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Generation and Characterization of Quantum Turbulence

Permanent Link: http://ufdc.ufl.edu/UFE0044099/00001

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Title: Generation and Characterization of Quantum Turbulence
Physical Description: 1 online resource (2 p.)
Language: english
Creator: Thompson, Kyle J
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: fluids -- helium -- superfluid -- turbulence
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Turbulence is a specific type of fluid ?ow. We all know what it is but it is difficult to define. Turbulence is the rule, rather than the exception is ubiquitous. It is prevalent in the interstellar dust clouds reaching between the stars and crucial to the plasma dynamics inside them. It can’t be ignored in the dynamics of planetary atmospheres, oceans, or cores. It is ubiquitous, from the observable environment down to the internal structure of the human body and every living organism known and unknown. We are composed of and encompassed in an ever changing mess of fluid motion. Nearly the entire universe is in a fluid state, and the majority of fluids are turbulent. Yet there exists no analytic solution to the equations that govern the chaos. This is incredible when pondered; almost everything that can be experienced is governed by the same rules, independent of size, shape or constituents. However, to describe our surroundings we are confined to brute force calculation and experiments. The theories of fluid dynamics are truly theories of everything, and yet, so many properties are still not understood.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Kyle J Thompson.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Ihas, Gary G.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044099:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044099/00001

Material Information

Title: Generation and Characterization of Quantum Turbulence
Physical Description: 1 online resource (2 p.)
Language: english
Creator: Thompson, Kyle J
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: fluids -- helium -- superfluid -- turbulence
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Turbulence is a specific type of fluid ?ow. We all know what it is but it is difficult to define. Turbulence is the rule, rather than the exception is ubiquitous. It is prevalent in the interstellar dust clouds reaching between the stars and crucial to the plasma dynamics inside them. It can’t be ignored in the dynamics of planetary atmospheres, oceans, or cores. It is ubiquitous, from the observable environment down to the internal structure of the human body and every living organism known and unknown. We are composed of and encompassed in an ever changing mess of fluid motion. Nearly the entire universe is in a fluid state, and the majority of fluids are turbulent. Yet there exists no analytic solution to the equations that govern the chaos. This is incredible when pondered; almost everything that can be experienced is governed by the same rules, independent of size, shape or constituents. However, to describe our surroundings we are confined to brute force calculation and experiments. The theories of fluid dynamics are truly theories of everything, and yet, so many properties are still not understood.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Kyle J Thompson.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Ihas, Gary G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044099:00001


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n r This is a License Agreement between Kyle Thompson ("You") and NRC Research Pres s ("NRC Research Press") provided by Copyright Clearance Center ("CCC"). The li cense consists of your order details, the terms and conditions provided by NRC Research Press,and the payment terms and conditions. !"# !$$! nrrr nrrr nrrrr rr r nrrr nrrrr #$%&'()('#*)+n',-'.$%n'-/''0%#1%%2345 .#$% n5/0.5 )'# nrrr 52.2021522$$r nrrrr 56 &rr 7 'rr 7 #"r!-r #r8.r r9r"r 5r + +:r8r r!!:r8r )rr!rrrr#r!"r8r ;rrr:r? # 2-(. #r General Terms & Conditions Permission is granted upon the requester's compliance with the following terms andconditions: A credit line will be prominently placed in your product(s) and include: for books theauthor, book title, editor, copyright holder, year of publication; for journals the author,title of article, title of journal, volume number, issue number, and the inclusive pages.The credit line must include the following wording: 2008 Canadian SciencePublishing or its licensors. Reproduced with permission," except when an author of anoriginal article published in 2009 or later is reproducing his/her own work. 1. The requester warrants that the material shall not be used in any manner that may be 2. Rightslink Printable License https://s100.copyright.com/App/PrintableLicenseFram e.jsp?publisherI... 1 of 2 7/18/2012 12:30 PM

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GENERATIONANDCHARACTERIZATIONOFQUANTUMTURBULENCEByKYLEJ.THOMPSONADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012KyleJ.Thompson 2

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ACKNOWLEDGMENTS Therearemanypeoplewithoutwhommyjourneytocompletingadissertationwouldbemuchmoredifcult.First,IwouldliketothankthedifferentstudentsIhaveworkedwith.Whentheywerepresenttherewasalwaysanadditionalenergyandurgencyinthelaboratory.SpecicallyIowe,LydiaMunday,Dr.WeiGou,AlexMarakov,RomanChapurin,AustinGrifth,JohnPilkey,ManishAmin,JiheeYang,ChadHopkins,IvanLozano,MilapPatel,TravisNelson,EmmaBorrowman,JoeTriana,CaseyLitton,JamesStankowicz,andCarleyNicolettifortheirtimeworkingwithme.Inparticular,IwouldliketothankLydiaforspendingherweekendsforthepastmonthreadingoverthisdocument,section,bysection,tomakeavailablethethesisthatishere.IwouldalsoliketothankWeiGuoforallowingmeusehisdesignforsecondsoundtransducersandforproductivediscussionsaboutthecontrolmotordesign,andthelongesttenuredundergraduate,Romanwhocalibratedtherstinductivepositionsensorandspentmanyhoursinthelab.InadditiontothelabmemberswithwhomIhaveworked,thetechnicalstaffinthedepartmenthavebeeninvaluable.Theelectronicsshophaveprovidedmewithadviceonthemanufactureofcircuitsandalwayswereabletoprovidethesparepartswhennecessary.Themachinistsinthemachineshopwerealwaysavailabletodiscussmylatestdesignideaandhowbesttomakeanapparatuswork.ThelastshopIwouldliketothankisthecryogenicshop.Theyalwayshadheliumavailablewhenitwasneeded,nightandday,weekendandweekday.Inadditiontotheirnormaljobofprovidingcryogenicstheyweremyprimarysourceofsoundengineeringadvice.Specically,IwouldliketothankGregLabbeforallofhistime,attention,andthoughtheputintomywork.Morethananyoneelse,hemadetheexperimentsworkandhasalwaysbeenaconstantsourceofinformation.Thisproject,likesomanyothers,hasturnedintoamulti-universitycollaborativeeffort.Innoparticularorder,Iwouldliketothank;Dr.WilliamVinenfromBirmingham 3

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Universityforcreatingandbeingaconstantpresenceinthedevelopmentoftheinertialmotor,Dr.PeterMcClintockfromtheUniversityofLancasterforbeginningthisresearchwithGaryanddevelopingtheideaforthepassingmotorandDrs.DanMcKinseyandWeiGuofromYaleUniversitywhohavebroughtthenewandexcitingexcimertechnologytoturbulenceresearch,Finally,Ioweaspecialdepttomychairmanandadvisor,Gary.G.Ihas.Overthepast5yearshehassteadilyguidedthisworkandbeenavailablewithenthusiasmatalltimes,particularlywhentheexperimentswererunning.InadditiontoallofthoseIworkedwith,IwouldliketothankmycolleagueswhowentthroughthisPh.D.processwithme;EvanDonoghue,MaureenPetterson,DanPajerowski,JoeGartner,DylanSweeny,RichOttens,JeffHoskins,andDanialArenas.IwouldalsoliketothankmyfriendsfromGainesvillewhohavedonetheirbesttokeepmefromgettingallmyworkdone,PhilNassoiy,SarahNassoiy,andMikeGreathouse.Finally,lastbutbynomeansleast,IwouldliketothankmypersonalsupportteamandgirlfriendDeniseBloom. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 12 CHAPTER 1INTRODUCTION ................................... 14 1.1GeneralBackground .............................. 15 1.2ClassicalFluidDynamics ........................... 15 1.3SuperuidHelium4 .............................. 21 1.4SecondSound ................................. 29 1.5QuantumDynamics .............................. 36 1.6Calorimetry ................................... 43 1.7ScopeofWork ................................. 44 2SENSORDEVELOPMENT ............................. 45 2.1SecondSound ................................. 48 2.2Thermistors ................................... 53 3MOTORDESIGN ................................... 59 3.1PassingMotor ................................. 62 3.2ImpulseMotor ................................. 66 3.3InertialMotor .................................. 71 3.3.1InitialResultsat4K .......................... 84 3.3.1.1OpticalCryostatResults ................... 88 3.3.1.2Resultswithinductivepositionsensor ........... 91 3.3.2InitialMotorTestsat1K ........................ 100 3.4ControlMotor .................................. 106 3.4.1ControlMotorDesign .......................... 106 3.4.2ControlMotorTheory .......................... 115 3.4.3ControlMotorExperiment ....................... 119 3.4.3.1Shuntresistance=0 ..................... 123 3.4.3.2Witha0.01shuntresistor ................ 128 3.4.3.3Shuntresistor=0.1and1 ............... 131 4MILLIKELVINEXPERIMENTS ........................... 134 5

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5ONGOINGANDFUTUREWORK ......................... 150 5.1SecondSoundandtheControlmotor .................... 150 5.2Nonconductingimpulsemotorcell ...................... 153 APPENDIX:MACHINEDRAWINGS ........................... 156 REFERENCES ....................................... 171 BIOGRAPHICALSKETCH ................................ 175 6

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LISTOFTABLES Table page 1-1KinematicViscosityofDifferentFluids ....................... 23 2-1Secondsoundtransducer .............................. 49 2-2QuantitiesusedtocalculatetheKapitzaResistance ............... 56 3-1InertialMotorParameters .............................. 79 3-2ControlMotorCoilParameters ........................... 108 4-1Relevantpropertiesoftheimpulsemotor ..................... 134 7

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LISTOFFIGURES Figure page 1-1Helium4phasediagram ............................... 27 1-2Poiseuilleowforsuperuidhelium ......................... 27 1-3Viscosityofsuperuidheliummeasuredbyarotatingviscomiter ........ 28 1-4Secondsoundvelocity ................................ 35 1-5Vortexlinereconnection ............................... 42 2-1Secondsoundtransducer .............................. 50 2-2Secondsoundmounts ................................ 50 2-3Secondsoundelectronicschematic ........................ 51 2-4Secondsoundresonance .............................. 51 2-5Secondsoundamplitudedependenceonbiasvoltage .............. 52 2-6Secondsoundamplitudedependenceonoscillationvoltageamplitude ..... 52 2-7Depositiondiagramofthethermistors ....................... 57 2-8Thermistorcalibration ................................ 57 2-9ThermistorKapitzacalculation ........................... 58 3-1Cartoonschematicof`passingmotor' ....................... 65 3-2Cartoonofcapacitivepressuresensordesign ................... 70 3-3Capacitivepositionsensorcalibration ....................... 70 3-4Inertialmotorschematic ............................... 81 3-5Diagramofcurrentineachcomponentof`inertial'motor ............. 82 3-6Idealmagneticeldfor`inertial'motor ....................... 83 3-7Inertialmotorcurrentdiagram ............................ 83 3-8Motorapparatusandarmatureforsuckstick .................... 86 3-9Photoofshuntapparatus .............................. 86 3-10Calculatedmagneticeldforinertialmotor ..................... 87 3-11PositionSensors ................................... 87 8

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3-12Imagesofthelevitatingarmature .......................... 90 3-13Motormotiontracesfromopticaldewarexperiments ............... 90 3-14Photoofassembledsuckstickapparatuswithaninductivepositionsensor ... 96 3-15Motiontracesofvirginniobiumcylinderat4.2K .................. 96 3-16Calculatedarmatureheightvs.measured ..................... 97 3-17Motiontracesofanexposedniobiumcylinderat4.2K .............. 97 3-18Motionfromseveralconsecutivecurrentsweeps ................. 98 3-19Evidencefortrappeduxintheniobium ...................... 99 3-20Criticalmagneticeldforniobium .......................... 99 3-21Currentinputforinertialmotor ........................... 103 3-22Typicalmotormotionat1K ............................. 103 3-23Positionvs.currentforinertialmotorat1.3Kwithnoshuntresistor ...... 104 3-24Positionvstimefortheinertialmotor ........................ 104 3-25Positionvstimefortheinertialmotor ........................ 105 3-26Currentstepusedtoproduce 3-25 ......................... 105 3-27Cartoonof`stepped'drivesolenoiddesign ..................... 109 3-28Magneticeldproleforsteppedmotordesign .................. 109 3-29Forceproleforsteppedmotordesign ....................... 110 3-30Calculatedeffectivespringconstantsfortwodifferentdrivecurrentsinsteppedmotor ......................................... 110 3-31Schematicdrawingsfortheanti-Helmholtzmagnets ............... 111 3-32Photooftheanti-Helmholtzmagnets ........................ 111 3-33Magneticeldproleforanti-Helmholtzmotordesign ............... 112 3-34Forceproleforanti-Helmholtzmotordesign ................... 112 3-35Thecalculatedarmatureequilibriumcurrentsvs.armatureposition ....... 113 3-36Calculatedmagneticeldinsideeachofthethreesolenoidsoftheanti-Helmholtzcircuit ......................................... 113 3-37Calculatedeffectivespringconstantsofthecontrolmotorvs.armatureposition 114 9

