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Quantum Turbulence: Decay of Grid Turbulence in a Dissipationless Fluid


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IamthankfulforallthehelpIhavereceivedinmyPh.D.program,inwritingthisdissertationandinlearningaboutthisresearchsubject.Firstofall,Iwouldliketothankmymentor,ProfessorGaryG.Ihas,whohasledmetotheamazingeldoflowtemperatureexperimentalphysics.Fromhim,IhavelearnedalotofphysicsandIwasabletocarryoutmyaccomplishments.Ireallyenjoythebeautyofquantumturbulencescience.Second,Iwouldliketothankmyparents,whohavegivenmealotofsupport,emotionally,nancially,andspiritually,andalotofencouragementbeforeandduringthepursuitofmyPh.D.Third,Iwouldliketothankthemembersofourresearchgroup,formerandcurrent,andthepeopleinthemachineandelectronicshops.Becauseofthem,Iwasabletostandonthosegiants'shouldersandlookhigherandfurther.Fourth,IwouldliketothankthewholeGainesvillecommunity,whichhasgivenmemuch. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 12 CHAPTER 1INTRODUCTIONTOQUANTUMTURBULENCE ............... 14 1.1Introduction ................................... 14 1.1.1BasicPropertiesofHeII ........................ 14 1.1.2Two-FluidModel,andLandau'sTwo-FluidEquations ........ 16 1.1.3QuantizationofVorticesandtheCriticalVelocities ......... 18 1.1.4KelvinWaves .............................. 20 1.2IntroductionofTowed-GridTurbulenceExperiments ............ 20 1.3ProposedTowed-GridSuperuidTurbulenceExperiment .......... 21 1.4HighResolution,FastRespondingMilikelvinThermometers ........ 25 1.4.1NeutronTransmutationDopedGermaniumBolometers ....... 25 1.4.2MiniatureGeFilmResistanceThermometers ............. 27 2SHIELDEDSUPERCONDUCTINGLINEARMOTOR .............. 34 2.1Introduction ................................... 34 2.2ModelsoftheShieldedandUnshieldedSuperconductingMotor ...... 35 2.3SimulationResultsandDiscussions ...................... 39 2.3.1SimulationAnalysis ........................... 39 2.3.2CriticalMagneticFieldsforNiobium ................. 41 2.3.3RequiredVoltageInputfortheSolenoid ................ 43 2.4UnshieldedMotorTestingExperiments .................... 45 2.4.1CapacitanceBridgeandLock-inAmplierforMonitoringArmatureMotion .................................. 46 2.4.2The555OscillatorforMonitoringArmatureMotion ......... 47 2.4.3TheQ-meterforMonitoringArmatureMotion ............ 49 2.4.4TheACBridgeCircuitsforMonitoringtheMotionoftheArmature 50 2.5ImprovedDesignoftheArmatureandtheTestCell ............. 57 2.6Conclusion .................................... 59 3CONSTRUCTIONOFSUPERCONDUCTINGSHIELDEDLINEARMOTORANDEXPERIMENTALCELL ........................... 84 3.1ConstructionofSuperconductingShield .................... 84 3.1.1ElectroplatingTheoryandElectrolyteRecipe ............. 84 5

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......................... 85 3.1.2.1Lead .............................. 85 3.1.2.2Methanesulfonicacid(MSA) ................. 86 3.1.2.3Leadcarbonate ........................ 86 3.1.2.4Labprotectiveequip ..................... 86 3.1.3LeadElectroplatingProceduresandResults ............. 86 3.1.3.1Proceduresteps ........................ 86 3.1.3.2Results ............................ 88 3.1.3.3Troubleshooting ........................ 88 3.2TestingtheExperimentalCellatLiquidHeliumTemperature ....... 90 3.2.1SimulationResults ........................... 90 3.2.2ExperimentalTestingResults ..................... 92 3.2.3ViscousDragandImpedanceForcesDiscussion ............ 92 3.2.4HeatDissipationDiscussion ...................... 93 3.3LeakTightElectricalFeedthroughDesign .................. 94 3.3.1ElectricalFeedthruConstruction .................... 95 3.3.2ThermistorCircuitBoardConstruction ................ 96 3.3.3Conclusion ................................ 97 4THERMISTORSELECTRONICS ......................... 111 4.1Introduction ................................... 111 4.2TheACBridgeCircuitAnalysis ........................ 111 4.3ResolutionMeasurementatRoomTemperature ............... 113 5EXPERIMENTALPROCEDURE,DATA,ANDANALYSIS ........... 118 5.1Introduction ................................... 118 5.2ExperimentalResults .............................. 118 5.2.1ThermistorCalibration ......................... 119 5.2.2BackgroundHeatingCheckat624mK ................ 119 5.2.3PerformingQTExperimentsat520mK ................ 120 5.3ExploringtheKelvinWavesintheEnergySpectra ............. 121 6CONCLUSIONSANDFUTUREWORK ...................... 127 6.1Conclusions ................................... 127 6.2FutureWork ................................... 127 APPENDIX ADERIVATIONANDNUMERICALANALYSISOFMAGNETICFIELDANDFORCE ........................................ 129 BSOMECCODE ................................... 131 CLABVIEWPROGRAMSHOTS .......................... 140 6

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............................... 144 REFERENCES ....................................... 145 BIOGRAPHICALSKETCH ................................ 148 7

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Table page 2-1Optimalparametersforthemotordesign. ..................... 40 2-2Parametersfortheunshieldedtestmotor. ..................... 45 3-1Parametersofsuperconductorshieldedsuperconductinglinearmotorsystem. .. 91 3-2Forcesonthearmaturewhenmovingupat1m/s(Downloadispositive). .... 94 4-1ParametersoftheACbridgeaectingthesensitivityandthepowerdissipation. 114 4-2SensitivitytestoftheACbridge. .......................... 115 8

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Figure page 1-1Viscosityofliquidhelium-4. ............................. 29 1-2Superuidandnormaluiddensitiesasafunctionoftemperature. ....... 29 1-3PhotographsofstablevortexlinesinrotatingHeII. ................ 30 1-4Dampingonasphereoscillatinginliquidhelium. ................. 30 1-5Donnelly-Glabersoninstabilityofaquantizedvortexline. ............. 31 1-6Energyspectrumforhomogeneousandisotropicturbulence. ........... 31 1-7Electricalconductionmechanismsinsemiconductors. ............... 32 1-8Calibrationplotsforthreetestthermistorsdevelopedforcalorimetry. ...... 32 1-9Designforthethermometertestcell. ........................ 33 2-1Gridturbulenceinaclassicaluid. ......................... 60 2-2Superconductingmotormodel. ........................... 60 2-3Armaturemotioninunshieldedandsuperconductingshieldedsolenoid. ..... 61 2-4Armaturemotioninsidesuperconductingshieldedsolenoid(linearacceleration). 61 2-5Armaturemotioninsidesuperconductingshieldedsolenoid(squareacceleration). 62 2-6Uppercriticaleldversustemperatureforniobium. ................ 62 2-7Voltageinputtothesuperconductingshieldedsolenoid. .............. 63 2-8Simulatedarmaturemotionforthesuperconductingshieldedsolenoid. ...... 64 2-9Drivingvoltage. .................................... 65 2-10Circuitoftheswitchbox. .............................. 66 2-11Machinedrawingsandphotosofthemotorsystem. ................ 67 2-12Experimentalapparatusforunshieldedmotor. ................... 68 2-13TheGR1616CapacitanceBridge .......................... 69 2-14Circuitryoftheexperimentalapparatusforunshieldedmotortestingexperimentsusingthe555oscillatorcircuitandLabViewcounterprogramtomonitorthemotionofthearmature. ............................... 69 2-15TestingcircuitofQ-meter. .............................. 70 9

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.............................. 70 2-17MeasuringthecapacitanceofthecapacitivepositionsensorwiththeGR1616capacitancebridgeandthelock-inamplier. .................... 71 2-18Circuitryandsetupfortestingthesuperconductingmotorat4.2K. ....... 72 2-19TheoreticalcalculationofthemagneticforceandRTsensorcalibrationcurve. 73 2-20Motionofthearmatureofthesuperconductingmotor. .............. 74 2-21Simulatedarmaturemotionwiththeinputpulsesenttothesolenoid. ...... 75 2-22Motionofthearmatureofthesuperconductingmotor. .............. 76 2-23Motionofthearmatureofthesuperconductingmotor(I). ............ 77 2-24Motionofthearmatureofthesuperconductingmotor(II). ............ 78 2-25Motionofthearmatureofthesuperconductingmotor(III). ........... 79 2-26Modiedsuperconductingmotorsystem. ...................... 80 2-27MachinedrawingofthegridandthecorrespondingReynoldsnumbers. ..... 80 2-28Calibrationcurveofthepositionsensorat4.2K. ................. 81 2-29Electronicscircuitsforthesuperconductingmotorsystem. ............ 82 2-30Motionofthearmatureofthesuperconductingmotor. .............. 83 3-1Machinerydrawingsforthecellcap. ........................ 98 3-2Machinerydrawingsforthecellbody. ........................ 99 3-3Electrodeforthecellcap. .............................. 100 3-4Electrodeforthecellbody. ............................. 101 3-5Procedurestepsfortheleadplatingforthecellcap. ................ 102 3-6Procedurestepsfortheleadplatingforthecellbody. ............... 103 3-7Masterpieceoftheleadcoatedcellcap. ....................... 104 3-8Masterpieceoftheleadcoatedcellbody. ...................... 105 3-9Towed-Gridexperimentcell. ............................. 106 3-10Simulationforsuperconductorshieldedsuperconductinglinearmotorsystem. .. 107 3-11Motionofthearmatureoftheshieldedsuperconductingmotor. ......... 108 10

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.......................... 109 3-13Leaktightelectricalfeedthrudesignforourcryogeniccell. ............ 110 4-1TheACbridgecircuitsimulation. .......................... 116 4-2ACbridgecircuitforthermistorsresistancemeasurement. ............ 117 4-3Roomtemperaturethermistorresolutioncurve. .................. 117 5-1Thermistorscalibrationcurves. ........................... 123 5-2Motionofarmatureandthermistorresponseatvacuumaround600mK. .... 124 5-3Quantumturbulenceat520mK. .......................... 125 5-4Fittingtheenthalpyofliquidheliumasafunctionoftime. ............ 126 C-1LabViewprogramsforcalculationofcurrent,magneticeld,magneticforceandarmaturemotion. ................................... 141 C-2LabViewprogramstolookforoptimalparameters. ................ 142 C-3LabViewprogramsfordataacquisitionanddataanalysis. ............ 143 11

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Weproducedgridturbulenceinliquidheliumat520mKtocomparewithclassicalexperimentsandtheories.AboveT=1K,withviscositypresent,ithasbeenshownthatgridturbulenceisequivalenttohomogeneousisotropicturbulenceinaclassicaluid.Weseektoinvestigatethenatureofgridturbulencewhenviscosityiszero.Specically,intheabsenceofviscosityinaquantumuid,throughwhatpathdoestheturbulencedecay?Toproducegridturbulence,anactuatorwasdesignedandbuiltthatcanaccelerateanddeceleratethegridrapidlyinashortdistance(1mm),andachieveglidespeedsofupto1m/s.ToavoidJouleandeddycurrentheatingoftheliquidhelium,amagneticallyshieldedsuperconductinglinearmotorwasbuilt.Thegridisattachedtotheendofaverylightinsulatingarmaturerodwhichhastwohollowcylindricalniobiumcansxedtoitabout26mmapart.Thispartoftherodisinsideasuperconductingsolenoidwhich,whendrivenwiththeproperlyshapedcurrentpulse,producesamagneticeldresultingintherequiredmotion. Detailedcomputersimulationsguidedthemotordesign.ThesimulationandmotorcontrolprogramswerewritteninLabViewwithanembeddedCcompiler.Usingthesimulator,variousdesignsofsolenoid(withandwithoutshielding)andarmaturewereinvestigated.Wecomparedthesimulationandtheexperimentalresultsinwhichcomplexcurrentpulseshapeswererequiredtoproducethedesiredmotion. 12

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1 ]anddiscoveredexperimentallybyHallandVinenayearlater[ 2 ]. Superuidquantumturbulenceproducedbytowed-gridexperimentsinliquid4Heatverylowtemperaturesispredictedtodecay,notthroughviscosity,asinaclassicaluid,butbyphononradiationwhentheenergyowsintothesmallerlengthscalesinaKelvin-wavecascade.Weproposeanewcalorimetrictechniquetoprobesuchadecaymechanismofsuperuidgridturbulenceatextremelylowtemperatures,say520mK,whilethenormaluiddensityisonly8.6ppm[ 3 ]. Thesuperuidityofhelium-4wasdiscoveredin1939byAllen,Misener,andKapitza[ 4 5 ]whileOshero,RichardsonandLeedidnotdiscoverthesuperuidityofhelium-3until1971[ 6 ].At2.17K,undersaturatedvaporpressure,thecurveofspecicheatversustemperaturefor4Heshowsadramaticspike,whichlooksliketheGreekletter`'.Thisiscalledalambdatransition,andcorrespondstoasecondorderphasetransition.In3He,thesuperuidtransitionoccursat0.9mKundersaturatedvaporpressure.Below 14

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ThethermaldeBrogliewavelengthofliquidheliumis at2.0K.Thisiscomparabletoorgreaterthanthemeaninterparticledistanceof3:6Aforhelium.SothedeBrogliewavelengthofeachatomislargerenoughtooverlapwithitsneighbor. Thisiswhyliquidheliumiscalledaquantumuid.Becausetherearetwoprotonsandtwoneutronsinthenucleus,thereisanevennumberofnucleons(totalnuclearspin=0),andthequantummechanicalbehaviorsof4HecanbeexplainedbyBose-Einsteinstatistics.Atomsof3HeobeyFermi-Diracstatisticsbecausetheirnucleiicontaintwoprotonsbutonlyoneneutron,totalinganoddnumberofnucleons(totalnuclearspin=1/2).(Thespinsofthetwoelectronscancelout.)Hence,4Heisabosonand3Heisafermion. Thesuperuid4He,alsocalledHeII,hasalmostzeroviscosity,whilethenormaluidof4HeaboveT,HeI,hasmuchhigherviscositywhichcandissipateenergyviainteractionswiththewallsofthecontainer.Theviscosityofliquid4HemeasuredbythemethodofoscillatingdiscviscometerisshowninFig. 1-1 [ 7 ].TwoofthemostfamousexperimentsdemonstratingthesuperuidpropertiesofHeIIarethebeakerexperimentandthefountain(thermo-mechanical)eect.IfyouputthebottomofanemptybeakerintheHeIIbath,a20-30nmthick4Hemobilelmformsonthewallsofthebeaker,andthenliquidHeIIowsalongthelmfromthebathintothebeakeruntilthelevelsareequal.Ifyouthenliftthebeakerabovethebathlevel,theliquidinsidethebeakerwillalsoowalongthelmoutofthewallsofthevesselintothebathuntilthebeakerisempty.NowifyouconnecttwovesselscontainingthesamelevelofHeIIatthesame 15

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8 { 10 ]toexplainthevariousinterestingphenomenathatoccurredinHeII.Inthismodel,theliquidheliumIIisconsideredasamixtureoftwointerpenetratinguids,calledthenormalcomponentandthesuperuidcomponent,withdensities,nands,respectively.Hence,thetotaldensityofliquidHeII[ 11 ]follows: Thesuperuidandnormaluidcomponentdensities,asafunctionoftemperaturebelowTunderthesaturatedvaporpressure,areshowninFig. 1-2 .Thesuperuidhaszeroentropy(Ss=0)andzeroviscosity(=0),whilethenormaluidexhibitsviscosity(n)andentropy(Sn),equaltotheentropyofalltheliquidhelium.Also,thesuperuidisconsideredtobeirrotational: where~vsisthevelocityofthesuperuid. ModifyingEuler'sequationsfortheclassical(Euler)uidsbasedonthecontinuityequation,andusingtherstandsecondlawsofthermodynamics,andhisownpostulatethatthechemicalpotential()isthedrivingforceforthesuperuid,LandauderivedthetwouidequationsforHeII[ 13 ]: @t+~r~v=0(1{5) 16

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@t+~rs~vn=0(1{6)(entropyconservation) andstresstensoris: InLandau'stwo-uidmodel,theelementarythermalexcitations,phononsandrotons,dependingonthewavenumber,ariseintheowofheliumIIthroughatubeorcapillaryatT6=0Kwhenthenormaluidcomponenthastheinteractionswiththewallscausingenergydissipationandviscousloss.Supposeanexcitationiscreatedwithenergy"andmomentumpduetothelossofenergyfromthetube(E=").ThenLandau'srelation(v"=p)givestheminimumvelocityofow requiredtoproduceanexcitation.However,actualcriticalowvelocitiesinexperimentsaremuchsmaller(mm/s)thanLandau'sprediction(60m/satthevaporpressureand46m/sathigherpressure[ 7 ]),duetothequantizedvortices. 17

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13 ]: Inaddition,theKelvincirculationtheorem, (byStokes'law)impliesthatthesuperuidcirculation staysconstant,andifatt=0thesuperuidvorticity!!s=!r!vsiszeroeverywhereitwillstayzero.If!r!vsdisappearseverywhere,thenthesuperuidcirculationis: Assumingthereissomecirculation,theremustbea\singular"regionwhereeither!r!vs6=0orthereisnosuperuid.Soweconsiderthesingularregionasaverythincylinder,calledasuperuidvortexlineorvortexcore.AccordingtoGauss'theorem:Rv(!r!v)d=Hs!vd!S,and!r!r!v0forany!v,wehaveRv(!r!r!vs)d=HS!r!vsd!S=H!vsd!`=0=.Soavortexlinecannotterminateintheuid,butmustendataboundaryorcloseinonitself(avortexring).Sincethevortexlineistheonlysourceofvorticityintheuidfor!r!vs=0everywhereexceptattheline,allpathintegralsencirclingthevortexlinehaveidenticalcirculations. AssoonasliquidheliumIIrotatesormovesbeyondacriticalvelocity,superuidvortexlinesappearanddemonstrateeitheranorderedarrayofvortexlinesbysteadyrotationordisorderedvortextanglesforcounterow(duetoheatow).ThatthesuperuidcirculationisquantizedwaspostulatedseparatelybyOnsagerandFeynman 18

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m9:97104ncm2=s,wherenisaninteger.Theradiusofthevortexcoreisabouta01A(atomicdimensions).ThestablequantizedvortexarraysinrotatingHeIIcanbevisualizedasshowninFig. 1-3 [ 14 ]:stablevortexlinesinrotatingHeIIinacylindricalbucketof2mmdiametertodepth25mmplacedattherotationaxisofarotatingdilutionrefrigeratorat100mK.8%3Hewasaddedtoprovidedampingwhichmaintainedstability.Thenegativeionsaretrappedonvortexcoresandareimagedonaphosphorscreenandrecordedoncinelm.Allsuperuidvortexlinesalignparalleltotherotationaxiswithorderedarraysofarealdensity(orlengthofquantizedvortexlineperunitvolume)asgivenbythefollowingequation: 2 (inlines=cm2),whereistheconstantangularvelocityfortherotation.Thiscanderivedasfollows:thecirculationaroundanycircularpathofradiusrconcentricwiththeaxisofrotation=H!vsd!`=H(!r!vs)d!S=2r2.Andthetotalcirculation=r2n0h=m,wheren0isthenumberoflinesperunitarea.Therefore,n0=2m=h=2=[ 7 ]. Theturbulentstate,describedasamassofvortexlines,usuallyhastwocriticalvelocitiessignalingtheonsetofturbulenceinsuperuidandinthenormaluidseparately,increasingthetotallengthofvorticitywiththeincreasingrelativevelocityofthetwouids.IntheexperimentonthedampingoftherotationofasphereoscillatinginliquidheliumIIasafunctionofthemaximumamplitude(orvelocity)oftheoscillation,theresultisshowninFig. 1-4 [ 7 ].AtregionA,thedampingisconstant,relatingtotheconstantnormalviscosityn.ThetwocriticalvelocitiesoccurredatthetransitionfromregionAtoB(whichistheonsetofturbulenceinthesuperuidcomponent),andCtoDcorrespondingtotheonsetofturbulenceinanordinaryclassicalliquid,wherethedampingincreasesdramatically.InregionsB,C,andD,bothsuperuidandnormaluidarecoupledandmovetogetherduetotheirmutualfriction.Thecriticalvelocityofthesuperuidriseswithreductionindiameterofthechannel. 19

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11 ](seeFig. 1-5 ).Theplaneofthesevibrations(calledKelvinwaves)precessaboutthecentercore,growingexponentiallyalongquantizedvortexlines.Thelengthsofthevortexlinesincrease,eventuallyresultinginavortextangleastheenergyistransferredfromthenormaluidtothesuperuid. Feynmansuggestedin1955thatvorticesapproachingeachotherverycloselyundergoreconnections.TheKelvinwavescanbegeneratedbyvortexreconnections,leavingkinksonthevortexlines,regardedassuperpositionsofKelvinwaves,leadingtothecontinuousgenerationofKelvinwaveswithawiderangeofwavenumbers. 1(whereUisthecharacteristicvelocityandisthekinematicviscosity)andenergydissipationbecauseofviscosityoccurs.However,iftheReynoldsnumberRe=inertialforce viscosityforce1(inan\inertialregime")sothattheviscositycanbeignored,thentheenergywillowinacascadefromlargescalestosmallerscales,asdescribedbytheKolmogorovspectrum[ 11 ]: whereE(k)dkistheenergyperunitmassforspatialwavenumbersintherangedk.ThefunctionE(k)hasdimensions[L3=T2](L:length,T:time).C1:5(theKolmogorovconstant,whichisdimensionless),and"=dE dtdk(thatistheaveragerateofkineticenergytransferperunitmassowingdownthecascade,dissipatedbyviscosityathighwavenumber,k>`1).Thedimensionsof"are[(L=T)2=T]=[L2=T3];thedimensions 20

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1-6 [ 11 ],thelargesteddieshavethemostturbulentenergy,anddecaymostslowly,whichdeterminestheenergydissipationrate(").Inthesteadystate,energyowsfromtheselargesteddiestothesmallesteddies. ThesuperuidgridturbulenceexperimentsinheliumIIabove1K[ 12 ]canalsobedescribedbytheKolmogorovspectrumintheinertialregimeonlengthscaleslargerthanthespacingbetweenvortexlines(`).Inthiscase,theenergydissipationbyviscosityoccursonthelengthscaleoforder`(thespacingbetweenvortexlines)duetothesignicantamountofnormaluidandthemutualfrictionbetweenthesuperuidandthenormaluid. 12 ]showthathomogeneousisotropicturbulenceisproducedbehindatowedgridmovingattheorder1m/s.Computersimulations[ 15 ]predictatzerotemperaturetheKelvinwavesonintersectingvortexlinesproducetheequivalentoftheviscousregimeinaclassicaluid.Thishasyettobeconrmedbyexperiments. Itissuggested[ 16 25 ]thatenergyowstothesmallestscalebyaKelvinwavecascadeonthevortices,leadingtoaKelvin-waveenergyspectrumforthewavenumber~kofKelvinwavesgreaterthantheinversevortexspacing`1.Kelvinwavesdonothaveanydampingatverylowtemperaturesuntilthewavenumbers~k(~k2=2108m1)[ 16 25 ]becomemuchgreaterthan`1.Ithasbeenpredicted[ 16 ]thatat0.46K,energydissipatesbyphononradiation;howeverthishasnotbeenconrmedexperimentally.TheresultsofthesimulationsdemonstratethecontinuousenergyowwithintheKelvinwavestowards 21

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ThecorrespondingKelvinwavespectrum,cutobydissipationat~k~k2,isproposedtobe:[ 25 26 ] ~E(~k)=A2~k1;(1{18) whereE(~k)d~kistheenergyperunitmassandunitlengthofvortexassociatedwithKelvinwaveswithwavenumbersintheranged~k,isthedensityofthehelium,andA2.TherateatwhichenergyowsintotheKelvinwavecascadeperunitmassofheliumisgivenby[ 26 ] ~"(2=2)3L20;(1{19) whereL0=`2isthelengthofthesmoothedvortexlineperunitvolume,and20:3.Theenergycontainedinthe\Kelvinwavecascade"perunitmassofheliumisgivenby[ 26 ] ~E=1 0)2L;(1{20) whereL=AL0ln(~k2`),Evistheenergyperunitlengthofvortexline,and0isthevortexcoreparameter.Thecut-owavenumber~k2isgivenbytheformula[ 16 ] ~k2`=(c` A1=3)3=4;(1{21) wherecisthespeedofsoundinhelium. Bymeasuringtheriseintemperatureoftheheliumaftercreatingturbulencewithahighresolutionthermometer,wecanprobetheturbulencedecayasafunctionoftimesincethetemperaturechangecorrespondingtothedecayofarandomvortextangleisproportionaltothechangeinthevortexlinedensity.Therefore,itispossibletoexploretheexistenceofaKolmogorovspectrumonlargelengthscales,aKelvinwavecascadeonsmalllengthscales,andthedissipationmechanism.Theenthalpyoftheheliumisgivenby 22

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3 ] (inJ=m3).Ifasmallamountofturbulenceenergy(E)isreleasedasthermalenergyinheliumattemperatureT0,theenthalpyvaluesofthetemperaturechangewouldbe (inK4),whichcanbemonitoredasafunctionoftime.Thegridturbulenceiscreatedbydrawingagridthroughtheheliumbyasuperconductinglinearmotorataconstantspeedasfastas1m/sfor10mmforapproximately10ms,whichisfasterthanthedecayspeedoftheturbulence(afewhundredmilliseconds).Anyenergydissipationintheheliumfrommovingthegridmustbemuchsmallerthanthereleasedthermalenergyfromthedecayofthesuperuidturbulence.Supposethesquarecrosssectionofthechannelisddandaverydensevortextangleaswellasquasi-classicalturbulence(onalargerscalethanthemeanvortexlinespacing`)areproducedbythetowed-grid.Theenergycantransfertoeitherlargerlengthscales(onthescaleofd)untilbecomingsaturated,ortosmallerlengthscales(lessthan`)whereenergydissipationoccurs,leadingtotheKolmogorovenergyspectrum.Thetimerequiredtobuildthisspectrumshouldbelessthantheturnovertime: u(d);(1{24) whered=eddysize,andu(d)=characteristicvelocityrelatingtothiseddysize,denedby[ 26 ] Fromthecluesinthepreviousexperimentsabove1K,itissurmisedthataKolmogorovspectrumjoinssmoothlytothe\quantumvelocity", `:(1{26) 23

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26 ] `)(d where0:25.Thetotalturbulentenergyperunitmass,mostofwhichcomesfromthelargesteddiesofsized,isgivenby 2u2(d)=3 2( `)2(d As`increases,turbulencedecaysandenergydissipatesviatheclassicalKolmogorovcascadeasdescribedbytheaboveequation.Attimetd,totalenergydecaywithtimeasdescribedbytheKolmogorovspectrumcanbethoughtofastheenergyowratewithtime[ 26 ], wheret0d,whichvarieswiththetowed-gridspeedsorinitialturbulentintensities.So: 2C3d2(tt0)2:(1{30) ComparingEquations 1{28 and 1{30 ,thetimedependenceof`isgivenby AnothercharacteristictimefortheKelvin-wavespectrum(Equation 1{18 )whichdescribesthefullydevelopedKelvin-wavecascadeisgivenby ~=~E 2)`2 EnergyowsintotheKelvin-wavecascadeattherate"=~",andeventuallydissipatesbyphononradiation.TheenergyperunitmasscontainedintheKelvin-wavecascadeisgivenas ~E=A2`2ln(~k2`):(1{33) 24

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2C3d2(tt0)2+A2`2ln(~k2`):(1{34) Since`increaseswithtime,thedecayofEwillbedominatedbythedecayofEclassforsmall`,butbythedecayof~Eforlarge`.Fortt0,Eclassisproportionaltot2,while~Eisproportionaltot3=2,whichwillbeexpressedbytheobservedrate-of-changeoftemperature. 1.Operatingtemperature:20mK-to1K(dilutionrefrigeratortemperatures). 2.Sensitivetotemperaturechange:T103K,orT=T0:05103. 3.Shortresponsetime:t103s.Theturbulenceenergydecayswithinafewhundredmilliseconds.Inordertohavegoodtimeresolutioninthedata,itisnecessarythatthethermistorsrespondwithin1ms. 4.Smallmass,smallheatcapacity,andgoodthermalconductivity. Sofarwehavefoundtwoexcellentcandidatestofullltheaboverequirementswhichcanbeusedinourcalorimetrictechnique:NeutronTransmutationDopedGermaniumBolometersandMiniatureGeFilmResistanceThermometers.Wehaveusedthelaterinourwork. 1-7 [ 18 ]) 1.Thermalgenerationofelectronsandholesacrossthebandgap,whichisnegligibleatlowtemperaturesincekTEgap. 2.Generationoffreechargecarriersbyionizationofshallowdonors,whichisnegligibleatlowtemperaturesincekTECED. 25

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4.Inheavilydopedandcompensatedsemiconductors,thecompensatingorminorityimpuritiescreatealotofmajorityimpuritieswhichremainionizeddowntoabsolutezero.Thechargecarrierhopsfromanoccupiedmajorityimpuritysitetoanemptysite,whichistheworkingprincipleforthelowtemperaturebolometer.Itisalsocalledhoppingmechanism. Thematerialweuseforourhigh-resolutiondilutionrefrigeratorthermometerisneutrontransmutationdoped(NTD)germanium[ 18 19 ],whichappliesthefourthelectricalconductionmechanismasdiscussedabove. ANTDGehasNAmajorityshallowacceptorimpuritiesandNDminorityshallowdonorimpurities(NA>ND).Atverylowtemperatures(kTEAwhereEAisthebindingenergyoftheelectronstotheacceptors),andinthedark,(NAND)acceptorshaveanelectronvacancyandareneutralwhileNDacceptorscaptureelectronsfromcompensatingdonors. Thetransmutationofstablegermaniumisotopesviathecaptureofthermalneutronsisaccomplishedbythefollowingprocedure: 1.Asingleultra-puregermaniumcrystalisgrowninahydrogenatmosphere(1atm)fromameltcontainedinapyrolyticcarbon-coatedquartzcrucibleusingtheCzochralskitechnique. 2.Six2mmthickslicesof36mmdiameterarecut,lappedandchemicallyetched. 3.Irradiatedwiththermalneutrons;doses7:510161:881018cm2. 4.Aftertenhalflivesof7132Ge(T1=2=11:2d),thesamplesareannealedat400Cfor6hoursinapureargonatmosphere(1atm)toremoveirradiationdamage. Somepapersdemonstratethat,evenatdilutionrefrigeratortemperatures,theNTDGethermometersstillhavesucientsensitivity.Forexample:at25mK,T=T4:8106,andresponsetime<20msforthermometersassmallas1mm1mm0:25mm[ 20 ].SomeverygoodcircuitryfortheNTDGebolometershasbeendeveloped[ 20 21 ]. 26

