HIGHVOLTAGEDEVELOPMENTANDLASERSPECTROSCOPYFORTHESEARCH OFTHEPERMANENTATOMICELECTRICDIPOLEMOMENTOFRADIUM-225 By RoyAnthonyReady ADISSERTATION Submittedto MichiganStateUniversity inpartialtoftherequirements forthedegreeof PhysicsŒDoctorofPhilosophy 2021 ABSTRACT HIGHVOLTAGEDEVELOPMENTANDLASERSPECTROSCOPYFORTHESEARCH OFTHEPERMANENTATOMICELECTRICDIPOLEMOMENTOFRADIUM-225 By RoyAnthonyReady Permanentelectricdipolemoments(EDMs)violateparity( P ),timereversal( T ),and combinedcharge-conjugationandparitytransformation( CP )assuming CPT symme- try.Radium-225isexpectedtohaveanenhancedatomicEDMbecauseitsnucleusis octupole-deformed.IntheRaEDMexperiment, 225 Raatomsarevaporizedinane usive oven,slowedandcollimatedbycoolinglasers,andtrappedbetweentwohighvoltage electrodes.Wemeasurethespinprecessionfrequencyofthetrappedradiuminuniform, appliedelectricandmagneticdsandsearchforafrequencyshiftcorrelatedwiththe electricd,thesignatureofanonzeroEDM. TherearetwogenerationradiumEDMmeasurements.Themostrecentmeasure- mentreducedtheupperlimitto1 : 4 10 23 e cm.Intheupcomingsecondgeneration measurements,wewillimplementkeyupgradestoimproveourEDMsensitivitybyup tothreeordersofmagnitude.Thisthesisfocusesonmyworkimprovingtheelectricd strengthandlasercoolinge ciencyforthesecondgenerationmeasurements. Additionally,TheFacilityofRareIsotopeBeamsisexpectedtoproduceRadium-225 thatcanbeharvestedforEDMmeasurements.Wearedevelopingalaserinducedu- orescenceexperimentthatwillmeasuretheabsoluteuxofadirectedbeamofatoms emittedfromane usiveoven.Theuxmeasurementwillusestablesurrogateisotopes tocharacterizeradiumharvestinge ciency.Iwillreporttheresultsofourinitiale orts modelingandmeasuringtheatomicbeamuorescenceofmultipleatomsources. Copyrightby ROYANTHONYREADY 2021 ACKNOWLEDGEMENTS Ihavebeenexceedinglyfortunateinhavingthesupportofwonderfulfriends,mentors, andfamily.Iamgratefultoallofthemforbeingwithmeonthisjourney. ThanksmyfriendsthatIgrewupwithandamnowgrowingoldwith:Joe,Jordan, Carissa,Johnny,Nick,andShea.Wheneverseeeachother,whichsometimesisnotso often,wepickthingsrightupasifIneverwento togradschool.It'sgreat. Thankstomyprofessorsandmentors,fromcommunitycollegetograduateschool: RonArmale,MattDietrich,JiyeongGu,MortenHjorth-Jensen,PrashanthJaikumar,KayKo- los,GlennOrton,GalenPickett,andMichaelSyphers.Iamabetter(andhappier)scien- tistforknowingthemandfromtheirsupport,guidance,andkindness. ThankstomyMichiganfriends:BakulAgarwal,DeannaAmbrose,LisaCarpen- ter,BoeColabewala,BrandonElman,AlecHamaker,MonicaHamaker,MaxHughes, JakeHuneau,AndrewLajoie,BrendenLongfellow,AndrewMiller,ElizabethMiller,Al- iceMills,DanielRhodes,SeanSweany,ZakTilocco,CorahWalker,LindsayWeinheimer, andErinWhite.Asimportantasacademicsuccesswas,sotoowasthejoyIfoundin commiserating,relaxing,laughing,andbuildingrelationshipswithyouall. ThankstoallmySpinlablabmates.YouareafantasticteamandI'mgladyouwere withmetoshareinthefunparts,thepainfulparts,thesetbacks,andthesuccessesthat comewithourlineofwork.BenLosethwasalwaysreadytolendahandwithcleanroom work,althoughhemayhaveoccasionallyruedthatstandingo erduringthelongershifts! TenzinRabgaisagreathangonlongdrivestofarawayworkshopsandanexcellentsoccer teammate.Foratime,ourroutinewastobattletemperamentallasersinthemorningand jokeaboutthefigoose-ianfldistributionofthelocalgeeseonthestrolltolunch. Thankstomyadvisor,JaideepSingh.IjoinedSpinlabforthelasers,butIstayed forthe...well,thelasers,butalsoallthefunIhadworkingwithJaideepandinthe supportivecommunityofstudentsandcollaboratorshehascultivated.Hemetmewhere iv Iwasatwhileguidingmewithwell-timedsupport,advice,andhumorousanecdotes. JaideeptaughtmeaninestimableamountaboutbeingagoodscientistandI'mexcitedto bringittobearinmyfutureendeavors. Lastly,I'mgratefulfortheloveandsupportofmyfamily:mysisters,Kristenand Brianna;myparents,CharisaandJohn;ourdog,Summer;mygrandparents,Joyand John;andmygreat-grandmother,Barbara.Youalwaysbelievedinmeandhelpedme buildthetofollowthroughwithmyeducation.Twentyyear-oldRoydidnot knowhewasgoingtostudyphysics!Itseemedtoturnoutokaythough,andthroughit allItooksolaceknowingthatyouwererootingformefromafar. Thankyou! v TABLEOFCONTENTS LISTOFTABLES ...................................... x LISTOFFIGURES ...................................... xiv KEYTOABBREVIATIONS ................................. xxiii CHAPTER1SYMMETRYVIOLATIONANDPERMANENTELECTRICDIPOLE MOMENTS ................................. 1 1.1TheStandardModel................................1 1.1.1Predictivepower..............................1 1.1.2Unsolvedpuzzles.............................2 1.2FundamentalSymmetries.............................2 1.2.1Timereversal................................2 1.2.2Paritytransformation...........................3 1.2.3Chargeconjugation............................4 1.2.4 CP transformation.............................4 1.2.5 CPT transformation............................6 1.3BaryonasymmetryoftheUniverse.......................6 1.4 CP ViolationBeyondtheStandardModel...................8 1.5Electricdipolemomentsearchesasaprobeof CP violation.........11 1.5.1Neutronelectricdipolemoment.....................13 1.6 CP ViolationinAtomsandMolecules......................14 1.6.1Theshieldingofthenucleusfromexternalds...........14 1.6.2Sensitivitytotheelectronelectricdipolemoment...........14 1.6.3Theelectron-nucleoninteraction....................14 1.7 CP ViolationinDiamagneticSystems......................19 1.8Thesisoutline...................................20 CHAPTER2INTRODUCTIONTOTHERAEDMEXPERIMENT ......... 22 2.1Motivation.....................................22 2.1.1Laser-cooledelectricdipolemomentsearches.............22 2.1.2Sensitivitytoexperimentalparameters.................25 2.2Overviewofexperimentalapparatus......................26 2.2.1LasercoolingandtheZeemanSlower..................27 2.2.2Lasertrapping...............................31 2.2.3The2015Radium-225measurement..................33 2.3Targetedupgradesforanimprovedelectricdipolemomentmeasurement.33 2.3.1AtomcoolingwithanimprovedZeemanslower............33 2.3.2Atomdetectione ciencywithStimulatedRamanAdiabaticPassage34 2.3.3Higherelectricdstrength.......................35 2.3.4IncreasingRadium-225availability...................36 vi 2.4Experimentalrequirements............................37 2.4.1Measurementtechnique.........................37 2.4.2MagneticJohnsonnoisecalculations..................39 2.4.3Paramagneticimpurities.........................42 2.4.4Leakagecurrentanddangle.....................43 2.4.5Polarityimbalanceintheelectricd.................45 2.5E ectofElectrodeMisalignments........................45 2.5.1Descriptionoftheelectricdelementanalysis........46 2.5.2Electricdresponsetoelectrodemisalignmentnearthecenter ofthegap..................................48 2.5.3Electricdbehaviorintheelectrodeedgeregion..........50 2.5.4Modelingtheelectricdbehaviornearthecenteroftheelec- trodegap..................................52 2.5.5Estimatinge ectsforrealisticmisalignmentsinthehighvoltage apparatus..................................54 2.6ElectrodeUpgradeStrategyandResults....................55 2.6.1Highvoltagedischarge-conditioning..................55 2.6.2Typicalsizeofdischarges.........................57 2.6.3Results...................................58 CHAPTER3HIGHVOLTAGEELECTRODEDEVELOPMENT ........... 60 3.1ElectrodePropertiesandPreparation......................61 3.1.1Legacyelectrodepreparation.......................61 3.1.2Considerationofmaterialsfornewelectrodes.............63 3.2ElectrodeResidualMagnetizationMeasurements...............65 3.3ReviewofHighVoltageSurfaceProcessingApplications...........68 3.3.1Secondgenerationelectrodesurfaceprocessing............69 3.3.2Cleanroomsandhighpressurerinsing.................71 3.4ElectrodeDischarge-Conditioning........................72 3.4.1Highvoltageteststation.........................72 3.4.2Opticalmeasurementsofelectrodesandgapsizes..........75 3.4.3Dataacquisitionandteringsettings.................76 3.4.4Identifyingelectrodedischarges.....................78 3.4.5Discharge-conditioningprocedure...................80 3.4.6ConditioningresultsforelectrodepairNb 56 .............82 3.4.7ConditioningresultsforelectrodepairNb 78 ..............85 3.4.8ConditioningresultsforelectrodepairTi 13 ..............86 3.4.9ConditioningresultsforelectrodepairNb 23 .............87 3.4.10Comparisonofoverallelectrodeperformance.............88 3.4.11Comparisonofelectrodeperformancewithothersystems......90 3.4.12Steady-stateleakagecurrentanalysis..................92 3.4.13TransportationandinstallationofelectrodesinRaEDMapparatus.93 CHAPTER4RADIUMBRANCHINGRATIOS .................... 95 4.1RadiumlasercoolingwiththeZeemanslower.................95 vii 4.2Lasersforthebranchingratiomeasurement..................101 4.3Radiumuoroscopyexperimentalsetup....................103 4.4Radiumuoroscopydataacquisition......................104 4.5Measurement....................................105 4.6Results.......................................107 4.7Analysis.......................................112 CHAPTER5CALIBRATINGTHEATOMICBEAMFLUXFROMANEFFUSIVE OVEN ................................... 117 5.1Motivation.....................................118 5.1.1Radiumsourceforelectricdipolemomentexperiment........118 5.1.2Rubidiumuxmeasurements......................121 5.2spectrum................................122 5.2.1Atomicstatenotation...........................122 5.2.2Atomictransitionintensity........................123 5.2.3Frequencyoftransitions.........................125 5.3Modelingthespectrallineofadirectedatomicbeam.........127 5.3.1TheABFapparatusandcalculatingthephotodetectorsignal....128 5.3.2Calculatingtheuorescencepoweronthephotodetector.......129 5.3.2.1Calculatingtheatomicux,vaporpressure,andtheatom rate................................130 5.3.3Thesingleatomuorescencerate....................132 5.3.4TheDoppler-freeexcitationrate.....................135 5.3.5Dopplerbroadeningforadirectedatomicbeam...........138 5.3.6Theatomicangulardistributionandphotodetectorsolidangle...140 5.3.7Atomicangulardistribution.......................141 5.3.8Solidanglecalculation..........................144 5.3.9Tyingeverythingtogetherintoanatomicbeamuorescencesim- ulation...................................145 5.4Comparingsimulationstodata.........................148 5.4.1Ybuorescenceandpowerbroadening.................148 5.4.2Rubidiumuorescence..........................153 5.4.3Simulationsofacalciumspectrum...................163 5.5Suggestedimprovementstomeasurementtechnique.............166 5.5.1Trackinglaserpolarizationandmagneticd.............166 5.5.2Increasingthesignalsizewithlightcollection.............167 5.5.3Increasingthesignalsizewithacalibratedpumpinglaserand atomicoven................................170 CHAPTER6PRECISIONGAMMA-RAYINTENSITYMEASUREMENTS ..... 172 6.1Introduction....................................172 6.1.1Gamma-rayspectroscopyandstockpilestewardship.........172 6.1.2Long-livedisotopes........................173 6.1.3HPGecalibration.............................178 6.1.4MonteCarlosimulation..........................181 viii 6.2Resultsandanalysis................................185 6.3ConclusionsandOutlook.............................188 CHAPTER7CONCLUSIONSANDOUTLOOK ................... 190 APPENDICES ........................................ 197 APPENDIXA:Constants,units,atomicandnuclearproperties........198 APPENDIXB:Codeanddataavailability.....................200 APPENDIXC:AvalanchePhotodiodeSettings..................200 APPENDIXD:Fluxgatemagnetometry......................201 BIBLIOGRAPHY ....................................... 203 ix LISTOFTABLES Table1.1:Even/odd-nessoftheelectricd( ~ E ),magneticd( ~ B ),intrinsic angularmomentum( ~ J ),andtheirdotproductsundertimereversal andparitytransformations...........................4 Table1.2:StandardModelestimatesofelectricdipolemomentsofdi erentpar- ticles.......................................5 Table1.3:EDMmeasurementsofdi erentsystems.UCN=ultracoldneutron. CL=level.PSI=PaulScherrerInstitute.JILA=JointIn- stituteforLaboratoryAstrophysics.Boulder=UniversityofColorado, Boulder.PTB=PhysikalischTechischeBundesanstalt.ANL=Argonne NationalLab.ILL=InstitutLaue-Langevin.................12 Table1.4:95%levelupperlimitcalculationsoflow-energy CP -violating parametersbasedonexperimentalmeasurementsusingaglobalap- proach[59,6]. C S and d e calculatedfrommeasurementsbypara- magneticsystems[60,61,62,63]. g (0) ˇ ; g (1) ˇ ;C T ; and d sr n calculated frommeasurementsindiamagneticsystemsandnucleartheoryasof 2019[64,65,43,66,67,68]..........................16 Table1.5:AcollectionofcalculationsofnuclearSchi momentcoe cientsfor Radium-225andMercury-199.Rangesarelistedinbrackets.......18 Table1.6:Experimental(even-even)andcalculated(odd-evenbetadeformation parametersforaselectionofisotopes.....................20 Table2.1:RadiumZeemanslowerpropertiesforthecurrentredcyclingtransi- tionandtheplannedbluecyclingtransition.................28 Table2.2:RaEDMsystematicrequirementsatthe10 26 e cmsensitivitylevel. Detailedsystematiclimitevaluationsfortheseparameterscanbefound inpreviouswork[48,95]. B isdeterminedbyEquation2.29.......38 x Table3.1:Electrodeinventory.Large-grain(LG)niobiumelectroderesidualre- sistanceratio(RRR) > 250.OF=oxygenfree.G2=grade-2.Simichrome polishbyhand.Diamondpastepolish(DPP)byhand.LPR=low pressurerinse.HPR=highpressurerinse.HF=huoricchemi- calpolish.EP=electropolish.BCP=bu eredchemicalpolish.SiC=silicon carbidemachinepolish.CSS=colloidalsilicasuspensionmachine polish.VB=420Œ450 Cvacuumoutgasbake.WB=150Œ160 Cwa- terbake.USR=ultrasonicrinseafterdetergentbath............62 Table3.2:Bulkmaterialpropertiesofelectrodes.Wefistrong B -impuritiesfl as ˜ m = (10 6 cm 3 mol 1 ) > +1000,where ˜ m isthemolarsusceptibil- ity. ˜ m (Nb)=+208...............................63 Table3.3:Surfacedecontaminationcomparison. P =rinsepressure, T =rinse time,CR=cleanroom,RR=rinseresistivity.................72 Table3.4:5 ˙ Dataacquisitionandteringsettings.Usedtersindicatedby circles.SR=samplerate.......................78 Table3.5:Electrodeconditioningsummary. E i =initialdstrength. E max =max dstrength. E f =validateddstrength. R i ( R f )=initial nal)dischargerate. ¯ I =steady-statecurrentat E f ..............89 Table4.1:Transitionsandwavelengthsforbranchingratiomeasurement.......101 Table4.2:MeasuredPMTsignalsofdecaysfrom 3 F o 2 ..................108 Table4.3:Calculatedbranchingfractions(BF)andoscillatorstrengthsfrom 3 F o 2 ..116 Table5.1:AselectionofatomictransitionsoftheYbgroundstate,S 1 0 .Val- uesfromNIST. I =intensity. =resonantwavelength,frequency. ˝ =lifetime. A =EinsteinA-coe cient....................121 Table5.2:YtterbiumtotalstrengthfactorsforS 1 0 ( F ) ! P 1 o 1 ( F 0 )...........123 Table5.3:Rubidiumrelativestrengthfactorsfor 2 S 1 = 2 ! P 2 1 = 2 .Wigner6- j val- uescalculatedwithanonlineversionoftheRoot-Rational-Fraction package[145]..................................124 Table5.4:Rubidiumtotalstrengthfactorsfor 2 S 1 = 2 ! 2 P 1 = 2 ..............125 Table5.5:LiteraturevaluesofthehconstantsofYb,Rb,andCaisotopes withnonzeronuclearspin...........................126 xi Table5.6:Calculatedhshifts E HF ofytterbium,rubidium,andcal- cium.Totalangularmomentum F = I + J ..................127 Table5.7:ValuesusedforYb 1 S 0 ! 1 P o 1 atomexcitationrate R ( ;~ r )........135 Table5.8:Calculated,measured,andliteraturevaluesofthe 1 S 0 ! 1 P o 1 transi- tionfrequencieswithrespecttoYb 174 ( I =0 ;F =1).............150 Table5.9:Saturationintensitiesandoscillatorstrengthsforselectedytterbium, rubidium,andcalciumtransitions. =frequency, A =EinsteinA- coe cient(NISTvalues). f a =oscillatorstrength. f a (Rb)from[146]. f a (Ca)from[152]. I 0 =saturationintensity..................151 Table5.10:AselectionofatomictransitionsoftheRbgroundstate,5 s S 2 1 = 2 .In- tensityvaluesandwavelengthsfromNIST,lifetimevaluesfrom[153]. I =intensity. =resonantwavelength,frequency. ˝ =lifetime. A =EinsteinA-coe cient...........................152 Table5.11:Calculatedrubidiumtransitionfrequencies(h+isotopeshifts) withrespecttothetransitionof 85 Rb, 0 85 Rb = S 2 1 = 2 ! P 2 o 1 = 2 = 377.107THz...................................153 Table5.12:AselectionofatomictransitionsoftheCagroundstate,4 s 21 S 0 .Inten- sityvaluesandwavelengthsfromNIST. 3 P o 1 lifetimefromDrozdowski et.al[154]. I =intensity. =resonantwavelength,frequency. ˝ =lifetime. A =EinsteinA-coe cient....................163 Table5.13:Calculatedandliteraturetransitionfrequencies(h+isotope shifts)withrespecttothetransitionof 40 Ca, 0 40 Ca = S 1 0 ! P 1 o 1 =709.078THz.Referencevaluefor 46 Ca from[155],allothers from[156]....................................165 Table6.1:Gamma-raydecaysfromaselectionoflong-livedisotopes. (BR) =branchingratiouncertainty..........................175 TableA1:Fundamentalphysicalconstants(fromtheNISTdatabase)........198 TableA2:Unit................................198 TableA3:Angularmomentum,masses,andabundancesofYb.ValuesfromNIST.199 TableA4:Vaporpressurecoe cientsforytterbium,rubidium,andcalcium....199 xii TableA5:Rubidiumproperties.Massnumber A ,nuclearspin I ,isotopeshiftIS. ValuesfromNIST................................199 TableA6:Calciumproperties.Massnumber A ,nuclearspin I ,isotopeshift(IS) forthetransition 1 S 0 ! 1 P o 1 . 47 CaatomicmassfromKramida[172]. 47 CaisotopeshiftbyAndl et.al [149].Allotherisotopeshiftsfrom Nörtershäuser et.al [155].AllothermasesfromNIST...........199 xiii LISTOFFIGURES Figure1.1:Ahierarchicaldiagramdepictingtherelationshipsbetween CP -violating phenomenaatthelow-energy(atomic)scaleuptothehigh-energy (theory)scale.Dottedlinesconnectparameterswiththehighestcou- plingstrength.Dashedlinesrepresentpotential CP -violatingpro- videdbyBSMphysics.............................8 Figure2.1:TheRaEDMexperimentalapparatus....................25 Figure2.2:AcartoonoftheradiumZeemanslower. ~ p = m~ v isthemomentum ofaradiumatomwithmass m andvelocity ~ v and ~ p = h= ‹ z isthe momentumofaslowinglaserphotonwithwavelength .........29 Figure2.3:Cloudofradiumatomstrappedbetweenhighvoltageelectrodesin opticaldipoletrap...............................31 Figure2.4:Left:assemblyoftheniobiumpairNb 56 at1mmgapinMacor holder.Right:aslitcenteredonthegapshieldstheelectrodesur- facesfromheatingbytheatomtrappingandpolarizinglasers......39 Figure2.5:Onepossibleelectrodedesignwhosevolumeisafactoroftensmaller thanthestandardRaEDMelectrode....................42 Figure2.6:Aplotofthemaximumalloweddmisalignmentoverarangeof leakagecurrentsforatargeted10 26 e cmsensitivity...........44 Figure2.7:Asoftwaremeshedmodeloftheelectrodepairandcoordinatesys- tem.The-meshedelectrodegapregionisshadedblue........46 Figure2.8:Aplotoftheelectricdangleasafunctionoftheverticalposition y .Inthisplot,theelectrodesareaxiallyalignedandtheangularmis- alignmentisvariedfrom0Œ16mrad.Thecenterofthegap,0.5mm belowthetopelectrode,correspondsto y =0................47 Figure2.9:Aplotoftheverticalelectricdforangularalignmentsintherange 0Œ16mrad.Theaxialmisalignmentis100 m.Thecenterofthegap, 0.5mmbelowthetopelectrode,correspondsto y =0...........48 Figure2.10:Acontourofthehorizontalelectricdmagnitudeformisaligned electrodesclosetothe8mmedgeregion...................50 xiv Figure2.11:Aplotoftheelectricdangleaswescanhorizontallyacrossthe electrodesurface(8mmradius)fromthecentertotheedgeregion....51 Figure2.12:Aplotoftheverticalcomponentoftheelectricdaswescanhor- izontallyacrosstheelectrodesurfaceintheedgeregion(radiusof 8mm)......................................52 Figure2.13:Astraightlinetothesimulatedpolarangleoftheelectricd foranangularmisalignmentof16mradandanaxialmisalignment of1mm.Thecenterofthegap,0.5mmbelowthetopelectrode, correspondsto y =0..............................53 Figure2.14:Aresidualplotofamodeloftheverticalelectricdfora16mrad angularmisalignmentand1mmaxialmisalignment.Themodelas- sumesthatthedisafunctionoftheangleoftheelectricd.....54 Figure2.15:Contourplotsoftheverticalcomponentoftheelectricdinthe xz (left)and xy (right)planeandwitha2mradtilt..............55 Figure2.16:Forty-minutesnapshotsoftheconditioningprocessinearly,middle, andstages.Positiveandnegativecurrentisplottedwithgreen crossesandredcirclesonalogarithmicscale.Leakagecurrentless than10pAisomittedforclarity.Therightverticalaxisistheapplied voltageandisplottedasablueline.....................56 Figure2.17:AschematicoftheperiodicEDMhighvoltagewaveform. A positive chargingupramp. B positivechargingdownramp. C negative chargingupramp. D negativechargingdownramp...........57 Figure3.1:(a)Cross-sectionalelectrodeschematic.Surfaceshaveatnesstol- eranceof25.4 mandaparallelismof50.8 m.Thetopsurfaceis polishedtoanaverageroughnessof0.127 m.Thebaseismounted bya10-32tappedhole.Copperrodsareusedtoconnecttotheelec- trodes'3.2mmdiametersideboretohighvoltagefeedthroughs.(b)A pairoflarge-grainNiobiumelectrodesinacleanroomstainlesssteel container....................................60 Figure3.2:Fromlefttoright:acopper,niobium,andtitaniumelectrode.......61 Figure3.3:Themagnetizationrailsystemsitsinsideamu-metalshield.......64 Figure3.4:Aschematicofthegradiometercircuit.Resistorandcapacitorvalues arelistedinFigureD1.............................65 xv Figure3.5:Simulated3kHzButterworthlowpasscurve(blueline)andmea- suredfrequencyresponsewithawaveformgeneratorinput(redcir- cles)ofthegradiometercircuitinFigure3.4.1.86kHzdashedverti- calline=measuredcuto frequency.16.4kHzdashedverticalline= uxgatefrequency,attenuatedby ˇ 53dB.................66 Figure3.6:Gradiometerresultsforaniobiumelectrode.Averagegradiometer signal= 440 : 8 1 : 6pT.Averagemonitorsignal=88 : 2 1 : 3pT.Av- eragenullsignal= 8 : 5 0 : 1pT.......................67 Figure3.7:Residualmagnetizationmeasurementsofgrade-2titaniumelectrodes usingcommercialuxgates(MSU)andacustommagnetometer(USTC).68 Figure3.8:(a)Ibuiltaportablecleanroomwitha2 0 2 0 HEPAter(SAM22 MSNCR).(b)TheNSCLdetectorcleanroom.IthasseveralHEPA unitsandaccommodatestheteststationanduptothreepersonnel...69 Figure3.9:Electrodehighpressurerinseequipment.(a)Theelectrodesaremounted onanacryliccylindricalshellcenteredonaturntable.Astheappa- ratusrotates,aconcentrichighpressurerinse`wand'rinsestheelec- trodes.(b)Theelectrodesaremountedsothattheprimarysurfaces facethewand.(c)Weswitchedtoarinsegunbecausethewaterqual- itywasbetter..................................70 Figure3.10:Electrodestorageandtransport.(a)Eachelectrodepairismounted fromthebaseinastainless-steelbin.(b)Theelectrodesarelabeled byetchingthematerialandelectrodenumberontheoutsideofthe bin.(c)Werecommendbucklinguptheelectrodesforcartrips.....71 Figure3.11:MSUHVtestapparatus. 1 9699334AgilentTurbo-Vvibrationdamper 2 Pfei erHiPace80turbomolecularpumpwithforelineEdwards nXDS10iA736-01-983dryscrollroughpumpandtwovalves 3 Matheson 6190Series0.01 mmembraneterandpurgeport 4 Ceramtec 30kV16729-03-CFfeedthrough 5 0 : 312in : 2 electrodesinPEEKholder (resistivity10 16 M cm) 6 20AWGKapton-insulated,gold-plated copperwire 7 MKS392502-2-YG-Tall-rangeconductron/iongauge 8 Shieldedprotectioncircuit:LittelfuseSA5.0Atransientvoltage suppressor,EPCOSEX-75Xgasdischargetube,Ohmite90J100E100 resistorinserieswithKeithley64822-channelpicoammeter 9 Ohmite MOX94021006FVE100M resistorsinserieswithAppliedKilovolts HP030RIP020HV...............................73 xvi Figure3.12:TheimagingcomponentsoftheHVapparatus.Thisisaview oftheapparatusafterrotatingtheschematicinFigure3.11by90 andremovingnon-imagingcomponents. 1 worm-driverailmount 2 ThorlabsMVTC23024tion(M)=0.243,4.06flworking distance(WD)telecentriclens 3 EdmundOpticsEO-2323monochrome CMOScamera,4 : 8 msquarepixels 4 AdjustableElectrodeGap Assembly:MDC660002linearmotion0.001flgraduated,1fltravel adjustabledriveandcustomPEEKmountinterfacewithangularad- justment.....................................76 Figure3.13:(a)Thelinearadjustableelectrodegapassembly(b)Aweightedline istoascatterplotofgapsizevs.drivepositionandaconversion frompixelstoinchesisdetermined.....................77 Figure3.14:Nb 23 at+22kV/mmoverone60secondcycleduringthehour ofconditioning.FromtheGaussian(solidredline)wedetermine themeantobe ¯ x ˙ =106 : 7 3 : 6pA.Thereare207totaldatapoints. Weiden7eventsexceedingthe ¯ x +5 ˙ =125pAthresholdas discharges....................................79 Figure3.15:Theo set-subtractedaverageleakagecurrentsforeachpositiveand negativehighvoltagetimeperiodduringconditioningat20kVwith Nb 56 atagapsizeof1mm..........................80 Figure3.16:Histogramsofallthedischargesforbothpolaritiesduringthethird hourofconditioningthetitaniumelectrodesonalog-logscale.....81 Figure3.17:Discharge-conditioningtimelineforNb 56 ata1mmgapsize......82 Figure3.18:InstallationofniobiumelectrodepairNb 56 inRaEDMapparatus. (a)ANLportablecleanroomwithaluminumbeams,plasticdrapes, anda4 0 2 0 HEPAter.(b)Theborosilicateglasstubewascleaned withaclean-roomgradewipewrappedaroundtheendofa pole.(c)Thecleanroomwaspositionedovertheelectrodeentrypoint beforeinstallingtheelectrodes(seeninthebottomcorner)........83 Figure3.19:AschematicofthewaterbakeoftheRaEDMexperimentalapparatus followingtheinstallationofthenewelectrodepair.............84 Figure3.20:Discharge-conditioningtimelineforNb 78 witha1mmgapsize.....85 Figure3.21:Discharge-conditioningtimelineforTi 13 ata0.9mmgapsize......86 Figure3.22:Discharge-conditioningtimelineforNb 23 ata1mmgapsize......88 xvii Figure3.23:Aplotofelectricdsreachedbyelectrodepairs.Bluedataare electrodesusedintheRaEDMapparatus.Greendataareelectron gunelectrodestestedwitha 100kVpowersupply[114].Reddata areelectrodestestedatMSU.Brighter,moreintensecolorsaremore recentresults..................................90 Figure3.24:Weightedaveragesofthesteady-stateleakagecurrentonlinearand logscales.Errorsareontheorderof0.1pA.................91 Figure4.1:Left:thecurrentfiredflZeemanslowingscheme. R 1=1429nm.Right: theenvisionedfiblueflZeemanslowerupgrade. R 1=698nm, R 2= 712nm, R 3=2752nm............................96 Figure4.2:TheMaxwell-Boltzmannspeeddistributionofradiumatomsexiting theoven.Theestimatedfractionofatomsthatcanbesu ciently slowedfortrappingareshadedaccordingtotheslowingscheme.....97 Figure4.3:AnenergyleveldiagramofthelowestenergylevelsandE1- allowedtransitionsof 226 Ra.Measuredlifetimes:7 s 7 p P 3 o 1 [137], 6 d 7 p F 3 o 2 [90],7 s 6 d D 3 1 [139],7 s 6 d D 1 2 [140].Calculatedlifetimes: 7 s 6 d D 3 2 [141],allothertransitions[89].Wavelengthsarelabeled alongtransitionlinesin[nm]invacuum/air................99 Figure4.4:Aschematicofthebranchingratiouoroscopysetup.Inset:energy diagramformeasuringthe 3 D 1 branchingratio..............100 Figure4.5:(a)NIRlaserdiodeinatemperature-controlledmount.Duringu- oroscopymeasurements,thepowermeterisremovedandlaserlight iscoupledtothebehindit.(b)Left:CustomNIRinterface boxcircuit.Right:Thecurrentsource,thermoelectrictemperature controller,andcustominterfaceboxusedfortheNIRlaserdiode....102 Figure4.6:Aofthenear-infrared(NIR)diodelaserwavelengthtothetemper- aturecontrollerresistancesetting......................103 Figure4.7:Left:thethreearecombinedwithdichroicsandsenttothe uorescencemirrorwithatelescopemirrorsetup.Right:atop-down viewofthebluelaserlightpassingthroughtheviewportintothe uorescenceregion..............................104 Figure4.8:AscreenshotoftheVIIwroteforrecordingPMTcountsforthe branchingratiomeasurements........................105 xviii Figure4.9:Fluorescencesignalofthe 3 F o 2 ! 3 D 1 transitionwhiledepopulating the 3 D 2 statewitha712nmprobelaser...................106 Figure4.10:8/8/2018Averageduorescencesignalofthe 3 F o 2 ! 3 D 1 transition whiledepopulatingthe 3 D 2 statewitha712nmprobelaser.......109 Figure4.11:8/8/2018Averageduorescencesignalofthe 3 F o 2 ! 3 D 1 transition whiledepopulatingthe 1 D 2 statewitha912nmprobelaser.......110 Figure4.12:8/9/2018Secondmeasurementof 3 F o 2 ! 3 D 1 transitionwhilede- populatingthe 3 D 2 statewitha712nmprobelaser............110 Figure4.13:8/8/2018Averageuorescencesignalwithpumpbeamandprobe beamsblocked.................................111 Figure4.14:8/9/2018Averageuorescencesignalofthe 3 F o 2 ! 3 D 3 transition withthepumpbeamtunedonresonance..................111 Figure4.15:8/9/2018Averageuorescencesignalofthe 3 F o 2 ! 3 D 3 transition withthepumpbeamtunedo resonance..................112 Figure4.16:Lineshapeforthe 3 F o 2 decaychannelsatdi erentprobelaser powers.....................................115 Figure5.1:Decayschemeof 225 Ra.Alphaandbeta-decayaredenotedby and ,respectively.Half-livesarefromtheNationalNuclearDataCenter. kyr=1000years.d=days.m=minutes..................118 Figure5.2:Aschematic(nottoscale)oftheatomicbeamuorescencesetup. Thisisgeneralizedtobeapplicabletoallthreesetupsdiscussedin thischapter...................................128 Figure5.3:Schematicoflasersystem...........................129 Figure5.4:Saturatedvaporpressurecurveforytterbium,calcium,andrubidium..131 Figure5.5:Calculateduorescencesignalastheoventemperatureisvariedus- ingalaserpowerof10mW..........................133 Figure5.6:Excitedstatepopulationofatwo-levelsystemfor R =2 10 8 s 1 and ˝ 0 =5ns : ....................................134 Figure5.7:TheweakpumpinglimitYbsingleatomlaserexcitationrateusing theparametersinTable5.7..........................136 xix Figure5.8:Fromlefttoright,inorderofincreasingnoodlediameter-to-length: bucatini,cannelloni,anellini noodles.ImagesobtainedundertheCC0 1.0Universal(CC01.0)PublicDomainDedicationLicense........141 Figure5.9:Theatomicangulardistributionofforarangeofnozzleratios.(a)80 degreerange,alllinesconvergetoanintensityofzeroat90degrees (b)Zoomedintowithin5degrees.Thelegendappearsintheor- derofdescendingintensity.Middlesolidline=ytterbiumandcal- ciumratio =0 : 25.Dashedline=radium =0 : 024.Bottomsolid line=rubidium =0 : 01...........................142 Figure5.10:Agridofthepointsusedtonumericallyintegratethesolidangleof acirculardetector.Westartwitha2 2squaremeshandcutouta circle(shownwithredsquares)toobtaintheresult............144 Figure5.11:Thephoton-atomyieldpercentchangeasthenumberofsubdivisions oftheuorescencevolumeisvaried.Themegacubesidelengthis 32mm,thelaserwidthis7mm........................146 Figure5.12:Theintegralof intheplane y =0.Inthisplane,thephotodetector at y =76 : 2mmviewingangleisconstrainedbytheinnerdiameter ofthevacuumcross(30.226mm).Thescanningareaavailabletothe photodetectoris15.52mmsquare......................147 Figure5.13:Yb5/15/2017ABFmeasurement.Yb-172TP=triplepeakconsisting ofYb 172 ,Yb 173 ( F =7 = 2),andYb 173 ( F =3 = 2).(a)Seven-peakVoigt +constanto settodata(b)Fractionalresidualof( y axistrun- catedforclarity)................................149 Figure5.14:SimulatedYbuorescencespectrumintheweakpumpinglimit.....151 Figure5.15:AmeasuredrubidiumABFspectrumwithalaserpowerof50 W. (a)Voigtlineshapetouorescencesignalvs.laserfrequency(b)Frac- tionalresidualofthe...........................155 Figure5.16:SimulatedRbuorescencespectrumintheweakpumpinglimit.Laser power=50 W,laserradius=2.7mm.(a)Collimatedbeamwith nozzleratio =0 : 01(b)uncollimatedbeamwithnozzleratio !1 ..156 Figure5.17:Voigttosimulateduorescence(redcircles)withcollimatedand uncollimatedangulardistributions.(a)Collimateddistribution,cor- respondingtooneofthepeaksinFigure5.16a(b)Uncollimateddis- tribution,correspondingtooneofthepeaksinFigure5.16b.......158 Figure5.18:ResidualsoftosimulatedRbtransitionsinFigures5.17b,5.17a...159 xx Figure5.19:Measuredtotalstrengthfactorratios S FF 0 = S 32 ofRb 85 .Thehorizon- tallinesareexpectedtotalstrengthfactorratiosforanunpolarized laserbeamusingthevaluesfromTable5.4.Dashedline S 23 = S 32 =1; dot-dashedline S 33 = S 32 =0.8;dottedline S 22 = S 32 =0 : 2857.....160 Figure5.20:Measuredtotalstrengthfactorratios S FF 0 = S 22 ofRb 87 .Thehorizon- tallinesareexpectedtotalstrengthfactorratiosforanunpolarized laserbeamusingthevaluesfromTable5.4.Dashedline S 21 = S 22 =1, equivalently S 12 = S 22 =1;dottedline S 11 = S 22 =0.2..........161 Figure5.21:MeasuredabundanceratioofRb 87 toRb 85 .Dashedline=0.3856is thecalculatedratiousingtheNISTdatabasevalueslistedinTableA5.162 Figure5.22:Simulatedcalciumuorescencespectrumintheweakpumpinglimit. Logscalecalciumuorescencespectrumsimulationtoshowtheweaker transitions.Thesmallsignaldiscontinuitiesat600MHzand1400MHz arenumericalartifacts.............................164 Figure5.23:(a)theAtomicFluxapparatus.(b)Schematicofin-vacuumlight collectionsetup(nottoscale).........................167 Figure5.24:Theatom-to-photonyieldifweusealight-focusinglens,or,equiv- alently,increasethedetectorarea.Thelaserwidthis7mminthis calculation.Assumingonlyraysperpendiculartothedetectorsur- facearefocusedontothedetector,wegetmaximumlightcollections foradetectorradiusofhalfthelaserwidth,or3.5mm..........168 Figure5.25:Theatom-to-photonyieldaswevarythelaserbeampower. cube sidelengthis0.5mm,megacubesidelengthis3.2cm. max =1 : 523 for w =7mm..................................169 Figure5.26:Theatomphotonyieldaswevarythesizeofthelaserbeamwidth. cubesidelengthis0.5mm,megacubesidelengthis3.2cm. max = 1 : 523for w =7mm..............................170 Figure6.1:Asimexampleofoneofthepossible 236 Udecaychains.Data from[158]...................................174 Figure6.2:LLNLgamma-raydetectorsetup.......................176 Figure6.3:AschematicoftheLLNLBEGedetector.ModelbyCanberra,Mirion Technologies.Usedwithpermission.....................177 Figure6.4:(a)Geant4modelofthegamma-raysource.