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3-381.5inchpositionsensor ............................... 121 3-391.5inchsolenoidcalibration ............................. 121 3-40Photographofquadruplemagnetsandtheirmounts ............... 122 3-41Measuredpositioncurrentrelationshipforcontrolmotor ............. 122 3-42Slowmotormotionswithspeedsrangingfrom1-20mm=s ............ 125 3-43Motormotionswithspeedsrangingfrom15-45mm=s .............. 125 3-44Twoidenticalcurrentinputstothecontrolmotorwithdifferentslewrates .... 126 3-45Atrainofslightlydifferentmotions ......................... 126 3-46Rapidcontrolmotormotionswithoutashunt ................... 127 3-47Slowrampofcurrentfor10mshuntcontrolmotor ................ 129 3-48Currentvs.positionforthe10mshuntmotor .................. 129 3-49Similarmotionprolesfordifferentanti-Helmholtzcurrents ............ 130 3-50Stepwiseincreaseinpositionshowingthedecelerationduetothefaradayvoltage ........................................ 130 3-510.1shuntresistormotionwithavelocityof19.7mm=s ............. 132 3-520.1shuntresistormotionwithavelocityof66mm=s .............. 132 3-530.1shuntresistormotion,velocitiesrangingfrom109)]TJ /F4 11.955 Tf 11.95 0 Td[(160mm=s ...... 133 3-541shuntresistormotion,velocitiesrangingfrom92)]TJ /F4 11.955 Tf 11.95 0 Td[(163mm=s ........ 133 4-1GridspeedandmeshReynoldsnumberduringthemeasurementat0.52K .. 143 4-2Emptyvslledcellthermistorresponse ...................... 144 4-3Comparingthemotormotionandmeasuredheatingatmillikelvintemperatures 144 4-44differentmotormotionsandtheheatingmeasured ............... 145 4-5Heatingincellvs.ohmicheatfromacarbonresistor ............... 146 4-6Measureddifferenceinheatingwithagridandwithoutagridpresent ...... 146 4-7Severalmeasurementsofheatafteragridpull. .................. 147 4-8Differenceinmeasuredenergybetweentwotraceswithsimilarmotionproles 147 4-9Differenceinmeasuredenergybetweentwotraceswithdifferentmotionproles 148 10

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4-10Changeinenergyfordifferentmotionproles ................... 148 4-11ThisgureshowsaturbulencecreatinggridmachinedfromG-10 ........ 149 5-1Secondsoundchannelandstainlesssteelgrids ................. 152 5-2Photographof0.5m2surfaceareasinterpopsicles ................ 155 A-1MachineDrawingofdrivesolenoid ......................... 157 A-2MachineDrawingofinductivepositionsensor ................... 158 A-3Schematicdrawingsfortheglassdewarinsert .................. 159 A-4MachineDrawingoftopplatefromglassdewarinsert .............. 160 A-5MachineDrawingofbafefromglassdewarinsert ................ 161 A-6MachineDrawingofbottomplatefromglassdewarinsert ............ 162 A-7MachineDrawingofquadruplemagnetmounts .................. 163 A-8MachineDrawingoftophalfoftheanti-Helmholtzmandrel ............ 164 A-9MachineDrawingofbottomhalfoftheanti-Helmholtzmandrel ......... 165 A-10MachineDrawingoftophalfofthesecondsoundtransducermount ...... 166 A-11MachineDrawingoftophalfofthesecondsoundturbulencecell ........ 167 A-12MachineDrawingofbottomoftheplasticimpulsemotorcell ........... 168 A-13MachineDrawingofmiddleoftheplasticimpulsemotorcell ........... 169 A-14MachineDrawingoftopoftheplasticimpulsemotorcell ............. 170 11

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyGENERATIONANDCHARACTERIZATIONOFQUANTUMTURBULENCEByKyleJ.ThompsonAugust2012Chair:GaryG.IhasMajor:PhysicsTurbulenceisaspecictypeofuidow.Weallknowwhatitisbutitisdifculttodene.Turbulenceistherule,ratherthantheexceptionandisubiquitousinnature.Itisprevalentintheinterstellardustcloudsreachingbetweenthestarsandcrucialtotheplasmadynamicsinsidethem.Itcan'tbeignoredinthedynamicsofplanetaryatmospheres,oceans,orcores.Itisubiquitous,fromtheobservableenvironmentdowntotheinternalstructureofthehumanbodyandeverylivingorganismknownandunknown.Wearecomposedofandencompassedinaneverchangingmessofuidmotion.Nearlytheentireuniverseisinauidstate,andthemajorityofuidsareturbulent.Yetthereexistsnoanalyticsolutiontotheequationsthatgovernthechaos.Thisisincrediblewhenpondered;almosteverythingthatcanbeexperiencedisgovernedbythesamerules,independentofsize,shapeorconstituents.However,todescribeoursurroundingsweareconnedtobruteforcecalculationandexperiments.Thetheoriesofuiddynamicsaretrulytheoriesofeverything,andyet,somanypropertiesarestillnotunderstood.Presentedinthisthesisare:anexperimentalimplementationofanexistinglinearmotortocreatetherststudiesofquantumturbulencecreatedbythismeansinthelowtemperaturelimit;identicationofthepossibleshortfallsofthispreliminarymotorsystem;thedevelopmentofanew,moresophisticated,driveapparatus,andthecreation,exploration,anddevelopmentofnewandexistingsensorsformeasuringthe 12

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propertiesofquantumturbulence.EachoftheabovepartsinthisthesisaredifferentprongsofamultifacetedapproachunderwayattheUniversityofFloridatounderstandquantumturbulenceacrossitsentiretemperaturerange.Theworkherebuildsonthepreviousprojectsinthegroup,todevelopmillikelvinturbulencemachineryandtechniques,butalsogoesforwardinanewdirectiontoexplorethehighertemperatureregimewithsomeexistingtechnology(secondsound)andpromiseforusingabrandnewheliummoleculevisualizationtechnique.Intheend,theworkinthisthesisisaimedatrevealingonemoretinysliverofinformationofafascinatingphenomenon. 13

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CHAPTER1INTRODUCTIONTurbulenceisaspecictypeofuidow.Weallknowwhatitisbutitisdifculttodene.Turbulenceistherule,ratherthantheexceptionandisubiquitousinnature.Itisprevalentintheinterstellardustcloudsreachingbetweenthestarsandcrucialtotheplasmadynamicsinsidethem.Itcan'tbeignoredinthedynamicsofplanetaryatmospheres,oceans,orcores.Itisubiquitous,fromtheobservableenvironmentdowntotheinternalstructureofthehumanbodyandeverylivingorganismknownandunknown.Wearecomposedofandencompassedinaneverchangingmessofuidmotion.Nearlytheentireuniverseisinauidstate,andthemajorityofuidsareturbulent.Yetthereexistsnoanalyticsolutiontotheequationsthatgovernthechaos.Thisisincrediblewhenpondered;almosteverythingthatcanbeexperiencedisgovernedbythesamerules,independentofsize,shapeorconstituents.However,todescribeoursurroundingsweareconnedtobruteforcecalculationandexperiments.Thetheoriesofuiddynamicsaretrulytheoriesofeverything,andyet,somanypropertiesarestillnotunderstood.Turbulenceisnotdenedbyanysinglepropertyorequation;thereisnocriticalparameteratwhichauidwillsuddenlyswitchfromapredictablelaminarowtoastochasticturbulentone.Rather,itisastringofeventsandaseriesofpropertieswhichdeneturbulence.TennekesandLumley[ 1 ]describeturbulenceascontainingmanyofthefollowingcharacteristics:Irregularityofow,whichmeansthattheowisstatisticallycharacterizedratherthandeterministically.Turbulenceisdiffuse,theowpropertiesspreadoutovermanymomentumandlengthscales.TheuidowhasalargeReynoldsnumberwhichseparatesthedissipativeandkinematicowregimes.Turbulenceis3dimensionalandcontainsvorticityineachdimension.Theowmusthaveasourceofdissipation.Turbulencerequirescontinuummechanics,implyingthatindividualatomsandmoleculesshouldnotfactorintothedynamics.Finally,turbulenceisaow 14

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phenomenon,notauidproperty.Amongtheseproperties,adeneddissipationsourceandthecontinuummechanicsrequirementareofparticularinteresttothequantumturbulenceinvestigatedinthisthesis. 1.1GeneralBackgroundFluidsareprimarilydescribedbyexaminingtheconservationofmomentum,mass,andenergyowingthroughandcontainedinanarbitrarytestvolume.Fromthesegoverningprinciples,theNaiver-Stokesequationcanbederived, @~v @t+r(~v~v)=rp+rT+~f.(1)Hereistheuiddensity,~visthevelocity,pisthepressure,Tisthedeviatoricstresstensor,and~fisthebodyforceontheuidvolume.Ingeneral,unlessspecicandconningassumptionsaremade,thisequationhasnoanalyticsolution.Themathematicalcomplexityofthisequationenforcesthatexperimentationtypicallyleadstheoryinourunderstandingofthedynamics.Thesimplestmanifestationofturbulencecontainshomogenousandisotropicvorticity,wheretheowisstatisticallyinvarianttobothcartesianshiftsinpositionandangularrotations,greatlysimplifyingtheNavier-Stokesequation.Inotherwords,forthisowpattern,thereisnopreferredpositionordirectionintheuidandonaveragetheuidlooksthesameineverydirection.Thisthesisconcentratesonboththecreationandcharacterizationofthisspecictypeofturbulence.Wehavechosentouseliquidheliumasourtestuid.Helium,inmanyregards,isthesimplestuid,yetinotherwaysthemostcomplex.Beforethespecicmaterialpropertieswhichmakeheliumspecialarediscussed,somebasicuiddynamicconceptsthatwillbepertinenttothisworkareexplained. 1.2ClassicalFluidDynamicsFromtheconservationofmass,momentumandenergy,thebasicformsofuiddynamicscanbederived.Allthreeconservationlawsareintuitive,butstillpowerful 15

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whenwrittenwithmathematicalrigor.However,priortoformulatingtheseequationsitisimportanttonotehowuiddynamicsdifferfromsolidmechanics.Astrikingdifferenceis,inuidsthefundamentalunitofvolumeusedtodescribeaowisastationaryandarbitraryvolumewhichtheuidows,ratherthanaspecicuidatomormolecule.Therefore,inaparticulartestvolumetherecanbebothanetacceleration@ @t,andaconvectivechangeoftheuidinthevolumewhichchangethemannerinwhichthedynamicsaredescribed.Inhydrodynamics,thispropertyistakenintoaccountbycreatingahydrodynamicderivative, D~A Dt=@~A @t+~Vr~A,(1)wherethechangeinthearbitraryvectorproperty~Aisexaminedbothintimeandspace.Thehydrodynamicderivative,Equation 1 ,reducesto@ @twhenthereferenceframeofthetravelinguidisused.Thistypeofderivativeisbestunderstoodbyexample.LetArepresentpressureforthisthoughtexperiment.Asensorissittinginauidwhenthepressureoftheuidisraised.Thesensorwouldmeasurethatchangeandthisisthe@ @tcomponentofthederivative.Nowthesamesensorisonceagainintheuidandacurrentcarriesextrauidintothetestvolume.Inthiscase,thesensorwillmeasureachangeinpressureasafunctionoftheamountofliquidtravelingintothesamplevolume,eventhoughtheoverallpressureproleoftheuidisleftunchanged.Thismeasurementrepresentstheconvectiveportionofthehydrodynamicderivative(~Vr~A).Whenbothtypesofpressurechangeareaccountedfor,thehydrodynamicderivativeforpressureisdened.Therstoftheconservationequations,theconservationofmass,isgivenbytheequation, @ @t+r(~v)=0.(1)Thisconservationlawisalsocalledthecontinuityequationanditstatesthatthenetsumoftheincomingtooutgoingmasshastoequalthechangeinmassofthetest 16

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volume.Thesecondequationistheconservationofmomentum,itisalsoknownastheNaiver-StokesequationandhasalreadybeenpresentedinEquation 1 .Finallythereistheconservationofenergy, @ @t(e+1 2v2)+r(~v(e+1 2v2))=r~q+r(T~v)+~v~f,(1)whereeistheinternalenergyandqisheatux.Thisequationcanbefurtherbrokenintoathermalandkineticcomponent.However,wewillnotrequirethisseparationtoeithermotivateourworkordescriberesultsinthisthesis.Therefore,furtherexplanationissuperuoushere.Fortheinterestedreader,moreinformationcanbefoundinmanyothersources[ 1 3 ].Together,thesethreeequationsformthebasisformostuiddynamictheory.Theseequationstaketheformofnon-linearpartialdifferentialequationswhichareanalyticallyunsolvableandcomputationallytaxing.Tobetterunderstanduiddynamics,simplifyingassumptionsarenecessaryandwillbeexplicitlymentionedastheyareusedinChapter 1 .Themostsevererestrictiontheoreticallyplacedonasystemistostipulatethatthevelocityeldoftheuidisequaltothegradientofascalareld.Thistypeofowiscalledpotentialow.Theentirevelocityeldisdenedbyascalarpotentialwhicheliminatescomplexowpatternsandvorticity.Itistheowtypethatisachievedwhenthereareno,orentirelynegligible,shearstresses.Descriptionsofrealphysicalsituationswiththissimplicationareusefulforbasicunderstandingofowpatterns.OneexampleofasuccessfulpotentialowexplanationistheBernoullieffectwhichisusedtounderstandthepressuresonaspinningbaseballorowoveranairplanewing.Thissimpleowtypeisalwaysagoodplacetoprepareinitialintuitiveassumptions.Forpotentialow,thevector~visconservativeandcanthereforebereplacedwithrintheEquations 1 1 ,and 1 .Onemotivationforusingsuperuidheliumasatestuidstemsfromthevelocityeldbeingconservativeduetoquantumeffects(seeEquation 1 ).Thewellknown 17

