdeviatesfrompreviousexperimentalresultsof 25 : 92(3)[52].Thereasonforthisdiscrepancyisnotknown.Anexaminationofourdatafor sourcesofpossiblesystematicthatcouldexplaintheinthisratiowere unsuccessful.Agreateruncertaintyintheabsolutevoltageofthesystem,thelaserfrequency, orevenananglebetweentheionandlaserbeamscannotaccountforthisdiscrepancybecause ofthefactthatitisaratio.Althoughthesevariablesdohaveanonthevaluesof A hf ( 2 S 1 = 2 )and A hf ( 2 P 3 = 2 ),theirratioishighlyinsensitivetothesesourcesofsystematic uncertaintyinourexperimentalsetup.Becauseofthisunexplaineddiscrepancy,an systematicuncertaintyisincludedwiththereportedhypecouplingconstantsfor 39 Ca. 3.2.3Results Theresultingof 36 ; 37 ; 38 ; 39 Ca,usingthedatacollectedfromtheupstreamdetector,are showninFigures3.1{3.4,alongwiththeresiduals.Atypicalexampleofthetwentyreference 48 Table3.1:Theobtainedhypecouplingconstantsof 37 ; 39 Caforthe4 s 2 S 1 = 2 and 4 s 2 P 3 = 2 states.Thestatisticalandsystematicuncertaintiesarereportedintheand secondparentheses,respectively.Thesystematicuncertaintycomesfromthehighvoltage calibration,andalsothedeviationofthe A hf -factorratiomeasuredfor 39 Cafromtheliter- aturevaluefrom 43 Ca.Thisunexplaineddiscrepancyisthedominantsourceofsystematic error. A (MHz) B (MHz) AI ˇ 2 S 1 = 2 2 P 3 = 2 2 P 3 = 2 373/2 + +1064.5(103)(08)+40.57(39)(27) 22.9(163)(05) 393/2 + +1457.20(14)(34)+55.53(9)(32)+5.79(26)(32) measurementsfor 40 ; 44 CaisshowninFigures3.5and3.6.Thesamewereperformed usingdatafromthedownstreamdetector,andtheresultswerecombinedusingaweighted average. 3.3NuclearMoments Whilethenuclearmomentsarenotthefocusofthisthesis,theirresultsarepresented here. Assuming B 0 and V zz tobeconstantbetweenisotopesandneglectingthehypne anomaly[37],asimpleratiobetweenisotopesusingEquations2.6and2.7providesthe followingdeterminationofthenuclearmoments: = R A hf A hf R I I R (3.5) Q = Q R B hf B hf R : (3.6) WherethesubscriptRreferstothereferencenucleus,whosehypcouplingconstantof thesameatomicstate,nuclearmoment,andnuclearspinareknown.Theknownvaluesof 49 Figure3.1:Thespectrummeasuredfor 36 Ca,alongwiththeresidualsfromtheusinga pseudo-VoigtfunctionasdescribedinSection3.2. Figure3.2:Thehypspectrummeasuredfor 37 Ca,alongwiththeresidualsfromthe usingsixpseudo-VoigtfunctionsasdescribedinSection3.2. 50 Figure3.3:Thespectrummeasuredfor 38 Ca,alongwiththeresidualsfromtheusinga pseudo-VoigtfunctionasdescribedinSection3.2. Figure3.4:Thehypspectrummeasuredfor 39 Ca,alongwiththeresidualsfromthe usingsixpseudo-VoigtfunctionsasdescribedinSection3.2. 51 Figure3.5:Atypicalspectrummeasuredforthereference 40 Ca,alongwiththeresiduals fromtheusingapseudo-VoigtfunctionasdescribedinSection3.2. Figure3.6:Atypicalspectrummeasuredforthecalibration 44 Ca,alongwiththeresiduals fromtheusingapseudo-VoigtfunctionasdescribedinSection3.2. 52 43 Ca ii ( I =7 = 2), A hf R ( 2 S 1 = 2 )= 806 : 40207160(8)MHz[53]and R = 1 : 317643(7) N [54], wereusedwiththeaboveratiotoderivethemagneticmomentsof 37 ; 39 Ca.Aprecise measurementof B hf R ( 2 P 3 = 2 )doesnotexisthowever,sothespectroscopicelectricquadrupole momentsof 37 ; 39 CawerederivedfromEquation2.7usingatheoreticalvalueof eV zz = 1 : 513(7)MHz = fm 2 [55].TheresultingnuclearmomentsareshowninTable3.2.Fora completediscussionoftheseresults,includingtheanexaminationofthedoubly-magicnature of 36 Ca,pleaserefertoref.[37]. Table3.2:Thenuclearmomentsof 37 ; 39 Caobtainedinthiswork.Thetotaluncertaintyis reportedintheparentheses. Q ( 39 Ca)isinagreementwiththepreviousexperimentalvalue of3 : 6(7) e 2 fm 2 [56],althoughthesignof Q ( 39 Ca)isexperimentallydeterminedhereforthe time. ( 39 Ca)reportedheredoesnottakeintoaccountthehfanomaly,whichthe previousandmoreprecise -NMRmeasurement,+1 : 0217(1) N [57],isinsensitiveto.For morediscussionoftheseresults,seeref.[37]. AI ˇ ( N ) Q ( e 2 fm 2 ) 373/2 + +0.7453(72)-15(11) 393/2 + +1.0204(3)+3.82(27) 3.4Chargeradii ThemeasuredspectrashowninSection3.2wereusedtoextractthechargeradiiof 36 ; 37 ; 38 ; 39 Ca relativeto 40 Ca.Variationinthetialmean-squarecharge-radiuscanbedetermined byacarefulevaluationoftheisotopeshift. 3.4.1Isotopeshift Whencomparinganatomictransitionintwoisotopesofthesameelement,theenergyor frequencyofthetransitionundergoesaslightchange,duetothechangeinthemassand sizeofthenucleus.Thisceinfrequencybetweenanisotope A and A 0 isknownas 53 theisotopeshift(IS), AA 0 = A 0 A : (3.7) Thevariationbetweentheseisotopesarisesfromtwoprimarysources,themassshift(MS) duetothechangeinmassofnucleus,andtheshift(FS)fromthechangeinthevolume oftheelectricatthenucleus AA 0 IS = AA 0 MS + AA 0 FS (3.8) AnexcellentexplanationofthesecontributionstotheIScanbefoundinref.[22],andare summarizedinFigure3.7.Toextractthechargeradii,theshifttermisofprimary interest.Here,thespfactorsusedforCawillbediscussed. Themassshiftcanbeexpressedas AA 0 MS = M A M A 0 M A M A 0 ( K NMS + K SMS )(3.9) where K NMS and K SMS arethenormalmassshiftandspmassshiftcotsrespec- tively,and M representsthemassoftheisotope.Inthecaseofmedium-massnuclei,where therelativisticcorrectionsandscreeningdon'tneedtobeconsidered[22],the shiftcanberepresentedas AA 0 FS = F h r 2 c i AA 0 (3.10) where h r 2 c i AA 0 istheofthemean-squarechargeradiusbetweennuclei A and A 0 . Theshiftcot, F ,mustbedeterminedforthespelementandtransitionbeing studied.Duetothenecessityofaninteractionbetweenthechargevolumeofthenucleusand 54 thewavefunctionoftheelectron,thisshiftcotislargestfortransitionsinvolving s typeelectronorbitals,astheyhavemorelikelihoodofbeingfoundwithinthenucleus.This cot F doeshavesomeslightvariationbetweenisotopes[58],howeverinthecaseof thesemedium-massnuclei,itissmallenoughtobenegligible,allowingtheisotopeshiftto beexpressedas AA 0 IS = M A M A 0 M A M A 0 ( K NMS + K SMS )+ F h r 2 c i AA 0 (3.11) 3.4.2Kingplot OnemajorbofstudyingtheCaisotopicchainisthethoroughresearchwhichhas beenperformedonthestableisotopes.BydividingeachsideofEquation3.11by( M A M A 0 ) =M A M A 0 alinearrelationshipcanbeobtainedbetweentheisotopeshiftandthems chargeradius AA 0 IS M A M A 0 M A M A 0 = k + F M A M A 0 M A M A 0 h r 2 c i AA 0 (3.12) allowingthemassshiftcot k = K NMS + K SMS tobeextractedfromtheintercept, andtheshiftcottobeextractedfromtheslope. Plottingisotopeswithaknownntialmean-squarechargeradiusandisotopeshiftin suchamannerisknownasaKingplot[60].Thistechniquecanbeusedevenwhenabsolute chargeradiiarenotknown,byallowingaratiooftheatomicfactorsbetweentwot transitionstobeobtained,asisshowninFigure3.8.Forsomeelements,theprecision obtainedbysuchalinearistduetofewisotopesorasmallvariationinisotope shiftorchargeradius.Thiswouldthenrequiretheuseofsemi-empiricalcalculationsto 55 Figure3.7:Diagramfromref.[22]describingthemassshiftandshiftprinciple.Theleft panelshowsthemassshift.Atthetop,isarepresentationofthenormalmassshift(NMS) whicharisesfromthebalanceofmomentumbetweenthenucleusandeachelectron,while themiddleandlowerdiagramsdepictthespmassshift(SMS)whichvariesdepending onthemomentumcorrelationbetweentheelectronsintheatomicsystem.Thecenterpanel describestheshift.ApureCoulombpotentialforapoint-likenucleusisshownbythe dottedline.Variationsinthechargevolumeofthenucleustheenergylevelsofthe electronsduetotheprobabilityoftheelectronsbeingwithinthenucleus.Thehorizontal dottedlinerepresentstheenergyofan s -electronforapoint-likenucleus,whiletheredand bluelevelsontheleftandrightshowtheenergylevelsshiftedduetothevolumeofthe nucleus A 0 and A respectively.Therightsectionprovidesanideaoftherelativecontribution totheisotopeshiftofeacht,versustheatomicnumber.Astheshiftisthe componentsensitivetothecharge-radius,thechargeradiusbecomestoextractfor lighterisotopes,wherethemassshiftdominates. 56 Figure3.8:Kingplotfromref.[59]comparingthemoisotopeshiftoftheD1andD2 transitionsinCa.Previousdatapointsareshowninred,whilethebluecirclesarethose measuredinthatwork.Boththeverticalandhorizontalaxesareplottingthe\mo isotopeshift,whichisthelefthandsideofEquation3.12.Here representstheexpression involvingthemasses.Thehighdegreeofprecisionandtheexcellentlinearityofthepoints givesgoodcausetoneglecthigherorderwhichmaycause F tovarybetweenisotopes. 57 deducetheatomicfactors,suchasinthecaseoftheFemeasuredrecentlyatBECOLA[21]. InthecaseofCa,however,theKingplothasbeenusedtoextracttheseatomicfactorswith muchhigherprecisionthanempiricalcalculationsareabletoprovide.Theatomicfactorsof the 2 S 1 = 2 $ 2 P 3 = 2 (D2)transitionweredeterminedtobe k =409 : 5(42)GHzamu(3.13) F = 284 : 7(82)MHz = fm 2 (3.14) froma3-dimensionalKingplotinvolvingboththeD1andD2transitions[59],andwereused inthepresentstudy. 