(b)Expandedschematic ofthegammaraysourcegeometry(nottoscale)..............179 xxi Figure6.5:Schematicofdetector-sampleation.................180 Figure6.6:Fitsforthe1173keVand1332keV 60 Cogamma-rayspectrum......181 Figure6.7:(a)E ciencyplotofHPGewithcalibratedgammasourcesanda sample-detectordistanceof164mm.(b)E ciencyplotofHPGewith asample-detectordistanceof95mm....................182 Figure6.8:Asimulationof1MeVgamma-raysemittingfromasource(rightside ofgraphic)abovetheLLNLHPGedetector(leftsideofgraphic).....183 Figure6.9:SnapshotofGeant4modeloftheHPGedetectorandbackgroundshield.184 Figure6.10:Fourth-orderempiricaltothemeasurede cienciesofasuiteof calibratedgammasourcesatadistanceof95mm.............185 Figure6.11:Fractionalresiduale ciencyscatterplotwithsample-detectordis- tanceof95mm.................................186 Figure6.12:Fractionalresiduale ciencyscatterplotwithsample-detectordis- tanceof164mm................................187 Figure6.13:custom-designedgamma-sourceholderfortheLLNLHPGedetector..188 FigureD1: C 1 =5 : 6nF ;C 2 =47nF ;C 3 =4 : 7nF ;C 4 =47nF ;C 5 =1 : 5nF ;C 6 = 5 : 6nF ;C 7 =100nF ;C 8 =2 : 2nF ;C 9 =15nF ;C 10 =0 : 82nF ;C N = 1 F ;R 1 =10k ;R 2 =100k ;R 3 + R 4 =10k ;R ref =10k ;R F = 10k ;R o =10k .............................201 FigureD2:BartingtonMag03IEL70uxgateschematicformagnetizationmea- surements....................................202 FigureD3:Fluxgate:BartingtonMag03IEL70.16kHzexcitationfrequency,noise ooris6pT rms = p Hz.Powersupply:BartingtonPSU1.5pT rms = p Hz noiseoor.Dataacquisition:NIPCie-6320.16-bit.2mVnoiseoor on10Vscale..................................202 xxii KEYTOABBREVIATIONS ANL.......... ArgonneNationalLab AOM.......... acousto-opticalmodulator APD.......... avalanchephotodiode AWG.......... Americanwiregauge BAU........... baryonasymmetryoftheUniverse BCP........... bu eredchemicalpolish BEGe......... broadenergygermanium BR............ branchingratio BSM.......... BeyondtheStandardModel C ............. chargeconjugationsymmetry CARIBU...... CaliforniumRareIsotopeBreederUpgrade CEDM........ quarkchromo-electricdipolemoment CKM.......... Cabibbo-Kobayashi-Maskawa CL............ level CMB.......... CosmicMicrowaveBackground CP ............ combinedchargeconjugationandparitysymmetry CPT .......... combinedchargeconjugation,parity,andtimereversalsymmetry CSS........... colloidalsilicasuspensionmachinepolish DAQ.......... dataacquisitioncard dph........... dischargesperhour DPP........... diamondpastepolish EDM.......... permanentelectricdipolemoment EP............ electropolish FRIB.......... FacilityforRareIsotopeBeams FWHM .......... fullwidthathalfmaximum xxiii G2............ grade-2 HEPA......... high-e ciencyparticulateair HF............ huoricchemicalpolish HPR.......... highpressurerinse HPGe......... highpuritygermanium HV............ highvoltage IS............. isotopeshift ISO........... theInternationalOrganizationforStandardization KEK.......... HighEnergyAcceleratorResearchOrganization LG............ large-grain LLNL......... LawrenceLivermoreNationalLaboratory LPR........... lowpressurerinse MJN.......... magneticJohnsonnoise mil............ thousandthofaninch MOT.......... magneto-opticaltrap MSU.......... MichiganStateUniversity NSCL......... NationalSuperconductingCyclotronLaboratory NSSC......... NuclearScienceandSecurityConsortium ODT.......... opticaldipoletrap OF............ oxygenfree ORNL......... OakRidgeNationalLaboratory P .............. paritysymmetry PEEK......... polyetheretherketone PMT.......... photomultipliertube RF............ radiofrequency rms........... rootmeansquare RRR........... residualresistanceratio xxiv SAM.......... SingleAtomMicroscope SiC............ siliconcarbidemachinepolish SM............ StandardModel STIRAP....... StimulatedRamanAdiabaticPassage SUS........... speciallystainlesssteel(fiClean-Zfl) T ............. timereversalsymmetry TEC........... thermoelectrictemperaturecontroller Ti:Saph....... titaniumsapphire TMP.......... turbomolecularpump TUM.......... TechnicalUniversityofMunich UCN.......... ultracoldneutron UHV.......... ultrahighvacuum UPW.......... ultrapurewater USR........... ultrasonicrinse USTC......... UniversityofScienceandTechnologyofChina VB............ vacuumoutgasbake WB........... waterbake WD........... workingdistance xxv CHAPTER1 SYMMETRYVIOLATIONANDPERMANENTELECTRICDIPOLEMOMENTS 1.1TheStandardModel TheStandardModel(SM)explainstheinteractionsbetweenallquarks,whichmake upbaryonssuchasprotonsandneutrons,andleptons,suchaselectrons.Theinter- actionsarecharacterizedbytheexchangeofforce-mediatingparticles:gluonsforthe strongforce,photonsfortheelectromagneticforce,and W and Z bosonsfortheweak force.Quarks,leptons,andtheirassociatedantiparticlesundergointeractionsinaccor- dancewithfundamentalsymmetryrulesestablishedbytheStandardModel. Thereismorematterthanantimatterintheuniverseduetoaminutedegreeofvi- olationoffundamentaldiscretesymmetriesthatotherwisetreatparticlesandantiparti- clesequally.Todate,theStandardModelisconsistentwithallexperimentallyobserved symmetry-breakingprocesses. 1.1.1Predictivepower Particleswithintrinsicangularmomentumwillprecessaboutanexternalmagneticd withafrequencythatischaracterizedbyitsgyromagneticratio g .Anelectronisapoint- likeparticlewithintrinsicspin J =1 = 2 : Inanemptyvacuum,theexpectationvalueofthe electron'sgyromagneticratiois g =2. Inreality,spaceispermeatedbyvirtualparticlesthatarespontaneouslycreatedand annihilated.Thedeviationsfromtheemptyvacuumexpectationvalueof g causedby theseparticlepairscanbecalculatedwithquantumelectrodynamics.Theelectron's gyromagneticratiohasbeenmeasuredtoaprecisionoflessthanonepartinatrillion (10 12 )[1,2].ThisisoneofthemostsensitivetestsoftheSMandturnsouttobein completeagreementwiththeory. 1 TheStandardModelhasalsopredictedtheexistenceofparticles,includingthetop quarkandtheHiggsboson. 1.1.2Unsolvedpuzzles Whileunifyingtheelectromagnetic,strong,andweakforces,theSMfailstodescribe dynamicsinvolvingthegravitationalforce.Italsocannotaccountformatterthatdoes notinteractthroughthethreeforces.Observable,radiativemattermakesup only5%ofthetotalmassneededtoexplaintheobservedkinematicsofgalaxiesandthe expansionoftheuniverse.Themissingmassisthoughttobebalancedby75%dark energyand20%darkmatter. 1.2FundamentalSymmetries 1.2.1Timereversal Therearethreefundamentaldiscretesymmetries:paritytransformation( P ),chargecon- jugation( C ),andtimereversal( T ).Fields,particles,andparticlepropertiesbehavedif- ferentlyunderapplicationofanyoneoranycompositeofthesetransformations.Their behaviorischaracterizedbyfieven-nessflorfiodd-nessflunderatransformation.Forex- ample,undertimereversaltheelectricdisevenandthemagneticdisodd: T ~ E ( ~ r ;t ) = ~ E ( ~ r ; t )= ~ E ( ~ r ;t )fieven 00 T ~ B ( ~ r ;t )= ~ B ( ~ r ; t )= ~ B ( ~ r ;t )fiodd 00 Here t istime, ~ r isthepositionvector, T isthetimereversaloperator, ~ E ( t )istheelectric d,and ~ B ( t )isthemagneticd.The P and T transformationsof ~ E , ~ B ,intrinsicangular momentum ~ J ,andtheirdotproductsaregiveninTable1.1. Thiscanbegeneralizedtoanyquantumsystem.Wecanwritethetimereversaltrans- 2 formationofanystatecharacterizedbythewavefunction i ( ~ r ;t ) h m 3 = 2 i : T 1 ( ~ r ;t ) = 1 ( ~ r ;t ) fieven 00 T 2 ( ~ r ;t ) = 2 ( ~ r ;t ) fiodd 00 1.2.2Paritytransformation Paritytransformation,orspaceinversion,invertsthecoordinatesofthestate.InaCarte- siancoordinatesystem( ~ r = x ‹ x + y ‹ y + z ‹ z ),aparitytransformationcanbewrittenas: ˇ ( ~ r ;t ) = ( ~ r ;t ) = 8 > > > > > < > > > > > : + ( ~ r ;t ) ,fieven 00 ( ~ r ;t ) ,fiodd 00 where ˇ istheparityoperator.Polarvectorssuchastheelectricd ~ E areodd,while pseudovectors(crossproductoftwopolarvectors)suchasthemagneticd ~ B areeven. Parityviolationwasmeasuredin1957byWuet.al[3],followingtheproposalof LeeandYang[4],inthebeta-decayofcobalt-60(1925-dayhalf-life)tonickel-60: Co 60 27 ! Ni 60 28 + e + e , where e isanelectron(betaparticle)and e isanantineutrino.Theypolarizedasample of 60 Coinamagneticdandmeasuredthebetaparticleintensityatanangle and 180 withrespecttothepolarizationaxis.Inthedorientation,thenuclei tendedtoemitbetaparticlesoppositethedirectionofnuclearspin.Wutheninverted thenuclearspinofthesamplebyswitchingthepolarizingddirection,simulatingthe paritytransformation,andrepeatedthemeasurement.Again,thebetaparticlespreferen- tiallyemittedoppositethenuclearspin.Thistestdemonstratedparityviolationthrough theobservationofthecorrelationbetweenthebetadecaydirectionandthenuclearspin. 3 Table1.1:Even/odd-nessoftheelectricd( ~ E ),magneticd( ~ B ), intrinsicangularmomentum( ~ J ),andtheirdotproductsundertimere- versalandparitytransformations. ~ J ~ B ~ E ~ J ~ B ~ J ~ E P +1+1 1+1 1 T 1 1+1+1 1 1.2.3Chargeconjugation Chargeconjugationchangesaparticletoitsantiparticleandviceversa,forexamplean electrontoapositron.Thistransformationcanbewritteninketnotationasfollows: C j e i ! e + E , where C isthechargeconjugationoperator.Unlike P and T symmetry,thestateis changedunlesstheparticleisitsownantiparticle,e.g.thephoton. 1.2.4 CP transformation TheCKMmatrixcharacterizestheapproximatepreservationofquarkgenerationnum- ber(up/down,charm/strange,top/bottom).Quarkinteractionsinvolveasmallamount fimixingflwhere,forexample,anupquarkmayundergoaninteractionandconvertto astrangequarkaverysmallpercentageofthetime.Violationofcombinedchargecon- jugation( C )andparity( P )symmetry,or CP ,isanecessaryingredientoftheobserved dominanceofmatteroverantimatter,orbaryonasymmetryoftheuniverse(BAU)[5]. CP violationisencodedintheStandardModel(SM)byacomplexphase intheCabibbo- Kobayashi-Maskawa(CKM)quarkmixingmatrix[6]. Todate, CP violationhasbeenmeasuredintwosystems.Theisfromtheindi- rectobservationofthe CP -forbidden2 ˇ decayofthelong-livedKmesonmixedstatein 1964[7].Thee ectissmall,afewpartsinathousand,butthisdecayprocessisquite common.Thiswaslaterdirectlyobserved(i.e.nostatemixing)[8]. 4 Table1.2:StandardModelestimatesofelectricdipolemomentsof di erentparticles. labelEDMsystemSMprediction 10 32 e cm d e electron0 : 000000000001[14] d q quark0 : 01[15] d n neutron1[16] d p proton1[16] d A 129 Xe xenonatomic0 : 005[6] d A 199 Hg mercuryatomic0 : 04[6] Thesecondmeasured CP -violatingprocessisthedecayofneutral B mesonpairs B 0 and ¯ B 0 in2001[9,10].Twocollaborations(thefi B -factoryflmeasurements)independently measuredasymmetricbranchingratiosinoneofthebaryon-antibaryondecaychannels. Themeasurementswereinitiallyindirectobservationsof CP violation.Theexperiment wasrepeatedbybothgroupsanddirect CP violationwasobservedin2004[11,12]. Experimentalinputfromthe B -factoryandothermeasurementsyieldSM-consistent CP -violationcalculationswiththesingle CP -violatingphaseparameterizationofthe CKMmatrix[13]. CP violationhasalsobeensearchedforinmeasurementsofthepermanentelectric dipolemoments(EDMs)ofleptons,nucleons,atoms,andmolecules.TheStandardModel predictionsforEDMsarefarsmallerthancurrentbestmeasurements,asshowninTa- ble1.2.Wewilldiscuss CP violationinthecontextofEDMsinSection1.5. CP -violatinginteractionsinquantumchromodynamicsarisefromthefithetatermfl [ dimensionless ] (alsocalled QCD )describedbyquarkvormixing[17].Aswewill seeinSection1.4,quarkandleptonEDMsscalelinearlywith .SMestimatesofEDMs ofelectrons,neutrons,andatomsarelistedinTable1.2. 5 1.2.5 CPT transformation The CPT theoremrosetonotorietyafter P violationwasobservedintheWuexperiment and CP violationwasobservedintheCronin&Fitchmeasurement. The CPT theoremarisesfromquantumdtheoryandstatesthatthecombineddis- cretesymmetrytransformationofcharge,parity,andtimereversalisconserved( CPT =+1) inallinteractions.Fromthisitfollowsthateachparticleanditsantiparticle,forexam- pleanelectronandapositron,musthaveexactlythesamemass. CPT conservationalso meansthatanyviolationof CP iscompensatedbyanequalviolationof T . Todate,thereisnoknowninteractionthatviolates CPT .Themoststringentexperi- mentaltestisthatofthemassdi erence m K 0 m K 0 [ GeV ] betweentheneutralkaonpair K 0 and K 0 [18,19,20,21,22,13]: m K 0 m K 0 < 4 : 0 10 19 GeV95%level Theneutralkaonmassis497 : 6MeV,sotheprecisionofthistestiseightpartsin10 19 . 1.3BaryonasymmetryoftheUniverse ThebaryonasymmetryoftheUniverse(BAU)istheextremelyhighabundanceof baryons,forexampleprotonsandneutrons,relativetoantibaryons.Baryondominance allowsmattertostickaround.Iffundamentalprocessesweightedbaryongeneration andantibaryongenerationequally,thenthesesandthekeyboardsneededwritethem wouldn'texistbecausethebaryonsneededtomakethosethingswouldannihilatewith anequalnumberofantibaryons. Antimatterabundancecanbedirectlysearchedforincosmicrays(atomstraveling nearthespeedoflight)andintheFaradayrotationoflightpassingthroughtheinter- stellarmedium,aswellasindirectlyinthedecayproductsofannihilationpairs[23]. Recentmeasurementsofantiproton/protonandpositron/electronratiosincosmicrays placeincreasinglystringentconstraintsonantimatterabundance[24,25,26,27]. 6 OnewaytheBAUcouldhavebeenestablishedisthroughbaryogenesis.Baryogen- esisproposesthatatsometimeaftertheearly,fiparticlesoupflphaseoftheUniverse, theUniversereachedacriticaltemperaturethatallowedsome CP -violatingmechanism switchedon,allowinganetgenerationofbaryons[5].AstheUniversecooledfurther,the netbaryon-generatingprocessrampeddown,preservingtheasymmetry[28]. Inelectroweakbaryogenesis, CP -violatingprocessesdrivebaryongenerationatacrit- icalelectroweakphasetransitiontemperatureofapproximately100GeV.However,Stan- dardModelcalculationsofthephasetransitioncannotreproducetheobservedBAU.This isprimarilyduetotheheavinessoftheHiggsboson(125GeV)andthesmall-nessofthe CKMmatrix-induced CP -violation[29]. This CP -violatingphaseisrelatedtotheobservedbaryon-to-photontoratio [ dimensionless ] : = n B n ¯ B n / sin ( ) , (1.1) where n B h m 3 i isthebaryondensity, ¯ n B h m 3 i istheantibaryondensity,and n h m 3 i istheearlyuniversephotondensity. Nuclearphysicsmodelsandastronomicalobservationsareusedtodeterminethemass fractionsoflightelementssuchasHelium-4.Thesemassfractionsareusedtoconstrain .Thenetbaryondensityisalsoinferredfrommeasurementsofthecosmicmicrowave background(CMB).BoththemassfractionsandCMBmeasurementsareinconcordance, resultingin ˇ 10 9 [30]. The CP -violatingphaseintheStandardModelyieldsabaryon-to-photonratioof ˇ 10 26 [31].Thisdiscrepancystronglymotivatesthesearchfornewsourcesof CP violation. 7 Figure1.1:Ahierarchicaldiagramdepictingtherelationshipsbetween CP - violatingphenomenaatthelow-energy(atomic)scaleuptothehigh-energy (theory)scale.Dottedlinesconnectparameterswiththehighestcoupling strength.Dashedlinesrepresentpotential CP -violatingprovidedbyBSM physics. 1.4 CP ViolationBeyondtheStandardModel Figure1.1showsasimhierarchyoftherelationbetweensubatomicEDMsand CP -violatinginteractionstoatomicandmolecularEDMs.Strongcouplingsbetween termsarehighlightedwithconnectinglines.ThepathfornonzeroEDMsintheStan- dardModelisthroughtheCKMmatrixand .BSMtheoriesprovidepotentialadditional CP -violatingchannelsthattlyincreasepredictedEDMmagnitudes. Supersymmetry(SUSY)isoneextensiontotheStandardModelthatproposesthat everyparticlehasitsownfisuperflparticle,doublingthenumberofparticlesintheStan- dardModel.TheminimalsupersymmetricStandardModel(MSSM)isoneversionof SUSYwhereallsupersymmetricmassesareequivalentto M SUSY [ TeV ] [32]. Inthesearchforatheoryunifyingtheelectromagnetic,weak,andstrongforces(figrand tionfl),particlespossessingbothquarkquantumnumbersandleptonquantum 8 numbershavebeenproposed.Theseleptoquarksarethoughttobeveryheavybosons thatcaninteractwithbothquarksandleptons[32].Ifobserved,leptoquarkswouldbea cleansourceofnewphysicsandprovideanadditionalcontributiontothetensorelectron- nucleoninteraction C T (discussedinSection1.6.3). SUSYprovidesacontributiontotheneutronEDMthroughthequarkEDMs d q and quarkchromo-EDMs Ÿ d q [33,34,17]: d n = 4 3 d d 1 3 d u m 2 ˇ e m N m 2 3 Ÿ d d + 1 3 Ÿ d u (1.2) m = m u + m d 2 ,(1.3) where m u =2 : 32 0 : 10MeVistheupquarkmass, m d =4 : 71 0 : 09MeVisthedownquarkmass, m N [ eV ] isthenucleonmass,and m ˇ [ eV ] isthepionmass. TheEDMoftheneutronandproton d n , d p dependsmoststronglyon andtheisoscalar pion-nucleoncouplingparameter g (0) ˇ .ThenucleonEDMshaveverysimilarexpressions, soforbrevityI'llshowjusttheneutronEDMdependence[35]: d n = d sr n eg A g (0) ˇ 8 ˇ 2 F ˇ 0 B B B B @ log m 2 ˇ m 2 N ˇm ˇ 2 m N 1 C C C C A ,(1.4) where e> 0istheelementarycharge, d sr n [ e cm ] istheshort-rangeneutronEDM, g A ˇ 1 : 27 [ dimensionless ] isthestrongpion-nucleoncouplingconstant,and F ˇ ˇ 92 : 4 [ MeV ] 1 isthepiondecayconstant. 1 Ihavealsoseenreportedvaluesof F ˇ ˇ 186MeV[36]and ˇ 130 : 2MeV[37],where thesevaluesdi erfromthemaintextvaluebyfactorsof2and p 2,respectively. 9 Recently,thedependenceof d n on and g (0) ˇ hasbeencalculatedusingLatticeQCD[35]: d n = ( 1 : 52 0 : 71 ) 10 16 e cm,(1.5) g (0) ˇ = ( 12 : 8 6 : 4 ) 10 3 (1.6) Theisovectorpion-nucleoncouplingconstant g (1) ˇ isrelatedtotheupanddownquark chromo-EDMs(CEDMs)bythefollowingexpression[34]: g (1) ˇ =3 10 12 Ÿ d u Ÿ d d 10 26 cm j < qq> j ( 225MeV ) 3 m 2 0 0 : 8GeV 2 ,(1.7) where j < qq> j = j 0 j qq j 0 j h MeV 3 i isthequarkgluoncondensateand m 2 0 ˇ m 2 N h MeV 2 i isthestrengthcoe cientof j < qq> j . Thepion-nucleoncouplingconstantsarerelatedto bythefollowingexpression[38, 39]: j g j ˇ 0 : 027 ,(1.8) g = g (0) ˇ + g (1) ˇ 2 g (2) ˇ (1.9) Thepion-nucleoncouplingconstantsarerelatedtotheCEDMsby[40,39]: g (0) ˇ + g (1) ˇ 2 g (2) ˇ = Ÿ d u Ÿ d d 10 14 cm (1.10) TheelectronEDMisalepton,doesnotparticipateinstronginteractions,andtherefore isnotexpressedintermsof orthepion-nucleoncouplingconstants.AsshowninFig- ure1.1, d e couplesstronglytoparamagneticsystems,whichI'lldiscussinSection1.6.2. IntheMSSMextension,theelectronEDM d e andthequarkEDM d q aregivenby[34,32]: d e ˇ em f 16 ˇ 2 M 2 SUSY 0 B B B B @ 5 g 2 2 + g 2 1 24 sin tan + g 2 1 12 sin A 1 C C C C A ,(1.11) d q ˇ Q q em f 16 ˇ 2 M 2 SUSY 2 g 2 s 9 sin [ tan ] 2 Q q +1 = 3 sin A ,(1.12) tan = v d =v u ,(1.13) 10 where Q q [ e ] istheelectricchargeofthequark, g 1 [ dimensionless ] isthe U (1) Y gaugetheorycoupling, g 1 [ dimensionless ] isthe SU (2) L gaugetheorycoupling, g s [ dimensionless ] istheQCDcoupling, A [ rad ] isa CP -violatingphase,and v u =v d [ dimensionless ] istheratioofthevacuumexpectationvaluesoftheupand downHiggsds. ThereisasimilarexpressionforCEDMs.Withreasonableassumptions,onecanestimate theelectronEDM d e ˇ 10 27 e cm,quarkEDM d q ˇ 10 25 e cm,andquarkchromo-EDM Ÿ d q ˇ 10 25 cmintheMSSMframework[32]. 1.5Electricdipolemomentsearchesasaprobeof CP violation Anelectricdipolemomentisthedistributionofchargealongthepositionvector pointingfromnegativetopositivecharge.Apermanentelectricdipolemoment ~ d [ e cm ] isalignedwiththeintrinsicangularmomentumoftheparticle, ~ J [6]: ~ d = Z ~ r ˆ Q d V = d ~ J J ,(1.14) where ~ r [ cm ] isthepositionofthecharge, ˆ h e m 3 i istheelectricchargedistribution, and V = R d V h m 3 i isthevolumeoftheparticle. Anonzeropermanentelectricdipolemomentviolatestime-reversal( T )and P sym- metry.Toseethis,weconsidertheHamiltonian H [ J ] ofasystemwithintrinsicspinin thepresenceofanelectricandmagneticd: H = ~ J ~ B J d ~ J ~ E J ,(1.15) where ~ = ~ J =J [ J = T ] isthemagneticmoment. TheterminEquation1.15isproportionaltothemagneticmoment.Aswecan seefromTable1.1, ~ J ~ B isevenunderboth P and T transformation.Thesecondterm, ~ J ~ E , 11 Table1.3:EDMmeasurementsofdi erentsystems.UCN=ultracoldneutron. CL=level.PSI=PaulScherrerInstitute.JILA=JointInstituteforLabo- ratoryAstrophysics.Boulder=UniversityofColorado,Boulder.PTB=Physikalisch TechischeBundesanstalt.ANL=ArgonneNationalLab.ILL=InstitutLaue- Langevin. particlesensitivity90%CL [ e cm ] 95%CL [ e cm ] Ref. UCN d n 1 : 8 10 26 PSI[41] UCN d n 3 : 0 10 26 3 : 6 10 26 ILL[42,43] 180 Hf 19 F + C S ;d e 1 : 3 10 28 JILA/Boulder[44] ThO C S ;d e 1 : 1 10 29 ACME[45] 199 Hg C T ; S 7 : 4 10 30 Seattle[46] 129 Xe C T ; S 1 : 4 10 27 HeXeEDMPTB[47] 225 Ra C T ; S 1 : 4 10 23 RaEDMANL[48] proton 205 TlF b d p 6 : 5 10 23c Yale[49,50] a EDMlimitinterpretedbysetting C S =0(solesource). b 199 EDMcurrentlygivesastrongerlimiton d p thanTlF.Thereportedlimitfor TlFinterpretsthe CP -violatingfrequencyshiftasane ectiveprotonEDM(sole source). c CalculatedfromsymmetricGaussianstatistics. isproportionaltotheEDMandisoddunder P and T .AnonzeroEDMthereforeviolates both P and T symmetry. Assuming CPT conservation,EDMsalsoviolate CP .Neutron,electron,molecular, andatomicEDMexperimentshavebeencarriedoutoverthelastsevendecadesinan e orttomeasureanonzeroEDMmagnitude.AnonzeroEDMhasnotbeenmeasured yet,buttheprecisionofEDMexperimentscontinuestoimprove.Observinganonzero EDMnearsensitivitiesoftoday'sleadingexperimentswouldprovideacleansignatureof BeyondtheStandardModelphysics[6]. ATableofEDMlimitsforneutron,proton,electron,andatomicEDMsisgivenin Table1.3.Theworld'smostsensitiveatomicEDMmeasurementuses 199 Hg. 12 1.5.1Neutronelectricdipolemoment NeutronsEDMsareprimarilysensitivetotheshort-rangeneutronEDM d sr n andpion- nucleoncouplingconstants g (0) ˇ ; g (1) ˇ . TheEDMexperimentwasabeamlineneutronmeasurementatOakRidgeNa- tionalLab(ORNL)[51].TheysentacollimatedbeamofneutronstravelingataMaxwellian velocityofapproximately2870m/sthroughauniformDCmagneticdandatuneable radiofrequency(RF)magneticd.Thespinprecessionfrequencywasdeterminedby measuringtheneutronintensitywithacounterasafunctionoftheRFfrequency. Tomeasurespinprecessionfrequenciescorrelatedwithanelectricd,theneutrons alsopassedbetweentwonickel-platedcopperelectrodes135cmlong.Thestaticelec- tricdwas25kV/3.49mm=7.2kV/mmandparalleltotheDCmagneticd.By measuringthespinprecessionfrequencyunderparallelandantiparallelDCds,they measuredtheupperlimitoftheneutronEDMtobe5 10 20 e cm. In1980theultracoldneutron(UCN)EDMmeasurementwasdemonstrated[52] attheLeningradNuclearPhysicsInstitute.Abeamofthermalneutronswasimpinged onaberylliumtargetcooledto30Kwithheliumgas.Theneutronswereguidedtoa precessionchamberwithareducedspeedofapproximately7m/s.TheUCNapproach allowedforlongerspinprecessiontimesandreducedthesystematicsourceofuncertainty duetomotionalmagneticds,orthefi ~ E ~v fle ect[6]. ThecurrentmostsensitiveneutronEDMmeasurementwasperformedin2020atthe PaulScherrerInstitute[41].Theyusea 199 Hgvaporasacomagnetometerdispersedwith theUCNstotracksystematicdriftsintheuniformmagneticd.Theyreportaneutron EDMupperlimitof1 : 8 10 26 e cm(90% 13 1.6 CP ViolationinAtomsandMolecules 1.6.1Theshieldingofthenucleusfromexternalds Thenucleusofaneutralatomisshieldedfromexternalelectricdsbythesurrounding electroncloud,whichpolarizestocancelthed.Theshieldingisexactandthenetd iszeroatthelocationofaclassicalpoint-likenucleus[53].Finite-sizednucleibreakthis perfectshielding.Thespinofthenucleusinteractswithafractionoftheexternald. Large,octupole-deformednucleiarelessshieldedthansmaller,moresphericalnuclei, enhancingthenuclearSchi moment[54,55]. 1.6.2Sensitivitytotheelectronelectricdipolemoment Inthepresenceofastaticelectricd,theatomicEDMcausesalinearStarkshift.The measurementoftheupperlimitoftheStarkshiftisinterpretedasanatomicEDM. Paramagneticatomsandmolecules,whichhaveanunpairedvalenceelectron,havean enhancedsensitivitytotheelectronelectricdipolemoment d e .Theenhancementcomes fromimperfectSchi shieldingduetorelativistice ectsoftheunpairedelectronand scaleswiththesizeofthenucleus[56,57]: d a d e ˇ 10 Z 3 2 , where d a istheatomicEDM, Z istheprotonnumberoftheatom,and =7 : 29735257 10 3 istheconstant. 1.6.3Theelectron-nucleoninteraction AnatomicEDMcanarisefrom CP -violatinginteractionsbetweenthenucleonsandelec- trons.Thesecouplingsarecharacterizedbythescalar,pseudoscalar,andtensorelectron- nucleoncouplings C S ;C P ,and C T [ dimensionless ] . 14 TheatomicHamiltoniancanbewrittenintermsoftheelectron-nucleoncouplings[17]: H TVPV = H S + H P + H T (1.16) TheHamiltoniansfollowsimilarforms,although H P issuppressedbyafactorof m N . Focusingon C T , H T showshowthe P -violatingand T -violatinginteractionbetweenelec- tronsandnucleonsgeneratesanatomicEDMthatdiamagneticsystemsareprimarily sensitiveto[58]: H T = 1 p 2 C T iG F X n;e n 5 ˙ n e ˙ e ,(1.17) where G F = ( ~ c ) 3 =1 : 16638 10 5 G e V 2 istheFermicouplingconstant, n ; e arethenucleonandelectronwavefunctions, 5 i 0 1 2 3 = 0 B B B B B B B B @ OI IO 1 C C C C C C C C A isthe4 4Diracgammamatrix,and ˙ aretheDiracmatricesgeneratedfromthePaulimatrices ˙ i in3+1dimensional notation. H T includescontributionsfromeverynucleon,sodiamagneticatomssuchas 129 Xe, 171 Yb, 199 Hg,and 225 Raareitsmostsensitiveprobes.Sensitivityto C T dependsbothon thenuclearandatomicstructureoftheatom. Theatomicpermanentelectricdipolemoment d A [ e cm ] canbeexplicitlywrittenas alinearcombinationof CP -violatingparameters[17]: d A = @d A @d e d e + @d A @ d sr n d sr n + @d A @ d sr p d sr p + @d A @C S C S + @d A @C P C P + @d A @C T C T + ::: + @d A @ g (0) ˇ g (0) ˇ + @d A @ g (1) ˇ g (1) ˇ + @d A @ g (2) ˇ g (2) ˇ , (1.18) wherethecoe cients @d A . @C j indicatethesensitivityoftheatomicEDMtoparameters 15 Table1.4:95%levelupperlimitcalculationsoflow-energy CP -violating parametersbasedonexperimentalmeasurementsusingaglobalapproach[59,6]. C S and d e calculatedfrommeasurementsbyparamagneticsystems[60,61,62,63]. g (0) ˇ ; g (1) ˇ ;C T ; and d sr n calculatedfrommeasurementsindiamagneticsystemsand nucleartheoryasof2019[64,65,43,66,67,68]. labeldescriptionprimarysensitivityglobalupperlimit d e electronEDMparamagnetic8 : 4 10 28 e cm C S scalarelectron-nucleoninteractionparamagnetic7 : 5 10 8 g (0) ˇ isoscalarpion-nucleoncouplingdiamagnetic1 : 5 10 8 g (1) ˇ isovectorpion-nucleoncouplingdiamagnetic2 : 4 10 9 C T tensorelectron-nucleoninteractiondiamagnetic1 : 1 10 6 d sr n short-rangeneutronEDMneutron2 : 4 10 22 e cm C j .Someofthecoe cientsareoftenwritteninamorecompactnotation: @d @d e ! e @d @C T ! C T b @d ˚ @ g (0) ˇ ! a 0 @d ˚ @ g (1) ˇ ! a 1 @d ˚ @ g (2) ˇ ! a 2 @d @C S ! k S @d @C P ! k P TheseparameterscouplefundamentaltheorypropertiessuchastheCKMmatrix,BSM physics,orthestronginteractionparameter tolow-energy,potentiallymeasurable EDMs. TosetthestageforthekeyparametersthatI'lldiscussinthefollowingsections,Iwill rewriteEquation1.18intermsoftheSchi moment,scalarandtensorelectron-nucleon interactions,andelectronEDM[6]: d A = e d e + k S C S + C T C T + S S ,(1.19) whereI'veomittedtermswithweakercouplingtoparamagneticanddiamagneticsys- tems. Paramagneticsystemsaremostsensitiveto d e and C S .Forexample,in 205 Tl,the tensorelectron-nucleoninteractionisahigher-ordere ect,and C S 205 Tl ˛ C T 205 Tl . b k T issometimesusedaswell. 16 Inthepasttenyears,stridesinmeasurementsensitivityhavebeenmadebyforming paramagneticsystemsfromdiatomicmolecules[69].Themoststringentlimiton d e and C S comesfromaglobalanalysisfromrecentEDMmeasurementsofThOand 180 Hf 19 F + , asshowninTable1.4. I'velistedglobal-sourcecalculationsofthelow-energy CP -violatingparametersfrom measurementsmadeinparamagneticanddiamagneticsystemsinTable1.4.Several parametersarenotincludedintheglobalanalysis.Thesole-sourcecalculationof theisotensorpion-nucleoncoupling g (2) ˇ < 1 : 1 10 12 andshort-rangeprotonEDM d sr p < 2 : 0 10 25 e cmarefoundfromthe 199 Hgmeasurement[46,6].Thepseudoscalar electron-nucleoninteraction C P isnotlistedbecauseitisahigher-ordere ectthatissup- pressedbyanadditionalfactorof1 =m N (thenucleonmass),giving C T ˛ k P [17,70]. Theleadingordertermoftheisoscalarpion-nucleoncoupling g (0) ˇ isgivenby[71]: 2 F ˇ g (0) ˇ = (0) m N m m ,(1.20) (0) m N = m d m u ,(1.21) = m d m u m d + m u ,(1.22) m = m u m d m s m s ( m u + m d )+ m u m d = m (1 2 ) 2+ mm 1 s (1 2 ) ,(1.23) where (0) m N [ MeV ] istheneutron-protonmassdi erenceand m s =92 : 9 0 : 7MeVis thestrangequarkmass. Usingthevaluesfromtheliteraturefor m =3 : 39 0 : 04MeV[30], m s = m =27 : 37 0 : 10MeV[30], (0) m N =2 : 39 0 : 13MeV[71],andthequarkmasses andEquation1.22andEquation1.23for =0 : 352 0 : 020, m =1 : 695 0 : 066MeV,I calculate g (0) ˇ ˇ ( 0 : 019 0 : 003 ) . Nuclearforcesindiamagneticatomsandmoleculesinducenuclearmomentsthatcan beseveralordersofmagnitudelargerthantheconstituentneutronandprotonEDMs[70]. Therefore,diamagneticatomicandmolecularEDMsarewrittenintermsofthenuclear 17 Table1.5:AcollectionofcalculationsofnuclearSchi momentcoe cientsforRadium- 225andMercury-199.Rangesarelistedinbrackets. System S cmfm 3 C T ( e cm ) a 0 e fm 3 a 1 e fm 3 Ref. 225 Ra+0 : 2 0 : 6 5 3[72] 225 Ra 8 : 5 10 17 +5 : 3 10 20 [ 6 ; 1 ][ +4 ; +24 ] [55,73,74,17,59] 199 Hg 2 : 8 10 17 +3 : 0 10 20 [ 0 : 005 ; 0 : 05 ][ 0 : 03 ; +0 : 09 ] [75,76,59] 199 Hg a +0 : 087+0 : 087[75,17] 199 Hg b +0 : 010+0 : 074[77] a Schematicmethod. b SkyrmeSkO'QRPA. Schi moment S h e fm 3 i : d A = S S k C T C T c ,(1.24) S = s N d N + m N g A F ˇ a 0 g (0) ˇ + a 1 g (1) ˇ + a 2 g (2) ˇ ,(1.25) where S h cmfm 3 i istheSchi momentsensitivityand k C T [ e cm ] isthe electron-quarktensorinteractionsensitivity.We'vedroppedthehigher-orderterms e ;@d A . @ d sr n ;@d A . @ d sr p ;@d A @C S ;@d A @C P : Theisotensorpion-nucleoncoupling g (2) ˇ mayalsobeneglected,asitissuppressedbyafactorof[78,6,71,17]: g (2) ˇ = g (1) ˇ = m 2 ˇ =M 2 QCD ˇ 0 : 007, where m ˇ =139 : 57MeVisthemassofthechargedpion[13]and M QCD ˇ 1GeVisthe hadronicenergyscale[78]. I'velistedestimatesoftheSchi couplingparametersofradiumandmercuryinTa- ble1.5.Notethatinthetoprow,recentcalculationsofradiumaidedbycorrelatedoc- tupolemomentmeasurementshavetlyreduceduncertaintiesof a i [72]. c k C T hasanisoscalarandisovectorcomponentwhichI'vesimforclarity. 18 1.7 CP ViolationinDiamagneticSystems TheatomicEDMisdirectlyproportionaltotheSchi moment,asshowninEqua- tion1.19.TheSchi moment S h e fm 3 i isgivenby[73]: S = D 0 ‹ S z 0 E (1.26) = X i , 0 D 0 ‹ S z i ED i ‹ V PT 0 E E 0 E i ,(1.27) ‹ S z = e 10 X p r 2 p 5 3 r 2 ch z p ,(1.28) where 0 h m 3 = 2 i isthegroundstatewavefunction, ‹ S z h e fm 3 i isthecomponentoftheSchi momentalongtheaxisofthenuclearspin, E 0 [ e V ] istheenergyofthegroundstate ‹ V PT [ e V ] isthe P and T -violatinginteraction, r 2 ch h m 2 i isthemeansquarechargeradius, r p [ m ] istheprotondistance,and z p [ m ] isthe z -componentoftheprotonposition. NuclearSchi moment-inducedatomicEDMsearchesarecarriedoutinthediamagnetic atomsMercury-199,Xenon-129,andRadium-225. ThenuclearSchi momentisenhancedbyoctupoledeformation(pearshape)of thenucleus.Thedeformationischaracterizedbytheoctupoledeformationparameter 3 [ dimensionless ] .TheSchi momentcanberewrittenas[6]: S = e 2 2 3 ZA 2 = 3 r 3 0 E ,(1.29) where [ e V ] isthestrengthcoe cientofthe P and T -oddinteraction, 19 Table1.6:Experimental(even-even)andcalculated(odd- evenbetadeformationparametersforaselectionofisotopes. isotope 2 3 S 10 8 e fm 3 E Ref. 129 Xe +1 : 75 [54] 199 Hg 1 : 4 [54] 223 Ra0.1250.100+50050 : 2[55,54] 224 Ra0.1540.097 [79] 225 Ra0.1430.099+110055 : 2[55,54] 229 Pa0.1760.082+3000000 : 22[55,54] a Calculatedusingone-tailedGaussianstatistics. e istheelementarycharge, 2 [ dimensionless ] isquadrupolemomentdeformationparameter, Z [ dimensionless ] isthenumberofprotonsinthenucleus, A [ dimensionless ] isthenumberofprotons+neutronsinthenucleus, r 0 =1 : 2fmistheinternucleondistance,and E [ e V ] isenergydi erencebetweentheparitydoubletstatesofthenucleus. ThedeformationparametersarefoundbyCoulombexcitationexperimentswithiso- topeswithevennumbersofneutronsandprotons(fieven-evenfl).Thedeformationpa- rametersofeven-oddisotopessuchas 225 Raareinferredfromthesemeasurementsor calculated.AselectionofdeformationparametersforseveralisotopesisgiveninTa- ble1.6. 1.8Thesisoutline Thisthesisdetailsmyworkonhighvoltagedevelopmentandprecisionspectroscopy. Sofar,I'vediscussedelectricdipolemomentsanddiscretesymmetryviolationingeneral. InChapter2,Iwilldescribetheradiumexperimentandassociatedsystematice ects. Chapter3isaboutmyworkindevelopinganewpairhighvoltageelectrodestouseinthe radiumexperiment.Idiscussmye ortsusinglaserinduceduorescence(LIF)tomea- surethebranchingratiosofacyclingtransitionforanimprovedlaser-coolingscheme 20 fortheradiumexperimentinChapter4.InChapter5,Iwilldetailisotopeharvesting studiesattheFacilityforRareIsotopeBeams(FRIB)forharvestingradiumforEDMmea- surementsinthefuture.ItakeadetourinChapter6todescribemycomputationaland experimentalworkinprecisiongamma-rayintensitymeasurementsfornuclearsecurity applications.Finally,Io erconcludingthoughtsandexplicitlylistmypersonalcontri- butionstotheexperimentsdescribedinthisthesisinChapter7. 21 CHAPTER2 INTRODUCTIONTOTHERAEDMEXPERIMENT 2.1Motivation TheatomicEDMof 225 Ra(nuclearspin I =1 = 2)isenhancedbytheoctupoledeforma- tion(fipearshapefl)ofitsnucleus.Radium-225hasa55keVparitydoubletgroundstate structure,comparedtoapproximately1MeVinsphericallysymmetricnuclei[80].The nuclearSchi momentof 225 Raispredictedtobeuptothreeordersofmagnitudelarger thanthatofdiamagneticatomswithsphericallysymmetricnuclei[54,73,74,76].These e ectsgreatlyenhanceexperimentalsensitivitytotheatomicEDM. 