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restrictiononthevelocityeldofsuperuidheliummatchespotentialowdenitionandmakesitanavelyperfectpotentialowuid.Ifthisweretheentirestory,superuidheliumwouldbethearchetypefortestsonpotentialow.PotentialownotonlysimpliestheNaiver-Stokesequations,butthisrestrictionalsodisallowsrotationalowpatternsandturbulence.Howcanthisbe?Thesuperuidvelocityisdenedbythegradientofascalar,butsuperuidshavebeenshowntosupportbothturbulenceandturbulentdecay.Thereisclearevidencetheapparentlysimpleuidisinterestingandunique.Anotherassumptioncommonlyusedinuiddynamicsisthattheowisincompressible.Careisneededhereasincompressibleowisnotthesamethingasanincompressibleuid.Heliumforexampleisaverycompressibleuid,buttheowsthatarediscussedinthisthesisaredescribedinincompressiblelanguage.Pantondescribesincompressibleowsasconstantdensity,viscosity,specicheat,andthermalconductivity.Withtheseassumptions,thevelocityeldcanbefoundusingthecontinuityandmomentumequationswithoutregardfortheenergyequationandequationsofstate[ 2 ].Thepreferenceofpotentialowhasmotivatedworkonthedynamicsofsuperuidhelium.Butduetounexpectedquantumeffects,superuidHe4isabletosupportacomplexandinterestingow.However,likealluidstherearemanydifferenttypesofow.Weelecttoinvestigatethesimplestturbulentsystem,homogenousandisotropicturbulence.Classically,themostcommonwaytoproducethisowtypeisachievedbypullingameshgridthroughastationaryuidorconverselyowingauidthroughastationarygrid.Wehaveelectedtodotheformerinsuperuidhelium.Withincompressible,nearlyNewtonianow,theNaiver-Stokesequationcanbesimpliedto, @~v @t+(~vr)~v=rP +4~v+~f .(1)Inthisequationisthekinematicviscositywhichisdenedastheratioofthedynamicviscosity,,andtheuiddensity,.ThelefthandsideofEquation 1 representsthe 18

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inertialbehavioroftheuidandtherighthandsideisresponsiblefordescribinguidstressanddissipation.Ontheleft,thereistheuiddynamicderivativedescribingtheuidmotion.@~v @tisthevolumeunitaccelerationand(~vr)~vistheconvectiveaccelerationofowthroughthesamplevolume.Therighthandsideoftheequationcontainsthebodyforce,~f,andthepressuregradient,rp,bothofwhichactontheuidperunitvolume.Thenaltermisthedissipativeviscousterm,4~v,responsiblefordissipation.Inthissimpliedform,theNavier-Stokesequationcanbemadenon-dimensionaltoproduceoneormorenon-dimensionalscalingparameterscharacteristicoftheow.Theexistanceoftheseparametersisduetodynamicsimilarityandispossiblythemostimportantconceptinuiddynamics.Thenon-dimensionalparameterrelevantfortheworkinthisthesisistheReynoldsnumber.Bytransformingeachvariabletoitsdimensionlesscounterpart,weareleftwithanewvariablethatisnormalizedtotheuidandow.Forthisprocess,thedoppelgangervariablesare,~v0=~v V,p0=p v2,~f0=D v2,@ @t0=D@ V@,r0=Dr.WiththesewecanrewritetheNavier-Stokesequationinanon-dimensionalform.Intheequationabove,Disthecharacteristiclength,andvisthemeanowvelocity.PluggingtheaboveparametersintotheNavier-Stokesequation,omittingtheprimesymbols,andmultiplyingthroughbyD=v2,thenon-dimensionalNavier-Stokesequationisexpressedas @~v @t+(~vr)~v=rP+1 Re4~v+~f(1)wheretheReynoldsnumber,Re,isdenedas Re=DV .(1)TheReynoldsnumberrepresentstheratiooftheinertialforcetothedissipativeviscousforceinaparticularow.Therefore,forhighReynoldsnumber,kineticenergyistransportedtodifferentlengthscaleswithoutdissipation,whileforlowReynoldsnumber 19

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theuidmotiondecaysintointernalheating.Thenon-dimensionalparametersandtheunitfreeReynoldsnumberandthetheoryofdynamicsimilaritystatethatowswithsimilarparameterscanbemeasuredinonesystem,andextrapolatedintoacompletelydifferentsystemwithoutlossofgenerality.SincethedissipativeforcesareinverselyproportionaltothecharacteristiclengthscaleD(Equation 1 andEquation 1 ),energyispredominantlylostbymotiononthesmallestscales.InotherwordshighReynoldsnumberowsarenotdissipative,whilethelowReynoldsnumberowsare.Oneoftherequirementsforturbulencementionedisthekineticenergyoftheowisspreadoutovermanylengthscales.However,forowswithhighReynoldsnumbertheremustbesomemechanismwhichtransportstheenergyfromlargelengthscalestothesmall.ThisthoughtledA.N.Kolmogorovin1941[ 4 ]toderiveaformulafortheenergyspectrumofturbulenceacrossthedifferentscales.HowevertheKolmogorovtheoryisasymptotic,itisvalidwhencertainconditionsaremet.SuchconditionsareahighReynoldsnumber,energyistransferredfromlargelengthscalestosmallerones,andthattheturbulenceisrandom.HeretheassumptionsandresultsoftheKolmogorovtheoryofturbulenceareoutlined.Thisgeneraloutlinecanbefoundinmanydifferenttextsonclassicalturbulence[ 1 3 ].WithFourierintegrals,theNavier-Stokesequationcanbewritteninspectralformmakingittransparenthowenergymaybetransferredamongdifferentwavenumbers.Fortherstorderofenergytransferitisassumedthatk1=k2+k3,wheresubsequentordersofapproximationseveralmorewavenumberscanbeinvolved.Forthisderivation,itisassumedthatthepredominantwavenumbermixingcomesfromsimilarkvalues,ork2k3ratherthank1k2.Thisassumptionpicksthewavenumbersthatmostrapidlyadvancetheenergyspectruminkspace,thisassumptionisalsoloosenedinmorerigorouscalculationsoftheenergyspectrum.Itisclearthatmixingofthewavenumbers,bysummation,leadstoeverincreasingwavenumbers,eventuallyresultingwithaverylargevalueofk1.Thisprocessisenvisionedtooccuruntilacutoffwave 20

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numberkmax,wheretheReynoldsnumberisapproximatelyequaltooneandviscousdissipationbecomesrelevant.Theintermediaterangeofvalidityink-space,from2=Lto2=~`,iscalledtheinertialrange,whereListhechannelsizeand~`isthedissipationlengthscale.AtthedissipationlengthscaletheReynoldsnumberwillbereducedtoRe`=O(1).Insidethisspecializedregimeoftheturbulentenergyspectrum,theenergydensityatanygivenkvalueshoulddependonlyonthewavenumberandtherateofenergydissipation,denedas.ThisassumptionisvalidbecauseweareexaminingtheturbulenceintheinertialregimewherethehighReynoldsnumberdictatesthatviscosityandinternalstressesarenotprimaryeffects.ThereforeEcanonlybefunctionallydependentonthesevariables,orE(k,)=Ck.HereandarearbitraryconstantswhichmustbeworkedouttomatchtheunitsofE.Adimensionalargumentcanbemadeforthespectrumoftheenergy,namelythatgiventhedimensionsofkand,onlycertainvaluesfortheexponentsandareallowed.Apossiblecongurationforthesystemis E(k,)=Ck)]TJ /F6 7.97 Tf 6.58 0 Td[(5=32=3.(1)ThisisthefamousKolmogorovspectrum,thespectrumwillprovetobeanimportantstartingpointforthestudyofenergydecayinquantum(aswellasclassical)turbulence.Tofurtherunderstandthequantumsystemasoundfoundationinliquidheliumisrequired. 1.3SuperuidHelium4Theelementofheliumisbothextremelylightandinert.Itbelongstothenobelgascolumnintheperiodictableandisnotreactive.Ithasaveryweakelectro-magneticinteractionwithotherelements,includingitself.Undernormalconditions,itdoesnot 21

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formmoleculesandonlyinteractsthroughtheVanderWaalsdipoleforces.HeliumhasalowmasswhichgivesitarelativelyhighRMSvelocityateverytemperature.Thesepropertiescombinetopushthesaturatedvaporliquefactiontemperatureofheliumdownto4.2K.Inadditiontothelowlignicationtemperature,itwillnotsolidifyatambientpressure,evenatabsolutezero.Toformasolid,pressurehigherthan25barisrequired.Below2.17K,wherethezeropointquantumenergyisgreaterthaneitherthethermalorelectricalinteractionenergyofhelium,theuidpassesthroughasecondorderphasetransitionfromaclassicallydeneduidintoanentirelynewquantumstate.Thesecondorderphasetransitionthatheliumpassesthroughisobservedbyadiscontinuityintherstderivativeofthethermodynamicfreeenergywhichdescribetheuidstate.Forsecondordertransitions,thereisnoreleaseofenergyintheformoflatentheat,buttherearesharpfeaturesinmanyproperties,suchasspecifyheat.Forhelium,thisisfamouslyseenasainnityinheatcapacityatthelambdapoint.ThephasediagramofheliumisshowninFigure 1-1 .ThelowtemperaturephaseofheliumiscommonlyreferredtoassuperuidheliumorasheliumII.Thenamesuperuidisduetohelium'svanishinglysmallkinematicviscositywhichmakesitowlikeasuperuid.TheminusculeorvanishingviscosityisthepropertythatproducesthehighReynoldsnumberowsrelativetoclassicaluids.AchartofkinematicandeffectivekinematicviscositiesisshowninTable 1-1 [ 5 ].HeliumIIsetsthebottomofthischart,butitsharesitsvanishingviscositywiththeotherquantumuids.Theseincludethelighterisotopeofhelium,helium3,andthevariousBose-Einsteincondensedgasses.Eachofthequantumuidshasavanishingclassicalviscosity,butcontainsanite`effective'viscosity.Forquantumuidsthesourceoftheviscosityisdifferentthanthatofaclassicaluidandisentirelyabsentduringlaminarow.ThelowviscosityofheliumII,forturbulentow,instigateditsuseasatestuid.TheinformationinTable 1-1 begsaninterestingquestion:howcanasuperuidshowanyviscosity?Thepreviousdenitionstates`asuperuidowswithoutresistance', 22

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Table1-1. KinematicViscosityofDifferentFluids MaterialKinematicViscosity Air1.9)]TJ /F4 11.955 Tf 11.95 0 Td[(25410)]TJ /F6 7.97 Tf 6.58 0 Td[(6m2=s[ 6 ]CompressedAir1.1)]TJ /F4 11.955 Tf 11.95 0 Td[(1,00010)]TJ /F6 7.97 Tf 6.59 0 Td[(7m2=s[ 5 ]Water2.9)]TJ /F4 11.955 Tf 11.95 0 Td[(17.8710)]TJ /F6 7.97 Tf 6.59 0 Td[(7m2=sFreon,1.9)]TJ /F4 11.955 Tf 11.95 0 Td[(3.110)]TJ /F6 7.97 Tf 6.58 0 Td[(7m2=sMercury6.7)]TJ /F4 11.955 Tf 11.95 0 Td[(12.410)]TJ /F6 7.97 Tf 6.59 0 Td[(8m2=sLiquidHelium1.8)]TJ /F4 11.955 Tf 11.95 0 Td[(2.610)]TJ /F6 7.97 Tf 6.58 0 Td[(6m2=sCompressedHelium5)]TJ /F4 11.955 Tf 11.96 0 Td[(10,00010)]TJ /F6 7.97 Tf 6.59 0 Td[(8m2=sSuperuidHelium1.0)]TJ /F4 11.955 Tf 11.95 0 Td[(20010)]TJ /F6 7.97 Tf 6.58 0 Td[(11m2=s[ 7 ] implyingzeroviscosity.Howcanbothofthesestatementsbetrue?Theansweris,liesinhowonemeasurestheviscosityandinwhatsituationitismeasured.TherstmeasurementsontheviscosityofheliumII,whichshowedazeroviscosity,weredonebyattemptingtomeasurethepressuredropofHeIIinPoiseuilleow(forcedowthroughapipe).TheseexperimentsproducedavalueofviscositywhichwasvanishinglysmallandareshowninFigure 1-2 [ 8 ].Theviscosityinthisgureiscalculatedbytakingthederivativeofvelocitywithrespecttotheappliedpressureacrossthechannel.ForPoiseuilleow,azeroslopemeansthattheowvelocitythroughthechannelisindependentofappliedpressureandthatthereiszeroshearforceontheliquidfromthewalls.However,whentheviscosityofheliumismeasuredbyadifferentmechanism,theresultsarestartlinglydifferent.TheseareshowninFigure 1-3 [ 9 ].Thesedataarefromthefamousexperimentwhereatorsionaldiskisrotatedinabathofsuperuidheliumandthechangeinnaturalfrequencyismeasured.Forthisexperimenttheviscosityisinferredfromtheamountofuiddraggedbytheoscillator.Thedifferencebetweenthesetwoexperimentsissubtle.Bothtrytomeasuretheensembleuidviscosity,butareinfactaremeasuringitintwodistinctcomponentsofsuperuidhelium.Figure 1-2 showsadiminishingviscositywithtemperaturewhileFigure 1-3 showsasharpriseinviscosityasafunctionofthetemperature.Theseseeminglyopposingmeasurementsareduetothedifferentmeasurementtechniques.InthePoiseuilleowexperiments,themeasuredquantityistheshearofthesuperuidwiththecapillarywalls,whileintherotatingdisc 23