3.4.3Results UsingthesepreciseevaluationsoftheatomicfactorsfortheD2transition,andthecentroid positionofeachisotope'sspectruminSection3.2.2,thetialmean-squarecharge radiuscanbeobtained.Theabsolutechargeradiiofeachisotopecanthenbeobtainedusing theexperimentallyknownradiusof 40 Ca.Thermschargeradiusof 40 Ca,3 : 4776(19)fm 2 [4], hasbeenobtainedfromacombinationofelectronscattering,muonscattering,andoptical isotopeshiftdataasdescribedinref.[4].Acombinationoftheelectronandmuonscattering datafor 40 Calistedinref.[61]wasperformedusingthe EXCEL formuladescribedinref.[61]. Followingthis,twomethodsofcombinedanalysisoftheabsolutescatteringmeasurements andopticalisotopeshiftsbetweenstableCaisotopeswereusedasdescribedinrefs.[62,63]. Theseallowtheabsolute,model-independentscatteringdatatobeconstrainedbytheprecise, relativeopticalisotopeshiftmeasurements. TheseresultsareshowninTable3.3,andplottedinFigure4.1alongwithseveralthe- 58 Table3.3:Isotopeshifts,deducedtialmean-squarechargeradii,andabsolutecharge radii.Thenumbersinparenthesesarethestatisticalandsystematicuncertainties,respec- tively.Thetialmschargeradiusof 39 Carelativeto 40 Caisconsistentwiththe previousvalueof 0 : 127(20)fm 2 [64].Rmschargeradiiwereobtainedusingthecharge radiusof 40 Ca3 : 4776(19)fmfromref.[4]. 40 ;A h r 2 i 40 ;A R A (MHz)(fm 2 )(fm) 36 1073 : 8(60)(43) 0 : 196(21)(16)3 : 449(3)(3) 37 766 : 1(47)(49) 0 : 205(15)(17)3 : 448(2)(3) 38 513 : 1(3)(17) 0 : 0797(11)(63)3 : 4661(2)(21) 39 230 : 5(2)(18) 0 : 1060(7)(64)3 : 4623(1)(21) Table3.4:Sourcesofuncertaintyinthetialmschargeradii, h r 2 i 40 ;A .Statisti- caluncertaintycomesfromtheofthespectra.Thecalibrationuncertaintyisfromthe intheextractedcalibrationvoltage,describedinSection3.1.Atomicfactor uncertaintyisduetothetotaluncertaintyfromtheatomicfactors, k and F asnotedin Equations3.13and3.14. StatisticalCalibrationAtomicfactors A (fm 2 )(fm 2 )(fm 2 ) 360 : 0210 : 01510 : 004 370 : 0150 : 01720 : 003 380 : 00110 : 00600 : 002 390 : 00070 : 00630 : 001 oreticalmodels.Thesystematicuncertaintiesforthentialmschargeradiicomepri- marilyfromthecalibrationuncertaintydescribedinSection3.1,andareshownindetailin Table3.4.Whendeducingtheabsolutechargeradii,theuncertaintyintheknownvalue of R ( 40 Ca)alsocontributes.While 36 ; 37 ; 38 Caaremeasurementsofthecharge radii, h r 2 i 40 ; 39 hasbeendeterminedpreviouslyas 0 : 127(16)fm 2 [64].Ourresultfor 39 Ca ( 0 : 1060(7)(64)fm 2 )isinagreementwiththepreviousmeasurement,andreducestheuncer- taintybyafactorof3. Thesechargeradiilighterthanthe N =20shellclosuretendtobecomesmalleras neutronsareremoved,whichisatbehaviortotheincreaseobservedbelowthe 59 N =28shellclosure.Althoughthistypeofsteadytrendacrosstheshellclosurehasbeen observedinneighboringelementsArandK[65][23],anunderstandingofthisweakening oftheshellclosurehasonlybeenexplainedqualitativelyasabalanceofthemonopoleand quadrupoleproton-corepolarizationsoneithersideof N =20,andwasnotmodeledwellby existingmodelsofcharge-radii[23]. 60 Chapter4 Discussion 4.1Modelingchargeradii NuclearDensityFunctionalTheory(DFT)hasbeenusedwithgreatsuccesstoreproduce importantfeaturesinground-stateobservables,includingmagicnumber\kinks"incharge radii,andbindingenergiesofnuclei[25,66].Theself-consistentmethodprovides avaluabledescriptionofcomplex,many-body,nucleiwhicharebeyondthereachof abinitio methodsduetocomputationallimitations.DFTmodelsemployaneinteraction whichservesasasubstituteforthetruenucleon-nucleoninteraction,andtheexpectation valueoftheenergyisthenminimized[67].TheapproachissimilartotheDFTusedin electronicsystems[68],however,inelectronicsystems, abinitio calculationshavebeenable toderiverobustenergyfunctionals.InnuclearDFT,thecompositestructureofthenucleon haspreventedausefuldirectderivationofenergyfunctionals,rathertheformofthemost eenergyfunctionalsisderivedbasedonvariousconsiderations,andtheirparameters aredeterminedusingtoempiricalnucleardata[66,69].Thisphenomenologicalapproach requirescarefulconsiderationoftheformofthefunctional,andtheexperimentalvaluesused toparametrizethemodel. SeveralDFTmodelsarecomparedtotheexperimentaldatainFigure4.1.Oneform istheSkyrmeenergydensityfunctional,whichhasbeenusedforover40years[70].Here, theSV-min(HFB)parametrization[69]ofthisfunctionalwascomparedtothenewexper- 61 imentalvalues.Thisparametrizationuses224experimentaldatapointsincludingbinding energies,radii,surfacethickness,chargeradii,andspin-orbitsplitting.Whilethis functionaldoesanadequatejobdescribingtheoveralltrendoftheradii,local suchastheodd-evenstaggeringandminimaatthemagicnumbers( N =20 ; 28)arenotwell described. ThesecondfunctionalusedisthemoremodernFayansmodel,whichaddssurfaceand pairingterms,whicharedependentonnucleardensitygradients[25,67].Inadditionto the224datapointsusedintheSV-minparametrization,tialmschargeradiiofCa isotopes( h r 2 i 48 ; 40 , h r 2 i 48 ; 44 ,and h r 2 i 52 ; 48 )werealsousedtoparametrizetheFayans functional, r ,BCS)[25].Throughtheseadditionalterms,andthedirectinputofdif- ferentialchargeradii,themicroscopicdetailsofthecalciumchaincanbereproducedquite well,despiteanexaggerationoftheodd-evenstaggering.Whenmovingtounstablenuclei however, r ,BCS)failstofullyreproducethesharpincreaseontheneutron-richside[26], anddivergesdramaticallyfromthesenewresultsreportedhereonthetside. Whenthisparametrizationwasdeveloped,itwasexaminedwithafocusonwell-boundnuclei, andsotheHartree-Fock(HF)+Bardeen-Cooper-Sc(BCS)approachwasused[25]. TheHF+BCSmethodisanapproximationoftheHartree-Fock-Bogolyubov(HFB)equa- tionwhichserveswellforlocalized,well-boundstates.Itreducesthenumberofcoupled equations,whichmustbesolvedintheHFBmethod,howeveritcannotbeusedwhendeal- ingwithexoticnucleinearthedriplines[66,71].Inthecaseoftheset (proton-rich)Caisotopes,theofloosely-boundprotonsbecomescrucialtoreproduce thechargeradii.The20protonsintheCanucleusproduceaclosed0 d 3 = 2 shellinthesingle- particlepicture,andthenextorbital,0 f 7 = 2 ,liesabovetheFermienergy.Whenmoving totheproton-richnuclei( 36 ; 37 ; 38 Ca),thisexcitedcanonicalHFBstaterisesabovethe 62 Figure4.1:Thechargeradiimeasuredinthiswork(redsquares)andpreviousexperimental values(blacksquares)arecomparedtoDFTpredictionsofSV-min(HFB), r ,BCS), r ,HFB),and r ,HFB)models.Thevaluesof r ,HFB)for A> 37arevery closeto r ,HFB)results;hence,theyarenotshown.Therms.chargeradiiwereobtained usingtheknownchargeradiusof 40 Ca(ref.[4]),anditserrorhasbeenincorporatedintothe systematicuncertainty(thegrayband).ThesevaluesareshowninTable3.3. 63 Coulombbarrier,becomingunboundasshowninFigure4.2(a)[36].Inthiscase,thepair densitiesgivenbytheBCSapproximationarenotlocalized,andnonzerooccupationsofthese unboundstatesleadtotheappearanceofanunphysicalgascomponent,givingdramatically increasingsingle-protonrmsradiiasshowninFigure4.2(b),andtheverylargeradiiand staggeringoftheproton-richCaisotopesinFigure4.1. Tocorrectlymodelthebehaviorinthisregion, r ,HFB)wascreatedusingthefull HFBapproach.Inaddition,while r ,BCS)usedneutronandprotonodd-evenbinding energystaggering, (3) E ,the r ,HFB)parametrizationuses ee E ,whichderivesfrom even-evennuclei[25].Thischangeinparametersisbetterforopen-systems[72],andwas tohaveverylittleonthe r ,BCS)model[36]. r ,HFB),producessingle-protonstateswhichdonotfromdrasticgrowthas theyriseabovetheFermienergy[36].Thesingle-protonradiiof 36 Castatesareshownin Figure4.2(b),andtheoccupationsin(c)ofthesameAsshown,thismodelreduces thesizeoftheunboundstates.While r ,BCS)showslargeradiiforthesestates,the occupationsaregreatlyreduced.Despitethisreductioninoccupation,thechargeradii predictedby r ,BCS)stillgreatlyoverestimatethetradiimeasured here.Amazingly,thenewmodelFy( r ,HFB)notonlyimprovestheagreementonthe tside,butalsoimprovesthebehaviorofthechargeradiiontheneutron-rich side,whererecentmeasurementsfoundtheradiitobeunexpectedlylarge[26]. Finally, r ,HFB)usesthesameexperimentalinputsas r ,HFB),butalsoin- cludesthenew h r 2 i 36 ; 40 measurementfromthiswork.Thechargeradiiobtainedusing thesenewHFBFayansmodelsareshowninFigure4.1. r ,HFB)improvesagreement for 36 ; 37 Ca,whiletherestofthechainislargelyidenticalto r ,HFB),soonlythesetwo valueshavebeenincludedinFigure4.1.Odd-evenstaggeringisconsistentlyoverestimated 64 Figure4.2:Graphsshowingtheenergylevels,radii,andoccupationsofthesingleproton statesintheCachain.Theuppergraph,a),highlightstheweakly-boundnatureofthe valenceprotons.Asshownbytheredtriangles,thesingleprotonenergyofthe0 f 7 = 2 orbital risesabovetheCoulombbarrierandbecomesunboundinthecaseof 36 38 Ca.Graphsb) andc)displaythesingle-protonrmsradiiandoccupations,respectively.Inthecaseofthe r ,BCS)approach,thesingle-protonradiusincreasesinanon-physicalwayasthestates riseabovetheFermienergyandcontinuumscomeintoplay(reddiamonds).With arealisticpairinginteraction,theseunboundstatesdonotgrowsodramaticallyinsize,as shownbythe r ,HFB)and r ,HFB)points(greencirclesandbluetriangles). 