2.1.1Laser-cooledelectricdipolemomentsearches Inthe 199 Hgand 129 XeEDMexperiments,theatomsarecontainedwithinvaporcells. Thistechniqueallowsthespinprecessionfrequencyofalargesampleofatomstobere- peatablymeasured.Thesamplesize N ,spinprecessiontime ˝ ,andintegrationtime T are large.Theleadingsystematicuncertaintyforboththemercuryandxenonexperimentsis relatedtohighvoltage-correlatedmotionofthevaporcells[46,47]. Thevaporpressureof 225 Raistoolowforaradiumvaporcell.Instead,atomsare heatedupinanoventogenerateanatomicbeam,andasystemoflasercoolingand trappingisneededtoplacetheatomsinanopticaldipoletrapbetweentwohighvoltage electrodesinabackground-shieldedfisciencechamberfl.TheODTisalinearlypolarized 1550nmlaserdetunedfarbelowtheatomresonanttransitionfrequency.Detuningthe laserfarbelowthetransitionfrequencyattractstheatomstotheODTintensitymaximum andreducesatomheatingfromscatteredlaserphotons[81]. TheRaEDMlasertraptheatomstoa100 mdiametercloud[48].Thesmall atomvolumeisadvantageousformaintainingahighlyuniformelectricandmagnetic 22 dthroughoutthecloud.Vaporcells,bycontrast,canbetensofmillimetersindiam- eter,andatomsmayexperiencetlydi erentdsatdi erentlocationsinthe cellvolume.AnewEDMmeasurementof 171 YbattheUniversityofTechnologyand ScienceofChina(USTC)willuseacoolingandtrappingsetupsimilartotheRaEDM experiment. Thelasertrappingapproachfacesthechallengeofe cientlycollectingatomsexit- ingtheoven.Thee ciencyoftrapping 225 Raatomsisontheorderofpartspermil- lion[82],resultinginanODTpopulationofonlyseveralhundredatoms.TheODTtrap lifetimelimitseachspinprecessionfrequencymeasurementtoapproximatelytwentysec- onds[48]. TherearethreeODTsystematice ectsthatmustbeconsidered.First,circularly- polarizedlightfromtheODTlasercausesZeemansplittingoftheatoms,causingavector lightshift[81].Thevectorlightshift V [ Hz ] isgivenbythefollowing: V = V ( P ) m f P h ‹ k ‹ B ,(2.1) where V ( P ) [ Hz ] isthevectorlightshiftscalefactoratlaserpower P [ W ] , m F [ dimensionless ] istheprojectionoftheintrinsicangularmomentumalongthe DCmagneticdaxis, P h [ dimensionless ] 2 [ 0 ; 1 ] isthefractionofcircularpolarization, ‹ k [ dimensionless ] istheODTaxis,and ‹ B [ dimensionless ] istheDCmagneticdaxis. Wetrytominimizethise ectbysuppressingresidualcircularpolarizationintheODT beamandbyaligningtheODTperpendiculartothemagneticd.Theupperboundof thise ectontheRaEDMexperimentiscalculatedtobe < 10 25 e cm[48]. Second,theDCelectricdcancausetmixingofopposite-parityatomic 23 states,orStarkmixing.Thise ectiscalledtheStarklightshift S [ Hz ] : S = 1 ‹ b ‹ ˙ ‹ ‹ E + 2 ‹ b ‹ E ( ‹ ‹ ˙ ) ,(2.2) where 1 ; 2 [ Hz ] aretheStarkinterferencescalefactors, ‹ b = ‹ k ‹ [ dimensionless ] istheACmagneticdoftheODT, ‹ ˙ = ‹ B [ dimensionless ] isthespinquantizationaxis,equivalenttotheDCmagnetic daxis, ‹ [ dimensionless ] isthelaserpolarizationdirection,and ‹ E [ dimensionless ] istheDCelectricdaxis. Thise ectissimilartothevectorlightshiftinthatitissuppressedbyappropriately orientedODTandDCmagneticldaxesandbyusingalinearly-polarizedODTlaser. AsIwilldiscussinSection2.4,wealign ~ E parallelorantiparalleltotheDCmagnetic d.TheRaEDMStarkshiftsystematicuncertaintyiscalculatedtobe < 10 25 e cm[48]. TheStarkmixingsystematicissensitivetothealignmentoftheDCelectricd.In Section2.5Iwilluseelementmodelingtodemonstratethee ectoftheelectricd forarangeofelectrodemisalignments. Atomswithnuclearspin I 1undergoaquadraticStarkshiftproportionaltothe squareoftheappliedelectricd[81,83]. 225 Raisspin-1/2,soitdoesnotexperiencea quadraticStarkshift. However,radiumatomsmayberepelledfromthecenteroftheODTby E 2 -proportional e ectsotherthanthequadraticStarkshift.Thedgradientwouldcausedi erentspin precessionfrequencies.Underanasymmetricelectricdreversal,thiswillintroducea systematicmimickinganEDMsignal.Inthemostrecent 225 Rameasurement,thed reversalasymmetrywas0.7%,resultinginasystematicuncertaintyof10 25 e cm[48]. Thissystematicscaleswiththestatisticalsensitivity,forexampleimprovingthesensitiv- ityofthemeasurementbyafactor X alsoreducesthe E 2 e ectbyfactor X .Thesystematic 24 Figure2.1:TheRaEDMexperimentalapparatus. willbesuppressedevenfurtherastheODTisimprovedtotheatomstoasmaller volume. 2.1.2Sensitivitytoexperimentalparameters TheRaEDMexperiment(ArgonneNationalLab,MichiganStateUniversity)searchesfor theEDMof 225 Rabymeasuringthespinprecessionfrequencyof 225 Rainacontrolled, uniformmagneticandelectricdbetweentwohighvoltageelectrodesinanoptical dipoletrap.EDMsearchesareperformedatArgonneNationalLab(ANL).O ineup- gradessuchasthehighvoltagedevelopmentdiscussedinthisthesisarecarriedoutat MichiganStateUniversity(MSU).Intheproofofprinciplemeasurement,theEDMupper limitwasmeasuredto5 : 0 10 22 e cmatthe95%level(CL)[65].Thiswas reducedto1 : 4 10 23 e cminthesubsequentrun[48].Hereafterwewillrefertothese asthegeneration'measurements. Thestatisticalsensitivityofourmeasurementisestimatedbythequantumprojection 25 noise-limitedEDMstandarderror ˙ EDM [ e cm ] : ˙ EDM = ~ 2 E p NT˝ ,(2.3) where E [ V = cm ] istheexternalelectricd, ~ [ eVs ] isthereducedPlanckconstant, [ unitless ] istheatomdetectione ciency, N [ unitless ] isthenumberofatomspersample, T [ s ] isthetotalmeasurementtime,and ˝ [ s ] isthemeasurementtimepercycle. AsseeninEquation2.3,theEDMstatisticalsensitivityscaleslinearlywiththeelectric dstrength.TheRaEDMexperimentwillbetlyimprovedwithtargeted upgradestotheexperimentalapparatusoverthenextseveral`secondgeneration'mea- surements.Inparticular,wewilluseanewatomdetectionmethodtoincrease andnew electrodestoincrease E . Wewillsurpassthe10 25 e cmsensitivitylevelduringthisphaseandthe 225 RaEDM limitwillconstrainhadronic CP -violatingparametersalongsideotherEDMexperiments. Forexample,aradiumEDMmeasurementof10 25 e cmwillimprovetheconstrainton thetensorelectron-nucleoninteraction C T bymorethananorderofmagnitudewithin theframeworkofaglobalEDManalysis[59,6]. 2.2Overviewofexperimentalapparatus AschematicoftheRaEDMexperimentalapparatusisshowninFigure2.1.Radium ispackagedasnitratesaltandloadedintotheovenwithmetallicbariumchips.Theoven isheatedto350Œ500 Ctoemitanatomicbeamfromtheovennozzle.Thenozzlehasa lengthof83mmandadiameterof2mm,oranozzleratioof =2 = 83=0 : 024[82]. 26 2.2.1LasercoolingandtheZeemanSlower Theatomicbeamiscollimatedinthechamberadjacenttotheovenbylaserlightfrom atitaniumsapphire(Ti:Saph)laser,orientedtransversetothebeamandtunedtothe S 1 0 ! P 3 o 1 transitionat714nm. ThentheatomsentertheZeemanSlowersection,wheretheyareslowed(ficooledfl) alongtheatomicbeamaxisbyacounterpropagatinglaserbeam.Inthecurrentfiredfl sloweration,atomsareslowedoveralengthofonemeterbylasersresonant withthe 3 P o 1 transition( =714nm, ˝ =420ns).TheDopplershiftoftheatomexcitation frequencyiscompensatedbyaZeemanshiftfromacalibratedmagneticdgradient thatisgeneratedbyataperedsolenoid. Forthefollowingdiscussionwewillworkinthedipoleapproximationlimit >>a 0 , where a 0 =5 : 292 10 11 mistheBohrradius.Whenatwo-levelatomwithstates j 1 i and j 2 i interactswithlightinanelectricd ~ E thatisresonantwiththetransitionfrequency betweenthetwostates,itwillexciteanddeexcitebetweenthelevels.Theexcitationrate occursattheRabifrequency [ rad = s ] [84]: = h 1 j e~ r ~ E 0 j 2 i ~ = e X 12 ~ E 0 ~ ,(2.4) X 12 = h 1 j x j 2 i ,(2.5) ~ E = ~ E 0 cos ( !t ) ,(2.6) where e [ C ] istheelectroncharge, ~ r [ m ] isthepositionoftheelectronwithrespecttothenucleus, ! [ rad = s ] istheangularfrequencyofthephoton,and ~ E 0 [ V = m ] istheelectricdamplitude. Thematrixelement X 12 canbeexpressedintermsofthelikelihoodofexcitingan 27 Table2.1:RadiumZeemanslowerpropertiesforthecurrentredcyclingtransitionand theplannedbluecyclingtransition. transitionwavenumberwavelengthlifetimerecoilscattersaturation (cm 1 ) (nm) ˝ (ns) v r (cm/s) R (s 1 ) I 0 (mW/cm 2 ) 1 S 0 ! 3 P o 1 13999.387144200.251 : 2 10 6 0.136 1 S 0 ! 1 P o 1 20715.714835.50.379 : 1 10 7 33.6 atomfrom j 1 i to j 2 i ,orEinsteinB-coe cient: B 12 = ˇe 2 j X 12 j 2 0 ~ 2 (2.7) Theratethatatomsabsorbandre-emitresonantlaserphotons,orthescatteringrate R ,isgivenby: R = 2 2 2 + 2 = 2+ 2 = 4 ,(2.8) where =1 =˝ h s 1 i isthedecayrateofthetransition.TheRabifrequencyisrelatedto thesaturationintensity I s ( ! ) h W = m 2 i : I s ( ! )= ~ !A 21 2 ˙ ( ! ) = I 0 2 2 2 ,(2.9) I 0 = I s ( ! 0 )(2.10) = ˇ 3 hc 3 ˝ ,(2.11) where A 21 h s 1 i isthespontaneousemissionrate(Einstein A -coe cient)from j 2 i ! j 1 i , ˙ ( ! ) h m 2 i istheabsorptioncrosssectionatangularfrequency ! , [ m ] isthewavelengthofthetransition,and ˝ =1 =A 21 [ s ] isthelifetimeofthetransition. Now R canbeexpressedintermsofthelaserintensity.Theforce F [ N ] exertedonan atombyacounterpropagatinglaserphotonwithmomentum ~ k [ kgm = s ] isgivenbythe following: F = ~ k R = ~ k 2 I=I 0 1+ I=I 0 +4 2 = 2 (2.12) 28 Figure2.2:AcartoonoftheradiumZeemanslower. ~ p = m~ v isthemomentum ofaradiumatomwithmass m andvelocity ~ v and ~ p = h= ‹ z isthemomentum ofaslowinglaserphotonwithwavelength . Whenanatomemitsanabsorbedphoton,itexperiencesafimomentumkickflcharac- terizedbytherecoilvelocity v r [ m = s ] : v r 2 ˝ = a max = F 0 m = ~ k m 2 ,(2.13) where a max h ms 2 i isthemaximumaccelerationand F 0 [ N ] isthemaximumforce exertedontheatom.FromNewtoniankinematicswecanestimatethelengthscale L 0 [ m ] overwhichanatomwithinitialspeed v 0 [ m = s ] canbestoppedbythelaserbeam: L 0 = v 2 0 a max (2.14) Theatomsareexposedtoacounterpropagating,circularly-polarizedlaserasthey movethroughthesolenoid,asshowninFigure2.2.Thesolenoidmagneticdistuned tocompensateforboththeatomDopplershifts(velocity-related)andtheZeemanshifts (quantumnumberrelated).Theatomsabsorbandemitthelaserphotonsataratede- terminedbythelifetimeoftheopticalcyclingtransition[85].Eachphotonabsorption- emissioncyclegivesasmallmomentumkicktotheatom,slowingitdown. 29 TheRaEDMZeemanslowerusestheS 1 0 ! P 3 o 1 opticalcyclingtransitiontotrap atomswithinitialspeedsupto55m/s.Aonemeter-long,taperedsolenoidgeneratesa magneticdgradientalongtheatomicbeamaxis.ThemagneticdcausesaZeeman shift E z [ J ] intheenergyoftheatom: E z = g F m F B B z ,(2.15) g F = F ( F +1)+ J ( J +1) I ( I +1) 2 F ( F +1) g J ,(2.16) g J = 3 2 + S ( S +1) L ( L +1) 2 J ( J +1) ,(2.17) where g F ;g J [ unitless ] aretheLandég-factors, B [ J = T ] istheBohrmagneton, F = I + J [ unitless ] isthetotalangularmomentum, J = L + S [ unitless ] istheelectrontotalangularmomentum, L;S [ unitless ] istheelectronorbitalangularmomentumandspin,and B z [ T ] istheZeemanslowermagneticd. Ingeneral,thetotalZeeman-shiftedenergy E z mustaccountfortheenergyshiftofthe excitedstateandthegroundstate.ForourcaseofanopticalcyclingtransitionwithS 1 0 atoms,theZeemanshiftisdominatedbytheexcitedstatecontributionandImakethe approximation E z ˇ E z . Theexcitationangularfrequency ! oftheatomisbyaDopplerandZeeman contribution,shiftingitfromtheunperturbedangularfrequency ! 0 : ! + kv = ! 0 + E z = ~ (2.18) Tokeeptheatomresonantwiththelaser,themagneticdisgivenby: B ( z )= B 0 1 z L 0 ! 1 = 2 + B bias ,(2.19) B 0 = hv 0 B (2.20) 30 Figure2.3:Cloudofradiumatomstrappedbetweenhighvoltageelectrodesinoptical dipoletrap. Forthe 3 P o 1 transition, B 0 =5 : 5mT=55G.ThisproducesaZeemanshift = E z =h ˇ 80MHz. 2.2.2Lasertrapping TheZeemanslowercoolsasmallfractionoftheatombeamtosu cientlylowspeeds. Thesearetrappedbyathree-dimensionalmagneto-opticaltrap(3DMOT)downstream oftheZeemanSlower(bottom-centerofFigure2.1).TheMOTisformedbythreelasers slightlydetunedfromthe 3 P 1 transition.Thelasersaremutuallyperpendicularand intersect.Thelaserpathsareimmersedwithinamagneticdgradientofapproximately 0.5Gauss/cm[86].Thetrappinge ciencyisafewpartspermillion[82]. A1550nmlaserisoverlappedwiththecenteroftheMOT.TheMOTlasersandd areswitchedo andtheatomsarenowattractedtothefocusofthe1550nmlaser.This laserisanopticaldipoletrapwitha500 Ktrappingdepth.Thelocationofthebeam focusiscontrolledbyalensonatranslationstage. 31 ThefiBusflODTtransportstheatomsfromtheMOTintoanonmagnetic,borosilicate glasstubechambercalledthefisciencechamber.flThetubeissurroundedbyconcentric nickel-alloyfimuflmetal,so-namedforitshighrelativepermeability r ˇ 20000.There arethreelayersofmu-metalsurroundingthetube.Themu-metalisde-Gaussedbyrun- ninga10HzACcurrentthroughwirescoiledaroundthemu-metal.Thede-Gaussed shieldsuppresseslow-frequencyexternaldsbyafactorof ˇ 10 4 .ThisreducesEarth's magneticdfrom500mGaussto50 Gaussinsidethesciencechamber. Theatomsaretransportedtothecenterofthesciencechamberbetweentwometal highvoltageelectrodes.TheelectrodesaremountedinaMacorholderwithinthetube, separatedbyadistanceontheorderofmillimeters.Theelectrodesarediscussedindetail inChapter3.Insidethesciencechamber,weapplyauniform10mGaussdinthe verticaldirection. TheBusODTisoverlappedwithasecond,perpendicularfiHoldingflODTintheelec- trodegap.TheHoldingODTisa1550nmlaserwitha100 mdiameteratthefocus.The BusODTisshutteredatthispointandtheatomcloudisinpositionforspinprecession frequencymeasurements. Figure2.3showsaschematicoftheEDMmeasurement.Theatomsarepolarized alongtheaxisoftheHoldingODTwithapulsefromthecollinearfiPump/Probeflbeam tunedtotheS 1 0 ! P 1 o 1 (483nm)transition.Theatomicspinsprecessatafrequencyof ˇ 20HzintheHoldingODT.Auniformelectricdisgeneratedparalleltotheapplied magneticdbychargingoneoftheelectrodeswithabipolarhighvoltagepowersupply. Theddirectionisreversedbyreversingthepolarityofthepowersupply.TheRaEDM measurementaimstodetectafrequencydi erenceintheatomspinprecessionwhenthe electricdisalignedandanti-alignedwiththemagneticd. Systematice ectsrelatedtolasertrappingandtheEDMapparatuswerestudiedby previousRaEDMgraduatestudentsMukutKalita,RichardParker,andIbrahimSulai. Theirarediscussedingreatdetailintheirtheses[86,87,88]. 32 2.2.3The2015Radium-225measurement Thesecondandmostrecent 225 RaEDMmeasurementwasperformedinthesummerof 2015.Thehighvoltageelectrodessuppliedanelectricdof15kV/2.3mm=6.5kV/mm[48]. Asingleradiumovenloadof9mCiwasused,simplifyingthepreviousexperimentwhich usedseparateovenloadsof3mCiand6mCi[65]. Theresultofthemostrecentmeasurement,anEDMupperlimitof d ( 225 Ra)=1 : 4 10 23 e cm), improvestheconstraintoftheformerlimitbyafactorof36[48].Generaltsin laserstability,dataacquisition,analysis,andtheapparatusbetweentheandsecond experimentcontributetotheimprovement.Thelargestsourceofimprovementwasa factorof10increaseintheatomtraplifetimeoftheatoms( ˝ inEquation2.3).Thiswas achievedbyimprovingthestabilityoftheHoldingODTandtlyreducingthe sciencechambervacuumpressure. I'llhighlighttargetedupgradestotheEDMexperimentalapparatusforthenextmea- surementinSection2.3.ThenI'lldescribetheEDMmeasurementschemeinSection2.4. 2.3Targetedupgradesforanimprovedelectricdipolemomentmea- surement 2.3.1AtomcoolingwithanimprovedZeemanslower WeusedtheS 1 0 ! P 3 o 1 transitiontoslowtheatomsinthegenerationmeasurements. ThiscyclingtransitiononlyrequiresasinglerepumplasertunedtotheD 3 1 ! P 1 o 1 transition.Thedrawbackisthatlessthan1%oftheatomsexitingtheovenaresu ciently slowedtobecaptured.ThedetailsoftheoperationofaZeemanslowerarediscussedin Section2.2.1. WewillimprovetheZeemanslowerbyusingthefaster-cyclingS 1 0 ! P 1 o 1 transition toslowmorethanhalfofalltheatomsexitingtheoven.Thiswillincreasethenumber ofatomsthatwecanmeasureinthesciencechamber( N inEquation2.3).Theimproved fiblueflslowerwilloperatesimultaneouslywiththecurrentfiredflslower.Theblueslower 33 opticalcyclingschemerequiresthreerepumplasers. Ibuiltauoroscopysetuptomeasuretheadditionalrepumpchannelstoverifythe feasibilityoftheblueslowerscheme.ThebranchingratiostotheseadditionalDstates werepredictedtobefavorable[89].Priortomywork,theDstatebranchingratioswere notyetexperimentallyv[89].I'mthethirdauthorofourpublicationdescrib- ingthebranchingratiomeasurementandresults[90].Mycontributiontotheradium branchingratiomeasurementisfurtherdiscussedinChapter4. 2.3.2Atomdetectione ciencywithStimulatedRamanAdiabaticPassage ForthegenerationEDMexperiments,wemeasuredtheatomspinpreces- sionbypulsingtheatomswithcircularlypolarized( ˙ + )laserlighttunedtothe S 1 0 ( F =1 = 2) ! P 1 o 1 ( F 0 =1 = 2)transition,where F = I + J isthetotalangularmomentum summingnuclearspin I andtotalelectronicangularmomentum J .Theatomsscatteran averageofthreephotonsbeforedecayingfromtheexcitedstatetoagroundstatethatwe cannotutilizeforspinprecessionfrequencydetection, i.e. adarkstate. i Toincreasethenumberofphotonscattersperatom,andthusthedetectione ciency inEquation2.3,wewilluseahmagneticsublevel-selectivemeasurementscheme withthe S 1 0 ( F =1 = 2) ! P 1 o 1 ( F 0 =3 = 2)transition.Withthismethod,theatomsare expectedtoscatteranaverageofonethousandphotons. Ourstrategytomeasurethespin-selective F 0 =3 = 2transitionistousethetechniqueof StimulatedRamanAdiabaticPassage(STIRAP).Thistechniqueusestwolasers:onetuned tothetransition j 1 i ! j 2 i ,andonetunedto j 2 i ! j 3 i .Bypulsingtheatomcloud withthetwolasersseparatedbythetimeinterval [ s ] ,wecantransferthepopulation ofstate j 1 i directlytostate j 3 i . i Atomsmayalsodecaytotheequallyunusablemetastable 3 D 1 state ˇ 0 : 1%ofthe time. 34 Thesearetheproposed 225 Radetectionschemestates: j 1 i =S 1 0 ( F =1 = 2; m F = 1 = 2 ) j 2 i =P 1 o 1 ( F =1 = 2; m F =+1 = 2 ) j 3 i =D 3 1 ( F =1 = 2; m F = 1 = 2 ) , Thespin-selectiveatomdetectionisthencarriedoutbyprobingthe S 1 0 ( F =1 = 2; m F =+1 = 2 ) ! P 1 o 1 ( F 0 =3 = 2; m F 0 =+3 = 2 ) transitionwith ˙ + circularly- polarizedlight. TenzinRabgaledtherecente ortresultingintprogresstowardsachieving spin-selectiveSTIRAPwithradium.Thisworkisdiscussedingreatdetailinhisthe- sis[91]. 2.3.3Higherelectricdstrength Thegenerationexperimentsusedanelectricdof6.7kV/mmand6.5kV/mm.We usedapairofoxygen-free,electropolishedcopperelectrodes.Thecopperelectrodeswere tested(conditioned)todsashighas10kV/mminatestapparatusatANL.However, theywereunstableatthosedswhentheywereinstalledintheEDMapparatus.The installationprocedureisinvasiveandrequiresateardownofthevacuumequipmenton theoppositesideoftheMOTchamberinFigure2.1. AfterthemostrecentEDMmeasurement,fourpairsofniobiumelectrodesandtwo pairsoftitaniumelectrodeswerepreparedatJe ersonLabandsenttoMSU.Ibuilta highvoltageteststationandconditionednewtitaniumandniobiumelectrodepairsat MSU.Iimprovedtheconditioningprocedureanddevelopedanalysiscodethatcanbe runconcurrentlywithconditioningtoinformthetesting.Tostore,transport,andinstall theelectrodesinhighvoltagesetups,Idesignedcleanrooms,storagecontainers,and installationprocedures. 35 ItransportedapairofconditionedniobiumelectrodestotheEDMapparatus.The electrodeswereconditionedto20kV/mmatMSU.Ibuiltacleanroomaroundthetear- downsectionoftheEDMapparatusandinstalledtheelectrodesintheEDMapparatus andrevalidatedthemto20kV/mm.Thiswillmorethantripletheelectricdstrength ( E inEquation2.3).SincetheEDMsensitivityislinearlyproportionaltotheelectricd strength,itshouldalsotriplethesensitivityofthenextEDMmeasurement.Thedetails ofthisworkisdiscussedinChapter3. 2.3.4IncreasingRadium-225availability 225 RawasprocuredfromOakRidgeNationalLab(ORNL)forthetwoEDMex- periments.ThenewFacilityforRareIsotopeBeams(FRIB)linearacceleratoratMSUis approachingfulloperation.IsotopeproductionwasrecentlybenchmarkedattheNSCL for 47 CatoinformfutureisotopeharvestingatFRIB[92].Whenfullyoperational,FRIB isexpectedtobecapableofsupplyingatleast4.9mCiof 225 Raperweek[93],andsignif- icantlymoreforadedicatedradiumharvestingcampaign. FRIB-harvestedradiumwillallowustoperformanEDMmeasurementwithlarger sourceloadsmorefrequently.Firstwewilldeveloptheextractionandsamplepreparation procedure,startingwithstablecalciumasaradiumsurrogate.Calcium,likeradium,has astrongP 1 o 1 cyclingtransitionandcanbeusedinatomicbeamstudies. We'redevelopinganatomicbeamuorescence(ABF)studyatMSUthatwillcalibrate theharvestingprocedure.Ourgoalistocomparetheinitialharvestedatomsourcesize totheatomrateitproducesinane usiveoventhatwillbemeasuredwithlaserinduced uorescence(LIF).ThedetailsofmyABFmeasurementsandanalysisareinChapter5. 36 2.4Experimentalrequirements 2.4.1Measurementtechnique TheEDMcouplestoanexternalelectricdanalogouslytothecouplingofthemagnetic dipolemomenttoanexternalmagneticd.TheHamiltonian H [ e V ] ofanatominthe presenceofaperfectlyuniformelectricandmagneticdisgivenby: H = 0 B B B B @ ~ I ~ B I 1 C C C C A d 0 B B B B @ ~ I ~ E I 1 C C C C A ,(2.21) where = 2 : 3 10 8 eV/Tisthenuclearmagneticmomentof 225 Ra[94], ~ I isthenuclearspin, ~ B [ T ] istheappliedmagneticd, d [ e cm ] istheatomicEDM,and ~ E [ V = cm ] istheappliedelectricd. The 225 Raatomicspinswillprecesswithfrequency ! + ( ! )when ~ E isparallel(antipar- allel)to ~ B : ! = 2 ~ ( dE ),(2.22) InthemostrecentRaEDMexperimentweapplieda2 : 6 Tmagneticdandmeasured aspinprecessionfrequencyof181 : 1 1 : 6rad/s[48]. Weuseapairofidenticalplane-parallelelectrodestoproduceastable,uniform,and symmetricelectricd.Thespinprecessionfrequencyoftheatomsismeasuredinthree ations:withtheelectricdparalleltothemagneticd,withtheelectric dantiparalleltothemagneticd,andwithnoappliedelectricd.Thed- o flsettingisusedtocontrolforasystematice ectgeneratedbyanimperfectreversal oftheelectricd.Wemeasuretheaccumulatedspinprecessionphaseforeachd ation.TheextractedEDMisproportionaltotheaccumulatedphasedi erence 37 Table2.2:RaEDMsystematicrequirementsatthe10 26 e cmsensitivitylevel. Detailedsystematiclimitevaluationsfortheseparameterscanbefoundinpre- viouswork[48,95]. B isdeterminedbyEquation2.29. descriptionsystematiclimitSection ~ E ; ~ B alignment E 2mrad2.4.4 polarityimbalance j E j E 0.7%2.4.5 electrodemagneticimpurity B 100fT a 2.4.3 steady-stateleakagecurrent ¯ I 100pA a 2.4.4 magneticJohnsonnoise r dB 2 n d 15 pT a p Hz 2.4.2 a permeasurementcycle betweentheparallelandantiparallelations, ˚ [ rad ] : d = ~ ˚ 4 E˝ (2.23) Withaperfectlyuniformandstaticmagneticdunderallations,thephase di erencebetweentheparallelandantiparalleldationsispurelyduetothe EDMinteractionwiththeelectricd.Ahigherelectricdwillresultinalarger accumulatedphaseandwillincreaseourEDMsensitivity. Duringeachmeasurementcycle,oneelectrodeischargedto +30kV(positivepo- larity)whiletheotherisgrounded.Theatomtraplifetimeiscurrentlyabouttwenty seconds.Weexpecttoincreasethetraplifetimetoonehundredseconds[81]asimprove- mentsaremadetotheODT.Thechargedelectrodeisthenrampedtozerovoltageand remainsgroundedforaperiodof60swhileanewsampleofatomsisprepared.Thecycle restartsandtheelectrodeischargedtothesamevoltagemagnitudeatnegativepolarity. Werepeatthisprocessuntiltheatomicovenisdepletedafterapproximatelytwoweeks. Nowwe'lldiscussEDMmeasurementsystematicsrelatedtothehighvoltagesystem. OurrequirementsforeachsystematicaregiveninTable2.2. Theelectricdbetweentheelectrodesmustbesymmetric,uniform,andreversible 38 Figure2.4:Left:assemblyoftheniobiumpairNb 56 at1mmgapinMacor holder.Right:aslitcenteredonthegapshieldstheelectrodesurfacesfrom heatingbytheatomtrappingandpolarizinglasers. tominimizesystematice ects.Thealignmentbetween ~ E and ~ B isaftermounting theelectrodestotheMacorholder,asshowninFigure2.4.Intheexperimentalapparatus, theholderandelectrodesrestwithinaborosilicateglasstube(seeFigure2.1).Wewill usevectoruxgateswithasystemofautocollimatorstoopticallydeterminethed uniformityandalignmentforthesecondgenerationEDMmeasurements[96].Thed reversibilityismeasuredwithacalibratedhighvoltagedivider(RossEngineeringV30- 8.3-A). 2.4.2MagneticJohnsonnoisecalculations Magneticductuationscausedbyrandomthermally-inducedcurrentsintheelec- trodes,ormagneticJohnsonnoise(MJN),limitsthechoiceofelectrodematerialsand geometriesthataresuitableforanEDMmeasurement[97,98]. Inthenexttwoyears,weareaimingforastatisticalsensitivityof d ˇ 10 25 e cm orbetterasimprovementsintheexternalelectricdandatomdetectione ciency areimplementedinthesecondgenerationmeasurements.TheRaEDMroadmapin- cludesupgradesoverthenexteyearsthatwillenableanEDMsensitivityashighas d ˇ 10 28 e cm. 39 Inthepresenceofperfectlyuniformmagneticandelectricds,theatomspinpre- cesseswithafrequencygivenbyEquation2.22.Withanappliedelectricdof30 kV/mm,IthefollowingfrequencyduetoanEDMmagnitude d ˇ 10 25 e cm: f (upperlimit)= 4 dE h = 4 10 25 e cm 30 10 4 V = cm 4 : 135 10 15 e V = Hz ˇ 2 : 9 10 5 Hz TheatomicspinsprecesswithaLarmorfrequencyofapproximately20Hzina10mGauss laboratorymagneticd.Thefractionalchangeinthespinprecessionfrequencydueto theEDMis2 : 9 10 5 = 20 ˇ 1 : 5ppm.Thereforesystematicsa ectingthespinprecession frequencysignalshouldbesuppressedtobelow150ppb. ThethermalorJohnsonnoiseinaconductoratposition ~ r isgivenbythefollowing: dB 2 n;q d = 2 0 k B T 4 ˇ 2 ˆ V u ,(2.24) V u = Z ( ~ r ~ u ) ‹ q ~ r ~ u 3 2 d 3 u ,(2.25) where dB 2 n;q =d h T 2 Hz 1 i isthemagneticdnoisedensityindirection ‹ q , 0 h NA 2 i isthevacuummagneticpermeability, k B [ J = K ] istheBoltzmannconstant, T [ K ] isthetemperature, ˆ [ m ] istheresistivityoftheconductor,and ~ u [ m ] isthelocationoftheconductorvolumeelement. Theresistivitiesofcopper,niobium,andtitaniumareshowninTable3.2.FortheRa EDMelectrodegeometry(Figure3.1), V x = V y =93 : 1cm 1 and V z =57 : 2cm 1 inthe verticaldirection[99]. Withimprovementstothethermalstabilityofthetransportbeam,weexpecttoachieve anatomtraplifetimeof ˝ =100s.Atroomtemperature( T =298K),Ithecorre- 40 spondingrmsmagneticdnoise q B 2 n;q [ T ] : q B 2 n;q = s dB 2 n;q d ˝ 1 = 2 = s V 1 : 647 ˝ˆ 0 10 12 T,(2.26) where ˆ 0 = ˆ= (10 8 m) : Forapairofniobiumelectrodes,themagnitudeofthemagnetic dnoiseintheverticaldirectionis: s dB 2 n;z d ( niobium ) =2 : 48 pT p Hz Withatraplifetimeof100s,thisgivesthefollowingrmsmagneticd: q B 2 n;z ( niobium ) = s dB 2 n;z d 1 p 100s =2 : 48 10 13 T=2 : 48 10 9 G Fromthermsdandthelaboratorymagneticd,Icanestimatetheper-shotfre- quencysensitivity f [ Hz ] : f ( niobium ) = 2 : 48 10 9 G 10 2 G ˇ 250ppb Thespinprecessionfrequencymeasurementisrepeatedmanytimesoverthecourseof theEDMexperiment.Thetotalnumberofspinprecessionfrequencymeasurements N reduces f totheintegratedfrequencysensitivity ˙ f [ Hz ] : ˙ f = f p N (2.27) Fora15-dayEDMmeasurement,Icalculatethefollowingnumberofspinprecession frequencymeasurements: N =15days 24hours day 60minutes hour 1measurement 2minutes ˇ 10 4 measurements : Ourintegratedfrequencysensitivitymustbebetterthanthefractionalchangeinthe spinprecessionfrequencyduetotheEDM,whichwefoundtobe ˇ 150ppb.Therefore theper-shotfrequencysensitivitymustbebetterthan150ppb p 10 4 =15ppm.This correspondstoaper-measurementnoiseof: 15ppm 10 2 G p 100s 10 12 pT 10 4 G =150 pT p Hz (2.28) 41 Figure2.5:Onepossibleelectrodedesignwhosevolumeisafactoroftensmaller thanthestandardRaEDMelectrode. Earlier,wesawthattheper-shotfrequencysensitivityofniobiumis 250ppb=2.48pT/ p Hz.Themagneticdnoisescalesas ˆ 1 = 2 ,fromwhich weestimatetheper-shotfrequencysensitivityofcopperandtitaniumtobe (250ppb) p 15 : 2 = 1 : 543 ˇ 780ppband(250ppb) p 15 : 2 = 39 ˇ 160ppb,respectively. Giventheseestimates,IexpectthatMJNwillremainaminorsourceofsystematic uncertaintyforEDMmeasurementsatthe d ˇ 10 25 e cmlevel.Asweapproachour long-termgoalof10 28 e cm,we'llneedtomodifytheelectrodestoreducemagnetic noise.Forexample,Figure2.5showsapossibledesignwherethevolumeoftheelectrode hasbeenreducedbyanorderofmagnitude,reducingthednoisebyapproximatelya factorof p 10. Anewelectrodegeometrycouldbeoptimizedbyrigorousmodelingoftheduni- formityunderavarietyofpotentialelectrodemisalignments.Idiscussthistypeofanal- ysisIperformedforthecurrentRaEDMelectrodegeometryinSection2.5. 2.4.3Paramagneticimpurities Weconsiderapotentialsystematicinwhichthemagnetizationofafractionoftheim- puritiesintheelectrodesdependsonthepolarityofthechargingcurrent.Asu - cientlyhighconcentrationofparamagneticimpuritiesnearanelectrodeprimarysur- facecouldperturbthemagneticdintheradiumcloudregion.Thiswouldinducea polarity-dependentspinprecessionfrequencymimickinganEDMsignal,orfifalseflEDM 42 d B [ e cm ] : d B = B E ,(2.29) where isthenuclearmagneticmoment, E isthemagnitudeoftheappliedelectricd,and B isthechangeinlocalmagneticdunderreversalof E . Impuritiesintheelectrodematerialareminimizedbyusinghigh-gradematerials,using machineshoptoolingthatdoesnotembedimpuritiesonthesurface,andusingpolishing andcleaningtechniquesthatremovesurface-levelcontaminants.Inthescenarioofan appliedelectricdof30kV/mmandamagneticdchangeof100fTunderreversal, IthefollowingfalseEDMmagnitude: d B = 2 : 3 10 8 eV = T 100 10 15 T 30 10 4 V = cm =7 : 7 10 27 e cm OtherEDMhighvoltagesystematicrequirementsaregiveninTable2.2.Table3.1and Table3.2listthematerialpropertiesandprocessingtechniquesthatweuse.I'lldiscuss electrodematerialselectionandsurfaceprocessingindetailinSection3.1. 2.4.4Leakagecurrentanddangle Ileakagecurrentasanycurrentowingbetweentheelectrodes.Thisincludescur- rentowingthroughtheinsulatingmountanddemissionflbetweenthetwoprimary surfacesacrosstheelectrodegap.Anelectrodegapof1mmisshowninFigure2.4.Leak- agecurrentinducesmagneticdswhosepropertiesdependonthemagnitude,path, anddynamicpropertiesofthecurrent. Ianelectrodedischargeasatransientsurgeindemissionbetweentheelec- trodesurfaces.DischargesarediscussedindetailinSection3.4.4.Intheeventofa 43 Figure2.6:Aplotofthemaximumalloweddmisalignmentoverarangeof leakagecurrentsforatargeted10 26 e cmsensitivity. dischargeclosetothelocationoftheatomcloud,theatomswillinteractwiththeinduced magneticd.Theinteractionwillmanifestasachangeinthespinprecessionfrequency oftheatoms.Ifthedischargerateiscorrelatedwiththepolarityoftheelectrodes,itwill introduceasystematicshiftintheEDMsignal. Tostudythee ectofleakagecurrentonthespinprecessionfrequency,wemodel thedischargeasathinwireofcurrenttravelingadistance r [ m ] fromthecloud.This consequentfifalseflEDMsignal d ¯ I [ e cm ] isgivenby[48]: d ¯ I = ~ B E ‹ B = E 0 ¯ I 2 ˇr sin E ,(2.30) where = 2 : 3 10 8 [ e V = T ] isthenuclearmagneticmomentof 225 Ra, ¯ I [ A ] isthesteady-stateleakagecurrent,and E [ rad ] istheanglebetweentheappliedelectricandmagneticds. ThedalignmenttoleranceforanEDMsensitivityof10 26 e cmisplottedasafunc- 44 tionoftheleakagecurrentinFigure2.6.Withanappliedelectricdof E =30kV = mm,a discharge-atomdistanceof r =50 m, d I =10 27 e cm,andaleakagecurrent I =100pA, Iamaximummisalignmentof30mrad. 2.4.5Polarityimbalanceintheelectricd AnychangeintheEDMspinprecessionfrequencyarisingfromadi erenceinthestrength oftheelectricdbetweennegativeandpolarityisproportionaltothesquareofthe electricd[48,46].Thisisapropertyofthehighvoltagesystem.IntheRaEDM measurement,thepolarityimbalanceissymmetrictowithin0.7%. 2.5E ectofElectrodeMisalignments Weuseidentical,plane-parallelelectrodessothatthereversibledisuniformand symmetricastheelectrodesalternaterolesascathodeandanodeeveryEDMmeasure- mentcycle.Theprimarysurface,seenasthetopsurfaceinFigure3.1,istand16mm indiameter.Theroundededgeshave4mmcircularradialcurvatures. TheRaEDMexperimentrequiresanappliedelectricdthatissymmetric,uniform, andreversibleinthecenteroftheelectrodegapwherethespinprecessionfrequencyof the50 mdiameterradiumcloudismeasured.Ourelectrodegeometryreliablymeets theserequirementsatdstrengthsof12Œ30kV/mm. Systematice ectsarisingfromasymmetricdreversalmustcontinuetobereduced astheexperimentalsensitivityimproves.Inthecurrentmeasurementscheme,oneelec- trodeispermanentlygroundedandtheotherelectrodeischargedbyabipolarpowersup- ply.Wewilldesignamoresymmetricapparatusthatallowsustoalternatethecharged andgroundedelectrodesusinghighvoltageswitchesandaunipolar50kVpowersupply inthenextphaseofhighvoltagedevelopment. Idemonstratedthee ectofsteady-stateleakagecurrentonthespinprecessionfre- quencywithasimplewiremodelinSection2.4.4.InSection2.5.1Iwillshowthatthe 45 Figure2.7:Asoftwaremeshedmodeloftheelectrodepairandcoordinatesys- tem.The-meshedelectrodegapregionisshadedblue. electrodeelectricdmatchesthatoftheidealcapacitorintheatomcloud regionusingelementmodeling.Wewillusethemethodsdevelopedheretoopti- mizeelectrodegeometriesastheexperimentsensitivityimproves. 2.5.1Descriptionoftheelectricdelementanalysis Imodeledthehighvoltageelectrodesintheelementanalysissoftware COMSOL Multiphysics (version5.3)tostudytheelectrostaticduniformityunderarangeof potentialelectrodemisalignments.Inthemodel,theelectrodesaresurroundedbyaper- fectvacuum.Theelectrodegapsizeissettoa1mmgapsizeandthetopelectrodeis chargedto 30kVforanominalelectricdof E 0 =30kV/mm. Mysimulationsusethe ExtremelyFine settingswith SizeExpression increasedto 4 10 4 inthegapregionand Resolution increasedto200alongtheuppercurvedelec- trodesurface.OnecanseethehighermeshelementdensityinFigure2.7.Theminimum meshelementsizeissetto20 m,whereIfoundthattheelectricddependenceonthe meshsizeconvergestonegligiblysmalluctuations. ThecoordinatesystemoftheelectrostaticmodeloftheelectrodesisshowninFig- 46 Figure2.8:Aplotoftheelectricdangleasafunctionoftheverticalposition y .Inthisplot,theelectrodesareaxiallyalignedandtheangularmisalignmentis variedfrom0Œ16mrad.Thecenterofthegap,0.5mmbelowthetopelectrode, correspondsto y =0. ure2.7,withtheoriginasthemidpointbetweenthetwoelectrodesalongtheir verticalaxisofthetopelectrode.Ithattheverticaldstrength E y changesbyless than6ppbper100 mwhentheelectrodesareperfectlyaligned.Thehorizontald magnitude E ? = q E 2 x + E 2 z changesbylessthan5ppbper100 mwithrespectto E 0 within0.5mmoftheorigin.Inpractice,wealignourelectrodestobetterthan4mradin thehighvoltageteststanddescribedinSection3.4.1. Themeshdensitywasoptimizedinthevolumebetweentheelectrodeprimarysur- faces.Wethemaximumandminimumelementsizetominimizedcalculation dependenceonmeshsettings.Thiswasdonebyconvergenceanalysis,decreasingthe minimumelementsizefrom120 mto18 : 5 mandrecordingthechangeinthemaxi- mumelectricdwithagapsizeof1mmandanappliedvoltageof 30kV.Thelower boundofthemeshsizeislimitedbytheRAMofourworkstationPC(32GB).Ithe maximumelementsizetobeafactorof4largerthantheminimumelementsize. 