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experimenttheincreasingmeanfreepathoftheexcitationsinthesuperuidbulkaremeasured.TheseexperimentsledtotheempiricaltwouidmodelwhereheliumIIiscomposedof2interpenetratinguidswithdifferentproperties.Theseparationofnormalandsuperisthefoundationofthetwouidmodel.Thiswellknownphenomenon(showninmanyheliumtextssuchasWilksandBetts[ 10 ])showsthatbelowthelambdatransition(at2.17K),therelativedensityofnormaluidmonotonicallydecreasesasthesuperuidcomponentincreaseskeepingthetotalrelativedensityalwaysequalto1.ThetheoreticalbasisofthetwouidmodelofliquidheliumwasrstproposedbyTizain1938toexplainseveralstrangebehaviors,suchasthelossofviscosity,thermo-mechanicaleffectandvarioussoundmodes.ThetwouidmodelresultsfromtheassumptionthatonlyeightvariablesareneededtodeneacompletesetofparameterstodescribethehydrodynamicpropertiesofsuperuidheliumII.ThedescriptionandsolutionforthemodelwasrstproposedbyLandau[ 11 ]andiswelldescribedbyS.J.Putterman[ 12 ].Theresultofthederivationisthefollowingsetof4equations, @ @t+r(n~vn+s~vs)=0(1) @s @t+r(s~vn)=0(1) D~vs Dt=0(1) @ @t(nvn+svs)i+@ @r(pi+nvn,ivn,+svs,ivs,)=0.(1)Thenandssubscriptsrepresentthenormalandsuperuidcomponentsrespectively,D() DtisdenedinEquation 1 ,isthetotaluiddensityincludingbothsandn,~visthevelocity,andEinsteinnotationisusedinthe4thequation.Thetwouidmodelhasfoundenormoussuccessinthedescriptionofthebehaviorofliquidhelium.Inessence,itallowseachuidcomponenttobehaveindependentlyoftheanotherwhilebeingcontainedbythesamevolume.Thetwouidmodelallowsforeachcomponenttohaveitsownvelocityeldandcarryadistinctentropy.Therefore,the 24

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superuidcomponentwillmoveaccordingtoapotentialowandwillcarryzeroentropy,andalloftheheatenergyandviscosityarecarriedinthenormaluidcomponent.Therearemanywelldescribedeffectsbythismodel,suchasthethermomechanicaleffectandseveralsoundmodes.Thethermomechanicaleffectassertsthatchangesintemperaturecandriveaverylargechangeinpressure,andisobservedintheformofthefountaineffect.Thetwouidmodelalsosupportsseveralnewsoundmodes,2nd,3rd,and4thsoundinadditiontothenormalpressurewavesof1stsound.Secondsoundisatransportmodeinwhichsuperuidandnormaluidmove180ooutofphase,suchthatthereisnoorverylittlechangeintheensembleuiddensity.Thirdsoundisamodeinwhichtheheliumisconnedtoatwodimensionallmandthenormaluidcomponentislockedtothesurfaces,allowingasurfacewaveofpuresuperuidtopropagate.Fourthsoundisthesameas3rd,butwiththenormaluidlockedintoaonedimensionalchannel.Itjusthappensthattheattenuationofsecondsoundwavesthroughvorticityisasensitivemeasurementofvortexlinedensity.Secondsoundisacommonmeasurementtoolabove1Kanditspropertieswillbereviewedinsec. 1.4 .Thetwouidmodeldescribesmanypropertiesofsuperuidhelium,butitfallsshortinsomesituations.Oneareainparticularexempliestheinsufcienciesofthemodel.Ifthesuperuidcomponentistrulydescribedbypotentialow,thenr~v0everywhereatsufcientlylowtemperatures.ThisleadstoacirculationintegralofH~vd`=0,implyingthatcircularmotionisnotallowedandthereisnovorticity.ExperimentalevidencecountertothiswasshowninanexperimentbyOsborne[ 13 ],whereaclassicalmeniscusisseenonarotatingsuperuid.Thecirculationintegraldictatedbythetwouidmodeldoesnotallowforaslopedmeniscusinasimplyconnectedvessel,suchastherotatingvesselinthisexperiment.Therefore,somethingmustbemissinginthis2uidmodel.AmoresophisticateddescriptionofthesuperuidHe4waspostulatedbyLondon.Thismoreformalapproachtounderstandingthebehaviorofsuperuidtreatstheuidas 25

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aquantumliquid,completewithawavefunction.Theexistenceofamacroscopicwavefunctionwiththeform =j jeiS(~r,t),(1)wasproposed.Here isthewavefunctionandSisthephase.Thisquantumwavefunctionobeysallthetypicalproperties,includingthedenitionofthevelocityoperator,)]TJ /F3 11.955 Tf 9.3 0 Td[(ir.Thereforethiswavefunctionnaturallygivesrisetoapotentialvelocityeld, ~v=~=mHerS(~r,t),(1)wheremHeisthemassofaheliumatom.Thewavefunctionproducesavelocityeldthatisagradientofascalar,inthiscasethephaseofthewavefunction.Thebrillianceofthisformulationisthatthevelocityeldforasuperuidisthenaturalresultofquantummechanics.FeynmanandOsangerin1955observedthatwiththisformulationthevelocityeldisnotrequiredtobesimplyconnected[ 14 ].Therefore,twodimensionallineardefectsintheuideldarepossibleandcircularowaroundthesedefectsdoesnotviolateanyphysicalpropertiesandstillallowsthesuperuidtobefreeofviscosity!Infactthecirculationthecirculationaroundavortexcorecanbecalculated.Giventhatthewavefunctionofsuperuidheliuminsinglevalued,ifthephaseSisintegratedinaclosedlooparoundavortexdislocationtheresultingvalueshouldbeanintegermultipleof2.Formallythisisexpressedbythefollowingintegral ISd~r=ImHe ~~vd~r=mHe ~=2,(1)whereH~vd~risthecirculation.FollowingfromEquation 1 =nh mHe,(1)wherenisthequantumnumberforthecirculation.Inallexperimentsnhasbeenfoundtobeequaltoexactly1.Vortexlinesandtheirdynamicsarediscussedinsection 1.5 .AfterthesecondsoundtechniqueisdescribedinSection 1.4 26

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Figure1-1. Thelowtemperaturephasediagramofhelium4.ReprintedfromLondon,F.London,Superuids:Macroscopictheoryofsuperuidhelium,Structureofmatterseries.[ 15 ]. Figure1-2. Poiseuilleheliumowinasmallcapillary.Thisgureshowsaclearincreaseinowvelocitiesthroughasmallchannelasafunctionofappliedpressurewhenthetemperatureoftheheliumbathislowered.Thelowertemperaturedata(1.164K)arepresentedinthetopofthegureandthetemperatureincreasesuntilthebottomsetofdata(2.162K).Thisimpliesthattheviscosityofheliumdecreasesasafunctionoftemperature.ReprintedfromMisener,J.Allen,D.Misener,RoyalSocietyofLondonProceedingsSeriesA172,467(1939)[ 8 ] 27

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Figure1-3. Thisplotshowsthenormaluidviscosityasafunctionoftemperatureinarotatingbodyofsuperuidhelium.Thisdataisindirectoppositiontothedecreasingviscosityshowninthepreviousgure.ReprintedwiththepermissionfromNRCresearchpress,CanadianJournalofPhysics41,596(1963)[ 9 ]. 28

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1.4SecondSoundPerhapsthebestuseforanytheoryistheabilitytopredictnewphenomenon.Forthetwouidmodel,therstinstanceofthiswastheobservationofsoundmodes.AfterthetwouidmodelwaspresentedbyTisza[ 16 ]in1938andlaterformalizedbyLandau[ 11 ]in1941,differentsoundmodesinheliumIIweredetectedbyPeshkov[ 17 ]in1944.Thisprovidedstrongevidencesupportingthetwouidtheoryinthisparticularsituation.Inadditiontoconrmationofearlierideasaboutthepropertiesofhelium,secondsoundhasturnedintoastrongtooltostudyvortexdynamics.Onceagain,followingPutterman,wecanlinearizeEquations 1 1 forthesituationofsmallpressureandentropychangesrelativetotheirabsolutevalue.Thisisthecaseforsound.Followingfromthisweobtainaclosedsetof4equationstodescribethesecondsoundmode, @ @t+r(n~vn+s~vs)=0(1) so@ @t+o@s @t+r(s~vn)=0(1) @~vs @t)]TJ /F3 11.955 Tf 11.95 0 Td[(sorT+1 orp=0(1) n,0@~vn @t+s,0@~vs @t=rp.(1)Thezerosubscriptdenotestheequilibriumstatesinwhichthesoundpropagates.Alloftheequationsareexpandedtotherstorderofthesmallpressureandentropyuctuations.Simultaneouslysolvingthesefourrelationsfortravelinglongitudinalwavesrevealstwodistinctsoundmodes,onewithavelocityu2=u21+u21u22 u21)]TJ /F11 7.97 Tf 6.59 0 Td[(u22(1)]TJ /F3 11.955 Tf 11.78 0 Td[(Cv=Cp)andtheother,u2=u22+u21u22 u21)]TJ /F11 7.97 Tf 6.58 0 Td[(u22(1)]TJ /F3 11.955 Tf 11.8 0 Td[(Cv=Cp).Theparameter1-Cv=Cpisalmostalwaysaverysmallnumber(10)]TJ /F6 7.97 Tf 6.59 0 Td[(3),asthedifferencebetweenconstantvolumeandpressurespecicheatissmall.Thereforethesoundmodesareoftenonlyexaminedintherstorder,oru=u1oru2.Asitturnsoutinu1thesuperuidandnormaluidcomponentstravelinphase,muchlikeclassicalsound,andinu2thetwocomponentstraveloutofphase.Thisisthenewanduniquesecondsoundmode. 29

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Theabsorptionofsecondsoundiscommonlyused[ 18 19 ]andisasensitivetool[ 20 ]todeterminethethevortexlinedensityinavolumeofquantumturbulence.Themeasuredlinedensity,L,canberelatedtothevorticity!ofthevolumebytheequation, !=L.(1)ThisequationisknownasFeynman'srule[ 21 ].Forexample,ifonevortexline(L=1cmline=cm3)withcirculation(=9.9710)]TJ /F6 7.97 Tf 6.59 0 Td[(4cm2=s),isstretchedacrossa1cmsamplevolume,thevorticityofthevolumeisL=9.9710)]TJ /F6 7.97 Tf 6.59 0 Td[(4Hz.Thisistheequivalentofclassicallyrotatingthesamecellonceevery17hours.Therefore,foranybuttheslowestofrotationsmanyvortexlamentswillspanasuperuidcontainer.Secondsoundisattenuatedbyseveraldifferentmeansotherthantheinteractionwithvorticity,suchasviscousbulkandsurfacelosses.Toaccountfortheselossesinthelinedensitymeasurements,acalibrationinquiescentuidisrequired.Thecalibrationisconductedbyexcitingastandingwaveintheexperimentalcavityandscanningfrequenciesaroundtheresonancepeak.Fromtheresonance,theQorqualityfactor,isdeterminedbytakingtheratiooftheresonantfrequencytothewidthoftheresonantpeak,measuredasthefullwidthathalfmaximumoftheresonanceresponse.Qisameasureoftheratioofenergystoredintheoscillatortotheenergydissipatedforeachoscillationcycle.MeasurementsandconclusionsfromourspecicexperimentswillbepresentedinthesecondsoundexperimentsectioninChapter 5 .AsfundamentalresearchintothepropertiesofheliumprogressedHansonandPellamcompletedanextensiveexperimentalstudyontheattenuationofsecondsoundin1954whichproducedamodelforbulkattenuationintheuidas[ 22 ] B=!2 2v32(4 3s n+IIs n+K C).(1)Hereistheattenuationofthesecondsound,!istheangularfrequency,isthedensity,v2isthesoundvelocity,istheviscosityofthenormaluidorordinary 30

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viscosity,IIisthesecondviscositycoefcient,sandnarethesuperandnormaluiddensities,Kisanalogoustoordinarythermalconductivity,andCisthespecicheatcapacityofheliumII.ThersttermintheequationisattributedtoPellam,andisduetotheordinarynormaluidviscosityofhelium[ 23 ].Thesecondtermisassociatedwithboththenormalandsuperuidcomponentsandistermedsecondviscosity[ 24 25 ].Thenaltermisofsecondorderinmagnitudeandisduetothermalprocessesinthesoundmode.Togethertheydenethebulkheliumsecondsoundattenuation.Oneofthethingstonoteaboutthisequationisthe!2dependenceonfrequency,whichmeansthatasfrequencyisincreased,theattenuationincreasesatamuchfasterrate.Also,althoughtemperatureisexplicitlyabsentfromtheequationitseffectsarestillobservedbythetemperaturedependenceofsoundvelocity,uidviscosity,andrelativesuperuiddensity.Therefore,itisfavorabletorunandcalibratethesecondsoundsensorsataplaceintheTvs.vplot(Figure 1-4 )thathasaverysmallderivative.Itisworthnotingthatthereisnosoundamplitudedependenceontheattenuation(atleastforthesoundamplitudesusedinthestudypresentedinRef.[ 22 ]),solargeramplitudewavesarenotnecessarilymoredampedinthelinearregime.Dissipationfromthevortexinteractionsandbulkinteractionsisjoinedbythesurfaceeffects.Thesurfacelossesarecausedbyhelium'sinteractionwithitscontainingvessel.FollowingfromHeisermanandRudnick[ 26 ],thedissipationfromthewallscanbeexpressedas, S=B 4As v2(1)whereisthecharacteristiclengthoverwhichaviscousdiffusionwavedecaysandAandBaregeometryspecicterms.Thisfactoristypicallyconsideredtobeasmallbutmeasurablecontributiontothebackgroundattenuationandthenalcomponenttotheattenuationofsecondsound.Measuringthetotalattenuationrepresentedbythesetermsyieldsthevortexlinedensity,v. 31