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22 { 24 ]havealreadybeendevelopedandtestedinourgroup.Thetypicalsizeforthesensitiveelementis300mindiameter,andthemountedgoldleadsare50mindiameter.Theconductionmechanismisvariablerangehopping.Fig. 1-8 showstheabsoluteresistancevaluesandthesensitivity(dR dT)overthetemperaturerangeofinterest,forthreetestthermistorsagainstarutheniumoxidecalibratedthermometer.WeusedtheLR-110picowattACresistancebridgeasthedetectioncircuit,andcalibratedthreeofouravailablethermometers.TheLR-110bridgecanmeasureresistancesbetween10and1.2Mwithhighresolution(betterthan0.1%)andgoodaccuracy(0.05%0:25%).Thecalibrationcurvesdemonstratingtheperformanceandthecharacteristicsofthethreetestthermometersarequitedierentduetothevariationsindopingandheattreatmentduringmanufacturing.Sensingpowerswerelessthan1013watts.Twoofthethermometersarenotidealaswecanseefromthecurves.Forthermistor1,theresistance7or8kunder100mKandwasnearlyconstantbelow38mKwithprettystablesensitivitywithinthemeasuredtemperaturerange.Forthermistor2,theresistancegoestoinnityatlowtemperaturesandonlybecomesmeasurableaboveabout49mK.Thermistor2demonstratesdramaticsensitivitychangeoverawiderangeoftemperatures.Forthermistor3,theresistanceis16kat88.7mKand526kat24.8mK.Thesensitivityrangesfrom1.7Meg=Kto140.3Meg=Kbetween50mKand20mK.Itsresponsetimeislessthan0.001s.Thismakesthermistor3agoodcandidate.Duringturbulencedecay,theresistanceofthermometer3isexpectedtochangefrom4kto1.75k,andthechangefrom3ktoapproximately2krepresentingtheKelvinwavedecayat520mK,whichappearsinournalexperimentalresults. ThemachinerydrawingforthethermometertestcellisshowninFig. 1-9 .Thethermometertestcellisusedtosimultaneouslytestthesensitivityandtheresolutionoftwothermometers(miniatureGelmresistancethermometers)separatedbyasmall 27

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28

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Viscosityofliquid4Hemeasuredinanoscillatingdiscviscometer[ 7 ]. Figure1-2: Superuidandnormaluiddensities(nands)asafunctionoftemperaturebelowlambdatransitionunderthesaturatedvaporpressure. 29

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PhotographsofstablevortexlinesinrotatingHeIIinacylindricalbucketof2mmdiametertodepth25mmplacedattherotationaxisofarotatingdilutionrefrigeratorat100mK. Figure1-4: Dampingonasphereoscillatinginliquidheliumat2.149Kwithaperiodof18.5s[ 7 ]. 30

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Donnelly-Glabersoninstabilityofaquantizedvortexlineoccursifthecomponentofthenormaluidvelocityparalleltothevortexlineexceedsacriticalvalue[ 11 ]. Figure1-6: Energyspectrumforhomogeneousandisotropicturbulence[ 11 ].(k:Kolmogorovwavenumber,whereviscousdissipationbecomessignicant;kC=2 d,d:thesizeofthecontainer;ke(t)=2 `e(t),`e(t):eddylengthscale)[ 11 ]. 31

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Electricalconductionmechanismsinsemiconductors[ 18 ]. Figure1-8: Calibrationplotsforthreetestthermistorsdevelopedforcalorimetry.(a)Resistanceversustemperature.(b)Sensitivityversustemperature. 32

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Designforthethermometertestcell.(a)Overview.(b)Topviewofcap. 33

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2-1 [ 27 ]),whichisthesimplestcaseamongthecomplexnonlineardynamicssystems.Inordertocomparewiththisclassicalcase,weintendtoproducehomogeneousisotropicquantumturbulence(HIQT)inliquidheliumIIbelow1K. InordertoproduceHIQT,wehavedesignedandbuilttheshieldedsuperconductinglinearmotorforourtowed-gridturbulenceexperiments.Firstofall,webuildamodel,showninFig. 2-2 [ 28 29 ]:asinglesuperconductingsolenoidmotorwithanarmaturemovingthroughitscenter.Thislightandhollowinsulatingarmatureisconstructedof3phenolictubesseparatedbytwohollowcylindricalniobiumcansplacedsomedistanceapart,withtheturbulence-producinggridattachedtooneend.Therequirementsandadvantagesofourmotorsystemareasfollows: 1.Thesuperconductingshieldcanavoideddycurrentheatinginthecellwalls. 2.ThesuperconductingsolenoidcanavoidJouleheatingfromthesolenoid. 3.Withanappropriatecurrentpulse,thegridcanbeecientlyacceleratedanddeceleratedfrom0to1.0m/swithin1mm.Andthegridcanbedrivenatanearlyconstantspeed1m/sfor10mm,producinghomogeneousisotropicturbulencewithin20ms. Intheresultinggridmotioninoursimulator,thegridwouldbeacceleratedfrom0to1m/sin1mm.Thenittravelsatalmostconstantspeed,1m/s,for10mm.Thenthegridwouldrapidlydeceleratetoceasewithin1mmwhenthethirdpulseisapplied.Thenweputoursimulationresultsintopractice.Webuildourtestcellguidedbythesimulation.Inourtestcell,wehaveonesuperconductingsolenoiddrivingthearmaturetomovethroughitscenter,withagridattachedattheend.Thislightinsulatingarmatureisconstructedof3phenolictubesseparatedbytwohollowcylindricalniobiumcansplacedexactly26mmapart,withthegridattachedtooneend.Aconductingsection 34

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Ourunshieldedtestmotorsystemhasbeentestedverysuccessfully.Weapplythepulsestothesuperconductingmotortodrivethearmatureinsidethesolenoid:twosinepulseswithDClevelinbetween,followedbysmallDClevelfor250ms.Intheresultingvelocityversustimecurve,wealreadycanacceleratethearmatureto1.1m/s.Thevelocityremainszeroforabout250mswhenthearmatureisheldonthetop.Inthevelocityversuspositioncurve,wecanseethatthearmaturemovesatalmostconstantvelocity1m/s0.1m/sforatleast8mm. Wealsoimprovethisdesigninoursuperconductorshieldedsuperconductingmotorsystem,discussedinthenextchapter.Oneoftheimportanttaskistobuildthesuperconductorshieldontheinteriorofthecell.Thedetailswillalsobediscussedinthenextchapter. Inchapterfour,wewouldliketodiscussabouttheaccomplishmentofthefollowings: 2-2 ,madeofthreecoaxialparts:onesuperconductingshieldwiththeradiusbandlengthh,onesuperconductingsolenoidwiththeinnerradiusrandlengthl,andalightinsulatingrodwithtwoniobiumcylindersattachedandseparatedbydistanceS.Thegridisattachedtotheendoftherodandhenceispushedbyit. Supposethediameterofthesuperconductingwireforthesolenoidisd,andthetotalnumberofturns,N.Wecanestimateapproximatelytheself-inductanceofsucha 35

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IN(0NI)r2 (inHenry,andallthelengthsareinmm),whereisthetotalmagneticuxowingacrossthesolenoid,Iisthecurrentowingthroughthesuperconductingwireofthesolenoid,Bistheapproximatemagneticeldatthecenterofthesolenoid,Aistheaveragecrosssectionareaofthesolenoid,andristheaverageradiusofthesolenoid: r=r+Nd2 Thezcomponentofthemagneticeldattheposition(,,z)nearthesolenoidisderivedasthefollowingbygeneralizingtheprobleminGriths[ 30 ]: where 2)d;z0=l 2)d:(2{5) Forthemagneticelddistributioninsidethesolenoidenclosedbythesuperconductingshield,weciteEq.12inSumner[ 31 ]: hlXkS1(krb)I0(k) where h(2{8) 36

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Supposetheniobiumisaperfectdiamagnet,thenthemagneticforceexperiencedbytheniobiumcylinder#1withthecenterofmassatz=zialongtheaxisofthesolenoid(thecenterofthesolenoidisdenedasz=0position): SinceM=Bz 32 ],themagneticforceisproportionaltothegradientofthemagneticeldsquare: wherem,M,0,ro1,l1arethemagneticmoment,themagnetization,thepermeabilityatvacuum,theouterradiusandthelengthofNbcylinder#1,respectively.Forthepurposeofcomputersimulations,thepracticalformulaforthenumericalanalysisandthecorrespondingCcodeareintheAppendixAandB,respectively. Inadditiontothemagneticforce,thegravityforce~Fgavisalsoconsidered;therefore,accordingtothesecondNewton'slaw,thenetforceexperiencedbythewholesystemwithmassMsystemproducetheacceleration,a:~Fmag+~Fgav=Msystem~a,or Thesystemincludestheinsulatingrod,thetwoniobiumcylindercansandthegrid.BecauseFmag(zi)/I2,wecanwriteFmag(zi)=fmag(zi)I2.Supposethesecondniobiumcylinderissomedistance,S,belowtherstone,thenthecurrentrequiredtoreachtheobjectiveaccelerationwouldbe: (unitinAmpere). 37

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Inaddition,wehaveseveraloptionsforthemathematicalformoftherelationshipbetweenthedestinedterminalvelocityv(inm/s)andthetravelingdistanceS1(inmm)duringtheperiodoftimeoftheaccelerationa(inm=s2),suchaslinear,squareandsinefunction.Foreachsmalltravelingdistance,dz(mm),theaccumulatedvelocity,accelerationandtimeattheithincrement,vi(m/s),ai(m=s2),andti(ms): S1dz(m=s)(2{13) 2)(v S1)2dz103(m=s2)(2{14) aS1)103(ms)(2{15) S21(idz)2(m=s)(2{16) 2)(v2 S1)(m=s)(2{19) 38

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S1)sin2[(i1) S1]g103(m=s2)(2{20) afsin(i S1)sin[(i1) S1]g103(ms)(2{21) Forthedeceleration,wesimplyhavethelinearmathematicalform.SupposethetravelingdistancewouldbeS3(inmm).Foreachsmalltravelingdistance,dz(mm),theresultantvelocity,accelerationandtimeatthejthdisplacementinterval,vj(m/s),aj(m=s2),andtj(ms): S3dz(m=s)(2{22) 2)(v S3)2dz103v2 aS3)103(ms)(2{24) 2.3.1SimulationAnalysis 2-1 (unitsoflengthsinmm). Fig. 2-3 (a)showstherequiredcurrentversustimecurvesfortheunshieldedandsuperconductingshieldedsolenoidwithasinefunctionacceleration(velocityisthesinefunctionoftheniobiumposition).Thecurvesshowthreepeaks:therstandthethirdpeaksaretoaccelerateanddeceleratetheniobiumcylinders.Themiddlepeakisduetothealmostbalancedmagneticforcesonthetwoniobiumcylindersatthatpositionsince 39

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2-3 (a),andseethatthevelocityversuspositioncurveinFig. 2-3 (b)(again-symbols)isvirtuallyunaected.Thedroopinvelocityafterz=10mmisunavoidablesinceallforces,includinggravity,aredirecteddownwardfortherestofthestroke. Table2-1: Optimalparametersforthemotordesign. ParameterDescriptionValue Notethatthesuperconductingshieldrequiresaslightlyhighercurrent(0.14Agreater)toproducethesamemotion.Theeectissmallbecausetheshieldissignicantlylargerthanthesolenoid. InthevelocityversuspositioncurvesinFig. 2-3 (b),weseethatthemotionisasexpected.Theniobiumcylinderstravelatalmostconstantspeed,1m/s,for10mm,butstarttoslowdownwhenthemiddlecurrentpeakoccurs,whichisquitereasonable.Aftertherstniobiumpassesz=10mm,thesecondniobiumisclosertothesolenoidandexperiencesastrongermagneticforceintheoppositedirection,resultingintheslightdeceleration.Therefore,applyingthethirdpulseproducesthedesireddecelerationtorapidlystopthegrid.Theevaluationresultsprovethatoursuperconductinglinearmotorisaveryfeasibledesign. 40

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2-4 ,Fig. 2-5 .Itrequiresaslightlyhighercurrent,0.149Aand0.8Agreater,forthesuperconductingshieldedsolenoidtoproducethelinearandsquareaccelerationmotion.Therefore,sinefunctionaccelerationisamoreecientwaythattakestheleastcurrentamongthethreeoptions. TheshotsofallthesimulationprogramsareinAppendixC. ThetheoreticalestimatesoftheloweranduppercriticalmagneticeldsforthetypeIIsuperconductorsare[ 32 ]: where0isthesuperconductinguxquantum,calledauxoidoruxon: 0=2~c=2e=2:0678107(2{27) (inGausscm2).andarethepenetrationdepthandthecoherencelength,respectively.Forexample,theniobiumhasthepenetrationdepthatabsolutezeroestimatedfromthemeasurementstobe470A[ 33 ],andthesuperconductingcoherencelengthis11nm 41

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34 ].SoHc12980Gauss,Hc254397Gauss.Thepenetrationdepthandthecoherencelengthareactuallytemperaturedependent. Experimentally,Hc1andHc2valueshavebeenmeasuredtobedependentonthepurityandtheresidualresistivityratios[RRR=R300=RN4:2]ofniobium[ 35 36 ],theeldorientation[ 37 ],andthetemperature[ 35 { 38 ].R300istheresistancemeasuredatroomtemperature.RN4:2isthenormalstateelectricalresistivitymeasuredat4.2Kinaeldof0.6Torhigherafterthemagnetizationdatameasurement.Thehigherpuritytheniobium,thehighertheRRRvalues,andthelowertheHc2values.Forexample,inFig. 2-6 [ 35 ],atthetemperatureT1.5K,theuppercriticaleldHc212.8KGauss,7.3KGaussand3.5KGaussforRRRoftheniobiumsamples=3.1,8.8and505.Forthetemperaturedependence,thedatahasshownthethermodynamiccriticaleld,Hc,withtheformHc(T)=1993[1(T Tc)2](inGauss)forRRR1600[ 36 ],thelowercriticaleld,Hc1(T)=1735[1(T Tc)2:13](inGauss)forRRR1400[ 36 ],andtheuppercriticaleldHc2=41001(T=Tc)2 38 ]. Inaddition,thecriticalsurfaceeld,Hc3(T)=1:695Hc2(T),hasbeenpredictedbySaint-JamesanddeGennes[ 39 ]andmeasured[ 36 38 ]fromtheonsetofzeroresistivityandtheACsusceptibility,whichisalsotemperature,purity,RRRvalueandsurfaceofthesamplesdependent.BetweenHc2andHc3,superconductivityandsurfacesupercurrentappearintheformofasurfacesheathwithathicknessabout(T)onsurfacesparalleltotheappliedmagneticeld. Experimentallywhatwewoulddoistomakethinhollowniobiumcylinderswithendcaps.Belowthesurfacecriticaleld,theenhanced\surfacesheath"magnetizationmakesthethinsurfacesheathstillsuperconductingwhilethebulkisinnormalstate.Bymakingthecylinderhollowwewouldenlargethetotalvolumeofniobiumtoexcludemoremagneticuxwithmuchsmallermassandreducingthetotalmassofthewholesystemmaketherequireddrivingmagneticforcemuchless.Evenwhentheeldisbeyondthe 42

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Innumericalanalysis,weuse: (involts),wheredtistheinnitesimaltimeintervalbetweenthetime(t-dt)andtwhenthecurrentowingthroughthesolenoidareI(t-dt)andI(t),respectively.TheLabViewsimulationprogramcalculatestherequiredvoltageinputtothesuperconductingshieldedsolenoidwithsine,linear,andsquarefunctionaccelerationasshowninFig. 2-7 .Sincethecurrentforthesinefunctionaccelerationhassmoothtransitionwithrespecttotime,ithasthebetterperformancethantheothertwo,i.e.lowerrequiredvoltage.Stillthevoltageistoohugetohavethepracticalapplicationinthelaboratory. Inordertosolvethisproblem,weneedanadditionalcapacitorwithcapacitanceC(F)inthecircuittoformanLRCcircuit.InanLRCcircuitunderthesinusoidallydrivenvoltage,V=V0ei!t,theKirchhorulerequiresthatthesumofthechangesinpotentialaroundthecircuitmustbezero,so dt+IR+Q C=LdI dt+IR+1 whereQ(Coulomb)isthetotalchargeonthecapacitor.Thesolutionforthecurrentwouldbe: 43

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and R)(2{33) Atresonance,for!=1 40 ]. Wecancalculatetherequiredvoltageinputtothesolenoidfromthecalculatedcurrentcurve: Inthewayofnumericalanalysis: NCNXi=1I(ti);(2{35) where N(i1 2)(2{36) Anotheralternativewayofsendingthepulsestothesolenoidistoinputtheperfectsinusoidalshapecurrentpulsemodiedaccordingtothepreviouscalculatedcurrentprole,whichwouldstillgivetheexpectedmotionforthearmatureofthemotor,asshowninFig. 2-8 SupposewesupplythecurrentI(t)=I0sin!t.Ifweonlyconsiderthesolenoidwithinductance,L,andtheresistanceforthewholecircuit,R,thentherequiredinputvoltagewouldbe ForI0=2.2A,R=0.1,!=1200rad/s,L=0.056H,thesameparametersasthoseinFig. 2-8 ,thentherequiredvoltageinputtothesolenoidversustimewouldbelikeFig. 2-9 .Somevoltagepulsesaslargeas150Vseemtobetoolargetobeapplicable. Experimentally,wemightneedC=11.2Ftotunethecircuitonresonance.Atresonance,i.e.!=1 R.Duringtheconstantcurrentperiod 44

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2-10 .Thedetailedcircuitryforthewholeexperimentalsetupwillbeintroducedveryshortly. 2-2 .Withthoseparameters,weruntheLabViewsimulationprogramstondouttherequiredcurrentproleforthesinefunctionacceleration,andthenwemodifythiscurrentproleintotwoperfectsinusoidalpulsesalongwiththeconstantDCcurrentinbetween.Fig. 2-11 showsthemachinedrawingsandthephotosforourunshieldedmotortestcell.Aswecanseefromthedrawingsthatwebuildthecapacitivepositionsensorinordertomonitorthemotionofthearmature. Table2-2: Parametersfortheunshieldedtestmotor.RefertoTable 2-1 ParameterValue 45

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2-12 .Wesendthenegativepulsefromthecomputer(analogoutput1:DAC0)throughourcurrentamplier.Thecurrentamplierampliesandinvertsthepulse.Thenthepulseisfedintotheswitchbox,wheretheswitchescanbecontrolledbytheTTLsignalsfromthecomputer(analogoutput2:DAC1).Eventuallythepulseissenttothesolenoidinthecryostatatheliumtemperature,4.2K.Wealsousethecomputertomonitorthepulsesenttothesolenoid(analoginput2:ACH2),andsimultaneouslythecapacitivepositionsensorismeasuringthepositionofthearmature(analoginput1:ACH1). Thesensorisconnectedtothelock-inamplierandcapacitancebridge(GR1616).Wesetthedrivingrmsvoltage1V,frequency1.01kHzfromthereferencesinewaveformoutputofthelock-inamplier,andthetimeconstant100ms.Atroomtemperaturethecapacitancebridgereadsthecapacitanceofthecapacitivepositionsensor3.32pF(themagnitude,R,oftheoutputonthedisplayofthelock-inamplierreadsminimum39V)whenthearmatureisatrestand3.00pF(themagnitude,R,oftheoutputonthedisplayofthelock-inamplierreadsminimum108V)whenthearmaturemovesallthewayuphittingthebrassplate12mmaboveintheair.Bysettingthesensitivityofthelock-inamplier1mVandthestandardcapacitanceofthecapacitancebridge3.32pF,thechannel1outputofthelock-inamplierreads0.44V(R=44.7V)asthearmaturesittingatrest,whileasthearmaturemovesup12mmthechannel1outputreads2.54V(R=255V).Withsuchsignicantvoltagechange,2.10V,whichcanbeeasilyrecordedbytheLabViewdataacquisitionprogram,wecanconvertthevoltageshiftintotheposition,oreventhevelocityofthearmaturemotion.Whencoolingdowntoliquidheliumtemperature,4.2K,someconditionsmightchange.Weneedtochangesomeofthesettingsofthelock-inampliertorecordthevoltageshiftfromchannel1 46

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Itturnsoutthatwedonotneedthecomputercontrolledswitchbox,asthegrayregionintheschematic,becausecurrentamplierltersoutthebackemfsfromsolenoid.Thedistortedpulsewithspikessenttothesolenoidwillbeproducedwithoutbothcurrentamplierandtheswitchbox,asmentionedpreviously.Weturnedupthecurrentsenttothesolenoidlittlebylittlebyturningupthegainofthecurrentampliergradually.Eventually,whenthepeakcurrentreachedtheexpected2.86A,weheardthesoundfromthedewarlikethearmaturehittingthebrassplate12mmaboveit.Whatasuccess!Verydisappointedly,thecapacitancebridgecannotrespondfastenoughtohaveanyobservablechange.Lateronwerealizedfromsomeexperimentalteststhatthecapacitancebridgetakesatleast100mstorespond.Itcouldbetheinductanceoftheratiotransformerslowingdowntheresponsetime.Abettermethodtomeasurequantitativelythepositionofthearmatureofthesuperconductingmotorsystemisnecessary.Sowehave555oscillatorcircuitbuiltandconnecttothesensorinparallelandtheLabViewprogramwiththecounterfunctioncountingthefrequencychangewithtimeisdevelopedasthesecondattemptforthealternativemeasurementtechnique. 2-14 .Nowinsteadofusingthecapacitancebridgeandthelock-inamplier,wehavethe555oscillatorcircuitbox,whichcanoscillateandoutputTTLpulseswith 47

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48

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2-15 (a).Ourtheoreticalsimulationprogramspredictthismethodfeasible,forthecapacitancechangeaccompanyingwithnoticeablecurrentchange,whichgivessomenoticeablevoltagechangeacrosstheresistor,capacitorortheinductor,especiallyforhigherQ(qualityfactor).SincethecurrentinanRLCcircuitinseriesis where R;(2{39) themaximumofthecurrentversusfrequencygraphisat!=!0=1=p ThetestingcircuitisshowninFig. 2-15 (b).Wetriedtotunethefrequencyofthereferenceoutput(sinewave)fromlock-inampliertogetthemaximumvoltageacrossanyoneelementofthecircuit,whichshouldfallatresonanceoftheRLCcircuit.Therefore,withalittlecapacitancechangefromthecapacitivepositionsensor,thecircuitwillbeoresonance,thenwearesupposedtoseethedramaticcurrentdecrease,therefore,dramaticvoltagechange.Unfortunately,wedidn'tobserveanynoticeablevoltagechange(alllessthan2%)eitheracrosstheresistor,capacitorortheinductor.Wealsoexchangedthepositionoftheresistorandinductor,orrearrangethecircuittomakethecapacitorsandinductorinparallel.Butnoneoftheaboveworkedwelltoshowanylargeenoughvoltageshift.Hence,weneedtogiveupthisapproach. 49

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Wehaveusedthecapacitancebridgealongwiththelock-inampliertomonitorthemotionofthearmaturebeforebymeasuringthecapacitancechangeofthecapacitivepositionsensor.Theoutputvoltageshiftfromlock-inamplierwhenmovingthearmatureupanddowncouldbeaslargeas2Vormore,whichgivesenoughsensitivity.Theonlyproblemistoovercometheslowresponsetime,100ms,oftheratiotransformer.Fig. 2-16 showsoneofthesimpleACbridgecircuitswebuilt.Aswhatwehadexpected,theACbridgecircuitcanrespondasfastaslessthanafewmicrosecondswithoutanydelay.Thefollowingsarethedetaileddescriptionsabouttheexperimentalapparatusandsetups. Thecapacitivepositionsensoriscomposedoftwocoppersemi-cylindricalsheetsalongwiththesecondniobiumcan.Inthemagneticeld,theslitsofthecoppercylindricalsensorcouldpreventtheeddycurrentandthereforetheheatdissipationfrombeingproduced.Weusethestrongersuperconductingwirefortheleadsofthesensorandthemoreexibleinsulatedbertubetoprotecttheleads.Epoxyisusedtogluethecoppersensortothecapacitorframemadeofphenolic.Fig. 2-17 showsthecircuitconnectiontomeasurethecapacitanceofthecapacitivesensorwiththecapacitancebridge.Atroomtemperaturethecapacitancebridgereadsthecapacitanceofthecapacitivepositionsensor2.11pF(themagnitude,R,oftheoutputonthedisplayofthelock-inamplierreadsminimum138.9V)whenthearmatureissittingallthewaydown,and1.59pF(themagnitude,R,oftheoutputonthedisplayofthelock-inamplierreadsminimum153.4V)whenthearmaturemovesallthewayuphittingthebrassplate12mmaboveintheair.Thecapacitancedierenceisassmallas0.52pF. NowweconnectthepositionsensortotheACbridgecircuitbox.ThecircuitdiagramisshowninFig. 2-18 (a).Insteadofusingthenewdigitallock-inamplier(SR830),weusetheantiqueanaloglock-inamplier(PAR119)togetridofthedigitizedproblem 50

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Wealsocarefullycalibrateandzerotheosetofthecurrentampliertowithinafewnano-amperes.Afteradjustingthebrassplateabovethetopofarmaturetobe21.8mm,thearmatureisexpectedtohaveenoughspacetomoveupandpassthesecondnetzeromagneticforcepositiontobeheldtherewithsomeDClevel.Fig. 2-19 (a)demonstratesthetheoreticalcalculationofthemagneticforceperunitofcurrentsquareexperiencedbythearmatureinsidethesuperconductingsolenoidasafunctionoftheposition,z,denedastheverticaldistancebetweenthecenteroftherstniobiumcanandthemiddleofthesolenoid.Ourplanistostartapplyingtheaccelerationpulseatz0=6.43mm,aroundthepeakpositionofthepositivemagneticforcezone,thenthevelocityofthearmaturecarriesthroughtherstnetzeromagneticforcepositionataboutz=12.5mm;afterwardsitwouldexperiencesomenegativemagneticforceuntilweapplythesecondpulsefordecelerationatthepeak,z=18.5mm.Ifoursecondbrakepulsedidn'ttotallystopthearmature,thentheresidualvelocitywouldbeabletobringthearmaturefurtherupuntilitpassthesecondnetzeromagneticforceposition,z=26mm.Afterthispoint,thearmaturewouldexperiencepositivemagneticforceagainandsomesmallDClevelpulsewouldbeenoughtocanceloutthedownwardgravityforceandthearmaturewouldbeabletooattherestillforhoweverlongweneed. Theroomtemperaturecalibrationcurvefortheconversionofthevoltageoutputfromchannel1ofthelock-inamplierintotheposition,z,ofthearmature,preciselymeasuredbythedepthmicrometergauge,isdemonstratedinFig. 2-19 (b).Theprobewassettobeverticaltobeasclosetothesituationsoftheactualexperimentsaspossibleandthevariablecapacitorwastunedtoabout3.8pFwhencalibrated.Thedipfortherst

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ln(b=a);(2{40) whereaninnercylindricalconductoroflengthLandradiusaissurroundedbyanoutercylindricalconductorofthesamelengthandradiusb,and"0isthepermittivityoffreespace.Hence,thecapacitivepositionsensorisexpectedtohavethecapacitanceproportionaltotheoverlappinglengthbetweenthesecondniobiumcanandthecoppersensor.Duringz=0and2mm,theoverlappinglengthisaboutthesame,15mm,sothecapacitanceissupposedtohavenochange.However,duetotheedgeorboundaryeect,weseethedipoccursatz=03mminthecalibrationcurveandinthedatawetookforthearmaturemotion.Inthedataanalysistheconversionofsignalsintopositionswillbemodiedaccordingly. Inordertomonitortheactualpulsessenttothesuperconductingsolenoid,wemeasurethevoltagedropacrossthe0.1resistorinserieswiththesolenoid.The0.1resistorboxisconnectedtotheoutputofthecurrentamplier(BOP)directlyviabananaconnectorsandtheothersideisconnectedtothesuperconductingsolenoidviathe12pinconnectoronthetopoftheprobedirectly.ThecircuitdiagramisshowninFig. 2-18 (b).Duetothealmostpurelyinductive(zeroresistance)motorcircuit,thepulsesfedtothesolenoidaresomewhatdistortedandasthegainofthecurrentamplierturnsuptoexceed3Aofcurrentoutput,thespikeswiththeoppositepolarityappear,resultingfromthebackemfduetothefastchangingcurrent.Therefore,weputcrossdiodesacrossthesolenoidinparalleltopreventthisoccurred. 52

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2-20 themotionofthearmatureofthesuperconductingmotorisclearlyexamined.Thepulseprolesenttothesolenoidisdistortedduetotheinductivebehaviorofthesuperconductorsolenoid.Asthegainofthecurrentamplieristurnedup,thehigherthegainthemorethecurrentwouldbefedtothesolenoid.Heretherstcurrentpulsepeakheightreachesaround-1.8A,thendecaysalittlebit,followedbythesecondpeakasmuchas-2.369Athendecayingtozerogradually.Thecurrentampliernotonlyampliesthepulses,butalsoinvertsthepulsesoutput;therefore,wehavethepulseprolewithnegativepolarity.Thecapacitivepositionsensorrespondstothemotionofthearmatureasthearmaturemovesupuntiloutofthereachofthesensor,thendropsbackdownbygravityforceasafunctionoftime.Thearmatureacceleratesto0.6m/satz=5mmwithinabout15ms(averageestimateacceleration40m=s2),thendeceleratestozeroatz=15mmin40ms(averageestimatedeceleration-15m=s2).Afterwards,duetothegravityforceitdropsbackdownandreachesthevelocityasfastas0.4m/satz=4mmwithin70ms(averageestimatedeceleration-5.7m=s2).Thearmatureprobablymovesfurtherupduringthetime60msand100ms,butoursensorcouldnotmeasureanychangesbecausethearmatureatthatmomentisalreadyoutofthedetectablerange.Fig. 2-20 (d)istheoriginalcurrentprolesendingtothecurrentamplierfromtheanalogoutputDAC0ofourLabViewdataacquisitionprogram. IfwecomparewiththetheoreticalpredictionfromoursimulatorwiththesamedistortedcurrentproleasinFig. 2-20 (a),thenthepredictedarmaturemotionislikeinFig. 2-21 .Thearmatureissupposedtoreachthevelocityasfastasmorethan1.0m/s,butitisalsopredictedtomoveasfarasapproximately10.5mmonly.Thedierencebetweenthetheoreticalpredictionandtheactualmotionisstillunderexploration. NowifwegetridofthesecondbrakepulseandincreasetheDClevelfollowingtherstaccelerationpulse,howwoulditaectthearmaturemotion? 53