65 Figure4.3:Nuclearchargeradiiofcadmiumisotopes.Figuretakenfromreference[73]. Experimentalvaluesandseveralmodelsareshown.Thegraybandrepresentsthesystematic uncertaintyoftheexperimentalvalues,arisingfromtheuncertaintyoftheconstant. Theinsetalsoshowsthecorrespondingone-neutronseparationenergies.Theagreementof theFayansfunctionalisquitegood,althoughsomeoverestimationoftheodd-evenstaggering ispresenthereaswell. acrossthechain,butthisisaminordiscrepancywhenconsideringthegoodagreementof themajorfeatures.Thetimprovementoverthe r ,BCS)predictionsonboth endsofthisisotopicchain,especiallygiventhechallengingbehaviorofthecalciumradii, highlightsthefactthatthesedensitygradientdependentusingthefullHFB formalismareacriticalsteptowardsdevelopingaglobalmodelofcharge-radii.Conversely, theseresultsalsoemphasizehowimportantchargeradiidataareinordertopindownthe formofthepairingenergydensityfunctional. 66 Figure4.4:Nuclearchargeradiioftinisotopes.Figuretakenfromreference[74].Ex- perimentalvaluesandseveralmodelsareshown.Theinsetalsoshowsthebehaviorofthe r ,HFB)modelinthevicinityof 208 Pb.TheabilityofthisFayansfunctionaltorepro- ducethesekink-structuresinheavyisotopeswithoutadditionaltuningisquiteremarkable. 67 ThisimprovementtotheFayansmodelisespeciallyusefulconsideringthebehaviorofthis functionalinotherisotopicchains.The r ,BCS)modelhasbeenshowntobeeat reproducingthemuchheavierchargeradii, 100 130 Cd( Z =48)[73],andthe r ,HFB) hasbeenshowntoreproducethekink-structurerecentlyobservedaround 132 Sn[74],aswell as 208 Pb.Experimentalchargeradiiandmodelpredictionsintheseregionsareshownin Figure4.3,andFigure4.4.ThisbroadapplicabilityoftheFayansfunctionalshowsgreat promiseforuseacrosstheentirechartofthenuclides.Thedemonstrationhereofitsability tocorrectlydealwithproton-richnucleiwhenpairingareaccountedforinaproper manner,showsthattheFayansmodelisenoughtoapplytoawiderangeofmasses andalsocanaccountforbehaviorswhichoccurfarfromstability. 4.2Empiricalextrapolation Anothertooltounderstandthebehaviorofthischainofchargeradiiistoexaminethe empiricalrelationshipsbetweenneighboringchargeradii.Inarecentstudy,fourprimary relationshipswereidenwhichallowchargeradiitobepredictedfromknownneigh- bors[75].These in jp ( i;j =1 ; 2)chargeradiirelationsareexplainedinFigure4.5. Ingeneral,theserelationshipsallowarobustpredictionofchargeradiiwhenneighboring nuclides,separatedby1or2protonsorneutrons,areknown.Inreference[75],manycharge radiiwerepredicted,anditwasnotedthatradiiwhichdeviatetlyfromthese empiricalrelationshipsmaysuggestinterestinglocalphysics,suchassuddenvariationsin shape.ThesepredictedvaluesareoverlaidwiththeexperimentalcalciumchaininFigure4.6. Acrosstheentirecalciumchain,thesepredictionsperformreasonablywell.Theareawith themostntdiscrepancyareisotopeslighterthan 40 Ca,whichareunder-predicted, 68 Figure4.5:Figurefromreference[75].Thisgureshowstherelativelocationsofthe nuclidesusedinthefourempiricalchargeradiirelationships.Inallfourcases, R in jp = R (1)+ R (2) R (3) R (4)=0,allowingthepredictionofachargeradiusfromtheother three.Thisrelationshipismostrobustfor(a),wherethereisonlyastepof1protonor neutronbetweenthefournuclidesbeingcompared,butholdsreasonablywellfortheother threerelationshipsaswell. 69 suggestingthatthesenucleiaregrowinglargerthanneighboringnucleiwouldsuggest,due totheweakly-boundprotonswhenmovingfurtherfromstability.Thestrongagreementof theseempiricalrelationshipsacrosstheentirecalciumchainhighlightsthefactthatthereis nouniquelocaloccurringinthecalciumisotopes,rathertheypresentacomplicated patternofchargeradiiwhichisalsosuggestedbyneighboringnuclei.Thisstrengthensthe argumentthattheseCaisotopescanbeusedasabenchmarkforchargeradiimodels,and suggeststhatamodelwhichperformswellacrossthisregioncanbeappliedtootherelements aswell. 70 Figure4.6:Empiricalpredictionsofchargeradiishownhereinpurplediamondsaretaken fromreference[75].Thesepredictionsusesomecombinationsofthefourrelationsshown inFigure4.5.Experimentalchargeradiiareshownusingtheredandblacksquares.Large deviationsfromtheempiricalpredictionscansuggestsuddenvariationsinshapeordensityof thenucleus.Itisinterestingtonotethatwhileprevioustheoriesdramaticallyoverpredicted thesizeoftheproton-richnucleiinthischain,theseempiricalpredictionsarereasonably accurate,andinfactunderpredictnucleibelow A =40,whichistobeexpectedconsidering theoftheweakly-boundprotonsintheseisotopes.Thishighlightsthefactthatthere isnosuddenchangeofbehaviorofthechargeradiiinthisregion,andstrengthensthebelief thatamodelwhichperformswellinthischainmaybeapplicableacrossthenuclearchart. 71 Chapter5 Conclusion Thehypspectraofcalciumisotopes 36 ; 37 ; 38 ; 39 Caweremeasured.Thesemeasurements haveadvancedthebunched-beamCLStechniqueatBECOLA;theuseofthenew PDS,productionofcalciumions,andcalibrationusingmultiplestableisotopeswere asuccess.Themeasurementof 36 Ca( T 1 = 2 =102ms)using50ions = sand33hofdata wasthemostsensitivemeasurementperformedatBECOLAtodate.Fortunately,itwas possibletoobtainalargefractionofcalciumionsfromthegascellintheirsinglyionized state,asopposedtodoublyionized.Thisallowedthemeasurementtobeperformedon theCa ii 4 s 2 S 1 = 2 $ 4 p 2 P 3 = 2 (393 : 3663nm)transition,usingfrequencydoubledlightfrom theTi:Sapphirelaser.Duetothefactthatchargeexchangewasnotrequiredtoobtain singly-chargedions,thebackgroundreductionofthenewPDSwase,andprovided anincreasedSNRovertheprevioussystem.Theadditionalstatisticsfromthefactthattwo identicalsystemswereinstalledalsohelpedtoachievethisnewlevelofsensitivity. Thehypspectrahadlinewidthsofapproximately75MHz,andtheisotopeshifts wereobtainedusingreferencemeasurementsof 40 Ca.Theseisotopeshiftswereextracted withlessthan6MHzofstatisticaluncertainty.Uncertaintyinthehighvoltageofthe cooler/buncherwasminimizedbyusingtwostableisotopes, 40 ; 44 Ca,andtheirknownIS forcalibration,leadingtolessthan5MHzofsystematicuncertaintyintheISforeachiso- tope.Fromtheisotopeshifts,thetialmean-squarechargeradiusrelativeto 40 Cawas obtainedforeachisotope.Thankstothecarefulstudieswhichthestablecalciumisotopes 72 havereceived,theatomicfactorsrequiredtodeducetheseradiiarepreciselyknown,allow- ingthechargeradiitobeextractedwithatotaluncertaintyoflessthan0 : 005fmfor 36 Ca, and0 : 002fmfor 39 Ca(seeTable3.3)[36].Thisisatimeresultforthechargeradii of 36 ; 37 ; 38 Ca,and 39 Caisinagreementwithpreviousexperimentaldatawhileincreasing precisionbyafactorofthree. Thesemeasurementshighlighttheimportanceofutilizingthebunched-beamCLStech- niqueinfacilitieswithmethodsofisotopeproduction.Thesuccessofthistechnique atISOLfacilitiesisbecomplementedbytheseexperimentsattheNSCL,asthet productionofrareisotopesprovidesaccesstoshort-livedtisotopeswhich aremoretoproduceatISOLfacilities[35].Inconjunctionwithmeasurementsof neutron-richcalciumisotopesfromsuchafacility[26],afullchainofthecalciumcharge radiifromttoneutron-richhasnowbeenmeasured.Suchalongchain, spanningacrossthedoubly-magicnuclei 40 ; 48 Caandclimbingoutofthevalleyofstabil- ityonbothsideshasprovidedanexcellentopportunitytochallengenucleartheorieswhich predictchargeradii.Indeed,eventhemostsuccessfulDFTmodeltodate r ,BCS)) showedpromisethroughoutthestableisotopesbutdidnotcorrectlycapturethebehaviorof theunstableisotopes[36].Thesharpincreasebeyond A =48wasunder-predicted,andthe isotopesbelow A =40wereseverelyoverpredicted,despitethefactthatempiricalrelation- shipcanpredicttheseradiiwell[75].ThisissuehighlightedthefactthattheBCSpairing approximationbreaksdownwhenappliedtolooselyboundnucleons[66,71].Duetothe factthattheexcitedprotonshell,0 f 7 = 2 ,risesabovethecoulombbarrierandintothe continuumasneutronsareremoved,theofprotonpairingintheseunboundorbitals becomest.ByincorporatingthemorephysicallycorrectHFBpairingapproxima- tion, r ,HFB)performsbetteracrosstheentirechain,accuratelymodelingthetrendof 73 thesenewexperimentalmeasurementsonthetside,andevenimprovingthe agreementontheneutron-richside[36].Suchadramaticimprovementandtheabilityto modelthiscomplicatedchainofisotopeshighlightstheilityoftheFayansmodel.This modelhasalsoshowngoodresultswhenappliedtotheheavierCd,Sn,andPbchains[73,74] andisafundamentalsteptowardsdevelopingaglobalmodelofnuclearchargeradii. Althoughobtainingnewmeasurementsofnuclearobservablescanbeusefulinitsown right,themeasurementsherehaveaccomplishedevenmore.Thesemeasurementshave pushedthesensitivityofthebunched-beamcollinearlaserspectroscopyevenfurther,and thesuccessfulresultshaveprovidedcriticalexperimentalvalueswhichhaveincreasedour understandingofnuclearstructure.Theextensionofthechainofcalciumchargeradiito theisotopes 36 ; 37 ; 38 Cahaschallengedthemostadvancedtheories,andbyreformulating theFayansmodelwithacarefulconsiderationofprotonpairinginweakly-boundstates, itsyandapplicabilityasaglobalmodelofchargeradiihavetakenacrucialstep forward.Inthefuture,astheFacilityforRareIsotopeBeams(FRIB)comesonlineatMSU, accesstounstableisotopesatBECOLAwillbegreatlyincreased[76].Byreachingtoeven moreexoticCaisotopes,measuringneighboringisotopicchainssuchasScandmirror-nuclei, BECOLAwilltocontinuetofurthertestandourunderstandingofnuclearstructure. 