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Heresecondsoundisproducedbythewellestablishedtechniqueofforcedoscillationofaporouslm.Thesesensorsweremadebythesametechniqueasmanyotherdetectors[ 27 28 ].Theoscillatinglmislitteredwithsmallpores(about0.2mindiameter[ 29 ])throughwhichthesuperuidowsunimpeded,whilethenormaluidviscositypreventsitstransit.Theresultisthousandsonpointsourcesecondsoundwavesthattraveloutintothetransducer.Afterashortdistancethewavesbegintointeractwithoneanotherand,byHuygens'principleaplanewaveisformedandpropagatesacrossthecell.Theplanewaveamplitudeiswrittenas, A(f,t)=Aoe)]TJ /F10 7.97 Tf 6.58 0 Td[(xei(kx)]TJ /F10 7.97 Tf 6.58 0 Td[(!t+').(1)InEquation 1 ,Aistheamplitude,isthetotaldissipation,xisthedistancetraveled,!isthefrequency,and'isthephase.SummarizingtheresultsfromStalp[ 30 ]andSwanson[ 31 ]wecandeterminethevorticityinaexperimentalvolumebytheattenuationofsecondsound.IfweacceptthatEquation 1 representsthesecondsoundwaveamplitude,thenafteronetripacrossthecelltheamplitudefallstoA(f,D=v2)=Aoe)]TJ /F10 7.97 Tf 6.59 0 Td[(Dei(kD)]TJ /F10 7.97 Tf 6.59 0 Td[(!D=v2+'),whereDisthedistancethewavetravels.Assumingthattheentirewavefrontisnotabsorbed,someofitwillreectoffofthedetectingsensorandcirculatebackacrossthecell.Ifthesumofallthebackandforthreectionsaretakenintoaccountwearriveatthet=1amplitudeofA(f,1)=Ao eR(erD+R)]TJ /F11 7.97 Tf 6.59 0 Td[(iD(!=v2)]TJ /F10 7.97 Tf 6.59 0 Td[(i))]TJ /F3 11.955 Tf 11.95 0 Td[(e)]TJ /F10 7.97 Tf 6.58 0 Td[(rD)]TJ /F11 7.97 Tf 6.59 0 Td[(R+iD(!=v2)]TJ /F10 7.97 Tf 6.59 0 Td[(i)).Inthisequationtheattenuationhasbeenseparatedintoitsdissipitivecomponentranditsimaginarycomponentthatshiftsthephaseoftheplanewavei,where=r+ii.Sinceitisnotthecomplexamplitudethatisexperimentallymeasuredbyalock-inamplier,themagnitudeofthesignalmustbetakenbymultiplyingAbyitscomplexconjugateAandtakingthesecondroot.Thefollowingequationrevealstheamplitudeofthesecondsoundasafunctionoffrequencyandattenuationafterthis 32

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operation, A(f,1)=AT(2cosh(2(rD+R)))]TJ /F3 11.955 Tf 11.96 0 Td[(cos(D(!=v2)]TJ /F5 11.955 Tf 11.96 0 Td[(i)))(1)whereATisdenedasequaltoAoeR.MoredetailonthesederivationscanbefoundintheworkofStalpandSwanson[ 30 31 ].Theattenuationofsecondsoundisthelinearsumofthethreeprocesses,=v+B+S.IfweassumesurfacelossesareinsignicantinthesamevainasStalp[ 30 ]orthatitisunchangingwithvorticityandcanthereforebegroupedwiththebulklosses,wecansolveforthevorticityasafunctionofsecondsoundamplitude.canbeexplicitlysolvedfor.Theamplitudeasafunctionofvortexlinedensityis A2=A2L=01)]TJ /F3 11.955 Tf 11.96 0 Td[(cos(2Dfr=Qv2) cosh(2(B+v)D+2R))]TJ /F4 11.955 Tf 11.96 0 Td[(1.(1)AL=0istheresonantamplitudewithzerovortexlinedensity.Thevortexlineattenuationamplitudecanbefound,givingthevorticityasafunctionofmeasuredamplitude[ 30 ] L=1 2Dln[1+(Ao=A)2C+p 2(Ao=A)2C+(Ao=A)4C2 1+C+p 2C+C2],(1)whereC=1)]TJ /F3 11.955 Tf 12.27 0 Td[(cos(2Do=u2).Inadditiontoacquiringthesecondsoundattenuation,therelationshipbetweenhomogenousandisotropicvorticityandsecondsoundattenuationiswellknownfromtheworksofHallandVinen[ 32 33 ], !rms=16v2v B.(1)HereBisthetemperaturedependentmutualfrictionparameterand!rmsistherootmeansquarevorticity,whatweseektodetermine.Mutualfrictionisthedissipativesourcethatoccursasaresultofexcitationsinhelium(rotonsandphonons)scatteringoffofvortexcores.Theequationdescribingmutualfrictionis[ 21 ], fD=)]TJ /F5 11.955 Tf 9.3 0 Td[(s~s0[~s0(~vn)]TJ /F5 11.955 Tf 14.13 .5 Td[(~vsl)])]TJ /F5 11.955 Tf 11.95 0 Td[(0s~s0(~vn)]TJ /F5 11.955 Tf 14.13 .5 Td[(~vsl).(1) 33

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Wherevslisthelocalsuperuidvelocity,~s0istheunitvectoralongthevortexcore,andand0areempiricalparameters.includesthetheparameterBinthesecondsoundequationbytherelation=Bn=2. 34

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Figure1-4. Thisplotshowsthedependenceofsecondsoundvelocityontemperature[ 34 ]. 35

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1.5QuantumDynamicsSecondsoundasameasurementtoolisimportantforoneofthestudiesreportedinthisdissertationandforseveralprojectsunderway(seeChapter 5 ).However,atmillikelvintemperatures,thistoolisunavailable,duetothelackofnormaluid,andadifferenttypeofexperiment,withdifferentdynamicsisinvestigated.Whilemanyaspectsofuidmotionchangeinthelowtemperaturelimit,notablythelossofmutualfriction,betweenthesuperuidcomponentandnormaluidcomponent,theuidkineticenergyisstillcontainedinsidethevortexbundleandismeasured,torstorder,bytheintegratedvortexlinelength.Thismeasurementistorstorderbecauseoftheoverlappingoweldsarounddifferentvortexlines.Thekineticenergyforasingle,isolated,quantumvortexlineperunitlengthisobtainedbyintegrationoftheuid'skineticenergyperunitlinelengthalongitsvortexcore,Ke=Rba1 2v2dr2.Forsuperuidheliumthisisbecomes K.E.=s2 4ln(b a),(1)whereisthecirculation,bisalength(suchastheexperimentalcelldiameter),andaisthecoreradius.Thevortexlineenergy,torstorder,isdirectlyproportionaltoitslength,withthehigherordertermsdependingontheinteractionbetweenthevelocityeldsofoneormorevortexlines.Therefore,forthevortexarraytobeinitsgroundstatethetotallinelengthofthesystemmustbeminimized.Thedecreaseinkineticenergywhenthevortextangleismovedintoitsshortestintegratedlengthstateisthedrivingforceforenergydecayinquantumturbulence.Theproportionalityoflineenergytolinelengthhastwoconsequenceswhichdrivethemotion.Thesearearestoringforceagainstdeformationsalongthevortexcoreandvortexlinereconnectionsastwovorticesinteractatcloserange.Bothofthesephenomenaarepresentinclassicaluids,butarenottypicallyimportantinthedecayofenergyduetotheviscousenergychannel.Avortexlinereconnectionistheinteractionbetweentwoseparatevorticesastheycomeintocloseproximitytoeachother.The 36

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linereconnectionisadistincttopologicalchangeintheuidwhichhasbeenrecentlyexperimentallyobservedbyagroupatMaryland[ 35 ].Itoccursastwodifferentvortexlinesortwopartsofthesamevortexlinecrosspaths.Inthisevent,itisenergeticallyfavorableforthetwolinestobreakapartattheirintersectionandreassociatewiththeotherline.Bothlinesbreakinhalf,changingthetopologyofthetangle,andreformwithtwolinesinterchangingaportionoftheirlengthandreducingthetotallinelengthofthesystemasawhole.ThisprocessisshownincartoonforminFigure 1-5 .Thisprocessproducesaverysharpkinkalongthevortexline,whichisparamountinthetransportofenergyinquantumturbulence.Thesharpkinkexciteswavemodesonthevortexcore,whichhavebeenshowntohaveinterestingproperties.Reconnectionsfundamentallychangethetopologyofthetangle,butmoreimportantlyforturbulentdecay,theyhavetheabilitytoproducesharpkinksonvortexlines.Thepropagationandeffectofthevortexwaveswillbeexpandedbelowwhenwediscusstherestoringforcesontheline.Althoughvortexlinereconnectionsareaknownphenomenon,theyarestillachallengefortheoristsandparticularlyforcomputationstocontendwithinturbulence.Thedifcultyforsimulationsisshownbythechangingcriticaldistancewithwhichlinesreconnect[ 36 38 ]whichisinsertedadhocdifferentlyfordifferentresearchgroups.Remarkablythough,iftheworkisdonewiththeentirenon-linearSchrodingerorGross-Pitaevskiiequationreconnectionsnaturallyarise[ 39 ].Duringreconnection,theoveralllinedensityofthetangledecreases,thelossofkineticenergythroughtheproductionofphonons.Leadbeateretal.studiedthisproblembysimulatingthereconnectionoftworingvorticesanddeterminedthattheenergygiventophononsdependsstronglyontheorientationofthevortexrings.Specically,thelossoflinelengthistan2(=2)[ 40 ],whereistheanglebetweencirculationandtheaxesofthering.Theangulardependenceisduetotheneedforthevortexlinesinvolvedinthereconnectiontolocallyorientthemselvessuchthattheconnectingpartsareparallelandthereforetheirvelocityeldswillexactlycancelduringthereconnectionevent.The 37

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energylosttophononsbythisprocessisnotlargerelativetootherpossibledissipationsources.Therefore,reconnectionsarenotthoughttodirectlyeffecttheturbulentdynamicsunlessthereisaverylargevortexlinedensityandaglutofreconnections.Vortexlinereconnectionsalsooccurinclassicaluiddynamics,althoughtheirpropertiesaresubstantiallydifferent.Forclassicalows,thereconnectionprocessseemstonotonlyinvolvebutdependontheuidviscosity,whichisabsentinasuperuid.AparticularreportonclassicaluidsstatesThisreconnectionprocesscannotbecapturedbyvortexmethodswhichusecontinuouslamentsforadescriptionofinviscidow[ 41 ],sincewithoutviscositythevortextuberetainsitsidentityforalltime.Inorderforreconnectionstooccurdissipativeeffectsmustbeconsidered.[ 42 ]Thisisamostinterestingpoint,sincemanyearlyquantumturbulencesimulationsexplicitlyusecontinuouslamentswithreconnectionsaddedinadhoc.Vortexreconnectionsinsuperuidheliumarenotentirelyunderstoodonthemostfundamentallevel,buttheireffectsinquantumturbulenceareimmenseand,withoutattentiontodetail,onemaylosetheconnectionwithclassicaluidresearch.Finally,thesharpdeformationsalongthevortexlineswillbeinvestigated.Thesekinksproducenon-linearwavemodesalongthevortexcoreitselfduetothestraighteningforceonthelines.Therestoringforceonavortexcoreisaspecialcaseoftheclassicallyknownmagnusforce,whereavortexisinuencedbyanexternalvelocityeldortwovorticesareinuencedbyeachother'svelocityeld.Themagnusforceisclassicallywrittenas~F=)]TJ /F5 11.955 Tf 9.3 0 Td[(~~V,whereisthecirculationandVisthevelocityeldaroundthecore.Forsuperuidheliumthisisequalto, ~F=~f+s^(~vL)]TJ /F5 11.955 Tf 13.12 .5 Td[(~vs),(1)whereish=m,^istheunitvectoralongthelengthofthevortex,~vLand~vsarethevortexlinevelocityandthesuperuidvelicityrespectively,and~fisthemutualfrictionforceofthevortexintheow[ 43 ].Whentheforceexperiencedbythevortexlineisdue 38