47 Figure2.9:Aplotoftheverticalelectricdforangularalignmentsinthe range0Œ16mrad.Theaxialmisalignmentis100 m.Thecenterofthegap, 0.5mmbelowthetopelectrode,correspondsto y =0. Isetthemaximumandminimumelementsizesinthegapbetweentheelectrodesto be80 mand20 m,respectively,wherethemaximumverticalcomponentoftheelectric dchangesbylessthan0 : 03%(about10V/mm)whenchangingthemeshsizeby10%. However,thesedeviationsarebasedondpointsveryclosetothemeshborderand electrodesurfaces.Whenweperformanidenticalconvergenceanalysiswhilerestricting themaximumdvaluetothehorizontalplanebisectingtheelectrodegap,thed changesbylessthan4partsperbillion. 2.5.2Electricdresponsetoelectrodemisalignmentnearthecenterofthegap Iinvestigatedthee ectofmisalignmentsbetweentheelectrodesontheelectricdan- gle,as E =arctan E ? =E y .Therearetwotypesofmisalignmentsweconsider. 48 Angularmisalignments,ortilts,areintroducedbyrotatingthebottomelectrodeabout the z axisintherange0Œ16mrad.Axialmisalignments,orshifts,translatesthebottom electrodealongthe x axisando setstheelectrodecenters.Shiftsofupto1mmdisplace- mentsareconsideredinthiswork.Whenthetiltandshiftarezero,theelectrodesare perfectlyalignedand E =0nearthecenterofthegap,correspondingtoauniformver- ticald.Whentheelectrodesareperfectlyaligned( E =0 ; =0) E ? uctuatesby ˇ 6 ppband ( E y ) ˇ 10ppbfromthenominalapplieddof 30kV = mmfor x \ z 100 m. I'veplottedtheelectricdangle E forarangeofelectrodetiltsinFigure2.8.From myofthecoordinatesystem, E iszeroat ~ r =+0 : 5 ‹ y mm,thesurfaceofthetop electrode.Foratiltof0mrad,theelectricdangleisperfectlyaligned( E =0)across theelectrodegap.Fornonzerotilts, E scaleslinearlywithdistance y fromthetopelec- trodesurfacetothevalueoftheelectrodetiltangleatthesurfaceofthebottomelectrode, ~ r = 0 : 5 ‹ y mm.Evenforthelargestangularmisalignmentof16mrad,theelectricd angle E islinearandappearsinsensitivetothee ectsfromtheelectrodeedge. Ialsostudiedthee ectofshiftingthebottomelectrodealong ‹ x withrespecttothe topelectrode'sverticalaxisintherange0Œ1000 m.InFigure2.9,Ishowthevertical componentoftheelectricdresponsefora100 mshift.Thebottomelectrodeiso set along ‹ x tomaximizetheconvolutionofangularandspatialmisalignment.Evenforlarge tilts,axialmisalignmentsintroduceaconstanto setin E y thatappearstobeindependent ofthetilt.Inthisworst-casescenario,themagnitudeoftheconstantdependsonboththe tiltandthehorizontalmisalignment,whichbringsonesideofthebottomelectrodecloser tothesurfaceofthetopelectrode. Theconstanttermin E y canbefoundbyconsideringtheelectrodesatareducedgap sizeof tan ,where isthespatialdisplacement.Foratiltof16mradandaspatial misalignmentof1mm(thelargestmisalignmentsimulated),thisresultsinashiftof +0 : 5kV/mmalongtheverticalaxis.Theshiftvaluewillbenegativeifwemovealong ‹ x adistancegreaterthantheo setbecausetheelectrodesurfaceswillbeangledawayfrom 49 Figure2.10:Acontourofthehorizontalelectricdmagnitudeformisaligned electrodesclosetothe8mmedgeregion. eachother.InallationsshowninFigure2.9,thecontributionstochangesinthe electricdduetoangularandspatialmisalignmentsareindependentofeachotherfor horizontaldisplacements x \ z 1mm. 2.5.3Electricdbehaviorintheelectrodeedgeregion Thecurvededgeregionofthe(Figure2.7)is8mmfromtheelectrodecenteraxis. Acontouroftheperpendicularcomponentoftheelectricd E ? isshowninFig- ure2.10.Thenominaldofthissimulation,likealltheothersIdiscussinthissection, is E 0 = V=d =30kV/mm.Withperfectlyparallelsurfaces(tilt E =0mrad)andanaxial misalignmentof1mm(shift =1000 m),thereisapproximatelya3 : 5%gradientin E ? about0.5mmfromtheedgeofthetopelectrode.Thehorizontaldisashighas 50 Figure2.11:Aplotoftheelectricdangleaswescanhorizontallyacrossthe electrodesurface(8mmradius)fromthecentertotheedgeregion. 7kV/mmastheedgeroundso tothesideoftheelectrode.Within7mmofthecenter oftheelectrode, E ? variesbylessthan3%. Tofurtherillustratetheedgebehavior,aplotofthedangle E isshownforper- fectlyparallelelectrodesandforatiltof16mradwitha1mmshiftinFigure2.11.The danglestartstochangetlyatahorizontaldistance x =6000 mfromthe origin.Interestingly,thedangleexponentiallyincreasesfortheparallellineseriesbut thereisadipinthedangleinthe16mradseries,leadingtoacrossingbetweenthe two.At x> 7000 m,thedangleofthe16mradlineincreasesmorerapidlythanthe parallelline. Finallywelookattheverticalcomponentofthedbehaviorneartheedgeregion inFigure2.12.Sincewe'reinterestedyintheedgebehavior,westartfrom ~ r ( m)=4000 ‹ x +0 ‹ y +4000 ‹ z andscanhorizontallyalong ‹ x . E y decreasesbyapproximately 0.9kV/mmoverbothcurvedsurfacesoftheelectrodes,from8mmto12mmand12mm 51 Figure2.12:Aplotoftheverticalcomponentoftheelectricdaswescan horizontallyacrosstheelectrodesurfaceintheedgeregion(radiusof8mm). to16mm.Inthisregion,thehorizontaldstrengthisonthesameorderofmagnitude astheverticaldstrength. 2.5.4Modelingtheelectricdbehaviornearthecenteroftheelectrodegap Theelectricdangle E scaleslinearlywiththeangularmisalignments,asshownin Figure2.8.Imodeledthechangein E asalinearfunctionofthepositioninboththe xy planeandthe xz plane.Thelinearchangein E y alongthehorizontal xz planeisveryweak. Ishowtheto E y forthemoreinterestingcaseofthevertical xy planeinFigure2.13. Thelinearmodelreproducesthechangeintheelectricdangletoanaccuracyofbetter than1 radinbothplanesupto1mmfromthecenterofthegap,evenforlargeangular andaxialmisalignments. Ialsomodeledthechangeinthehorizontalcomponentoftheelectricd E ? asthe fractionofthemaximumangle.Liketheto E inFigure2.13, E ? scaleslinearlywith thedistancefromtheorigin.Fora16mradtiltand E 0 =30kV/mm,Ithefollowing 52 Figure2.13:Astraightlinetothesimulatedpolarangleoftheelectricd foranangularmisalignmentof16mradandanaxialmisalignmentof1mm. Thecenterofthegap,0.5mmbelowthetopelectrode,correspondsto y =0. fractionallinearchange: E ? =E 0 ˇ 0 : 1% = mrad Imodeledthebehavioroftheverticalelectricd E y usingthesamemisalignment settingsasthe E modelinFigure2.14.Themisalignmentcausesa1.6%reductioninthe dstrengthandthechangein E y along ‹ y isnonlinear.Icomparedthebehaviorwitha cos ( E ) model,whichassumesthatthechangein E y ispurelyduetothechangingd direction. Thereductionin E y islargerthantheamountreproducedbythecosinemodel.For a16mradtilt,thecosinemodelaccountsforachangeof E y =E 0 ˇ 100ppmper500 m fromtheoriginwhilethetotale ectis230ppm/500 m. 53 Figure2.14:Aresidualplotofamodeloftheverticalelectricdfora16mrad angularmisalignmentand1mmaxialmisalignment.Themodelassumesthat thedisafunctionoftheangleoftheelectricd. 2.5.5Estimatinge ectsforrealisticmisalignmentsinthehighvoltageapparatus Acontouroftheverticalcomponentoftheelectricdfora2mradtiltisshownin Figure2.15.Thetotalchangein E y acrossa1000 mrangein x is2.1%.Evenfor largehorizontalradialdisplacements( x \ z> 100 m),theelectricdisuniform.This canbeseenintheleftpanel,wherethechangeintheelectricddependson x andis insensitivetodisplacementalongtheaxisofrotation( ‹ z ). Foralarger4mradtilt,theelectricdgradientbecomes( E y =E y ) = 25 m ˇ 100ppm. ThisgradientwouldcauseanEDMsystematicontheorderof10 29 e cmacrossa100 m radiumcloud.Thebehaviorof E ? isidenticalbutthestrengthofthegradientismore thantwoordersofmagnitudeweaker. Inthemorerealisticcaseofa2mradtilt,wethat E changesby0.2 radper100 m intheverticalplaneand0.02 radper100 minthemidplane.EDMsystematice ects arisingfromdanglechangesofthismagnitudearefarbelowourcurrentstatistical 54 Figure2.15:Contourplotsoftheverticalcomponentoftheelectricdinthe xz (left)and xy (right)planeandwitha2mradtilt. sensitivity. 2.6ElectrodeUpgradeStrategyandResults 2.6.1Highvoltagedischarge-conditioning Wedischarge-conditioningastheprocessofapplyingiterativelyhighervoltagesto theelectrodestosuppresssteady-stateleakagecurrentanddischargeratesbetweenthem. Leakagecurrentreferstoanycurrentowingbetweentheelectrodesandismeasuredby apicoammeterinserieswithoneoftheelectrodes,asshowninFigure3.11. Figure2.16showsseveralforty-minutesnapshotsofasubsetofshiftsoverthelifetime ofconditioningandvalidatingapairofniobiumelectrodes.Istartedwithmanually controlledvoltagestepsbeforeusingaperiodic,polarity-alternatingvoltagetosimulate EDMmeasurementconditionsfortheelectrodes. AschematicofonefullperiodoftheconditioningvoltagewaveformisshowninFig- ure2.17.Theperiodis280seconds(60secondspositivepolarity,80secondsnovoltage, 60secondsnegativepolarity,80secondsnovoltage).Thisperiodischosentothe measurementtechniquediscussedinSection2.4.1. Wedi erentiateourmethodfromthestandardficurrent-conditioningflmethod[100] 55 Figure2.16:Forty-minutesnapshotsoftheconditioningprocessinearly,mid- dle,andstages.Positiveandnegativecurrentisplottedwithgreencrosses andredcirclesonalogarithmicscale.Leakagecurrentlessthan10pAisomit- tedforclarity.Therightverticalaxisistheappliedvoltageandisplottedasa blueline. becausewecharacterizeelectrodeperformancebycountingdiscretedischargesovertime andweuseaperiodicvoltagewaveform.Iwillinterchangeablyusetheshorthandterm ficonditioningflwhenreferringtodischarge-conditioning. Intheabsenceofsurfaceparticulatecontamination,electrodedischargesarecaused bychargebuilduponmicroprotrusionsontheelectrodesurfaces[101],whichwewill refertoaschargeemitters.WeprocessandhandleourelectrodesinClass100orbetter environmentstominimizeparticulatecontamination.Theheightofchargeemittershave beenmeasuredontheorderof1 minbu erchemical-polishedlarge-grainniobium electrodespreparedsimilarlytoourelectrodes[102].Ifthechargeemitterisnearthe edgeoftheelectrode,weexpectthehighergradientswillincreasethelikelihoodofa 56 Figure2.17:AschematicoftheperiodicEDMhighvoltagewaveform. A posi- tivechargingupramp. B positivechargingdownramp. C negativecharging upramp. D negativechargingdownramp. discharge. Controlleddischargeselectricallypolishaway,orablatechargeemittersovertime, allowingtheelectrodestoperformreliablyathighervoltages[100].AsshowninSec- tion3.4,itmaytaketenstomorethanonehundredhoursofdischarge-conditioningto suppresschargeemitters.Weexpecttherequiredconditioningdurationmaytakelonger ifthesurfaceisinsu cientlypolishedorcontaminated.Bulkproperties,suchasthe workfunction,resistivity,orhardnessoftheelectrodemayalsoplayaroleinthecondi- tioningtime.Thesebulkpropertiesarelistedforaselectionofcommonlyusedelectrode materialsinTable3.2. 2.6.2Typicalsizeofdischarges TheelectrodegeometryisshowninFigure3.1.We'llconsiderthemainhigh-gradient surfacesoftheelectrodesastheonlysurfaceswheredischargesoccur.Forapairofcircu- lar,parallelplatecapacitorswhosediameter2 R ismuchgreaterthantheelectrodegap 57 d ,theelectrodecapacitance C [ F ] isgivenbythefollowing: C = Q V = o ˇR 2 d ,(2.31) where Q [ C ] isthechargeoneachelectrode, V [ V ] istheelectricpotentialbetweenthe twoelectrodes,and o h Fm 1 i isthevacuumelectricpermittivityconstant. Foranelectrodegapof d =1 10 3 m,aplateradius R =8 10 3 m,andanapplied voltageof V =30kV,wegetacapacitanceof1.78pF.However,fromthemodelof theelectrodesinCOMSOL,Igetacapacitanceof3.3728pF.Thisgivesatotalchargeof Q =3 : 3728pF 30kV=1 : 01 10 7 C. Iintegratethedischargecurrent I ( t ) [ A ] toestimatetheamountofchargeejectedfrom anelectrodesurfaceinadischarge, Q dc [ C ] : Q dc = Z + 1 I ( t ) dt (2.32) Thedischargecurrentwaveformvariesindurationandamplitude,butareasonable estimateisanamplitudeof100nAovera1mstimescale.I'llapproximatethewaveform asGaussian: Q dc = Z + 1 (100nA)exp ( t 2 2 ˙ 2 !) dt ,(2.33) ˙ =1ms(2.34) Insuchadischargewewouldexpecttosee ˇ 2 : 51 10 10 Cor1 : 56 10 9 electrons. Thisis ˇ Q dc =Q 100%=0 : 25%ofthetotalchargestoredoneachelectrode. 2.6.3Results Fourpairsofniobiumelectrodesandtwopairsoftitaniumelectrodesweresurfacepro- cessedasdescribedinTable3.1.Afterhigh-pressurerinsingtheyarepreservedinclean roomenvironmentsofClass100(ISO5)orbetter.Iconditionedpairsofelectrodesina custom,Class100-ratedhighvoltageteststationatMSUbyapplyingDCvoltagesashigh 58 as 30kVatgapsizesintherange0.4Œ2.5mm.Maximumdsof+52 : 5kV/mmand 51 : 5kV/mmweretestedandarediscussedinSection3.4.7. Onepairoflarge-grainniobiumelectrodeswasvalidatedtooperatereliablyat 20kV/mmatMSU.Imountedtheelectrodesinastainlesssteelcontainerandsealed thecontainerwithtubing.Thecontainerwaswithtered,drynitro- genandtransportedtoANL.ThenIconstructedandvalidatedaClass100cleanroom thatcoveredtheelectrodeentrypointtotheRaEDMexperimentalapparatus.Theelec- trodeswereremovedfromtheirpackagingandinstalledintheexperimentalapparatus inMay2018,whereIrevalidatedthemto20kV/mm. 59 CHAPTER3 HIGHVOLTAGEELECTRODEDEVELOPMENT InSection3.1Iwilldescribepastandpresentconsiderationsinelectrodematerialand surfaceprocessing.Westartwiththepreparationofthepreviouselectrodepairused forthegenerationEDMmeasurementsinSection3.1.1.Materialselection,surface processing,andelectrodedecontaminationforthenewelectrodestestedinthiswork aredetailedinSections3.1.2and3.3.1.Iwillpresentourmethodofbenchmarkingthe performanceoftheelectrodesinSection3.4.Finally,we'llcomparetheperformanceof allthetestedpairsinSection3.4.10. (a) (b) Figure3.1:(a)Cross-sectionalelectrodeschematic.Surfaceshaveatnesstol- eranceof25.4 mandaparallelismof50.8 m.Thetopsurfaceispolishedto anaverageroughnessof0.127 m.Thebaseismountedbya10-32tappedhole. Copperrodsareusedtoconnecttotheelectrodes'3.2mmdiametersidebore tohighvoltagefeedthroughs.(b)Apairoflarge-grainNiobiumelectrodesina cleanroomstainlesssteelcontainer. 60 Figure3.2:Fromlefttoright:acopper,niobium,andtitaniumelectrode. 3.1ElectrodePropertiesandPreparation 3.1.1Legacyelectrodepreparation ThegenerationEDMmeasurementsusedapairofelectropolishedoxygen-freecop- perelectrodes[65,48].Theirgeometryisidenticaltothenewelectrodesdiscussedin thiswork(Figure3.1).Surfaceprocessingoftheseelectrodes,labeledCu 12 ,isdetailedin Table3.1. ThelegacyelectrodeswereconditionedatANLwithaunipolar 30kVpowersupply (GlassmanPS/WH-30N15-LR)inaMacorholderata2mmgapsizein2008[103].The electricdwasreversedbyturningthesystemo andmanuallyswitchingthepower supplyterminationsatthehighvoltagefeedthroughs.Voltagewasincreasedfrom1Œ 20kVin1kVstepswhilemonitoringthesteady-stateleakagecurrent.Conditioningwas declaredcompleteiftheelectrodescouldhold20kVwithasteady-stateleakagecurrent of < 100pAfortenhours. Fourpairsofelectrodestotalweretestedinthismanner,includingtwopairsoftita- 61 Table3.1:Electrodeinventory.Large-grain(LG)niobiumelectroderesidualresis- tanceratio(RRR) > 250.OF=oxygenfree.G2=grade-2.Simichromepolishbyhand. Diamondpastepolish(DPP)byhand.LPR=lowpressurerinse.HPR=highpressure rinse.HF=huoricchemicalpolish.EP=electropolish.BCP=bu eredchemi- calpolish.SiC=siliconcarbidemachinepolish.CSS=colloidalsilicasuspensionma- chinepolish.VB=420Œ450 Cvacuumoutgasbake.WB=150Œ160 Cwaterbake. USR=ultrasonicrinseafterdetergentbath. batchmaterialpairsurfaceprocessingrecipe 1OFcopperCu 12 a Simichrome ! EP ! USR ! WB 2LGniobiumNb 14 SiC ! BCP ! DPP ! CSS ! USR VB ! LPR ! HPR 2LGniobiumNb 23 SiC ! BCP ! USR ! VB ! HPR resurface ! BCP ! HPR 2G2titaniumTi 24 SiC ! HF ! USR ! VB ! HPR 2G2titaniumTi 13 SiC ! HF ! EP ! USR ! VB HPR 3LGniobiumNb 56 b SiC ! BCP ! USR ! HPR ! WB 3LGniobiumNb 78 SiC ! BCP ! USR ! HPR a Legacyelectrodesusedfortwomeasurements[65,48]. b Secondgenerationelectrodes,currentlyinstalledintheRaEDMapparatus. niumelectrodesandonepairofcopperelectrodeswithoutelectropolishing.Thelegacy titaniumelectrodesallexhibitedleakagecurrenthigherthan100pAat20kV.Flooding thetestchamberwithargongasandplasmadischarge-conditioningthetitaniumelec- trodeswasattemptedwithoutanobservableBothcopperelectrodepairswere conditioned,withtheelectropolished(EP)electrodestakingtlylesstime. ThelegacyelectrodepairCu 12 wasmountedinaMacorholderata2.3mmgapsize andinstalledintheRaEDMexperimentalapparatus[86].Thetwopublished 225 RaEDM resultsemployedelectricdsof 6 : 7kV/mmand 6 : 5kV/mm[65,48].Thepairwas retestedat20kV/2.3mm=8.7kV/mmbutexceededthe100pAlimit.Thiswasreme- diedbyreducingtheelectricdby25%to6.5kV/mmfortheEDMmeasurement. Wesuspectthattheprimarysurfaceofoneorbothoftheselegacyelectrodeswascon- taminatedduringinstallation.Thiswasamotivatingfactorinthedevelopmentofthe decontaminationtechniquesforthenewelectrodesdiscussedinsubsequentsections. 62 Table3.2:Bulkmaterialpropertiesofelectrodes.Wefistrong B -impuritiesflas ˜ m = (10 6 cm 3 mol 1 ) > +1000,where ˜ m isthemolarsusceptibility. ˜ m (Nb)=+208. material ˚ strong B densityresistivityhardnessoutgasrate ( e V)impurity(%) kg m 3 ! ( cm) a kgf mm 2 ! TorrnL scm 2 Nb b 4.32 : 7 10 2 857015.2134 : 630 Cu c 4.652 : 5 10 7 89601.54335 : 016 : 3 Ti d 4.335 : 5 10 1 45063999 : 0184 SS e 4.348 : 1 10 +1 800069.017642 : 8 Mo f 4.61 : 4 10 2 102004.85156 : 036 : 7 References [104,105][106][106,107][108,109][110,107][111,112] a Resistivitymeasuredat273K. b Hardnessmeasuredat473K.OutgasrateestimatedfromthecorrelationbetweenCu, SS,andNbdesorption. c Hardnessmeasuredforsinglecrystal(III)at293K.Outgasratemeasuredforunbaked OFhigh-conductivityaftertenhours. d Hardnessmeasuredforiodide-annealed,99.99%purityat293K.Outgasratemea- suredforunbakedOFhigh-conductivityaftertenhours. e SS=stainlesssteel.Hardnessmeasuredfordesignationtype304.Outgasratemea- suredforunbaked,electropolishedNS22Saftertenhours. f Hardnessmeasuredat293K. 3.1.2Considerationofmaterialsfornewelectrodes Weselectedlarge-grainniobiumandgrade-2titanium(middleandrightelectrodesin Figure3.2)fortestingafterreviewingacceleratorphysicsliterature.Thebulkproperties ofthesemetalsandothercommonlyusedhighvoltagemetalsarecataloguedinTable3.2. Ourgoalistousethematerialthatsustainsthehighestelectricdstrengthwhilemin- imizingleakagecurrentandmagneticimpuritiesthatcouldintroduceEDMsystematic e ects.Stainlesssteelwasexcludedfromourtestingduetoitsrelativelyhighferromag- neticcontentbutitspropertiesareneverthelessincludedforreference. Large-grainniobiumelectrodeswithacathodeareaof3170mm 2 havebeentestedto dsashighas18.7kV/mm[102].Fine-grainappearstoperformslightlyworse,per- hapsbecausethehighergrainboundarydensityincreasesparticulateadherencetothe 63 Figure3.3:Themagnetizationrailsystemsitsinsideamu-metalshield. electrodesurface[113].Thehighestreportedelectricdforgapsizesnear1mmthat wefoundis130kV/mmusinganasymmetrictitaniumanodeandmolybdenumcathode withane ectiveareaof7mm 2 [114].Thee ectiveareaoftheRaEDMelectrodeis200 mm 2 ,approximatelyafactorofthirtylarger.Thereisevidencethatlargerstressedareas arepronetolowerbreakdownvoltages,suggestingthataminiaturizedRaEDMelectrode geometrycouldimprovethemaximumstableelectricd[115]. Inthepresenceofhighelectricds,anoxidelayeronanelectrodesurfacecould beatsourceofparticleemission.Niobiumoxidizesatahigherratethan titaniumandoxygen-freecopper[116,117,118,119,120].However,toxi- dationratesforthesematerialshaveonlybeenobservedattemperaturesinexcessof 500 C[116,120,119,121,122].TheRaEDMexperimentalapparatusispumpedtoul- trahighvacuum( < 10 11 Torr)atroomtemperature.Wethereforeexpectthatoxidation ratesarenegligiblylowforanyselectionoftheconsideredelectrodematerials. 64 Figure3.4:Aschematicofthegradiometercircuit.Resistorandcapacitorvalues arelistedinFigureD1. 3.2ElectrodeResidualMagnetizationMeasurements WehaveconsideredapotentialEDMsystematicarisingfrommagneticimpuritiesin theelectrodesthatchangepolarizationwitheachelectricdreversal.Asu ciently highconcentrationofsuchimpuritiescouldperturbthemagneticdintheradium cloudregion.Idescribepotentiale ectsfromthissystematicinSection2.4.3. Imeasuredtheresidualmagnetizationofcopper,niobium,andtitaniumelectrodes inamagneticallyshieldedmu-metalenclosureshowninFigure3.3.Themu-metalbox isaprototypethatweborrowedfromourgenerousHeXeEDMcolleaguesattheTech- nicalUniversityofMunich(TUM).Iusedcommerciallow-noiseuxgates(Bartington Mag03IEL70)withamaximumnoiseoorof6pT/ p Hz. Theresidualmagnetizationmeasurementrecordsdatafromeachofthethreeux- gates.Foreachmeasurement,theelectrodeisalternatedbetweentheandthird uxgate.Theuxgatecenteredontheelectrodeisthefisignalfluxgate;theuxgate furthestfromtheelectrodeisthefibackgroundfluxgate.Thesesignalsareinputstoa 65 Figure3.5:Simulated3kHzButterworthlowpasscurve(blueline)andmea- suredfrequencyresponsewithawaveformgeneratorinput(redcircles)ofthe gradiometercircuitinFigure3.4.1.86kHzdashedverticalline=measuredcut- o frequency.16.4kHzdashedverticalline=uxgatefrequency,attenuatedby ˇ 53dB. gradiometercircuit,whichinputstwosignals V A and V B toadi erentialopamp(shown inFigure3.4).Thesignalsaresubtractedandamtoisolatetheresidualmagnetiza- tionduetotheelectrode.Theresultingambackground-subtractedsignalisthen sentthroughaseventh-orderlow-passter. Thelow-passtercircuitisshownininFigure3.5.BecauseIuseslightlylarger capacitancesthanwhatisfora3kHzlowpasster,mycuto frequencyis lower(1.86kHz).Idesignedtheterfor60dBattenuationattheuxgatefrequencyof 16.4kHz.Inpractice,Ithe16.4kHzsignalisattenuatedbyapproximately53dB. Thepassbandisverytuptoabout200Hz,andthenstartstoslopedownward. Imeasuredtheresidualmagnetizationofcopper,aluminum,stainlesssteel,Macor, niobium,andtitanium.Agradiometermeasurementofaniobiumelectrodeisshown inFigure3.6.Ourgradientsignalswereallontheorderofapproximately400pTdue 66 Figure3.6:Gradiometerresultsforaniobiumelectrode.Averagegradiometer signal= 440 : 8 1 : 6pT.Averagemonitorsignal=88 : 2 1 : 3pT.Averagenull signal= 8 : 5 0 : 1pT. totheuxgatepottinglimitingtheminimumsensor-surfacedistanceto ˇ 15mm(see FigureD2).Titaniumwasthemostmagnetic,inagreementwiththemagneticproperties listedinTable3.2. WealsosentapairoftitaniumelectrodestocolleaguesattheUniversityofScience andTechnologyofChina(USTC).Theymeasuredtheresidualmagnetizationofapairof titaniumelectrodesto 5nTwithacustom5mmatomicvaporcellmagnetometerthat allowedthemtoplacetheirsensorapproximately8mmfromthesurface.Theresidual magnetizationmeasurementswiththeMSUuxgatemeasurementsandUSTCmagne- tometermeasurementsareshowninFigure3.7. Becauseofthehigherresidualmagnetizationofthetitaniumelectrodes,wedecided touselarge-grainniobiumforradiumspinprecessionfrequencymeasurements. 67 Figure3.7:Residualmagnetizationmeasurementsofgrade-2titaniumelec- trodesusingcommercialuxgates(MSU)andacustommagnetometer(USTC). 3.3ReviewofHighVoltageSurfaceProcessingApplications Electrodeperformancedependsonthematerial,geometry,gapsize,vacuumpressure, appliedvoltagemagnitude,voltagepolarity,voltagefrequency,andtheelectrodesurface condition.Theanalysisdescribedinthisthesisbenchmarkstheofconventional highgradientsurfaceprocessingtechniquesforouruniqueelectrodegeometryandop- erationalrequirements.InthissectionIwillydescribesomeoftheusecasesof high-voltageacceleratorapplicationsthatinspiredus. Radiofrequency(RF)cavitiesaredesignedtoaccelerateand,insomecases,bunchan incomingbeamofparticles.TheACacceleratingpotentialistypicallyappliedacrossa largegap( > 5mm)[113].Theyareusuallymadeoflarge-grainorainniobiumand areoftencooledtosuperconductingtemperaturestoreduceresiduallosses. Electrongunsprovideelectronsourcesforbeamexperiments.Theseapplicationstyp- icallyuseaconical(fipointfl)small-areacathodeandarelativelylarge-areat(fiplanefl) anodetogeneratehigh-intensitycurrent.ElectrongunscanbeACorDCandprovide 68 (a) (b) Figure3.8:(a)Ibuiltaportablecleanroomwitha2 0 2 0 HEPAter(SAM22 MSNCR).(b)TheNSCLdetectorcleanroom.IthasseveralHEPAunitsand accommodatestheteststationanduptothreepersonnel. astableelectronbeamforhundredsofhours.Forlong-pulse(DC)guntypes,applied voltagesreachhundredsofkilovoltsandgapsizesoftensofmillimeters[102]. Electrodegeometryandoperatingvoltageisoptimizedtosteerchargedparticlesand simplifytheirmotioninstorageringEDMexperiments.Theparticlespinsprecessin multipleplanesthroughtheirelectricandmagneticdipoleinteractionswiththeapplied electricandmagneticds.Theparticlespinprecessionfrequencycanbeconstrainedto asingleplanerelativetothemomentumvectorbyappropriatelychoosingthestrengthof theappliedds.AppliedvoltagescanrangefromafewkVto ˇ 240kVandelectrode gapsrangefrom30Œ120mm[123,124]. 3.3.1Secondgenerationelectrodesurfaceprocessing Wefabricatedfourpairsoflarge-grainniobiumelectrodesandtwopairsofgrade-2ti- taniumelectrodesintwoseparatebatches.Surfacetreatmentproceduresforeachelec- trodepairarecataloguedinTable3.1(batches2and3).Forthisphaseofthehighvoltage development,ourgoalwastoincreasetheelectrodedstrengthfrom6.5kV/mmto 15kV/mmorbetter. 69 (a) (b) (c) Figure3.9:Electrodehighpressurerinseequipment.(a)Theelectrodesare mountedonanacryliccylindricalshellcenteredonaturntable.Astheap- paratusrotates,aconcentrichighpressurerinse`wand'rinsestheelectrodes. (b)Theelectrodesaremountedsothattheprimarysurfacesfacethewand. (c)Weswitchedtoarinsegunbecausethewaterqualitywasbetter. Withthisinmind,weusedprocessingproceduresinformedbydiscussionswithJef- fersonLabacceleratorphysicistsandareviewoftheliterature.Thegeneralrecipeisto mechanicallypolish,chemicalpolish,thenhigh-pressurerinse(HPR)theelectrodes. ChemicalpolishingandHPRwithultrapurewater(UPW)tlyimproveselectric dstrengthandstability[102,114,125,126].Foranoverviewofchemicalpolishing, includingelectropolishingandbu eredchemicalpolishing(BCP),wereferthereaderto [127,128,129,130,131,132,133].Comparingthemachineddimensionstoopticalmea- surementsofthepolishedelectrodes(discussedinSection3.4.2),Ithatmechanical andchemicalpolishingreducesourelectrodedimensionsbyapproximately100 m. Recently,centrifugalbarrelpolishinghasbeenshowntoreducetherequiredcondi- tioningtimecomparedtochemicaletching[134].Thisisanencouragingprospectfor conditioningRaEDMelectrodestotlyhigherdsinafuturephaseofdevel- opment. Thefourtitaniumelectrodes(Ti 1 ,Ti 2 ,Ti 3 ,andTi 4 )weremechanicallypolishedwith siliconcarbideafterfabrication.Theirmeansurfaceroughnessaveragesweremeasured intherange16Œ23nmusingaometer(MicroXAM)inacleanroom.Weelectropol- 70 (a) (b) (c) Figure3.10:Electrodestorageandtransport.(a)Eachelectrodepairismounted fromthebaseinastainless-steelbin.(b)Theelectrodesarelabeledbyetching thematerialandelectrodenumberontheoutsideofthebin.(c)Werecommend bucklinguptheelectrodesforcartrips. ishedpairTi 13 commerciallyandremeasuredtheelectrodesurfaces.Weobservedan increaseinthesurfaceroughnessoftheelectropolishedtitaniumelectrodesby ˇ 50% andmicroprotrusionsintherange1Œ10 m. 3.3.2Cleanroomsandhighpressurerinsing Wedevelopedourcleanroomtechniquebybuildingacleanroominthelabandinstalling apairofelectrodes.TheSpinlabportablecleanroomisshowninFigure3.8a.TheHEPA terissafelysecuredoverhead.Itaped2milpolyethylenesheetinginapleatfashion. ThenIusedPVCpipestoframea5 0 4 0 areaundertheter,overwhichthesheetingwas draped.IvalidatedthecleanroomtoClass100withaNIST-calibratedparticledetector. WedecontaminatetheelectrodesincleanroomsattheFacilityforRareIsotopeBeams (FRIB)afterpolishing.Theelectrodesarecleanedwithdetergentandrinsedwithpure waterinanultrasonicbathinastagingarea.Theyarerinsedinasecondultrasonicbath withUPWinsideaClass100cleanroom.Theelectrodesarethenhighpressure-rinsed withUPWat1200psifortwentyminutes. 71 Table3.3:Surfacedecontaminationcomparison. P =rinsepressure, T =rinsetime,CR=cleanroom,RR=rinseresistivity. Lab PT RRCRRef. (psi)(min)(M cm)(Class) CERN15003018100[127] JLab120020 > 18-[102] KEK1100580100[114] MSU12002018.1100Thiswork WeusedtwoHPRmethods,showninFigure3.9.First,werinsedtheelectrodessi- multaneouslywithaturntableandhighpressurerinsewand.Whenwelaterrinseda repolishedniobiumelectrodepair(Nb 23 ),thewaterqualityoftheturntablesetupde- graded.WeinsteadusedahighpressurerinsegunasshowninFigure3.9c.AfterHPR, theelectrodesdryinthecleanroomforseveraldaysbeforebeingsealedinpolytubing withdry,terednitrogen.AsummaryofcleanroomandHPRparameters fromseveralhigh-gradientdevelopmentgroupsisgiveninTable3.3. Thecleanedelectrodesrestineitherinthehighvoltageteststationorinasealed container,asshowninFigure3.10.Theelectrodeshaveapproximately1flclearancefrom thecontainerwallsonthesideandbottom,and2flbelowtheupperedge.Forstorage andtransport,theyaresealedintwolayersofcleanroomtubingandarewith drynitrogen.Thenitrogenisteredatthepointofusebya0.2 mmembrane ter. 3.4ElectrodeDischarge-Conditioning 3.4.1Highvoltageteststation AschematicoftheMSUhighvoltageteststationisshowninFigure3.11.Electrodepairs aremountedtoapolyetheretherketone(PEEK)holderinsideasix-waycrossvacuum chamber.Toestimatethesteady-stateleakagecurrentowingthroughtheholderitself, wemodelthefourfiarmsflasresistorsinparallelwithlength ` =2 1 : 6+0 : 1=3 : 3cm 72 Figure3.11:MSUHVtestapparatus. 1 9699334AgilentTurbo-Vvibra- tiondamper 2 Pfei erHiPace80turbomolecularpumpwithforelineEdwards nXDS10iA736-01-983dryscrollroughpumpandtwovalves 3 Matheson6190 Series0.01 mmembraneterandpurgeport 4 Ceramtec30kV16729-03- CFfeedthrough 5 0 : 312in : 2 electrodesinPEEKholder(resistivity10 16 M cm) 6 20AWGKapton-insulated,gold-platedcopperwire 7 MKS392502-2-YG-Tall-range conductron/iongauge 8 Shieldedprotectioncircuit:LittelfuseSA5.0Atransientvoltage suppressor,EPCOSEX-75Xgasdischargetube,Ohmite90J100E100 resistorinseries withKeithley64822-channelpicoammeter 9 OhmiteMOX94021006FVE100M re- sistorsinserieswithAppliedKilovoltsHP030RIP020HV. andcross-sectionalarea A =(1 : 27cm) 2 .FromOhm'slaw,thesteady-stateleakagecurrent I [ A ] is: I = V R =4 VA ˆ` ,(3.1) where V [ V ] istheappliedvoltageand ˆ =10 16 cmisthePEEKholderresistivity.The factorof4comesfromtheequivalentresistanceoftheparallelleakagepaths.Withan appliedvoltageof30kV,Iestimate I max ˇ 6pA. Thevacuumchamberismaintainedat10 7 Torrwithaturbomolecularpump(Pfeif- ferHipace80).Atthispressurethemeanfreepathofresidualgasmoleculesisovera meter,tlylargerthanthedimensionsofthechamber.TheRaEDMapparatus typicallyoperatesatultrahighvacuum(UHV)pressures( < 10 11 Torr)intheregionof theelectrodesandtrappedatoms[48].Theteststationdoesnotinvolveanytrapping ofatomsandsoweonlyrequireapressurelowenoughsuchthattheatmosphericcon- 73 stituentsdonotcollideonalengthscaleclosetoourgapsizeofafewmillimeters.The meanfreepathofanatomormolecule [ m ] isgivenby[135]: = 1 ˙n (3.2) = k B T ˙ 1 P ,(3.3) where ˙ h m 2 i isthecollisionalcrosssection, k B [ J = K ] istheBoltzmannconstant, T [ K ] isthetemperatureinthevacuumchamber, P [ Pa ] isthepressure,and n h m 3 i = P= ( k B T )isthenumberdensityfornon-interactingparticles,i.e.inthe limitoftheidealgasequation. Foranoxygenmoleculewithacrosssectionof ˇ 5 10 20 m 2 [136]andavacuum pressureof ˇ 5 10 5 Torr,themeanfreepathisoverameter.Witharoughingpump andturbomolecularpump(TMP)wetypicallyoperateatpressures ˇ 2 10 7 Torr,well belowminimumrequirements. Theteststationisfrequentlybroughttoatmosphericpressureforupgradesandelec- trodeinstallations.WeperformthisworkincleanroomsthatarevalidatedtoClass100or betterwithaNIST-calibratedparticlecounter(LighthouseHandheld3016).Thechamber iswithdry,high-puritynitrogenthrougha0.01microngasmembraneparticle ter(Matheson6190Series)whileventingthechamberandaftercleanroomoperations. Duringinitialevacuationthepumprateiscontrolledat1Torr/swithforelinevalvesto reducetheriskofdisturbingvacuumchambersurfaces. Weusepolishedcoronaballconnectionsinsideandoutsidethetestchambertomin- imizedischargeriskbeyondtheelectrodegapregion.Thepowersupply(AppliedKilo- voltsHP030RIP020)andfeedbackresistorsaremountedinsideagroundedhighvoltage 74 cage.Thefeedthroughsareenclosedbygroundedfisoupcanflstyleshieldsthatcanbe oodedwithdrynitrogentoreducehumidity. Weusea2-channelpicoammeter(Keithley6482)tomeasurethecurrentowingbe- tweentheelectrodes.Onechannelisnotconnectedandisusedtotrackcorrelateddrifts betweenthechannels.Aprotectioncircuitbetweentheelectrodeandpicoammetersup- presseshigh-powertransientsthatcoulddamagethepicoammeter.Typicaldischarges betweentheelectrodesdonottriggertheprotectioncircuit.Wecalibratedthepicoam- meterwiththeprotectioncircuittowithin10pA. 3.4.2Opticalmeasurementsofelectrodesandgapsizes TheRaEDMexperimentrequiresagap-measuringprecisionof0.1mmorbetter.Idevel- opedanimagingsystemtomeasureelectrodedimensionsandgapsizeswithoutmaking contactwiththeelectrode.ThesystemusesaCMOScameraandbi-telecentricmachine lens(ThorlabsMVTC23024).AschematicoftheopticalimagingsystemisshowninFig- ure3.12. Totesttheelectrodesatdi erentgapsizes,weadjustthegapsize insitu bytranslating thebottomelectrodeverticallywithahigh-precisionlineardrive(MDC660002).The assemblyisshowninFigure3.13a. Icalibratedtheopticalsystembyimagingthegapsizeoverarangeoflengthscor- respondingtodrivepositions.Thegapsizeismeasuredinpixelsandconvertedtomil- limeters.AplotofonegapcalibrationmeasurementisshowninFigure3.13b.Theo set parameterisrelatedtotheinitialgapsizeandcanvarybetweencalibrationsifthelinear drivedirectionisreversed.Wecalibratedtheopticalimagingsystemusingthelinear drivetoagap-measuringprecisionof1%ofitspixel-conversiontionof19.8 m/px. Itestedelectrodeperformanceovergapsizesranging0.4Œ2.5mmbeforeremoving thelineardriveandstandardizingthegapsizeto1mmfortesting.Thisgapsizewas 75 Figure3.12:TheimagingcomponentsoftheHVapparatus.Thisisaviewof theapparatusafterrotatingtheschematicinFigure3.11by90 andremovingnon- imagingcomponents. 1 worm-driverailmount 2 ThorlabsMVTC23024 tion(M)=0.243,4.06flworkingdistance(WD)telecentriclens 3 EdmundOpticsEO- 2323monochromeCMOScamera,4 : 8 msquarepixels 4 AdjustableElectrodeGap Assembly:MDC660002linearmotion0.001flgraduated,1fltraveladjustabledriveand customPEEKmountinterfacewithangularadjustment. chosentoensureelectricduniformityoverthelengthscaleoftheatomcloudand avoidaccidentalheatingoftheelectrodesbytrappinglasers.TheODTbeamdiameter constrainstheatomclouddiameterto100 m. WeusedtheopticalimagingsystemelectrodemeasurementstofabricatetheEDM electrodeholderfortheniobiumelectrodepairNb 56 ,showninshowninFigure2.4.The EDMelectrodeholderisdesignedfora1 : 0 0 : 1mmgapsize. 