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2-22 (d),thenthecurrentpulseproleinthesolenoidactuallylookslikeasmoothtransitionfromzerotosomesaturatedDClevelandthecapacitivepositionsensorrespondscorrespondinglybyoscillatingaroundtheequilibriumpositionabout0.9Vanddampingwithtime,Fig. 2-22 (a).FromFig. 2-22 (a)and(b),thearmatureseemstobeacceleratedtoalmost1.0m/swithin10msand4.5mm,thenitslowsdownuntilstopsatthehighestpoint,about6.8mm,anddropsbackdowntoasfastasabout-0.47m/s,asfarasthepositionabout4.3mm.Afterwards,itoscillatesbackandfortharoundtheequilibriumpositionabout5.2mm,likeaharmonicoscillator.Duetotheviscosityandimpedanceoftheliquidheliumandthefrictiononthebearingscontactingwiththearmature,theoscillationmotionalsodampsin80ms,thenthearmaturestaysstationary.AssoonastheDClevelisturnedo,thearmaturedropsbackdowntotheoriginalposition. IfwecomparewiththemagneticforceversuspositiongraphinFig. 2-19 (a),thiskindofoscillationmotionisquiteunderstandable.Weapplytheaccelerationpulseatthemaximumpeakofthemagneticforcezone,z6.5mm.Assoonthearmaturetravelspassingthroughthezeromagneticforceposition,z=12.5mm,itexperiencesnegativemagneticforce,deceleratingandpushingitbackdown.Andthenitexperiencespositivemagneticforceagainbelow12.5mmposition,whichwillpushitup.Suchacyclerepeatsuntilitdamps,thenitwilloatstillatthepositionwherethegravityforceandthemagneticforceallcancelout,resultinginzeronetforces. Ifweapplytheaccelerationpulsefollowedby2.0VDCpulsefor112.5ms,Fig. 2-22 (d),thenthecurrentpulseproleinthesolenoidactuallylookslikearapidtransitionfromzerotothesamesaturatedDClevelandthecapacitivepositionsensorrespondscorrespondinglybyoscillatingaroundtheequilibriumpositionabout0.85Vanddampingquicklywithtime,Fig. 2-22 (a).FromFig. 2-22 (a)and(b),thearmatureseemstobeacceleratedtoalmost0.9m/swithin10msandabout1.3mm,thenitkeepsalmost 54

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WithhigherDClevel,thearmaturecanbeacceleratedtohighervelocityfollowedbymorerapidlydampedoscillationafterwardsbecausetheactualcurrentpulseinthesolenoidisrampedfasteruptothesaturatedcurrentofBOP.Sincewereallyneedtohavethearmaturetravelwithconstantspeedforatleast10mm,thisresultwithourcurrentdesigncannotsatisfyus.Sothearmaturemustbemodied.Withthedistancebetweenthecentersofthetwoniobiumcansabout35mm,wecouldjustsimplyapplyonesinewavepulsefollowedbyaDCpulse,thenweareabletoacceleratethearmatureandholditaround10mmpositionforaslongasweneed.ThedrawbackofthisdesignistheunavoidableoscillationwhichmightmoreorlessaecttheturbulencejustproducedandtheDCpulsetoholdthearmaturemustbeatleastafewamperes. Nowifwereversethepolarityoftheinputpulsegoingtothecurrentamplier,howwoulditaectthearmaturemotion?Weapplytheoriginalcurrentpulsetothesuperconductingsolenoid,Fig. 2-24 (d),thenthemeasuredpulseproleispositiveandtheinductiveeectsarenotsosignicant,Fig. 2-24 (a).Theactualpulsepeakssaturateupto7.5A,morethanwhatwehaveexpected.Thearmatureisacceleratedtoabout0.7m/swithinabout10msand1.5mmandupto0.85m/sat4.2mmpositionandatthetime15ms.Thearmaturegoesupashighas8.7mm,andthendropsbackdownwiththevelocityashighas0.2m/s.Thearmatureseemstogetstumbledforalittlebitwhileatabout4.8mmposition,whereisestimatedtobethezeronetmagneticforcesposition.WegureoutthattheDClevelpulseashighas2Acouldhavedeceleratedthearmature 55

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WelengthentheDClevelto25mswhilekeepingtherestoftheoriginalcurrentprolethesame,Fig. 2-25 (d),thenthemeasuredpulseprolefedtothesuperconductingsolenoidispositiveandtheinductiveeectsarenotsosignicantaswell,Fig. 2-24 (a).Theactualpulsepeakssaturateupto7.5and10.5A,whilewedokeepthegainofthecurrentamplierthesameasinthepreviouscase.Thearmatureisacceleratedtoabout0.72m/swithinabout10msand1.2mmandupto0.9m/sat4mmpositionandatthetime12.5ms.Itseemstogetstumbledandslowdownbetween4and6mmposition,butspeedsupto0.95m/safterwards.Thearmaturegoesupashighas15.2mm,beyondwhichthearmatureisactuallyoutofthedetectablerangeofthecapacitivepositionsensor.Itshowszerovelocityduringthetime30msand110ms,whenthearmatureiseitheroutofthereachofthesensororitisoatingabovethesecondzeronetmagneticposition,z=26mminFig. 2-19 (a),or19.5mmequivalentlyinFig. 2-25 (c).Itthendropsbackdownwiththeprettyconstantvelocity,0.15m/s,allthewaydown.TheDClevelpulseashighas2Aislongenoughnottodeceleratethearmatureintime,soitcanmoveupbeyondthedetectableposition15.2mmorhigherbeforethebrakepulseapplies.Thisprovidesusoneoftheimportantsuccessfulexperiencesofhowwecouldachieveourgoal.Thearmatureisproposedtobeaccelerated,travelforabout10mmwithalmostconstantvelocity,bedeceleratedthenpassthesecondpointofthezeronetmagneticforcepositionandthenbeheldaboveitforhoweverlongweneed.Thedrawbackofthisdesignisthatwedon'tknowifwearegoingtomakethearmaturehitthetopofthecell.TheDCpulselevelshouldbecarefullyadjustedtobejustasmuchaswhatweneedandthewholetrajectoryofthemotionshouldbemonitored.Inordertobeabletomonitorthewholetrajectoryofthemotionofthearmature,thecapacitivepositionsensorneedstobelengthenedtoappropriatelengthwithoutmodifyingourcurrentarmature.Alsoweshouldlowerdownthesensorforafewmm(23mm)toavoidthedipproblem. 56

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2-26 (a)[ 41 ]).Thiscapacitor,coupledtoabridgecircuit,measuresthearmatureposition.Thetotalmassofarmatureisnow2.60g.Thegrid,madeof0.125mmthickspringsteelwith1mmsquareholesand70%transparency,isattachedtothelowerendofthearmature(seeFig. 2-26 (b),(c)andFig. 2-27 (a)).ThecorrespondingReynoldsnumberforthedierentvelocitiesofthegridmotionisshownFig. 2-27 (b).Let'sndouttheReynoldsnumberofourmotorsystemintheliquidhelium-4bathat4.2K.Reynoldsnumberisdenedastheratioofinertialforceandviscosityforce,orthevelocityscalemultipliedbythelengthscale,dividedbythekineticviscosity: =inertialforce viscostyforce:(2{41) Atmotortestingtemperature,4.2Kinliquidhelium,gridReynoldsnumber: (inm/s).Ifweconsiderthesizeofthemeshofthegridasthelengthscale,velocityofthegridmotionasthevelocityscale,thenwehaveReynoldsnumberrangingfrom4000to40,000whenthegridvelocityvariesfrom0.1to1m/s,abovetheonsetofturbulence. Theroomtemperaturecalibrationcurveshowsthemonotonicincreaseinvoltageasmovingupthearmaturewithoutanydip.Theentireassemblyismountedinaheliumtestcellandcooledinatransportdewarcryostat[ 42 ].Thearmaturepositionsensorwas 57

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2-28 [ 41 ]showsthecalibrationcurveatliquidheliumtemperature,4.2K. Intheelectronics,wehavetwoindependentcircuitryforthesuperconductingsolenoidandforthecapacitivepositionsensor(Fig. 2-29 ).Wedrivewiththeproperlyshapedcurrentpulseproledeterminedbysimulation,thenamplifythecurrentviathebipolarpowersupply.Theinputpulsetothesolenoidismonitoredbymeasuringthevoltagedropacrossthe0.1ohmresistorshunt.Twocrossdiodesaretoprotectthesolenoidfromthespikesduetofastchangingcurrent.Forthepositionsensor,weusetheACbridgecircuit.Twoarmsare100ohmresistors,andtheothertwoarevariablecapacitorandthepositionsensor.Whilethearmaturemoves,thecapacitancewouldchange,givingdierentvoltageresponse.Weconvertthevoltageintothepositionofthearmaturewithourcalibrationcurveperformedat4.2K. TheexperimentalresultsareshowninFig. 2-30 [ 41 ].Weapplythepulseprole:twosinusoidalshapepulsesfollowedby-0.1VDClevelfor250ms(Fig. 2-30 (e)).Thesolenoidisprotectedbytwocrossedsilicondiodeswhichhavecut-involtagesof4Vand18Vat4.2K.Therefore,reversingthepolarityofthesolenoidcurrentproducesanasymmetryinresponse(Fig. 2-30 (a)andFig. 2-30 (b)).InFig. 2-30 (c)and(d),withthecurrentpulselikeFig. 2-30 (a)thearmatureisseentoaccelerateto0.8m/swithin40msoveradistanceof9mm(estimatedaverageacceleration20m=s2).Thenitundergoesanearconstantdecelerationtozerovelocitywithin33msinanother12mm(estimatedaveragedeceleration24m=s2).Itwasheldatthepositionof22mmforabout250msuntilthecurrentwasturnedo;itthendroppedundergravity,asfastas0.48m/sin75msover10mmposition(estimatedaverageacceleration6.4m=s2).AsseeninFig. 2-30 (c)and(d),withthecurrentpulseinFig. 2-30 (b)thearmatureacceleratesto1.09m/swithin20msover9mm(estimatedaverageacceleration55m=s2).Thenitslowsdowntozerovelocitywithin38msovertheremaining12mmofstroke(estimatedaveragedeceleration29m=s2).Itwasheldat22mmforabout250msuntilthecurrentwasturnedo,then 58

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59

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Gridturbulenceinaclassicaluid[ 27 ]. Figure2-2: Superconductingmotormodelwithgrid,thermistorsandsuperconductingshield[ 29 ]. 60

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Armaturemotioninunshielded(pinkcurves)andsuperconductingshielded(blackcurves)solenoidunderthesinefunctionacceleration:(a)I(t)curve;(b)v(z)curve.(:modieddatawithoutthecentralpeakinthecurrentprole)[ 29 ] Figure2-4: Armaturemotioninsidethesuperconductingshieldedsolenoidunderthelinearfunctionacceleration:(a)I(t)curve;(b)v(z)curve. 61

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Armaturemotioninsidethesuperconductingshieldedsolenoidunderthesquarefunctionacceleration:(a)I(t)curve;(b)v(z)curve. Figure2-6: UppercriticaleldversustemperatureforniobiumsampleswithdierentRRR.[ 35 ] 62

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Voltageinputtothesuperconductingshieldedsolenoidwith:(a)sine;(b)linear;(c)squarefunctionacceleration. 63

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(a)SinusoidalcurrentinputtothesuperconductingshieldedsolenoidwiththedimensionsasstatedinTable1.;(b)thecorrespondingvelocityversuspositionoftheshaftmotiongraphs;(c)thecorrespondingmagneticeld,andmagneticforceversuspositionoftheshaftmotiongraphs.[ 29 ] 64

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Requireddrivingvoltage(inV)inputtothesolenoidversustime(inms)ifthesolenoidandtheresistoraretheonlyelementsinthecircuit. 65

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Schematicdrawing(a)andthephoto(b)forthespeciallydesignedswitchbox.(c)Detailedcircuitdrawingfortheswitchbox. 66

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Machinedrawingsandphotosofthemotorsystem:(a)machinedrawing[ 29 ];(b)photoofthetestcell;(c)photoofthetestcellinstalledattheendofthesuck-stickprobe. 67

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Experimentalapparatusforunshieldedmotortestingexperimentsusingthecapacitancebridgetomonitorthemotionofthearmature. 68

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TheGR1616CapacitanceBridge:(a)Theapparatusappearanceandrelatedcircuits.(b)Theschematiccircuits. Figure2-14: Circuitryoftheexperimentalapparatusforunshieldedmotortestingexperimentsusingthe555oscillatorcircuitandLabViewcounterprogramtomonitorthemotionofthearmature. 69

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TestingcircuitofQ-meter. Figure2-16: SimpleACbridgecircuit. 70

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MeasuringthecapacitanceofthecapacitivepositionsensorwiththeGR1616capacitancebridgeandthelock-inamplier. 71

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Circuitryandsetupfortestingthesuperconductingmotoratliquidheliumtemperature,4.2K.(a)Monitoringthemotionofthearmaturewiththecapacitancepositionsensor;(b)monitoringthepulsesenttothesuperconductingsolenoid. 72

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(a)Theoreticalcalculationofthemagneticforceperunitofcurrentsquareexperiencedbythearmatureinsidethesuperconductingsolenoidasafunctionoftheposition,z.(b)Roomtemperaturecalibrationcurveofthevoltageoutputfromchannel1ofthelock-inamplierversustheposition,z. 73

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Motionofthearmatureofthesuperconductingmotor.(a)Pulseprolesenttothesolenoidandcapacitivepositionsensorresponseasafunctionoftime.(b)Velocityofthearmatureasafunctionoftime.(c)Velocityversuspositionofthearmature.(d)OriginalcurrentprolefromanalogoutputDAC0ofLabView.[ 29 ] 74

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(a)Distortedcurrentproleduetothealmostpurelyinductivebehaviorofthesolenoid(smallresistance1)inthesuperconductingshieldedsolenoidwiththedimensionsasstatedinTable2.;(b)correspondingvelocityversuspositionoftheshaftmotiongraphs;(c)correspondingmagneticeld,andmagneticforceversuspositionoftheshaftmotiongraphs;(d)correspondingmagneticeld,andmagneticforceversustimegraphs.[ 29 ] 75

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Motionofthearmatureofthesuperconductingmotor.(a)Pulseprolesenttothesolenoidandcapacitivepositionsensorresponseasafunctionoftime.(b)Velocityofthearmatureasafunctionoftime.(c)Velocityversuspositionofthearmature.(d)OriginalcurrentprolefromanalogoutputDAC0ofLabView. 76

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Motionofthearmatureofthesuperconductingmotor.(a)Pulseprolesenttothesolenoidandcapacitivepositionsensorresponseasafunctionoftime.(b)Velocityofthearmatureasafunctionoftime.(c)Velocityversuspositionofthearmature.(d)OriginalcurrentprolefromanalogoutputDAC0ofLabView.[ 29 ] 77

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Motionofthearmatureofthesuperconductingmotor.(a)Pulseprolesenttothesolenoidandcapacitivepositionsensorresponseasafunctionoftime.(b)Velocityofthearmatureasafunctionoftime.(c)Velocityversuspositionofthearmature.(d)OriginalcurrentprolefromanalogoutputDAC0ofLabView. 78

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Motionofthearmatureofthesuperconductingmotor.(a)Pulseprolesenttothesolenoidandcapacitivepositionsensorresponseasafunctionoftime.(b)Velocityofthearmatureasafunctionoftime.(c)Velocityversuspositionofthearmature.(d)OriginalcurrentprolefromanalogoutputDAC0ofLabView.[ 29 ] 79

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(a)Newmachinedrawingsofthemodiedmotorsystem[ 41 ].(b)Pictureofthearmaturewiththegridmountedattheend.(c)Pictureofthetestcellshowingthegridmounted. Figure2-27: (a)Machinedrawingforthegrid.(b)Reynoldsnumberversusgridvelocityat4.2K. 80

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(a)Thecalibrationofthearmaturepositionsensor(4.2K).Positionsensorbridgevoltageoutputfromlock-inamplierversusthepositionofthearmature[ 41 ].(b)Thecapacitanceofthecapacitivepositionsensorversusthepositionofthearmature,measuredbyusingthecapacitancebridge,GR1616. 81

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Electronicscircuitsforthesuperconductingmotorsystem. 82

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Motionofthearmatureofthesuperconductingmotor[ 41 ].(a)Currentthroughthesolenoidandcapacitivepositionsensorresponseasafunctionoftime.(b)Sameas(a)withpolarityreversed.(c)Velocityofthearmatureasafunctionoftime.(d)Velocityversuspositionofthearmature.(e)OriginalcurrentprolefromanalogoutputDAC0ofNationalInstrumentsboardsenttoKepcoBOPcurrentamplierdrivingsolenoidcurrent. 83

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32 ].Leadplatingisalsoapproachableandeconomictobedoneinthelaboratory.Wehavedecidedtoelectroplate,ratherthanchemicallydepositthelead,becausechemicalvapordeposition(CVD)requireshightemperaturesandisdiculttoperforminthelaboratory.Forreference,leadmetalhasveryhighboilingpoint,about2022K,andmeltingpoint,ashighas600K,usingphysicalvapordepositionmethodrequiringvaporizationofthemetalelementatveryhightemperatures,whichisalsonotapplicableforourlab.Anothersuperconductingmetal,niobium,withcriticaltemperature9.5K[ 32 ]hasmuchhighermeltingpoint,2740K,andevenhigherboilingpoint,about5017K.Therefore,consideringtheoptionswedecidetoelectroplatealeadmagneticshieldingenclosingthemotoratverylowtemperature. 84

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WegotanicerecipefortheelectrolytesolutionmakeupfromTechnicInc.: Sincethecoherencelengthofleadis0.083m[ 32 ],wewouldliketodeposit25moflead,whichismorethanenough.Forourpurposes,ourcellismadeofpurecopperfreeofoxygen,Fig. 3-1 andFig. 3-2 .Forthecellcapandcellbody,theinteriorsurfaceareasareabout68.3cm2and78.9cm2.Weneed1.469Aand1.695Aofcurrenttodeposit1mofleadperminuteonthem,respectively. 3.1.2.1Lead 85

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44 ]. 45 ]. 46 ]. 3.1.3.1Proceduresteps 86

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Thefollowingsaretheactualprocedurestepsofleadelectroplatingperformedinthelaboratory. 1.CleaningthecellwiththeElectronicshop'shelp.Theyusethecommercialcoppercleanerandultrasonicvibrationequipmentforcleaningthecell.Fig. 3-3 (b)andFig. 3-4 (b)showthecopperinteriorofthecellcapandcellbody,andthewell-builtsiliconewall,readyfortheleadplating. 2.Sealingtheholeswithsilicone[?]andbuildingaboutoneinchhighofsiliconewallleaningagainstthepaperwallaboutafewmmextendedoutwardalongtheedgesofthetoprimofthecellcaporcellbody(Fig. 3-3 (b)andFig. 3-4 (b)). 3.Makingtheleadelectrodeforleadplatingofthecellcapandcellbody(thedrawingoftheexactsizeshowninFig. 3-3 (a)andFig. 3-4 (a)).Theleadelectrodeiswelltrimmedandcarefullyadjustedtobeabout2mmclearanceawayfromeverycellwall.Withtheaidofamagnifyingmirror,anohmmeter,andaroundrubberpadattachedtothecenteroftheturntable,wecandoanalmostperfectjob,asshowninFig. 3-3 (c),(d)and(e)andFig. 3-4 (c),(d)and(e).Thecellcapissupportedbyarubberpistonunderneathtostandmorestably.Thecellbodyhasgoldplatedatthebottom.Whilerotatingtheturntable,wemakesuretheresistancebetweentheelectrodeanodeandcellcathodeisinnity. 4.Theelectrolyteisinjectedtothecellcaporcellbodyabouthalftooneinchhigherthantherimenclosedbythesiliconewall(seeFig. 3-5 (a)andFig. 3-6 (a)).Theresistancebetweenthetwoelectrodes(theleadelectrodeastheanodeandthecellasthecathode)nowisonlyafewohms.Thecellisrotatedsteadilyandslowlyontheturntableofabrokenmicrowaveoven.Therefore,theleadanodecanalsoagitatetheelectrolytetomaketheleadionsinthesolutiondistributeveryuniformlywhileleadplatingisperformed.Westarttosupplythecurrentofabout1.47A(Fig. 3-5 (b))forthecellcapandabout1.70A(Fig. 3-6 (b))forthecellbody.Theleadelectrode(anode)connectstothepositiveterminalofpowersupply,whilethecell(cathode)isincontactwiththethinstainlesssteelpiececonnectingtothenegativeterminalofthepowersupply.Nowhitebubblesoccurwithhighconcentrationoflead,30grams/literofleadintheelectrolyte.Theexteriorpartofthecellwallincontactwiththethinstainlesssteelpiece,reducestoshinypurecopperfromtheoxidizedcopper(Fig. 3-5 (c)).Undertheliquidsurface,itiscleartoseethattheleadelectrodeiswell 87

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3-5 (d)andFig. 3-6 (d)).Thedissolvedleadprecipitatesatthebottomofthecell. 5.Thepowersupplyisturnedontosupplythecurrentforapproximately25minutes.Thecurrentuctuatesslightly,butoverallverystable.Thevoltageoutputfromthepowersupplyis8.2Vwhenleadplatingthecellbody.Note:Donottouchtheelectrodesduringtheelectroplating;otherwise,youwillgetburnbecausetheyareveryveryhotduetohighcurrentowingthroughthem.Thewholeleadplatingprocessisdoneintheventhood. 6.RinsewithampleDIwater(aboutonegallon).Drythecellwithcleantissueandushwithahighowofnitrogengasorcoolwindfromtheheatgunimmediatelyandquickly. 7.Theelectrochemicalequivalentforthereaction:Pb2++2e!Pbis3.86g=(Amperehour)[ 47 ].Byleadplatingonthecellbodywithcurrent1:70Afor27minutes,thedepositedleadmassis2:95g.Sincethedensityofleadis11:34g=cm3,thesurfaceareaofthecellbodyis78:85cm2,thethinknessofthedepositedleadwouldbe33.0m. 3-7 andFig. 3-8 showthemasterpiecesafterleadelectroplatingwithshinyleadofsilverwhitecolorintheinteriors,exceptthatcoveredbythesilicone.Acloseviewofthesilverwhiteinteriorofthecellcapshowstherims,thewallsandtheedgesofthewelldowntothecellbottomfullycoveredbythelead.Fig. 3-7 (c)andFig. 3-8 (c)showstheresidueleadelectrodeswhilethepartsimmersedundertheliquidsurfacebecomedarker,dissolvedtoalmostonlyhalfleft. 88

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3.Whiteanddelicatebubblesshowuponthesurfaceoftheelectrolytewithleadconcentrationonly15gperliterofelectrolytewhenturninguptotheoperatingcurrentduringleadelectroplating.Thewhitebubblesonthesurfaceofthesolutionoccurredathighcurrentdensityisne 4.\ModernElectroplating"bySchlesingerandPaunovic(2000)onpage366[ 47 ]:\Insolubleanodescannotbeusedinleadplatingelectrolytesasleaddioxide,PbO2,willformonthesurfaceoftheanodes.Thepurityofthesolubleleadanodesuseddeterminestheextenttowhichalmformsonthesurfaceoftheanodes."Onpage369[ 47 ]:\Duringthedepositionofthickleadcoatings(upto200m)formationofnodulesor\growths"canoccur.Thisfailuredoesnotgenerallyoccurwithafreshlymade-upsolution,andwhenitdoesoccur,itcaninmostcasesberectiedbyapuricationoftheelectrolytewithactivatedcarbon.contaminationoftheelectrolytebybreakdownproductsoftheorganicadditives.togetherwitharapiddecreaseoftheleadconcentrationintheelectrolyte.Theanodiccurrenteciencywasreduced,whichcausedthedropoftheleadcontentinthebath.Bythepassivationoftheanode,leaddioxidewasformedontheanodesurface,whichcausedapartialoxidationoftheorganicadditives." Therefore,weshoulduse100%pureleadastheanodetopreventtheleaddioxidefromformingonthesurfaceoftheelectrode,causinganodicoxidationoftheorganicadditivesontheanodesurface,ortheoxidationoftheorganicadditivesbytheleaddioxide,resultingintheroughnessofleaddeposition.Alsoiftheelectrolytesolution 89

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5.\ModernElectroplating"bySchlesingerandPaunovic(2000)onpage373[ 47 ]:\Discolorationoftheleaddeposittoabrownorblackcoloroccursduetodepositionofcopperontotheleadsurfacebyadisplacementreactionthatcanhappenintheelectrolyteifthecurrentisleftswitchedowhilethepartsarestillimmersed,orintherinsesiftheyareheavilycontaminatedwithcopper." Therefore,afternishingtheleadplating,weshoulddumpouttheelectrolyteimmediatelyafterthepowersupplyisswitchedo.AndrinsethepartswithplentyofDIwaterimmediately. 3-9 showsthemachinedrawingwiththedimensionsandthepicturesofourexperimentalcell.Wehaveonesuperconductingsolenoiddrivingthearmaturetomovethroughitscenter,withagridattachedattheend.Thislightinsulatingarmatureisconstructedof3phenolictubesseparatedbytwohollowcylindricalniobiumcansplaced26mmapart,withtheturbulence-producinggridattachedtooneend.Aconductingsectiononthearmature,composedofoneoftheNbcylindersandsilverpaintcoatingpartofthephenolicrod,isinsideacloselyttingcapacitormadeoftwosemi-cylindricalcoppersheets.Thiscapacitor,coupledtoabridgecircuit,measuresthearmatureposition.ThedimensionparametersofoursuperconductorshieldedsuperconductinglinearmotorsystemarelistedinTable 3-1 .Theelectronicscircuitsarethesameasdescribedinchapter2,e.g.Fig.2-29,excepttheleadresistance0.7.Theexperimentcellismountedtothe0.25inchdiameterprobeandtestedinatransportdewarcryostat[ 42 ]atliquidheliumtemperature. 3-10 (a)),thenthemagneticforcedistributionalongthez-axisofthesolenoidarecalculated,asinFig. 3-10 (b).Tobemoreecient,wewouldapplytheaccelerationpulsewhenthecenterofmassoftheniobiumcan#1islocatedatz=6.5mmposition,anddecelerationpulseatz 90

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Parametersofsuperconductorshieldedsuperconductinglinearmotorsystem. ParameterValue =18.5mmposition.Afterthearmaturepassesthez=26mmposition,wewouldapplytheholdingpulse,justenoughtobalanceoutthegravity. Inthecurrentprolefromoursimulationprogram(Fig. 3-10 (c)),therstpulseproducessinefunctionacceleration,andthethirdpulsedealswithlinearfunctiondeceleration.Thecentralpeakisduetothealmostbalancedmagneticforcesonthetwoniobiumcansatthatposition,whereeachisalmostequidistantfromtheendsofthesolenoid.Wecanremovethecentralpeakandtheinertiawillservetocarrythroughit.Ittakesslightlymorecurrent,0.147Amore,forthesuperconductingshieldedmotorsystemthantheunshieldedone.Intheresultinggridmotion(Fig. 3-10 (d)),thegridwouldhavesinefunctionaccelerationfrom0to1m/sin1mm.Thenittravelsatalmostconstantspeed,1m/s,for10mm.Inthemid-way,thegridstartstoslowdownat12.5mmpositionaftertheniobiumcan#2becomesclosertothesolenoid,meaningstrongermagneticforceintheoppositedirection.Thenthegridwouldrapidlydeceleratetoceasewithin1mmwhenthethirdpulseisapplied.Theemptycirclesrepresentthegridmotionwithoutthecentralpeakinthecurrentprole.Withoutthecentralpeakinthecurrentprole,thegridmotionisnotsignicantlydierent. 91

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3-11 showstheexperimentalresults.Consideringtheinductivebehaviorofthesolenoid,aninductor,thetimeconstantofanLRcircuitisL=R=16:34mH=0:723:3milliseconds.Evenifweapplyasquarepulse,say1V,ittakesvetimesthetimeconstant(116.5ms)tosaturateto1V.Therefore,weneedtotakethisintoaccountwhenweapplythepulsestothesuperconductingsolenoid.Weapplytheappropriatepulseprole:DC2.89Vfor14ms,followedbyDC-0.3Vfor10ms,then2.5msdelay,eventuallytheholdingpulse-0.02Vfor100ms(Fig. 3-11 (d)).Thesolenoidisagainprotectedbythesametwocrossedsilicondiodesasdescribedinchapter2.InFig. 3-11 (b)and(c),thearmatureisseentoaccelerateto0.7m/swithin6.5msoveradistanceof2.5mm(estimatedaverageacceleration98m=s2).Thenitundergoesanearconstantdecelerationtozerovelocitywithin66msinanother14mm(estimatedaveragedeceleration17.5m=s2).Itwasheldabovetheposition28mmforabout100msuntilthecurrentwasturnedo;itthendroppedundergravity,asfastas0.4m/sin60msover12mmposition(estimatedaverageacceleration6.7m=s2).Atthepositionbetween8and17mm,thevelocityofthearmaturewasabove0.6m/sandbetween0.6and0.7m/s. 1.At4.0K,dynamicviscosity=36micropoise=36107Pascalsecond= 36107Ns=m2.(Kinematicviscosity:==;1strokes=1cm2s1)[ 48 ]. 2.Atsaturatedvaporpressure,at4.20K,theliquidhelium-4hasdensity, 49 ]. 92

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50 ]. 4.Totallengthofarmature(Fig. 3-12 )=L=3:1875"=8:10cm=8:10102m.Thediameterofthearmature=D=(1=4)"=0:635cm=0:635102m. 5.Re`=UsL==(1m=s)125:4075kg=m38:10102m 2U2sCf=1 2125:4075kg=m3 Viscousdrag=dragsurfacearea=DragDL=0:2326N=m20:635 50 ]. 6.Forceexertedupontheuppersideplatesofthearmature(Fig. 3-12 (a))=Fext:d=0:3"=0:762cm=7:62103m;totalcrosssections=A0=(d 50 ]. 7.Forceexerteduponthebottomsideplatesofthearmature(Fig. 3-12 (b))=F0ext:Bernoulli'sequation:p=1 2U2s=62:70N=m2[ 50 ];F0ext=pA0=4:607103N. 8.Buoyancyforce=gV=125:4075kg=m39:8m=s22:565106m3= 3:1524103N. 9.Gravityforce=2:60g=0:02548N. 10.Ifthegridismountedattheendofthearmature,thetotalnon-transparentsurfaceareaofthegrid,excludingtheareaunderneaththearmature,isA"=1:7765cm2.Forceexertedupontheuppersideofthegrid=A"v2=22:28103N;forceexerteduponthebottomsideofthegrid=pA"=11:14103N.Sothetotalforcesexerteduponthegridis33:42103N. Sothetotalnetforce(downward)whenthearmaturetravelsupwardatthevelocityof1m/s=0.06994N=2.745Gravityforce(Table 3-2 ).Inoursimulationprogram,weonlyconsiderthegravityforceandthemagneticforce.Asthearmaturemovesathighervelocity,theviscousdragforceandimpedanceforcesbecomesignicant,prettycomparabletothegravityforce. 93