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toadifferentpartofthesamevortexlament,thelocalvelocityeldcanbegivenbytheBiot-Savartlaw, ~L(~ro)= 4Z(~r)]TJ /F5 11.955 Tf 12.92 .5 Td[(~ro)dr j(~r)]TJ /F5 11.955 Tf 12.93 .5 Td[(~ro)j3.(1)Theposition~randtheintegralaretakenalongthelengthofthevortex.Theintegraldivergesfor~r=~ro,butisaccountedforbythenitesizeofthevortexcore.Theself-inducedowisimportantinthedynamicsofsuperuidturbulence,becauseitcanbeshownthattheBiot-Savartforcesupportsthepropagationofhelicalwavesalongavortexcore.CalledKelvinwaves,theseareaveryimportantpathforthetransportofkineticenergyforsuperuidturbulencedecay.ThiswaveformanditsdispersionwereexpressedbyLordKelvinin1880[ 21 44 ].Equation 1 canbeexpandedforlocaldeformationsofthevortexcorewhere~rand~roaresimilar.Inthiscasetheintegralreducestoonlythelocalcontributions ~vs(~ro)=(=4(R))ln(R=ao),(1)HereRisthelocalradiusofcurvatureofthevortexcoreandaoisthecorediameter.Equation 1 showsthatsmallvaluesofRareassociatedwithlargelinevelocities.Therefore,thetwoprocessesdescribedinthissectionstronglyinteractinthedynamicsoftheuid.Specically,areconnectioneventbetweentwoseparateortwopartsofthesamevortexlamentcreatesasharpkinkinthevortextube,whichtheninducesahighvelocitymotiononthevortexcoreduetothemagnuseffect.Thisinturninstigatesmorereconnectionsduetotheincreasedmotionandsoonasacascadeprocess.Oncethemotionisestablished,non-linearKelvinwavesareabletotransporttheenergytodifferentwavenumbersalongthevortexcoreinaprocesseerilysimilartotheRichardsoncascadeintheKolmogorovspectrum.Theseriesofprocessesoutlinedarebelievedtobeveryimportanttothedecayofenergyinquantumturbulence,particularlyinthezerotemperaturelimit.AswasdescribedinSec. 1.2 ,theenergyspectrumofclassicalturbulenceisthoughttobe 39

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spreadoutallovertheinertialregime.Uponreachingahighwavenumber,viscouseffectsbecomedominant.Butinasuperuid,viscouseffectsshouldbeirrelevant.Thismeansthatturbulenceisunabletodecay.However,thereareotherprocessesuniquetoquantumuidsthattakeplace.Thegeneralconsensusinthecommunityisthatformosttypesofturbulenceasecond,different,cascadetakesoverattheendoftheRichardsoncascadeandcarriestheenergytoevenhigherwavenumbers.TheRichardsoncascadeistheprocessbywhichtheKolmogorovspectrumarises.TheideaofacascadeofKelvinwaveswasrstproposedbyB.V.Svistunovin1995[ 45 ],andwasfollowedbyW.F.Vinen'ssuggestionthatatextremelyhighwavenumberstheenergyonthevortexcouldberadiatedasphonons[ 46 47 ].Thesecondenergycascadepresentinquantumturbulenceisanenergytransportalongthevortexlinesthemselves.Themechanismthatcarriesthekineticenergyisthekelvinwavedistortionsofthevortexline.Thecascadeproceedsaccordingtothefollowingscheme:Bundlesofvorticitytravelthroughthesuperuiduntiltheyencounteranothertravelingvorticitybundle.Uponmeeting,severalvortexlamentscrosspathsandperformreconnectionoperations.Sharpkinksformalongthevortexcoreandinitiatemotionofindividualvorticesinsidethevortexbundleduetothemagnusforce.Thevorticesinasinglebundlethenacquiretheimpulsetomoveandbeginreconnectingwiththeirneighboringvortices.Thisprocessfurtherkinksthelaments,sendingenergytoincreasinglyhigherwavenumbers,untilthevortexlinesreachapointwheretheyhavesuchalargeamplitudeofcurvaturethatreconnectionswiththemselvesbecomethemostimportantprocess.Oncethecascadeofreconnectionshasseededthevortexcorewithmanyandhighmomentumwavemodes,thenon-linearkelvinwavesareabletomovetheenergyintostillhighervaluesofktoeventuallyberadiatedasphonons.ForeachreconnectioneventandthesubsequentKelvinwavecascade,thevortexcoreissetinoscillationalongwiththesuperuidsurroundingit.Thisprocesscreatessoundwavequantainthesuperuid,orphonons,whichareabletoradiativelyreduce 40

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thekineticenergyofthesystem.Itisonlyattheextremelyhighwavenumbersthatthephononradiationisefcientenoughtocarryawaysubstantialenergy.Itisnotsurprisingthatlargewavenumbersareneededtoefcientlyradiatetheenergybecauseinsuperuidheliummostoftheenergycontainedinphononsareassociatedwiththehighestenergymodes[ 10 ].Theabovetheoreticalframeworkisnearlyuniversallyacceptedformoderatetangledensities.However,forlowdensityturbulence,suchasthatduetovibratingwiresorquartztuningforks,diffusivedecaythroughtheemissionofvortexringsisalsoapossibility[ 48 ].Forhighdensityturbulencesoundemissiondirectlyfromreconnectioneventscouldalsobearelevantdecaychannel.Thereislittlecoconsciousaboutthepropertiesofthemomentumregimewherequantumenergyandclassicalenergyhavecomparablemagnitudes.Thisprocessisstillupfordebatewithnostrongconsensusinthecommunity[ 49 50 ]. 41

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Figure1-5. Thisshowsacartoonoftwovortexlinesreconnecting.Afterthereconnectiontwonewvorticesareformed,eachwithasharpkinkatthelocationofthevortexcrossinglocation. 42

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1.6CalorimetryOnemeasurementtoolthatisunavailableatlowtemperaturesisthermometers(seeCapter 2 ).Duetotheexceptionaldecreaseintheheatcapacityofheliumbelow1K,theheatingthatarisesasaresultofturbulentmotioncanbemeasuredasatemperaturerise.ThiswasrstpointedoutbySamuelsandBarenghi[ 51 ]foranon-classicalow,buttheideaiswellsuitedforgridturbulence.Tounderstandthisitisimportanttoknowhowtheenergytravelsthroughtheturbulenceasafunctionoftime.SimilartothederivationofKolmogorovspectruminmomentumspace,anequivalentdimensionalargumentcanbemadefortheenergytransportinrealspace.Forturbulencegeneratedatascalemuchlargerthanthedissipationscale,itcanbeassumedthatthemajorityoftheenergyresidesineddieswithacharacteristicsizeDwhichisontheorderofthesizeoftheeddies.TheseeddieswillhaveacharacteristicvelocityU,andtheenergyperunitmassofthesystemwillthereforebe E=U2=2,(1)IfweassumethethedissipationofenergyatlowscalesintheformdescribedbyVinenandNeimela[ 43 ], "=v02L2.(1)ThisequationcanbecomparedtothethederivativeofEquation 1 toshowthatthevortexlinedensityshoulddecaywiththeformof L=2D p v0(t+to))]TJ /F6 7.97 Tf 6.59 0 Td[(3=2.(1)Ifalloftheseequationsarecombinedweareleftwiththeenergyasafunctionoftime, E=2D2 (t+to))]TJ /F6 7.97 Tf 6.58 0 Td[(2.(1) 43

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Theseequations,basedontheknowninformationaboutquantumturbulence,shouldbeapproximatelyaccurate.Thisimpliesthatthedissipationofenergyshouldbe, Q=)]TJ /F4 11.955 Tf 9.3 0 Td[(4D2 (t+t)]TJ /F6 7.97 Tf 6.59 0 Td[(3o).(1) 1.7ScopeofWorkThisdissertationconcentratesonthecreationandmeasurementofquantumturbulencebynovelprocesses.Thersthalfofthedissertation,comprisingofChapters 2 and 3 ,describeworkcompletedtowardsthegoalofcreatingandmeasuringturbulence.ManyofthetechniquesandapparatusdescribedinChapters 2 and 3 weredevelopedascollaborativeeffortsbetweentheIhasresearchgroupandvariousotherresearchgroupsaroundtheworld.Theywerecreatedwiththehopethatthedevelopedtechnologywouldbeimplementedinawiderangingsetofdifferentexperiments.Thelatterportionofthedissertation,comprisingofChapter 4 ,isareviewofthemeasurementsonquantumturbulencecarriedoutatmillikelvintemperatures.Thedissertationwillconcludewithnalthoughtsonongoingandpossiblefuturedirectionsforcontinuedworkonthetechniquesdescribedinthiswork. 44

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CHAPTER2SENSORDEVELOPMENTThedeningequationsofuiddynamicsaremathematicallyintractable,thereforeprogressinexperimentaluiddynamicsisdictatedbythepaceofinnovation.Fluiddynamicsisasensordriveneldwheresophisticatedprobesareusedtoprobeboththesmall,andlargescalestructuresofparticularowpatterns.Tostudyclassicalviscousuids,arraysofpressure,temperature,andvelocitysensorsarecommonplace.Byusingmanysensors,structuralinformationabouttheowisgleanedfromuidpropertycorrelationsintimeandspace.Thecorrelationsareanalyzedto`visualize'theowonalllengthscales.Inadditiontotheseindirectmeasurements,testuidscanbeseededwithdyesorneutrallybuoyantparticleswhichcanbedirectlyvisualizedandtrackedastheuidtravels.Entireconferencesandjournalsarededicatedtonotonlytheuidsbeingstudied,butalsothetechniquesusedtostudythem.However,verylittleoftheinstrumentationcommontostudyingclassicaluidscanbedirectlyappliedtotheinviscidsuperuidHelium4.Manyoftheinterestingfeaturesofheliumdynamicsoccuratextremelysmalllengthscalesandarethereforedifculttomeasure.Forexample,thelargeKolmogorovinertialrange(discussedinSection 1.2 )isonlyofexperimentalinterestifwecanprobethesmalllengthscaleandlargewavenumberproperties.Otherwise,theadditionaldynamicrangeisoflimiteduse.Inaddition,theinterestingquantumphysicsoccursonthelengthscalesofthevortexcoresandtheirrespectivebundles.Therefore,forthein-depthstudyofquantumturbulence,verysmallsensorswithshorttimeconstantsareneeded.Inadditiontothesizerequirementsofthesensors,therigorsoflowtemperatureresearcharequitedemandingontheinstrumentation.Theprimaryconcernwithinstrumentationatmillikelvintemperaturesisheatload.Specically,atverylowtemperaturesthermalheatcapacitiesofmaterialstendtodiminish,andthereforeallofthesensorsusedtomeasureheliumneedtoberunonverysmallpowerloads.In 45

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additiontothespecicheatoflowtemperaturematerials,superuidheliumisanearlyprefectconductorofheat,itisthereforeimpossibletomeasureuidowvelocitiesbyclassicalhotwiretechniques.Heliumisalsooneofthelightestelements,makingitverydifculttoproducesmallneutrallybuoyantparticlestobeusedinlightscatteringandothervisualizationtechniques.Overall,heliumisadifcultmaterialtoworkwithandtostudy.Traditionally,bulkratherthanlocaluidpropertiesaremeasuredinsuperuidhelium,usingcomparablylargerandmorewell-developedsensors.Forexample,secondsoundisameasureoftheaveragevortexlinedensityofaheliumsampleacrosstheentirecrosssectionalareaofasensor,typicallyoforder1cm2.Atlowertemperatures,asimilaraveragedvortexlinedensitymeasurementisaccomplishedusingionizedparticles.Theparticlespropagatethroughasamplelledwithturbulence,andthetotalscatteringoftheionsbytheuidowismeasured.Theintegratedscatteringandtrappingoftheionsareconvertedintotheaveragelinedensity.Sensorslikethesearenotabletoresolvethevorticityonasmalllocalscale,butrathermeasuretheaveragepropertiesovertheentirevolumeencompassedbythesensor.Thesearepowerfultools,butmicroscopicscaledetail,observedinclassicalows,isabsent.However,wehavecontinuedwiththehelpfromourcollaborationtobuildandtestglobalpropertysensors,suchasmodularsecondsoundtransducers.Thermometersareanothersensorusefulatlowtemperatures.Inmanyways,thermometryisamorepowerfultoolforquantumuidsthanclassicalones.Thisisbecauseatlowtemperatures,smallchangesinenergycanbemeasuredascomparablylargechangesintemperature,whereasathighertemperaturestheelevatedheatcapacitiesofmaterialsrequirelargerenergychangestoalterthetemperature.However,inotherwaysthermometryisnotwellsuitedfordynamicmeasurementsatmillikelvintemperaturesbecausecommerciallyavailable,typicalthermometersrequireintegrationtimesof10'sofsecondstoupwardsofseveralminutestoobtainprecisereadings. 46