74 APPENDICES 75 APPENDIXA Hardware A.1Photondetectionsystem A.1.1Opticalsimulations PriortothemanufacturingofthenewPDS,simulationswereperformedusingFRED,a raytracingsimulationforopticalsystems.Inthesesimulations,anarbitraryamountofsignal andbackgroundweregenerated,andthefractionofeacharrivingatthePMTwereevaluated. Signalwasapproximatedasraysemanatinginrandomdirectionsfromacylindricalvolume, wherethelaserbeamandionbeamoverlap.Backgroundphotonsweresimulatedusing raysoriginatingatrandompointswithinacircleplacedjustoutsidethePDS,emanatingin randomdirectionswithinthehemispheretowardsthePDS. ThissimulationwasperformedusingbothamodeloftheoldPDS,aswellasthenew design.Whilethesimulatedsignalcaneasilybeassumedtobeidenticalbetweeneachsys- tem,thereissometyindecidinghowtoideallycomparethesimulationofbackground photonsbetweenthetwosystems,astheoldandnewsystemshavetdiameteropen- ingsateringdistancefromthecenterofthePDS.Thebackgroundsourcewasplacedat theedgeoftheellipticalregionforthenewsystem,howeverwhenthisgeometryisdirectly copiedtotheoldsystem,thereisagapbetweenthesourceandtheshorttubeleadingto theellipsoidalregion.Toaccountforthissimulationsoftheoldsystemwerealso 76 performedwiththebackgroundsourcemovedtowardthecenterofthePDS,tosit withthetube.Thesethreesimulations,onewiththenewsystem,andtwovariationsusing theoldsystemwereusedtogiveanestimatedrangeforthecomparisonoftheoldandnew systems. FortheexistingPDS,therayswerecollectedattheplaneofthePMT,andtheresults areshowninFigureA.2.Inthenewsystem,therayswereanalyzedatthefocalplaneofthe ellipticalsectiontodeterminetheappropriatepositionoftheaperturetoselectthefocused signal,andtherayswerethenalsocollectedatthePMTfollowingtheCPC,asshownin FigureA.3.Thefractionalacceptanceofsignalandbackgroundweredeterminedforeach systeminthesimulations,andarelistedinTableA.1. Thedecisiontousea20 ° CPCwasmadebyexaminingtheangulardistributionofphotons atthefocalplaneofthenewsystem.AsshowninFigureA.4,thebackgroundandsignal anglesarefairlywellseparated.Thesamecannotbesaidfortheoldsystem,soaCPC addedtotheexistingsetupwouldnotbehelpful,asshowninFigureA.5. FigureA.6showsthesimulatedacceptanceofthesignalfromvariouspointswithinthe detectionregion.Whiletheoldsystemcollectsalargeamountofsignalatthecenter,the newsystemhasamoreconsistentacceptancethroughoutthedetectionregion.Ultimately, thisvariationinthepositionalongthebeamlinewheresignaliscollectedisnotimportant, theprimaryinterestisthetotalsignalandbackgroundacceptanceofeachsystem. InordertocomparetheSNRofeachsystem,thesignalandbackgroundcountsofine Carunstakenwiththeexistingsystemwereused.Fromthesimulations,theratiobetween thesignalacceptanceofthenewandoldsystemswasusedtoscalethesignalcounts,while theratioofthebackgroundwasusedtoscalethebackgroundcounts.Inthisway,thesignal andbackgroundfromactualmeasurementsusingtheoldsystemcanbecomparedtothe 77 FigureA.1:Schematicshowingthethreesimulationsetups.Raytracingsimulationswere performedinFREDusingperfectlyresurfaces.Thenewsystemwassimulatedwith acylindricalvolumeemittingraysinrandomdirections,whilethebackgroundwassimulated asraysemittedinrandomdirectionsfromwithintheareaofacircleplacedattheentrance totheellipticalregion.Theoldsystemwassimulatedtwice,oncewiththesourcespositioned identicallytothenewsystem,shownontheright,andagainwiththebackgroundsource movedclosertobewiththesystem,asshownontheleft.Thesetwosimulationsof theoldsystemwereusedtoevaluatethestabilityofthesimulationsandobtainarangeof expectedvalues. 78 FigureA.2:Simulationshowingthegeometricdistributionofphotonsarrivingatthefocal planeusingtheexistingPDS.Thebluedashedcircleindicatesthegeometricacceptance ofthePMT,collectingthemajorityofthesignal(redpoints),andexcludingaportionof thebackground(yellowpoints).Inthissimulation,44 : 8%ofthesignalphotonsreachthe focalplane,while25{40%ofthebackgroundphotonssimulatedarriveatthefocalplane.In theareacoveredbythePMT,13 : 4%ofthesignal,and1 : 5{2 : 6%ofthebackgroundwere collected.Therangeofvaluesforthebackgroundareduetoavariationinthepositionof thesimulatedbackgroundsourceasdescribedinSectionA.1.1. 79 FigureA.3:Simulationshowingthegeometricdistributionofphotonsarrivingatthefocal plane(left)andthePMT(right)withthenewPDS.Thebluedashedlineindicatesthe geometricoftheapertureatthefocalplane,collectingthemajorityofthesignal (redpoints),andexcludingaportionofthebackground(yellowpoints).Inthissimulation, 42 : 9%ofthesignalphotonspassthroughtheapertureofthefocalplane,comparedtoonly 15 : 3%ofthebackgroundphotons.FollowingtheCPC,thePMTaccepts8 : 3%ofthesignal, andonly0 : 21%ofthebackgroundphotons. 80 FigureA.4:Simulationresultsdescribingtheangulardistributionofphotonsarrivingat thefocalplaneofthenewPDS.Signalisshowninred,andbackgroundinyellow.Thepolar angleof0 ° correspondstotheperimeterofthehemisphere,whileanazimuthalangleof0 ° or 180 ° correspondstothebeamaxis.Theangulardistributionisalsoshownprojected ontoahemisphereaboveeachgraph,thecenterofthehemispherefacesthePMT.Theblue dashedlineindicatestheangularchosenfortheCPC.Photonsarrivingwithapolar anglelessthan70 ° arerejectedexcludingalargepartofthebackground 81 FigureA.5:Simulationresultsdescribingtheangulardistributionofphotonsarrivingat thefocalplaneoftheoldPDS.Signalisshowninred,andbackgroundinyellow.Thepolar angleof0 ° correspondstotheperimeterofthehemisphere,whileanazimuthalangleof0 ° or 180 ° correspondstothebeamaxis.Theangulardistributionisalsoshownprojectedonto ahemisphereaboveeachgraph,thecenterofthehemispherefacesthePMT.Asshown,the primaryintensityofboththesignalandbackgroundlieatthesamepolarangle,preventing anyimprovementfromaddingaCPCtotheexistingsystem. 82 FigureA.6:Thissimulationhighlightstheceinsignalcollectionofthetwosystems. Whiletheoldsystemcollectsalargefractionofthesignalatthecenterofthedetection region(thefocusoftheellipsoid),thenewsystemcollectssignalmoreconsistentlyacross theentireregion.Thetotalacceptanceofthenewsystem(areaunderthiscurve)isstillless thantheoldsystem,however,theincreaseinnoiserejectionallowsittooutperformtheold systemfortheCa ii measurement. 83 simulatedperformanceofthenewsystemforthemeasurementofCaions. Whilethissimpleratioofsignalandbackgroundsimulationtakesintoaccountback- groundfromstraylaserlight,morecareisrequiredwhenthecharge-exchangecellisused inanexperiment.Toexaminethectsintheseconditions,datatakenatBECOLAusing 53 Fewereused.Thesignalandbackgroundinthiscaseweredividedintothreecategories: signalfromthelaser-atominteraction,backgroundfromatomicdecayssubsequenttothe chargeexchange,andbackgroundfromstraylaserlight.Usingbunchedbeamspectroscopy, theratesfromeachofthesecategoriescanbemeasuredexperimentally.Whentheatom bunchisnotpresent,therecordedratesindicatethebackgroundduepurelytothestray laserlight.Atthewindowoftimewherethebunchispresent,yetthescanningvoltageis the\glow"fromthechargeexchangeprocesscanbecountedabovethelaser background.Finally,wheretheresonantpeakofthespectrumismeasured,theratesofall threecombinetogether. Thisallowstheextractionofeachcategoryofphotoncountsseparatelyand,eachcanthen betreatedthemaccordingtotheappropriatesimulationmethod.Thelaser-atominteraction andthedecaysfromthechargeexchangewillbeacceptedbythesystemaccordingtothe signalinthesimulation,astheyareproduceddirectlybythebeampassingthroughthe system,whilethestraylightismodeledusingthebackgroundinthesimulation. FortheCaexperiment,becauseatransitioninCa ii isbeingused,thechargeexchange cellisnotnecessary,andthusthereisnobackgroundrelatedtothedecayfromhigheratomic statesproducedinthechargeexchangeprocess. Thesignalandbackgroundlevelsfromdatatakenfor 56 Feusingchargeexchange,and 40 Cawithnochargeexchangewereusedinconjunctionwiththesimulationsinorderto estimatetheSNRwhichcouldbeachievedwiththenewsystem.Whenchargeexchangeis 84 beingperformed,comparisontothesimulationbecomeslessdirect,asanon-trivialamount ofbackgroundisproducedbytheionsduetoexcitedatomicstatesdecayingfromthecharge exchangeprocess.Thisbackgroundisbeingintroduceddirectlyfromthebeampath,and soisscaledbasedonthesimulatedacceptanceofthesignal,whilethebackgroundfrom thelasermustbescaledaccordingtothesimulatedbackground.Becausethisbeam-related backgroundcannotbereducedusingthisdesign,therelativereductioninsignalcollection comparedtotheexistingsystemleadstoareductioninSNRforameasurementinvolving charge-exchange.WhenexaminingstableisotopesofCawhichweremeasuredat BECOLAwithoutchargeexchange,thenewsystem'sbackgroundrejectionismuchmore e,andinthesimulationsitperformedbetterthantheoldsystemdespitetheslight reductioninsignal.TheseresultsareshowninTableA.1. Basedonthesesimulations,thenewdesignincludinga20 ° CPCwasconstructedand installedatBECOLA,insertedintothebeamlinejustfollowingtheexistingdetectionre- gion.Thenewsystemincludestwoseparatedetectionregions,eachwiththeirownelliptical CPC,andPMT,furtherincreasingthemeasurementcapabilitycomparedtothe previoussystem. A.1.2Installation Inordertoinstallthesystemintothebeamline,acustombeampipecrosswasfabricatedto provide8inCFoneachend,whilefouronthesidesallowforthehighvoltage passthroughtocontrolthedetectionregion,andthewindowswhichallowthephotonsto becollectedoutsideofthevacuum.Aschematicofthissectionofbeampipeisshownin FigureA.7.Thetwonewdetectionregionsuseidenticaldesigns,withoneorientedupward, andtheotherdownwardfromthebeamline. 