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Forcesonthearmaturewhenmovingupat1m/s(Downloadispositive). ParameterValue Viscousdrag3:758104NFext9:214103NF0ext4:607103NBuoyancyforce3:1524103NGravityforce0:02548NExternalforcesonthegrid33:42103N 43 ]areconcerned,weestimatetheresultingheatingduetothevaryingmagneticeld,whichisnotsignicant.Weuseepoxy-resinimpregnationtopreventfrictionaldisplacementwithoutnoticinganydeformationofthecoil.Wedonothaveanysplicelossproblemwithoutusinganyslices,weassumethattheheatingduetotheabovethreefactorscouldbeneglected.Whenwerunthemotorinthedilutionrefrigerator,wewillrunitwithoutanyliquidheliumandmeasurethebackgroundtemperatureuctuation.PossibleAClossesofthesuperconductingsolenoidduetothevaryingcurrentcanbemeasuredbysubstractingfromtheblankbackgroundwithoutanycurrentinthesolenoid.Andthenwewillrunthemotoragainwithgridimmersedintheliquidheliumbath.Wecanthendothesubtractiontogetthenettemperaturechangesimplyduetothequantumturbulenceenergydecay. 3-13 showsthedesignofourexperimentalcellforthetowed-gridstudiesofquantumturbulenceexperiments. 94

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Fig. 3-13 showsthewholeexperimentalcellassemblyforthetowed-gridexperimentsandthecircuitboardforthethermistorswiththeelectricalfeedthrupinsmounted.Theelectricalfeedthrupinsaregoldplatedonallsurfacestobenetthebestelectricalconductivity Beforeputtingeverythingtogetherandapplyingtheepoxyforseal,thedegreaseandcleaningprocessisveryimportant: 95

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2.Cleanthedirtordustwiththesoftironbrush,roughspongeandsandpaperontheexteriorsurfaceofeachfeedthruholeonthecellwall. 3.SpraytheacetonetowettheQ-tipandthenapplytheQ-tiptocleantheinteriorofeachfeedthruholeonthecellwall. 4.SpraytheacetonetowettheKimwipeswipersandthenapplythewiperstocleantheexteriorsurfaceofeachfeedthruhole. Applyingtheepoxytothejointsurfacesbetweenthecoppercellwallandthefeedthruplugneedsenoughsurfaceedgesfortheepoxytoadherewell,sowemakethepolycarbonateplugshigherthanthecellwallsurface,stickingfromoutsideandthenon-connectionpartoftheeveryelectricalfeedthrupinislongerthaneveryplug(Fig. 3-13 (a)).Itusuallytakesadaytodryouttheepoxy;however,wecanilluminatewiththeinfraredlighttoshortenthetimetoaboutahalfday.Shinningthislightalsohelpsstrengthentheepoxy.Weshouldbeverycarefulwhencoolingdownandwarmingupthecellveryslowly,avoidchemicaltouchingoralsoputtinglargeforcesonthejoints.Inthisway,thesealcanbeusedforthermalcyclesformanytimeswithoutanycrack. 51 ].Theyarelessthan300mdiameterandwillbeimmersedintheturbulentheliumallowfastcalorimetricmeasurementstobemade.Therefore,wehavethethermistorcircuitboardspeciallymade(Fig. 3-13 (b)).Thecircuitboardhasradius0.5incharcononesidetomatchtheinteriorarcofthecell;ontheotherside,anextensionpartismadefora0-80slottedatscrewtomounttheboardtothebottomofthecellandalsofortheeaseofourngershandling.Twominiaturegermaniumlmthermometersaremountedtothecircuitboardbyindiumsolderandattached/gluedtotheboardbythecryogenicgrease.Threeelectricalfeedthrupinsaremountedinparalleltothecircuitboardbyindiumsolderandthetwothermistorsare 96

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3-13 (b):electricalfeedthrupin#1)andtoslideinto(e.g.Fig. 3-13 (b):electricalfeedthrupin#1')anotherthreecounterelectricalfeedthrupinsimbeddedinthecellwall(e.g.Fig. 3-13 (b):electricalfeedthrupin#2).Atroomtemperature,theresistancesforthetwothermometersmeasuredfromtheelectricalfeedthrupinsoutsidethecellare60.0and57.5,respectively. Wehaveleakcheckedthecellatroomtemperatureandatliquidnitrogentemperature,77K,withourhomemade\Redeye"instrument.Thisprovesthatourelectricalfeedthrudesignisasuccessfuldesign. Outsidethecell,wewillplugtheelectricalfeedthrupin#2withtheelectricalfeedthrupin#3connectedtoaminiaturecoaxialcable 97

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Machinerydrawingsforthecellcap. 98

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Machinerydrawingsforthecellbody. 99

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(a)Thedrawingoftheleadelectrode.(b)Thecellcapsupportedbyarubberpistonunderneathbeforeleadplating.(c),(d)and(e)Adjustingtheleadelectrodepositiontobe2mmclearanceawayfromeverywallwiththeaidofamagnifyingmirrorandanohmmeterwhenrotatingtheturntable.(Note:Thepicturesof(c),(d)and(e)areforthesecondattempt.) 100

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(a)Thedrawingoftheleadelectrode.(b)Thecellbodybeforeleadplating.(c),(d)and(e)Adjustingtheleadelectrodepositiontobe2mmclearanceawayfromeverywallwiththeaidofamagnifyingmirrorandanohmmeterwhenrotatingtheturntable.(Note:Thepicturesof(c),(d)and(e)areforthesecondattempt.) 101

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Procedurestepsfortheleadplatingforthecellcap.(a)Fillingupthecellcapwithelectrolyteabouthalftooneinchhigherthantherim.(b)1.47Aofcurrentissuppliedduringplating.(c)Closeviewofthecellwhenleadplating.(d)Theleadelectrodeis2mmawayfromeverywallofthecell. 102

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Procedurestepsfortheleadplatingforthecellbody.(a)Fillingupthecellbodywithelectrolyteabouthalftooneinchhigherthantherim.(b)1.70Aofcurrentissuppliedduringplating.(c)Closeviewofthecellwhenleadplating.(d)Theleadelectrodeis2mmawayfromeverywallofthecell. 103

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(a)Themasterpieceshowingshinysilverwhitecolorofleaddepositontheinteriorwalls.(b)Closelookatthecellcap.(c)Electrodeafterleadplating.(d)Cellcapgettingridofthesiliconewall. 104

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(a)Themasterpieceshowingshinysilverwhitecolorofleaddepositontheinteriorwalls.(b)Cellbodygettingridofthesiliconewall.(c)Electrodeafterleadplating.(d)Cellcapandcellbodyhasleaddepositontheinteriorwalls. 105

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(a)Machinedrawingoftheexperimentalcell.(b)Pictureofthesuperconductingmotorsystem,composingthesuperconductingsolenoid,thecapacitivepositionsensor,thegridandthearmature.(c)Pictureoftheexperimentalcellmountedonthe0.25inchdiameterprobefortestingattheliquidheliumtemperature. 106

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(a)Schematicdrawingofthesuperconductingmotorsystem.(a)Calculatedmagneticforceperunitofthecurrentsquareexperiencedbythearmatureversusthepositionoftheniobiumcan#1.(c)Simulatedcurrentprole.(d)Simulatedgridmotion. 107

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Motionofthearmatureoftheshieldedsuperconductingmotor.(a)Currentthroughthesolenoidandcapacitivepositionsensorresponseasafunctionoftime.(b)Velocityofthearmatureasafunctionoftime.(c)Velocityversuspositionofthearmature.(d)OriginalcurrentprolefromanalogoutputDAC0. 108

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(a)Externalforceonthetopsurfacesofthearmaturewhenitmovesupwithvelocity1m/s.(b)Externalforceonthebottomsurfacesofthearmaturewhenitmovesupwithvelocity1m/s. 109

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Themachinedrawingfortheexperimentalcelldesign.(a)Experimentalcellassembly.(b)Circuitboardwiththethreeelectricalfeedthrupinsandtwothermistorssolderedwithindiumandattachedbythecryogenicgrease. 110

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4-1 (a)).Theadvantagesofthecircuitisthatithasenoughresolutionandsensitivitytomeasureslightresistancechange,i.e.,temperaturechange,ofthethermistors,andinthemeantime,withoutputtingtoomuchheatonthethermistors.Our300mindiameterGermaniumlmthermistorswithgoldleads50mindiameterareverydelicate;therefore,wemustbeverycarefulnottohavetoomuchheatdissipationonthem.TheorderofmagnitudeofthepowerdissipationinnWwouldbeappropriate,eitherfornotburningoutthethermistors,orfornotaectingthetimedependenttemperaturerisemeasurementfromtheproducedturbulence.Hence,wecarefullycalculatetheoptimalcongurationsofthesetuptoachieveourgoal. AsshowninFig. 4-1 (a),alock-inamplier(PrincetonAppliedResearch,PAR)istheheartofthethermistormeasurementcircuit.Itsuppliesthe70kHzbridgesignalatVref=0.1VthroughaR=100kOhmresistortomakethebridgedrive1A. InFig. 4-1 (a),Rsistheresistanceoftheresistorconnectedinseries.Rthistheresistanceofthedelicatethermistor.Xistheresistanceoftheadjustablevariableresistor.andRistheresistanceofthetworesistorsonthebottomtwoarms.Sothetotalresistanceforthewholecircuitwouldbe: 111

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Thepotentialaftertheresistor,Rs,becomes: Therefore,thecurrentI1owsthroughthethermistor,Rth,is: whilethecurrentI2owsthroughthevariableresistor,X,is: Theresultingpowerdissipationonthethermistorwouldbe: ThepotentialatpointAis: whilethepotentialatpointBis: Thepotentialdierencedetectedbythenulldetectoris: Rs(X+2R+Rth)+(Rth+R)(X+R) (4{9) 112

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[Rs(X+2R+Rth)+(R+Rth)(R+X)]2:(4{10) ForVref=0:1Vrms,Rs=100k,Rth=100,R=10k,theLabViewprogramforourACbridgecircuitanalysisasshowninFig. 4-1 (b)tellsthattheclosertheXtoRth,theclosertheVABtozeroinVABversusXgraphandjdVAB Table 4-1 listshowthefourparameters,Vref,Rs,RthandR,wouldaectthesensitivity,dVAB=dXjX=Rth,andthepowerdissipation,Pth,onthethermistors.Basically,dVAB=dXjX=Rth/Vref,Pth/V2ref.WehavechosentherstsetofparametersforourACbridgecircuitsettingssinceithasenoughsensitivity,andthepowerdissipationislimitedtotheorderofmagnitudeofnW.Rthrangesfrom100to200k,representingtheresistancechangeofthethermistorsfromroomtemperaturetolowtemperatures,aslowasafewdecadesofmilliKelvins.Thecalculationofthetemperaturesensitivityofthethermistorissimplythethermalresolutiongivenontheresistancebridge,whichwillbeexplainedingreaterdetailinthenextchapter. 4-2 isthecircuitryfortheresistancemeasurementofthethermistors.Thethermistor#1hasresistance133.8,whilethethermistor#2hasresistance130.8.Table 4-2 isthesensitivitytestingresultsatroomtemperature.ThenulldetectorPAR116(PrincetonAppliedResearchanaloglock-inamplier),isadjustedtohavethesensitivityrange50V)astheadjustabledecaderesistorturnsto133.6forthermistor#1.Sotheresolutionisasmuchas2.8mV=(Fig. 4-3 ). 113

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ParametersoftheACbridgeaectingthesensitivityandthepowerdissipation. 0:1100k10010k0:47130:022650:1100k100010k0:43080:22460:1100k10k10k0:22732:0660:1100k50k10k0:064107:3970:1100k100k10k0:0293310:410:1100k200k10k0:0116111:900:1100k1001k0:45210:024730:1100k10001k0:24750:24510:1100k10k1k0:043082:2460:1100k50k1k0:0078127:9360:1100k100k1k0:00328911:040:1100k200k1k0:00124112:440:1100k1001000:24980:024950:1100k10001000:045210:24730:1100k10k1000:0047132:2650:1100k50k1000:00079817:9940:1100k100k1000:000332911:100:1100k200k1000:000124912:490:1010010k9:8039:8030:10100010k8:26482:640:1010k10k2:500250:00:1050k10k0:2778138:90:10100k10k0:0826482:640:10200k10k0:0226845:350:101001k82:64826:450:1010001k25:002500:00:1010k1k0:8264826:40:1050k1k0:03845192:20:10100k1k0:00980398:030:10200k1k0:00247549:500:10100100250:025000:00:1010001008:2648264:50:1010k1000:09803980:30:1050k1000:003984199:20:10100k1000:000998099:800:10200k1000:000249849:95 114

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Table4-2: SensitivitytestoftheACbridgecircuitatroomtemperature. DecadeResistorChannel1Output(mV)Box()Thermistor#1 33.6283283.61452133.632183.61343233.62732283.64132333.65552383.66932433.68352 115

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TheACbridgecircuitsimulation:(a)Schematiccircuitdrawing.(b)LabViewprogramfortheACbridgecircuitanalysis. 116

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ACbridgecircuitforthermistorsresistancemeasurement.PrincetonAppliedResearch(PAR)lock-inamplierisusedtosupplytheACvoltageoutputanddetectthesignal. Figure4-3: Roomtemperatureresolutioncurveforthermistor#1. 117

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Thequantumturbulenceisproducedinliquidhelium4inthebottomoftheexperimentalcellbyagridtowedbythesuperconductinglinearmotorenclosedinthesuperconductingshieldedcell.Theturbulence,producedbythetowedgridinafewmilliseconds,isthenmonitoredwithasensitivethermistorwhichresponsesveryrapidlytomeasureanyheatdissipation.Thesourcesofheathavealreadybeendescribed.OuranalysisidentiestheheatassociatedwiththedecayoftheturbulenceandcomparestheresulttotheKelvinwavetheorythatwehavediscussed. 118

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5-1 (a).TemperatureisttedasafunctionoftheresistanceofathermistorinFig. 5-1 (b)usingasinglepowerlawindicativeofhoppingconductioninanamorphousmaterial Holdingthetemperatureconstant,wechangethedecaderesistorsettingandmeasurethebridgeoutputfromthelock-inamplierforthetwothermistors.TheplotsandquadratictstothesecalibrationsareshowninFig. 5-1 (c).Wefurthertthethermistor#1resistance,Rtherm,asafunctionoftheoutput(V)ofthelock-inamplierPAR124A: Thisallowsustoconvertthechangingbridgesignalintotemperatureastheturbulencedecays. 5-2 (a).Weuseanaccelerationsquarewavepulse(10ms)toacceleratethearmatureup,whilethedecelerationpulselastsonlyfor2.5ms,justenoughtostopthearmature,followedbya-0.1VDCholdingpulsewhichisexpectedtoholdthearmatureonthetopfor250ms.WecanseethethermistorresponseinFig. 5-2 (b).Thebridgeoutputrisesfromabout 119

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5-3 (a)showstherawdataresults.Weseethethermistorandthecalibratedthermometerwarmupafterthegridmotion. Theperformanceofourshieldedsuperconductinglinearmotorat520mKisnearlyasexpected.Weapplyonesquareaccelerationpulse,2.86Vfor10msandonesquaredecelerationpulse,2.86Vfor2.5ms,withsomeDClevel0.32Vfor10msinbetween,followedby-0.1Vholdingpulsefor250ms.Inthevelocityversustimegraph,Fig. 5-3 (b),weseethegridmovesasfastas1m/sandwaszeroforabout40mswhenbeingheldatthetop.Inthevelocityversuspositioncurve,Fig. 5-3 (c),weseethegridwasmovingatalmostconstantspeedabove0.90.1m/sforabout9mm. InFig. 5-3 (b)and(c),thearmatureisseentoaccelerateto1m/swithin20msoveradistanceof7mm(estimatedaverageacceleration50m=s2).Thenitundergoesanearconstantdecelerationfrom0.8m/stozerovelocitywithin32msinanother8.5mm(estimatedaveragedeceleration25m=s2).Itisheldupat28.5mmforabout40msuntilthecurrentwasturnedo;itthendropsundergravity,asfastas0.6m/sin38msat13.5mm(estimatedaverageacceleration15m=s2).Between12and20.5mm,thevelocityofthearmaturewasabove0.8m/sandnearlyconstantat0.9m/swithabout10%uncertainty.Ifweconvertthethermistorandthermometerresponseintothetemperaturesasafunctionoftime,wefoundthedramatictemperatureriseoftheliquidheliuminthecell,from520mKto1.5Kafterthegridmotion,whiletheexperimentalcellonlychangesforafewdecademilliKelvins.Sincetheexperimentalcoppercellhasheatcapacity3.9 120

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5-3 (d),theliquidheliumtemperaturerisesafterthearmaturemotionindicatesthedecayofturbulence.Afterthearmaturedropsbackdowntotheoriginalposition,wecanseethattheliquidheliumtemperaturerisesdramatically,fromabout700mKto1.5Kwithinthenext120ms.Thiscouldbecausedbythedecayofadditionalturbulencecreatedbythefallinggrid.Duringtherst400mswhenthethermistorhasalargetemperaturechange,weseeuctuationswhichmustbeinvestigated. Wethenconvertthetemperatureintotheenthalpyofliquidheliuminthecellasafunctionoftime.TheenthalpyoftheliquidheliumvaryingwithtimeisshowninFig. 5-3 (e),obtainedfromasplineinterpolationfromTable7.6inreference[20].Sinceturbulenceisanonlinearchaosdynamicssystem,thetemperaturechangeduetotheturbulencedissipationuctuatesirregularly.Fortheerraticpart,wesimplyaverageevery400adjacentpointsinourdataanalysistogettheredcurves. Wewilltthiscurveintwodierenttimedomainswiththettingequation: ,tocomparetotheKelvinwavetheoryinthenextsection. 5-4 .WettheredcurveseparatelyintwodierenttimedomainswiththefunctionH(t)=a+btc.InFig. 5-4 (a)and(b),theenthalpyiscalculatedfromthesplineinterpolationoftheTablevaluesinRef.[20]. InFig. 5-4 (a)theenthalpy(H)isttedasafunctionoftime(t)duringtherst230and275ms: 121

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InFig. 5-4 (b)theenthalpy(H)isttedasafunctionoftime(t)duringthelater295and360ms: ,meaningthatthedecayofturbulenceenergypermoleisproportionaltot3:608atlaterstage.Fortheenthalpyduringthesecondtimedomain,wegettheexponentcorderof-3.6,whichmustbestudiedmore. Fromtheexperimentsofthevisualizationofquantizedvorticesinliquidhelium4achievedbyLathrop'sgroupusingsmallsolidhydrogenparticlesandParticleImageVelocimetry(PIV)technique[ 52 ],thequantumturbulenceisproducedattensofmillikelvinbelowthetransitiontemperature(2.172K).Sinceourexperimentsareperformedatmuchlowertemperatures(0.5K)wheretheviscosityisevenlowerandtheReynoldsnumberisevenlarger,itisverypromisingthatthequantizedvorticesinsuperuidhelium4wouldbeproducedwithourtowed-gridmoreeasily. 122

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Thermistorscalibrationcurves:(a)Thermistor#1and2arecalibratedagainstthecalibratedthermometerRuO2848.(b)Temperatureisttedasafunctionoftheresistanceofthermistor#1.(c)Calibrationofthermistor#1and2attheliquidheliumtemperatureabout500mK:varyingtheresistanceofthedecaderesistorasafunctionofthefunctionoutputoftheanaloglock-inamplierPAR124A.(d)Fittedcurveofthermistor#1resistanceasafunctionofthefunctionoutputoftheanaloglock-inamplierPAR124A. 123

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Motionofarmatureandthermistorresponseatvacuumatabout600mK.(a)Pulseprolesenttothesolenoid.(b)Rawdatatakenforthemonitoredpulse,positionsensorandthermistorresponses.(c)Thevelocityofthearmatureasafunctionoftime.(d)Thevelocityofthearmatureasafunctionofthepositionoftheniobiumcan1. 124

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Quantumturbulenceat520mK:(a)Rawdata.(b)Velocityofarmatureasafunctionoftime.(c)VelocityofarmatureasafunctionofpositionofNbcan#1.(d)Zoomintoseethethermistorresponseattherst400ms.Liquidheliumtemperaturerisesafterthearmaturemotionasanindicationofproducedturbulencebeingdissipated.(e)Enthalpyoftheliquidheliumasafunctionoftime. 125

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Fittingtheenthalpyofliquidheliumcalculatedfromsplineinterpolationasafunctionoftime:(a)Fittingenthalpycurveduringtherst230and275ms.(b)Fittingenthalpycurveduringthelater295and360ms. 126

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InclassicalgridturbulenceandsuperuidgridturbulenceaboveoneKelvin,eddieseithergrowaslargeasthechannelordiminishassmallastheviscousdissipationlength(classicaluid)ortothescaleofvortexlinespacing(quantumuid).Classically,theenergywilleventuallydecayviaviscositythroughthesmallestofeddiesfollowingtheKolmogorovspectrum.Thisisnotthecase,however,insuperuidquantumturbulenceclosetozeroKelvinwherethereisnoviscosity. TheonlytheorypredictstheenergywilldecayviaaKelvinwavecascade,ultimatelyresultinginphononradiation.Kelvinwaves,likerotatinghelicalwaves,havetransverseandcircularlypolarizedwavemotionandaresupportedbyquantizedvortices.KelvinwavescanbeproducedbytheDonnelly-Glabersoninstabilityorvortexreconnections.Eddieswillgrowordiminishoncetheturbulenceisproduced.Theenergyoftheturbulenceisinjectedatalowfrequencyanddissipatesbyphononemissionatahighfrequency. Theenergydecayrateofgridturbulenceat520mKisexploredandcomparedwiththeKelvinwavecascadetheory.Ourexperimentalresultsatthisearlystagesupportthistheory. 127

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Turbulenceisaverycomplexsystemandthehomogeneousisotropicturbulenceisjustthesimplestcase.Quantumturbulenceusinglowtemperatureheliumasatestuidhasveryuniquefeaturesandthersttaskistoreproduceourexperimentalresultsattemperaturesdownto20mK. 128

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Thefreeenergyoftherstniobiumcylinderwiththecenterofmassatz=ziinteractingwiththeexternalmagneticeldproducedbythesolenoid: (A{1) Themagneticforceexperiencedbytherstniobiumcylinderwiththecenterofmassatz=zi: z(A{2) zf1 z(Zz=zi+l1=2z=zil1=2Zz=zi+l1=2+zz=zil1=2+z)Z20Z=ro1=0~B2z(;;z)dddz=1 z(Zz=zi+l1=2z=zil1=2Zz=zi+l1=2+zz=zil1=2+z)f(z)dz; where z(Zz=zil1=2+zz=zil1=2Zz=zi+l1=2+zz=zi+l1=2)f(z)dz(A{6) 129

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NNXi=1f(xi);(A{7) 2)(ba N)(A{8) zZz=zil1=2+zz=zil1=2f(z)dz=1 zz NNXi=1f(xi)=1 where 2)(z N)(A{10) 2)(z N);xj=zi+l1 2)(z N0)(A{12) Considertwoniobiumcylinderswithsomedistanceapart,dS: where 2)(z N);(A{14) 2)(z N0);(A{15) 2)(z N00);(A{16) 2)(z N000):(A{17) 130

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53 ]

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Fig. C-1 (a)isoursimulationprograminLabViewtocalculatetherequiredcurrent,voltage,magneticeldandmagneticforcebychangingdierentparameters.Justnoticeherethatthecurrentneedstodrivethemotorarecalculatedforagivencoilandarmaturedesign. Fig. C-1 (b)isanothersimulationprograminLabViewtondoutthevelocityversusthepositionofthearmaturebyinputtingthecalculatedcurrentprole. Inordertondouttheoptimalparametersforourdesign,wehavewrittenanotherLabViewprogram,Fig. C-2 (a).Bychangingoneparameteratonce,youcanndoutwhichdesignneedstheleastcurrent,voltage,ormagneticeld. WehaveimprovedtheLabViewprogram,Fig. C-2 (b),tolookfortheoptimalparameters.Thisprogramcanchangeasmanyparametersatonceaspossible. WehavewrittenthedataacquisitionprograminLabView,Fig. C-3 (a),tooutputspecialpulsestothesuperconductingsolenoid,acquirethedataforthearmatureposition,andtheactualpulsesendingtothesolenoid. WehavehomemadedataanalysisLabViewprogram,Fig. C-3 (b),toconverttherawdatafromthepositionsensorandthelowtemperaturecalibrationcurveintothevelocityversustimeandvelocityversuspositiondiagrams. 140

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LabViewprogramsforcalculating:(a)thecurrent,magneticeldandforce.(b)armaturemotion. 141

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LabViewprogramstolookforoptimalparameters. 142

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LabViewprogramsfordataacquisitionanddataanalysis. 143

PAGE 144

Feynman'sspeculationsaboutthenatureofturbulenceinthesuperuid:"Inordinaryuidsowingrapidlyandwithverylowviscositythephenomenonofturbulencesetsin.Amotioninvolvingvorticityisunstable.Thevortexlinestwistaboutinanevenmorecomplexfashion,increasingtheirlengthattheexpenseofthekineticenergyofthemainstream.Thatis,ifaliquidisowingatauniformvelocityandavortexlineisstartedsomewhereupstream,thislineistwistedintoalongcomplextanglefurtherdownstream.Totheuniformvelocityisaddedacomplexirregularvelocityeld.Theenergyforthisissuppliedbypressurehead.Wemayimaginethatsimilarthingshappeninhelium.Exceptfordistancesofafewangstromsfromthecoreofthevortex,thelawsobeyedarethoseofclassicalhydrodynamics.Asinglelineplayingoutfrompointsinthewallupstream(bothendsofthelineterminateonthewall,ofcourse)cansoonllthetubewithatangleofline.Theenergyneededtoformtheextralengthoflineissuppliedbythepressurehead.(Theforcethatthepressureheadexertsonthelinesactseventuallyonthewallsthroughtheinteractionofthelineswiththewalls).Theresistancetoowsomewhataboveinitialvelocitymustbetheanalogueinsuperuidheliumofturbulence,andacloseanalogueatthat." 144

PAGE 145

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V.F.Mitin,J.McFarland,G.G.Ihas,andV.K.Dugaev,Physica(Amsterdam)B284,1996(2000). [23] C.M.McKenney,V.K.Dugaev,G.G.Ihas,V.V.Kholevchuk,V.F.Mitin,I.Yu.Nemish,E.A.Soloviev,andM.Vierra,inProceedingsofIEEESensors2002(Piscataway,NewJersey,2002),p.1275. [24] N.S.Boltovets,V.K.Dugaev,V.V.Kholevchuk,P.C.McDonald,V.F.Mitin,I.Yu.Nemish,F.Pavese,I.Peroni,P.V.Sorokin,E.A.Soloviev,andE.F.Venger,inProceedingsofEighthInternationalTemperatureSymposium{Temperature:ItsMeasurementandControlinScienceandIndustry,editedbyD.C.Ripple,AIPConf.Proc.No.684(AIP,NewYork,2003),p.399. [25] W.F.Vinen,Phys.Rev.B61,1410(2000). [26] W.F.VinenandJ.J.Niemela,J.LowTemp.Phys.128,167(2002). [27] U.Frisch,Turbulence(CambridgeUniversityPress,Cambridge,1995). [28] S.C.Liu,Y.Zhou,andG.G.Ihas,inProceedingsofthe24thInternationalConfer-enceonLowTemperaturePhysics,Orlando,FL,Aug2005,editedbyY.Takanoetal.,AIPConf.Proc.No.850(AIP,NewYork,2006),p.213. [29] S.C.Liu,G.Labbe,andG.G.Ihas,J.LowTemp.Phys.145,165(2006). [30] D.J.Griths,IntroductiontoElectrodynamics(Prentice-Hall,NewJersey,1989),p.213. [31] T.J.Summer,J.Phys.D20,692(1987). [32] C.Kittel,IntroductiontoSolidStatePhysics(JohnWileySons,NewJersey,1996). [33] B.W.MaxeldandW.L.McLean,Phys.Rev.139,A1515(1965). [34] A.Homann,DoctoralThesis,UniversityofCalifornia,SanDiego(1999). [35] E.S.Rosenblum,S.H.Autler,andK.H.Gooen,Rev.Mod.Phys.36,77(1964). [36] D.K.Finnemore,T.F.Stromberg,andC.A.Swenson,Phys.Rev.149,231(1966). [37] D.E.Farrell,B.S.Chandrasekhar,andS.Huang,Phys.Rev.176,562(1968). [38] S.Casalbuoni,E.A.Knabbe,J.Kotzler,L.Lilje,L.vonSawilski,P.Schmuser,andB.Steen,Nucl.Instr.andMeth.A538,45(2005). [39] D.Saint-James,P.G.deGennes,Phys.Lett.7,306(1963). [40] E.M.Purcell,ElectricityandMagnetism(McGraw-HillBookCompany,NewYork,1965)p.282. 146

PAGE 147

S.C.Liu,G.Labbe,andG.G.Ihas,\ProducingTowedGridQuantumTurbulenceinLiquid4He,"inProceedingsoftheInternationalSymposiumonQuantumFluidsandSolids,Kyoto,Japan,Aug2006,J.LowTemp.Phys.,inpress. [42] B.N.Engel,G.G.Ihas,E.D.Adams,andC.Fombarlet,Rev.Sci.Instrum.55,1489(1984). [43] Y.Iwasa,CaseStudiesinSuperconductingMagnets(DesignandOperationalIssues),(PlenumPress,NewYork,1994),Chap.7. [44] TeckComincoMetalsLtd.,LeadMetalMaterialSafetyDataSheet,2005. [45] Technic,Inc.,MethaneSulfonicAcidMaterialSafetyDataSheet,2003. [46] FisherScienticCompany,LeadCarbonateMaterialSafetyDataSheet,2005. [47] M.SchlesingerandM.Paunovic,ModernElectroplating(JohnWileySons,NewYork,2000). [48] R.D.TaylorandJ.G.DashPhys.Rev.106,398(1957). [49] R.J.DonnellyandC.F.Barenghi,TheObservedPropertiesofLiquidHeliumattheSaturatedVaporPressure(Website:http://darkwing.uoregon.edu/rjd/vapor1.htm).c2004. [50] J.F.Douglas,J.M.Gasiorek,andJ.A.Swaeld,FluidMechanics(LongmanScienticandTechnical,England,1995),Chap.11.9,p.344. [51] Y.Zhou,V.F.Mitin,S.C.Liu,I.Luria,M.Padron,R.Adjimambetov,andG.G.Ihas,inProceedingsofthe24thInternationalConferenceonLowTemperaturePhysics,Orlando,FL,Aug2005,editedbyY.Takanoetal.,AIPConf.Proc.No.850(AIP,NewYork,2006),p.1631. [52] G.P.Bewley,D.P.Lathrop,andK.R.Sreenivasan,Nature441,588(2006). [53] W.H.Press,B.P.Flannery,S.A.Teukolsky,andW.T.Vetterlig,NumericalRecipesinC:TheArtofScienticComputing(CambridgeUniversityPress,Cambridge,England,1992). 147

PAGE 148

Shu-chenLiuwasborninChuang-huaCounty,Taiwan,theRepublicofChina,in1974.ShegraduatedfromthegiftedclassinTaichungGirls'SeniorHighSchoolinTaichungCityinTaiwan,andthengotguaranteedadmissiontoNationalTaiwanNormalUniversityin1993.AfterbeingaphysicsandchemistryteacherintheShin-YijuniorhighschoolinTaipeiinTaiwanfrom1997-1998,shereceivedaB.S.inPhysicsfromNationalTaiwanNormalUniversityin1998.ShebecametheteachingassistantoftheDepartmentofPhysicsattheNationalTaiwanNormalUniversityin1998-2000,andthenwenttoUniversityofFloridaintheUnitedStatestopursueherPh.D.inphysicssince2000.Hercurrentresearcheldislowtemperaturephysicsinthespecialityofquantumturbulenceinsuperuidhelium-4atmillikelvintemperaturesundertheinstructionofProfessorIhas.HerfavoriteproverbisDescartes'"Cogitoergosum." 148


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Title: Quantum Turbulence: Decay of Grid Turbulence in a Dissipationless Fluid
Physical Description: Mixed Material
Copyright Date: 2008

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Source Institution: University of Florida
Holding Location: University of Florida
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QUANTUM TURBULENCE :
DECAY OF GRID TURBULENCE IN A DISSIPATIONLESS FLUID


















By
SHU-CHEN LIU


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007




































S2007 Shu-Ch.! i. Liu




































To my home country, Taiwan, R.O.C. To my mentor, Professor Gary G. Ihas. To my

parents. To my two sisters and my brother. To my grandparents.