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Thelongequilibriumtimerequiredforthesesensorstendtomaskthelocaldynamicnatureoftheuid,andthereforethequicktemperaturechangeswhichcorrespondtofastuidmotionsandsmallscaledynamicsarelost.Thisprocessmakesindustrialthermometrynon-idealforourworkandhasledtothedevelopmentofourownsensorsincollaborationwiththeInstituteofSemiconductorPhysicsinKiev,Ukraine,discussedinChapter 2 .Aslightlymorelocalizedprobeisapressuresensor.Thesearecommontechniquesinclassicalhydrodynamics,buthavenotbeenfullydevelopedandmadecommerciallyavailableforworkinsuperuidheliuminthelowtemperaturelimit.Pressuresensorshavebeensuccessfullyusedinpitottubegaugesattemperaturenearandbelowthesuperuidtransition(1.4K)[ 52 53 ].Thesesensorsareexpensiveandstillverylargecomparedwiththesmallestscalemotionofinterestbuthaveshownpromise.IncollaborationwiththeChanresearchgroup,formallyattheUniversityofFlorida,wehavedesignedandtestedminiatureMEMSpressuresensors.Otherlocalizedprobesaresmalloscillatingobjectsforthedetectionofturbulentuidproperties.Bothvibratingquartzcrystalsandvibratingwiresareusedasameasureofvorticity,andthesesensorshavefoundcommonuseinhelium3turbulenceexperimentsaswellashelium4.Thevibratingwireswererstusedtodetectthequantizationofcirculation[ 32 ]andhavebeenusedasasensitiveprobeeversince.ThesehavefounduseinmanylocationsincludingtheUniversityofLancaster,OsakaCityUniversity,andCharlesUniversity[ 54 56 ]Thenaltechniqueusedtoobserveturbulenceisthroughvisualization.Heliumisoftenvisualizedbyinterminglingsolidhydrogenglobulesorhollowglassballsinliquidheliumandscatteringlaserlightfromtheimpuritysurfacestoformanimageoftheuidow.Thesetechniquesarederivedfromtheimagingtechniquesofclassicaluidsandhaveobservedsomeinterestingbehaviors[ 35 57 ].Manyproblemsusingthistechnique,withliquidhelium.Twooftheproblemsare;matchingseedparticledensity 47

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tothatofheliumandarranginganopticalsystemandcryostatsothatthreedimensionalimagescanberecorded.Thereforeanewvisualizationtechniqueisbeingdeveloped,incollaborationwithYaleUniversityandtheManchesterUniversity,usingopticallyexcitedheliummoleculestotracetheuidow[ 58 ]. 2.1SecondSoundSecondsound,asmentionedisacommontechniqueusedtoprobethebulkpropertiesofhelium.Itisasensitiveprobetostudyturbulentenergydecay.Theliteratureislledwithsecondsoundexperimentsandtransducerdesigns[ 27 59 60 ].ThemethodofoscillatingaporousmembranewasrstproposedandtestedbySherlockandEdwards[ 61 ]toproduceanddetectthepresenceof2ndsoundinhelium.AmodularsecondsoundtransducerwasdevelopedthroughourcollaborationwithYaleUniversityandisshowninFigure 2-2 .Thesesecondsoundtransducersproducethesoundmodebyoscillatingametalcoatedporouslmbyanalternatingcurrent.Theporouslmisa5-25mthickpolycarbonatelterpaperwith0.2mholeswhichiscoatedwithathinlayerofmetal.ThelmsaremountedonapolycarbonateringandstretchedacrossthetransducersshowninFigure 2-1 .Pairsofthesetransducers,mounted1cmapartacrossaowchannel,havebeenusedintwoexperiments:opticalexperimenttovisualizesuperuidheliumowandagrid-pullexperimenttomeasurethevortexlinedensityproduced.ThemountforthissecondsoundtransducerisshowninFigure 2-2 alongwiththemachinedrawingsfortheexperimentalcell(intheappendix),Figure A-10 ,andtheelectricaldiagramFigure 2-3 .Thetransducersusedinaresonantcongurationwithoneasadroverandtheotherasdetectoroftheplanewaveactivityinside,wascooledbelowthelambdatransitiontoscanforsecondsoundresonances;atypicalfrequencyscanisshowninFigure 2-4 .ThepertinentparametersofthesecondsoundsensorsaresummarizedinTable 2-1 .Therelativesignalmagnitudefortheresonantpeakat26,000HzisshowninFigure 2-5 asafunctionofd.c.biasvoltageandinFigure 2-6 asafunctionofpeaktopeak 48

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Table2-1. Secondsoundtransducer ParameterValue Poresize0.2mSensordiameter0.16inchGapbetweenspeakerandmicrophone0.8inch oscillatordrivevoltage.Thesecalibrationdatawereusedtopicktheparametersforanexperimentonthedecayofvorticityafteralineargridpullabove1K.ThisexperimentisintroducedinChapter 5 49

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Figure2-1. Thisgureshowstwosecondsoundtransducers,onewilloperateasaspeakerofsound,whiletheotherisamicrophone.Aporuslmisstretchedacrossthebrasselectrodetoproducethesound. Figure2-2. Thisisaphotographoftheframesinwhichthesoundtransducersaremounted.Thesquarechannelinsidethemountsisthesamplespacewhereturbulenceiscreated. 50

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Figure2-3. Thisisaelectricdiagramofthesecondsoundgenerationanddetectionapparatus. Figure2-4. Thisgureshowsascanacrossawiderangeoffrequencieswithmanysecondsoundresonances.TheblackcurveisforaDCbiascurrentof150vandtheredisforabiasof100V.Thepeaktopeakdrivevoltageusedforbothofthesescanswas2V. 51

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Figure2-5. Thisplotshowsthesecondsoundsignaldependenceonthebiasfortwoseparateharmonicsnear26000Hzforapeaktopeakdriveof6V. Figure2-6. Thisgureshowsthesecondsoundresonanceresponseat26KHzvs.peaktopeakdrivevoltagedependencefordifferentbiasvoltages. 52

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2.2ThermistorsTheinitialmotivationforturbulenceresearchinourgroupwasthemeasurementofheatreleaseafteraturbulenteventinasampleofhelium.Thismeasurementwastobetakenasafunctionoftimewithquickrespondingthermometers.Tomeasurethisheatreleasethethermometersneededtobespecicallydesignedtomeettheexperimentalneeds.ForthisreasonacollaboratedbetweentheIhasresearchgroupandV.F.MitinattheInstituteofSemiconductorPhysicsinKiev,Ukraine,wasopened.ThegoalofthiscollaborationwastoproduceminiatureGelmthermometers[ 62 ].Thecharacterizationofthethermometerswasconductedinsituwithourcalorimetryexperiments.Becauseoftherequirementsoflowtemperaturecalorimetry,thethermistorsweredesignedtohaveaveryweakheatlinktoeverythingexcepttheexperimentalhelium.Specically,thesensorswereimmersedinsuperuidhelium4andelectricallywiredwith50mgoldwiretoanelectricalconnector,whichwasheatsunktothemixingchamberofthedilutionfridge.Thewirewasthintoensurethesensorsprimaryheatlinkwasthehelium.Duringthecalibrationonthesesensors,nosinterwasusedtoheatsinkthecelltothemixingchamber,sotheminimumoperatingtemperaturewasabout65mK.Theexperimentalcellwasheatsunktothemixingchamberby41=4"copperrodsmountedonagoldplatedmount.ThefullexperimentalsetupisdescribedinChapter 4 .Thethermistorsarebasedona2mthickthermosensitivelmofGe,depositedon150mthicksubstratemadefromsemi-insulatingGaAs.Layeringofthematerialsbythistechniqueproducesadiffusivedoping,whichmoveschargecarriersfromtheGelayerintothebulkGaAslm.ElectriccontactstoGeareproducedbyasequentialdepositionofMoandAuonthelm[ 63 ].ThesensorisshowninFigure 2-7 withalloftheindividuallayers.ThesensorareaandmassaredenedbytheGaAssubstrateratherthanthelayersoftheGeandothermetals,asthisisthematerialthatisheatsunktothehelium. 53

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TheGe/GaAsthermistorsaretypically0.3mmsquareby0.15mmthick,yieldinganestimatedmassofthethermosensitiveportionofthethermistorof10)]TJ /F6 7.97 Tf 6.59 0 Td[(5g.Forthenavecalculationstofollow,thepropertiesoftheGaAslayeralonewillbeinvestigatedandtheothermetalliclmswillbeignored.Theresultwillbealimitonthetherealresponse.Therearetwomainreasonswhythermometryrequiresmanysecondsforaprecisereading.Therstisthatthepowerusedtorunaresistivemeasurementhastobekeptextremelylowtoavoidjouleheatinginthesensor.Thispowerisoforderpicowattsformeasurementsbelow100mK.Atsuchalowexcitationcurrentanddetectionvoltage,electricalwhitenoiseisaconcernandneedstobeminimized.Thisistypicallydonebyexcitinganddetectingthethermometeralongaverysmallbandwidth.Thistechniqueiscommontolowtemperatureapplicationandrequiresalock-inamplier.TheothermitigatingfactorfordeterminingthetemperatureofaheliumsampleistheKapitzaboundaryresistancebetweenthethermometerandtheliquidhelium.Intheeventthattheresistanceofthethermometercanbereadquickly,thisfundamentalmaterialpropertycharacterizesabarriertoheattransfer,bothintoandoutofthethermometer.IftheKapitzaresistanceislarge,thethermometer,takesalongtimetoreachanequilibriumtemperaturewiththeliquidanddoesnotreadilyradiateheatintotheliquidwhichlowerstheallowedexcitationcurrent.EvidencefortheeffectsofinternalheatinginthethermistorsisshowninFigure 2-8 .Asthetemperatureoftheapparatusisloweredfrom4Kto50mKtheresistanceofthethermistorincreasesinapredictablemanner.Howeverasthetemperatureisloweredpast100mK,aclearindicationofselfheatingisobservedbysomeofthecalibrationcurvesceasingtochange.Thisisavoidedbyusinglowerexcitationpowersandlongeraveragingtimeinthemeasurement.Thiseffectcanbelessenedbyusingalargersensorwhichincreasesthesurfaceareaofthesensorandtheheatexchangebetweentheheliumandthethermometerorusinglongerintegrationtimesonthelock-in.Inthisexamplethesecondmethodwasused.Forsteadystatequantumturbulenceexperimentslongerintegrationtimeanda 54

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largermeasurementareahavethesameeffectofenlargingthevolumeofturbulencethatismeasuredanddelocalizingtheprobe.ThisequivalenceisduetotheTaylorfrozenturbulencelaw[ 64 ].TheKapitzaboundresistanceispresentinallresistivelowtemperaturethermometers,butitseffectcanbeestimatedbyexaminingthesensorsused.Kapitzaboundaryresistancesetsalimittothetransferofenergyfromonematerialtoanother.Forthermometry,theKapitzaresistanceisduetoamismatchofphononmodesbetweentheheliumliquidandthesensorbody.Thedifferentmaterialspectrumsdonotalign,andmanyoftheenergycarryingphononsarereectedbythesensorinterfaceandtheirenergyisnottransmitted.ThisfundamentalmaterialpropertyissummedupinthematerialspecicKapitzaboundaryresistance,Rk.ItcanbecalculatedwithdetailedmodelsofthephononspectrumofthetwointeractingmaterialsasduetoKhalatnikov[ 65 66 ], RK=15h3sc3t 165k4c1F(c`=ct)T3,(2)wheresandarethesolidandliquiddensities,kistheBoltzmannconstant,c`andctarethelongitudinalandtransversesoundwavespeedsinthesolid,Fisafunctionofc`=ctwhichisusuallyaround1.5-2,andc1isthespeedofrstsoundintheliquid.TheKapitzaboundaryresistanceisalsoheavilydependentonsurfacenishandroughness,sothevaluesforRkaretypicallytakenfrommeasurements.Theboundaryresistanceismeasuredinasimilarwaytothermalconductivities,where RK=AT .Q.(2)Inthisequation.Qistheheatow,Aisthecrosssectionalarea,andrTisthetemperaturegradient.Fromthisequation,itbecomesclearthatthesmallsensorshaveverypoorabsolutethermalcontactwiththeheliumbath.Thisisperhapsthereasonwhythethermistorsdidnothavegoodheatdissipationbelow100mK.Thesmall 55

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Table2-2. QuantitiesusedtocalculatetheKapitzaResistance ParameterValue Mass(m)7.210)]TJ /F6 7.97 Tf 6.59 0 Td[(8KgArea(A)910)]TJ /F6 7.97 Tf 6.59 0 Td[(8m2KapitzaResistance,(Rk)at1K[ 66 ]8.122m2K=WSpecicHeatwhendopedwithAs,(c)[ 67 ]1.479910)]TJ /F6 7.97 Tf 6.59 0 Td[(4J=KgKSpecicHeatwhendopedwithGa,(c)[ 67 ]1.872310)]TJ /F6 7.97 Tf 6.59 0 Td[(4J=KgK sizeofourthermistorsgivesthemtheadvantageofacomparablylowerheatcapacitytootherthermometersbutthedisadvantageofnotbeingabletodissipatetheohmicheatingfrommeasurement.Thereactiontimetotemperaturevariationsintheliquidcanbeestimatedbycomparingtheheatcapacityofoursensorstotheirmaximumthermalconductivity,determinedbytheKapitzaboundaryresistance.WecanestimatetheamountofenergyitwouldtaketochangethetemperatureofthethermometerswiththeinformationinTable 2-2 .ThespecicheatforthematerialinourthermistorsisnotexplicitlystatedsoestimatesfortheirvaluesweretakenfromP.H.KeesomandG.Seidel[ 67 ].TheestimatesareforAsandGadopedGeat500mKratherthantheexactdopingofmaterialinthesensors.Inaddition,theKapitzaresistancefordopedgermaniumwasnotknown,sovaluesofelementswithsimilarstructureandperiodictablelocation(Si,Sn,andPb)wereaveragedtogetherandtheRk/T)]TJ /F6 7.97 Tf 6.59 0 Td[(3scalingwasused.IntegratingthelinearizedspecicheatfromTable 2-2 andapplyingtheKapitzaresistancethemaximumexcitationcurrentasafunctionoftemperaturecanbecalculated.TheseresultsareplottedinFigure 2-9 wherethex-axisisthethermometertemperatureandthey-axisistherequiredtemperaturegradientforasensorwiththespecicationsinTable 2-2 todissipatethe2.510)]TJ /F6 7.97 Tf 6.58 0 Td[(12Watwhichthethermometerisdriven.ThesethermistorswereusedtocollectthedatapresentedinChapter 4 56