85 TableA.1:ThedesignofthenewPDSwascomparedtotheexistingsystemusingsimula- tionswithinFRED,aray-tracingtool.Signalphotonsweresimulatedusingraysemittedin alldirectionsfromrandompointswithinthevolumeofacylinder,approximatingtheover- lappinglaserandionbeam.Backgroundphotonswereapproximatedbyraysoriginating fromrandompointswithinacircle,andemanatinginarandomdirectioninthehemisphere facingthePDS.Fortheoldsystem,thisbackgroundsourcewassimulatedintwodit positions(seeSectionA.1.1)leadingtotherangesofbackgroundacceptanceandSNR.Both systemsweresimulatedwithperfectThefractionofsignalandbackground reachingthedetectorwerecomparedtoseveralactualmeasurementsusingreferenceions andtheexistingsystem,inordertodeterminetheappropriateratioofincomingsignaland backgroundphotons.TheSNRinthetableareextrapolatedfromthesimulationsandthese referencemeasurements,assigningthemeasurementfromtheexistingsystemanSNRof1. Duetothefactthatthe 56 Femeasurementrequiresthecharge-exchangecell,thereisan additionalbackgroundsourcecomingfromtheionbeamitself,whichiscollectedinthesame wayasthesignal.Becauseofthis,thestrategyemployedbythenewsystem,tominimize noiseevenatthecostofsomesignal,isnotase,duetothefactthatthenoisefrom thecharge-exchangeprocesscannotbereducedthisway. SignalBackground SystemAcceptanceAcceptanceSNRfor 40 CaSNRfor 56 Fe Existingsystem13 : 36%1 : 45{2 : 65%11 Newsystem8 : 42%0 : 24%1.5-2.00.94-0.97 Installationofthesystemrequiredspecialcareduetothetightspaceoftheassembly withinthebeampipe.A\dirty"assemblywascompletedoutsideofthebeamline inordertoevaluatewhichcomponentscouldbeinstalledfromthesideandwhich neededtobeputinthebeampipecrossbeforeitwasinsertedintotheBECOLAbeam- line.Followingthis,thecomponentsweredisassembledcompletelyandwipeddownwith methanol,inordertocleanthemforvacuum.Duringthereassembly,thinesheets wereinstalledontheopticalsurfaces,allowingahighlytivesurfacewithoutthelengthy manufacturingtimeofhand-polishingtechniques.Theyofthesesheetsisshown inFigureA.8. ThemainsupportofthePDSusesfourrodsthatrunthelengthofthebeamcross.The framesupportingtheserodsandinsulatingthemfromthebeampipeisshowninFigureA.9. 86 FigureA.7:Aschematicofthebeampipeusedtoenclosethetwonewdetectionregions.The 8inonthetopandbottom,whicharefromoneanother,arethelocationofthe twonewdetectionregions.Asmalloppositeeachallowsahighvoltagefeedthrough inordertoapplythescanningpotentialtothedetectionregion,asshowninFigureA.12. Thisframeassemblywasinstalledintothebeamcross,andthebeamcrosswastheninserted intotheBECOLAbeamline.Alignmentwasthenperformedusingatelescopeattheendof thebeamlinetoensurethatthefrontandbackofthisnewPDSwerecollinearwiththeold PDSandtheapertureswithintheBECOLAbeamline.Adjustmentsweremadebyreaching throughthelargesideofthecross,androtatingthedouble-threadedbrassscrews. Thesescrewshaveoneendthreadedleft-handed,andtheotherendright-handed.Rotating thesescrewsthenallowsthe3Dprintedinsulatingfeettobeshiftedoutwardorinward, allowingthefoursupportrodstobealignedproperlywiththebeamaxisandelectrically isolatedfromthebeampipe. Withthefoursupportrodsinstalledandaligned,theellipticalregionsweretheninstalled throughthesideinthebeampipe.Eachellipticalregionwasassembledasshownin FigureA.10andthenattachedtotwoofthesupportrodsasshowninFigureA.11.Ashort 87 FigureA.8:dataforthealuminumsheetingusedfortheesurfacesofthe newPDS.TheMIRO4300UPwasused,inordertoallowforgoodythroughout theUVrange(250{450nm). 88 FigureA.9:Lookingfromtheendofthebeamcross,theinternalsupportsofthePDSare shown.Fourwhiteinsulatingfeetsupportitwithinthebeampipe,andthefourmainsupport rodsarepointingawayfromtheviewer,fromtheendsofthealuminum inthecenterof thebeampipe.Thesmalltubeatthecenterofthe iswherethelaserandionbeams travel.Alignmentofthismainframewasaccomplishedbyrotatingthedouble-threaded brassconnectorsnexttotheinsulators. lengthofinterlockingpipewasalsoattachedtothesidesofeachellipticalregion,sothat therearenogapsinthepotentialalongthebeamaxis.Ahighvoltagefeedthroughwasthen installedinordertoapplythescanningpotentialtotheregions,asshowninFigureA.12. Next,thewindowswereattachedtothesideandthebeamlinewaspumpeddown tovacuumonceagain.TheCPCswereassembledasshowninFigureA.13andattached tothefrontofeachsidewindow,alongwithtwoplatesfortheadjustableaperture.The completeexteriorassemblyincludingtheaperture,CPC,andPMTmountinghardwareis showninFigureA.14. Finally,eachexteriorassemblywasenclosedwithalight-proofcloth,topreventlight leakagereachingthePMTs.Brassthreadedrodswhichallowforadjustmentoftheapertures protrudefarenoughthattheycanbeadjustedwithoutremovingthelight-proowrap. 89 FigureA.10:PhotographsshowingtheassemblyofanellipticalOnthetop,the partsareshownpriortoassembly.Theleftimageshowstheassemblyminusonesideplate, andtheendplate.Whentheendplateisattached,itpushestheesheetinwith theendoftheellipticalplates,causingittobowfurtheroutwardandlayagainstthe ellipticalplates.Therightphotoshowstheassembly,lookinginthroughthewindow oftheendplate.Notshownhereisthemetalmeshwhichcoversthiswindowinorderto ensurethattheentiredetectionregionisenclosedbythedesiredpotentialwhenscanning. 90 FigureA.11:Photographsshowinghowtheellipticalareattachedtothesupport rods.Onthelefttheinteriorofthebeampipeisshown.Asmall\J"shapedhookholds theellipticalectorassemblyupagainstthesupportrods.Itislockedinplacewithaset screwonthesidefacingoutthesideandsoitcanbeaccessedfromoutsidewith carefuluseoftweezersandalonghexwrench,asshowninthephotoontheright. FigureA.12:Photographsshowingtheinstallationofthehighvoltagefeedthroughusedfor thescanningpotential.Asmallspringallowscontactwiththebackofoneoftheelliptical Theleftphotoshowsthehighvoltageconductorincontactwiththesystem,while therightshowsthefeedthroughremovedfromthesystem. 91 FigureA.13:AphotographoftheCPCassembly.TwohalvesofoneCPCareshowninthe foreground,whileintheback,afullyassembledCPCcanbeseen.Thewidesideattaches tothesidewindowofthebeampipe,andthePMTisattachedtothenarrowend. FigureA.14:AphotographshowingtheassemblyoftheexteriorcomponentsofthePDS.An adjustableapertureofaluminumplatesisattachedtothewindow,withtheCPCfollowing. AttherightendoftheCPC,themountinghardwareforaPMTisshown.Thefaceofthe PMTpressesupagainstarubberO-ringontheCPC,whilethebackendisheldtightly usingaplateheldbythreadedrodsprotrudingfromtheCPC. 92 TworowsofVelcrostripswereusedtoattachthelightprocloth,anddarkcountsin eachPMTwerelessthan100 = s. A.1.3Furthersimulations Inordertounderstandthetwopeakstructureseenatthefocalplane(seeSection2.3.2.2), severalmoresimulationswereperformed.Threepotentialsourcesofthisunexpected wereidentAmisalignmentofthePDSwiththesourceofthesignal(possiblyonthe orderof0 : 5mm),amisalignmentofthePDSrelativetothewindowonthebeamline(onthe orderof3mm),orlly,aslightdeviationfromtheellipticalshapeatthepointwiththe strongestcurvature(duetoaslightcreaseneededinthefoiltoallowittobendintoshape). Thesimulationwasrepeatedusingeachofthesepossibledeviationsfromtheidealdesign ofthesystem.Itwasfoundthatwhileallthreepossibilitiescancauseaslightreductionin signalatthecenterofthefocalplane,thesimulationmostcloselyresemblingtheobserved structureinvolvedacombinationofdeformationintheellipse,andanofthebeamfrom thefocusoftheellipse.TheseresultsareshowninFigureA.15.Althoughtheefoil isconvenientandonlyasmallsectionatthesharpestpointoftheellipseshowsat deformation,thisregiondoesrepresentatsolidangleofthesignalduetoits proximitytothebeams.Inthefuture,theseellipticalsurfacesmaybehandpolished,rather thanusingthefoil.Thismayprovideamoreaccurateellipticalgeometry,andallowabetter focusedsinglepeakofsignalatthefocalplane. 93 (a) (b) (c) (d) FigureA.15:Severalsimulationsincorporatingpossiblesystematicwhichmay bepresentintheactualdetectionsystem.(a)Showstheofaddingaslightcreaseto thepointoftheellipse,somethingwhichoccursintheactualsystemduetothethicknessof thefoilusedtocreatetheesurface.