ACKENOWLED G1\ENTS

I am thankful for all the help I have received in my Ph.D. program, in writing this

dissertation and in learning about this research subject. First of all, I would like to

thank my mentor, Professor Gary G. Ihas, who has led me to the amazing field of low

temperature experimental physics. Front hini, I have learned a lot of physics and I was

able to carry out my accomplishments. I really enjoy the beauty of quantum turbulence

science. Second, I would like to thank my parents, who have given me a lot of support,

emotionally, financially, and spiritually, and a lot of encouragement before and during

the pursuit of my Ph.D. Third, I would like to thank the nienters of our research group,

former and current, and the people in the machine and electronic shops. Because of them,

I was able to stand on those giants' shoulders and look higher and further. Fourth, I

would like to thank the whole Gainesville coninunity, which has given me much.











TABLE OF CONTENTS


page

ACK(NOWLEDGMENTS ......... . .. .. 4

LIST OF TABLES ......... ... . 8

LIST OF FIGURES ......... .. . 9

ABSTRACT ......... ..... . 12

CHAPTER

1 INTRODUCTION TO QITANTITA TITRBITLENCE ... .. .. 14

1.1 Introduction ......... . .. .. 14
1.1.1 Basic Properties of He II . . .. .. 14
1.1.2 Two-Fluid Model, and Landau's Two-Fluid Equations .. .. .. 16
1.1.3 Quantization of Vortices and the Critical Velocities .. .. .. .. 18
1.1.4 K~elvin Waves ....... .. .. 20
1.2 Introduction of Towed-Grid Turbulence Experiments .. .. .. 20
1.3 Proposed Towed-Grid Superfluid Turbulence Experiment .. .. .. 21
1.4 High Resolution, Fast Responding Milikelvin Therntoneters .. .. .. 25
1.4.1 Neutron Transmutation Doped Gernianiunt Bolonieters .. .. .. 25
1.4.2 Miniature Ge Film Resistance Therntoneters .. .. .. 27

2 SHIELDED SITPERCONDITCTING LINEAR MOTOR .. . :34

2.1 Introduction ... .. . .. ... .. :34
2.2 Models of the Shielded and Unshielded Superconducting Motor .. .. :35
2.3 Simulation Results and Discussions ...... .. :39
2.3.1 Simulation All ll-k- ........ ... :39
2.3.2 Critical Magnetic Fields for Nichium ... .. .. 41
2.3.3 Required Voltage Input for the Solenoid .. .. .. .. 4:3
2.4 Unshielded Motor Testing Experiments .. .. .. .. 45
2.4.1 Capacitance Bridge and Lock-in Amplifier for Monitoring Armature
Motion ............. .. ...... 46
2.4.2 The 555 Oscillator for Monitoring Armature Motion .. .. .. .. 47
2.4.3 The Q-nieter for Monitoring Armature Motion .. .. .. .. .. 49
2..4Te Aic Bridge circuits for Monitoring the Motion of the Armature 5

2.5 Improved Design of the Armature and the Test Cell .. .. .. 57
2.6 Conclusion ......... .. . 59

:3 CONSTRUCTION OF SITPERCONDITCTING SHIELDED LINEAR MOTOR
AND EXPERIMENTAL CELL ......... ... .. 84

:3.1 Construction of Superconducting Shield .... .. . 84
:3.1.1 Electroplating Theory and Electrolyte Recipe .. .. .. .. 84











:3.1.2 Properties of C'I. !! I- ....... .I .. .. .. .. 85
:3.1.2.1 Lead ........... ...... ..... 85
:3.1.2.2 Methanesulfomic acid (j!HA) . . 86
:3.1.2.3 Lead carbonate . ..... .. .. 86
:3.1.2.4 Lab protective equip ..... .. . 86
:3.1.3 Lead Electroplating Procedures and Results .. . .. 86
:3.1.3.1 Procedure steps . ..... .. .. 86
:3.1.3.2 Results ......... ... .. 88
:3.1.3.3 Troubleshooting . . ... .. 88
:3.2 Testing the Experimental Cell at Liquid Helium Temperature .. .. .. 90
:3.2.1 Simulation Results ....... ... .. 90
:3.2.2 Experimental Testing Results ..... .. . 92
:3.2.3 Viscous Drag and Impedance Forces Discussion .. .. .. .. .. 92
:3.2.4 Heat Dissipation Discussion ..... .. .. 9:3
:3.3 Leak Tight Electrical Feedthrough Design .... .. .. 94
:3.3.1 Electrical Feedthru Construction .... ... .. 95
:3.3.2 Thernxistor Circuit Board Construction .. .. .. .. 96
:3.3.3 Conclusion ......... . .. 97

4 THERMISTORS ELECTRONICS . ...... .. 111

4.1 Introduction ......... . .. .. 111
4.2 The AC Bridge Circuit Analysis . .... .. .. 111
4.3 Resolution Measurement at Room Temperature ... .. . .. 11:3

5 EXPERIMENTAL PROCEDURE, DATA, AND ANALYSIS .. . 118

5.1 Introduction ......... . .. 118
5.2 Experimental Results ......... .. .. 118
5.2.1 Thernxistor Calibration . .... .. 119
5.2.2 Background Heating C'I. I 1: at 624 n1K .. .. .. .. 119
5.2.3 Performing QT Experiments at 520 n1K .. .. .. .. 120
5.3 Exploring the K~elvin Waves in the Energy Spectra ... .. . .. 121

6 CONCLUSIONS AND FITTIRE WORK( .... .. .. 127

6.1 Conclusions ........ . .. 127
6.2 Future Work ........ . .. 127

APPENDIX

A DERIVATION AND NITMERICAL ANALYSIS OF MAGNETIC FIELD AND
FORCE ........ .... ......._ .. 129

B SOME C CODE ........ .. .. 1:31

C LABVIEW PROGRAM SHOTS ....... ... .. 140











D FEYN1\AN'S SPECULATIONS ABOUT THE NATURE OF TITRBITLENCE
IN THE SITPERFLITID ........_ .. 144

REFERENCES ..........._ ..........._. 145

BIOGRAPHICAL SK(ETCH ......... .. .. 148










LIST OF TABLES


Table page

2-1 Optimal parameters for the motor design. .... ... . 40

2-2 Parameters for the unshielded test motor. .... ... . 45

3-1 Parameters of superconductor shielded superconducting linear motor system. 91

3-2 Forces on the armature when moving up at 1 m/s (Download is positive). .. 94

4-1 Parameters of the AC bridge affecting the sensitivity and the power dissipation. 114

4-2 Sensitivity test of the AC bridge. ........ ... .. 115










LIST OF FIGURES


Figure page

1-1 Viscosity of liquid heliuni-4. ......... .. 29

1-2 Superfluid and normal fluid densities as a function of temperature. .. .. .. 29

1-3 Photographs of stable vortex lines in rotating He II. ... .. .. :30

1-4 Damping on a sphere oscillating in liquid helium. .... .. :30

1-5 Donnelly-Glaherson instability of a quantized vortex line. ... .. .. :31

1-6 Energy spectrum for homogeneous and isotropic turbulence. .. .. .. :31

1-7 Electrical conduction mechanisms in semiconductors. .. .. .. :32

1-8 Calibration plots for three test thernxistors developed for calorintetry. .. .. :32

1-9 Design for the therntoneter test cell. . ...... .. 3:3

2-1 Grid turbulence in a classical fuid. . ..... .. 60

2-2 Superconducting motor model. ......... ... .. 60

2-3 Armature motion in unshielded and superconducting shielded solenoid. .. .. 61

2-4 Armature motion inside superconducting shielded solenoid (linear acceleration). 61

2-5 Armature motion inside superconducting shielded solenoid (square acceleration). 62

2-6 IUpper critical field versus temperature for niohium. ... ... .. 62

2-7 Voltage input to the superconducting shielded solenoid. ... .. .. 6:3

2-8 Simulated armature motion for the superconducting shielded solenoid.. .. .. 64

2-9 Driving voltage. . .. .... .. 65
2-10 circuit of the switch box. ..........6


2-11 1\achine drawings and photos of the motor system. ... .. .. 67

2-12 Experimental apparatus for unshielded motor. .... ... .. 68

2-1:3 The GR 1616 Capacitance Bridge ....... ... .. 69

2-14 Circuitry of the experimental apparatus for unshielded motor testing experiments
using the 555 oscillator circuit and LahView counter program to monitor the
motion of the armature. ......... . .. 69

2-15 Testing circuit of Q-nleter. ......... . 70











2-16 Simple AC bridge circuit.

2-17 Measuring the capacitance of the capacitive position sensor with the GR 1616
capacitance bridge and the lock-in amplifier.

2-18 Circuitry and setup for testing the superconducting motor at 4.2 K(.

2-19 Theoretical calculation of the magnetic force and RT sensor calibration curve.


2-20 Motion of the armature of the superconductingf motor.

2-21 Simulated armature motion with the input pulse sent to the sol

2-22 Motion of the armature of the superconductingf motor. ....


enoid. .. .. 75

76

. 77

. 78

. . 79

80

umbers.. .. 80

81

. . 82

. 8:3

98

99

.. 100

.. 101

.. 102

. 10:3

.. 104

.. 105

.. 106


2-2:3

2-24

2-25

2-26

2-27

2-28

2-29

2-:30

:3-1

:3-2

:3-3

:3-4

:3-5

:3-6

:3-7

:3-8

:3-9

:3-10

:3-11


Motion of the armature of the superconducting motor (I). ..

Motion of the armature of the superconducting motor (II). ..

Motion of the armature of the superconducting motor (III). .

Modified superconducting motor system. .....

Machine drawing of the grid and the corresponding Reynolds n

Calibration curve of the position sensor at 4.2 K(. ....

Electronics circuits for the superconducting motor system. ..

Motion of the armature of the superconducting motor. ....

Machinery drawings for the cell cap. .....

Machinery drawings for the cell body. ......

Electrode for the cell cap. .....

Electrode for the cell body. .....

Procedure steps for the lead plating for the cell cap. .....

Procedure steps for the lead plating for the cell body. .....

Masterpiece of the lead coated cell cap. .....

Masterpiece of the lead coated cell body. .....

Towed-Grid experiment cell. .....


11


Simulation for superconductor shielded superconducting linear motor system. .. 107

Motion of the armature of the shielded superconducting motor. .. .. .. .. 108











:3-12 External forces on the armature. ........ ... .. 109

:3-13 Leak tight electrical feedthru design for our cryogenic cell. .. .. .. .. 110

4-1 The AC bridge circuit simulation. ........ ... .. 116

4-2 AC bridge circuit for thernxistors resistance measurement. .. .. .. .. 117

4-3 Room temperature thernxistor resolution curve. .. ... .. 117

5-1 Thernxistors calibration curves. ....... ... .. 12:3

5-2 1\otion of armature and thernlistor response at vacuum around 600 niK. .. 124

5-3 Quantum turbulence at 520 n1K. ........ ... .. 125

5-4 Fitting the enthalpy of liquid helium as a function of time. .. .. .. .. 126

C-1 LahView programs for calculation of current, magnetic field, magnetic force and
armature motion. .. ... . .. 141

C-2 LahView programs to look for optimal parameters. .. .. .. 142

C-:3 LahView programs for data acquisition and data analysis. .. .. .. .. 14:3









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

QUANTUM TURBULENCE :
DECAY OF GRID TURBULENCE IN A DISSIPATIONLESS FLUID

By

Shu-Ch.! i. Liu

May 2007

C'I I!1-: Gary G. Ihas
Major: Physics

We produced grid turbulence in liquid helium at 520 mK( to compare with classical

experiments and theories. Above T = 1 K(, with viscosity present, it has been shown that

grid turbulence is equivalent to homogeneous isotropic turbulence in a classical fluid. We

seek to investigate the nature of grid turbulence when viscosity is zero. Specifically, in the

absence of viscosity in a quantum fluid, through what path does the turbulence decay?

To produce grid turbulence, an actuator was designed and built that can accelerate and

decelerate the grid rapidly in a short distance (~ 1 mm), and achieve glide speeds of up

to 1 m/s. To avoid Joule and eddy current heating of the liquid helium, a magnetically

shielded superconducting linear motor was built. The grid is attached to the end of a very

light insulating armature rod which has two hollow cylindrical niobium cans fixed to it

about 26 mm apart. This part of the rod is inside a superconducting solenoid which, when

driven with the properly shaped current pulse, produces a magnetic field resulting in the

required motion.

Detailed computer simulations guided the motor design. The simulation and motor

control programs were written in LabView with an embedded C compiler. Using the

simulator, various designs of solenoid (with and without shielding) and armature were

investigated. We compared the simulation and the experimental results in which complex

current pulse shapes were required to produce the desired motion.










The motor and grid (1 mm square hole array with '71I' transparency) were mounted

in a copper cell containing a pool of liquid helium cooled to 520 mK( by dilution

refrigeration. We measured the decay of the turbulence produced, after one 28 mm stroke

of the grid, using calorimetry. Doped germanium thermometers less than 300 micrometer

diameter immersed in the turbulent liquid helium allowed fast calorimetric measurements

limited by the electronic time constant of 1 ms. The decay of turbulence was detected by

the rate of temperature rise in the isolated cell after the grid was pulled. Recent theory

-II- -_ -r ;the decay occurs through a K~elvin-wave cascade on the vortex lines which couples

the initially large turbulent eddies to the short wavelength phonon spectrum of the liquid,

yielding a characteristic rate of temperature rise. Initial measurements support the K~elvin

wave cascade theory.









CHAPTER 1
INTRODUCTION TO QUANTUM TURBULENCE

1.1 Introduction

Quantum turbulence is a very interesting and amazing field in physics. Richard

Feynman said that turbulence is the most important unsolved problem of classical physics.

At temperatures below 2.17 K( (the A- transition temperature), the turbulence produced in

superfluid helium II demonstrates quantization of vortex lines (i.e., quantized circulation).

Therefore, this turbulence is called quantum turbulence. The idea of quantized circulation

in superfluid helium was proposed by Onsager and Feynman in 1955 [1] and discovered

experimentally by Hall and Vinen a year later [2].

Superfluid quantum turbulence produced by towed-grid experiments in liquid 4He

at very low temperatures is predicted to decay, not through viscosity, as in a classical

fluid, but by phonon radiation when the energy flows into the smaller length scales in a

K~elvin-wave cascade. We propose a new calorimetric technique to probe such a decay

mechanism of superfluid grid turbulence at extremely low temperatures, wi 520 mK(,

while the normal fluid density is only 8.6 ppm [3].

1.1.1 Basic Properties of He II

Helium is the only known element that remains liquid at extremely low temperatures,

even down to absolute zero, under saturated vapor pressure. Its remarkable superfluid

properties have been a drawn to scientists and they have begun studying its other

hydrodynamic properties as well.

The superfluidity of helium-4 was discovered in 1939 by Allen, Misener, and K~apitza

[4, 5] while Osheroff, Richardson and Lee did not discover the superfluidity of helium-3

until 1971 [6]. At 2.17 K(, under saturated vapor pressure, the curve of specific heat versus

temperature for 4He shows a dramatic spike, which looks like the Greek letter 'A'. This

is called a lambda transition, and corresponds to a second order phase transition. In

3He, the superfluid transition occurs at 0.9 mK( under saturated vapor pressure. Below









the transition temperature, 3He and 4He liquids show superfluid phases with low to zero

viscosity.

The thermal de Broglie wavelength of liquid helium is


Ar= 8.9A1 (1-1)


at 2.0 K(. This is comparable to or greater than the mean interparticle distance of a 3.6P4

for helium. So the de Brogflie wavelength of each atom is larger enough to overlap with its

neighbor.

nA~ = 2.1 x 1028 1773 x (8.9 x 10-ton)3 = 14.8 > 2.61 (1-2)

This is why liquid helium is called a quantum fluid. Because there are two protons and

two neutrons in the nucleus, there is an even number of nucleons (total nuclear spin=

0), and the quantum mechanical behaviors of 4He can he explained by Bose-Einstein

statistics. Atoms of "He obey Fermi-Dirac statistics because their nucleii contain two

protons but only one neutron, totaling an odd number of nucleons (total nuclear spin

=1/2). (The spins of the two electrons cancel out.) Hence, 4He is a hoson and 3He is a
fermion.

The superfluid 4He, also called He II, has almost zero viscosity, while the normal

fluid of 4He above T He I, has much higher viscosity which can dissipate energy via

interactions with the walls of the container. The viscosity of liquid 4He measured by the

method of oscillating disc viscometer is shown in Fig. 1-1 [7]. Two of the most famous

experiments demonstrating the superfluid properties of He II are the beaker experiment

and the fountain (thermo-mechanical) effect. If you put the bottom of an empty weaker

in the He II hath, a 20-30 nm thick 4He mobile film forms on the walls of the beaker,

and then liquid He II flows along the film from the hath into the beaker until the levels

are equal. If you then lift the beaker above the hath level, the liquid inside the beaker

will also flow along the film out of the walls of the vessel into the hath until the beaker

is empty. Now if you connect two vessels containing the same level of He II at the same










temperature, wi 2 K(, with one very narrow capillary which can only pass superfluid, and

then increase the temperature of one vessel by 10-3 K(, the level in the other will rise, by

9.6 cm. In addition, if you apply extra pressure on one vessel, then the fluid level will fall

because superfluid flows into the other vessel. These are both manifestations of superflow

without loss.

1.1.2 Two-Fluid Model, and Landau's Two-Fluid Equations

Landau and Tisza proposed the Two-Fluid Model in 1941 [8-10] to explain the

various interesting phenomena that occurred in He II. In this model, the liquid helium II

is considered as a mixture of two interpenetrating fluids, called the normal component and

the superfluid component, with densities, p, and p,, respectively. Hence, the total density

of liquid He II [11] follows:

p = p, + p, a 0.145 (1-3)

g/cm3 (at Saturated vapor pressure).

The superfluid and normal fluid component densities, as a function of temperature

below Tx under the saturated vapor pressure, are shown in Fig. 1-2. The superfluid has

zero entropy (S, = 0) and zero viscosity (rl = 0), while the normal fluid exhibits viscosity

(r7;) and entropy (S,), equal to the entropy of all the liquid helium. Also, the superfluid is
considered to be irrotational:

V x v4 = 0, (1-4)

where v6 is the velocity of the superfluid.

Modifying Euler's equations for the classical (Euler) fluids based on the continuity

equation, and using the first and second laws of thermodynamics, and his own postulate

that the chemical potential (p) is the driving force for the superfluid, Landau derived the

two fluid equations for He II [13]:

dp
+ pil = 0 (1-5)










(mass conservation)


8ps
,+ psy, = (1-6)

(entropy conservation)



Dt dt


+ = (1-8)


(momentum conservation)
where the total mass flow is:

7 pil py,v + ps@~ (1-9)

and stress tensor is:

Pij = p6ij + pay,,iva,, + psy,4vs,,4, (1-10)

In Landau's two-fluid model, the elementary thermal excitations, phonons and rotons,

depending on the wave number, arise in the flow of helium II through a tube or capillary

at T / OK when the normal fluid component has the interactions with the walls causing

energy dissipation and viscous loss. Suppose an excitation is created with energy E and

momentum p due to the loss of energy from the tube (AE = E). Then Landau's relation

(v > E/p) gives the minimum velocity of flow


vc = E/p (1-11)


required to produce an excitation. However, actual critical flow velocities in experiments

are much smaller (~ mm/s) than Landau's prediction (60 m/s at the vapor pressure and

46 m/s at higher pressure [7]), due to the quantized vortices.









1.1.3 Quantization of Vortices and the Critical Velocities

W;lith the assumption =D Vp in the liquid He II, the superfluid velocity

obeys a K~elvin circulation theorem [13 :


-nv d e = 0. (1-12)
Dt

In addition, the K~elvin circulation theorem,

D, d A1 = 4, (V x i ) S (1-13)
Dt Dt

(by Stokes' law) implies that the superfluid circulation

n = d (1-14)


stays constant, and if at t = 0 the superfluid vorticity = V x is zero everywhere it

will stay zero. If V x disappears everywhere, then the superfluid circulation is:


i = -de i (V x ) dS = 0. (1-15)

Assuming there is some circulation, there must he a gl region where either

V x / 0 or there is no superfluid. So we consider the singular region as a very thin

cylinder, called a superfluid vortex line or vortex core. According to Gauss' theorem:

f,(V 7)~dv = J 7i d S, and V V x 7 -- 0 for any 7, we have f,,(V Vx ~)d-r=

J,~ Vx dS = J d -e = 0 = m. So a vortex line cannot terminate in the fluid, but
must end at a boundary or close in on itself (a vortex ring). Since the vortex line is the

only source of vorticity in the fluid for V x = 0 everywhere except at the line, all path

integrals encircling the vortex line have identical circulations.

As soon as liquid helium II rotates or moves hevond a critical velocity, superfluid

vortex lines appear and demonstrate either an ordered array of vortex lines by steady

rotation or disordered vortex tangles for counterflow (due to heat flow). That the

superfluid circulation is quantized was postulated separately by Onsager and Feynman









in 1955, a = J v, .97 x 10-48 C112 8, Where n is an integetr. The radius of

the vortex core is about ao ~ 1P1 (atomic dimensions). The stable quantized vortex arrays

in rotating He II can be visualized as shown in Fig. 1-3 [14]: stable vortex lines in rotating

He II in a cylindrical bucket of 2 mm diameter to depth 25 mm placed at the rotation axis

of a rotating dilution refrigerator at 100 mK(. >' 3He was added to provide damping which

maintained stability. The negative ions are trapped on vortex cores and are imaged on a

phosphor screen and recorded on cinefilm. All superfluid vortex lines align parallel to the

rotation axis with ordered arrays of areal density (or length of quantized vortex line per

unit volume) as given by the following equation:

220 Vx
~20000 (1-16)


(in lines/cm2), Where R is the constant angular velocity for the rotation. This can derived

as follows: the circulation around any circular path of radius r concentric with the axis of

rotation = J -, d e = J(V x v) dS = 2xrr20. And the total circulation = Kr2no0 m,

where no is the number of lines per unit area. Therefore, no = 20m/h = 20/s [7].

The turbulent state, described as a mass of vortex lines, usually has two critical

velocities signaling the onset of turbulence in superfluid and in the normal fluid separately,

increasing the total length of vorticity with the increasing relative velocity of the two

fluids. In the experiment on the damping of the rotation of a sphere oscillating in liquid

helium II as a function of the maximum amplitude (or velocity) of the oscillation, the

result is shown in Fig. 1-4 [7]. At region A, the damping is constant, relating to the

constant normal viscosity rls. The two critical velocities occurred at the transition from

region A to B (which is the onset of turbulence in the superfluid component), and C

to D corresponding to the onset of turbulence in an ordinary classical liquid, where the

damping increases dramatically. In regions B, C, and D, both superfluid and normal fluid

are coupled and move together due to their mutual friction. The critical velocity of the

superfluid rises with reduction in diameter of the channel.









1.1.4 Kelvin Waves

Quantized vortex lines in helium II at high temperature can vibrate parallel to the

vortex lines under the influence of the normal fluid if they have large enough velocity.

This is called the Donnelly-Glaberson instability [11] (see Fig. 1-5). The plane of these

vibrations (called K~elvin waves) process about the center core, growing exponentially along

quantized vortex lines. The lengths of the vortex lines increase, eventually resulting in a

vortex tangle as the energy is transferred from the normal fluid to the superfluid.

Feynman -II---- -1h I1 in 1955 that vortices approaching each other very closely undergo

reconnections. The K~elvin waves can he generated by vortex reconnections, leaving kinks

on the vortex lines, regarded as superpositions of K~elvin waves, leading to the continuous

generation of K~elvin waves with a wide range of wave numbers.

1.2 Introduction of Towed-Grid Turbulence Experiments

In classical grid turbulence, the eddy motion can range from large length scales,

(as large as the mesh of the grid or the size of the channel), to small length scales (or

larger wave numbers than the inverse of the vortex line spacing, -e-1). In the later case,

the Reynolds number R, = ~ 1 (where U is the characteristic velocity and n is the

kinematic viscosity) and energy dissipation because of viscosity occurs. However, if the

Reynolds number R, =-~ ~ inril oc 1 (in an "inertial regime") so that the viscosity can

he ignored, then the energy will flow in a cascade from large scales to smaller scales, as

described by the K~olmogorov spectrum [11]:

E /) C2/3k_-'l" (1-17


where E(k)dk is the energy per unit mass for spatial wave numbers in the range dk. The

function E(k) has dimensions [L3/T2](L: length, T: time). C ~ 1.5 (the K~olmogorov

constant, w~hic~h is dimensionless), and E idk (that is the average rate of kinetic

energy transfer per unit mass flowing down the cascade, dissipated by viscosity at high

wave number, k > -e-1). The dimensions of E are [(L/T)2/T] = [L2 T3]; the dimensions










of k are [L-l]. As -II_t---- -1.. by the energy spectrum shown in Fig. 1-6 [11], the largest

eddies have the most turbulent energy, and decay most slowly, which determines the

energy dissipation rate (E). In the steady state, energy flows from these largest eddies to

the smallest eddies.

The superfluid grid turbulence experiments in helium II above 1 K( [12] can also be

described by the K~olmogorov spectrum in the inertial regime on length scales larger than

the spacing between vortex lines (e). In this case, the energy dissipation by viscosity

occurs on the length scale of order -e (the spacing between vortex lines) due to the

significant amount of normal fluid and the mutual friction between the superfluid and

the normal fluid.

1.3 Proposed Towed-Grid Superfluid Turbulence Experiment

We seek to verify or eliminate the K~elvin-wave cascade as the mechanism for

the decay of quantum turbulence produced by a towed-grid in liquid 4He at very low

temperatures. At these temperatures, the component of the normal fluid is negligible, so

mutual friction may be neglected. measurements above 1 K( [12] show that homogeneous

isotropic turbulence is produced behind a towed grid moving at the order 1 m/s.

Computer simulations [15] predict at zero temperature the K~elvin waves on intersecting

vortex lines produce the equivalent of the viscous regime in a classical fluid. This has yet

to be confirmed by experiments.

It is ell----- -r1 11 [16, 25] that energy flows to the smallest scale by a K~elvin wave

cascade on the vortices, leading to a K~elvin-wave energy spectrum for the wave number k

of Kelvin waves greater than the inverse vortex spacing -e-1. Kelvin waves do not have any

damping at very low temperatures until the wave numbers k (> k2 = 2 x 10s m-l) [16, 25]

become much greater than -e-1. It has been predicted [16] that at 0.46 K(, energy dissipates

by phonon radiation; however this has not been confirmed experimentally. The results of

the simulations demonstrate the continuous energy flow within the K~elvin waves towards









highest wave numbers at which the phonon emission dissipates the energy, and the steady

state (a K~elvin-wave cascade) with balanced energy input and dissipation.

The corresponding K~elvin wave spectrum, cut off by dissipation at k ~ k2, iS proposed

to be: [25, 26]

E(k) = Aps2 -1, 18

where E(k)dk is the energy per unit mass and unit length of vortex associated with K~elvin

waves with wave numbers in the range dk, p is the density of the helium, and A ~ 2. The

rate at which energy flows into the K~elvin wave cascade per unit mass of helium is given

by [26]

E" (X12/2x)K3L (1-19)

where Lo = e-2 is the length of the smoothed vortex line per unit volume, and X2 ~ 0.3.

The energy contained in the "K~elvin wave cascade" per unit mass of helium is given by

[26]
1 K2 e
E =-E,AL =AL Inb( ) a ri2AL, (120)
p 4xr fo
where aL = ALoln(k2e), E, is the energy per unit length of vortex line, and (o is the

vortex core parameter. The cut-off wave number k2 is giVen by the formula [16]


k2e = ( 3/4, _21)
A1/3~

where c is the speed of sound in helium.