3.4.3Dataacquisitionandteringsettings Acompletedescriptionofacquisitionandteringsettingsusedforeachtestedpairof electrodesisgiveninTable3.4.Werecordthepowersupplycurrent,powersupplyvolt- 76 (a) (b) Figure3.13:(a)Thelinearadjustableelectrodegapassembly(b)Aweightedline istoascatterplotofgapsizevs.drivepositionandaconversionfrompixels toinchesisdetermined. age,vacuumpressure,leakagecurrent,androughpumpforelinepressurewitha16-bit, 250kS/sdataacquisitiondevice(NIDAQUSB-6218)connectedtoano cemodeldesk- topPC.Eachinputchannelusesa 10Vrange.Theanalogsignalsaredigitallytered toremove60Hzoutletnoiseandmechanicalvibrationsfromthevacuumpumps.We initiallysampleddataat16kHzbutlaterincreasedthesamplerateto30kHzafterup- gradingtheRAMandharddiskoftheDAQPC.Themeanandstandarddeviationfor eachrecordeddatapointiscalculatedfrom8192samples. Weremovedtheoutletnoisetersafterconditioningseveralpairsofelectrodesbe- causetheyintroducedshapesinthesignalwaveform.Comparingtheleakage currentdataofelectrodepairswithdi erentteringsettings,wefoundthatthedig- italtersdidnottlya ectthedistributionofthedatasetdiscussedinSec- tion3.4.4.AsIwilldiscussinSection3.4.10,wearesensitivetoabsolutecurrentsas smallas ˙ ˇ 25pAwiththeacquisitionsettingsdescribedinTable3.4. 77 Table3.4:5 ˙ Dataacquisitionandteringsettings.Usedtersindicatedby incircles.SR=samplerate. DAQDigitalters pairSRsamples25Œ35Hz55Œ65Hz109Œ113Hz115-125Hz7.5kHz (kHz) pointbandstopbandstopbandstopbandstoplowpass Nb 56 168192 Nb 78 168192 Ti 13 308192 Nb 23 308192 3.4.4Identifyingelectrodedischarges Weaverage8192samplesatsampleratesof16kHzforelectrodepairsNb 56 ,Nb 78 and 30kHzforTi 13 ,Nb 23 ,correspondingtoarecordeddatapointevery512msand273ms, respectively.Sinceeachhighvoltagemagnitudeandpolaritysettinglastsforatimepe- riodof60seconds,thiscorrespondstoabout120samplescollectedpertimeperiodat 16kHzand220samplescollectedpertimeperiodat30kHz. Thedistributionofeach60stimeperiodinaconditioningshiftismodeledasaGaus- siandistribution.AnexampleofaGaussiantoone60stimeperioddistributionis showninFigure3.14.Theisreasonableanddischargesareclearlydistinguishedfrom therestofthedata. IafidischargeflasanystandarddeviationexceedingtheGaussian5 ˙ threshold abovethemeanvalue ¯ x .Wetypicallyobservedischargesizesintherange100Œ1000pA, butitisnotuncommontoobservelargerdischargesizesaround1Œ10nA. Toestimatedischargemagnitudes,Ireportthemedianvalueofthestandarddevia- tionsineach60stimeperiod.Weexpecttoseehighratesofdischargesduringcondi- tioning.Smalldischargesoccurringatastablerateareanddonotdamagethe electrodesurfaces. Ourdischargecountingmethoddoesnotexcludedischargesthatcouldoccuratan- otherpartoftheteststation,forexamplethehighvoltagefeedthroughs.Therefore,we 78 Figure3.14:Nb 23 at+22kV/mmoverone60secondcycleduringthehour ofconditioning.FromtheGaussian(solidredline)wedeterminethemeanto be ¯ x ˙ =106 : 7 3 : 6pA.Thereare207totaldatapoints.Weiden7events exceedingthe ¯ x +5 ˙ =125pAthresholdasdischarges. expectourreporteddischargeratesareconservativeoverestimatesofthetrueelectrode dischargerate. Additionally,wecalculatethesteady-stateleakagecurrent ¯ I byaGaussianto thesamplemeans,ratherthanthestandarddeviations,ofeach60stimeperiod.Elec- tronico setsanddriftsareremovedbysubtractingthemeancurrentmeasuredbythe picoammeterduringthezerovoltagetimeperiodsneighboringeachhighvoltagetime period.Anexampleof ¯ I measurementsmeasuredoverafour-hourconditioningshiftfor Nb 56 isshowninFigure3.15. Sincetheintegrationtimeforeachdatapointat30kHzis273ms, ¯ I isinsensitive todischarges,whichtypicallylast ˇ 1ms.Toillustrate,wecancomparethedischarges idenbythemeandataandthestandarddeviationinthethirdhourofa26.2kV conditioningshiftofTi 13 .Histogramsofthedischargesidenbythestandardde- viationsareshowninFigure3.16.Icountapolarity-combined111dischargeswiththe standarddeviationdataoveronehour.Usingasimilaranalysiswiththesamplemeans,I 79 Figure3.15:Theo set-subtractedaverageleakagecurrentsforeachpositive andnegativehighvoltagetimeperiodduringconditioningat20kVwithNb 56 atagapsizeof1mm. countonly11eventsoutsideofthe5 ˙ thresholdduringthesametimeperiod. Myanalysiscodemodelstheleakagecurrentandcalculatesthediscussedperfor- mancemetrics.TheaccuracyoftheanalysiswasindependentlyvCodeanddata availabilitymaybefoundinAppendixB. 3.4.5Discharge-conditioningprocedure Ourgoalismaximizetheelectricdstrengthwhileminimizingthedischargerateand dischargesize.WeconditiontheelectrodesatDCvoltage,alternatingthevoltagepolarity every60s.Thevoltageisappliedtothetopelectrode.Theperiodicvoltagewaveform ischosentosimulatetheEDMmeasurementandismorechallengingtostabilizethan holdingo astaticunipolard.Eachconditioningshiftatonehighvoltagemagnitude 80 (a) (b) Figure3.16:Histogramsofallthedischargesforbothpolaritiesduringthethird hourofconditioningthetitaniumelectrodesonalog-logscale. withthisperiodicwaveformtypicallylasts3Œ5hours 1 . Intheconditioningphasewevalidatetheelectrodesatsomefractionofthemax- imumvoltagetoreducethedischargerate.Thevalidationvoltageistypically80Œ95%of themaximumtestedvoltage,consistentwiththeliterature[100,113]. Ishowtheaveragedischargeratesanddischargesizesforbothpolaritiesforallthe electrodesinFigures3.17,3.20,3.21,and3.22.Ia`baseline'valueastheaverage dischargerateanddischargesizeduringtheconditioningshift.Eachofthese hasabaselinevaluethatissubtractedfromthedata. InSections3.4.6,3.4.7,3.4.8,and3.4.9,Iwilldiscussthedischarge-conditioningre- sultsofeachelectrodepair.InSection3.4.10,Iwillcomparetheoverallperformanceof theelectrodes. 1 Wefoundthatthehighestdischargeratestendedtooccurduringthesecondand thirdhoursofashift,soweschedulede-hourshiftswhenpossible. 81 Figure3.17:Discharge-conditioningtimelineforNb 56 ata1mmgapsize. 3.4.6ConditioningresultsforelectrodepairNb 56 Theaveragedischargerateoverthecourseofconditioningtheniobiumelectrodepair Nb 56 isshownintheupperpanelsofFigure3.17.Ateachvoltage,thedischargerates, expressedindischargesperhour(dph),tendtodecreaseaswecondition.Thereisa step-likeincreaseindischargerateswhenthevoltageisincreased.Nb 56 wasvalidated at20kV/1mmwithanaveragedischargerateof98 19dphafterapproximatelythirty hoursofconditioning. Atnegativepolarity,thedischargerateincreasesmoreslowlywitheachvoltagestep. However,theoverallcurvedoesnotttenataminimumcountrateasitdoesatpositive polarity.Thissuggeststhatadditionalconditioningcouldfurthersuppressdischargesat negativepolarity.It'salsopossiblethattheteststationdesignfacilitatesahigherdis- chargerateatnegativepolarity.Wewillexplorethisinthenearfuturebyconducting 82 (a) (b) (c) Figure3.18:InstallationofniobiumelectrodepairNb 56 inRaEDMapparatus. (a)ANLportablecleanroomwithaluminumbeams,plasticdrapes,anda4 0 2 0 HEPAter.(b)Theborosilicateglasstubewascleanedwithaclean-roomgrade wipewrappedaroundtheendofapole.(c)Thecleanroomwasposi- tionedovertheelectrodeentrypointbeforeinstallingtheelectrodes(seeninthe bottomcorner). conditioningtestswhiletheelectrodesareremovedfromtheteststation. Nb 56 dischargesizesareshowninthelowerpanelsofFigure3.17.Aswewillsee withallthedischargeplots,thedischargesizebehaviordoesnotscalewiththedischarge rate.Thelargestmediandischargesizeoverthecourseofconditioningis60pA,which isrelativelysmallcomparedtothetypicaldischargesizesoftheotherelectrodepairs. Inthelasthourofconditioningthedischargesizesare20pAsmallerthanthestarting dischargesizes. AsmentionedinSection3.1.1,thelegacycopperelectrodeswereconditionedto 10kV/mmbutcouldonlybeoperatedat6.5kV/mmafterinstallingthemintheRaEDM apparatus.Forthesecondgenerationelectrodes,Imadetwomajorimprovementstothe techniquetopreventasimilarreductionindstrength.First,ourelectrodesarenow preservedinClass100(ISO5)orbettercleanroomenvironmentsduringbothcondi- tioningandtransportasdescribedinSections3.3.1and3.4.1.Second,weusedthenew, rigorousdischarge-conditioninganalysisdescribedSection3.4.4forNb 56 andtheelec- 83 Figure3.19:AschematicofthewaterbakeoftheRaEDMexperimentalappa- ratusfollowingtheinstallationofthenewelectrodepair. trodesdiscussedinthesubsequentsections. Nb 56 wasinstalledintheRaEDMapparatusonMay15,2018intheconditionsshown inFigure3.18.DetailswillbediscussedinSection3.4.13. Followinginstallation,IfibakedfltheEDMapparatusbyheatingtheborosilicatetube andhighvoltagevacuumcomponentsat120 Cforthreedaysandthenincreasedthe temperatureto150 Cforapproximatelyeweeks.Thistemperatureishighenough toremovemoisturethatwasintroducedduringtheelectrodeinstallation.Thevacuum pressuredroppedbymorethanthreeordersofmagnitudeoverthecourseofthewater bake. AschematicofthewaterbakeisshowninFigure3.19.Toequalizethetemperature throughoutthesystem,Iusedheatertape,foil,andseveralheatguns.Imonitoredthe temperaturewiththermocouples. Afterthewaterbake,IrevalidatedelectrodesNb 56 at20kV/mm.Thiselectrodepair willbeusedforupcomingsecondgenerationEDMmeasurementsandisexpectedafactor 84 Figure3.20:Discharge-conditioningtimelineforNb 78 witha1mmgapsize. of3.1improvementtoourEDMmeasurementsensitivity. 3.4.7ConditioningresultsforelectrodepairNb 78 DischargeratesandsizesforthesecondpairofniobiumelectrodesNb 78 aregiveninFig- ure3.20.WestartedconditioningNb 78 at12kV/1mm,thesameelectricdasNb 56 . Theinitialdischargeratesareoccasionallyinexcessof1000dph,oraboutonceevery threesecondsforseveralhourswithdischargesizesof50pA.Thehighdischargerate coupledwithlowdischargesizeisanindicationthatweareoperatingatanidealvoltage fordischarge-conditioning.Duringthelast10hoursofconditioningthedischargerates decreasetolessthantheinitialrates.Theconditioningshiftwasperformedat17.8 kV/mm. TheseelectrodeswerepackagedaccordingtoourproceduredescribedinSection3.3.2 85 Figure3.21:Discharge-conditioningtimelineforTi 13 ata0.9mmgapsize. andshippedtoUSTC,wheretheyarebeingusedinanewytterbiumEDMmeasurement. 3.4.8ConditioningresultsforelectrodepairTi 13 WechangedourdataacquisitionanddigitaltersettingsforTi 13 andthepairthat wewilldiscussinSection3.4.9(seeTable3.4).Toreachelectricdshigherthan 20kV/mm,weconditionedthetitaniumelectrodesfor110hours,fourtimeslongerthan thepreviouspairs. DischargeratesandsizesforthetitaniumelectrodesareshowninFigure3.21.We startedconditioningtheelectrodesat14.9kV/0.9mm=16.5kV/mm.Theinitialdis- chargesizesareapproximately100pA,tlyhigherthanNb 56 andNb 78 .The dischargeratesdidnotconsistentlydecreaseoverthecourseofseveralshiftsat19.4kV. Athour12,wereducedthevoltageto0.7kVforoneshifttoverifythatthedischarge 86 ratesdecreasebeforeresumingtestingathighervoltages. Thedischargerateincreasesfrom290dphto5550dphwhensteppingthevoltage from 26 : 2kVto 27 : 6kV.Thisstep-like`switchingon'ofleakageemissionsitesiscon- sistentwithourexpectations,giventhephysicalpictureofconditioningwedescribein Section2.6.Inprinciple,theemissionsites,whichmaybethoughtofasmicroprotru- sions,areablatedafterspendingsu cienttimedischarge-conditioningtheelectrodes. Thefactorsuencingtherequiredamountoftimeincludethesmoothnessofthehigh- gradientsurfaces,thegapsize,andtheappliedvoltage.Wewereunabletotly reducethedischargeratesat27.6kV/0.9mm=30.7kV/mmdespitemorethantwenty hoursofconditioningatthatdstrength. Duringtheshift,wereducedthevoltageto14.7kV/0.9mm=16.3kV/mmand againobservedthedischargeratesreturningtothebaseline.Ti 13 canlikelybecondi- tionedtoperformstablyat ˇ 24kV,or85%ofthemaximumappliedvoltagewithaddi- tionalconditioning.However,theconcentrationofmagneticimpuritiesinourtitanium electrodes(showninTable3.2)islikelytoohightobeusedforanEDMmeasurement. 3.4.9ConditioningresultsforelectrodepairNb 23 IinitiallytestedNb 23 ata0.4mmgapwithdsashighas+52 : 5kV/mmand 51 : 5kV/mmusingthetraditionalhold-o orficurrent-conditioningflmethod[100]. ThenIdischarge-conditionedtheelectrodeswiththeperiodicwaveformdescribedinSec- tion3.4.4to27.5kV/mm.However,alargedischargeof ˇ 100nAduringa30kV/mm conditioningshifttriggeredacurrentavalanchethatrapidlyincreasedtheleakagecur- rentanddamagedtheelectrodes. Iwasunabletorecovermeaningfulperformanceafterseveralconditioningattempts. Theelectrodeswereremovedfromtheteststationandrepolishedthesurfaceaccording toTable3.1.ThenIreinstalledNb 23 andrepeatedtheconditioningata1mmgap. RepolishedNb 23 dischargeratesandsizesareshowninFigure3.22.Theratesstay 87 Figure3.22:Discharge-conditioningtimelineforNb 23 ata1mmgapsize. nearthebaseline,about200dphforbothpolaritiesupto20kV.Whenweincreasedthe voltagefrom20kVto22kV,thedischargeratesatnegativepolaritybecomeashighas 3000dph(aboutonceeverysecond).Thedischargesizeswerelow,lessthan500pA,sowe continuedconditioningatthisvoltage.Despiteconditioningtheelectrodesat22kV/mm formorethantwentyhours,thedischargerateremainedhigh.Iexpectthatreducingthe voltageby ˇ 1kVwillrestorethebaselinedischargerate. Forthispairofelectrodes,repolishingandreconditioningallowedustorecover80% oftheoriginalelectricdperformance. 3.4.10Comparisonofoverallelectrodeperformance Table3.5comparestheelectricdstestedanddischargeratesobservedforallofthe conditionedelectrodepairsaveragedoverbothpolarities. E i istheelectricdstrength 88 Table3.5:Electrodeconditioningsummary. E i =initial dstrength. E max =maxdstrength. E f =vali- dateddstrength. R i ( R f )=initialdischargerate. ¯ I =steady-statecurrentat E f . pair E i E max E f E f E i R f R i ¯ I (kV/mm)(kV/mm)(kV/mm)(pA) Nb 56 11.919.819.81.71.6 < 10 Nb 78 12.017.917.91.50.9 < 10 Ti 13 19.832.329.11.52.2 < 30 Nb 23 12.022.021.91.81.3 < 25 atthestartoftheconditioning,while E max istheelectricdstrengthattheendofthe conditioning. E f isthevalidateddstrength,andmaybereducedto ˇ 85%of E max formorestableperformance. Theinitialelectricdis12kV/mmfortheniobiumelectrodesand16.5kV/mm forthetitaniumelectrodepair.Ichosethesedstrengthsbecausethedischargerate anddischargesizewassu cientlylow.Themaximumandelectricdsdepended entirelyontheperformanceoftheelectrodes. Alltheelectrodesweredischarge-conditionedtohigherthanouroriginalgoalof 15kV/mm.Thesteady-statecurrents ¯ I at E f areundertheEDMexperimentalthreshold of100pA. Onemetricforcharacterizingdischarge-conditioningistocomparetheelectricd strengthratio E f =E i versusthedischargerateratio R f =R i .Myexpectationforsuccessful conditioningisthattheelectricdscalingshouldbecomparable,andideallylarger thanthedischargerateratio.Wewereabletoscaletheelectricdmorequicklythan thedischargeratesforalltheniobiumelectrodes.ForNb 78 ,thepolarity-averaged dischargerateswerelowerthantheinitialdischargerates.AsdiscussedinSection3.4.8, theTi 13 dischargerateoutpacedtheelectricdstrengthincreaseat30.7kV/mm. Ofparticularnoteisthepolaritydependenceoftheelectrodedischargerates.Inall casesexceptforNb 23 ,thenegativepolaritydischargeratesaretlyhigherthan 89 Figure3.23:Aplotofelectricdsreachedbyelectrodepairs.Bluedataare electrodesusedintheRaEDMapparatus.Greendataareelectrongunelec- trodestestedwitha 100kVpowersupply[114].Reddataareelectrodestested atMSU.Brighter,moreintensecolorsaremorerecentresults. thedischargeratesatpositivepolarity.Polarity-correlateddischargeratescouldbeacon- sequenceoftheasymmetryofoursetup,asillustratedbyFigure3.11.Ourgoalforthe nextphaseinhighvoltagedevelopmentistodesignamoresymmetricsetupthatwillal- ternatetheroleofgroundedandchargedelectrode.Inadditiontoincreasingtheelectric dstrengthcapability,weexpectthenewsetuptoaidtheinvestigationofpolarity- correlateddischargerates. 3.4.11Comparisonofelectrodeperformancewithothersystems Acomparisonofthemaximumstableelectricdperformanceofelectrodepairspre- paredatANL[65,48],theHighEnergyAcceleratorResearchOrganization(KEK)[114], andMSUisshowninFigure3.23.Wetestedamaximumelectricdof+52 : 5kV/mm 90 Figure3.24:Weightedaveragesofthesteady-stateleakagecurrentonlinearand logscales.Errorsareontheorderof0.1pA. and 51 : 5kV/mmwithoneofourniobiumpairsofelectrodes(Nb 23 )atagapsizeof 0.4mm.TheelectrodeconditioningresultsdescribedbyFurutaet.alareauseful,if imperfectcomparison.Theirelectrodesizeisandelectrodegapsizeissimilar,butthey useanasymmetricelectrodegeometryandaunipolarpowersupply.TheKEKgroup pairsastainlesssteelsphericalanodewithfibutton-shapedflcathodesmadefrom high-purity316Lstainlesssteel(fiClean-Zfl),titanium,andmolybdenum. ThedatainFigure3.23suggeststhatdsintherealmof100kV/mmareattainable 91 inprinciple.Electrodegeometry,high-gradientsurfacearea,highvoltagepolarity,and thenatureofthevoltagewaveformaresomeoftheparametersthatmaylimittheultimate performanceofourelectrodes. Atpresent,themainlimitingfactoroftheRaEDMelectricdstrengthisthe30kV maximumoutputofthehighvoltagepowersupply. 3.4.12Steady-stateleakagecurrentanalysis Weplottheweightedaveragesteady-stateleakagecurrentforeachappliedvoltageforall theelectrodesinFigure3.24.Eachdatapointrepresentstheaverageofonepolarityat onevoltagesettingoveranentireshift.Forexample,Figure3.15showsthefirawdatafl that'saveragedinto 20kVdatapointsforNb 56 andplottedinFigure3.24. InSection3.4.4,Istatedthatthesteady-stateleakagecurrentiscalculatedbysub- tractingthesignalmeasuredduringthezero-voltagetimeperiodsthatoccurbetween eachhighvoltagetimeperiod.Thiszero-voltagebackgroundsubtractionremovesthe instrumento setsandlineardrift,butdoesnotaccountforhigh-voltagecorrelatedsys- tematics. Thepowersupplyvoltagemonitorandcurrentmonitorsignalsscalewithvoltage magnitude.BecausewesampledataononeDAQcard,asmallamountofsignalfrom adjacentandnon-adjacentinputchannelscana ecteachother.Eachleakagecurrent value I fromFigure3.24canbewrittenasthetrueleakagecurrent I (0) pluscorrelated systematics: I (0) + I VMON ( V )+ I IMON ( V ),(3.4) where I VMON ( V )and I VMON ( V )arevoltagemagnitude-correlatedsystematics. Iestimatethevoltagemagnitude-correlatedsystematicsassuminganappliedhigh voltageof30kV.Atthisvoltage,thesignalsizeofthevoltagemonitor V VMON ˇ 10V and V IMON ˇ 0 : 4V.Thecurrentmonitorchannelisadjacenttotheleakagecurrentsignal andthevoltagemonitorsignalisnon-adjacent.FromtheDAQdatasheet(NIDAQUSB- 92 6218),Iassumetheadjacentchannelcrosstalkis 75dBandthenon-adjacentcrosstalk is 95dB.Thisgivesatotalcrosstalksignalof V CT ˇ ( +0 : 4V ) 10 (75 = 20) + ( 10V ) 10 (95 = 20) ˇ +0 : 3mV Thepicoammeterrangeis200nAandtheanalogvoltageoutputisinvertingona10V scale,sothecrosstalkvoltageshiftstheleakagecurrentsignalatbothhighvoltagepolar- itiesby +0 : 3mV ( 200nA ) = 10V= 6pA Themagnitudeofvoltage-correlatedsystematicsisontheorderofthesamplingresolu- tionofthe16-bitDAQcard.Thereisalsoapolarity-dependentcrosstalksystematic,but thise ectisnegligibleatthe ˇ 1pAlevel. Takinghigh-voltagecorrelatedsystematicsintoaccount,thesteady-stateleakagecur- rentsinFigure3.24aresubjecttosystematicuncertaintyatthe ˇ 10pAlevel. Thesteady-stateleakagecurrentversusvoltagetrendismodestlylinearwithanohmic resistanceof40kV = 10pA ˇ 10 16 .Weobservelargeleakagecurrents > 100pA,corre- latedwithhighdischargerates,forTi 13 andNb 23 beyond22kV.Inthemostextreme case,wemeasured ¯ I ˇ 670pAduringconditioningTi 13 at 27 : 6kV. Thesteady-stateleakagecurrentmustbelessthan100pAtoavoidsystematicsthat couldmimicanEDMsignalatourcurrentstatisticalsensitivity.Thiscriterionissimilar tometricsusedinotherelectrodedevelopmentgroups[102,114].Oursteady-stateleak- agecurrentsensitivityislimitedto ˇ 25pA.AttheNb 56 validateddstrengthof 20kV/mm,themeasuredsteady-stateleakagecurrentisbelowthisupperlimit. 3.4.13TransportationandinstallationofelectrodesinRaEDMapparatus MovingtheNb 56 electrodesfromtheMSUhighvoltageteststationtotheANLEDM apparatusposedthehighestriskofsurfacecontamination.Beforetransport,Ipackaged theminaClass100cleanroomattheNationalSuperconductingCyclotronLaboratory 93 (NSCL).Theelectrodesweremountedinacleanedstainlesssteelcontainerbytheirbase tominimizetheirriskofcontactwithanysurface,asseeninFigure3.1.Idouble-bagged andsealedthecontainerandelectrodesincleanroomtubingandwithparticle- tereddrynitrogen. ItransportedtheelectrodesfromMSUtoANLinDecember2017. StartinginFebruary2018,IconstructedacustomportablecleanroomatANL.Iused aNIST-calibratedparticlecountertoverifythatthecleanroomwaswithinClass100 limits.TheportablecleanroomisshowninFigure3.18a.OnMay13,2018,Iunpacked Nb 56 intheportablecleanroomandassembledinanewMacorholderdesignedforan electrodegapof1mm.IpackagedtheelectrodesandMacorholderinpoly tubing,asbefore. OnMay15,2018,theRaEDMsciencechamberfrontendwasdisassembled(shown inFigure3.18b)andthelegacycopperelectrodeswereremoved. Theportablecleanroomwaspositionedoverthesciencechamberopening.Theclean roomandsciencechamberwerecleanedandvalidatedatClass100standard.Aview frominsidethecleanroomisshowninFigure3.18c.Finally,theelectrodeswereremoved fromtheirpackagingandinstalledinthesciencechamberonMay15,2018. TheEDMapparatuswasvacuumpumpedandwaterbakedasdescribedinSection3.4.6. OnJuly22,2018,IrevalidatedtheNb 56 electrodeperformanceto20kV/mm. 94 CHAPTER4 RADIUMBRANCHINGRATIOS IworkedatArgonneNationalLaboratory(ANL)fromFebruary2018throughAugust 2018.DuringthattimeIworkedontheupgradeforthelongitudinalatomslower(fiBlue slowerfl)project.I'mthethirdauthoronthepublicationdescribingouruorescence measurementsbranchingfractionmeasurements[90]. First,I'lldescribetheBlueSlowerprojectinthecontextoftheRaEDMexperiment. ThenI'lldescribetheexperimentalsetupforthemeasurements,includingthelasers needed.Thenwe'lldiscusstheuoroscopymeasurementsoftheatomictransitionsof interest.Finally,I'lltalkabouttheanalysisthatweusetotheintensityoftheBlue slowertransitions. 4.1RadiumlasercoolingwiththeZeemanslower TomeasureanEDM,weneedtotrapatomsbetweentwohighvoltageelectrodesto performspinprecessionfrequencymeasurements(seeFigure2.1).FromEquation2.3, thestatisticalsensitivityoftheEDMmeasurementsscalesas p N ,where N isthenumber ofatomsprecessingbetweentheelectrodes.Ourgoalistointerrogateasmanyatomsas possible,i.e.maximize N . Radiumatomsexitane usiveovenwithsomeangulardistribution j ( ),where istheanglefromthelongitudinalaxis,andvelocitydistribution g ( v ),where v [ m = s ] is thespeed.Afterradiumatomsexittheoven,theyarecollimatedwitha transverselasertoreducetheangularspread. Next,theatombeampropagatesthroughaZeemanslowingsection.Thedetailsofthe ZeemansloweraregiveninSection2.2.1.Ataperedsolenoidcoilaroundthebeamline Zeeman-shiftsthetransitionfrequencytocompensatefortheDopplere ect.Theresult isafractionofatomsthataresu cientlyslowedfortrapping. 95 Figure4.1:Left:thecurrentfiredflZeemanslowingscheme. R 1=1429nm. Right:theenvisionedfiblueflZeemanslowerupgrade. R 1=698nm, R 2= 712nm, R 3=2752nm. Wecurrentlyusethe 1 S 0 ! P 3 o 1 ,orfiredflcyclingtransitiontodeceleratetheradium atoms.Inthisscheme,showninside(a)Figure4.1,radiumatomsareexcitedto 3 P o 1 with aTi:Saphlaserat714nm.Theydecayto 3 D 1 withahalf-lifeof422(20)ns[137,138]. Tocircumventtherelativelylonglifetimeofthismetastablestate,anadditionallaseris usedtofirepumpflatomsto 1 P o 1 ,wheretheydecaytothegroundstateafterapproximately 5ns.Thisschemeissimple,requiringonlyasinglerepumplaser.Usingtheredcycling transition,wecanslowatomswithaninitialvelocityof 60m = s,orabout0.2%ofallthe atomsexitingtheoven.Themomentumofanyatomsexitingtheovenataspeedgreater than60m/sistoolargetosu cientlyslowfortrapping. ForthenextphaseoftheradiumZeemanslower,we'lluseanadditional,stronger cyclingtransitiontoslowdownalargerfractionoftheatomsexitingtheoven.The 1 S 0 ! P 1 o 1 ,orfiblueflcyclingtransition,deliversastrongermomentumkicktotheatom andcanbecycledabout80timesquickerthantheredcyclingtransition.Theblueslower upgradewillbeassembledupstreamoftheredslowerandcalibratedtoslowatomsto 96 Figure4.2:TheMaxwell-Boltzmannspeeddistributionofradiumatomsexiting theoven.Theestimatedfractionofatomsthatcanbesu cientlyslowedfor trappingareshadedaccordingtotheslowingscheme. 60m = s,sotheredslowercanbeusedwithorwithoutthebluecyclingtransition. TheMaxwell-Boltzmannspeeddistributionofradiumatomsexitinga500 Coven isshowninFigure4.2.Theshadedareaofthecurverepresentstherelativeamountof atomsthatcanbetrappedfortheredslower(shadedred)andtheblueslower(shaded blue).Theblueslowerupgradeisexpectedtotrapmorethan50%ofatomsexitingthe ovenandwillyieldapproximately100timesmoretrappableatomsthantheredslower alone. Theblueslowercyclingschemeismorecomplexandrequiresconsiderationofaddi- tionaldecaychannelsthantheredcyclingscheme.Once 1 P o 1 ispopulated,therearefour non-cyclingdeexcitationpathsthattheatomscantake.Electricdipole,orE1transitions ofthelowestatomicexcitedstatesofradiumareshowninFigure4.3.Newrepumplasers 97 arerequiredfordecaystateswithtbranchingfractions.Theyneedsu ciently highintensitytosaturateeachofthesetransitions.Thefractionalrateofatomsdeexcit- ingfrominitialstate j i i toonepossibledecaystate j k i isknownasthebranchingratio BR ( j i i ! j k i ) [ unitless ] : a BR ( j i i ! j k i ) = g k f ik 2 ik ˚ X ` g ` f i` 2 `k ,(4.1) where g 0 k [ unitless ] isthedegeneracyof j k i , f ik [ unitless ] istheoscillatorstrengthofthetransition j i i ! j k i ,ortheratioof powerabsorbedbytheatomtothatabsorbedbyaclassicaloscillator,and ik [ m ] isthetransitionwavelengthfrom j i i ! j k i . Thedecaystrengthsaresometimesexpressedintermsofthetransitionmatrixelement j D ik j ratherthan f ik intheorycalculations.Thetwoarerelatedby f ik / j D ik j 2 g i . E 1 E 1 E 1 -allowedatomictransitions: J =0 ; 1exceptforgs ! gstransitions M =0 ; 1exceptforgs ! gstransitionswhen J =0 oneelectronjumpwith ` = 1 LS LS LS coupling: S =0 L =0 ; 1exceptforgs ! gstransitions TheenvisionedrepumpingschemeisshowninFigure4.1.Oneofthepossiblenon- cyclingstates, 3 D 3 ,ispredictedtohaveasurprisinglyweakbranchingratio.Another state, 1 D 2 ,isnormallyaforbiddentransition(E1, S =0),butispredictedtohavea a Wearetechnicallycalculatingbranchingfractions,butthisisthenomenclatureused inourpaper.Thetruebranchingratioistheratioofonebranchingfractiontoanother branchingfraction. 98 Figure4.3:AnenergyleveldiagramofthelowestenergylevelsandE1- allowedtransitionsof 226 Ra.Measuredlifetimes:7 s 7 p P 3 o 1 [137],6 d 7 p F 3 o 2 [90], 7 s 6 d D 3 1 [139],7 s 6 d D 1 2 [140].Calculatedlifetimes:7 s 6 d D 3 2 [141],allother transitions[89].Wavelengthsarelabeledalongtransitionlinesin[nm]invac- uum/air. 99 Figure4.4:Aschematicofthebranchingratiouoroscopysetup.Inset:energy diagramformeasuringthe 3 D 1 branchingratio. favorablystrongbranchingratio[89].Thisisduetothetotalangularmomentum J = L + S couplingoftheparentstate 3 F o 2 .Thehighintensityof 1 D 2 willallowustoflthespin ofthe 3 F o 2 stateandrepumpalongthetransition 1 D 2 ! P 1 o 1 . Mygoalwastomeasurethe 1 P o 1 decaychannelsusinglaser-induceduoroscopyto experimentallyverifythepredictedbranchingratiosfortheBlueslowerupgrade.This wasnecessarybecauseimplementingtheblueslowingschemerequireslasersthatcan providesu cientpowerateachtransition.Therequisitepowerdependsonthebranch- ingfractionintensityof 1 D 2 andthe 3 D J states.Inthecaseof 3 D 3 ,thebranchingratiois predictedtobelowenoughthatwecanneglectrepumpingthatstatewithoutt atomloss. Tomeasuretheuorescencefromasignalstate,wepopulatealltheDstateswitha 483nmpumplaserresonantwiththe 1 S 0 ! 1 P 1 transition.Thenwedepopulateoneof thestrongdecaystates,either 3 D 1 or 3 D 2 ,withasecondprobelaser.Finally,wedetect theuorescencetothedecaychannelofinterestwithaPMTandanappropriatelychosen 100 Table4.1:Transitionsandwavelengthsforbranching ratiomeasurement. transitionwavelength(nm)laser 1 S 0 ! 1 P o 1 482.7254blueimaginglaser 1 D 2 ! 3 F o 2 912.6919NIRdiodelaser 3 D 1 ! 3 F o 2 698.2168tunableTi:Saphlaser 3 D 2 ! 3 F o 2 712.0438tunableTi:Saphlaser bandpasster.AschematicisshowninFigure4.4.Alistofthetransitionsandthelasers usedtoexcitethestateto 3 F o 2 isgiveninTable4.1.Bymeasuringtheuorescenceofthe transitionsusingallthepossibleations,wecanconstructasystemofequations thatallowsustosolveforindividualbranchingratios. 4.2Lasersforthebranchingratiomeasurement Thepumpingtransitionandthreeofthefourtransitionsofinterestwereaccessible withtwoexistingRaEDMlasers.Iusedtheimaging&polarizinglaser(Moglabsexternal cavitydiode)forthe 1 P o 1 transitionat483nm.IusedtheZeemanslowerlaser(Spectra- PhysicsMatissering-cavityTitanium:Sapphire)forthe 3 D 1 (698nm) 3 D 2 (712nm)and 3 D 3 (750nm)transitions. Iassembledadiodelaserfromtoprobe 1 D 2 .It'saTO-can300mW-ratedlaserdiode (ThorlabsM9-915-0300)activelycooledwithatemperaturemount,asshowninFig- ure4.5a.Thelightpassesthroughanopticalfree-spaceisolatorandananamorphicprism pair. Thelaserwavelengthistunedwithathermoelectrictemperaturecontroller(ILXLDT- 5412)andpoweredwithaprecisioncurrentsource(ILXLDX-3525).Iassembledacircuit thatinterfacesthethermoelectrictemperaturecontroller(TEC)andcurrentsourcetothe laserandconnectstheinstrumentstothelaboratorysafetyinterlock(Figure4.5b). Apicko feedslaserlightintoaspectrometer(OceanOpticsFLAME-VIS-NIR-ES).I usedthespectrometertocalibratethelaserwavelengthasafunctionofTECsetpoint.I 101 (a) (b) Figure4.5:(a)NIRlaserdiodeinatemperature-controlledmount.Duringu- oroscopymeasurements,thepowermeterisremovedandlaserlightiscoupled tothebehindit.(b)Left:CustomNIRinterfaceboxcircuit.Right:The currentsource,thermoelectrictemperaturecontroller,andcustominterfacebox usedfortheNIRlaserdiode. 102 Figure4.6:Aofthenear-infrared(NIR)diodelaserwavelengthtothetem- peraturecontrollerresistancesetting. plottedthewavelengthvs.setpointandthedatainFigure4.6.Thedataisreasonably modeledwithalinear 4.3Radiumuoroscopyexperimentalsetup Thelasersinteractwiththeradiumatomsjustoutsidetheoveninasix-wayvacuum cross.First,theatomstraversethe483nmpumpingbeamtopopulateallthesinglet andtripletDstates.Afterthepumpingbeam,oneofthestatesD 2 S +1 J ! F 3 o 2 isdriven withafiprobefllaser,whichistheNIRorTi:Saphdependingonthetransition. Tocollimatethebeamsandorientthemparalleltoeachother,theyareall- coupledtoasmallstageshowninFigure4.7.Isetthebeamdiameterswithlensespo- sitionedattheexits.Aseriesofdichroicmirrorscombinethebeamssothatthey areparallelandcloselygrouped.Aperiscopemirrordirectsthebeamabovetheview- port,whileamirrorsteersthemverticallydownthroughtheuorescenceregionap- 103 Figure4.7:Left:thethreearecombinedwithdichroicsandsenttothe uorescencemirrorwithatelescopemirrorsetup.Right:atop-downviewof thebluelaserlightpassingthroughtheviewportintotheuorescenceregion. proximately2maway(Figure4.7).An8mm 6mmphotomultipliertube(Hamamatsu R2949)ispositionedperpendiculartotheatomicbeamandlaseraxes.Acollectionlens focusestheuorescenceontothePMTsensor.Weplaceabandpassterappropriatefor thetransitionwavelengthofinterestbetweenthecollectionlensandPMT. 4.4Radiumuoroscopydataacquisition ThePMTcountsarerecordedbyaUSBdataacquisitioncard(DAQ)(NationalInstru- mentsUSB-6341)withanonboardtimereverysecond.Toscanthewavelengthovera transition,weuseasignalgeneratortosendawaveformtoanacousto-opticalmodulator (AOM).ThewaveformgeneratorfrequencyisalsosenttotheDAQ. IcreatedaLabViewprogramthatlogsthePMTcountsandAOMfrequencyasafunc- tionoftime.AscreenshotisshowninFigure4.8.Onthemaingraph,therawPMTcount isplottedwithauser N -sampleaverage.Thebottomgraphplotstheacousto- opticalmodulatorfrequencysetting.Theusercanrunalasersweepwithafrequency 104 Figure4.8:AscreenshotoftheVIIwroteforrecordingPMTcountsforthe branchingratiomeasurements. stepsizeoftheirchoosing.ThetersinstalledonthePMTandthelasersbeingusedare intheboxontheleftandthesettingsaremappedtointegerswhicharesaved toatextalongwiththePMTcountsandAOMsettings. 4.5Measurement Tomeasuretheuorescenceofoneofthe 3 F o 2 decaychannels,Iidenthe 1 P o 1 resonantwavelength.Iinstalledthe698nmbandpassteronthePMTsensor andlookedforpeakcountswhichwouldindicatethat 3 D 1 ispopulated.Theliterature excitationwavenumberis20715 : 71cm 1 [142].Weusuallymeasurethetransitionat around20715 : 6042cm 1 ,about = k=k 2 ˇ 0 : 0025nmdi erence.Thisiswithinthe rangethatourwavemetertendstodrift. 105 Figure4.9:Fluorescencesignalofthe 3 F o 2 ! 3 D 1 transitionwhiledepopulating the 3 D 2 statewitha712nmprobelaser. Inaddition,weshifttheblueimaginglaserwavelengthwithtwoAOMs,adouble-pass setto 447MHzandasingle-passsetto+80MHz.Thismeansthatwelookforresonance ataround: 20715 : 6042cm 1 ( 2 447 80 ) MHz 3 10 10 cm = s =20715 : 5771cm 1 Toscanthepumplaserfrequency,Imanuallychangedthecurrentsourcedrivingthe blueimaginglaserandreadthewavemeter.Ifoundpeakuorescenceat20715 : 5756cm 1 , showninFigure4.8. AschematicoftheuorescencemeasurementtechniqueisshowninFigure4.4.After identifyingthepumptransitionfrequency,IinstalledthePMTbandpassterthatgates onthefisignalfltransition 3 F o 2 ! D 2 S +1 J .ThenIusedaprobelasertodepopulatea di erentDstatewithtotalangularmomentum J 0 .Irepeatedthesearchforauorescence signalcorrelatedwiththesignaltransition. Itookaseriesoffltripletflmeasurementsbyintegratingtheuorescencesignalinal- 106 ternatingcyclesofblockingandunblockingthepumpbeam.Thisallowsustoseparate thesignaltransitionfrombackgroundlight.InFigure4.9,Iidentifyasignalpeakusing thisscanningmethod. 