(b)Showshowthe\splitting"ofthesignalis exaggeratedfurtherifthelocationoftheapertureisslightlyfromthetruefocalplane. (c)Showsanotherpossiblesourceofthetwo-peakstructure,byslightlythesource ofthesignalfromthefocusoftheellipticalregion.(d)Showsthesimulationwhichwas bestabletomatchtheexperimentallyobservedsignalusingacombinationofa(a) and(c).Inafuturemotothesystem,theellipticalregionsmaybehandpolished ratherthanusingtheefoils,inordertomoreaccuratelyproducetheellipticalshape. 94 A.2PIGsource ThePenningIonizationGauge(PIG)sourceisaversatileplasmasputteringsourcethatwas constructedatBECOLAin2015.Ananode,withalargeopening,sitsatthecenterofthe plasma,whileacathodeandanti-cathodesitoneitherside.Thesearesurroundedbya magneticandagasisfedintothesourceinordertosustainaplasma.Asmall holeinthecathodeallowstheionsgeneratedinthesource(bothionsfromthegas, aswellasthosesputteredofthecathodes)tobeextractedinabeam.Anyconductive elementthatcanbemachinedtothedimensionsrequiredforthecathodecanbeusedin thesource.Severalelementshavebeenusedascathodestoproducereferenceionssofar includingFe,Ni,Sc,Zr,andnowinthisexperiment,aSn-Caalloywasused. RecentupgradestothePIGsourcewerementionedinSection2.3.1.1andafewmore technicaldetails,whichmaybeusefulforfutureoperationorexperiments,aredescribed here. A.2.1Masswcontroller Priortothisexperiment,amasswcontroller(MFC)wasobtainedinordertoallowabetter controlofthewofgasintothePIGsource.Thedeviceinuseatthetimeofthiswriting isfromBronkhorst,modelnumberF-200CV-002-AGD-22-V.TheMFCandsurrounding componentscanbeseeinFigureA.16.AnidenticalMFChasalsobeenpurchasedasa backup,andisinstorage. ThisMFCusesa9-pinD-subconnectorforbothpowersupplyandcontrolinterface.It hastwoprimarycontrolmodes,digitalandanalog.Becauseitutilizesasingleconnectorfor devicepowerandbothofthesecontrolmodes,acustom\T"connectorwascreated.One 95 endofthisconnectorgoesintotheMFC,andtwo9-pinconnectorsareontheotherend. Oneoftheseconnectorsallowsconnectiontotheanalogcontrolpins,andthepowersupply forthedevice.TheothercanbeconnectedtoaPCforRS232communication(ofcourse thiscanonlybeconnectedwhenthehighvoltagesystemisOFF).Aschematicindicating thewiringofthisconnectorisshowninFigureA.17. Whenoperatinginthedigitalmode,andconnectedviaRS232(onlywhenthehigh voltageisOFF)thegaswcanbecontrolledusingtheBronkhorstcontrolsoftware.Inthe analogcontrolmode,thedevicecontrolsthewofgasbasedonacurrentappliedtopin 3,whileareadoutofthegaswisprovidedbyacurrentoutputonpin2.Thesecurrents areinarangefrom4{20mAandareinusenowtocontrolthedeviceviathePLCsystem intheelectronicsrackwithinthehighvoltagecage.Thesecurrentscanbecontrolledand readasapercentageonthePIGsourcecontrolpage. Inordertoenablethisanalogcontrol,thecontrolmodewasselectedusingthe urationsoftwarewhilethedevicewasconnectedtothecomputerviaRS232.Thiscontrol modeselectionhasalreadybeenperformedforthebackupMFCaswell,soadirectswapof thetwoshouldnotyieldanychangeinbehaviororoperationofthesystem. TwochallengesarosewheninstallingtheMFC,isolatingthehighvoltage,andpreventing gasleakagewhiletheMFCwasclosed.Toremedytheseissues,theMFCsitsonarubber insulator,onabracketprotrudingfromthewall,sothatthereisnoleakagecurrentwhenthe systemisbroughtuptohighvoltage,andashvalveisinstalleddirectlyfollowingthe MFC.WhennotoperatingthePIGsource,thisgreenshvalvecanbeclosedtoprevent anygasleakingthroughintothesystem. WhenchangingthegasattachedtotheMFC,itisimportanttowthegasthroughthe longrubbertubeforafewminutestocleartheline(3minseemst).Whilethegas 96 FigureA.16:PhotographoftheMFCandsurroundingcomponents.Theopen9-pincon- nectorseenonthe\T"connectorcanbeattachedtoacomputerviaanRS232serialcable whenthesystemisnotonhighvoltage.Theinsulatorisnecessarytopreventcurrentleakage whenthePIGsourceisraisedonthehighvoltage.Originally,thebentaluminumbracket wasthoughttobet,howevertherewasanexcessiveamountofleakagecurrentwhen theMFCwasconnectedtheredirectly.WhennotoperatingthePIGsource,thesh valvecanbeclosedtoallowabettervacuuminthePIGsource. 97 FigureA.17:Schematicshowingthewiringofthecustom\T"connectorbeingusedforthe MFC.ThediagramistakenfromthemanualprovidedbyBronkhorst,andcolorlabelsare addedtoindicatetheactualwiresusedwithintheconnector.Theleftsideofthis connectstotheMFC,andtheconnectiononthelowerrightallowsforpowerandanalog controlstobeattached.Theupperrightconnectorisusedforserialcommunicationwitha PC,andisnotusuallyconnected. 98 isstillouttheline,itcanthenbeconnectedtotheMFC.Ifthelineisnot out,changesinpressure,despitetheMFCwappearingstable,willbeseenforthe hoursofthePIGsourceoperation.Thisisduetothegraduallychangingcompositionofthe gasenteringtheMFC. WhenoperatingtheMFC,asmallspikeinpressuremaybeseenwhengaswis initiated.Thisisnormal,andshouldquicklystabilize.ThecontrolrangeoftheMFCis giveninpercentofthemaximumw.Itisratedforamaximumof0 : 7ml n = min,butthe actualwratedependsonthegasbeingused.Theminimumcontrollablewisaround 2%ofthemaxrange. AdditionaltechnicalspfromBronkhorstandthecalibrationateofthe MFCpurchasedbyBECOLAcanbefoundinmydocumentationintheBECOLAdirectory. A.2.2Higherchargestates Althoughsingly-ionizedcalciumwasusedasareferenceintheonlineexperiment,several testswereperformedtoexaminedoubly-chargedcalciumionsfromthereference source,incasethemajorityoftherarecalciumionsdeliveredtoBECOLAturnedoutto bedoubly-charged.TestswereperformedwiththecoolerRFsetto5 : 34MHz,900V pp ,and thebunchingregionsetto1 : 2MHz,56V pp .Atthissetting,nosingly-chargedionswere seenusinglaserspectroscopy,andsoitwasassumedthatthebeamconsistedentirelyof doubly-chargedions. FollowingtheCEC,theionbeamcurrentwasmeasuredusingaFaradaycup(FC)with andwithouta50Vrepellerpotential.Theinthesecurrentsallowsameasurement relatedtothenumberofparticlespresent,regardlessofcharge(assumingeachchargestate producesthesamenumberofsecondaryelectronswhenimpingingontheFC).Bycomparing 99 this\totalparticle"measurementtothecurrentmeasuredwithavoltageappliedtotheion- kickerwhichislocatedjustaftertheCEC,thefractionofthebeamconsistingofneutral atomsisobtained.Thetotalbeamcurrentofthechargedparticlescanbefoundusing theFCmeasurementwithoutthekicker,andwithoutthe50Vrepellerpotential.Using thebeamcurrentmeasuredbeforetheCEC,assumingatransmissionof98%(measured priortoheatingtheCEC),andtakingintoaccountthefractionofneutralatomsmeasured, thefractionofsingly-anddoubly-chargedionscanbeobtained.Thismeasurementwas performedastheCECwaswarmedup,andtheresultsareshowninFigureA.18.Large errorbarshavebeenplacedontheseresults,astheassumptionsabovethatthetransmission wasconsistent,andallchargestatesproducethesameamountofsecondaryelectronswhen strikingtheFCarefarfromperfect.Indeed,someextrapolationssuggestanegativefraction ofdoubly-chargedions,anobviousimpossibility,butthegeneraltrenddoesrevealthatthe crosssectionsforbothsingleanddoublechargeexchangedoincreasewithtemperature,and thepresenceofbothsingly-chargedionsandneutralCawasusingCLS. A.3DVMinterface Twodigitalvoltmeters(DVMs)areusedtopreciselyrecordvoltagesintheBECOLAsys- tem.Oneisusedinconjunctionwiththelarge10 ; 000:1voltagedividerconnectedtothe cooler/buncherhighvoltage(theFuGpowersupply).Thesecondisconnectedtothesmall chipbased1 ; 000:1voltagedividerwhichisconnectedtothescanningpotential(theMatsu- sadapowersupply).ThesetwometersareanAgilent34410A,andanewer,butcompatible model,Agilent34465A. Thelab'sEPICSsystemisusedtocommunicatewiththeseDVMsandtheir 100 FigureA.18:Datatakenusingdoubly-charged 40 Caions.Thecooler/buncherwasset toallowdoubly-charged 40 Caionstopassthrough,andsingly-chargedionswererejected (vusingspectroscopy).AstheCECtemperaturewasincreased,ionbeamcurrents weremeasuredaftertheCECwithandwithouttheionkicker.Bymeasuringthetotal currentfromthedoubly-chargedionbeampriortotheCEC,assumingatotaltransmission throughtheCECof98%,andmeasuringtheneutralandchargedbeamcurrent,thefraction ofatomsineachchargestatecanbedetermined.Whilethismethodisnotveryprecise,it doesshowthatthecrosssectionforthedoubleandsinglechargeexchangeprocessincreases withtemperature. 