By measuring the rise in temperature of the helium after creating turbulence with

a high resolution thermometer, we can probe the turbulence decay as a function of time

since the temperature change corresponding to the decay of a random vortex tangle is

proportional to the change in the vortex line density. Therefore, it is possible to explore

the existence of a K~olmogforov spectrum on large length scales, a K~elvin wave cascade on

small length scales, and the dissipation mechanism. The enthalpy of the helium is given by











H(T) = 747T4 _122)

(in J/m3). If a Small amount of turbulence energy (AE) is released as thermal energy in

helium at temperature To, the enthalpy values of the temperature change would be


T4 TO4 = 1.34 x 10-3AE (1-23)


(in K4), Which can be monitored as a function of time. The grid turbulence is created by
drawing a grid through the helium by a superconducting linear motor at a constant speed

as fast as 1 m/s for 10 mm for approximately 10 ms, which is faster than the decay speed

of the turbulence (a few hundred milliseconds). Any energy dissipation in the helium from

moving the grid must be much smaller than the released thermal energy from the decay

of the superfluid turbulence. Suppose the square cross section of the channel is d x d and

a very dense vortex tangle as well as quasi-classical turbulence (on a larger scale than the

mean vortex line spacing e) are produced by the towed-grid. The energy can transfer to

either larger length scales (on the scale of d) until becoming saturated, or to smaller length

scales (less than e) where energy dissipation occurs, leading to the K~olmogorov energy

spectrum. The time required to build this spectrum should be less than the turnover time:


re = ,(1-24)

where d= eddy size, and u(d)= characteristic velocity relating to this eddy size, defined by

[26]

u(d) = CE2/3k-2/3. (1-25)

From the clues in the previous experiments above 1 K(, it is surmised that a K~olmogforov

spectrum joins smoothly to the "quantum v. I ~1, n


u() = .(1-26)









which is characteristic of motion on the scale e. So [26]


u(d) = p1/2 1/3 _127)
e 2x-

where p ~ 0.25. The total turbulent energy per unit mass, most of which comes from the

largest eddies of size d, is given by


Eclass = 32() __ 2a 2/3s. (1-28)
2 2 P 2x-

As -e increases, turbulence decays and energy dissipates via the classical K~olmogorov

cascade as described by the above equation. At time t > -r, total energy decay with time

as described by the K~olmogorov spectrum can be thought of as the energy flow rate with

time [26] ,

E = 27C3 2( t0 -3, (1-29)

where to ~ 9d, which varies with the towed-grid speeds or initial turbulent intensities. So:

27
Class = -C3 2( t0 -2. 30)


Comparing Equations 1-28 and 1-30, the time dependence of e is given by

P3/8 ~3/4
= (t to)3/4 1 )
33/4 (2xT)1/4 C9/8 1/2

Another characteristic time for the K~elvin-wave spectrum (Equation 1-18) which

describes the fully developed K~elvin-wave cascade is given by

E 2; A 2
-r ( ) I(k2). (1-32)


Energy flows into the K~elvin-wave cascade at the rate E = E", and eventually dissipates by

phonon radiation. The energy per unit mass contained in the K~elvin-wave cascade is given


E = Ais2e-2 2 k~)* (33)










The total turbulent energy per unit mass is


27
E = Eclass + E = C3 2" t0 -2 + Ais2e-21ke) 2 )*)


Since -e increases with time, the decay of E will be dominated by the decay of Eclass for

small but by the decay of E for large For t > to, Eclass is proportional to t-2, While

E is proportional to t-3/2, Which will be expressed by the observed rate-of-change of

temperature.

1.4 High Resolution, Fast Responding Milikelvin Thermometers

Thermometers must meet the following requirements to be used in our quantum

turbulence experiment:

1. Operating temperature: 20 mK to 1 K( (dilution refrigerator temperatures).

2. Sensitive to temperature change: 6T ~ 10-3K, or 6T/T ~ 0.05 10-3

3. Short response time: 6t ~ 10-3s. The turbulence energy decays within a few hundred
milliseconds. In order to have good time resolution in the data, it is necessary that
the thermistors respond within 1 ms.

4. Small mass, small heat capacity, and good thermal conductivity.

So far we have found two excellent candidates to fulfill the above requirements which

can be used in our calorimetric technique: Neutron Transmutation Doped Germanium

Bolometers and Miniature Ge Film Resistance Thermometers. We have used the later in

our work.

1.4.1 Neutron Transmutation Doped Germanium Bolometers

The basic electrical conduction mechanisms in semiconductors can be classified into

four categories: (The diagram is shown as Fig. 1-7 [18])

1. Thermal generation of electrons and holes across the bandgap, which is negligible at
low temperature since kT
2. Generation of free charge carriers by ionization of shallow donors, which is negligible
at low temperature since kT









3. C'!I. ge carrier movement from one impurity to the next in heavily doped semiconductors
(metal-insulator transition) also called banding mechanism.

4. In heavily doped and compensated semiconductors, the compensating or minority
impurities create a lot of ill s .int il,y impurities which remain ionized down to absolute
zero. The charge carrier hops from an occupied 1!! lint ~l~y impurity site to an empty
site, which is the working principle for the low temperature bolometer. It is also called
hopping mechanism.

The material we use for our high-resolution dilution refrigerator thermometer is

neutron transmutation doped (NTD) germanium [18, 19], which applies the fourth

electrical conduction mechanism as discussed above.

A NTD Ge has NVA 1!! I li~ ~ry Shallow acceptor impurities and NDo minOrity Shallow

donor impurities (NVA > ND). At very low temperatures (kT
binding energy of the electrons to the acceptors), and in the dark, (NVA 1VD) acceptors

have an electron vacancy and are neutral while NVD acceptors capture electrons from

compensating donors.

The transmutation of stable germanium isotopes via the capture of thermal neutrons

is accomplished by the following procedure:

1. A single ultra-pure germanium < li -r I1 is grown in a hydrogen atmosphere (1
atm) from a melt contained in a pyrolytic carbon-coated quartz crucible using
the Czochralski technique.

2. Six 2 mm thick slices of 36 mm diameter are cut, lapped and chemically etched.

3. Irradiated with thermal neutrons, doses 7.5 x 1016 ~ 1.88 x 10lscm-2

4. Ater en hlf lves f ne(T1/2 11.2d), the samples are annealed at 400 oC for 6
hours in a pure argon atmosphere (1 atm) to remove irradiation damage.

Some papers demonstrate that, even at dilution refrigerator temperatures, the NTD

Ge thermometers still have sufficient sensitivity. For example: at 25 mK(, 6T/T ~ 4.8 x 10-6,

and response time < 20 ms for thermometers as small as 1mm x 1mm x 0.25mm [20].

Some very good circuitry for the NTD Ge bolometers has been developed [20, 21].









1.4.2 Miniature Ge Film Resistance Thermometers

Microsensors based on Ge film on semi-insulating GaAs substrates [22-24] have

already been developed and tested in our group. The typical size for the sensitive

element is 300 pm in diameter, and the mounted gold leads are 50 pm in diameter.

The conduction mechanism is variable range hopping. Fig. 1-8 shows the absolute
resistance values and the sensitivity (R ) over the temperature range o intret,,t for

three test thermistors against a ruthenium oxide calibrated thermometer. We used the

LR-110 picowatt AC resistance bridge as the detection circuit, and calibrated three of our

available thermometers. The LR-110 bridge can measure resistances between 10 R and

1.2 M R with high resolution (better than 0.1 .) and good accuracy (0.05'~ 0.25' .).

The calibration curves demonstrating the performance and the characteristics of the

three test thermometers are quite different due to the variations in doping and heat

treatment during manufacturing. Sensing powers were less than 10-13 WittS. TWO Of

the thermometers are not ideal as we can see from the curves. For thermistor 1, the

resistance ~ 7 or 8 kR under 100 mK( and was nearly constant below 38 mK( with

pretty stable sensitivity within the measured temperature range. For thermistor 2, the

resistance goes to infinity at low temperatures and only becomes measurable above about

49 mK(. Thermistor 2 demonstrates dramatic sensitivity change over a wide range of

temperatures. For thermistor 3, the resistance is 16 kR at 88.7 mK( and 526 kR at 24.8

mK(. The sensitivity ranges from 1.7 M~egR/K to 140.3 M~egR/K between 50 mK( and

20 mK(. Its response time is less than 0.001 s. This makes thermistor 3 a good candidate.

During turbulence decay, the resistance of thermometer 3 is expected to change from 4 kR

to 1.75 kR, and the change from 3 kR to approximately 2 kR representing the K~elvin wave

decay at 520 mK(, which appears in our final experimental results.

The machinery drawing for the thermometer test cell is shown in Fig. 1-9. The

thermometer test cell is used to simultaneously test the sensitivity and the resolution

of two thermometers (miniature Ge film resistance thermometers) separated by a small










distance. By applying current to the heater located next to these two thermistors, we

can also measure how fast heat is transported in the superfluid or measure how fast the

thermometers respond to heating.






















2 4
T/K


Viscosity


Tj.

~"C~-


Figure 1-1: Viscosity of liquid 4He measured in an oscillating disc viscometer [7].







Fluid density


P


56 % P


T (K)


0 2.0 Th


Figure 1-2: Superfluid and normal fluid densities (p,z and p,)
below lambda transition under the saturated vapor pressure.


as a function of temperature


































Figure 1-3: Photographs of stable vortex lines in rotating He II in a cylindrical bucket of 2
mm diameter to depth 25 mm placed at the rotation axis of a rotating dilution refrigerator
at 100 mK(.


A B









1 2 3 4
Amrplitudelrad

Figure 1-4: Damping on a sphere oscillating in liquid helium at 2.149 K( with a period of
18.5 s [7].





















Ici


Figure 1-5: Donnelly-Glaberson instability of a
component of the normal fluid velocity parallel
[11].


quantized vortex line occurs if the
to the vortex line exceeds a critical value


k (Airbinry U~n rdr


Figure 1-6: Energy spectrum for homogeneous and isotropic turbulence [11]. (k,:
K~olmogorov wavenumber, where viscous dissipation becomes significant; kc 2Z, d: the
size of th~e con~ta~iner; ke(t) 7, -e,(t): eddy length? scale) [11].


\; "i~
(a)














_ r


d' '~'


IU
^il
g~i-ir iix
1111
YR L~PIIICE B~NC]


I UU
Test Thermistor 1

100
Test Ther-
-200 Illistor 3

-300
'Test Tiermistor 2
-400

-500

-600
10 100 101
Temperature (mK)


cONDUCTION BAND~


~---- --


- 6


6 DYONCS


~-t~f


Eg


0 a-CEPTORs


Figure 1-7. Electrical conduction mechanisms in semiconductors [18].


Temperature (mK)


Figure 1-8: Calibration plots for three test thermistors developed for calorimetry. (a)
Resistance versus temperature. (b) Sensitivity versus temperature.


~t~,~8~b~~


I
I
I
I
I
I
--- ---














Thermometer


Test


Port #1






Part #2

Part #3
thread rod

Part
circui-t aboard


0.75 I


0.5.:






f 1,0750
1.5500


,l-ier, silver powder,
thlcknessi 0.1250'


(b) tl-roucgh on 1.3120* ECD
(2) 4-40 toppFed the ougjh Top V/Eewn
( jack screws) on 2 21?0' BCD

00 6685' capi ary he e 00.125' clear hole through on
-thrmagn~~ on 0,750' BC 0,75C' BCD C(t:li fit -to par-t #6)
Ut~~ 6-32 tappeob alepth7 0.12t5"
~ r --- n000' BCD
DO :100 thermometer hle 2 -8 42pe, eph 012'
dept: a7:.i', n I loi~i:n on 0 700" BCD under surfaee






Figure 1-9: Design for the thermometer test cell. (a) Overview. (b) Top view of cap.


Cell










CHAPTER 2
SHIELDED SITPERCONDITCTING LINEAR 1\1TOR

2.1 Introduction

Grid turbulence in the classical fluid produces homogeneous isotropic turbulence (e.g.,

Fig. 2-1 [27]), which is the simplest case among the complex nonlinear dynamics systems.

In order to compare with this classical case, we intend to produce homogeneous isotropic

quantum turbulence (HIQT) in liquid helium II below 1 K(.

In order to produce HIQT, we have designed and built the shielded superconducting

linear motor for our towed-grid turbulence experiments. First of all, we build a model,

shown in Fig. 2-2 [28, 29]: a single superconducting solenoid motor with an armature

moving through its center. This light and hollow insulating armature is constructed of

:3 phenolic tubes separated by two hollow cylindrical niohium cans placed some distance

apart, with the turbulence-producing grid attached to one end. The requirements and

advantages of our motor system are as follows:

1. The superconducting shield can avoid eddy current heating in the cell walls.

2. The superconducting solenoid can avoid Joule heating front the solenoid.

:3. With an appropriate current pulse, the grid can he efficiently accelerated and
decelerated front 0 to 1.0 nt/s within 1 nin. And the grid can he driven at a nearly
constant speed 1 nt/s for 10 nin, producing homogeneous isotropic turbulence within
20 ms.

In the resulting grid motion in our simulator, the grid would be accelerated front 0

to 1 nt/s in 1 nin. Then it travels at almost constant speed, 1 nt/s, for 10 nin. Then

the grid would rapidly decelerate to cease within 1 nin when the third pulse is applied.

Then we put our simulation results into practice. We build our test cell guided by the

simulation. In our test cell, we have one superconducting solenoid driving the armature to

move through its center, with a grid attached at the end. This light insulating armature

is constructed of :3 phenolic tubes separated by two hollow cylindrical niohium cans

placed exactly 26 nin apart, with the grid attached to one end. A conducting section










on the armature, composed of one of the Nb cylinders and silver paint coating part of

the phenolic rod, is inside a closely fitting capacitor made of two semi-cylindrical copper

sheets. This capacitor, coupled to a bridge circuit, measures the armature position.

Our unshielded test motor system has been tested very successfully. We apply the

pulses to the superconducting motor to drive the armature inside the solenoid: two sine

pulses with DC level in between, followed by small DC level for 250 ms. In the resulting

velocity versus time curve, we already can accelerate the armature to 1.1 m/s. The

velocity remains zero for about 250 ms when the armature is held on the top. In the

velocity versus position curve, we can see that the armature moves at almost constant

velocity 1 m/s + 0.1 m/s for at least 8 mm.

We also improve this design in our superconductor shielded superconducting

motor system, discussed in the next chapter. One of the important task is to build

the superconductor shield on the interior of the cell. The details will also be discussed in

the next chapter.
In chapter four, we would like to discuss about the accomplishment of the followings:

* Run the superconducting shielded linear motor system at 520 mK( in the dilution
refrigerator.

* Explore the energy spectra and the decay mechanism of homogeneous isotropic
quantum turbulence at 520 mK(.

2.2 Models of the Shielded and Unshielded Superconducting Motor

The motor is shown schematically in Fig. 2-2, made of three coaxial parts: one

superconducting shield with the radius b and length h, one superconducting solenoid with

the inner radius r and length 1, and a light insulating rod with two niohium cylinders

attached and separated by distance AS. The grid is attached to the end of the rod and

hence is pushed by it.

Suppose the diameter of the superconducting wire for the solenoid is d, and the

total number of turns, NV. We can estimate approximately the self-inductance of such a









solenoid:
4 NBA N1oNVx22r2
L 4x,2 x 10-10 x (2-1)
I I ll 1

(in H. ,:, a and all the lengths are in mm), where # is the total magnetic flux flowing

across the solenoid, I is the current flowingf through the superconducting wire of the

solenoid, B is the approximate magnetic field at the center of the solenoid, A is the

average cross section area of the solenoid, and r is the average radius of the solenoid:

NVd2
F = r +(2-2)


The z component of the magnetic field at the position (p, 8, z) near the solenoid is

derived as the following by generalizing the problem in Griffiths [30]:

PolR~ 27 pCOSeCOs@ painesin~
B,(l : -) C~4~ol4x [P2 + 2 _( 0 XO2 2pRcosHcos~ 2pRainesin@]3/2
(2-3)



Bz(p, 8, z) = s(p, 8, z)I, (2-4)

where
2n 1 1 2m 1
R = r+( )d, zo = + ( )d. (2-5)
2 2 2

For the magnetic field distribution inside the solenoid enclosed by the superconducting

shield, we cite Eq. 12 in Sumner [31]:

41-oN1Vr S(r lo(kp) kl
Bz (p, z)=-Sxkb ik) sin( 2)cos(kz) = x(p, z)I, (2-6)

where

Sl(kst) = li(ks)Kl(kt) Kl(ks)II(kt), (2-7)

II and K1 are the Bessel functions of the first and the second kind, and

(m +1/2) x
k = (2-8)










with m zero or a positive integer.

Suppose the niobium is a perfect diamagnet, then the magnetic force experienced by

the niobium cylinder #1 with the center of mass at z = ze along the axis of the solenoid

(the center of the solenoid is defined as z = 0 position):

/zi+11/2 2 o
ma z) [771 z ,(p, 0, z)]z= z ,[M z p ,z,-d e. (2-9)


Since M~ = -~i for a perfect diamagnet [32], the magnetic force is proportional to the

gradient of the magnetic field square:

zi Izi+11/2 r2~ yrol ao
0 zi-11/2 0 0

where m, M, I-o, rol, 11 are the magnetic moment, the magnetization, the permeability

at vacuum, the outer radius and the length of Nb cylinder #1, respectively. For the

purpose of computer simulations, the practical formula for the numerical analysis and the

corresponding C code are in the Appendix A and B, respectively.

In addition to the magnetic force, the gravity force E;',, is also considered, therefore,

according to the second Newton's law, the net force experienced by the whole system with

mass M.1 ... produce the acceleration, a: E;ma,, + E;,,,=M ...Eo





The system includes the insulating rod, the two niobium cylinder cans and the grid.

Because Fma,(ze) cm I2, We CR1 Write Fmayxi) ma f,(i 2. Suppose the second niobium

cylinder is some distance, AS, below the first one, then the current required to reach the

objective acceleration would be:

(a +9.8)i1 M.,
I = (2-12)
f mas ( z ) +f m,,( z A S)

(unit in Amp~ere).









By applying a properly shaped current pulse to the solenoid, the magnetic interaction

with the niobium cylinders produce the required motive forces, with the upper cylinder

accounting for the initial acceleration and the lower cylinder the deceleration of the grid.

In addition, we have several options for the mathematical form of the relationship

between the destined terminal velocity v (in m/s) and the traveling distance S t(in mm)

during the period of time of the acceleration a (in m/s2), Such as linear, square and

sine function. For each small traveling distance, dz (mm), the accumulated velocity,

acceleration and time at the ith increment, I (m/s), ai (m/s2), and ti (ms):

Linear acceleration: v oc S1


S= s dz (m/s) (2-13)
S1



as=(i-1)v)2dz x 103 (l2) (2-14)
2 S1


vdz
ti = ti_l + ( ) x 103 (ms) (2-15)
aS1

Square acceleration: vU oc S,


= S(idz)2 (m/s) (2-16)



as= 2i _2 + i-)()dz3 x 103 (l2) (2-17)



ti = ti-1 + (2i 1) ( aS) x 103 (ms) (2-18)

Sine function acceleration: v oc sin(S1)

.x dz
=vain(i- ) (m/s) (2-19)
2 S1











ai = -sin2(2 si2[(i 1) ]} x 103 (l2) (2-20)
2dz 2 S1 2 S1



is 41 si(2 -sin[(i 1) ]} x 103 (ms) (2-21)
a 2 S1 2 S

For the deceleration, we simply have the linear mathematical form. Suppose the

traveling distance would be S3 ill mm). FOr each small traveling distance, dz (mm), the

resultant velocity, acceleration and time at the jth displacement interval, vj (m/s), aj

(m/s2), and tj (ms):



vy = v j S3dz (m/s) (2-22)



as = (j )()2dz x 103 -x 103 (l2) (2-23)


vdz
tj = tj_, (aS3 x 103 (ms) (2-24)

2.3 Simulation Results and Discussions

2.3.1 Simulation Analysis

In order to find the best motor design, we have developed a LabView simulator

program, in which the various design parameters may be varied while maintaining the

required motion of the grid. Optimization in general yielded the lowest voltage and

current applied to the solenoid, hence minimizing the magnetic field produced. This

process yielded the optimized parameters of Table 2-1 (units of lengths in mm).

Fig. 2-3(a) shows the required current versus time curves for the unshielded and

superconducting shielded solenoid with a sine function acceleration (velocity is the sine

function of the niobium position). The curves show three peaks: the first and the third

peaks are to accelerate and decelerate the niobium cylinders. The middle peak is due to

the almost balanced magnetic forces on the two niobium cylinders at that position since










each niobium cylinder is almost equidistant from the ends of the solenoid. We can run the

simulator removing the middle peak (o- symbols) in Fig. 2-3(a), and see that the velocity

versus position curve in Fig. 2-3(b) (again 0-symbols) is virtually unaffected. The droop

in velocity after z = 10 mm is unavoidable since all forces, including gravity, are directed

downward for the rest of the stroke.

Table 2-1: Optimal parameters for the motor design.
Parameter Description Value
d the diameter of the superconducting wire 0.2
NV the number of turns of the solenoid 1500
r the inner radius of the solenoid 5.0
1 the length of the solenoid 10.0
zo the initial position of Nb cylinder #1 3.5
rol the outer radius of Nb cylinder #1 4.0
To2 the outer radius of Nb cylinder #2 4.0
11 the length of Nb cylinder #1 9.0
12 the length of Nb cylinder #2 11.0
as the distance between the center of masses of two Nb cylinders 21.0
& the length of the superconducting shield 50.0
b the radius of the superconducting shield 20.0
Ml. .,.. the mass of the whole system 2.0g
(the armature, including the insulating rod,
two Nb cans and a grid)


Note that the superconducting shield requires a slightly higher current (0.14 A

greater) to produce the same motion. The effect is small because the shield is significantly

larger than the solenoid.

In the velocity versus position curves in Fig. 2-3(b), we see that the motion is as

expected. The niobium cylinders travel at almost constant speed, 1 m/s, for 10 mm, but

start to slow down when the middle current peak occurs, which is quite reasonable. After

the first niobium passes z = 10 mm, the second niobium is closer to the solenoid and

experiences a stronger magnetic force in the opposite direction, resulting in the slight

deceleration. Therefore, applying the third pulse produces the desired deceleration to

rapidly stop the grid. The evaluation results prove that our superconducting linear motor

is a very feasible design.









For comparison, the required current versus time curves for the superconducting

shielded solenoid with a linear and square function accelerations and the corresponding

velocity versus position curves are shown in Fig. 2-4, Fig. 2-5. It requires a slightly

higher current, 0.149 A and 0.8 A greater, for the superconducting shielded solenoid to

produce the linear and square acceleration motion. Therefore, sine function acceleration is

a more efficient way that takes the least current among the three options.

The shots of all the simulation programs are in Appendix C.

2.3.2 Critical Magnetic Fields for Niobium

Niobium (Nb) belongs to transition metal (group IIIB). It has atomic number 41,

atomic weight 92.9 g/mol, density 8.58 g/cm3, melting point 2741 K( and remain solid

at room temperature. Niobium is type II superconductor, which forms a vortex stable

state mixed with normal and superconducting regions over the range between the lower

and upper critical magnetic field strengths, H,1 and Hc2, for partial penetration of the

magnetic flux, called the incomplete Meissner effect. Below the lower critical magnetic

field, it exhibits the same phenomenon as type I superconductor to expel the whole

magnetic flux, i.e. complete Meissner effect.

The theoretical estimates of the lower and upper critical magnetic fields for the type

II superconductors are [32]:

Hei~ (2-25)



Hc2 2 (2-26)

where @o is the superconducting flux quantum, called a flux~oid or flux~on:


Go = 2xh~c/2e T2 2.0678 x 10- (2-27)


(in Gauss -cm2). A and ( are the penetration depth and the coherence length, respectively.

For example, the niobium has the penetration depth A at absolute zero estimated from

the measurements to be 470 P1 [33], and the superconducting coherence length ( is 11 nm









[34]. So H, a 2980 Gass Hc 54397 G~1S, H ~ causs. The penetration depth and the coherence

length are actually temperature dependent.

Experimentally, H,1 and Hc2 ValueS have been measured to be dependent on the
purity and the residual resistivity ratios [RRR = R30 4 2N Of Il-bu [35,- 36], the field

orientation [37], and the temperature [35-38]. R300 iS the resistance measured at room

temperature.I~ R4N2 is the normal state electrical resistivity measured at 4.2 K( in a field

of 0.6 T or higher after the magnetization data measurement. The higher purity the

niobium, the higher the RRR values, and the lower the Hc2 ValueS. For example, in Fig.

2-6 [35], at the temperature T s 1.5 K(, the upper critical field Hc2 m 12.8 K( Gauss, 7.3K(

Gauss and 3.5 K( Gauss for RRR of the niobium samples = 3.1, 8.8 and 505. For the

temperature dependence, the data has shown the thermodynamic critical field, He, with

the form He(T)= 1993[1 (g)2] (in Gauss) for RRR = 16300 [36], the lower critical field,

HeI(T) = 1735[1 (g)2.13] (in Gauss) for RRR = 1400 [36], anld th~e upper critical field

Hc2 = 4100- (in Gauss) for RRiR 300 [38].
In addition, the critical surface field, Hc3(T) = 1.695Hc2(T), has been predicted by

Saint-James and de Gennes [39] and measured [36, 38] from the onset of zero resistivity

and the AC susceptibility, which is also temperature, purity, RRR value and surface of

the samples dependent. Between Hc2 and Hc3, Superconductivity and surface supercurrent

appear in the form of a surface sheath with a thickness about ((T) on surfaces parallel to

the applied magnetic field.

Experimentally what we would do is to make thin hollow niobium cylinders with

end caps. Below the surface critical field, the enhanced -IIn! II -1,- I11, magfnetization

makes the thin surface sheath still superconducting while the bulk is in normal state. By

making the cylinder hollow we would enlarge the total volume of niobium to exclude more

magnetic flux with much smaller mass and reducing the total mass of the whole system

make the required driving magnetic force much less. Even when the field is beyond the









upper critical magnetic field and below the critical surface field, the surface sheath can

still form a quasi-perfect diamagnet.

2.3.3 Required Voltage Input for the Solenoid

If the resistance and the inductance of the solenoid are the only elements in the

circuit, and at the cryogenic temperature of 20 mK( the total resistance of the whole circuit

is R (Ohm), then the required voltage to supply the circuit would be:

dl(t)
V(t) = L + I(t)R (2-28)

In numerical analysis, we use:

[I(t) I(t dt)]
V(t) = Lx+ I(t)R (2-29)


(in volts), where dt is the infinitesimal time interval between the time (t-dt) and t when

the current flowing through the solenoid are I(t-dt) and I(t), respectively. The LabView

simulation program calculates the required voltage input to the superconducting shielded

solenoid with sine, linear, and square function acceleration as shown in Fig. 2-7. Since the

current for the sine function acceleration has smooth transition with respect to time, it has

the better performance than the other two, i.e. lower required voltage. Still the voltage is

too huge to have the practical application in the laboratory.

In order to solve this problem, we need an additional capacitor with capacitance

C (p-F) in the circuit to form an LRC circuit. In an LRC circuit under the sinusoidally
driven voltage, V = Voeiet, the K~irchhoff rule requires that the sum of the changes in

potential around the circuit must be zero, so

dl Q dl 1
V(t) = L + IR + L + IR + Id = ii Voeiwt, (2-30)
dt C dtC

where Q(Coulomb) is the total charge on the capacitor. The solution for the current would

bne:

I = Ioei(wt+ P, (2-31)









where
LcCVo Vo
lo =(2-32)
R29g2 (1- Lu2 @2+ (wL 1/wC)2~
and
1 LCw2 1 wL
p tn-() = tan- ( )(2-33)
RwC RwC R

At resonance, for w =iO = Wo 7= e"" [40].

We can calculate the required voltage input to the solenoid from the calculated

current curve:
dl(t) 1 Pt
V(t) = L + I(t)R + I(t)dt (2-34)
dt C o
In the way of numerical analysis:

[I(t) I(t dt)] t
V(t) = L x dt + I(t)R + N (),(2-35)
i= 1

where
1 1
ti = -( )(2-36)

Another alternative way of sending the pulses to the solenoid is to input the perfect

sinusoidal shape current pulse modified according to the previous calculated current

profile, which would still give the expected motion for the armature of the motor, as

shown in Fig. 2-8.

Suppose we supply the current I(t) = Iosinwt. If we only consider the solenoid with

inductance, L, and the resistance for the whole circuit, R, then the required input voltage

would be
dl(t)
V(t) = L +I(t)R = L7,,... ..!: + IoRaincut. (2-37)

For Io = 2.2 A, R = 0.1 R, w = 1200 mad/s, L = 0.056 H, the same parameters as those

in Fig. 2-8, then the required voltage input to the solenoid versus time would be like Fig.

2-9. Some voltage pulses as large as 150 V seem to be too large to be applicable.

Experimentally, we might need C = 11.2 pF to tune the circuit on resonance. At

resonancei iWe. 11"' = o te = ~eime = ~. During the constant current period










for 10 ms, if the R = 0.1 R, L = 0.056 H, and we cut off the voltage, exclude and ground

the capacitor, then the current flowing through the solenoid would decay very negligibly:

I = Ioe-Rt/L o -Rt/L = -1.786t. It WOuld take 560 ms to decay to I = Ioe-l = 0.368Io-

The key is to find out the appropriate micro-electronics analog switches (IRF 7311, PVI

1050N) which can be controlled by the TTL signals from the computer to include or

exclude and discharge the capacitor simultaneously with the input pulse for the solenoid.

The schematic drawing for such a specially designed circuit is shown in Fig. 2-10. The

detailed circuitry for the whole experimental setup will be introduced very shortly.

2.4 Unshielded Motor Testing Experiments

Now it is time to put our simulation results into practice. Here comes the unshielded

motor testing mission. The dimension parameters of our unshielded motor system are

listed in Table 2-2. With those parameters, we run the LabView simulation programs to

find out the required current profile for the sine function acceleration, and then we modify

this current profile into two perfect sinusoidal pulses along with the constant DC current

in between. Fig. 2-11 shows the machine drawings and the photos for our unshielded

motor test cell. As we can see from the drawings that we build the capacitive position

sensor in order to monitor the motion of the armature.

Table 2-2: Parameters for the unshielded test motor. Refer to Table 2-1.
Parameter Value
d 0.1905
NV 1246
r 5.0
1 10.0
zo 6.43
rol 3.175
To2 3.175
11 13.0
12 15.0
as 25.9
M. ... .2.0









2.4.1 Capacitance Bridge and Lock-in Amplifier for Monitoring Armature
Motion

The primary circuitry and the experimental setup for the motor test cell are shown

in Fig. 2-12. We send the negative pulse from the computer (analog output 1: DAC 0)

through our current amplifier. The current amplifier amplifies and inverts the pulse. Then

the pulse is fed into the switch box, where the switches can be controlled by the TTL

signals from the computer (analog output 2: DAC 1). Eventually the pulse is sent to

the solenoid in the cryostat at helium temperature, 4.2 K(. We also use the computer to

monitor the pulse sent to the solenoid (analog input 2: ACH 2), and simultaneously the

capacitive position sensor is measuring the position of the armature (analog input 1: ACH

1).

The sensor is connected to the lock-in amplifier and capacitance bridge (GR1616).

We set the driving rms voltage 1 V, frequency 1.01 kHz from the reference sine waveform

output of the lock-in amplifier, and the time constant 100 ms. At room temperature the

capacitance bridge reads the capacitance of the capacitive position sensor 3.32 pF (the

magnitude, R, of the output on the di pl w~ of the lock-in amplifier reads minimum 39

p-V) when the armature is at rest and 3.00 pF (the magnitude, R, of the output on the

display of the lock-in amplifier reads minimum 108 pV) when the armature moves all

the way up hitting the brass plate 12 mm above in the air. By setting the sensitivity

of the lock-in amplifier 1 mV and the standard capacitance of the capacitance bridge

3.32 pF, the channel 1 output of the lock-in amplifier reads 0.44 V (R = 44.7 pV) as the

armature sitting at rest, while as the armature moves up 12 mm the channel 1 output

reads 2.54 V (R = 255 pV). With such significant voltage change, 2.10 V, which can be

easily recorded by the LabView data acquisition program, we can convert the voltage

shift into the position, or even the velocity of the armature motion. When cooling down

to liquid helium temperature, 4.2 K(, some conditions might change. We need to change

some of the settings of the lock-in amplifier to record the voltage shift from channel 1










output in a timely manner: the driving rms voltage 1 V, frequency :3.01 kHz from the

reference sine waveform output; time constant 1 ms (18 dB); sensitivity 1 mV. For the

capacitance bridge, we set the standard capacitance to be :3.00 pF while the channel 1

output of the lock-in amplifier reads 0.55:3 V (R = 0.28 mV) as the suck-stick probe is in

the liquid helium dewar and the armature is surely sitting all the way down. As long as

we adjust the standard capacitance to :3.30 pF, then the channel 1 output reads 2.38 V.