4.6Results Foreachtransition,Iintegratedfor100secondsforeachcycleofblockedandun- blockedpumpbeam.Thisgaveusenoughstatisticstosu cientlyreducetheuncertainty ofthesamplemean.ThePMTreportscounts,sowemodelpopulationwithaPoissonian distribution.Thestandarddeviationscaleswiththerootofthenumberofsamplestaken, or ˙ = p N .Integratingfor100secondsisashortenoughtimescaletomanuallystabilize laserfrequencyandwashoutshort-termfrequencydrifts. Each100sintegrationperiodisreducedtoasingleweightedaverageandshownin Figures4.10,4.11,4.12,4.13,4.14,4.15.Togetacountdueonlytothesignal,Isubtracta weightedaverageofthebackgroundmeasurementsbeforeandaftereach(signal+back- ground)measurement: S i = A i B i 1 t i t i 1 t i +1 t i 1 B i +1 t i +1 t i t i +1 t i 1 ,(4.2) where S i [ unitless ] isthe i th signal-onlyterm, A i [ unitless ] isthe i th measurementandisa(signal+background)measurement, B i [ unitless ] isthe i th measurementandisabackground-onlymeasurement,and t i [ s ] isthemediantimeofthe i th measurement. Thebackground-subtractedaveragePMTcountsforsixationsandanindex oftheirassociatedplotsaresummarizedinTable4.2.Wewereabletomeasuretwoofthe fourtransitionsduringmyANLfellowship. OurPMTistlylesssensitivetothe 3 F o 2 ! D 1 2 912nmtransition.Initial e ortsdidnotyieldaconvincinguorescencemeasurementof 1 D 2 .Iattemptedusing 107 Table4.2:MeasuredPMTsignalsofdecaysfrom 3 F o 2 . probesignalnotesignalFig.pumpprobe T count(mW)(mW)( C) 3 D 2 D 3 1 80 : 8(24)4.102.54.8441 D 1 2 3 D 1 2 : 3(18)4.112.54.8441 3 D 2 D 3 1 57 : 5(36)4.122.66.0491 3 D 2 D 3 3 beamsblocked 1 : 9(45)4.132.54.8441 3 D 2 D 3 3 pumpresonant42 : 0(58)4.142.66.0491 3 D 2 D 3 3 pumpdetuned45 : 6(70)4.152.66.0491 theNIRlasertodepopulate 1 D 2 forameasurementofthe 3 D 3 signalinFigure4.11.The signalwasconsistentwithzero,suggestingthattherewasinsu cientprobepower.The excitationpowerfor 1 D 2 waslaterincreasedbyreducingthelinewidththeNIRlaser. Istartedthesearchforthe 3 F o 2 ! D 3 2 transition,butbackgroundPMTcountswere greaterthan4 10 5 countss 1 .Theatomicovencouldhavecontributedtothehigh background.Theexpectedsignalstrengthwas < 100countss 1 ,soIprioritizedmea- surementsoftheothertransitions. Inonemeasurement,Icarriedoutthemeasurementprocedurewithbothbeamsblocked usingasecondbeamblockdownstreamofthepumplaserbeamblock(Figure4.13).The 750nmPMTbandpassterwasinstalledfor 3 D 3 measurements.Asexpected,the signal-onlymeasurementsareindistinguishablefromthe(signal+background)measure- ments.Thetotalweightedaverageforthesignalmeasurementisconsistentwithzero counts(1 ˙ ).Theno-beammeasurementmeasurestheambientlightinginsidethecham- ber,e.g.fromtheatomicoven,andalsoprovidesabaselinemeasurementofthecounts fortheinstalledbandpasster.Finally,overthecourseofthemeasurementthePMT countsdrifteddownby ˇ 100counts.Fromthiswecanexpectanapproximatelylinear driftof1Œ2countshr 1 . Tomeasurethe 3 D 3 transition,Iinstalledthe750nmPMTbandpassterandmade twomeasurements.Inthemeasurement,Itunedthepump(483nm)andprobe 108 Figure4.10:8/8/2018Averageduorescencesignalofthe 3 F o 2 ! 3 D 1 transition whiledepopulatingthe 3 D 2 statewitha712nmprobelaser. (712nm)beamstotheirresonantwavelengths(Figure4.14).Inthesecondmeasurement, IdetunedthepumpbeamsothattheDstateswouldnotbepopulated(Figure4.15).The di erencebetweenthetwomeasurements,whichgivesustheuorescencesignalofthe 3 D 3 transition,is42 : 0 5 : 8 (45 : 6 7 : 0)= 3 : 6 12 : 8countss 1 .Ourmeasuredresult isconsistentwithzero.Asexpected,thistransitionistooweaktobemeasuredbyour method. Tomeasurethe 3 D 1 transition,Iinstalledthe698nmPMTbandpasster.Itook measurementsfortwodi erentations.Intheation,Idepopulated the 3 D 2 statewiththe712nm.Imeasuredtheuorescencetwiceforthisation overtwodays(Figures4.10,4.12).Inbothcases,Imeasurenonzerocountrates,butthere isa ˇ 30countshr 1 discrepancybetweenthetwodays.Forthe(8/8)measurement, weusedapumpbeampowerof2.5mWandaprobebeampowerof4.8mW.Forthe second(8/9)measurement,weusedapumpbeampowerof2.6mWandaprobebeam powerof6.0mW.Despitethehigherpower,the(signal+background)measurementsare 109 Figure4.11:8/8/2018Averageduorescencesignalofthe 3 F o 2 ! 3 D 1 transition whiledepopulatingthe 1 D 2 statewitha912nmprobelaser. Figure4.12:8/9/2018Secondmeasurementof 3 F o 2 ! 3 D 1 transitionwhilede- populatingthe 3 D 2 statewitha712nmprobelaser. 110 Figure4.13:8/8/2018Averageuorescencesignalwithpumpbeamandprobe beamsblocked. Figure4.14:8/9/2018Averageuorescencesignalofthe 3 F o 2 ! 3 D 3 transition withthepumpbeamtunedonresonance. 111 Figure4.15:8/9/2018Averageuorescencesignalofthe 3 F o 2 ! 3 D 3 transition withthepumpbeamtunedo resonance. smallerby ˇ 50countss 1 .Thisislikelyduetodepletionoftheatomsource. 4.7Analysis Afterconstructingtheuoroscopysetupandcalibratingthelasers,lasersweeps wereperformedtocaptureatomiclineshapesofeachofthetransitions.Inthissection,I willshowhowweextractedbranchingfractionsfromthelineshapedata. Wemeasureauorescencesignalbysweepingthelaserfrequencyacrossthereso- nancefrequency.Thenwethemeasureddatatoafunction L characterizingtheline- shape,givenbythefollowing: L = D n max X n =1 p n 1 (1 p )[1 CDF ( ;n )]+ C 0 ,(4.3) n =0 ; 1 ; 2 ;::: (4.4) where n [ dimensionless ] isthenumberofphotonsscatteredbeforeanatomdeexcites, 112 p [ dimensionless ] isthebranchingratiototheprobestate, D [ arbitrary ] istheamplitudeofthelineshape,and C 0 [ arbitrary ] isano set. Theprobabilitythat n photonsarescatteredbeforedecayingtothesignalstateis givenbyaPoissoniancumulativedistributionfunction CDF ( ;n ): CDF ( ;n )= [ n +1 ; ] n ! ,(4.5) ( `; )= Z 1 t ` 1 exp ( t ) dt , ` = n +1 > 0, (4.6) where ( `; )istheincompleteuppergammafunction.Thelineshapeischaracterized thePoissonianweight : = X y ˝ ¯ n ( y ) f˙ 0 V ( ! = ! 0 ; ;˙ D ),(4.7) ¯ n ( y )= P ~ ! y 2 P x I ( x;y ) P x;y I ( x;y ) ,(4.8) V ( ! = ! 0 ; ;˙ D )= 1 ˙ D Z 1 0 exp 2 6 6 6 6 4 0 a ˙ D ! 2 3 7 7 7 7 5 = (4 ˇ 2 ) ( 0 ) 2 +( = 4 ˇ ) 2 d 0 ,(4.9) =2 X i A ki = 8 ˇ˙ 0 g k X i g i 2 ik f ik ,(4.10) ˙ 0 = e 2 = (4 0 m e c )=2 : 65400886 10 6 m 2 = s,(4.11) where A ki [ Hz ] istheEinsteinspontaneousdecayrate(fiA-coe cientfl)for j k i ! j i i , [ Hz ] istheLorentzianwidth, ! 0 [ rad = s ] istheresonantfrequencyoftheatomictransition, m e [ kg ] istheelectronmass, e [ C ] istheelementarycharge, c [ m = s ] isthespeedoflightinvacuum, 0 [ F = m ] isthevacuumelectricpermittivity, 113 P [ W ] isthepoweroftheexcitationlaser, y [ m ] isthebeamimagepixelwidth, I ( x;y ) h Wm 2 i istheintensityofthelaseratapixelwithcoordinates(x,y), g i =2 J i +1isthedegeneracyof j i i withtotalangularmomentum J i , ˝ [ s ] isthelaser-atominteractiontimeastheatomtraversesonepixellength,and ¯ n ( y ) h Wm 2 i isthephotonintensityonanatomatcoordinate y . Thebranchingratiooftheatomictransition i ! k isgivenbythefollowing: p ik = A ki = 2 ,(4.12) where p ik istheprobabilitythatanatominstate i decaystostate k . Theparametersoftheare f ik , D i , ! 0 ;i , ,and C i . k isthe 3 F o 2 state. i ={1,2,3} correspondstostates 1 D 2 , 3 D 2 ,and 3 D 1 ,respectively.Thereare3+3+3+1+3=13 totalparameterstoWemeasuredtheDoppler-broadenedlinewidthonthe 1 S 0 ! 3 P 1 (714nm)tobe = 2=2 : 32MHz. Thebranchingratio(orbranchingfraction) R ki istheintensityof j i i $ j k i relative toallotherallowedtransitionsfrom j k i : R ki = A ki P i A ki (4.13) Thebranchingratiocanberelatedtodipole-allowedtransitionamplitudematrixele- mentsthroughtheEinsteinA-coe cient: A ki = 1 g k 16 ˇ 2 3 3 0 hc 3 j D ik j 2 ,(4.14) where [ Hz ] isthefrequencyofthetransitionand j D ik j [ Cm ] isthedipole-allowedtran- sitionamplitudematrixelementonthetransition j i i ! j k i . Theoscillatorstrength f ik theratioofthepowerabsorbedbyanatomon j i i ! j k i to thepowerabsorbedbyaclassicaloscillatorwitheigenfrequency ! ik =( E k E i ) = ~ .The oscillatorstrengthisrelatedtotheEinsteinA-coe cientbythefollowing: A ki = 2 ˇe 2 m e c 0 2 g i g k f ik ,(4.15) 114 Figure4.16:Lineshapeforthe 3 F o 2 decaychannelsatdi erentprobelaser powers. where m e [ kg ] isthemassoftheelectron. Wemeasuredtheuorescencesignalfrom 3 D 1 , 3 D 2 ,and 1 D 2 overarangeofprobe laserpowersasshowninFigure4.16.The 3 D 3 uorescenceistooweaktoaccurately alineshape.Forthistransitionwetookaratiooftheuorescencefor 3 D 3 to 3 D 1 and determinedabranchingratioupperlimitof0.4%[90]. Thetheoreticalbranchingratioscalculatedfromthedipoletransitionamplitudeand ourmeasuredbranchingratiosderivedfromthelineinEquation4.3aregivenin Table4.3.Wemeasuredthe 3 D 1 and 3 D 2 branchingratiostobeafactoroftwosmaller andafactoroftwolargerthanthepredictedvalues,respectively.The 1 D 2 branchingratio measurementisafactorofthreesmallerthanthepredictedvalue,butat5%thebranch- ingfractionintensityislargeenoughfortheblueslowerschemedepictedinFigure4.1. Themeasurementsarequalitativelyconsistentwiththetheoreticalvalues,andwe 115 Table4.3:Calculatedbranchingfractions(BF)andoscillatorstrengthsfrom 3 F o 2 . transitionwavelength(nm) f ik (measured)BF(theory)%BF(measured)% 3 F o 2 ! 3 D 1 698.215100 : 25 0 : 0854.031 11 3 F o 2 ! 3 D 2 712.043410 : 32 0 : 1231.864 24 3 F o 2 ! 3 D 3 750 0.0359 < 0 : 4 3 F o 2 ! 1 D 2 912.682770 : 041 914.25 : 0 1 : 1 References [90][90][89][90] concludethat 3 D 3 isweakenoughtoneglectrepumpingand 1 D 2 isstrongenoughtouse asarepumpingchannel. 116 CHAPTER5 CALIBRATINGTHEATOMICBEAMFLUXFROMANEFFUSIVEOVEN TheRaEDMexperimentusesane usiveoventogenerateadirectedbeamofradium atoms.Afractionoftheatomsarelaser-cooledandtrappedforspinprecessionfrequency measurements.InthephaseoftheSingleAtomMicroscope(SAM)experiment,an atomicbeamofneutralatomsisimplantedinasolidnoblegasBothprojectsrequire preciseknowledgeoftheatomicbeamintensityanddistributiontoaccuratelycountthe rateofatomsexitingtheoven. IntheMSUatomicbeamuorescence(ABF)measurement,weilluminatethedirected atomicbeamwithalaserbeamtunedtotheatomictransitionfrequency.Thelaseris scannedoveranappropriatefrequencyrangespanningtheatomictransition(s)ofinter- est.Theresultinglaserinduceduorescence(LIF)oftheatomicbeamiscapturedby aphotodetectorpositionedperpendiculartotheplaneformedbytheatomicbeamaxis andlaserbeamaxis.Fromthisdatawecanplottheatomicabsorptionlineand determinetheovenatomrate. IwillmotivatetheABFexperimentinSection5.1.Then,Iwilldescribetheh structureandhtransitionstudiesforrelevantisotopesinSection5.2Thiswill befollowedbyadiscussionofatomicabsorptionlineforthecaseofadirected atomicbeamintersectingaweak-pumpinguorescentlaserinSection5.3.Iwilldescribe previousABFmeasurementsinSection5.4.InSection5.4,Iwillcomparethemeasure- mentstosimulatedspectra.Iwillconcludethechapterwithsuggestionsforimproved ABFmeasurementsinSection5.5. 117 Figure5.1:Decayschemeof 225 Ra.Alphaandbeta-decayaredenotedby and ,respectively.Half-livesarefromtheNationalNuclearDataCenter. kyr=1000years.d=days.m=minutes. 5.1Motivation 5.1.1Radiumsourceforelectricdipolemomentexperiment TheFacilityforRareIsotopeBeams(FRIB)islinearacceleratoratMSUthatisplanned tobefullyoperationalin2022.Exoticnucleiwillbegeneratedbyimpingingauranium beamonawater-cooledgraphitetarget.Thiswillcreateprimary,desiredisotopes,along withmanysecondaryisotopes.Theprimaryisotopeswillbedirectedtoexperimental halls,whilesecondaryisotopeswillbeextractedfromthetargetwaterreservoir(forde- tails,seePaigeAbel'sthesis[92]).Theprocessofextractingtheisotopeofinterestfrom theFRIBtargetcoolantandpreparinganoven-loadableatomsampleisthefiisotopehar- vestingflprocess. TheRaEDMexperiment(ANL,MSU)used 225 Ra( I =1 = 2)preparedatOakRidge NationalLab(ORNL)inthetwoEDMmeasurements[65,48].Adecayschemeof radiumisshowninFigure5.1.RadiumisproducedatORNLfroma 229 Thstockpile andsenttoANLasradiumnitratesalt.Thenwedissolvethesaltinnitricacidandadd 118 metallicbariumtothesolutionbeforewrappingtheradium-bariumsolutioninfoiland placingitintheovencrucible[48]. Radium-225hasbecomeincreasinglysoughtafterformedicalresearchinrecentyears. Thedaughterisotope,Ac 225 ,hasbeenidenasane ectiveisotopefortargetedalpha therapy[143].Thenewdemandfor 225 Raoversubscribesthecurrentavailablesupply andmakesanEDMcampaignwithRa 225 fromORNLunlikelyfortheforeseeablefuture. We'readdressingthischallengeintwoways.Asastop-gapmeasure,wewillacquire commercially-availableRa 223 (nuclearspin I =3 = 2,half-life11.43days)andrecalibrate theEDMlasercoolingandtrappingsetupatANLforRa 223 EDMmeasurements. Simultaneously,wearedevelopingaRa 225 harvestingprogramatFRIB.Theharvest- inge ciencyofa 225 RasourceforanewRaEDMexperimentwillbeevaluatedwith ABFmeasurements.Idiscussmyworkine usiveovenatomuxcalibrationsforisotope harvestingandnoblegasimplantationinthischapter. Inthemostrecent2015RaEDMexperiment,weusedanovenloadof10mCi(dis- cussedindetailinSection2.2.3): 10mCi=10 10 3 Ci 3 : 7 10 10 Bq = Ci =3 : 7 10 8 Bq, whereBqaredecayspersecond. 225 Rahasahalf-life t 1 = 2 =14 : 9days=1 : 29 10 6 s, orequivalentlyameanlifetimeof ˝ = t 1 = 2 = log ( 2 ) =1 : 86 10 6 s.Thedecaycon- stantis =1 =˝ =5 : 38 10 7 s 1 .Thatcorrespondstoaninitialovenloadof N 0 =10mCi = ˇ 7 10 14 atoms. FRIBisexpectedtoproducesecondaryradiumisotopes,including 225 Ra.Wewillbe abletoextractradiumfromthetargetcoolantandchemicallypurifyanoven-loadable sampleanalogouslytotheORNLmethod.Thenewsourcewilldeliver 225 Ramorefre- quentlyandinlargerquantitiesthanthetwopreviousRaEDMmeasurements. OurgoalistodevelopanABFmeasurementwiththeaidofcomputationaltoolsto measuretheovenatomratetowithin20%.Wecancomparethisratetotheinitialsource 119 sizetoquantifyourisotopeharvestingability.Thecriticalatomandgeometry-dependent propertyisthenumberofphotonsemittedbytheatomduringtheuorescenceinterac- tion,orfiphoton-atomyieldfl [ dimensionless ] : = dN =dt dN a =dt ,(5.1) where dN a =dt h s 1 i istheatomovenexitrateand dN =dt h s 1 i istherateofphotons emittedfromtheatoms. Thephoton-atomyieldisdependentonthepropertiesoftheisotope,electronictran- sition,pumping(excitation)laser,atomicangulardistribution,andphotodetector. In2017,IworkedontheABF-commissioningstudyofstableytterbiumisotopes.We weresuccessfulinmeasuringanytterbiumspectrumandestablishingtheproperopera- tionofvacuumchambers,laserequipment,anddataacquisition.Weusedalaserpower ofapproximately800mWfortheP 1 o 1 (398.8nm)transition.In2019,theABFsensitiv- itywasimprovedwithameasurementoftherubidium 2 P o 1 = 2 transition(795nm)forthe SAMproject(fordetails,seeBenLoseth'sthesis[144]).Usingalowerlaserpowerrange of10 Wto10mW,theSAMteamimprovedthesensitivityofthemethodandiden allthehtransitionsinthespectrum. TheYbapparatuswasdisassembledtobuildtherubidiumSAMABFsetup.After therubidiumABFmeasurement,IassembledanewABFsetup.Thenewsetup,which IwillrefertoasthefiAtomicFluxflapparatus,willbeusedforisotopeharvestingABF measurements.Idesignedanin-vacuumlight-collectingsetupthatwillbeinstalledin theAtomicFluxapparatustoimprovethelightcollectione ciencybyanestimatedtwo ordersofmagnitude(detailsinSection5.5.2). Ourtimelinefortheisotopeharvestinge ciencymeasurementbeginswithanew ABFmeasurementofstableytterbiumintheAtomicFluxapparatus.Thiswillcalibrate thenewsetupandwillbeaidedbycomputationalmodeling(discussedinSection5.3) tomakeanaccurateatomratecount.ThenwewillrepeattheABFmeasurementwith commercially-availablecalciumchips.Thiswillallowustocalibratethesetupforcal- 120 Table5.1:AselectionofatomictransitionsoftheYbgroundstate,S 1 0 . ValuesfromNIST. I =intensity. =resonantwavelength,frequency. ˝ =lifetime. A =EinsteinA-coe cient. excitedstate I (arb.) (nm) (THz) ˝ (ns) A (MHz) 6 s 6 p P 1 o 1 1000398.799751.535.21192 6 s 6 p P 3 o 1 130555.6466539.387869.61 : 15 2 F o 7 = 2 5 d 5 = 2 6 s 2 ( 7 = 2 ; 5 = 2 ) o 130346.437865.1115.768 : 3 ciumandimprovethesensitivityofthemeasurement.Next,wewilldissolvecommercial calciuminwater,simulatingtheinitialconditionsofanFRIBharvest.Thedissolvedcal- ciumwillbeharvestedandpreparedasanitratewithbariuminfoil,identicaltothe ORNL/ANLsourcepreparationprocedure.We'llmeasuretheuorescenceofthedis- solvedcalciumanddeterminetheharvestinge ciencybycomparingtheinitialsource sizeandmeasuredatomrate. 5.1.2Rubidiumuxmeasurements TheSingleAtomDetection(SAM)projectaimstomeasurerarenuclearreactions,on theorderofoneeventperday,relevanttonuclearastrophysicsbycapturingreaction productsinatransparent,frozennoblegasandcountingtheproductswithLIF. Aprototypemicroscopewasbuilttodemonstratethemethodbyimplantingrubidium atomsinakryptonandcountingtherubidiumatomsbeforeandafterimplantation. First,ane usiveovensourcewasusedasarubidiumsource.Thentheprototypewas placedontheReA3beamlineandtwoacceleratorexperimentswerecarriedout:krypton ionsimplantedinakryptonfollowedbyrubidiumionsimplantedinanewkrypton [144]. Intheo inerubidiumABFmeasurement,arubidiumsourcewasplacedinanoven similartotheYbandRaovens,butwithamuchnarrowernozzle(discussedinSec- tion5.3.7).Theovenwasheatedtooventemperaturesrangingfrom25Œ220 Ctogener- ateadirectedatomicbeam.Thebeampassedauorescencechamber,liketheYbsetup. 121 Aftertheuorescencechamber,theatomswereimplantedinanoblegasfrozentar- get. Liketheisotopeharvestingproject,thee usiveovenrubidiummeasurementrequires acarefulmeasurementoftheatomicangulardistribution.Ipresentanalysisoftheru- bidiumABFmeasurementsinthecontextofmodelingthelineshapeoftheuorescence spectrumtoderiveanabsolutecalculationoftheatomicuxforisotopeharvestingstud- ies. 5.2spectrum 5.2.1Atomicstatenotation TheelectronicationofthegroundstateofneutralytterbiumfiYb(I)flisexplicitly labeledinthefollowingmanner: 1 s 2 2 s 2 2 p 6 3 s 2 3 p 6 3 d 10 4 s 2 4 p 6 4 d 10 5 s 2 5 p 6 4 f 14 6 s 21 S 0 Theelectronshellscanbeabbreviatedinthe LS -couplingschemewithprincipal quantumnumber n ,theangularmomentum,andspinwiththefollowingnotation: n 2 S +1 L J , L= ( S ; P ; D ; F ::: ) 7! L = ( 0 ; 1 ; 2 ; 3 ::: ) , where S isthespin,Lthetotalelectronicorbitalangularmomentum L ,and J = L + S isthetotalelectronicspinoftheatom.Inthisnotationthegroundstatebe- comes6 1 S 0 ,oftenshortenedto 1 S 0 .Fortheuorescencemeasurement,we'reprobing thestrongtransition 1 S 0 ! 1 P o 1 .Alistofselectedgroundstatetransitionsisgivenin Table5.1. Nucleonswithinthenucleusanalogouslytoelectronorbitals.Whenthere areoneormoreunpairednucleons,thereisanet,nonzeronuclearspin I .Thetotal angularmomentum F oftheatomisthendescribedbythesumofelectronicandnuclear 122 Table5.2:YtterbiumtotalstrengthfactorsforS 1 0 ( F ) ! P 1 o 1 ( F 0 ). isotope S 1 = 2 S 3 = 2 S 5 = 2 S 7 = 2 171 Yb 2 = 6 4 = 6 173 Yb 4 = 18 6 = 18 8 = 18 angularmomentum: F = I + J;I + J 1 ;:::; j I J j Tocompletelycaptureanatomictransition,weneedtolabelthetotalangularmomentum aswell.Inthecaseof 171 Yb( I =1 = 2),oneofthepossibletransitionscanbewritteninthe followingmanner: 1 S 0 ( F =1 = 2) ! P 1 o 1 ( F 0 =3 = 2), where F and F 0 istheinitialandtotalangularmomentum.Thegroundstateofthis transitionhas J = L = S =0 ; so F canonlybe1/2.Ingeneral,boththeinitialand totalangularmomentumcanbothtakeonmultiplevalues. 5.2.2Atomictransitionintensity splittingispresentinatomswithnonzeronuclearspinandshiftsthetran- sitionfrequencyofthehtransition j iF i ! fF 0 relativetothetransition j i i ! j f .TosimulatethehspectrumofYb,weneedtodistributethepopula- tionsofthenonzeronuclearspinisotopesamongitsdegeneratestates.Wewillassume thatthemagneticsublevels m F areequallypopulatedandthatthepumpinglaserisun- polarized.Theunpolarizedassumptionimpliesthatthetransitions m F m F 0 =+1 ; 0 ; 1 areequallylikely. We'lllookatthecaseofYb 171 ( I =1 = 2),wherethereisnohstructureinthe groundstate.Thetransitionofinterest,S 1 0 ( F ) ! P 1 o 1 ( F 0 ),hasaelectronicangular momentum J 0 =1.Thetotalangularmomentumoftheexcitedstatecantakeonpossible valuesof F 0 =3 = 2 ; 1 = 2 : 123 Table5.3:Rubidiumrelativestrengthfactorsfor 2 S 1 = 2 ! P 2 1 = 2 . Wigner6- j valuescalculatedwithanonlineversionoftheRoot- Rational-Fractionpackage[145]. isotope S 33 S 32 S 23 S 22 S 21 S 12 S 11 85 Rb 4 = 9 5 = 9 7 = 9 2 = 9 87 Rb 1 = 2 1 = 2 5 = 6 1 = 6 Theallowedmagneticsublevelsare m F 0 = F 0 ;F 0 1 ;:::; F 0 ,whichgivesadegeneracy of g F 0 =2 F 0 +1.Forunpolarizedlaserlight,wecancharacterizetheintensityofthehy- transitionbythedegeneracy.Itheintensity,orfitotalstrengthfactorfl S F 0 asthefollowing: S F 0 = g F 0 P i g F 0 i (5.2) TotalstrengthfactorsfortheYbisotopeswithnonzeronuclearspinaregiveninTable5.2. Nextweconsiderthegeneralcaseofhstructureinboththegroundstateand excitedstate.Thestrengthofatransition j Fm F i ! F 0 m F 0 drivenbyaresonantphoton (e.g.,fromalaser)isproportionaltothedipolematrixelement: D Fm F e~ r F 0 m F 0 E ,(5.3) where e istheelectronchargeand ~ r isasphericaltensorofrank1.IfollowSteck's method[146]tocalculatetherelativestrengthofeachbranchinatransitionwithhy- splitting.Therelativehtransitionstrengthfactor S FF 0 [ unitless ] isgiven by: S FF 0 = 2 F 0 +1 ( 2 J +1 ) 8 > > > < > > > : JJ 0 1 F 0 FI 9 > > > = > > > ; 2 ,(5.4) X F 0 S FF 0 =1,(5.5) wherewehaveusedtheWigner6- j symbol,whichisderivedfromtheClebsch-Gordon coe cient Fm F F 0 m F 0 .For 85 Rb F =3 ! F 0 =2,Icalculatethefollowingstrength 124 Table5.4:Rubidiumtotalstrengthfactorsfor 2 S 1 = 2 ! 2 P 1 = 2 . isotope S 33 S 32 S 23 S 22 S 21 S 12 S 11 85 Rb 28 = 108 35 = 108 35 = 108 10 = 108 87 Rb 15 = 48 15 = 48 15 = 48 3 = 48 factor: S 32 = ( 5 ) ( 2 ) 0 B B B B @ 1 3 r 1 2 1 C C C C A 2 = 5 9 Table5.3liststherelativehtransitionstrengthfactorsfortherubidiumisotopes. Therelativestrengthfactorsprovidefractionalstrengthsforthedi erentbranches F 0 ,givenaninitial F .However,weneedanadditionalfactortodistinguishtherelative intensitiesfordi erentinitial F .Forexample,in 85 Rb,weneedtoknowtherelative intensitybetweenthetransitionsfor F =3versus F =2. Foratransitiondrivenbyunpolarizedlaserlight,thetotalstrengthfactorforeach transitionisbyfoundmultiplyingeach S FF 0 bythedegeneracy g F andnormalizingtoa weightedsum: S FF 0 = g F S FF 0 P FF 0 g F S FF 0 (5.6) where S FF 0 isthetotalstrengthfactorwithinitialandangularmomentum F and F 0 .Forexample, S 32 = 35 = 108 for 85 Rb.Thetotalstrengthfactorsforthetransition 2 S 1 = 2 ! P 2 o 1 = 2 aregiveninTable5.4. 5.2.3Frequencyoftransitions Theresonancefrequencyofanatomdependsontheisotopenumberandhstruc- ture.Inthecaseofytterbium,therearesevenstablenaturally-occurringisotopes.Two oftheseisotopeshavehstructurewith2+3=5htransitions.Intotal, therearetenresonancepeaks.Abundance,mass,andnuclearspinvaluesarelistedin TableA3. 125 Table5.5:LiteraturevaluesofthehconstantsofYb,Rb,andCaisotopes withnonzeronuclearspin. isotopelevel A HF (MHz) B HF (MHz)Ref. 171 Yb 1 P 1 214 : 173(53)0 : 0[147] 173 Yb 1 P 1 57 : 682(29)+609 : 065(98)[147] 87 Rb 2 S 1 = 2 +3417 : 341305452145(45)0 : 0[148] 87 Rb 2 P 1 = 2 +407 : 25(63)0 : 0[148] 85 Rb 2 S 1 = 2 +1011 : 9108130(20)0 : 0[148] 85 Rb 2 P 1 = 2 +120 : 527(56)0 : 0[148] 43 Ca 1 P o 1 15 : 46(15) 9 : 7(7)[149] 47 Ca 1 P o 1 16 : 20(23)+4 : 1(6)[149] Toorder,therearetwocontributionstoshiftofanatom'senergylevelduetoa nonzeronuclearspin:amagneticdipolemomentandelectricquadrupolemomentterm. Themagneticdipolehshiftterm E m1 [ MHz ] isasthefollowing[150]: E m1 = 1 2 A HF K ,(5.7) K = F ( F +1) J ( J +1) I ( I +1),(5.8) where A HF [ MHz ] isthemagneticdipolehconstant. Theelectricquadrupoleinteractiontermisasfollows: E e2 = 1 4 B HF 3 2 K ( K +1) 2 I ( I +1) J ( J +1) I (2 I 1) J (2 J 1) ,(5.9) where E e2 [ MHz ] istheelectricquadrupolehshifttermand B HF [ MHz ] isthe electricquadrupolehconstant. A HF and B HF areexperimentallymeasuredpa- rametersforeachisotopewithnonzero I .Together,wehavethetotal-orderh structureshift, E HF [ MHz ] : E HF = E m1 + E e2 (5.10) Equation5.10givesthehshiftwithrespecttothestateL 2 S +1 J ( I;F ).Literature valuesofhconstantsforYb,Rb,andCaisotopesarelistedinTable5.5.Icalcu- lated E HF foreachstatewithhstructureinYb,Rb,andCainTable5.6. 126 Table5.6:Calculatedhshifts E HF of ytterbium,rubidium,andcalcium.Totalangu- larmomentum F = I + J isotopestate IF E HF (MHz) 171 Yb6 s 6 p P 1 o 1 1/21/2+214 : 2 171 Yb6 s 6 p P 1 o 1 1/23/2 107 : 1 173 Yb6 s 6 p P 1 o 1 5/23/2+224 : 5 173 Yb6 s 6 p P 1 o 1 5/25/2 544 : 9 173 Yb6 s 6 p P 1 o 1 5/27/2+296 : 5 85 Rb5 s S 2 1 = 2 5/22 1770 : 8 85 Rb5 s S 2 1 = 2 5/23+1264 : 9 85 Rb5 s P 2 1 = 2 5/22 210 : 88 85 Rb5 s P 2 1 = 2 5/23+150 : 62 87 Rb5 s S 2 1 = 2 3/21 4271 : 7 87 Rb5 s S 2 1 = 2 3/22+2563 : 0 87 Rb5 s P 2 1 = 2 3/21 508 : 75 87 Rb5 s P 2 1 = 2 3/22+305 : 25 43 Ca4 s 4 p P 1 o 1 7/29/2 57 43 Ca4 s 4 p P 1 o 1 7/27/2+22 43 Ca4 s 4 p P 1 o 1 7/25/2+64 47 Ca4 s 4 p P 1 o 1 7/29/2 56 47 Ca4 s 4 p P 1 o 1 7/27/2+13 47 Ca4 s 4 p P 1 o 1 7/25/2+75 Thereisalsoanfiisotopeshiftflinthetransitionfrequencyduetothedi erentatomic masses.Thecalculationoftheisotopeshiftiscomplex(see,forexample,Woodgate[150]). Theisotopeshiftofytterbiumisapproximatelylinearfortheevenisotopes.Iestimated thehisotopeshiftsbyinterpolatingtheeven-nucleonshiftsasinputsforthe computationalmodelingdiscussedinSection5.3. 5.3Modelingthespectrallineofadirectedatomicbeam Theatomicangulardistributionmustbewell-characterizedtoaccuratelycountatoms inadirectedbeamfromane usiveoven.IdevelopedaPythonprogramthatsimulates alasersweepandgeneratesauorescencespectrumforagivensetofisotopes.The 127 Figure5.2:Aschematic(nottoscale)oftheatomicbeamuorescencesetup. Thisisgeneralizedtobeapplicabletoallthreesetupsdiscussedinthischapter. programmodelsthevacuumgeometry,laserphotodetector,andatomicovenge- ometry.Thesimulationalsoacceptsanangulardistributioninput,whichwewillvaryto matchthesimulationtoLIFmeasurements. 5.3.1TheABFapparatusandcalculatingthephotodetectorsignal AschematicofthebeamlineisshowninFigure5.2.Anatomsource,suchasametal ingot,isloadedintoane usiveoven.Theovenisheatedandemitsatomsfromtheoven nozzlewithageometry-dependentangulardistribution j ( ),where istheanglewith respecttothenozzleaxis ‹ z . Theatomsenterauorescencechamberwherearesonantlaserpropagatesperpen- diculartothenozzleaxisalong ‹ x .Thechamberisasix-waycross.Perpendiculartothe zx plane,aphotodetectorismountedatthewindowofthecrossarm.Forthemeasurements discussedinthischapter,weuseanavalanchephotodiode(Thorlabs410APD2)witha 0.5mmdiametercircularactivearea.Thelaserisscannedoveranappropriatefrequency rangeandafractionofthelightemittedbyatomsabsorbingthelaserlightiscaptured 128 Figure5.3:Schematicoflasersystem. bytheavalanchephotodetector(APD).AschematicoftheABFlasersetupisshownin Figure5.3. ThevoltageoutputoftheAPD V ( ) [ V ] isgivenbythefollowing: V ( )= P d ( ) R M ( ) G ,(5.11) where P d ( ) [ W ] istheincidentuorescentlightpoweratfrequency , R M ( ) [ A = W ] isthedetectorresponsivityatwavelength , M(thefiM-factorfl)isthegain,and G [ V = A ] isthetransimpedancegain. WewilluseaNIST-traceablepowermetertocalibratetheAPD(Thorlabs410-APD2) wavelengthresponseandgainfortheuxmeasurements.Forsimulations,Iestimatethe responseusingthemanufacturertions(seeAppendixC). 5.3.2Calculatingtheuorescencepoweronthephotodetector TheAPDvoltagesignalisproportionaltotheuorescentpowerincidentonthephotode- tector.Givenanactivesensorarea A d h m 2 i ,theincidentpower P q d ( ) [ W ] isgivenby 129 thefollowing: P q d ( )= ZZ v a a ( ~ r ) F q ( ;~ r ) dV a dA d 4 ˇ ~ d ~ r 2 ,(5.12) where q =1 ; 0 ; 1for ˙ + , ˇ ,and ˙ polarizedlight,respectively, F h s 1 i isthesingleatomuorescencerate, ~ d [ m ] isthepositionofthecenterofthephotodetectorsurface, v a [ m = s ] isthecomponentoftheatomvelocityalongthe z axis,and a ( ~ r ) h m 2 s 1 i istheatomuxatposition ~ r . Aswewillseeinthenextsection,theatomphotonemissionrateisrelatedtotheuores- centpowerthroughtheatomicuxoftheoven. 5.3.2.1Calculatingtheatomicux,vaporpressure,andtheatomrate Weconsideraposition ~ r ,wheretheoriginisastheovennozzleexitinFigure5.2. Atthislocation,theatomicux a ( ~ r ) h m 2 s 1 i isgivenbythefollowing: a ( ~ r )= dN a dt j ( ) r 2 ,(5.13) dN a dt = n o v a A o 4 ˇ ,(5.14) n o = P k B T ,(5.15) where j ( ) [ unitless ] istheatomicangulardistributionatpolarangle , n o h m 3 i istheatomnumberdensity, A o h m 2 i isthecross-sectionalareaoftheovennozzle, P [ Pa ] isthesaturatedvaporpressureoftheatoms,and T [ K ] istheoventemperature. 130 Figure5.4:Saturatedvaporpressurecurveforytterbium,calcium,andrubidium. Thevaporpressureisthepressureatwhichthegaseousatomsareinthermodynamic equilibriumwithitssolidphase.Thevaporpressureissaturatedwhenthevaporiza- tionandcondensationratesareequivalent.Weusethefollowingempiricalequationto determinethesaturatedvaporpressure P oftheoven: log 10 P P 0 = + A + B T + C log 10 T + D T 3 ,(5.16) = 8 > > > > > < > > > > > : 2 : 881 ;P 0 =1Torr, 5 : 006 ;P 0 =1Pa Theconstants A;B;C; and D arepropertiesoftheovenatomspecies.Foranoventemper- atureof300 C, P =9 : 4 10 4 Pa=7 : 1 10 6 Torr: n o =1 : 2 10 17 atomsm 3 , dN a dt =1 : 7 10 13 atomss 1 IplottedvaporpressurecurvesforYb,Rb,andCainFigure5.4.Vaporpressurecoe - cientsarelistedinTableA4. Isimulatedthephotodetectorsignalforarangeoftemperatureswithanytterbium ovensourceandovennozzleratio =0 : 25inFigure5.5.At ˇ 250 Cthecalculated 131 signalis10 V,whichisthelimitofthemeasurementsensitivityoftheABFmeasurement withoutlight-collectingoptics.Thiscorrespondstoanatomovenrateofapproximately 10 12 atomspersecond. Foranoventemperatureof300 C,nozzleratio =0 : 25,withanovennozzleradius of1.5875mm,andatomstravelingat v a =232 : 3m/salong ‹ z 13.2cmdownstreamfrom thenozzle,Icalculateanatomicuxof: ˇ 10 15 atomsm 2 s 1 Wecannowestimatethetotalpoweronthephotodetectorusingthecalculatedatom rateandEquation5.12.InthecaseoftheYb S 1 0 ! P 1 o 1 transition,aperfectlyon- resonancelaseryieldsanorder-of-magnitudeestimate: P d ˇ 6 : 63 10 34 JHz 1 751 : 5 10 12 Hz 4 ˇ ( 230m = s ) 1 : 0 10 15 m 2 s 1 4 : 1 10 6 s 1 ˇ (3 : 5 10 3 m) 2 (30 10 3 m) 3 : 3 10 5 sr ˇ 2 : 7 10 11 W Thisisreasonablyclosetothefullyintegratedsolutionof3 : 47 10 11 W.FromEqua- tion5.11,theconvertedphotodetectorvoltageis0.196mV. 5.3.3Thesingleatomuorescencerate Thefollowingdiscussionpresumesatwo-levelsystemofatomicstates a (theground state)and b (theexcitedstate)inaradiationdsuchastheelectricdproducedbya laser.Thesingleatomuorescencerate F ( ;~ r ) h s 1 i inthelaserinteractionregionfrom Equation5.12istherateatwhichanatominstate b emitsaphotonanddecaysto a : F ( ;~ r )= b ( ;~ r ) A = b ( ;~ r ) ˝ 0 ,(5.17) where [ Hz ] isthefrequencyofthelaser, 132 Figure5.5:Calculateduorescencesignalastheoventemperatureisvaried usingalaserpowerof10mW. ~ r [ m ] isthepositionoftheatom, A [ Hz ] isthespontaneousemissionrateEinsteinA-coe cient,and ˝ 0 [ s ] isthelifetimeoftheatomicstatewhenaphotonofwavelength 0 isabsorbed bytheatom. Thefractionofatomsintheexcitedstate b ( ;~ r ;t ) [ unitless ] isderivedfromthepop- ulationrateequationswithequalstimulatedexcitationandemissionrates R h s 1 i andthe spontaneousdecayratefrom b to a ,givenbytheEinsteinA-coe cient A =1 =˝ 0 h s 1 i : da dt = Ra + Rb + b ˝ 0 ,(5.18) da dt =+ Ra Rb b ˝ 0 ,(5.19) a + b =1(5.20) 133 Figure5.6:Excitedstatepopulationofatwo-levelsystemfor R =2 10 8 s 1 and ˝ 0 =5ns : Solvingtheseequationsyieldsthefractionofatomsin b attime t : b ( t )= b 0 exp t ˝ + R˝ 0 1+2 R˝ 0 ! 1 exp t ˝ ,(5.21) ˝ = ˝ 0 1+ R˝ 0 ,(5.22) where b 0 [ unitless ] isthepopulationfractionof b at t =0. Asanexampleofthetimetoreachpopulationequilibrium,I'lluseroundedP 1 o 1 num- bers: R =2 10 8 s 1 , ˝ 0 =5ns.Inthelimit t !1 ,wegetthesteady-stateexpressionfor thefractionofatomsin b : b ( ;~ r )=lim t !1 b ( ;~ r ;t )= R ( ;~ r ) ˝ 0 1+2 R ( ;~ r ) ˝ 0 Inourexample, b ( ;~ r )=1 = 3.Figure5.6plotstheexcitedstatefractionvs.timesfor factorsofthemeanlifetime ˝ 0 .Afterlessthanthreelifetimes,thefractionhasconverged to1 = 3towithin1%.Duringthistimeanatompassingthroughalaser-generatedelectric dwouldtraveladistance: ˇ 300m = s 15ns=4 : 5 m Wetypicallyuselaserdiametersof5Œ10mm,makingthesteady-stateapproximation quitereasonableforoursetup.UpcomingABFmeasurementswillbeperformedinthe 134 Table5.7:ValuesusedforYb 1 S 0 ! 1 P o 1 atomexcitationrate R ( ;~ r ). parametervalue T oventemperature300 C a resonanttransitionfrequencyoftheatom7 : 51526 10 14 Hz FWHM fullwidth-halfmaxofthelaser5 : 0 10 6 Hz P laserpower1 : 0 10 2 W w ( z )beamradius1 : 0 10 2 m ˆ radialdistancefromlaseraxis0 : 0m f a atomictransitionoscillatorstrength1 : 37 R M detectorresponsivityat398.8nm11.3A/W G transimpedancegain5 10 5 V = A r interaction-sensordistance7 : 74 10 2 m v p;z mostprobablespeedatomspeedalong ‹ z 232 : 3m = s A det sensorarea1 : 96 10 8 m 2 F atomuorescencerate4 : 1 10 6 s 1 0 emittedphotonfrequency7 : 51526 10 14 Hz V interactionvolume8 10 9 m 3 `weakpumpinglimit,'or R ( ;~ r ) ˝ 0 << 1.Inthislimit, b ( ;~ r ) ˇ R ˝ 0 andthesingleatom uorescencerateisequivalenttotheexcitationrate: F ( ;~ r )= 1 ˝ 0 R ˝ 0 = R ( ;~ r )(weakpumpinglimit)(5.23) I'veplottedthesingleatomexcitationrateinFigure5.7usingytterbiumtransitionvalues listedinTable5.7. 