101 TableA.2:ThesechannelsareusedtocontroltheDVMs.Identicalchannelsexistfor MTER N0002 . V RD isthechannelwherethevoltageisactuallydisplayed. POLL TXT isthe SCPImessagesenttopollthedevice.Usingacommandotherthantheoneshownheremay inadvertentlyresetthemeasurementsettingsonthedevice. V RD.SCAN isusedtodetermine howoftenthevoltageisread. RST CMD mustbesetto 1 eachtimetheDVMispowercycled, inordertorunthe RST TXT commands.The SCPI channelscanbeusedtosendcommands andqueriestotheDVM,ifaqueryissenttheresponseisshowninthereplychannel.The commandsinthe RST TXT channelshereresettheDVM,thenthevoltagerange andprecision,andlyenablethehighimpedancemode.Notethatahigherprecisionand thusslowerpollingrateisusedfortheFuG,whilealowerprecisionandfasterpollingrate isusedfortheMatsusada.FormoredetailsontheSCPIcommandswhicharerecognized bytheDVMs,seetheSCPIprogrammingguidefromAgilent. Channel TypicalSetting DBEC_BCLS:MTER_N0001:V_RD readchannel _DBEC_BCLS:MTER_N0001:POLL_TXT READ? DBEC_BCLS:MTER_N0001:V_RD.SCAN 5 (FuG) .5 (Matsusada) DBEC_BCLS:MTER_N0001:RST_CMD 0 _DBEC_BCLS:MTER_N0001:SCPI_CMD sendSCPIcommands _DBEC_BCLS:MTER_N0001:SCPI_QUERY sendSCPIqueries _DBEC_BCLS:MTER_N0001:SCPI_REPLY readchannel _dev_DBEC_BCLS:MTER_N0001:RST_TXT.AA *RST _dev_DBEC_BCLS:MTER_N0001:RST_TXT.BB CONF:VOLT:DC10,3E-7 (FuG) CONF:VOLT:DC10,3E-6 (Matsusada) _dev_DBEC_BCLS:MTER_N0001:RST_TXT.CC VOLT:IMP:AUTOON _dev_DBEC_BCLS:MTER_N0001:RST_TXT.DD empty settings.Thisallowsthesevoltagestobearchived,aswellasmakingthemavailableto bereadonachannelthatisrecordedintothedatabytheBECOLADAQduringex- periments.Thecontrolmethodusedisaserialcommunicationinterface,SCPI.Inorder toobtainprecisemeasurementswhicharerefreshedattheappropriaterate,severalcon- optionsshouldbechangedfromthedefaults.Thehigh-impedancemodeshould beturnedon,andthemeasurementrangeandprecisionshouldbesetaccordingly.Higher precisionmeasurementsrequireintegrationoveralargernumberofpower-linecycles(the 60HzACpoweringthedevice). Forarecordingofthestablehighvoltageforthecooler/buncher,theprecisionismore 102 importantthanthetimebetweenmeasurements,sothemeasuringmodewiththeleast uncertainty(100power-linecycles)isusedtorecordanewmeasurementevery1 : 6s.In thecaseofthescanningvoltage,wherethevoltageischangedeveryfewseconds,ahigher polling-rateisrequired,soahigheruncertaintyispreferred. ThesesettingstoandreadfromtheDVMsarecontainedintheEPICSsys- tem.ThemainchannelforthetwoBECOLADVMsare DBEC_BCLS:MTER_N0001 and DBEC_BCLS:MTER_N0002 .Thereareanumberofrelatedchannelswhichcanbeusedforcon- ofhowthesystempollsthedevice,andtheexamplechannelsfromhereforward willuse MTER_N0001 ,butreplacingthiswith MTER_N0002 willyieldthesamechannelsfor thesecondDVM.Thevoltagecanbereadusingthesub-channel :V_RD ,and :V_RD.SCAN determineshowfrequentlythedeviceispolled.Aresetofthemetercanbeperformedby settingthe :RST_CMD channelto 1 .Thischannelwillquicklysetitselfbackto 0 ,butthe fourcommandsstoredinchannels _dev_DBEC_BCLS:MTER_N0001:RST_TXT.AA , :RST_TXT.BB , :RST_TXT.CC ,and :RST_TXT.DD willbesenttothevoltmeter.Theparametersstoredinthesetextchannelsshouldbethe SCPIcommandsdesiredtoresetthedeviceandit.Anexplanationofthesettings usedcurrentlyforthetwoDVMsisshowninTableA.2. Therearefourotherchannelswithat _DBEC_BCLS:MTER_N0001 (notethe _ ).Thesechannelsare :POLL_TXT ,whichholdstheSCPIcommandusedtopoll thedevice, :SCPI_CMD and :SCPI_QUERY ,whichcanbeusedtosendacommandorquery tothedevice,and :SCPI_REPLY whichwillupdatetoshowtheresponsefromthedevicein thecaseofaquery. SeveralAgilentDVMsareusedbytheelectronicsgroupintemporarysetupsunder thechannels HPTEST1 - HPTEST4 .Thechannelstheyusefortakeaslightly 103 tformatbuthavesimilarpurposes.ThechannelsfortheseDVMs,aswellasall theBECOLADVMchannelsmentionedinthissectioncanbeseenintheQTparam locatedintheBECOLAfolderat controls\dvm_settings.prm .TheHPTESTchannels shouldnotbeusedasexamples,astheSCPIcommandstheyusearetypicallynotwell Theyuse MEAS:VOLT:DC? astheSCPIcommandusedtopollthedevice,which resetsthemeasurementsettingstodefaultseachmeasurement.Moredetailsaboutthe HPTESTchannels,includingreviewingtheirlogsthroughtheEPICSsystem,canbefound inthe\HighVoltageMonitoring"documentinmyfolderintheBECOLAdirectory. Whenrunninganexperiment,itisimportanttocheckthatbothDVMsare properly.Afterpower-cyclingtheDVM,besuretosendthevalue 1 totheresetchannel torunthecommands.Afterthishasbeendone,theDVMsdisplayshould indicatethatitisusingamanualrange,andthe\Hi-Z"impedancemode.Thevoltage readoutscanthenbemonitoredwithQtChartlive.Theywillalsoberecordedforeachscan pointintheexperiment,andsavedintothedataasdescribedinAppendixB. A.4Waveformgeneratorinterface InordertoproducetheRFfortheRFQtrapsinthecooler/buncher,twowaveform generatorsareused.Thelowerwaveformgeneratorintherack,aBKPrecision4078B,has twoseparateoutputchannels,whichareusedtopowerthetwoout-of-phaseelectrodepairs inthedownstreambunchingsection.Tocontrolthisdevice,becauseitisraisedonthehigh voltagewiththeentirecooler/buncher,er-opticnetworkcablesaretheonlywaytoenter thehighvoltagecage. Unfortunately,the4078BcanonlybecontrolledviaUSB.Thedevicehasanethernetport 104 FigureA.19:PhotographoftheRaspberryPi3B+anditsconnectionsusedtointerface withthewaveformgenerator.Thisisintherackinsidethehighvoltagecage.Ratherthan connectthewaveformgeneratordirectlytothePi,aUSBhubmustbeplacedinbetween, otherwisethePidoesnotrecognizethedevice. onthebackpanel,butasfarasIcandeterminefromthedocumentationandcontactwith themanufacturer'ssupport,itisanon-functionalport.Thepredecessortothiswaveform generatorpossessedaserialcommunicationport,andsoanethernettoserialconverterwas usedtocontrolthedeviceremotely.Unfortunately,thatwaveformgeneratorwasdamaged whenahighvoltagesparkoccurred,andthecommunicationthroughtheserialportwas broken.Tobringthesamefunctionalitytothisnewmodel,theethernetcableisconnected toaRaspberryPi,whichessentiallyfunctionsasanethernettoUSBadapter. ARaspberryPi3B+isusedwhichhastheRaspbianoperatingsystem(Debian-based 105 Linuxdistribution),andasmallPythonscriptrunsconstantlytolistenonthenetwork forthepacketsfromtheIOCwhichwerepreviouslysenttotheethernettoserialadapter, andthecommandsareinterpreted,andsenttothewaveformgeneratorwiththeUSBTMC protocol.Ifaqueryissenttothewaveformgenerator,theresponseissentbackonthe ethernettotheIOC.BecauseofsomesloppinessintheUSBspecadherenceofeither thePiortheBK4078B,USBcommunicationisnotpossiblewithouttheuseofasimple, non-poweredUSBhubbetweenthePiandthewaveformgenerator.Theseconnectionsare showninFigureA.19. ThePiiswithastartupscriptwhichrunsautomaticallyandinitiatesthe Pythonscript,soasidefromperhapsoccasionallypower-cyclingthePiandthewaveform generator,nostepsshouldbeneededtoinitiatetheconnection.Forthefulldetailsofhow theRaspberryPiiswithregardtothenetwork,andthePythonpackagesused fortheUSBconnection,aREADMEisincludedalongsidealltherelevantscriptsinmy directory,under "Documentation\BK4078BInterface" .AspareRaspberryPiisinone ofthecabinets,buthasnotyetbeenco 106 APPENDIXB Softwaretools Forthisexperiment,IperformedmyanalysisusinganumberofscriptsinROOT.Other experimentershaveusedscriptingsuchasMathematica,Python,andOrigin.Themethods usedtocalibrateandanalyzethedatahavebeendescribedwithinthemainchapters,however thesetechnicaldetailsmaybeusefultofutureexperimenterswhomaybeutilizingorbuilding ofthecodesIhaveused.Thescriptsdetailedbelowarealllocatedwithinmyfolderin theBECOLAdirectory,under "ROOTAnalysisTools\" ,butareintendedtoberunusing ROOTonthetank. B.1BECOLAanalysiswithROOT B.1.1ROOTstructure TheDAQatBECOLAstoresthedatain .mda whichuseabinaryformat.Prior toanalysiswithanyscriptorprogram,itisusefultoconvertthesedataintoASCII allowingthemtobereadastext.TheseASCIIarelargeandtakesometimetoread,so toperformtheanalysisinROOT,Iconvertthedataintoanotherbinaryformat,aROOT Tree,toallowittobereadquicklywhencompilingthehistogramsandperforming Toquicklyconvertfrom .mda thereisaPERLscriptcalled mdaconvert_ajm_regions.pl whichlooksfor .mda withina data directory,andruns mda2ascii_64 oneachus- 107 ingtheoptions -mt1f .Aftercreating .asc foreachrun,itthenusesROOTtorun rootscripts/asc_to_root_v5.C ,whichwillcreatea .ROOT foreachrun,containingthe datainaTree.ThesescriptsarebuiltontheonesdevelopedbyDominicRossi. Afterconvertingthedatausingtheabovemethods,analysiscanbeperformedusingthe toolsdescribedinSectionB.1.2.HereIwilldescribethebasicformatoftheASCIIas wellastheROOTTree. DatafromtheDAQistakeninseveraldimensions.Currently,thedataisstoredina 5-Dformat.ToreadtheappropriatevaluesfromtheASCIIalinewithmorethan12 numbersisidenasthestartofaspmeasurement.Thereare31columnsinthese lines,andtheyholdthefollowinginformation: ‹ 1: 5-Dindex(beginningfrom1)thescanorrunnumber(savedtotheROOTtreeas run ). ‹ 2: 4-Dindex(beginningfrom1)theregionnumber(savedtotheROOTtreeas region ). ‹ 3: 3-Dindex(beginningfrom1)thevoltagestepnumber(savedtotheROOTtree as vstep ). ‹ 4: 2-Dindex(beginningfrom1)unuseddimension,always1(savedtotheROOT treeas p4 ). ‹ 5: 1-Dindex(beginningfrom1)thetimebinnumber(1-1024). ‹ 6: 5-Dpositioner(beginningfrom0)similarto 1 butcountingfrom0(savedtothe ROOTtreeas run_bin ). ‹ 7: 4-Dpositioner(beginningfrom0)similarto 2 butcountingfrom0(savedtothe ROOTtreeas region_bin ). ‹ 8: 3-Dpositioner(beginningfrom0)similarto 3 butcountingfrom0(savedtothe ROOTtreeas vbin ). ‹ 9: 3-Ddetector1,DACvoltagesetting(savedtotheROOTtreeas dac_set ). ‹ 10: 3-Ddetector2,DACrawvoltagereading(savedtotheROOTtreeas raw_dac ). ‹ 11: 3-Ddetector3, VD_D1285 theDACvoltagereading(savedtotheROOTtreeas dac_read_plc ). ‹ 12: 3-Ddetector4, MTER_N0001 Matsusadavoltagedividerreading(savedtothe ROOTtreeas dac_read_dvm ). ‹ 13: 3-Ddetector5, HPTEST1 unusedDVMreading(savedtotheROOTtreeas hptest1_read ). 