Therefore, we could expect some dramatic voltage increase around 1.8 V to be detected if

the armature moves all the way up to hit the brass plate.

It turns out that we do not need the computer controlled switch box, as the gray

region in the schematic, because current amplifier filters out the back emfs from solenoid.

The distorted pulse with spikes sent to the solenoid will be produced without both current

amplifier and the switch box, as mentioned previously. We turned up the current sent

to the solenoid little by little by turning up the gain of the current amplifier gradually.

Eventually, when the peak current reached the expected 2.86 A, we heard the sound from

the dewar like the armature hitting the brass plate 12 mm above it. What a success! Very

disappointedly, the capacitance bridge cannot respond fast enough to have any observable

change. Later on we realized from some experimental tests that the capacitance bridge

takes at least 100 ms to respond. It could be the inductance of the ratio transformer

slowing down the response time. A better method to measure quantitatively the position

of the armature of the superconducting motor system is necessary. So we have 555

oscillator circuit built and connect to the sensor in parallel and the LahView program with

the counter function counting the frequency change with time is developed as the second

attempt for the alternative measurement technique.

2.4.2 The 555 Oscillator for Monitoring Armature Motion

The improved circuitry and the experimental setup for the motor test cell is shown

in Fig. 2-14. Now instead of using the capacitance bridge and the lock-in amplifier,

we have the 555 oscillator circuit box, which can oscillate and output TTL pulses with










any destined frequency, which is determined by the resistances of the resistors in the

circuit and the capacitance of the capacitors in the circuit along with the capacitive

position sensor connected in parallel. The output frequency from 555 oscillator is inversely

proportional to the total capacitance of the whole circuit, and we would like the frequency

measurement in the LahView counter program to have the time resolution as small as

possible, 11- 1 ms; therefore, we set the gate width 0.5 ms. With the gate width 0.5 ms,

the uncertainty of the rising or falling edges counting during this short period of time

would be + 1, which means the frequency calculation would have the uncertainty +

2000 Hz. Hence, in order to have higher frequency resolution, we tune the 555 oscillator

as high as possible, ;?-- ~ 1 1\Hz. The capacitance change of the capacitive position

sensor for the armature at rest and moving up for 12 mm is roughly about 0.5 pF,

way too small if compared with all the other cables in the circuit, 11- 200 p-FF, which

gives the capacitance change the same as the frequency change about 0.25 So the

frequency change would be expected to be 2.5 kHz, not too much above our frequency

measurement uncertainty. We also have tried to decrease the gap between the sensor

and the niohium can to around half, but it still doesn't improve the capacitance change

too much! Since it takes time for the SuhVIs in the LahView to execute, about 1 ms for

reading one frequency measurement, therefore about 1.5 ms for one frequency data point,

and the frequency uncertainty is so high, we consider to run the experiments for dozens

to one hundred times and then do data averaging analysis. The signal-to-noise ratio is

proportional to the square root of the number of data sets for averaging. By averaging

100 runs of data sets, we can increase the signal-to-noise ratio by ten times, reducing the

time resolution down to 0.05 ms, and the frequency resolution to 200 Hz. We have the

LahView program developed to do continuous data acquisition and the averis?~ine-: however,

averaging one hundred times of experimental data is too much burden for our tiny, fragile,

and resistless armature. So we had to give up this way of measurement method. The

testing results for our unshielded motor test cell to monitor the motion of the armature










using the 555 oscillator circuit and the LabView counter program for the frequency

measurement show no frequency change with time within the frequency uncertainty.

2.4.3 The Q-meter for Monitoring Armature Motion

We also spent a week to build the Q-meter, i.e. resonant RLC circuits driven by a

sinusoidal voltage, like in Fig. 2-15 (a). Our theoretical simulation programs predict this

method feasible, for the capacitance change accompanying with noticeable current change,

which gives some noticeable voltage change across the resistor, capacitor or the inductor,

especially for higher Q (quality factor). Since the current in an RLC circuit in series is


I(t) =cos(wt + cp), (2-38)


where
1 wL
tany = (2-39)
RwC R '
the maximum of the current versus frequency graph is at w = w0 = 1/& onreoane

The higher the Q, the sharper the peak will be due to the full width at half maximum

(FWHM) of I v.s. w curve, wo/Q, where Q = woL/R. In order to have more sensitive

measurement, we need to design the circuit with higher Q.

The testing circuit is shown in Fig. 2-15 (b). We tried to tune the frequency of the

reference output (sine wave) from lock-in amplifier to get the maximum voltage across any

one element of the circuit, which should fall at resonance of the RLC circuit. Therefore,

with a little capacitance change from the capacitive position sensor, the circuit will be off

resonance, then we are supposed to see the dramatic current decrease, therefore, dramatic

voltage change. Unfortunately, we didn't observe any noticeable voltage change (all less

than 2 .~) either across the resistor, capacitor or the inductor. We also exchanged the

position of the resistor and inductor, or rearrange the circuit to make the capacitors and

inductor in parallel. But none of the above worked well to show any large enough voltage

shift. Hence, we need to give up this approach.










2.4.4 The AC Bridge Circuits for Monitoring the Motion of the Armature

Eventually we find the excellent way to monitor the motion of the armature, the AC

bridge circuits.

We have used the capacitance bridge along with the lock-in amplifier to monitor the

motion of the armature before by measuring the capacitance change of the capacitive

position sensor. The output voltage shift from lock-in amplifier when moving the armature

up and down could be as large as 2 V or more, which gives enough sensitivity. The only

problem is to overcome the slow response time, 100 ms, of the ratio transformer. Fig. 2-16

shows one of the simple AC bridge circuits we built. As what we had expected, the AC

bridge circuit can respond as fast as less than a few microseconds without any delay. The

followingfs are the detailed descriptions about the experimental apparatus and setups.

The capacitive position sensor is composed of two copper semi-cylindrical sheets along

with the second niobium can. In the magnetic field, the slits of the copper cylindrical

sensor could prevent the eddy current and therefore the heat dissipation from being

produced. We use the stronger superconducting wire for the leads of the sensor and the

more flexible insulated fiber tube to protect the leads. Epoxy is used to glue the copper

sensor to the capacitor frame made of phenolic. Fig. 2-17 shows the circuit connection to

measure the capacitance of the capacitive sensor with the capacitance bridge. At room

temperature the capacitance bridge reads the capacitance of the capacitive position sensor

2.11 pF (the magnitude, R, of the output on the display of the lock-in amplifier reads

minimum 138.9 pV) when the armature is sitting all the way down, and 1.59 pF (the

magnitude, R, of the output on the di pl w~ of the lock-in amplifier reads minimum 153.4

p-V) when the armature moves all the way up hitting the brass plate 12 mm above in the

air. The capacitance difference is as small as 0.52 pF.

Now we connect the position sensor to the AC bridge circuit box. The circuit diagram

is shown in Fig. 2-18 (a). Instead of using the new digital lock-in amplifier (SR830), we

use the antique analog lock-in amplifier (PAR119) to get rid of the digitized problem










of repetitive ripples with periods about :3 ms superimposed to the signals. Here are the

settings for the analog lock-in amplifier: signal channel- frequency 92.25 kHz, mode

handpass; sensitivity 500 pV. We tune the variable capacitor to about :3.5 pF, and the

channel 1 output from the lock-in amplifier becomes 0.116 V when the armature sits all

the way down.

We also carefully calibrate and zero the offset of the current amplifier to within a few

nano-amperes. After adjusting the brass plate above the top of armature to be 21.8 mm,

the armature is expected to have enough space to move up and pass the second net zero

magnetic force position to be held there with some DC level. Fig. 2-19 (a) demonstrates

the theoretical calculation of the magnetic force per unit of current square experienced

by the armature inside the superconducting solenoid as a function of the position, z,

defined as the vertical distance between the center of the first niohium can and the middle

of the solenoid. Our plan is to start applying the acceleration pulse at xo = 6.4:3 mm,

around the peak position of the positive magnetic force zone, then the velocity of the

armature carries through the first net zero magnetic force position at about z = 12.5 mm;

afterwards it would experience some negative magnetic force until we apply the second

pulse for deceleration at the peak, z = 18.5 mm. If our second brake pulse didn't totally

stop the armature, then the residual velocity would be able to bring the armature further

up until it pass the second net zero magnetic force position, z = 26 mm. After this point,

the armature would experience positive magnetic force again and some small DC level

pulse would be enough to cancel out the downward gravity force and the armature would

be able to float there still for however long we need.

The room temperature calibration curve for the conversion of the voltage output from

channel 1 of the lock-in amplifier into the position, z, of the armature, precisely measured

by the depth micrometer gauge, is demonstrated in Fig. 2-19 (b). The probe was set to

be vertical to be as close to the situations of the actual experiments as possible and the

variable capacitor was tuned to about :3.8 pF when calibrated. The dip for the first ~










3 mm has two possible sources. One is due to the deviation of the second niobium can

from the central vertical axis of the sensor, therefore, resulting in more or less smaller

readings. Usually the armature will be brought to the center by the magnetic force as soon

as the pulses are applied to the solenoid. The other reason is due to the longer sensor of

the copper sheets (about 17 mm) than the length of the second niobium can (15 mm) by

around 2 mm. The capacitance of a coaxial cylindrical capacitor is

27rEoL
C =(2-40)
In(b/a) '

where an inner cylindrical conductor of length L and radius a is surrounded by an outer

cylindrical conductor of the same length and radius b, and Eo is the permittivity of

free space. Hence, the capacitive position sensor is expected to have the capacitance

proportional to the overlapping length between the second niobium can and the copper

sensor. During z = 0 and 2 mm, the overlapping length is about the same, 15 mm, so the

capacitance is supposed to have no change. However, due to the edge or boundary effect,

we see the dip occurs at z = 0 ~ 3 mm in the calibration curve and in the data we took

for the armature motion. In the data on~ ll-k- the conversion of signals into positions will

be modified accordingly.

In order to monitor the actual pulses sent to the superconducting solenoid, we

measure the voltage drop across the 0.1 R resistor in series with the solenoid. The 0.1 R

resistor box is connected to the output of the current amplifier (BOP) directly via banana

connectors and the other side is connected to the superconducting solenoid via the 12

pin connector on the top of the probe directly. The circuit diagram is shown in Fig. 2-18

(b). Due to the almost purely inductive (zero resistance) motor circuit, the pulses fed to

the solenoid are somewhat distorted and as the gain of the current amplifier turns up to

exceed 3 A of current output, the spikes with the opposite polarity appear, resulting from

the back emf due to the fast changing current. Therefore, we put cross diodes across the

solenoid in parallel to prevent this occurred.










Let's look at some of the data and testing results. For instance, in Fig. 2-20 the

motion of the armature of the superconducting motor is clearly examined. The pulse

profile sent to the solenoid is distorted due to the inductive behavior of the superconductor

solenoid. As the gain of the current amplifier is turned up, the higher the gain the more

the current would be fed to the solenoid. Here the first current pulse peak height reaches

around -1.8 A, then decays a little bit, followed hv the second peak as much as -2.369 A

then decaying to zero gradually. The current amplifier not only amplifies the pulses, but

also inverts the pulses output; therefore, we have the pulse profile with negative polarity.

The capacitive position sensor responds to the motion of the armature as the armature

moves up until out of the reach of the sensor, then drops back down by gravity force as

a function of time. The armature accelerates to 0.6 m/s at z = 5 mm within about 15

ms (average estimate acceleration 40 m/s2), then decelerates to zero at z = 15 mm in

40 ms (average estimate deceleration -15 m/s2). Afterwards, due to the gravity force

it drops back down and reaches the velocity as fast as 0.4 m/s at z = 4 mm within 70

ms (average estimate deceleration -5.7 m/s2). The armature probably moves further up

during the time 60 ms and 100 ms, but our sensor could not measure any changes because

the armature at that moment is already out of the detectable range. Fig. 2-20 (d) is the

original current profile sending to the current amplifier from the analog output DAC 0 of

our LahView data acquisition program.

If we compare with the theoretical prediction from our simulator with the same

distorted current profile as in Fig. 2-20 (a), then the predicted armature motion is like in

Fig. 2-21. The armature is supposed to reach the velocity as fast as more than 1.0 m/s,

but it is also predicted to move as far as approximately 10.5 mm only. The difference

between the theoretical prediction and the actual motion is still under exploration.

Now if we get rid of the second brake pulse and increase the DC level following the

first acceleration pulse, how would it affect the armature motion?










If we apply the acceleration pulse followed by 1.5 V DC pulse for 112.5 ms, Fig.

2-22 (d), then the current pulse profile in the solenoid actually looks like a smooth

transition from zero to some saturated DC level and the capacitive position sensor

responds correspondingly by oscillating around the equilibrium position about 0.9 V

and damping with time, Fig. 2-22 (a). From Fig. 2-22 (a) and (b), the armature seems

to be accelerated to almost 1.0 m/s within 10 ms and 4.5 mm, then it slows down until

stops at the highest point, about 6.8 mm, and drops back down to as fast as about -0.47

m/s, as far as the position about 4.3 mm. Afterwards, it oscillates back and forth around

the equilibrium position about 5.2 mm, like a harmonic oscillator. Due to the viscosity

and impedance of the liquid helium and the friction on the hearings contacting with the

armature, the oscillation motion also damps in 80 ms, then the armature -r li- stationary.

As soon as the DC level is turned off, the armature drops back down to the original

position.

If we compare with the magnetic force versus position graph in Fig. 2-19 (a), this

kind of oscillation motion is quite understandable. We apply the acceleration pulse at the

maximum peak of the magnetic force zone, z ~ 6.5 mm. As soon the armature travels

passing through the zero magnetic force position, z = 12.5 mm, it experiences negfative

magnetic force, decelerating and pushing it back down. And then it experiences positive

magnetic force again helow 12.5 mm position, which will push it up. Such a cycle repeats

until it damps, then it will float still at the position where the gravity force and the

magnetic force all cancel out, resulting in zero net forces.

If we apply the acceleration pulse followed by 2.0 V DC pulse for 112.5 ms, Fig. 2-22

(d), then the current pulse profile in the solenoid actually looks like a rapid transition

from zero to the same saturated DC level and the capacitive position sensor responds

correspondingly by oscillating around the equilibrium position about 0.85 V and damping

quickly with time, Fig. 2-22 (a). From Fig. 2-22 (a) and (b), the armature seems to

be accelerated to almost 0.9 m/s within 10 ms and about 1.3 mm, then it keeps almost










constant speed until at the position about 4.7 mm it slows down to zero and stops at the

highest point, about 6.3 mm. Afterwards, it drops back down to as fast as about -0.3 m/s,

as far as the position about 4.6 mm. It oscillates back and forth around the equilibrium

position about 5 mm, like a harmonic oscillator. Due to the viscosity and impedance

of the liquid helium and the friction on the hearings contacting with the armature, the

oscillation motion damps very soon in :30 ms, then the armature cr li- stationary. As soon

as the DC level is turned off, the armature drops back down to the original position.

With higher DC level, the armature can he accelerated to higher velocity followed

by more rapidly damped oscillation afterwards because the actual current pulse in the

solenoid is ramped faster up to the saturated current of BOP. Since we really need to

have the armature travel with constant speed for at least 10 mm, this result with our

current design cannot satisfy us. So the armature must he modified. With the distance

between the centers of the two niohium cans about :35 mm, we could just simply apply one

sine wave pulse followed by a DC pulse, then we are able to accelerate the armature and

hold it around 10 mm position for as long as we need. The drawback of this design is the

unavoidable oscillation which might more or less affect the turbulence just produced and

the DC pulse to hold the armature must he at least a few amperes.

Now if we reverse the polarity of the input pulse going to the current amplifier,

how would it affect the armature motion? We apply the original current pulse to the

superconducting solenoid, Fig. 2-24 (d), then the measured pulse profile is positive and

the inductive effects are not so significant, Fig. 2-24 (a). The actual pulse peaks saturate

up to 7.5 A, more than what we have expected. The armature is accelerated to about 0.7

m/s within about 10 ms and 1.5 mm and up to 0.85 m/s at 4.2 mm position and at the

time 15 ms. The armature goes up as high as 8.7 mm, and then drops back down with

the velocity as high as 0.2 m/s. The armature seems to get stumbled for a little bit while

at about 4.8 mm position, where is estimated to be the zero net magnetic forces position.

We figure out that the DC level pulse as high as 2 A could have decelerated the armature










rapidly beyond the position z = 5 nin before the brake pulse. That is the reason why the

armature could not have gone farther and reached the expected height, 12 nin.

We lengthen the DC level to 25 nis while keeping the rest of the original current

profile the same, Fig. 2-25 (d), then the measured pulse profile fed to the superconducting

solenoid is positive and the inductive effects are not so significant as well, Fig. 2-24 (a).

The actual pulse peaks saturate up to 7.5 and 10.5 A, while we do keep the gain of the

current amplifier the same as in the previous case. The armature is accelerated to about

0.72 nt/s within about 10 nis and 1.2 nin and up to 0.9 nt/s at 4 nin position and at

the time 12.5 ms. It seems to get stumbled and slow down between 4 and 6 nin position,

but speeds up to 0.95 nt/s afterwards. The armature goes up as high as 15.2 nin, beyond

which the armature is actually out of the detectable range of the capacitive position

sensor. It shows zero velocity during the time 30 nis and 110 nis, when the armature is

either out of the reach of the sensor or it is floating above the second zero net magnetic

position, z = 26 nin in Fig. 2-19 (a), or 19.5 nin equivalently in Fig. 2-25 (c). It then

drops back down with the pretty constant velocity, 0.15 nt/s, all the way down. The DC

level pulse as high as 2A is long enough not to decelerate the armature in time, so it can

move up beyond the detectable position 15.2 nin or higher before the brake pulse applies.

This provides us one of the important successful experiences of how we could achieve our

goal. The armature is proposed to be accelerated, travel for about 10 nin with almost

constant velocity, he decelerated then pass the second point of the zero net magnetic force

position and then he held above it for however long we need. The drawback of this design

is that we don't know if we are goingf to make the armature hit the top of the cell. The

DC pulse level should be carefully adjusted to be just as much as what we need and the

whole trajectory of the motion should be monitored. In order to be able to monitor the

whole trajectory of the motion of the armature, the capacitive position sensor needs to be

lengthened to appropriate length without modifying our current armature. Also we should

lower down the sensor for a few nin (2 ~ 3nin) to avoid the dip problem.










2.5 Improved Design of the Armature and the Test Cell

In order to be able to monitor the whole trajectory of the motion of the armature,

and solve the dip problem at the initial positions, we actually lengthen the copper sensor

to 27 mm and move it down by 7 mm. Conducive silver paint is also applied to the

armature for total conductive length 31 mm. A conducting section on the armature,

composed of one of the Nb cylinders and silver paint coating part of the phenolic rod, is

inside a closely fitting capacitor made of two semi-cylindrical copper sheets (see Fig. 2-26

(a) [41]). This capacitor, coupled to a bridge circuit, measures the armature position. The

total mass of armature is now 2.60 g. The grid, made of 0.125 mm thick spring steel with

1 mm square holes and 70 transparency, is attached to the lower end of the armature

(see Fig. 2-26 (b), (c) and Fig. 2-27 (a)). The corresponding Reynolds number for the

different velocities of the grid motion is shown Fig. 2-27 (b). Let's find out the Reynolds

number of our motor system in the liquid helium-4 bath at 4.2 K(. Reynolds number is

defined as the ratio of inertial force and viscosity force, or the velocity scale multiplied by

the length scale, divided by the kinetic viscosity:

U/L inertial force
Re (2-41)
V v .o~lu force

At motor testing temperature, 4.2 K( in liquid helium, grid Reynolds number:

U x 1.016 x 10-3m
Re 3.927 x 104 x U (2-42)
2.587 x 10-sm2 s-1

(in m/s). If we consider the size of the mesh of the grid as the length scale, velocity of the

grid motion as the velocity scale, then we have Reynolds number ranging from 4000 to

40,000 when the grid velocity varies from 0.1 to 1 m/s, above the onset of turbulence.

The room temperature calibration curve shows the monotonic increase in voltage as

moving up the armature without any dip. The entire assembly is mounted in a helium

test cell and cooled in a transport dewar cryostat [42]. The armature position sensor was









calibrated at 4.2 K( using a micrometer drive mounted at room temperature. Fig. 2-28 [41]

shows the calibration curve at liquid helium temperature, 4.2 K(.

In the electronics, we have two independent circuitry for the superconducting solenoid

and for the capacitive position sensor (Fig. 2-29). We drive with the properly shaped

current pulse profile determined by simulation, then amplify the current via the bipolar

power supply. The input pulse to the solenoid is monitored by measuring the voltage drop

across the 0.1 ohm resistor shunt. Two cross diodes are to protect the solenoid from the

spikes due to fast changing current. For the position sensor, we use the AC bridge circuit.

Two arms are 100 ohm resistors, and the other two are variable capacitor and the position

sensor. While the armature moves, the capacitance would change, giving different voltage

response. We convert the voltage into the position of the armature with our calibration

curve performed at 4.2 K(.

The experimental results are shown in Fig. 2-30 [41]. We apply the pulse profile: two

sinusoidal shape pulses followed by -0.1 V DC level for 250 ms (Fig. 2-30(e)). The solenoid

is protected by two crossed silicon diodes which have cut-in voltages of 4 V and 18 V at

4.2 K(. Therefore, reversing the polarity of the solenoid current produces an .I-i-mmetry

in response (Fig. 2-30(a) and Fig. 2-30(b)). In Fig. 2-30(c) and (d), with the current

pulse like Fig. 2-30(a) the armature is seen to accelerate to 0.8 m/s within 40 ms over

a distance of 9 mm (estimated average acceleration 20 m/s2). Then it undergoes a near

constant deceleration to zero velocity within 33 ms in another 12 mm (estimated average

deceleration 24 m/s2). It WaS held at the position of 22 mm for about 250 ms until the

current was turned off, it then dropped under gravity, as fast as 0.48 m/s in 75 ms over

10 mm position (estimated average acceleration 6.4 m/s2). As seen in Fig. 2-30(c) and

(d), with the current pulse in Fig. 2-30(b) the armature accelerates to 1.09 m/s within

20 ms over 9 mm (estimated average acceleration 55 m/s2). Then it slows down to zero

velocity within 38 ms over the remaining 12 mm of stroke (estimated average deceleration

29 m/s2). It WaS held at 22 mm for about 250 ms until the current was turned off, then










dropped under gravity as fast as 0.45 m/s in 49 ms over a movement of 12 mm (estimated

average acceleration 9.2 m/.s2). At the position between 4 and 14 mm, the velocity of the

armature was between 0.8 and 1.09 m/s, constant within about 15

2.6 Conclusion

We have written the code in LahView to calculate the current and voltage required

to supply the solenoid in the magnetically shielded superconducting linear motor

system to produce the homogeneous isotropic turbulence for our low temperature

quantum turbulence experiments. This kind of magnetic driven motor shielded by the

superconductor can avoid the heating due to eddy current on the wall resulting from the

changing magnetic field from the surroundings, the Joule heating from the solenoid with

current and the friction during the process when the grid is towed through the liquid

helium at 520 mK(. We also find the optimal parameters for our design demanding the

least voltage, current or magnetic field. The evaluation results from the velocity versus

position curves demonstrate that the grid can he efficiently accelerated to 1 m/s over

1 mm, then proceed with that about constant speed for 10 mm and slow down quickly

within 1 mm, which is a very reasonable and feasible design. We plan to implement our

successfully developed superconducting linear motor to our quantum turbulence studies.
















































superconducting shield


hollow
niobium
can \


Figure 2-2: Superconducting motor model with grid, thernxistors and superconducting
shield [29].


Classical Homogeneous

Isotropic Turbulence


Figure 2-1: Grid turbulence in a classical fluid [27].


superconduc-
1;a solenoid


light insulated
hollow rod


capacitive
NI1r
position
E Sensor

liquid helium
level
mm grid
thernaistors



















_lah


I


2.0-




0.5-
0.0-
-2


0 2 4 6 8 lb1012
Time (ms)


Position of Nb #1 (mm)


114 ~ 16


Figure 2-3: Armature motion in unshielded (pink curves) and superconducting shielded
(black curves) solenoid under the sine function acceleration: (a) I(t) curve; (b) v(z) curve.
(o: modified data without the central peak in the current profile) [29]


(8) 2.5

2.0



S1.0

0.5

0.0


4 6 8 10 12 14
Position of Nb #1 (mm)


Time (ms)


Figure 2-4: Armature motion inside the superconducting shielded solenoid under the linear
function acceleration: (a) I(t) curve; (b) v(z) curve.


14 16 18
















(a) 3.0

2.5-


; 2.0--




1.5-

1.0 -


0 2 4 6 8 10 12 1416 1820 2224 26 28
Time (ms)


4 6 8 10 12 14
Position of Nb #1 (mm)


Figure 2-5: Armature motion inside the superconducting shielded solenoid under the
square function acceleration: (a) I(t) curve; (b) v(z) curve.











123.
10 N 4 T '8T"



P6 64









TEIPO-RATURE (')I

Figure 2-6: Upper critical field versus temperature for niobium samples with different
RRR. [35]



























(a) 2.5


1)(b i 0


E
v
,X
o
o
P


0.8-

0.6- .

0.4-

0.2-

0.0-

-0.2-


0.5 *


nn


4 6 8 10 12
Position, z (mm)


14 16


2 4 6810 12 1'4 1'6
Time (ms)


S(c) (d)
2.0 -2.0 -
-* Magnetic field at r-rol, z=0
1.5 Magneticforce1. -

1.0 -i 1.0-

~b0.5-~ 0.5-

m m
-0. -. -05

S-1.0 -I -1.0
tR-1.5- -F -1.5-


0,, 2 4 6 8 10 12 14 16 4 0
Z ~Time (ms) E


*Magnetic field at r=rol, z=0
SMagnetic force


6b 81 1'2
Position, z (mm)


114 1'6


Figure 2-8: (a) Sinusoidal current input to the superconducting shielded solenoid with
the dimensions as stated in Table 1.; (b) the corresponding velocity versus position of
the shaft motion graphs; (c) the corresponding magnetic field, and magnetic force versus

position of the shaft motion graphs. [29]
























































Figure 2-11: Machine drawings and photos of the motor system: (a) machine drawing [29];
(b) photo of the test cell; (c) photo of the test cell installed at the end of the suck-stick
probe.


(a)

0,9524


fv

pl
bi


9
tl



























-ground
Capacit ive
position se~nsor


Figure 2-15: Testing circuit of Q-meter.


Capacithie podition
sensor


null
)scillator
Detector

Rr7 PR






Figure 2-16: Simple AC bridge circuit.
















(a) r7111CUUy IU~reft- rlll fi I

IOscilator

Fleference f=91kHz
Sne Out ~LVrms-5V
Probe

Variable I Capacitive
capacitor /I / position
I I sensor

Analog lock-in Amp

A B
CH1 Otu
100X 1 100
Ohm Ohm

LabeUrm

ground Probe, Crost~at at 4.2K
(b)
,Analog output Irpoolpolar output R-0.1ohm I r=1 ohlm
Oxnputer DAC O input Ftueaapy
Lab~eur -- llferaal 4
ground rr 56~dutg
Analog input ACHO ACH8 lni
Oxnputer = sm
Labvour


Figure 2-18: Circuitry and setup for testing the superconducting motor at liquid helium
temperature, 4.2 K(. (a) monitoring the motion of the armature with the capacitance
position sensor; (b) monitoring the pulse sent to the superconducting solenoid.























*Current pulse profile
(a) Capacitive position sensor response


200 points adjacent normal averaging


(b) 0.6


J I I I I 10.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
0.00 0.05 0.10 0.15 0.20
Time (s)
Time (s)
200 points adjacent normal averaging


)3.0-

2.5

2.0-


1.5


(c) 0.6

0.4


S0.2

'~0.0


-0.2


0.5


0 2 4 6 1 12 1'4
Time (ms)


10 12 14 16 18 20
Position, z (mm)


Figure 2-20: Motion of the armature of the superconducting motor. (a) Pulse profile sent
to the solenoid and capacitive position sensor response as a function of time. (b) Velocity
of the armature as a function of time. (c) Velocity versus position of the armature. (d)
Original current profile from analog output DAC 0 of LabView. [29]













































0.e


(a) .


2.0


~1.5




0.5


0.0



(c)


3 10 12 14 11
Position of Nb #1 (mm)

Magnetic Field at r=rol, z=0
Magnetic Force


Time (ms)
Magnetic Field at r=rol, z=0
Magnetic Force


I
0.4 -


-0 4 -

F4-0.9 -
-10
O -.


o

i)


8 1 12 14
Position of Nb #1 (mm)


10 20 30 40 50 60
Time (ms)


Figure 2-21: (a) Distorted current profile due to the almost purely inductive behavior of
the solenoid (small resistance ~ 10) in the superconducting shielded solenoid with the
dimensions as stated in Table 2.; (b) corresponding velocity versus position of the shaft
motion graphs; (c) corresponding magnetic field, and magnetic force versus position of
the shaft motion graphs; (d) corresponding magnetic field, and magnetic force versus time
graphs. [29]


50.


F4 -0.4 -
~ -0.6-


























100 points adjacent normal averaging
(b)
1.0-
0.8-
0.6-

S0.4

S0.2


-0.0
-0.2



0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
Time (S)


(d) 3.0-

2.5-

2.0


S1.5

S1.0

0.5

o n ,,


-*-Pulse profile
(a) -- Capacitive position sensor response
One pulse, DC level=2V for 112.5ms
1.-.in the original pulse profile
1.2-
Actual pulse peak height= ~ -0.55V


0.6 -

a 0.4 -

-0.2 -



-0.6
0.00 0.05 0.10 0.15 0.20


100 points for adjacent normal averaging
1.0 -


0.4 :

0.2

0.0


~-0.2-
-0.4 -( '

,


Position, z (mm)


1'2 1'3


20 40 60 80
Time (ms)


100 120


Figure 2-23: Motion of the armature of the superconducting motor. (a) Pulse profile sent
to the solenoid and capacitive position sensor response as a function of time. (b) Velocity
of the armature as a function of time. (c) Velocity versus position of the armature. (d)

Original current profile from analog output DAC 0 of LabView. [29]
























Pulse profile
(a) Capacitive position sensor response
4.0-
DC level= 0.32V for 25ms
3.5 -
in the original pulse profile
3.0-1 Actual pulse peak height=
2.5-1 1 1.05V






0.5
0.0

0 00 0.05 0.10 0.15 0.20 0.25 0.30
Time (s)

(c) 1~100 points adjacent normal averaging



0.4-



S0.2-



-0.2 -

-0.4
8 10 12 14 16 18 20 22
Position, z (mm)


(b)1.0100 points adjacent normal averaging
l.


a .0.8 -

S0.6-


0.00 0.05 0.10 0.15 0.20
Time (s)


0.25


(d) 3.0-

2.5-

2.0-




S1.0-

0.5-


5 10 1'5
Time (ms


20 2'5 30


Figure 2-25: Motion of the armature of the superconducting motor. (a) Pulse profile sent
to the solenoid and capacitive position sensor response as a function of time. (b) Velocity
of the armature as a function of time. (c) Velocity versus position of the armature. (d)

Original current profile from analog output DAC 0 of LabView. [29]













































40000


Figure 2-26: (a) New machine drawings of the modified motor system [41]. (b) Picture of
the armature with the grid mounted at the end. (c) Picture of the test cell showing the
grid mounted.