5.3.4TheDoppler-freeexcitationrate TheDoppler-freeatomabsorptioncrosssection ˙ ( ; a ) h m 2 i isasfollows: ˙ ( ; a )= c B a L ( ; a ;A ),(5.24) B a = ˇr e c 2 f a ,(5.25) L ( ; a ;A )= A= (4 ˇ 2 ) ( a ) 2 +( A= 4 ˇ ) 2 = n = (2 ˇ ) ( a ) 2 +( n = 2) 2 ,(5.26) where B a h s 1 i istheEinsteinabsorptionB-coe cient, 135 Figure5.7:TheweakpumpinglimitYbsingleatomlaserexcitationrateusing theparametersinTable5.7. L ( ; a ;A ) h Hz 1 i istheprobabilityofatomictransitionperunitfrequency, f a [ unitless ] istheatomictransitionoscillatorstrength,and n = A= 2 ˇ [ Hz ] isthenaturallinewidth. Thenaturallinewidth,alsoknownasthehalfwidth,isthewidthofthelineof anatomictransitionatwhichtheamplitudeisonehalfthecentralfrequencypeakmax- imum.TheLorentziannaturallinewidthissometimesreferredtoasafull-widthhalf- maximum,butwewillreservethattermforreferringtothelaser FortheYb 1 P 1 transition, A =1 : 92 10 8 s 1 and n =30 : 6MHz. TherateatwhichaDoppler-freesingleatomabsorbsaresonantphotonforagivenatomic 136 transitioninaradiationdisastheatomexcitationrate R ( ; a ;~ r ): R ( ; a ;~ r )= Z 1 0 ˚ ( ; ; FWHM ;~ r ) ˙ ( ; a ) d ,(5.27) ˚ ( ; ; FWHM ;~ r )= P S ( ~ r ) G ( ; ; FWHM ),(5.28) S ( ~ r )= I ( ~ r ) P = 2 ˇw 2 ( z ) exp " 2 ˆ 2 w 2 ( z ) # ,(5.29) G ( ; ; FWHM )= 2 p log2 =ˇ FWHM exp 2 6 6 6 6 4 4log ( 2 ) ( ) 2 FWHM 2 3 7 7 7 7 5 ,(5.30) where a [ Hz ] istheresonanttransitionfrequencyoftheatom, FWHM [ Hz ] isthefullwidth-halfmaxofthelaser, ˚ ( ; ; FWHM ;~ r ) h m 2 i isthelocalphotonux, S ( ~ r ) h m 2 i isthefractionofallphotonsperunitarea, P [ W ] isthelaserpower, I ( ~ r ) h Wm 2 i isthelaserintensity, w ( z ) [ m ] isthebeamradius, ˆ [ m ] istheradialdistancefromthelaserbeamlongitudinalaxis,and G ( ; ; FWHM ) h Hz 1 i isthefractionofallphotonsperunitfrequency. Nowwe'reinapositiontoplugeverythingintoEquation5.27andseparate R ( ; a ;~ r ) intoaprefactorandanintegraloverfrequency: R ( ; ;~ r )= P 2 ˇr e cf a ˇw 2 ( z ) exp " ˆ 2 w ( z ) # 2 ˇA L ( ; ;A; FWHM ),(5.31) L ( ; ;A; FWHM )= ˇA 2 Z 1 0 p 4log ( 2 ) =ˇ FWHM exp 2 6 6 6 6 4 4log ( 2 ) ( ) 2 FWHM 2 3 7 7 7 7 5 L ( ; a ;A ) d ,(5.32) wherewehavetheintegralfactor L ( ; ;A; FWHM ) [ dimensionless ] asthefiline- shapeoverlapflfunction. L canbesolvednumerically,butIfoundthatthenarrowwidthofthepeakwas notproperlyintegratedbysomethestandardPythonandMatLabsolvers,suchas 137 theFortran-based QUADPACK .Iinsteadusedthesimplercompositetrapezoidalin- tegrationroutine numpy.trapz() .Iapproximatedtheintegrationlimitsof to h 3( FWHM ) ; +3( FWHM ) i ,beyondwhichtheexponentialtermrapidlydrives R to0. AplotofthesingleatomexcitationratefortheYb 1 S 0 ! P 1 o 1 transitionisshownin Figure5.7. 5.3.5Dopplerbroadeningforadirectedatomicbeam TheSAMandFluxABFsetupsuseane usiveoventogenerateanatomvaporbeam.As seenintheexperimentallayoutinFigure5.2,we'vechosentheorigintobetheexitpoint ofthenozzle,sothat ~ r and ~ v haveidenticaltrajectories.Thelaserisorientedperpendicu- lartotheatombeamaxis,along ‹ x .Atomtrajectoriesatsomeangle fromtheatombeam axis ‹ z willalsohaveavelocitycomponentalignedwiththelaser. Itheanglebetweenthelaseraxisandtheatomvelocityatposition ~ r as [ rad ] : cos ( ) = ~ r ‹ x ~ r (5.33) When = ˇ= 2,theDopplershiftbetweentheatomandlaseriscos ( ˇ= 2 ) =0. Weassumetheatomsinourdirectedbeamarenon-interactingparticlesatthermody- namicequilibrium.Forthisscenariowemodelthespeeddistributionoftheatombeam asaMaxwell-Boltzmanndistribution g ( v ) [ m = s ] 1 atoventemperature T [ K ] ,givenby thefollowing: g ( v )= r 2 ˇ m k B T ! 3 = 2 v 2 exp 2 6 6 6 6 4 v v p ! 2 3 7 7 7 7 5 ,(5.34) v p = r 2 k B T m , Z 1 0 g ( v ) dv =1,(5.35) where v [ m = s ] istheatomspeed, v p [ m = s ] isthemostprobableatomspeed, 138 k B [ J = K ] istheBoltzmannconstant,and m [ kg ] isthemassoftheatom. For 174 Ybexitinga300 Coven,themostprobablespeedis v p =234m/s.Themost probablespeedofFor 85 Rbexitinga100 Covenis v p =270m/s. TheDopplerbroadeninge ectdependsonboth andthevelocityoftheatom.To order,theDopplertermisgivenbythefollowing: 1 v c cos ( ) The-orderDopplertermisincorporatedbymodifyingtheatomabsorptioncross section(Equation5.24) ˙ ( ; a ) ! ˙ D ( ; a ): ˙ D ( ; a ;~ r )= c B a L D ( ; a ;A ),(5.36) L D ( ; a ;A;~ r )= A= (4 ˇ 2 ) [ a (1 cos ( ) v=c ) ] 2 +( A= 4 ˇ ) 2 (5.37) Theexcitationrateincludestheadditionalspeedintegralfrom ˙ D : R ( ; a ;~ r )= Z 1 0 Z 1 0 ˚ ( ; ; FWHM ;~ r ) ˙ D ( ; a ;~ r ) g ( v ) ddv (5.38) Equation5.38hasthepracticale ectofbroadeningthespectralwidth. ThelinearDopplerfullwidthathalfmaximum FWHM D [ Hz ] isgivenby: FWHM D = p 8 k B T log2 =m c a sin ( ) ,(5.39) =2 : 92 p T=m a sin ( ) 10 20 ,(5.40) where c [ m = s ] isthespeedoflightinvacuumand [ rad ] isthepolarangleoftheatom relativetothebeamaxis. Thenaturallinewidth n oftheYb 1 P o 1 transitionisapproximately30.6MHz.I calculateaDopplerbroadenedlinewidthof D =121MHzusinganoventemperature of300 C=573 : 15Kandamaximumangleof =0 : 12435rad. 139 Ouruorescencemeasurementusesadirectedatomicbeamwithalargeangulardis- tributionandrequiresthegeneralformoftheexcitationrateinEquation5.38.Acommon alternativeuoroscopysetupusescollimationdownstreamoftheoventosuppressangu- lardependenceonatomintensity.Inthiscasetheatomsmoveuniformlyalong ‹ z ,so is smalland ˇ ˇ= 2.ThebroadeningterminthedenominatoroftheLorentzian reducesto a (1 cos ( ) v=c ) ! a ,tlysimplifyingthecalculationoftheatom excitationrate. 5.3.6Theatomicangulardistributionandphotodetectorsolidangle Nowthatwecancalculatetheatomicux,excitationrate,andphotodetectorpower,it's naturaltoreexaminethephoton-atomyieldshowninEquation5.1. Thephotonemissionrate dN =dt canbewrittenasfollows: dN dt = 4 ˇd 2 P d ( ) A d ,(5.41) where ~ d [ m ] isthepositionofthecenterofthephotodetectorsurface, v a [ m = s ] isthecomponentoftheatomvelocityalong ‹ z ,and A d h m 2 i isthephotodetectoractivearea. UsingEquation5.41,wecanrewritethephoton-atomyield intermsof P d ( ): = ZZ 1 v a d 2 y A d j ( ) F ( ;~ r ) dV a r 2 dA d ~ d ~ r 2 ,(5.42) where d y [ m ] isthedistancealong ‹ y fromthecenteroftheuorescencevolumetothe photodetectorsurface. Inthenextsection,Iwilldescribethemodelfortheangulardistribution j ( ).Then I'lldetailthesolidanglecalculationinSection5.3.8. 140 Figure5.8:Fromlefttoright,inorderofincreasingnoodlediameter-to-length: bucatini,cannelloni,anellini noodles.ImagesobtainedundertheCC01.0Uni- versal(CC01.0)PublicDomainDedicationLicense. a 5.3.7Atomicangulardistribution TheovennozzlegeometryisshownschematicallyinFigure5.2.Thedistributionofthe atomsowingthroughthenozzledependsonthenozzleaspectratio.Longnozzlescolli- matethebeam,whileshorternozzlespermitahigheratomux.Wecancharacterizethe nozzlebytheratiooftheradiustolength,ortheovennozzleratio : = 2 a L ,(5.43) where a [ m ] isthenozzleradiusand L [ m ] isthenozzlelength. It'snatural(anddelicious)tothinkoftheovennozzlesasdi erentkindsofnoodles.A diverseselectionofnoodlesisshowninFigure5.8.As !1 ,onecanimagineashorter andwidernoodle,forexample anellini .For ! 0,thenozzleisverylongcomparedto itsdiameter,similarto bucatini .TheYbnozzleratioisclosesttothe cannelloni geometry, with =0 : 1250 00 = 0 : 5000 00 =0 : 2500.Thecalciumovennozzleratiowillbeidenticalto theytterbiumnozzleratio. Collisionsbetweentheatomsexitingthechannela ecttheresultingangulardistri- bution.WeusethetubelengthKnudsennumber K nL tocharacterizethedensityofatoms intheovenchannel[151]: K nL = L ,(5.44) a https://creativecommons.org/publicdomain/zero/1.0/deed.en 141 (a) (b) Figure5.9:Theatomicangulardistributionofforarangeofnozzleratios.(a)80de- greerange,alllinesconvergetoanintensityofzeroat90degrees(b)Zoomedinto within5degrees.Thelegendappearsintheorderofdescendingintensity.Middle solidline=ytterbiumandcalciumratio =0 : 25.Dashedline=radium =0 : 024.Bot- tomsolidline=rubidium =0 : 01. 142 where [ m ] isthemeanfreepathoftheatomsinthechannel.Atom-ovenstateswith Knudsennumbersintherange K nL > 10areasthefimolecularowflregime, wheretheonlytatominteractionsarecollisionsalongthenozzlechannelwall. Atomsinintermediateregime, K nL 10musttakeatomiccollisione ectsintoaccount. Fortheytterbiumnozzle( =0 : 25), K n > 10 4 foroventemperatureslessthan330 C.For therubidiumnozzle( =0 : 01), K n > 10foroventemperatureslessthan100 C. Inthemolecularowlimit,theangulardistributionofatomsexitingtheoven j M ( ) [ unitless ] atsomeangle [ rad ] withrespecttothebeamaxis ‹ z isgivenby[151]: j M ( )= 8 > > > > > > < > > > > > > : 0 cos + 2 ˇ cos " (1 0 ) R ( p )+ 2 3 ( 1 0 ) 1 (1 p 2 ) 3 = 2 p # , p 1 0 cos + 4 3 ˇ ( 1 0 ) cos 2 sin ;p 1 (5.45) 0 = 1 2 1 3 2 2 6 6 6 6 4 1 2 3 +(2 2 1) p 1+ 2 p 1+ 2 2 sinh 1 (1 = ) 3 7 7 7 7 5 ,(5.46) 1 =1 0 ,(5.47) R ( p )=cos 1 ( p ) p q 1 p 2 ,(5.48) p = 1 tan ,(5.49) where 0 [ dimensionless ] isthechannelexitcollisionparameter, 1 [ dimensionless ] isthechannelentrancecollisionparameter, R ( p ) [ dimensionless ] isthenoodleparameter,and p [ dimensionless ] isthenoodleangle. Aplotofthenormalizedangulardistributionisoverawiderangeofanglesandmag- towithinseveraldegreesinFigure5.9aandFigure5.9b.Therubidiumnozzle ( =0 : 01)isdesignedtocollimatethedistributiontowithinseveraldegrees.Bycontrast, theintensityofatomsexitingtheytterbiumandcalciumnozzle( =0 : 25)ist evenat50degrees.Inthecaseofanozzlewidthmuchlongerthanthenozzlelength 143 Figure5.10:Agridofthepointsusedtonumericallyintegratethesolidangleof acirculardetector.Westartwitha2 2squaremeshandcutoutacircle(shown withredsquares)toobtaintheresult. (think anellini ), j M approachesacosinedistribution: lim !1 j M ( )=cos 5.3.8Solidanglecalculation Atomicbeamuorescenceismeasuredwithanavalanchephotodiodewitha0.5mm diameteractivesurface(Thorlabs410-APD2).Iinvestigatedsolidanglecoveragecalcu- lationsusingbothapproximationandanumericalmethodfordi erentdetectorsizesand distancesfromtheuorescenceregion. The dA d = ~ d ~ r 2 terminEquation5.12canberewrittenasthesolidanglecoverage 144 ofthephotodetector: d det = dA d ~ d ~ r 2 (5.50) Thefi1 =r 2 flapproximation ˇ A det =d 2 y =3 : 3 10 5 srisreasonablyaccurateforthis geometry,butthisbreaksdownforlargerdetectorsorshorteruorescence-detectordis- tances. AmapoftheverticesofeachdetectorsurfaceareaelementisshowninFigure5.10. Tomakeuseofparallelprocessing,Iinitializethedetectorelementsasa squaregrid.ThenIapplyaboundaryconditiontouseelementswithintheradiusofthe detectortocalculatethesolidangle.Icalculate =3 : 27 10 5 srwiththeAtomicFlux detectorusing441elementswithsidelength R det = 10=0 : 25mm = 10=25 m. Whenthedetectorislarge( R det ˇ 12 : 7mm)orclose( d y ˇ 40mm)totheuorescence region,thesolidanglecalculationishighlydependentonthedetectorshape.Thisisalso trueforthelightcollectionimplementationthatIdiscussinSection5.5.2.Thesolidangle ofasquaredetector,suchasthatusedintheSAMsolidnoblegasmeasurement[144], deviatesfromthesolidangleofanequivalent-surfaceareacirculardetectorbytensof percentasthesolidangleiscomparedatdi erentpositionsintheuorescenceregion. 5.3.9Tyingeverythingtogetherintoanatomicbeamuorescencesimulation Theprevioussectionsofthischapterdescribethenecessarycalculationsthatareinputs foraABFsimulationcodeIdevelopedinthePythonprogramminglanguage.InSec- tion5.3.7,Ishowedanalyticexpressionsfortheatomicangulardistribution j ( ).In practice,thefunctionsaswrittendonotcaptureallthefeaturesofameasuredspectrum. TheABFsimulationprovidesacomparisontoameasuredspectrumandallowsusto numericallyderive j ( )andthee ectiveovennozzlegeometry. Thesimulationintegratesovertheinteractionvolumewheretheatomicbeampasses throughthelaserradiationd.Iapproximatethetruevolumeasasimplerightrectan- 145 Figure5.11:Thephoton-atomyieldpercentchangeasthenumberofsubdivi- sionsoftheuorescencevolumeisvaried.Themegacubesidelengthis32mm, thelaserwidthis7mm. gularprism,whichIasthefimegacubefl V h m 3 i : Z d V = ZZZ dxdydz = V fimegacube 00 Eachvolumeelement dxdydz iscalledafimicrocube.flThesetermsarede- pictedinFigure5.2. Istudiedthee ectofvaryingthemegacubeandmicrocubesize.InFigure5.11,I themegacubeto32 32 32mmcubefortheFluxABFovennozzle( =0 : 25)and variedthemicrocubesize.Ifoundthatthechangeinthecalculatedphoton-atomyield changedbylessthan.1%whenusingamicrocubesizeof1mmorless. Fortherubidiumoven( =0 : 01),theangularintensitychangestlyovereven onedegree,asshowninFigure5.9b.Thisrequiresanappropriatelysmallmicrocubeand iscomputationallyexpensive. 146 Figure5.12:Theintegralof intheplane y =0.Inthisplane,thephotode- tectorat y =76 : 2mmviewingangleisconstrainedbytheinnerdiameterofthe vacuumcross(30.226mm).Thescanningareaavailabletothephotodetectoris 15.52mmsquare. Theatomicbeamuorescencesimulationsdiscussedinthefollowingsectionsare computedwith68921microcubesofsidelengthof0 : 3902mminamegacubewitha 16mmsidelength. Figure5.12shows zx planecontourplotsoftheintegrandofthesolidangle,uores- cencerate,angulardistribution,andphoton-atomyieldforanytterbiumABFsimulation. ThesetermsrepresenttpiecesoftheABFsimulationandarecombinedinthe mannerprescribedbyEquation5.42.Each zx fisliceflof ;F ( ;~ r ) ; and j ( )isintegrated overaverticalrangecorrespondingtothemegacubesidelengthtodetermine . 147 5.4Comparingsimulationstodata 5.4.1Ybuorescenceandpowerbroadening ThecommissioningytterbiumABFmeasurementwasperformedin2017.Laserpower datawasrecordedbyaThorlabspowermeterthatmeasuredlaserintensitysampledfrom a8:92pelliclebeamsplitter.Weusedalaserscanstepsizeof9MHzattheTi:Sapphire outputwhichisfrequency-doubledto17.9MHzattheexternaldoublingcavity. IusedasumofsevenVoigt(convolutionofaGaussianandLorentziandistri- bution)plusaconstanto set C + P 7 i =1 V i tothespectruminFigure5.13a.Thedashed linesareindividualpeaks,andthesolidlineistheoverall ThetriplepeakconsistingofYb 172 andYb 171 ( F =3 = 2 ; 7 = 2)isdi culttodecouple giventherelativecoarsenessofthescansize.TheVoigtfunctionpreferstounder- weighttheamplitude 173 Yb( I =3 = 2)andover-weightthetwomorepopulatedstates. Toaconvergent,IcondensedthetriplepeakintooneVoigtThefrac- tionalresidualofthespectrumisshowninFigure5.13b.Themodelsthedatato within10%exceptfortheboundariesofthelaserscanandintheregionbetweenYb 174 andthetriplepeak. Atableofthecalculated,measured,andliteraturetransitionfrequenciesaregivenin Table5.8. TheGaussianwidths ˙ areallowedtovaryindependentlyandrangefrom40Œ80MHz. Thesaturationintensity I s ( ; a ) h Wm 2 i ofasingleatomisgivenbythefollowing: I s ( ; a )= A 2 ˙ ( ; a ) ,(5.51) where ˙ ( ; a )isthecrosssectionoftheatomgivenbyEquation5.24.InthecaseoftheYb P 1 o 1 transition, a =7 : 515 10 14 HzandIcalculatethefollowingresonantcrosssection 148 (a) (b) Figure5.13:Yb5/15/2017ABFmeasurement.Yb-172TP=triplepeakconsistingof Yb 172 ,Yb 173 ( F =7 = 2),andYb 173 ( F =3 = 2).(a)Seven-peakVoigt+constanto set todata(b)Fractionalresidualof( y axistruncatedforclarity). 149 Table5.8:Calculated,measured,andliteraturevaluesofthe 1 S 0 ! 1 P o 1 transition frequencieswithrespecttoYb 174 ( I =0 ;F =1). isotopeCalc. (MHz)Meas. (MHz)Ref.[147](MHz) 168 Yb( I =0 ;F =1)+1887 : 4 170 Yb( I =0 ;F =1)+1192 : 4+1183(18)+1192 : 393(66) 171 Yb( I =1 = 2 ;F =1 = 2)+1077 : 0+1106(67)+1153 : 696(61) 171 Yb( I =1 = 2 ;F =3 = 2)+755 : 76+849 : 6(59)+832 : 436(50) 172 Yb( I =0 ;F =1)+533 : 3+554 : 9(16)+533 : 309(53) 173 Yb( I =5 = 2 ;F =3 = 2)+491 : 11 173 Yb( I =5 = 2 ;F =5 = 2) 278 : 28 261(10) 253 : 418(50) 173 Yb( I =5 = 2 ;F =7 = 2)+563 : 12 176 Yb( I =0 ;F =1) 509 : 3 526 : 7(44) 509 : 310(50) andsaturationintensity: ˙ 0 = ˙ ( a ; a )=7 : 58 10 14 m 2 , I 0 = I s ( a ; a )=63mWcm 2 SaturationintensitiesfortheYb,Rb,andCatransitionsofinterestarelistedinTa- ble5.9. Iestimateabroadenedlinewidthof260MHzforFigure5.13a.Withalaserintensityof approximately I =863mW =ˇ (0 : 35cm) 2 =2240mW = cm 2 ,thesaturationfactoris: I =I 0 =36 Thepower-broadenedtransitionlinewidth A s [ Hz ] cannowbecalculated: A s 2 ˇ = A 2 ˇ 1+ I I 0 ! 1 = 2 ˇ 6 : 1 A 2 ˇ (5.52) FromthisIestimatealinewidthof A s = 2 ˇ ˇ 190MHz.Thelinewidthsrangefrom 150Œ245MHzinFigure5.13a.Theclosest-matchingtransitionisYb 176 withalinewidth of206 16MHz. InFigure5.14Ishowasimulatedytterbiumspectrumwithalaserintensityof: 10mW = ˇ 0 : 35cm 2 =26mW = cm 2 150 Table5.9:Saturationintensitiesandoscillatorstrengthsforselectedytterbium,ru- bidium,andcalciumtransitions. =frequency, A =EinsteinA-coe cient(NIST values). f a =oscillatorstrength. f a (Rb)from[146]. f a (Ca)from[152]. I 0 =satura- tionintensity. transition (THz) A (MHz) f a I 0 h mW = cm 2 i Yb6 s 2 S 1 0 ! 6 s 6 p 1 P o 1 751 : 531921.3763 Rb5 s S 2 1 = 2 ! 5 p P 2 o 1 = 2 377 : 1074336 : 00.34231(33)1 : 5 Ca4 s 2 S 1 0 ! 4 s 4 p P 1 o 1 709 : 0782352201.7561 Figure5.14:SimulatedYbuorescencespectrumintheweakpumpinglimit. Thisiswellbelowthesaturationintensityandintheweakpumpinglimit(Equation5.23. TheDopplerbroadeningistlyreducedandthe 170 Yband 171 Yb( F =1 = 2)peaks areeasilyresolved.Thepeakvoltageisontheorderofhundredsof V,whichwe'reeasily sensitiveto. InthenextYbABFmeasurement,wewillreducethelaserintensitytothesimulated intensityanduseasmallerlaserscanstepsizeof5MHztocontrolDopplerbroadening andimproveoursensitivitytoindividualtransitionsintheclusterpeak. 151 Table5.10:AselectionofatomictransitionsoftheRbgroundstate,5 s S 2 1 = 2 . IntensityvaluesandwavelengthsfromNIST,lifetimevaluesfrom[153]. I =intensity. =resonantwavelength,frequency. ˝ =lifetime. A =Ein- steinA-coe cient. excitedstate I (arb.) (nm) (THz) ˝ (ns) A (MHz) 5 p P 2 o 3 = 2 1000780 : 027384 : 2303526 : 25(8)38 : 1 5 p P 2 o 1 = 2 500794 : 760377 : 1074327 : 75(8)36 : 0 Allthenumbersusedforcalculatingtheux,excitationrate,andphoton-atomyield aregiveninTable5.7.NowIwillshowexplicitcalculationsforaselectionofthevalues. Origin-to-photodetectordistance r : ThedistancefromthecenteroftheatomicbeamtothefrontsurfaceoftheAPDdetector r [ m ] isthesumofthedistancesof(1)thecenterofthe2.75fl6-waycrosstothetopofthe e(KurtJ.LeskerC6-0275)(2)thewidthofthecageplate(ThorlabsLCP01)(3)the distancefromtheAPD(ThorlabsAPD410A2)etotheactivesurfaceofthedetector: r =62 : 484mm+12 : 7mm+2 : 2 0 : 3mm=77 : 4 0 : 3mm Mostprobableatomspeedalongnozzleaxis v p;z : Tothemostprobablespeed v p;z [ m = s ] ,Iusedanoventemperatureof T =573 : 15K andthemassof 174 YbfoundinTableA3.Thisgives v p =234 : 08m = s.Forauxcalcula- tion,weareinterestedinthecomponentofthevelocitythatisparalleltotheaxisofthe ovennozzle.Thereforeweneedtoknowthemaximumdivergenceangleoftheatomic beamexitingthenozzle.Theytterbiumovennozzlehasalengthof1/2flandadiameter of1/8fl.Ifwebisecttheconeformingtheboundaryofbeam,thedivergenceangleis: =arctan 0 : 5 0 : 1250 00 0 : 5000 00 =0 : 12435rad Thelongitudinalcomponentofthemostprobablevelocityisgivenby: v p;z =cos ( ) v p =232 : 3m = s 152 Table5.11:Calculatedrubidiumtransitionfrequen- cies(h+isotopeshifts)withrespecttothe transitionof 85 Rb, 0 85 Rb = S 2 1 = 2 ! P 2 o 1 = 2 = 377.107THz. isotope 5 s S 2 1 = 2 ! 5 s P 2 o 1 = 2 0 85 Rb [ MHz ] F ! F 0 85 Rb2 ! 2+1560 : 0 85 Rb2 ! 3+1921 : 5 85 Rb3 ! 2 1475 : 8 85 Rb3 ! 3 1114 : 3 87 Rb1 ! 1+3840 : 5 87 Rb1 ! 2+4654 : 5 87 Rb2 ! 1 2994 : 2 87 Rb2 ! 2 2180 : 2 Laser-atominteractionvolume V : I'massumingthattheinteractionvolume V h m 3 i isa2mmcube: V = ` 3 =8 10 9 m 3 The2017Ybmeasurementusedahighlaserpowerthatdoesnotsatisfytheweak pumpinglimit(Equation5.23).TheABFsimulationcodeisintendedforweakpumping limitanalysis. 5.4.2Rubidiumuorescence Atableofrelevantpropertiesofthe5 2 S 1 = 2 ! P 2 o 1 = 2 transitionareshowninTable5.10. SeventeenABFmeasurementswereperformedwitha0.54cmlaserdiameteratpow- ersrangingfrom10 Wto9.8mWandoventemperaturesrangingfrom25Œ220 C[144]. ATi:SapphireidenticaltothatoftheAtomicFluxlaser(Figure5.3)isusedtogenerate the795nmlaserbeam.Thebeamispickedo attheTi:Saphoutputandbeforeany frequencymixingordoubling.Thelaserlightislinearlypolarizedbutis-coupledto 153 theuorescencechamber.Thedoesnotconservepolarizationandweassumethat thelightisunpolarized,orequalparts ˙ + , ˇ ,and ˙ components. IperformedofeachrubidiumisotopepeakinthespectrawithVoigtlinepro- Ialsoincludedaconstanto settothebackground.EachVoigtpeak V i has fouradjustableparameters:theGaussianstandarddeviation ˙ i ,thepeakcenter i ,the Lorentzian FWHM i ,andtheamplitude C i .Foreachdataset,IallowedtheLorentzian FWHM oftheRb 85 ( F =3 ! F 0 =2)varywithinbounds,thenthatvalueforthere- mainingpeaks.Thisleavesatotalof1peak 4+7peaks 3+1background=26free parametersforeachdataset. IcalculatedthetotaltransitionfrequencygivenforeachrubidiumisotopeinTa- ble5.11.ThelaserfrequencyaxisoriginisbysettingtheRb 85 ( F =3 ! F 0 =2)peak center 1475 : 8MHzfromtheorigin. ArepresentativerubidiumABFspectrumandassociatedresidualisshowninFig- ure5.15aandFigure5.15b.Peakwidthsrangefrom350Œ500MHz,tlylarger thanexpected.Theresidualshowsgeneralagreementtowithin20%,withdiscrepancies aslargeas300%inthepeak-freeregions.Atlowerlaserpowers(tensof W),thedis- crepancyinpeakheightismorepronouncedasthepeaksaresharper.Asweincrease thelaserpowertowards9.8mW,thebroaderpeaksaremorecloselymatchedbyaVoigt curve. Isimulatedarubidiumspectrumwithalaserpower50 Wcorrespondingtoalaser intensityof0.22mW/cm 2 inFigure5.16a.Atthislaserintensity,Icalculateanon- resonanceatomexcitationrateof R ( 0 ) ˇ 1 : 3 10 6 s 1 .ThelifetimefromTable5.10is 27.8ns,sothissimulationsatheweakpumpingrequirement R˝<< 1. Ichoseanoventemperatureof100 Ctoconstraintheangulardistributiontothe molecularowregime.Thepeakwidthsare20MHz,afactoroftwentysmallerDoppler broadeningthanthemeasureddata.Notethatabove100 C,theKnudsennumberfor therubidiumovenis K n < 10andtheMaxwellianandmolecularowtreatmentthatwe 154 (a) (b) Figure5.15:AmeasuredrubidiumABFspectrumwithalaserpowerof50 W.(a)Voigt lineshapetouorescencesignalvs.laserfrequency(b)Fractionalresidualofthe 155 (a) (b) Figure5.16:SimulatedRbuorescencespectrumintheweakpumpinglimit.Laser power=50 W,laserradius=2.7mm.(a)Collimatedbeamwithnozzleratio =0 : 01 (b)uncollimatedbeamwithnozzleratio !1 . 156 usebecomesanincreasinglycrudeapproximation. Figure5.16aassumesthemachineddimensionsoftherubidiumnozzleratioof =0 : 01. Therangeofoventemperaturesinthemeasureddataisreportedintherangeof25Œ220 C. Becausethemeasuredlinearewiderthanexpected,itraisesthenefariouspossi- bilitythatadatarunat ˇ 220 Csomeofthemetallicrubidium.Inthisscenario, someoftherubidiumcouldhavefileakedfloutoftheovencrucibleandsomefractionof thewaydownthenozzle[144].Indeed,Inotedacolorlessonthesurfaceoftheoven cruciblewhentroubleshootingtheSAMABFsetup. Wecaninterpretthepotentialleakagedistanceoftherubidiumasafreeparameterof thenozzleratio.Forexample,iftheliquidtraveledhalfwaydownthenozzle,thiswould doublethee ectivenozzleratio.Iftheliquidtraveled100%downthenozzle,thiswould e ectivelybeacompletelyuncollimatedovensource( !1 ). Toinvestigatetheuorescenceforanuncollimatedatomicbeam,Irepeatedthesimu- lationwiththesameovenandlasersettingswhilesettingthenozzleratioto !1 .The fullwidthofthetransitionsinFigure5.16bis130MHz,aboutathirdofthemeasured peakwidths.Thisisclosertowhatismeasured,thoughthepeaksarestillafactorof ˇ 3 narrowerthanthemeasureddata. MovingbeyondtheABFsimulation,Ifurtherinvestigatethee ectofrubidiumleak- agewithanoventemperatureof220 C.Iassumearubidiumnozzleratiothat allowsnonzero j ( )at ˇ 20degrees,forexample ` ˇ 0 : 25.Withtheseparameters,Ies- timateamaximumbroadeningof225MHzusingEquation5.40.Thisisstilltly smallerthanthemeasuredspectrumpeakwidths,suggestingthatadditionalfactorsare contributingtothebroadening. Nozzle-laseralignment,oventemperature,laserintensity,backgroundlight,andlaser polarizationdi erencesbetweenexperimentandsimulationareallpossiblecontributing factorstothediscrepancyinlinewidth.Thenozzleandlaseraxisarenominallyperpen- diculartoeachother.Amisalignmentwouldintroducelargeranglesbetweentheatom 157 (a) (b) Figure5.17:Voigttosimulateduorescence(redcircles)withcollimatedanduncol- limatedangulardistributions.(a)Collimateddistribution,correspondingtooneofthe peaksinFigure5.16a(b)Uncollimateddistribution,correspondingtooneofthepeaksin Figure5.16b. 158 Figure5.18:ResidualsoftosimulatedRbtransitionsinFigures5.17b,5.17a. andlaserphotontrajectories( )andincreaseDopplerbroadening. Assumingtheexcessbroadeningispurelyduetoamisalignmentbetweenthenozzle axisandlaseraxis,Icalculateamisalignmentof ˇ 20degreeswouldaccountforthe observedpeakwidths. We'veseenthatthemeasuredrubidiumtransitionsaren'tcompletelycapturedbythe Voigtlineshape.ThetruelineshapeofadirectedatomicbeamisthegeneralizedDoppler- broadenedexpressiondiscussedinSection5.3.5. WiththegeneralexpressionforDopplerbroadening,Isimulatedasinglerubid- iumpeakwithanozzleratio =0 : 01(Figure5.17a)and !1 (Figure5.17b).Then IVoigttothepeaks.TheovenandlasersettingsareidenticaltoFig- ure5.16a.Thecollimatedtransitionissharplypeakedandrequiredalaserscan steptocapturetheshape.Fromthe,Ithatthecollimatedlinewidthisnar- rowwith FWHM =23 : 88+ = 0 : 10MHzandtheuncollimatedtransitionisbroadwith FWHM =120 : 69 0 : 79MHz. 159 Figure5.19:Measuredtotalstrengthfactorratios S FF 0 = S 32 ofRb 85 .Thehor- izontallinesareexpectedtotalstrengthfactorratiosforanunpolarizedlaser beamusingthevaluesfromTable5.4.Dashedline S 23 = S 32 =1;dot-dashed line S 33 = S 32 =0.8;dottedline S 22 = S 32 =0 : 2857 TheVoigtstrugglestosimultaneouslyreproducethetransitionpeakandtailsofa uorescencespectrumofadirectedatomicbeam.Forthecasesofahighlycollimatedand uncollimatedovennozzle,thetransitionpeakandtailsareunderestimatedintheformer andoverestimatedinthelatter.Thepeakmismatchisclearlyseeninthefractional residualsforbothangulardistributionsinFigure5.18.Theuncollimatedisaccurateto withinapproximately4%within10MHzofthepeak,owingtothetransitionbroadness. O scale,theis10%-accuratewithin75MHzoftheresonancebutthendivergesby morethan+1000%asonemovesfurtherout.Thecollimatedisaccuratetowithin approximately5%within6MHzoftheresonanceandthensharplyconvergesto ˇ 100% fartherfromresonance. Iintegratedthepeakareasoftheseventeenmeasuredspectratothetotal strengthfactorsandplotted S FF 0 = S 32 0 (Figure5.19)and S FF 0 = S 22 0 (Figure5.20)asa 160 Figure5.20:Measuredtotalstrengthfactorratios S FF 0 = S 22 ofRb 87 .Thehor- izontallinesareexpectedtotalstrengthfactorratiosforanunpolarizedlaser beamusingthevaluesfromTable5.4.Dashedline S 21 = S 22 =1,equivalently S 12 = S 22 =1;dottedline S 11 = S 22 =0.2 functionoflaserpower.Thestandarddeviationsarecalculatedfromtheuncertainty inthespectrumThesemeasuredratiosarecomparedwiththepredictedstrength factorsforperfectlyunpolarizedlightfromTable5.4. In 85 Rbplot,thestrengthfactors S 23 , S 22 ,and S 33 aredividedby S 32 .Dashedhori- zontallinesarethepredictedratios S FF 0 = S 32 whicharecalculatedfortheconditionthat thelaserlightisperfectlyunpolarized.Theplottedratiosareallsmallerthanexpectedup tolaserpowersof1mW,indicatingthat S 32 islarge.Theintensitiesofthe F =2 ! F 0 =3 and F =3 ! F 0 =3transitionsincreaseasthelaserpowerisincreasedfrom1Œ10mW, invertingtherelationshipbetweenthetransitionsataround5mW. FortheRb 87 plot,Idividethestrengthfactors S 21 , S 12 ,and S 11 by S 22 .Theratios S 21 = S 22 and S 12 = S 22 arepredictedtobeunity.Instead,thetworatiosareanticorre- lated,with(2 ! 1)athigherintensitythanexpectedand(1 ! 2)is20Œ40%lowerthan 161 Figure5.21:MeasuredabundanceratioofRb 87 toRb 85 .Dashedline=0.3856 isthecalculatedratiousingtheNISTdatabasevalueslistedinTableA5. expected.Startingatalaserpowerofapproximately300mW, S 21 approachesunitylog- arithmicallyand S 12 deviatesbyasimilaramount.Atthemaximumpowerof10mW, S 21 ˇ 1 : 4and S 21 ˇ 0 : 5 InbothFigure5.19andFigure5.20,theweakeststrengthfactorsaretheleastsensitive tolaserpower. Similarly,Icalculatedtheisotopicabundancesandplottedtheisotoperatioasafunc- tionoflaserpowerinFigure5.21.Atthelowestlaserpowerof10 W,themeasured abundanceratioisconsistentwiththeNISTvalueofRb 87 = Rb 85 =0 : 3856.Theratio increasesto0.46whenthelaserpowerisincreasedto30 W.Asthelaserpowerisin- creased,theratioincreasesapproximatelylogarithmicallyto0.53at10mW. Theseresultssuggestthattherearelaserpower-correlatede ectsthatuenceboth thestrengthfactorsandtheisotopicratios.FromTable5.9,thesaturationintensityof rubidiumisaround2mW/cm 2 .Assumingthelaserradiusis0.27cm[144],Iexpectthe 162 Table5.12:AselectionofatomictransitionsoftheCagroundstate,4 s 21 S 0 .Inten- sityvaluesandwavelengthsfromNIST. 3 P o 1 lifetimefromDrozdowskiet.al[154]. I =intensity. =resonantwavelength,frequency. ˝ =lifetime. A =Einstein A-coe cient. excitedstate I (arb.) (nm) (THz) ˝ (ns) A (MHz) 4 s 5 p P 1 1 o 140272 : 16451101 : 18613.7 10 3 0 : 27 4 s 4 p P 1 1 o 1000422 : 6727709 : 0782354.5220 4 s 4 p P 1 3 o 500657 : 2777455 : 9862175.7(3) 10 5 0 : 0018 ratiostobeinsensitivetolaserpowersupto ˇ 200 W.However,weseethesurprising resultthattheratiosarea ectedbythelaserpowerwellbelowthisthreshold. 5.4.3Simulationsofacalciumspectrum Isimulatedacalciumuorescencespectrumofthe4 s 4 p P 1 o 1 transition( ˝ =4 : 5ns)inFig- ure5.22.AtomicpropertiesofthistransitionandseveralothersarelistedinTable5.12. Iusethesameovennozzledimensions( =0 : 25)andlasersettings( P =10mW, r =3 : 5mm, I =26mW/cm 2 )astheytterbiumsimulationinFigure5.14.TheCasimula- tionalsosatheweakpumpinglimitrequirement R ( ;~ r ) ˝ 0 << 1. CalculatedandliteraturetransitionfrequenciesarelistedinTable5.13.Thetransition intensityofthemostabundantisotope, 40 Ca,willbemorethananorderofmagnitude largerthanthenextmostabundantisotope.Thebottompanelisalog-scaleplotthat showsthesmallerpeaks. Thesimulatedcalciumsignalofthedominantpeakisapproximately50nV,severalor- dersofmagnitudelowerthantheytterbiumuorescence.Thisisbecauseoftherelatively lowvaporpressureofcalcium.Theoventemperaturecanbeincreasedtocompensatefor thelowvaporpressure,butthiswillalsoincreasetheDopplerbroadening.Becauseof thesmalleratomicmass,thecalciumpeakswillbetlywiderthantheytterbium peaksforequivalentoventemperatures. WewillusetheAtomicFluxapparatus,picturedinFigure5.23a,tomeasurethe 163 (a) (b) Figure5.22:Simulatedcalciumuorescencespectrumintheweakpumpinglimit.Log scalecalciumuorescencespectrumsimulationtoshowtheweakertransitions.Thesmall signaldiscontinuitiesat600MHzand1400MHzarenumericalartifacts. 164 Table5.13:Calculatedandliteraturetransitionfrequencies (h+isotopeshifts)withrespecttothetransitionof 40 Ca, 0 40 Ca = S 1 0 ! P 1 o 1 =709.078THz.Reference valuefor 46 Ca from[155],allothersfrom[156]. A 4 s S 1 0 ! 4 p P 1 1 0 ( 40 Ca) [ MHz ] F ! F 0 CalculatedReference 420 ! 1+393.5+393.1(4) 439 = 2 ! 9 = 2+555.3 9 > > > = > > > ; +611.8 a +610.7(6) a 437 = 2 ! 7 = 2+634.2 435 = 2 ! 5 = 2+676.2 440 ! 1+773.8+773.8(2) 460 ! 1+1159.8+1159.8(7) 480 ! 1+1513.0+1513.1(4) a ficenterofgravityflvalues,whichistheaverageofthe hpeakfrequenciesweightedbytheirrelative isotopicabundances. atomicbeamuorescenceofcalcium.IexpecttoimprovetheABFmeasurementsen- sitivitybyafactorof100ormorewiththeadditionofalightcollectionsetup(discussed inSection5.5.2).ThelightcollectiongainwillamplifythecalciumAPDsignaltoap- proximately10 V,areadilymeasurableuorescencesignal.Detectingthenextmost abundantisotope 44 Capeak,ontheorderof1nVwithoutlightcollection,wouldbea powerfuldemonstrationoftheABFmeasurementsensitivity. Theoventemperatureandlaserpowercanalsobecautiouslyincreasedtoboostthe uorescencesignal.Atanoventemperatureof250 C,thecalciumlinewidthisal- readytlyDoppler-broadenedwith FWHM =200MHz.However, 40 Caisapprox- imately400MHzfromtheneighboring 42 Capeak,soonecantradeo theadditional broadeningifahigheruorescencesignalisneeded.Thesimulatedlaserintensityis 10mW/ ˇ (0 : 35cm) 2 =26mW = cm 2 .Thiscouldbeincreasedbyuptoafactorof2and stillremainbelowthesaturationintensity. 165 5.5Suggestedimprovementstomeasurementtechnique 5.5.1Trackinglaserpolarizationandmagneticd CompetingpropertiesoftheABFmeasurementcouldbedrivingthehtransition strengthfactordependenceonpumpinglaserpower,forexampletheRb 85 S 32 transition inSection5.4.2. TheweakpumpingthresholdfromEquation5.23isapproximately200 W,assum- ingapumpinglaserradiusof0.27cm.Thefourlowest-powerratiosinofeachofthe Figures5.19,5.20,and5.21rangefrom10Œ200 W.Withaperfectlyunpolarizedlaser beamandnegligiblemagneticds,Iexpecttherubidiumtransitionratiostobeinsen- sitivetolaserpowersbelowthe200 Wthreshold.Instead,weseelaserpower-correlated trendsinthedatastartingatthe10Œ50 Wrange. The795nmTi:Sapphirelaseroutputislinearlypolarized.Theoutputis-coupled totheABFchamberwithanopticalthatisnotpolarization-maintaining,sowe assumethepumpinglaserlightisperfectlyunpolarized.Itmaybethatsomeresidual polarizationispresentaftercoupling.Atdegreeoflinearorcircular polarizationwouldmodifythemagneticsubleveltransitionamplitudes.Mystrength factorcalculationsareonlyvalidforanunpolarizedlaserbeam. It'salsopossiblethattheambientmagneticd,whichwetaketobeonthescale ofEarth'sd( ˇ 60 T),istlya ectingthetransitions.However,thelitera- turesuggeststhattransitionprobabilitydependenceonexternalmagneticdsarenot tbelow ˇ 1mT[157]. SeveralimprovementscanbemadetofutureABFmeasurements.Tocontrolmag- neticd-correlatede ects,wecanscreenexternalmagneticdsandmeasurethe magneticduniformityintheuorescenceregion,e.g.withuxgates.Wecanmea- surethepumpinglaserpolarizationtoverifyourassumptionsaboutthestrengthfactor calculationsandteroutanyresiduallinearorcircularpolarization,ifnecessary. 166 (a) (b) Figure5.23:(a)theAtomicFluxapparatus.(b)Schematicofin-vacuumlight collectionsetup(nottoscale). Inthecaseoftherubidiumexperiment,themajorityofthemeasuredratiosdonot satisfytheweakpumpingcondition R˝<< 1.OurnextABFexperimentshouldconcen- tratedatacollectionatlowlaserintensitiesintheweakpumpingregime.Forthecase ofa0.27cmlaserdiameter,mostoftheuorescencemeasurementscouldbemadebe- low200 W.Weobservedlaserpower-correlatede ectsaslow50 Wcorrespondingto 0.22mW/cm 2 .Itwouldbeveryinterestingtoseemoremeasuredratiosatbelowthis intensity. 