108 ‹ 14: 3-Ddetector6, MTER_N0002 FuGDVMreading(savedtotheROOTtreeas bcb_voltage_read ). ‹ 15: 2-Dpositioner,(beginningfrom0)similarto 4 butcountingfrom0(savedtothe ROOTtreeas p15 ). ‹ 16: 1-Ddetector1,Thetime(inseconds)ofthissptimebin(savedtotheROOT treeasanarraycalled time ). ‹ 17: 1-Ddetector2,channel0countsrecordedinthisspbin,typicallythePMT fromtheolddetectionsystem(savedtotheROOTtreeasanarraycalled phot ). ‹ 18: 1-Ddetector3,channel1countsrecordedinthisspbin,typicallythePMT fromtheupstreamnewdetectionsystem(savedtotheROOTtreeasanarraycalled sciup ). ‹ 19: 1-Ddetector4,channel2countsrecordedinthisspbin,typicallythePMT fromthedownstreamnewdetectionsystem(savedtotheROOTtreeasanarraycalled sci_dn ). ‹ 20-31: 1-Ddetector5-17,channel3-15countsrecordedinthisspbin,typically unused(savedtotheROOTtreeasarrayscalled scal4-scal16 ). 1024rowsofthisformatarereadintogiveasingleentryintotheROOTtree.From these1024rows,asindicatedabove,the1-Ddetectorand1-Dindexvaluesarecombined intoanarray.Inthisformatthedatacaneasilybecompiledinto2Dhistograms,aseach entryintheTreecontainsthecompletedatafromasinglevoltagestep. Asimpleexamplescript,showinghowoneoftheseROOTtreescanbecompiledintoa 2D(time,voltage)histogramcanbefoundinmyfolder: "ROOTAnalysisTools\ReadingfromASCII\becolatree.C" . ThiscanbecompiledinROOT,andthe becolatree_run functioncanthenbeexecuted. Althoughthisdoesnotcarefullybinthedata,itcanbeusefultounderstandhowROOT Treeswork,andhowtocompilethedatatobeginperforming B.1.2Analysistools Thecompletesetofscriptsusedforanalysisislocatedin "ROOTAnalysisTools\e16003" . Adescriptionoftheprocedureusedtoperformtheanalysiscanbefoundin Procedure.txt . 109 EachscriptismeanttobecompiledandrunusingROOTonthe fishtank .Forexample, thesteptoprocessthedatauses caSections.C .Thisprogramhasalistgroupingthe intosections,andwillperformthetimecut,andthesectionintoa1Dhistogram. Themacro SectionMacro1NoDVM and SectionMacro2NoDVM containalistofcallsto thefunctionin caSections.C .Thewaythattheselistsweregeneratedcanbeseenin the FinalAnalysis Excelspreadsheet,locatedinthe ExcelFiles folder.Onthe\Run Summary"page,severalcolumnsatthefarrightgeneratelistsoffunctioncallstoberunon eachsection.Notethatsometimecuarehardcodedherebasedondatafrommultiple sectionscombined,orforchannel2,shiftedbasedonthetimecutsfromchannel1. Toperformthestepdescribedin Procedure.txt ,theprogramshouldbecompiled andloadedintoROOT,andthenthemacroscanbeexecuted,usingthefollowingcommands withintheROOTcommandprompt: >.LcaSections.C++ >.xSectionMacro1NoDVM >.xSectionMacro2NoDVM Thisproducesa1Dhistogramforeachsection,andfromthis,thecalibrationusingthe 40 ; 44 Cadatacanbeperformed.Using calib40_44.C and Macro4044.C ,calibratedofthe referenceisotopescanbeobtained.Inthe\SectionCalibration"pageoftheExceleach setof40and44datagivesa\SpCalibration"incolumnF,neighboringcalibrations arethenaveragedtodeterminean\AverageCalibration"incolumnG. This\AverageCalibration"voltagewasthenusedfortherareisotopedataforeach section,itcanbefoundinthe\CalibFitMacro"Nowthateachsectionhasbeen calibrated,thesectionsofeachrareisotopecanthenbecombinedtogether.Therearemany scriptspresenttodothesecombinations,ingeneral,theyarenamedaccordingtotheisotope 110 (36,37,or38)andthenthechannel(1or2),andinsomecaseswillalsohaveC1orC2in thename,toindicatethattheyuseacalibrationobtainedfromchannel1,orchannel2. Whenthesesectioncombinationsareperformed,theresultinghistogramsarecalibrated andshiftedsothatthe0pointisthecentroidof 40 Ca.Theresultsoftheseandthe combinationofchannel1andchannel2datatodeterminetheisotopeshiftanduncertainty areintheExcelonthe\FinalResults"page. TherearealsoseveralROOTscriptsthatcanbeusedtocreatebetterplots(showingthe residualsofthealongsidethedata).Thesearenamedinastraightforwardmanner. AnotherExcelisalsoincluded,\ChargeRadiiPlot"whichtakestheisotopeshifts, andconvertsthemintothetialandabsolutechargeradiiusingtheatomicfactors. 111 APPENDIXC Personalcontributions C.1Publications ‹ Chargeradiioft 36 ; 37 ; 38 K Phys.Rev.C92,014305,July7,2015 Iwasresponsiblefortheoperationofapairofelectro-opticmodulatorsforthis experiment.Isimulatedtheresultingfrequencyspectrumofthelaserlightforseveral tfrequencythensetuptherequiredelectronicstodrivethem atthesefrequenciesandusedaninterferometertoevaluatetheirreallifebehavior. Ialsowroteadetailedsetofinstructionsfortheiroperationduringtherunofthe experiment,andtookshiftsduringtheexperiment. ‹ Populationdistributionsubsequenttochargeexchangeof29 : 85keVNi + onsodium vapor SpectrochimicaActaPartB,113,16-21,November1,2015 Iparticipatedinthedatacollectionforthisexperiment,includingtheoperationof thePIGsource,andprovidedcommentsonthepaper. ‹ Chargeradiiofneutront 52 ; 53 Feproducedbyprojectilefragmentation Phys.Rev.Lett.117,252501,December15,2016 112 Forthiswork,Iwasresponsibleforthelasersystem.Throughseveralmo totheopticalsystemwherethelaserisintroducedtothebeamline,Iwasableto improvethecollimationofthelasersystemandreducethelaserbackgroundbyafactor of3.DuringtherunoftheexperimentIwasinchargeofchangingfrequenciesand aligningthelaserasneeded,aswellasleadingashift.Ialsoperformedanindependent dataanalysisforthisexperimentinordertoverifytheresults. ‹ Firstdeterminationofground-stateelectromagneticmomentsof 53 Fe Phys.Rev.C96,054314,November16,2017 Thispaperusesthesameexperimentaldataasthepublicationmentionedabove.I preparedthedraftoftheExperimentandResultsection,aswellastheDiscussion sectionsregardingtheBuck-PerezsystematicrelationandIsoscalarspinexpectation value.Ialsohelpedpreparetheandeditedthedraftforsubmission. ‹ Protonsupyandchargeradiiinproton-richcalciumisotopes NaturePhysics,15,432-436,February,2019 IwasresponsibleforsimulatingandinstallingtheupgradedPDSasdescribedin thisthesis,includingleadingthedatacollectionandanalysisusedtocharacterizethe newdetectoranddecidetheusedduringtheonlineexperiment.Iwas responsibleforoperatingtheion-sourceduringthisexperiment,includingthe installationandsetupofanMFC,andpreliminarytestsleadinguptotheexperiment usingvariousgases,chargestates,andtheCEC.IalsoperformedtheSNRcal- culationsusedtoselectthebunchreleaseperiodfortheshort-livedisotopes.Following theonlineexperiment,Iperformedthecalibrationofthedataandtheextractionof 113 thechargeradii.Iwrotetheportionsofthepaperdescribingtheexperimentand extractionofthechargeradii,ledtheeditingprocess,andhelpedpreparethe ‹ Ground-stateelectromagneticmomentsof 37 Ca Phys.Rev.C,99,061301(R),June,2019 Thispaperexaminesthenuclearmomentsextractedfromthesameexperimental dataasthepapermentionedabove.Iprovidedthecalibrateddata,andperformeda parallelanalysisofthenuclearmomentstocontheresults.Ialsohelpedprepare theandprovidedcommentsonthemanuscript. C.2Presentations ‹ productionoftransitionmetalionsforcollinearlaserspectroscopyatBECOLA/NSCL 7 th InternationalConferenceonLaserProbing(LAP2015),EastLansing,MI,June 7,2015. (Poster/Talk) ‹ ChargeradiimeasurementsatBECOLA LowEnergyCommunityMeeting,EastLansing,MI,August21-22,2015. (Talk) ‹ Populationdistributionfollowingatomicchargeexchangeof29.85keVNi + onasodium vapor DivisionofNuclearPhysicsMeetingoftheAmericanPhysicalSociety,SantaFe, NM,October28-31,2015. (Talk) ‹ Chargeradiiandnuclearmomentsof 52 ; 53 Fe,andtheshell-closuresignatureat N =28 NuclearStructure,Knoxville,TN,July24-29,2016. (Poster/Talk) 114 ‹ TowardchargeradiimeasurementsoftCa DivisionofNuclearPhysicsMeetingoftheAmericanPhysicalSociety,Vancouver, BC,Canada,October13-16,2016. (Talk) ‹ CollinearlaserspectroscopyofstableZrinpreparationforfuturepulsedlasertech- niques NNSAStewardshipScienceAcademicPrograms,Naperville,IL,April12-13,2017. (Poster) ‹ Nuclearmomentsof 53 Fe,andtheshell-closuresignatureat N =28 AdvancementsinRadioactiveIsotopeScience,Keystone,CO,May28-June2,2017. (Poster) ‹ ChargeradiimeasurementsoftCa DivisionofNuclearPhysicsMeetingoftheAmericanPhysicalSociety,Pittsburgh, PA,October25-28,2017. (Talk) ‹ Firstdeterminationofnuclearground-stateelectromagneticmomentsof T =3 = 2 37 Ca NuclearStructure2018,EastLansing,MI,August5-10,2018. (Poster) ‹ RFQiontrapforlaserspectroscopymeasurementsatBECOLAfacility InternationalConferenceonTrappedChargedParticlesandFundamentalPhysics (TCP2018),TraverseCity,MI,September30-October5,2018. (Talk) ‹ Chargeradiioft 36 ; 37 ; 38 Ca JointMeetingoftheNuclearPhysicsDivisionsoftheAPSandtheJPS,Waikoloa Village,HI,October23-27,2018. (Talk) 115 REFERENCES 116 REFERENCES [1]E.Rutherford.Thestructureoftheatom. TheLondon,Edinburgh,andDublinPhilo- sophicalMagazineandJournalofScience ,27(159):488{498,1914. [2]R.Hofstadter.Electronscatteringandnuclearstructure. 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