Top View










0.008'
0.040'


30000 1

S20000 1
-

S10000-


2-56 clear holes through
(smooth fit to pittar rodl)


0.0 0.2 0.4 0.6
Grid Velocity, U (m/s)


0.8 1.0


Figure 2-27: (a) Machine drawing for the grid. (b) Reynolds number versus grid velocity
at 4.2 K(.


"I i;r-
nobum ar
11515 1 IT ------a



ii K

t-


011.1000
00.22500
00,0890
F0.00890.0400
































(b) 2.0 -

2.1

2.2-

~ a2.3-

2.4 -
v,2.5-

2.6-

S0 2.7

U 2.8 -
-


.*


Position, z (mm)


0 2 4 6 8 10 12 14 16 18 20 22
Position, z (mm)


Figure 2-28: (a) The calibration of the armature position sensor (4.2 K(). Position sensor
bridge voltage output front lock-in amplifier versus the position of the armature [41]. (b)
The capacitance of the capacitive position sensor versus the position of the armature,
measured by using the capacitance bridge, GR 1616.


























I~o pols lbuor simply
I~npllkr6M


Probe Cryostat at 42K


Figure 2-29: Electronics circuits for the superconducting motor system.

















<+.U
3.5-

3.0Capacitive position
2.5- sensor response
2.0-
1.5-
1.0-
0.5
Pulse profile (b)
0.0
-0.5-11 Actual current pulse peak
height= 6A
-1.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.r

Time (s)

1.2 '- -Pulbs (bi.











-0.6




0 2 4 6 8 10 12 14 16 18 20 22 2


4.5 -
4.0 -
3.5
3.0-
2.5-
2.0 -
1.5 -
1.0 -
0.5-
0.0 --
-0.5 -
0.0


0.2 0.3
Time (s)


-0.6
Time119F (s)


POSitiOn, z (mm)



DC level -0 10V for 250 ms


( 0.5

0.0

-0.5



-,1.5

p -2.0


10 ms delay
-


10 ms DC level 0.32V


2. ims Sine pulse with peak 2.86V1



2.5 ms Sine pulse with peak 2.86V
)0 0.05 0.10 0.15 0.20 0.25 0.30
Time (s)


Figure 2-30: Motion of the armature of the superconducting motor [41]. (a) Current
through the solenoid and capacitive position sensor response as a function of time. (b)

Same as (a) with polarity reversed. (c) Velocity of the armature as a function of time.

(d) Velocity versus position of the armature. (e) Original current profile from analog
output DAC 0 of National Instruments board sent to K~epco BOP current amplifier driving
solenoid current.









CHAPTER 3
CONSTRUCTION OF SUPERCONDUCTING SHIELDED LINEAR MOTOR AND
EXPERIMENTAL CELL

3.1 Construction of Superconducting Shield

For our quantum turbulence research, we are building a shielded superconducting

linear motor to tow a grid through superfluid helium. Hence, one of our tasks is to

construct a superconducting shield on the interior of the experimental cell. To accomplish

this, we have decided to electroplate our cell with lead. Lead has the advantage over other

possible superconductors of a high critical magnetic field, 803 Gauss, and high critical

temperature, 7.2 K [32]. Lead plating is also approachable and economic to be done in

the laboratory. We have decided to electroplate, rather than chemically deposit the lead,

because chemical vapor deposition (CVD) requires high temperatures and is difficult to

perform in the laboratory. For reference, lead metal has very high boiling point, about

2022 K, and melting point, as high as 600 K, using physical vapor deposition method

requiring vaporization of the metal element at very high temperatures, which is also not

applicable for our lab. Another superconducting metal, niobium, with critical temperature

9.5 K [32] has much higher melting point, 2740 K, and even higher boiling point, about

5017 K. Therefore, considering the options we decide to electroplate a lead magnetic

shielding enclosing the motor at very low temperature.

3.1.1 Electroplating Theory and Electrolyte Recipe

Basically, the idea of lead electroplating is to have lead deposited on the desired parts

while a pure lead electrode is dissolved in the electrolyte. The electrode is used to supply

and keep the lead concentration in the solution at a sufficient level. The process is as

follows.

Anode : Pb Pb2+ + 2e- (Oxidation) (3-1)



Cathode : Pb2+ + 2e- Pb(Reduction) (3-2)









We used methanesulfonic acid (ilr;A) as our electrolyte in lead plating. Recently,

(since the 1980's) MSA has been used in industry for electre~lpta filr_ rather than lead

fluoroborate (Pb(BF4)2), fluOroboric acid (HBF4) and boric acid (H3BO3) because it

is less corrosive and the produced effluent is easier to deal with. Besides, the organic

additives aid to produce a smooth, fine-grained deposit and to increase throwing power.
We got a nice recipe for the electrolyte solution makeup from Technic Inc.:

* Techni-Solder NF Acid (70 .~): 20 .~ or 200.0 mL in one liter of electrolyte.
Ingredients: 70 .~ or 10 M methane sulfonic acid (CH4SO3 -

* Techni-Solder 700 HS MakeUP: 2.5 or 25.0 mL in one liter of electrolyte.
Ingredient: two non-ionic surfactants, which can reduce the surface tension to
have the lead deposited more delicate and fine-grained.

* Lead carbonate (PbCO3): 15.0 g of lead or 19.3 g of lead carbonate in one liter of
electrolyte. Lead can be dissolved in MSA very slowly. It would be more efficient to
put lead carbonate into MSA resulting in lead (II) methanesulfonate solution, carbon
dioxide (CO2) and water (H20) produced.

* DI water: for balance and adding up to one liter of solution.

* The optimal current density would be 20 ASF (ampere per square feet) for deposit
1 p-m of lead per minute. Agitation is required to make the lead ions in the solution
uniformly distributed.

Since the coherence length of lead is 0.083 pm [32], we would like to deposit 25 pm of

lead, which is more than enough. For our purposes, our cell is made of pure copper free of

oxygen, Fig. 3-1 and Fig. 3-2. For the cell cap and cell body, the interior surface areas are

about 68.3 cm2 and 78.9 cm2. We need 1.469 A and 1.695 A of current to deposit 1 pm of

lead per minute on them, respectively.

3.1.2 Properties of Chemicals

3.1.2.1 Lead

Lead has a wide range of applications in our daily life, like lead paint, solder joints on

PCB in electronics, car batteries, etc. However, over the past few decades, more and more

information has come ut about the hazards of lead and lead compounds and it has seen a

gradual decline in use. Lead accumulates in bone and body, primarily through inhalation










or ingestion of dust and fumes, resulting in headache, nausea, vomiting, abdominal

spasms, fatigue, sleep disturbances, weight loss, anemia, and pain in legs, arms, and joints;

an acute, short-term dose of lead could cause acute encephalopathy with seizures, coma,

kidney damage, anemia, and even death [44].

3.1.2.2 Methanesulfonic acid (MSA)

The appearance of methanesulfonic acid (CH4SO3) is ClarT, traUSparent liquid and it

has a faint sulfur oxide odor. Thermal decomposition of MSA may release sulfur dioxide

and sulfur trioxide. It is corrosive to skin and causes severe burns if contacted, swallowed

or inhaled. Therefore, it should be handled carefully. If released or spilled, MSA can be

neutralized by slow and careful applications of a solution of soda ash and water [45].

3.1.2.3 Lead carbonate

Lead carbonate (PbCO3), alSO known as cerussite, is a white
powder without any odor. It is harmful if inhaled or swallowed. Lead is a cumulative

poison and exposure even to small amounts can raise the body's content to toxic levels.

Risk of cancer depends on level and duration of exposure. Target organs: K~idney, central

nervous system, blood, reproductive system [46].

3.1.2.4 Lab protective equip

CI.- --1. s and face shield; lab coat and acid resistant apron; vent hood; rubber gloves

(vinyl gloves are the best, and latex gloves are acceptable); respirator or NIOSH/j\!SllA4

approved mask.

3.1.3 Lead Electroplating Procedures and Results

3.1.3.1 Procedure steps

Making new electrolyte solution:

* 100 mL of Techni- Solder NF Acid (70 ~) is added to about 200 mL of DI water in
the beaker.

* Add 12.5 mL of Techni- Solder 700 HS MakeUp to the beaker.









* Add 15 grams of lead (or 19.34 grams of PbCO3) to the solution. Note: this is double
amount of lead as that in the recipe. Now the solution becomes turbid and a lot
of exquisite white bubbles of CO2 float above the surface filling up the 1000 mL of
beaker. After a while, the solution becomes clear again.

* Add DI water to the beaker up to 500 mL of electrolyte solution.

The followingfs are the actual procedure steps of lead electroplatingf performed in the

laboratory.

1. Cleaning the cell with the Electronic shop's help. They use the commercial copper
cleaner and ultrasonic vibration equipment for cleaning the cell. Fig. 3-3 (b) and
Fig. 3-4 (b) show the copper interior of the cell cap and cell body, and the well-built
silicone wall, ready for the lead plating.

2. Sealing the holes with silicone [? ]and building about one inch high of silicone wall
leaning against the paper wall about a few mm extended outward along the edges of
the top rim of the cell cap or cell body (Fig. 3-3 (b) and Fig. 3-4 (b)).

3. Making the lead electrode for lead plating of the cell cap and cell body (the drawing
of the exact size shown in Fig. 3-3 (a) and Fig. 3-4 (a)). The lead electrode is well
trimmed and carefully adjusted to be about 2 mm clearance away from every cell
wall. With the aid of a magnifying mirror, an ohmmeter, and a round rubber pad
attached to the center of the turntable, we can do an almost perfect job, as shown in
Fig. 3-3 (c), (d) and (e) and Fig. 3-4 (c), (d) and (e). The cell cap is supported by
a rubber piston underneath to stand more stably. The cell body has gold plated at
the bottom. While rotating the turntable, we make sure the resistance between the
electrode anode and cell cathode is infinity.

4. The electrolyte is injected to the cell cap or cell body about half to one inch higher
than the rim enclosed by the silicone wall (see Fig. 3-5 (a) and Fig. 3-6 (a)). The
resistance between the two electrodes (the lead electrode as the anode and the cell as
the cathode) now is only a few ohms. The cell is rotated steadily and slowly on the
turntable of a broken microwave oven. Therefore, the lead anode can also agitate the
electrolyte to make the lead ions in the solution distribute very uniformly while lead
plating is performed. We start to supply the current of about 1.47 A (Fig. 3-5 (b))
for the cell cap and about 1.70 A (Fig. 3-6 (b)) for the cell body. The lead electrode
(anode) connects to the positive terminal of power supply, while the cell (cathode) is
in contact with the thin stainless steel piece connecting to the negative terminal of
the power supply. No white bubbles occur with high concentration of lead, 30 grams/
liter of lead in the electrolyte. The exterior part of the cell wall in contact with the
thin stainless steel piece, reduces to shiny pure copper from the oxidized copper (Fig.
3-5 (c)). Under the liquid surface, it is clear to see that the lead electrode is well










adjusted- 2 mm clearance away from every cell wall (Fig. 3-5 (d) and Fig. 3-6 (d)).
The dissolved lead precipitates at the bottom of the cell.

5. The power supply is turned on to supply the current for approximately 25 minutes.
The current fluctuates slightly, but overall very stable. The voltage output from
the power supply is 8.2 V when lead plating the cell body. Note: Do not touch the
electrodes during the electlll rfpiing. Otherwise, you will get burn because they are
very very hot due to high current flowing through them. The whole lead plating
process is done in the vent hood.

6. Rinse with ample DI water (about one gallon). Dry the cell with clean tissue and
flush with a high flow of nitrogen gas or cool wind from the heat gun immediately
and quickly.

7. The electrochemical equivalent for the reaction: Pb2+ + 2e- Pb is 3.86
g/(Almpere hour) [47]. By lead plating on the cell body with current 1.70A for
27 minutes, the deposited lead mass is 2.95g. Since the density of lead is 11.34g/cm3,
the surface area of the cell body is 78.85cm2, the thinkness of the deposited lead
would be 33.0 pm.

3.1.3.2 Results

Fig. 3-7 and Fig. 3-8 show the masterpieces after lead electroplating with shiny lead

of silver white color in the interiors, except that covered by the silicone. A close view of

the silver white interior of the cell cap shows the rims, the walls and the edges of the well

down to the cell bottom fully covered by the lead. Fig. 3-7 (c) and Fig. 3-8 (c) shows

the residue lead electrodes while the parts immersed under the liquid surface become

darker, dissolved to almost only half left.

3.1.3.3 Troubleshooting

1. If the lead anode were too small, then it might all get dissolved into the electrolyte,
resulting in zero current before the lead plating process is complete. Especially, the
dissolution of lead electrode happens at the interface between the electrolyte and air,
where oxygen is available. If the current were too small, like 90 mA, then it would
take 7 or more hours to achieve 25 p m of lead thickness, and it is possible that the
lead deposition process is much slower than lead dissolution into the electrolyte,
hence, the actual thickness of deposited lead could not be sure. Therefore we make
the lead electrode as large as 2 mm clearance away from the cell wall, so that we
could turn up the current up to the optima.










2. Deposited lead on the cell interior could be oxidized to yellow-colored and powder like
lead oxide (PbO) after drying out naturally without some special care, which could
be due to the residual acid or just water causing the oxidation. The lead oxide can
he easily wiped off to make the copper on the rim edge exposed. After sanding off
the oxide or sponge on the interior surface of the cell, the shiny metal lead appears.
Therefore, we know that oxide on the outer surface 1,o ;r can protect the inner
1.w-;r of lead metal to be further oxidized. To prevent this situation, we need to use
abundant DI water to clean thoroughly the acid residue and dry the wet immediately
and quickly.

:3. White and delicate bubbles show up on the surface of the electrolyte with lead
concentration only 15 g per liter of electrolyte when turning up to the operating
current during lead electroplatingf. The white bubbles on the surface of the solution
occurred at high current density is fine 1 When the current efficiency is not 100' .~
the hydrogen ions, competing with the lead ions, capture the electrons instead, then
the hydrogen gas would be produced, forming the bubbles. When the current is
high, as high as 1.70 A like the above process, and the current density is not the
same everywhere on the cell, the lead might he deposited more or less at different
positions. If the bubbles occur, then it could result in uniform thickness throughout
the whole cell walls. The sponge like, dark color lead forms rough surfaces at high
current density when the lead concentration in the electrolyte is too low. We solved
this problem by doubling the amount of the lead in the electrolyte.

4. "1\odern Electroplatingt by Schlesinger and Paunovic (2000) on page :366 [47]:
lIs-u~llah!.- anodes cannot he used in lead plating electrolytes as lead dioxide, PbO2,
will form on the surface of the anodes. The purity of the soluble lead anodes used
determines the extent to which a film forms on the surface of the anodes." On page
:369 [47]: "During the deposition of thick lead coatings (up to 200 pm) formation
of nodules or ,y ~.--IlI!-" can occur. This failure does not generally occur with a
freshly made-up solution, and when it does occur, it can in most cases he rectified
by a purification of the electrolyte with activated carbon. contamination of
the electrolyte by breakdown products of the organic additives. together with
a rapid decrease of the lead concentration in the electrolyte. The anodic current
efficiency was reduced, which caused the drop of the lead content in the hath. By the
passivation of the anode, lead dioxide was formed on the anode surface, which caused
a partial oxidation of the organic additives."

Therefore, we should use 100 .pure lead as the anode to prevent the lead dioxide
from forming on the surface of the electrode, causing anodic oxidation of the organic
additives on the anode surface, or the oxidation of the organic additives by the lead
dioxide, resulting in the roughness of lead deposition. Also if the electrolyte solution




1 Private communication with Chuhua Wan--_ from Technic, Inc.










is made for a while, it should be discarded following the hazardous waste disposal
rules. The solution should be made just before we attempt the lead pI rfl-,Hi and the
work should be done within two or three d we~ before the solution becomes old.

5. "Modern Electroplatingt by Schlesinger and Paunovic (2000) on page 373 [47]:
"Discoloration of the lead deposit to a brown or black color occurs due to deposition
of copper onto the lead surface by a displacement reaction that can happen in the
electrolyte if the current is left switched off while the parts are still immersed, or in
the rinses if they are heavily contaminated with copper."

Therefore, after finishing the lead plating, we should dump out the electrolyte
immediately after the power supply is switched off. And rinse the parts with plenty
of DI water immediately.

3.2 Testing the Experimental Cell at Liquid Helium Temperature

Fig. 3-9 shows the machine drawing with the dimensions and the pictures of our

experimental cell. We have one superconducting solenoid driving the armature to move

through its center, with a grid attached at the end. This light insulating armature is

constructed of 3 phenolic tubes separated by two hollow cylindrical niobium cans placed

26 mm apart, with the turbulence-producing grid attached to one end. A conducting

section on the armature, composed of one of the Nb cylinders and silver paint coating part

of the phenolic rod, is inside a closely fitting capacitor made of two semi-cylindrical copper

sheets. This capacitor, coupled to a bridge circuit, measures the armature position. The

dimension parameters of our superconductor shielded superconducting linear motor system

are listed in Table 3-1. The electronics circuits are the same as described in chapter 2,

e.g. Fig. 2-29, except the lead resistance 0.7 R. The experiment cell is mounted to the

0.25 inch diameter probe and tested in a transport dewar cryostat [42] at liquid helium

temperature.

3.2.1 Simulation Results

If we define the center of the solenoid as the position, z = 0 (see Fig. 3-10 (a)), then

the magnetic force distribution along the z-axis of the solenoid are calculated, as in Fig.

3-10 (b). To be more efficient, we would apply the acceleration pulse when the center of

mass of the niobium can #1 is located at z = 6.5 mm position, and deceleration pulse at z










Table 3-1: Parameters of superconductor shielded superconducting linear motor system.
Parameter Value
d 0.1905
NV 1535
r 5.0
1 10.0
zo 6.43
rol 3.175
To2 3.175
11 13.0
12 15.0
as 25.9
& 50.0
b 20.0
Ml. ... 2.6


=18.5 mm position. After the armature passes the z = 26 mm position, we would apply

the holding pulse, just enough to balance out the gravity.

In the current profile from our simulation program (Fig. 3-10 (c)), the first pulse

produces sine function acceleration, and the third pulse deals with linear function

deceleration. The central peak is due to the almost balanced magnetic forces on the

two niobium cans at that position, where each is almost equidistant from the ends of

the solenoid. We can remove the central peak and the inertia will serve to carry through

it. It takes slightly more current, 0.147 A more, for the superconducting shielded motor

system than the unshielded one. In the resulting grid motion (Fig. 3-10 (d)), the grid

would have sine function acceleration from 0 to 1 m/s in 1 mm. Then it travels at almost

constant speed, 1 m/s, for 10 mm. In the mid-way, the grid starts to slow down at 12.5

mm position after the niobium can #2 becomes closer to the solenoid, meaning stronger

magnetic force in the opposite direction. Then the grid would rapidly decelerate to cease

within 1 mm when the third pulse is applied. The empty circles represent the grid motion

without the central peak in the current profile. Without the central peak in the current

profile, the grid motion is not significantly different.









3.2.2 Experimental Testing Results

Fig. 3-11 shows the experimental results. Considering the inductive behavior of

the solenoid, an inductor, the time constant of an LR circuit is L/R = 16.34mH/0.70

~ 23.3milli seconds. Even if we apply a square pulse, ;?i 1 V, it takes five times

the time constant (116.5 ms) to saturate to 1 V. Therefore, we need to take this into

account when we apply the pulses to the superconducting solenoid. We apply the

appropriate pulse profile: DC 2.89 V for 14 ms, followed by DC -0.3 V for 10 ms, then

2.5 ms delay, eventually the holding pulse -0.02 V for 100 ms (Fig. 3-11(d)). The solenoid

is again protected by the same two crossed silicon diodes as described in chapter 2. In

Fig. 3-11(b) and (c), the armature is seen to accelerate to 0.7 m/s within 6.5 ms over a

distance of 2.5 mm (estimated average acceleration 98 m/s2). Then it undergoes a near

constant deceleration to zero velocity within 66 ms in another 14 mm (estimated average

deceleration 17.5 m/s2). It WaS held above the position 28 mm for about 100 ms until the

current was turned off, it then dropped under gravity, as fast as 0.4 m/s in 60 ms over 12

mm position (estimated average acceleration 6.7 m/s2). At the position between 8 and 17

mm, the velocity of the armature was above 0.6 m/s and between 0.6 and 0.7 m/s.

3.2.3 Viscous Drag and Impedance Forces Discussion

In our theoretical calculation, we have tried to simplify the real physical situation by

only considering the magnetic force and gravitational acceleration force. However, inside

the fluid, even in our quantum fluid- less viscous liquid helium-4, there are still some other

significant forces, e.g. viscous dr I_ Buoi- maly force, impedance force due to the flow based

on Bernoulli law, exerted upon the armature and the grid. Now we are going to estimate

the sizes of these forces.

1. At 4.0 K(, dynamic viscosity p S;:I,,.:. I opoise 36 x 10-7Pascal second =
36 x 10-71N s/m2. (K~inematic viscosity: v p/~p, strokes 1 cm2 -1)[4

2. At saturated vapor pressure, at 4.20 K(, the liquid helium-4 has density,
p 0.1254075g/cm3 125.4075kg/m3 [49].









3. Reod ubr Rg Up/ lm/s x25 075iksm 2m3 = .4835 x 10 ( in meter).
So it must be turbulent since very small length scale [50].

4. Total length of armature (Fig. 3-12) L = 3.1875" = 8.10cm 8.10 x 10-2m. The
diameter of the armature D (1/4)" = 0.635cm = 0.635 x 10-2m

5. Rer UsipL/pL = (1/sx1 475g 3 1010 = 2.82 x 106. Skin friction coefficient:
Cf 0.455(logloRee)-2.58 = 3.7094 x 10-3. Drag = plf, xa 12405/3
(1m/s)2 x 3.7094 x 10-3 0.2326N/lm2
Viscous drag drag x surface area Drag x xD x L = 0.2326N/lm2 x xrx 0.635
x 10-2m x 8.10 x 10-2m 3.758 x 10-4NV [50].

6. Force exerted upon the upper side plates of the armature (Fig. 3-12 (a)) = Fext:
d 0.3" 0.762cm 7.62 x 10-3m; t0181 CTOSS SeCtiOUS A' x( )2+
,[( )2\ Do\l 2] x 2 ing 7.347 x 10-sm2
Fext d(M~omentum)/dt pA'v2 9.214 x 10-3NV [50].

7. Force exerted upon the bottom side plates of the armature (Fig. 3-12 (b)) F'me:
Bernoulli's equation: p p= -~i 62.70N/m2l [50]; Fit, pA' 4.607 x 10-31V

8. Buoi- oi-i force = pgV = 125.4075kg/m3 x 9.8m/s2 x 2.565 x 10-6 3
3.1524 x 10-31V

9. Gravity force = 2.60g = 0.02548N\1.

10. If the grid is mounted at the end of the armature, the total non-transparent surface
area of the grid, excluding the area underneath the armature, is A" 1.7765cm2
Force exerted upon the upper side of the grid pA"v2 22.28 x 10-31V; foTCe
exerted upon the bottom side of the grid pA" 11.14 x 10-3NV. So the total forces
exerted upon the grid is 33.42 x 10-31V

So the total net force (downward) when the armature travels upward at the velocity

of 1 m/s = 0.06994 N = 2.745 x Gravity force (Table 3-2). In our simulation program,

we only consider the gravity force and the magnetic force. As the armature moves at

higher velocity, the viscous drag force and impedance forces become significant, pretty

comparable to the gravity force.

3.2.4 Heat Dissipation Discussion

The Joule heating dissipation is avoided by using superconducting wires winding

the magnet. Building the superconductor shield enclosing the magnet is to prevent the









Table 3-2: Forces on the armature when moving up at 1 m/s (Download is positive).
Parameter Value
Viscous drag 3.758 x 10-41
Fezt 9.214 x 10-31
F'z 4.607 x 10-31
Buonne1y force -3.1524 x 10-31
Gravity force 0.02548N\1
External forces on the grid 33.42 x 10-31


eddy current heat dissipation on the cell wall. As far as the AC loss (including hysteresis,

coupling and eddy-current losses), splice and mechanical losses [43] are concerned, we

estimate the resulting heating due to the varying magnetic field, which is not significant.

We use epoxy-resin impregnation to prevent frictional displacement without noticing

any deformation of the coil. We do not have any splice loss problem without using any

slices, we assume that the heating due to the above three factors could be neglected.

When we run the motor in the dilution refrigerator, we will run it without any liquid

helium and measure the background temperature fluctuation. Possible AC losses of the

superconducting solenoid due to the varying current can be measured by substracting

from the blank background without any current in the solenoid. And then we will run

the motor again with grid immersed in the liquid helium bath. We can then do the

subtraction to get the net temperature change simply due to the quantum turbulence

energy decay.

3.3 Leak Tight Electrical Feedthrough Design

The electrical feedthrough for the cryogenic cell has been designed and constructed.

We also check the leak tight of the feedthru pins imbedded in our experimental cell at the

liquid nitrogen temperature, 77 K(, with our homemade "R. II. i-n (Roving Experimental

Device Investigator) apparatus. Fig. 3-13 shows the design of our experimental cell for the

towed-grid studies of quantum turbulence experiments.









3.3.1 Electrical Feedthru Construction

One of our tasks for building our superconductor shielded superconducting linear

motor system is to construct the circuits, such as for the superconducting solenoid, the

position sensor and thernxistors leads. When designing these feedthroughs, we need to

concern ourselves with keeping the cell leak tight, especially when the wires come out

front the inside of the experimental cell to the exterior circuit connection. An extra

challenge occurs when different jointed materials cool down to the cryogenic temperatures,

the distinct thermal contractions could cause a crack. We have designed and built the

electrical feedthroughs and a novel way of mounting the wires through the cell walls.

Fig. :3-13 shows the whole experimental cell assembly for the towed-grid experiments

and the circuit board for the thernxistors with the electrical feedthru pins mounted.

The electrical feedthru pins are gold plated on all surfaces to benefit the best electrical

conductivity 2 and tight fit to the homemade polycarbonate plugs. Polycarbonate

material has very close thermal expansion coefficient to that of the copper, so the plugs

and the copper cell wall will have nmininial differential thermal contractions when cooling

down to nxilliiE~levin temperatures. We have our cell made of pure copper free of oxygen.

Each individual electrical feedthru pin is imbedded in each plug, and then inserted to each

individual hole on the cell wall. In order to have leak tight, we seal each feedthru with

epoxy. Epoxy is mixed with 10 g of Stycast 2850FT and about 0.65 ~ 0.75 g of Catalyst

24LV (proportion 100:7).

Before putting everything together and applying the epoxy for seal, the degrease and

cleaning process is very important:




2 Electrical feedthru connector pins: Mouser Electronics (Order Number 575-11:3164)
Manufacturer's (11.1141ax) Part Number: :310-1:3-164-41-001000. Each individual machined
pin is gold plated on all surfaces. Each strip contains 64 pins.










1. Clean the polycarbonate plugs and electrical feedthru pins by immersing them under
the liquid surface of methanol or ethanol in a ultrasonic vibration equipment for a
few minutes.

2. Clean the dirt or dust with the soft iron brush, rough sponge and sand paper on the
exterior surface of each feedthru hole on the cell wall.

:3. Spray the acetone to wet the Q-tip and then apply the Q-tip to clean the interior of
each feedthru hole on the cell wall.

4. Spray the acetone to wet the K~imwipes wipers and then apply the wipers to clean the
exterior surface of each feedthru hole.

Applying the epoxy to the joint surfaces between the copper cell wall and the

feedthru plug needs enough surface edges for the epoxy to adhere well, so we make the

polycarbonate plugs higher than the cell wall surface, sticking from outside and the

non-connection part of the every electrical feedthru pin is longer than every plug (Fig.

:3-13 (a)). It usually takes a day to dry out the epoxy; however, we can illuminate with

the infrared light to shorten the time to about a half d w-. Shinning this light also helps

strengthen the epoxy. We should be very careful when cooling down and warming up the

cell very slowly, avoid chemical touching or also putting large forces on the joints. In this

way, the seal can he used for thermal cycles for many times without any crack.

3.3.2 Thermistor Circuit Board Construction

We have doped germanium thermometers [51]. They are less than :300 pm diameter

and will be immersed in the turbulent helium allow fast calorimetric measurements to be

made. Therefore, we have the thermistor circuit board specially made (Fig. :3-13 (b)). The

circuit board has radius 0.5 inch are on one side to match the interior are of the cell; on

the other side, an extension part is made for a 0-80 slotted flat screw to mount the board

to the bottom of the cell and also for the ease of our fingers handling. Two miniature

germanium film thermometers are mounted to the circuit board hv indium solder and

attached/glued to the board by the cryogenic grease. Three electrical feedthru pins are

mounted in parallel to the circuit board hv indium solder and the two thermistors are









soldered to those pins for electrical connections. This design allows the circuit board to

be independent (e.g. Fig. 3-13 (b): electrical feedthru pin # 1) and to slide into (e.g.

Fig. 3-13 (b): electrical feedthru pin # 1') another three counter electrical feedthru

pins imbedded in the cell wall (e.g. Fig. 3-13 (b): electrical feedthru pin # 2). At room

temperature, the resistances for the two thermometers measured from the electrical

feedthru pins outside the cell are 60.0 and 57.5 R, respectively.

We have leak checked the cell at room temperature and at liquid nitrogen temperature,

77 K(, with our homemade "R. II. i- instrument. This proves that our electrical feedthru

design is a successful design.

Outside the cell, we will plug the electrical feedthru pin # 2 with the electrical

feedthru pin # 3 connected to a miniature coaxial cable 3 Which on the other end

connects to a ultra miniature (Lepra/Con) connector plug 4 to be mounted to the dilution

refrigerator.

3.3.3 Conclusion

We use epoxy to vacuum seal the electrical feedthru, tight fit to the polycarbonate

plugs, imbedded in the cell walls. We also built the circuit board to mount our thermistors.

This design promises the flexibility of mounting the experimental cell and is leak tight.
















3 COTO COnductor: 30 AWG copper; stainless steel braided shield; insulator: PFA jacket
(Janis <.l~l Ilr .n).

4 ManufaCtuTOT: TyCO Electronics; distributor: March Electronics.