5.5.2Increasingthesignalsizewithlightcollection Alimitingfactoronthephoton-atomyield isthesolidanglecoverageofthephotode- tector.Icalculateasolidangleof ˇ 33 rad(Figure5.12)inthecenteroftheuorescence regionoftheAtomicFluxapparatuswithour0.5mmdiameteractivesurfaceAPDposi- tioned77mm(Table5.7)away. Toimprovethesolidangle,Iconsideredtwoscenarios.Inthescenario,theAPD canbemovedclosertotheuorescenceusingasmallervacuumchamber.Ifwereplace 167 Figure5.24:Theatom-to-photonyieldifweusealight-focusinglens,or,equiv- alently,increasethedetectorarea.Thelaserwidthis7mminthiscalculation. Assumingonlyraysperpendiculartothedetectorsurfacearefocusedontothe detector,wegetmaximumlightcollectionsforadetectorradiusofhalfthelaser width,or3.5mm. our2.75flwindowvacuumcrosswithacommercial1.33flsetup,thedistancefromthe uorescencecentertotheAPDwouldbereducedbyapproximatelyhalf,or d y ˇ 30mm. Iestimateafactorofapproximately5increaseinthesolidanglebyreducingtheAPD distanceinthismanner. Themorepromisingscenarioistoplacealightcollectionlensinthevacuumchamber betweentheuorescenceregionandtheAPD.Idesignedanin-vacuumlightcollection setup,showninFigure5.23b.Thelensisa1/2flbiconvexlenswitha20mmfocaldistance thatismountedtoa16mmcageplate.Thecageplateismountedtothebottomvacuum eusingthreadedrods.Thedistanceofthelightcollectionlenstothecenterofthe uorescencevolumeisadjustablesothepositioncanbeoptimizedformaximumsignal 168 Figure5.25:Theatom-to-photonyieldaswevarythelaserbeampower. cube sidelengthis0.5mm,megacubesidelengthis3.2cm. max =1 : 523for w =7 mm. onthephotodetectoractivesurface. Thephoton-etayieldincreaseswithincreasingsolidanglecoverageofthephotode- tector(Equations5.42and5.50).Toestimatethegaininthesolidangle,Imakethe assumptionthatalluorescencephotonsreachingthelightcollectionlenssurfaceisfo- cuseddirectlyontothephotodetector.AtthecenteroftheuorescencevolumeIcalculate anenhancedsolidangleof78mrad,afactorof2400higherthanasetupwithoutlight collection. Since isavolumeintegral,thesolidangleenhancementwillvarybyuorescence location.Tosimulatetheenhancementin Iapproximatethelightcollectiongainby increasingthephotodetectorarea.Aplotoftheatom-to-photonyieldasweincreasethe detectorareaisshowninFigure5.24.FromtheplotIestimatethatthelight-collecting lenswillincreaseuorescencesensitivitybyatleastafactorof100. 169 Figure5.26:Theatomphotonyieldaswevarythesizeofthelaserbeamwidth. cubesidelengthis0.5mm,megacubesidelengthis3.2cm. max =1 : 523for w =7mm. 5.5.3Increasingthesignalsizewithacalibratedpumpinglaserandatomicoven Istudied dependenceonarangeoflaserpowerandbeamwidthsforasimulatedyt- terbiumABFmeasurementwithovennozzleratio =0 : 25.InFigure5.25Iplotted forlaserpowersrangingfrom1 WŒ1W,keepingthelaserwidthat1mm.Ialso plottedthephoton-atomyieldforlaserwidthsrangingfrom1mmŒ10mm,keepingthe powerinFigure5.26.Iexpectthatthetrendsinthesearealsorelevant forthecalciumABFmeasurement,whichwillusethesameovennozzleratioanda 1 P o 1 atomictransition. Theseplotsshowthatthelaserwidthcanbecalibratedtomaximize foragivenlaser intensity.Thesimulateddatadependsontheatomicdistribution j ( ).We'veseenfrom therubidiumanalysisinSection5.4.2thatourunderstandingoftheatomicdistribution 170 islimited,sothesimulatedoptimizedlaserwidthshouldonlybeusedasastartingpoint. AcarefulcalibrationoftheABFmeasurementparametersisespeciallyimportantfor calcium,whichisexpectedtohaveaverysmalluorescencesignal.Thephoton-atom yielddependsontheoverlapbetweentheatomicangulardistributionandthepumping laser.Theatomicangulardistributioniscontrolledbytheovennozzleratio.Thelaser beampoweranddiametershouldbechosentomaximize whilesatisfyingtheweak pumpingrequirement. 171 CHAPTER6 PRECISIONGAMMA-RAYINTENSITYMEASUREMENTS AsafellowoftheNuclearScienceandSecurityConsortium(NSSC),Ihadtheopportu- nitytoresearchnuclearsecurityapplicationsinnuclearphysics.IwenttoLawrenceLiv- ermoreNationalLaboratory(LLNL)todevelopanewprecisiongamma-rayspectroscopy experimentfromJanuary12019toMarch292019. 6.1Introduction 6.1.1Gamma-rayspectroscopyandstockpilestewardship TheUnitedStatesandothernuclearpowersminimizenuclearweaponthreatsbynego- tiatingtreatiesthatlimitnuclearweaponstockpileinventoryandbanintrusiveweapons testing,includingabove-groundandundergrounddetonation.Thesetreatieshavemech- anismsthatprovidememberswithrightstolimitedinspectionseachother'sstockpiles andvalidatethenumberofstockpiledwarheads. Fissilematerialoccasionallyfallsundertheinvestigativepurviewofnuclearsecurity, forexampleinthecaseofstockpilevtion,recentweaponstesting,orweaponstraf- Thesizeandnatureofasamplecanbedeterminedbymeasuringtheintensity oftheradiationofthedaughterisotopesinthesample.Ahighprecisionradiationmea- surementcanprovideinsightonthecompositionandtimescaleoftheoriginalmaterial thesampleisderivedfrom. Gamma-rayspectroscopyisusedtocharacterizenucleardecayspectratoderivenu- clearpropertiesfromisotopesofinterest.Onefacetofgamma-rayspectroscopyisstudy- ingthepossibledecaypathsanexcitednucleuscantakeinordertodecaytoamorestable nucleus.Thisisparticularlyusefulapplicationfornuclearforensicsandnuclearsecurity, asthenumberofthatoccurredinanuclearsamplecanbequanbymea- 172 suringtheintensityofthephotons,orgamma-rays,emittedbythenucleiofthe daughterisotopes. 6.1.2Long-livedisotopes Nuclearinducedcanbeinitiatedbyimpinginganeutronona 235 Unucleus.To alessere ect,canalsobeinitiatedbyotheruraniumisotopesandsomethorium andplutoniumisotopes.We'lllimitthescopeofthisdiscussiontoa 235 Unucleus. Aftertheincomingneutroniscapturedbythe 235 U,anexcitedstateof 236 Uis formed.The 236 Unucleuswillprimarilyintotwounstableisotopesofmassnum- ber A 1 ˇ 90and A 2 ˇ 145,roughlya2:3ratio.236 A 1 A 2 freeneutronswill beejectedaswell.Theprocessisstatisticalsothenumberofnucleonsineachisotope daughteructuates. Figure6.1isasimexampleofasim A =147decaychainofanuclear isotope. 147 Ceisthe 236 Udescendentinthisdecaychain. 147 Cehasa half-lifeof56secondsandarelativeproportionofdecaysfromitsparentnucleus,or independentyield(IY),of1 : 9%.Thedaughterisneutron-richandwilldecaytoamore stablenucleusbyconvertingoneofitsneutronstoaprotonviatheweakforce: A Z X ! A Z +1 X 0 + e + e ,(6.1) where A Z X istheoriginalnucleuswith A nucleonsand Z protons, A Z +1 X 0 istheproductnucleuswith A nucleonsand Z +1protons, e isanelectron,and e isanelectronantineutrino. 173 Figure6.1:Asimexampleofoneofthepossible 236 Udecaychains.Data from[158]. 174 Table6.1:Gamma-raydecaysfromaselectionoflong-livedisotopes. (BR)=branchingratiouncertainty. isotope -ray(keV)Q-value(keV) (BR)(%)half-life(days)Ref. 95 Zr724 : 192 0 : 0041123 : 6 1 : 80 : 5064.03[159] 95 Zr756 : 725 0 : 0121123 : 6 1 : 80 : 4064.03[159] 156 Eu811 : 77 0 : 052449 58 : 215.19[160] 147 Nd531 : 016 0 : 022896 : 0 0 : 92 : 210.98[161] 147 Nd91 : 105 0 : 002896 : 0 0 : 92 : 510.98[161] 144 Ce133 : 515 0 : 002318 : 7 0 : 81 : 7284.91[162] 161 Tb74 : 56669 0 : 00006593 : 0 1 : 34 : 96.89[163] 127 Sb685 : 5 0 : 51581 55 : 63.85[164] 111 Ag342 : 13 0 : 021036 : 8 1 : 44.97.45[165] 147 Cebeta-decaystoanewisotopewithalongerhalf-life.Afterseveralbeta-decays downthechaininFigure6.1,thehalf-lifeapproachesanorderofhoursordays.Thisis longenoughthatasampleofsuchmaterialcouldbetransportedfromascenetoalabo- ratoryforspectroscopyanalysis.Theseisotopesareknownaslong-livedisotopes. Inordertousefullyquantifythenumberofnuclearthatoccurredinasample ofdecayedmaterial,anuncertaintyof2%orbetterinthebranchingratiooftheisotopeof interestisdesired.Table6.1showsarepresentativelistoflong-livedisotopes.The primarybeta-decaybranchingratiosof 95 Zrareknowntosub-percentprecision.The branchingratiouncertaintiesoftheotherisotopesarerelativelylarge,rangingfrom1 : 7% to8 : 2%.Thereasonsfortherelativelyimprecisemeasurementsoftheseisotopes'decay propertiesvary.Itmaybeduetousingimpuresamples,havinginsu cientcounting statistics,orinternalconversioncompetingwith decay.Sometimesthesourcesare di culttofabricate,forexampleiftheacceleratorusedtoproducethesourcecannot deliverasu cientlypureand/orintensebeam.Thesubstratetheisotopeiscollectedon mayattenuatethesignalifthegamma-rayofinterestislow-energy. Aprecision branchingratiomeasurementforlong-livedisotopesisbeing developedatLawrenceLivermoreNationalLaboratory.Thenewmethodusesthinsam- plesproducedattheCaliforniumRareIsotopeBreederUpgrade(CARIBU)andanearly 175 Figure6.2:LLNLgamma-raydetectorsetup. 100%-e cient4 ˇ betacounter.Aproof-of-principlemeasurementwasperformedusing ahighpuritygermanium(HPGe)detectormeticulouslycalibratedatTexasA&MUniver- sity[166,167]. In2017theymeasuredthetwoprimarybranchingratiosof 95 Zr[168].Thetwopri- mary decays, 1 (keV)=724 : 2and 2 (keV)=756 : 7,weremeasuredwith0.6%preci- sion,inagreementwiththeliteraturevaluesshowninTable6.1[169]. TheZrmeasurementvalidatedthenewsamplepreparation,calibratedgamma-ray detection,and -coincidencemeasurement.ThegoaloftheLLNLgamma-rayspec- troscopygroupistoimprovethebranchingratiomeasurementsofadditionallong-lived isotopessuchasthoselistedinTable6.1tobetterthan1%precision. Forthenextphaseofthelong-livedisotopegamma-rayspectroscopyexperi- ment,theisotopesamplewillbemeasuredinanewdetectionsystematLLNL,shown inFigure6.2.Thenewly-assembleddetectorisabroadenergygermanium(BEGe)de- 176 Figure6.3:AschematicoftheLLNLBEGedetector.ModelbyCanberra,Mirion Technologies.Usedwithpermission. tector(schematicshowninFigure6.3),so-namedforitssensitivitytogamma-raysinthe range ˇ 0 : 01Œ3MeV.Thismodelhasaparticularlythinfrontlayerofinactive,orfideadfl germaniumofabout0.3 m,morethananorderofmagnitudethinnerthanastandard HPGedesign.Unlikestandardficoaxialfldetectors,theBEGedoesnothaveat bullet-holedesignonthebottomsurfaceofthedetector,whichpreservesthevolumeof activegermaniumandsimmodeling. TheLLNLBEGedetectorismountedonanultralow-backgroundpream(Can- berraiPA)insidealeadshieldwithanadditionalinnerlayerofhighpuritycopper(Can- berra777series),showninFigure6.2.Theshieldblocksbothexternale ectslikecosmic radiationandemissionfromtheleadliningitself.DataacquisitionishandledbyaCAEN DT5780dualdigitalmultichannelanalyzer. 177 6.1.3HPGecalibration Inordertomeasuretheintensityofagamma-ray,thee ciencyofthedetectormustbe knownatthatgamma-rayenergy.E cienciesatseveralenergiesaretypicallyprovided bythemanufacturer,andinmanycasesinterpolationissu cienttocalculatereasonable gamma-rayintensitiesofwell-understoodisotopes.However,highprecisionmeasure- mentsofhard-to-measureisotopesrequirecorrespondinglywell-calibrateddetec- tors.Adetectorcanbecalibratedbyusingagamma-raysourceofaknownintensity andderivingthee ciency.We'reinterestedinthee ciencyofthefullgamma-rayde- position,orthefull-energypeake ciency[170].Thisignoresgamma-raysthatpartially deposit,orscatter,inthedetector.Standardizedsamplesofreferencesources,suchas 152 Eu, 241 Am,and 60 Co,arereadilyavailableo theshelf. Thefull-energypeake ciencyofadetectoratgamma-rayenergy E ,or ( E ),is givenbythefollowing: = R S P ,(6.2) R = NT 1 ,(6.3) S = A 0 exp ( ) ,(6.4) where R h s 1 i isthefull-energypeakcountrate, T [ s ] isthedetectorlivetime, N [ dimensionless ] isthedetectorcount, S [ Bq ] isthesourcestrength, A 0 [ Bq ] isthesourceinitialactivity, h s 1 i isthesourcedecayconstant,and P istheprobabilitythatthesourceemitsaphotonat E ,i.e.itsbranchingratio. Forthisexperimentit'susefultousethehalf-liferatherthanthedecayconstant.The 178 (a) (b) Figure6.4:(a)Geant4modelofthegamma-raysource.(b)Expandedschematic ofthegammaraysourcegeometry(nottoscale). twoshareaninverserelationship: t 1 = 2 = ln2 ,(6.5) where t 1 = 2 [ s ] isthehalf-life.Wemeasuredtheabsolutegamma-raydetectione ciency ofthedetectorusingasetoftenstandardizedsourceswithgamma-rayemissionspanning 14keVŒ1 : 4MeV.Thesourcesarecalibratedisotopeswithwell-understoodgamma-ray branchingratiosarehigh-intensityandwell-measuredatusefulenergies.Thegeometry isaTypeMfiscatterlessflgeometry(showninFigure6.4). Eachsamplehasathin3mmdiameterdepositionofanisotopewhoseprimarygamma- raybranchingratioismeasuredtosub-percentlevel,exceptfor 241 Am,whoseprimary gamma-raybranchingratio( =59 : 54keV)ismeasuredtowithin1 : 1%.Theinitialactiv- itiesofthesamplesaremeasuredtowithin3%bythemanufacturer(Eckert&Ziegler). Tomeasurethee ciencyspectrumofthedetector,thesamplesweremountedona plasticholderasshowninFigure6.5.TheholderissimilartoastackableCDholder.The plasticholderhasatraydesignforloadingsamplesatdi erentheights.Thereisnobase soitcanbecenteredoverthedetector. 179 Figure6.5:Schematicofdetector-sampleation. Wetookasuiteofgamma-raysourcemeasurementsfortwodi erentsample-source distances.Measuringe cienciesattwodi erentdistanceso ersarobustwaytobench- markanumericalmodelofthedetectorsystem,sinceanon-axisdisplacementwould resultinane ciencychangeduesolelytogeometry.Foreachmeasurement,weplaced thesampleatoneoftwodistancesandcollectedstatisticsuntilenoughcountswerein thepeakscorrespondingtotheprimary decaybranchingratioenergiesofthesample. Anexampleofameasuredspectrumofa 60 CosourcesisshowninFigure6.6. Thegamma-raycollectione ciencyofthedetectorscalesas d 2 ,where d isthedis- tancefromthesourcetothefrontsurfaceofthedetector.Thesampleshouldbeclose enoughtocollectenoughstatisticsinareasonableamountoftime( ˇ hourstodays)but farenoughawaytoavoidpileupe ectsonthedetector.Ourgoalistoachieveameasure- mentuncertaintyofbetterthan0.1%,correspondingtoaminimumdetectorpeakcount N : ˙ Poisson N = p N N 10 3 ! N 10 6 (6.6) 180 Figure6.6:Fitsforthe1173keVand1332keV 60 Cogamma-rayspectrum. FortheLLNLHPGedetector,thiscorrespondstoadetector-sampledistanceof10Œ15mm. Themeasurede cienciesandtheiruncertaintiesoftheHPGedetectorareshown inFigure6.7aandFigure6.7b.Theshapeischaracteristicofmanye ciencyspectra ofsemiconductordetectors.Thereisalogarithmicincreasinge ciencyfromlowenergy ( < 100keV),apeakatthefikneeflataround90keV,andalogarithmicdecreasinge ciency fromthekneeonwards.Thee ectivebandwidthofthedetectorisabout2MeV. 6.1.4MonteCarlosimulation IstartedwithaboilerplatemodelofaHPGedetectorwithinabackgroundshieldin Geant4.Theoriginalmodelgeneratesanumberofgamma-raysatanenergy bytheuserwithrandominitialvectorsfromapointsource.Anexampleofadetector measurementofagammasourceisshowninFigure6.8.Toclearlyviewthegamma-ray 181 (a) (b) Figure6.7:(a)E ciencyplotofHPGewithcalibratedgammasourcesanda sample-detectordistanceof164mm.(b)E ciencyplotofHPGewithasample- detectordistanceof95mm 182 Figure6.8:Asimulationof1MeVgamma-raysemittingfromasource(right sideofgraphic)abovetheLLNLHPGedetector(leftsideofgraphic). trajectories,Ionlysimulatedthreehundredgamma-rays.ForasimulationthatIusefor analysis,onemillionormoregamma-raysaresimulated.Thenumberofgamma-raysthat aredepositedintheactivegermaniumvolumeofthedetectorarerecorded.Ahistogram isgeneratedwithacalltothesciencodingtoolkitROOT.Thehistogrambinheightis scaledtothenumberofhitsinthedetector. Theboilerplatedetectormodelincludestheactivegermanium,thefrontdeadlayer, theshell,thedetectorwindow,andtheshield.Itmodelsamoreconventionaldetec- torwithabullet-holedesignandastandard( ˇ 600 m)frontdeadlayerthickness.The sourceisapointsourcewithnosourceholdergeometry. Toimprovetheaccuracyofthesimulation,Iupdatedthedetector-sourcemodelto morecloselymatchthenewLLNLsystem.Tostart,Ithedetectormodelto matchtheparametersprovidedbythemanufacturerforournewdetectorasshownin Figure6.3.Overthecourseofthepracticum,Iupdatedthemodeltoincludeadditional componentsmodifyexistingdesigns.InoteaselectionsofimprovementsImadetothe 183 Figure6.9:SnapshotofGeant4modeloftheHPGedetectorandbackgroundshield. MonteCarlomodelinthefollowinglist: Ł reducethebullet-holedepth Ł addsidedeadlayersandbackdeadlayerswhichcanbevariedindependentlyofthe frontdeadlayer Ł addaninfraredbetweenthedetectorwindowandgermaniumcrystal Ł addaplasticconcentricsampleholder Ł createaType-MsourcegeometrywiththelayersshowninFigure6.4 Ł modifythegamma-raysourcetobeauniformplanarcirculardistributionconsis- tentwiththegeometryshowninFigure6.4 184 Figure6.10:Fourth-orderempiricaltothemeasurede cienciesofasuiteof calibratedgammasourcesatadistanceof95mm. InFigure6.9IshowasnapshotoftheupdatedBEGeGeant4MonteCarlomodel.The detectorissurroundedbythreelayersofshielding(ingray)andaconcentricsample holder(inyellow).Thegermaniumcrystalisshadedingreenandissurroundedbya vacuum-sealedshell.Thefrontoftheshell,whichfacesthegammasource,isacarbon compositecryostatwindow. 6.2Resultsandanalysis IwroteaBashscriptthatwrapsthesingle-energye ciencycalculationcodeandre- peatsthesimulationforarangeofenergies.Thee ciencycalculatedforeachiterationis storedinaseparatearray.IwroteaROOTscriptthatahistogramwiththestoredef- aftertheenergyrangeisswept.Thebinheightsarescaledbytheire ciencies. Thesimulatede cienciesforagivenenergyisthenumberofgamma-raysdepositedin thedetectordividedbythenumberofgeneratedgamma-rays. 185 Figure6.11:Fractionalresiduale ciencyscatterplotwithsample-detectordis- tanceof95mm. OnewaytomodelHPGedetectore ciencyistoanempiricalfunctiontothemea- surede ciencies.Theformoftheempiricalisgivenbythefollowing: log ( E )= a 0 + X n =1 a n (log ( E ) ) n Afourth-orderusuallythedatawell.Fornon-precisionmeasurements,thisis enoughtocharacterizetheHPGee ciencycurve. Theempiricalisalsousefulforcomparingsimulatede cienciestomeasurede - ciencies.IwroteaROOTscriptthatafourth-ordercurvetoa2Darrayofsimulated e ciencyvs.energy.ThefunctionisminimizedwiththeFMINUITroutine.InFig- ure6.10theempiricalcurveisdrawnandthemeasureddataisplottedoverit. OurcalibrationmethodfollowstheprocedureofHelmeret.alforcalibratinganHPGe detectortothesub-percentleveloveranenergyrangeof3.5MeV[166,167]. 186 Figure6.12:Fractionalresiduale ciencyscatterplotwithsample-detectordis- tanceof164mm. TheHPGedetectorMonteCarlocalibrationischaracterizedbycomparingthefrac- tionalresidualbetweenthemeasurede ciencyandthesimulatede ciencyofthedetec- toratagivenenergy.Thefractionalresidualofthedetectore ciencyatagamma-rayof energy E ,or R ( E ),isgivenbythefollowing: R ( E )= m ( E ) s ( E ) s ( E ) ,(6.7) where m ( E )isthemeasurede ciencyand s ( E )isthesimulatede ciencyofthe detectoratgamma-rayenergy E . Figure6.11andFigure6.12areplotsofmeasuredvs.simulatede cienciesoveran energyrangeof2MeVforasample-sourcedistanceof95mmand164mm,respectively. Weighted-averagefractionalresidualsarequotedforeachplot.Weobtaina5 : 25%av- 187 Figure6.13:custom-designedgamma-sourceholderfortheLLNLHPGedetector. erageresidualforthe95mmplotanda2 : 99%residualforthe164mmplot.Thistook approximatelytenweeksofmodelingwork. Ingeneral,theresidualsappearrandomlydistributedaboutzero.However,when lookingateachsourceindividually,inmanycasestheresidualsappearconsistentlyun- derestimatedoroverestimatedbythesimulation.Forexample,inFigure6.11,the 57 Co datapointsappearunderestimated,butthe 152 Eusampleappearsoverestimated. Wesuspectthatthesystematicshiftsinthee ciencyareduetosmalldisplacements thatareintroducedbyremovingandinstallingthegammasources.Iwilldiscussour designofanew,highlyrepeatablesampleholderinthenextsection. 6.3ConclusionsandOutlook ThenewhighpuritygermaniumdetectorsystematLLNLwillmeasuregamma-ray intensitiesoflong-livedisotopestosub-percentprecision.Istartedmypracticum shortlyafterthenewHPGewasassembled.Wemeasuredthegamma-rayintensitiesofa suiteofstandardizedgammasourcestoobtainthee ciencycurveofthedetector.Then IdevelopedaMonteCarlosimulationcodetoreproducethemeasuredintensities.The 188 codeiswritteninGeant4andmodelstheHPGedetectionsystem,simulatesgamma-ray e ciencyoftheHPGeasafunctionofenergy,andcomparesthesimulationdeviation frommeasuredgamma-rayintensities.Imeasuredgamma-rayintensityofasuiteof calibratedgamma-raysourcesatdi erentdistancesandcomparedtheresultstothesim- ulation. IcalibratedtheHPGedetectorMonteCarlomodeltowithin3%ofthemeasured gamma-raye cienciesovera2MeVrange.Webelievethattheleadingsourceofuncer- taintyinthecalibrationisinthevarianceingamma-raysourcepositionwhenchanging samples. IdesignedasampleholderthatmountsdirectlytotheHPGeshelltosuppressshifts inthesourcepositionbetweenmeasurements.Thegamma-raysampleholderis showninFigure6.13.AsheathinterfaceswiththeendcapedgeoftheHPGe.Thesample holderdistanceiswithstackingspacerscalibratedtowithina0.2mmtolerancethat willletusvarythesample-detectordistancefrom2Œ30cminincrementsassmall1cm. Wehavetwosampleholdingadaptersthatwillallowustomounteitherasourcedisk forcalibrationmeasurements(leftofFigure6.13)orthe4 ˇ betacounterforlong-lived isotopeintensitymeasurements. Weexpectthatanewmeasurementofknowngammasourceswillcalibratethede- tectortowithin1%usingtheposition-repeatablesampleholder.Afterthedetectoris su cientlycalibrated,precisionmeasurementsoflong-livedisotopeswillbepos- sible.Thegoalistomeasurethegamma-rayintensitiesof 147 Ndandotherlong-lived isotopestodeterminetheprincipalbeta-decaybranchingratiostowithin1%. 189 CHAPTER7 CONCLUSIONSANDOUTLOOK TheRaEDMexperimentmeasurestheatomicelectricdipolemomentof 225 Ra.Atoms arevaporizedinanovenandarecollimatedandcooledwithresonantlasers.Theyare trappedinamagneto-opticaltrap,thentransportedbetweentwohigh-voltageelectrodes usingopticaltweezers.Duringthemeasurement,theatomsprecessbetweenapairof identicalplane-parallelelectrodesthatgenerateauniformandstableDCelectricd thatreversesdirectioneverymeasurementcycle.Weusedapairofoxygen-freecopper electrodesthatoperatedat 6 : 7kV/mmata2.3mmgapsizeandmeasuredanEDM upperlimitof1 : 4 10 23 e cminthegenerationofmeasurements.Forthesecond generationmeasurements,wewilluseanewpairoflarge-grainniobiumelectrodeswhose systematice ectshavebeenevaluatedtothe10 26 e cmlevel. Iconstructedahighvoltageteststationtoconditionhighvoltageelectrodesatgap sizesof0.4Œ2.5mmwitha30kVbipolarpowersupplyatMSU.Theteststationwas commissionedwithapairofcopperelectrodes,andthefollowinghighvoltagetestswere performedwithniobiumandtitaniumelectrodes.Ivariedtheelectrodegapsizewith ahigh-precisionlineardriveandvthattheelectrodescouldperformreliablyat 1mm.Then,agapholderwasfabricatedandreplacedtheadjustablegapassembly. Subsequentconditioningwasperformedwithelectrodepairsatagapof1 : 0 0 : 1mm. Toreachdshigherthan10kV/mm,Idevelopedhardwareandprocedurestoclean andpreservethesurfacepurityoftheelectrodes.Ibuiltaportablecleanroomvalidated toClass100(ISO5)withaNIST-calibratedparticlecounteranddevelopedacleanroom electrodeswapprocedure.ThenIworkedwithengineersatFRIBtodevelopaprocedure ofhigh-pressurerinsingtheelectrodeswithultrapurewater.Idesignedapackaging methodforstorageandtransportofdecontaminatedelectrodes.Subsequentteststation workandelectrodeswapswereperformedintheNSCLdetectorcleanroom. 190 Twopairsofgrade-2titaniumandfourpairsoflarge-grainniobiumelectrodeswere fabricatedandpolishedaccordingtosurfacepreparationtechniquesthatweremodi- fromacceleratorphysicsliterature.Wedischarge-conditionedthreepairsofnio- biumelectrodesandonepairoftitaniumelectrodes,alternatingthepolarityofthe appliedDCdevery60stomimictheEDMmeasurement.Electricdswere testedashighas+52.5kV/mmand 51.5kV/mm.Alltheelectrodesexhibitedlessthan 100pAsteady-stateleakagecurrentwhenoperatedunder22kV.Wevalidatedapairof large-grainniobiumelectrodes(Nb 56 )at20kV/mmwithanaveragedischargerateof 98 19dischargesperhourandasteady-stateleakagelessthan25pA(1 ˙ ). Thelarge-grainniobiumelectrodes(Nb 56 )weretransportedtoANL,whereIcon- structedaportablecleanroomandpositioneditattheEDMapparatus.Nb 56 wasin- stalledintheRaEDMapparatusallwhilepreservingtheelectrodesinClass100environ- ments.IrevalidatedtheperformanceofNb 56 to20kV/mmaftertheapparatusvacuum pressurewasrestored. SeveraltargetedupgradeswillbeimplementedinthesecondgenerationEDMmea- surementstocollectivelyimprovesensitivitybyuptothreeordersofmagnitude.The newniobiumelectrodesincreasetheelectricdstrengthandwillinitiallycontribute anenhancementfactorof3.1inEDMstatisticalsensitivity. Weplantofurtherincreasetheelectricdtoafactorof7.7duringafuturephase ofhighvoltagedevelopment.Thiswillbeachievedbydesigningamoresymmetrichigh voltagetestchamberusingaunipolarpowersupplythatalternatestheddirectionby switchingconnectionsbetweentheelectrodes.Ourgoalistodischarge-conditionelec- trodestooperatereliablyat 50kV/mmovera1mmgap. ThenewBlueSlowerwillusethe 1 S 0 ! P 1 o 1 opticalcyclingtransitiontoallowus traptwoordersofmagnitudemoreatomsforspinprecessionfrequencymeasurements. Ibuiltauoroscopysetuptocharacterizetheadditionaldecaychannelsthatwewill needtorepumptousethebluecyclingtransition.Thesetupcombinesseveral- 191 coupledlaserstopopulateandprobethedeexcitationpathsfromthe 3 F o 2 atomicstateto severalDstatesandmeasurestheuorescenceintensityofeachofthechannels.From theuorescenceintensitywethendeterminethebranchingratioforeachdecaychannel. InthismannerwevthequalitativebehavioroftheDstatebranchingratiosinthe contextoftheblueZeemanslowingscheme. TheZeemanslowerupgradeisexpectedtoimproveEDMsensitivitybyanorderof magnitude.Inaddition,aspin-selectiveSTIRAPatomdetectione ciencyupgradeis beingdeveloped.STIRAPisalsoexpectedtoimprovesensitivitybymorethanoneorder ofmagnitude.Takentogetherwithimprovementstotheelectricd,weexpecttoreach anEDMsensitivityatthe10 26 e cmlevelorbetter. Inthenearfuture,wewillbeabletoharvest 225 RafromtheFRIBbeamline.FRIBis expectedtoproducelargerquantitiesof 225 RamorefrequentlyfortheRaEDMexper- imentthanpastsources.OurgoalistodeveloptheFRIBharvestingprocedurewitha seriesofprogressivelysophisticatedmeasurementswithstableisotopesthathavesimilar spectroscopicpropertiestoradium. Themeasurementisalaserinduceduorescencemeasurementofadirectedbeamof atomsfromane usiveoven.Wewillinitiallyusecommercial,stableytterbiumasara- diumsurrogatetocommissiontheuxmeasurement.Asthemeasurementiswe willprepareasampleofstablecalciumextractedfromwater,simulatingaradiumhar- vestingprocess.Thecalciumsamplewillbemeasuredwiththeatomicbeamuorescence techniqueandallowustobenchmarktheharvestinge ciency. Iassembledanatomicbeamuorescencesetupthatgeneratesadirectedbeamof atomsfromane usiveoven,illuminatestheatomswithalaserbeam,andcapturesthe uorescencewithaphotodetector.Iwroteaprogramthatsimulatesthephotodetector signalasafunctionofthelaserfrequency,laserpower,andoventemperatureforagiven atomicangulardistributionandatomspecies.Thesimulationwillallowustooptimize andinterpretfutureatomicbeamuorescencemeasurementsofytterbium,calcium,and 192 otherisotopesrelevantforprecisionnuclearphysicsexperiments. Personalsciencontributions ThefollowinglistsummarizesthetasksIperformedformythesiswork. 1Electrodematerialmagnetizationmeasurement Ł Designedandassembledanelectrodemagnetizationmeasurementsetupusingan externald-shieldingmu-metalboxandacustommechanicaltranslationstage. Ł Cuxgatemagnetometersabovethetranslatingstageandperformeda suiteofgradiometermeasurementswithcopper,stainlesssteel,Macor,titanium, aluminum,andniobiumelectrodesandelectrode-sizedcylindersurrogates. Ł Builtaconditioningcircuitwithadi erentialopampinputandlow-passterto amplifythemagnetizationsignal. 2Highvoltageelectrodepreparation,testing,andoperation Ł Designedandassembledahighvoltageteststationtodischarge-conditionsixpairs ofhighvoltageelectrodes. Ł Builtdataacquisitioninterfacecircuitryandhousingunitsforaunipolar 30kV unipolaranda 30kVbipolarpowersupplyforthehighvoltageteststation. Ł Leadmorethan80conditioningshiftsrangingfrom3Œ6hourseach. Ł Wroteanalysissoftwareforcharacterizingelectrodeconditioningperformance. Ł Designedandassembledhighvoltagecomponents,includingHVfeedthroughshield- ingandin-vacuumelectrodegapalignmentcomponents. Ł Calibratedgapalignmentwithcustomopticalsystem. Ł Designed,built,andassembledcleanroomsforhighvoltageteststationworkat MSUandANL. 193 Ł Performedhighvoltageteststationmaintenanceandelectrodeinstallationandpack- agingincleanroomsatMSU,FRIB,andANL. Ł TransportedapairofconditionedniobiumelectrodesfromMSUtoANL,assembled electrodesinholder,andassistedininstallationoftheelectrodesintheANLsetup. Ł RevalidatedtheelectrodeperformanceatANL. Ł Firstauthorofthepublicationbeingpreparedforthiswork(submittedOctober 2020). 3LasercoolingZeemanslowerupgrade Ł Builtuoroscopysetupthat-couplesthreelasersandcombinesthebeamswith dichroicmirrorsforradiumlaserinduceduorescencestudyatANL. Ł Builtnear-infrareddiodelaserandfocusingcomponentsforradiumuorescence. Ł Builtnear-infraredlaserinterfaceboxwhichconnectsthethermoelectrictempera- turecontrollerandcurrentsourcetothediodelaserandinterlocksthesetuptothe laboratorysafetysystem. Ł WrotedataacquisitionlaserscanningLabViewsoftwarefortheradiumbranching ratiomeasurement. Ł Manuallysearchedforandfoundresonancefrequenciesforpumptransitionand excitedstate. 4Long-livedisotopegamma-raybranchingratios Ł CreatedGeant4modelofnewhighpuritygermaniumgamma-raydetectoratLLNL forthelong-livedisotopegamma-rayspectroscopyexperiment. Ł Assistedwithmeasurementofstandardizedgamma-emittingsources. 194 Ł ComparedGeant4MonteCarlosimulationofgammasourcedetectore ciencyand measurede ciencyandmatchedsimulationtowithin3%ofexperiment. Ł Designedposition-repeatableprecisiongammasourceand4 ˇ betacounterdetector mountstopreciselysource-detectordistances. 5Atomicbeamuorescence Ł Assembledatomicbeamuorescenceapparatusatforlaserinduceduorescence studiesofFRIB-harvestedisotopesatMSU. Ł Assembledvacuumhardwareandatomicoven. Ł Tunedtitaniumsapphirelaserwithfrequency-doublingcavitytoytterbiumexcita- tionwavelength. Ł Wroteanalysissoftwarethatsimulatesanatomicbeamuorescencespectrumfor useratomicspeciesandtransition,ovengeometry,atomicangulardistribu- tion,photodetector,andlaser. Ł Simulatedytterbium,rubidium,andcalciumuorescencespectra. Ł Developedcalculationoftotalatomratecountforagivenphotodetectoruores- cencesignal. Ł Designedin-vacuumlight-collectionsetuptoamplifyphotodetectoruorescence signal. Support MyheartfeltthankstothePhysics&Astronomydepartment,andtothemanyop- erationsfolksinadministration,electronics,IT,andsafetywhohelpedmakethiswork possible.ThankstotheNSCLandFRIBengineersforhelpingmebuildcoolstu todo somecoolscience. 195 ThankstoKayKolos,NickScielzo,andKeenanThomasforintroducingmetoapplied nuclearspectroscopy. Thankstomygraduatecommittee:MortenHjorth-Jensen,KeiMinamisono,DaveMor- rissey,JohannesPollanen,andJaideepSinghfortheguidanceandsupport. ThankstotheArgonneRaEDMteam:KevinBailey,MichaelBishof,MattDietrich, PeterMueller,andThomasO'Connorforarewardingcollaborativeexperience. Thisworkwassupportedby:MichiganStateUniversity;USDOEO ceofScience,Of- ofPhysicsunderDE-AC02-06CH11357;DOEOakRidgeInstituteforScienceandEd- ucationDE-SC0014664;DOENationalNuclearSecurityAdministrationthroughNSSC DE-NA0003180;andUSDOEO ceofScience,O ceofNuclearPhysicsundercontract DE-SC0019455. 196 APPENDICES 197 APPENDIXA:Constants,units,atomicandnuclearproperties TableA1:Fundamentalphysicalconstants(fromtheNISTdatabase) constantvalue h Planckconstant6 : 62607015 10 34 JHz 1 4 : 135667696 10 15 e VHz 1 k B Boltzmannconstant1 : 380649 10 23 JK 1 uatomicmassunit1 : 66053906660(50) 10 27 kg c speedoflightinvacuum2 : 99792458 10 8 ms 1 r e classicalelectronradius2 : 8179403262 10 15 m o vacuumelectricpermittivity8 : 8541878128(13) 10 12 Fm 1 e elementarycharge1 : 602176634 10 19 C N nuclearmagneton5 : 0507837461(15) 10 27 J = T B Bohrmagneton9 : 2740100783(28) 10 24 J = T 5 : 7883818060(17) 10 5 e V = T B =h 1 : 39962449361(42) 10 10 Hz = T 0 vacuummagneticpermeability1 : 25663706212(19) 10 6 NA 2 m e electronmass9 : 1093837015(28) 10 31 kg a 0 Bohrradius= ~ 2 ( e 2 = 4 ˇ 0 ) m e =5 : 29177210903(80) 10 11 m G F = ( ~ ) 3 Fermicouplingconstant1 : 1663787(6) 10 5 G e V 2 TableA2:Unit unit Pascal(Pa)1Pa=1Nm 2 atmosphere(atm)1atm=101325Pa Torr1Torr=101325 = 760=133 : 3Pa bar1bar=10 5 Pa Tesla(T)1T=10 4 gauss elementarycharge( e )1 e =1 : 602176634 10 19 C 198 TableA3:Angularmomentum,masses,andabundancesofYb.Val- uesfromNIST. massnumber A Nuclearspin I mass( 10 25 kg)abundance(%) 16802.78860780.123(3) 17002.82183312.982(39) 1711/22.838464514.09(14) 17202.855070921.68(13) 1735/22.871706616.103(63) 17402.888322832.026(80) 17602.921595212.996(83) TableA4:Vaporpressurecoe cientsforytterbium,rubidium, andcalcium. atom A [ 1 ] B [ K ] C [ 1 ] D h K 3 i Ref. Yb9 : 111 8111 : 0 1 : 08490 : 0[171] Rb4 : 857 42150 : 00 : 0[108] Ca10 : 127 9517 1 : 40300 : 0[108] TableA5:Rubidiumproperties.Massnumber A ,nuclearspin I ,iso- topeshiftIS.ValuesfromNIST. Rb A I mass( 10 25 kg)abundance(%)IS 0 ( 85 Rb) [ MHz ] 873/21.443161027.8377 : 583(12) 855/21.409993572.170 : 0 TableA6:Calciumproperties.Massnumber A ,nuclearspin I ,iso- topeshift(IS)forthetransition 1 S 0 ! 1 P o 1 . 47 Caatomicmassfrom Kramida[172]. 47 CaisotopeshiftbyAndl et.al [149].Allother isotopeshiftsfromNörtershäuser et.al [155].Allothermasesfrom NIST. AI mass( 10 25 kg)abundance(%)IS 0 ( 40 Ca) [ MHz ] 4000.6635944496 : 9410 : 0 4200.696739240 : 647393 : 5 437/20.713347090 : 135611 : 8 4400.729897912 : 086773 : 8 4600.763078960 : 0041159 : 8 477/27.7969848synthetic1348 : 7 4800.796270880 : 1871513 : 0 199 APPENDIXB:Codeanddataavailability Thecodeusedtoanalyzethehighvoltagedataandgeneratethecurrentdischarge plotsisavailableforuseathttps://zenodo.org/badge/latestdoi/294766922.Thedata usedforthehighvoltageanalysismaybemadeavailableforreasonablerequestssentto singhj@frib.msu.edu . APPENDIXC:AvalanchePhotodiodeSettings Thevoltageoutputoftheavalanchephotodiode V ( ) [ V ] isgivenby: V ( )= P d ( ) R M ( ) G , where P d ( ) [ W ] istheincidentuorescentlightpoweratfrequency , R M =11 : 3(24 : 0)A = Wfor =398 : 8(555 : 6)nmforM=50isthedetectorrespon- sivityatwavelength, M 2 [ 5 ; 50 ] isthegainorfiM-factorfl,and G =500kV = Aisthetransimpedancegain. V out 4 : 1(2 : 0)Vathigh-Z(50 )termination.Thedetectorareais A det = ˇ (0 : 25mm) 2 =0 : 196mm 2 : Thedistancebetweenthesurfaceoftheactivedetec- tiveareaandthee=2 : 2 0 : 3mm.Theopticaldamagethreshold=1mW. 200 APPENDIXD:Fluxgatemagnetometry FigureD1: C 1 =5 : 6nF ;C 2 =47nF ;C 3 =4 : 7nF ;C 4 =47nF ;C 5 =1 : 5nF ;C 6 = 5 : 6nF ;C 7 =100nF ;C 8 =2 : 2nF ;C 9 =15nF ;C 10 =0 : 82nF ;C N =1 F ;R 1 =10k ;R 2 = 100k ;R 3 + R 4 =10k ;R ref =10k ;R F =10k ;R o =10k 201 FigureD2:BartingtonMag03IEL70uxgateschematicformagnetizationmeasurements. FigureD3:Fluxgate:BartingtonMag03IEL70.16kHzexcitationfrequency,noiseooris 6pT rms = p Hz.Powersupply:BartingtonPSU1.5pT rms = p Hznoiseoor.Dataacquisi- tion:NIPCie-6320.16-bit.2mVnoiseooron10Vscale. 202 BIBLIOGRAPHY 203 BIBLIOGRAPHY [1] B.Odom,D.Hanneke,B.D'Urso,andG.Gabrielse.Newmeasurementofthe electronmagneticmomentusingaone-electronquantumcyclotron. Phys.Rev. Lett. ,97:030801,Jul2006. 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