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Figure2-7. Thisschematicofathermistorshowsthelayersofdepositedmetalpresentineachthermistorandtheirrelativesizes. Figure2-8. Thisshowsaresistancevs.temperaturecalibrationforathermistorproducedbytheMitingroup[ 63 ].Thedivisionbetweenthecurvesatlowtemperaturesisduetoaselfheatingeffectinthethermistor.Theatteneddataisfortheexcitationneededtomeasurethethermistorsontimescaleslessthanonesecond(30ms)andthelongintegrationtime(3s)dataisforalowerexcitationcurrentandlongermeasurementtimes. 57

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Figure2-9. Thisplotshowsthetemperaturegradientbetweenthethermistorandheliumbathrequiredtoavoidselfheatingasafunctionoftemperature.Thepowerusedtomeasurethethermistorinthisplotis1.5pW. 58

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CHAPTER3MOTORDESIGNAfterdesigningnewsensorsnecessarytomeasurethepropertiesofquantumturbulence,whattypeofturbulenceshouldbecreatedandhowisitmade?Thisquestionarisesnaturallybecauseoftheplethoraofdifferentuidsystemsdescribedintheliterature.Turbulentsystemsareubiquitousinnaturalprocesses,andsuperuidsarenodifferent.Infact,superuidssupportatypeofturbulencethatisnotseeninclassicaluids,thermalcounterowturbulence.Withalloftheoptionsavailablethesimplestwaschosenforthisresearch,homogenousandisotropic.Previouslyinthisthesis,itwasmentionedthattherearetwotypicalmethodstocreatethiskindow.Firstisowthroughachannelintoastationarymeshgridandtheotherisagridpulledthroughastationaryuid.Insuperuidhelium,duetothelowkinematicviscosity,itisincrediblydifculttoproduceanon-turbulentpipeowwithaowofanysignicantvelocity.Therefore,pullingagridthroughastationaryuidismoreappealing.HeliumowingthroughagridistheGalileantransformofpullingagridthroughastationaryuidandthereforethesetwoareequivalent.Inthescenarioinvestigated,theuidisstationarysoitwillnotbeturbulentwhenitcomesincontactwiththegrid.Severalmotorsystemswereconceived,designed,andbuilttopullthegridthroughtheheliumoperatingatarangeofvelocities.Thenalmotordesigniscapableofmovingthegridacrossalargedynamicrangeinvelocities,from0.01-100mm=s.Thismotorisdesignedtoproducearelativelylargepullrange,sinceanitedistanceisrequiredtoestablishhomogenousisotropicturbulence.Themotorhasastableoperationrangeofmorethan1.5",muchlongerthanthe10meshspacingsrequiredfortheowbehindagridtobehomogenousandisotropic[ 68 ].Thespeedatwhichthegridispulledislargeenoughtocreateturbulencebutslowenough,relativetothespeedofsound(200m=s),sothatwecantreattheowasincompressible.Thisisaprojectwhichreceivedcontributionsfrommanyresearchgroups,inparticularLancasterand 59

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Florida.Allofthemotorsfromourgrouphaveallworkedontheprincipleofmagneticlevitationduetoeldexpulsionfromasuperconductingcylinder.Theprocedureisusedtocreatemotionwithminimalheatinputtothesystemandnophysicalconnectionstotheoutside.Theoperatingproceduresandexperimentaldesignshavematuredovertheyears,becomingmoresophisticatedandproducingmoreconsistentmotionsoverincreasinglylargepulldistances.AlloftheseandtheirresultsarediscussedinChapter 3 .Thedevelopmentofthisapparatusposedaseriousexperimentalchallengebecauseoftheseverephysicalenvironmentofultralowtemperaturesandstringentrequirementsofthemotion.Forexperimentstobeofscienticuse,thedimensionalscalesneedtobewelldened.Thismeansthatamongotherthings,thegridmustbepulledataconstantvelocity,V.Producinglargescalemotion,ontheorderofinches,atmillikelvintemperatures,withoutalsoproducingcopiousheatingrequireselectro-magneticlevitation(forlinearlymovinggrids)orgas/liquidbellows(forowingliquid).Thisisbecauseatmillikelvintemperaturesitisdifculttotransfermotionfromaroomtemperatureapparatustothebottomofacryostat.Thistypeofsystemcanbecreatedforhightemperatureexperimentsabove1K[ 30 ].Standardmachineryisdifculttooperateatlowtemperatures;thegearscreatefrictionalheatingandallresistiveelectroniccomponentsaresourcesofheat.Inthepast,resistivemotorshavebeenusedatmillikelvintemperatures,buttheinternalwiringofthemotorwasremovedandreplacedwithsuperconductingwire[ 61 ].Eveninthissituation,longwaittimesarerequiredaftertheirusefortheheattodissipate.Inaddition,mostmotorswillfreezeatthesetemperaturesunlessmodiedforlowtemperatureoperation,i.e.removingthelubricants.Tocreateturbulenceforcalorimetricmeasurements,verylargemotionsarerequired,sooscillatingstructuresarenotideal.Althoughthereareinterestingquestionsforoscillatingowtypes,thereislittlereasontobelievetheyproducehomogenous 60

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isotropicturbulence.Forallofthesereasons,wechosetoinitiateourturbulencewithanelectro-magneticlevitationdevice,henceforthreferredtoasamotor.Inadditiontotheinaccessibilityoftheexperimentaluidvolume,theimportantheliumparametersrelevantforthisworkarestronglytemperaturedependent.Therefore,anyspuriousheatgeneratedfromnon-uidmotioncanseverelychangetheexperimentparameters.Forourexperimentsabove1K,onesuchpropertyisthefractionalsuperuiddensity,andforthemillikelvinwork,theheatcapacityofheliumisproportionaltoT3.Forcomparison,metallicheatcapacitiestypicallydiminishlinearlywithtemperature.Oncetheelectronmodesinmetalshavefrozenout(TTDebye),theratioofheatcapacitiesCHe=CMetal/T2.Thisnotonlymakestheuseofmetalintheconstructionofourapparatusunfavorable,italsomeansthatourexperimentissensitivetoverysmallthermalenergysourceswhichareunrelatedtotheturbulentdecay,includingohmic,eddycurrentandfrictional.Overthepast20yearstherehasbeenaurryofworkonturbulenceinsuperuidhelium.However,otherthanthemotordesignsdiscussedhere,noapparatushasattemptedtocreateisotropicandhomogenousturbulenceatmillikelvintemperaturesbythismechanism.Allturbulencegenerationapparatuscanbecategorizedashighlypolarized(rotating)[ 7 ],counterow[ 69 ],pipeow[ 53 ],oroscillatingobjects[ 55 70 ].Thegenerationofowinthesesystemsisnothomogenousandisotropic,butforsuchexperiments,thereisevidencethatthelatetimebehavioris.Tobesure,gridturbulenceisnotgeneratedashomogenousandisotropic,butthereissubstantialevidencethatitbecomessoafewmeshspacingsromthegrid.Thenearestexperimentstolineargridturbulenceatmillikelvintemperaturesaretherecentworkonoscillatingstructuresatlowandnearlyzerofrequencies,whereacloseapproximationtolinearmotionispresent[ 71 ].Thisworkhasotherproblemsaswell,suchasowaroundtheoscillatinggridinadditiontotheowthroughit.Fortheworkonoscillatingstructures,itisnotclearifthequasi-linearmotionisofhighenoughqualitytoreproducelineargridow,norisit 61

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clearthatthemostimportantowlengthscalesareduetotheowthroughthegridoraroundtheouterregionofthegrid.Workonquantumhomogenousisotropicturbulenceabove1Kforbothapulledgrid[ 72 ]andforastationarygridwithatravelinguid[ 53 ]hasbeenreported.Bothoftheseworksshownearlyclassicalbehavior,sinceat1Kthenormalandsupercomponentsarerigidlylockedbymutualfriction.Forthehightemperatureuidow,astrongclassicalinuenceispresentduetomutualfriction,andthisdifferentiatestheowthatwestrivetocreatefromthepreviousworkdoneinthisarea.Wewouldliketouseonesystem,toexploregridturbulenceacrossthewiderangeoftemperaturesinsuperuidwithseveralmeasurementtechniques.Overthepast10yearswehavedevelopedaprogressionofdifferentmotordesigns,beginningwiththe`passingdesign'andendingwiththe`controlmotor'.EachdesignwillbereviewedinChapter 3 withvaryingdegreesofemphasis. 3.1PassingMotorTherstlinearmotor,basedonanoperatingprinciplesimilartothatofaparticleacceleratorconceptwasproducedoutofacollaborationbytheUniversityofFloridaandtheUniversityofLancaster.Inaparticleaccelerator,manydifferentmagneticdrivecoilsworkwithoneanothertomoveachargedparticlealongaringorline,eachofthemproducingawelltimedimpulsetopushtheion.Inthecaseofalevitatingmotor,theparticleisasuperconductingNbcylinder(partofarodshapedarmature)andtheacceleratorissubmergedinliquidhelium.Eachcoilinthearrayisresponsibleforproducingasmalllocalizedmagneticeldtopushthearmatureashortdistanceintoitsneighboringcoil.Withprogrammedinputcurrents,thisapparatuspassesthesuperconductingarmaturefrompointtopointandmayproducenearlyanymotionprole.Aturbulencecreatinggridcanbeattachedtothesuperconductorforthecreationofturbulence.Thismotorisdescribedintheliterature[ 73 ].Thepassingmotorwasdesignedtobedynamicinthemotionpatternsitproduces,aswellasscalableinsize.Forlongertraveldistance,morecoilsareaddedtotheseries 62

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toincreasethelengthofthetrack.Thedesignhasadisadvantageofbeingawkwardanddifculttocontrolasmanyindividualactuatingcoilsrequireindividualcircuitryandtiming.Forconsistentoperationallthecoilsshouldbenearlyidentical.Withmanycoils,andperhapsslightlydifferentcoils,aprecisecontrolcanbeexerted(withdifculty)overshortdistancesofroughlythelengthoftheindividualsolenoids.Thecontrolofsuchasystemrequiresanunderstandingofhowalloftheindividualeldsandsuperconductorinteract.Inparticular,attentionneedstobegivenduringtheperiodoftimewhenthesuperconductingcylinderismovingthroughthegapsseparatingthecoils.Asthesuperconductortravelsacrossthegapsinthecoils,themagneticeldgradientalongthez-axiswillswitchfromnegativetopositive.TheMeissnerforce,whichwillbeexpandedoninSec. 3.3 ,isproportionalto)]TJ /F3 11.955 Tf 9.3 0 Td[(BdB dzsothereisaninherentdifferencebetweenpositiveandnegativeeldgradients.Negativegradientsaredynamicallystable,whereaspositivegradientsareunstableequilibriums.Thisisclearlyvisualizedbylookingattheintegralofforceorthepotentialenergy.FordF dz<0,thepotentialenergyatthatpointcanbeapproximatedasapositiveparabola,wherethestablelocationistheminimumofthepotential.OntheotherhandifdF dz>0,thepotentialisapproximatedasanegativeparabolawheretheequilibriumpointisthemaximum.Thispointisunstableduetotheneighboringlowerenergystates.ThissituationisillustratedinFigure 3-1 .Herethesuperconductingactuatorisshownasayellowborderedbluebox,andtheindividualsolenoideldsareshownasblackcurvesabovethesolenoidswhicharethegreyboxes.Thisgureshowsatimelapsewiththreedifferentmotorpositions:intherstandtopguretheactuatorispushedtotherightbythemagneticeld,howeverinthesecondimageitisunclearwhichdirectiontheforceontheactuatorispointing,anditisnotuntilthesuperconductingcylindertravelsallthewaytoitslocationinthethirdimagethattheeldgradientonceagainpushestheactuatorintheproperdirection.Thisguredemonstratesthedifculty 63

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withcreatingadriveprogramforthisdesignandexempliesthatprecisetiminginthecircuitisrequired.Stackingseveralidenticalsolenoidsontopofoneanotheralmostguarantiesthatatsomepointthelevitatedcylinderwillpassthroughoneofthesetransitions.Forthismotordesigntowork,themomentumoftheactuatorneedstobereliedontocarrythemotionalongthroughtheseunstableareas.Thisisnotimpossible,butitaddsalayerofdifcultytothedesignandmakestheelectronictimingofcriticalimportance.Todate,noworkingversionofthisdesignisinoperationforquantumturbulenceduetothesecomplications. 64

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Figure3-1. Thisplotshowsthemotionoftheniobiuminacartoonpassingmotor.Thegreybarsatthebottomoftheplotrepresenttwoofthedrivesolenoidsusedinthisdesign.Theblackcurvesabovethemagnetsdisplaythemagneticeldsofthesolenoidsandthebluebaristheniobiummovingthroughtheelds.Thismotoroperatesbypreciselytimingthecurrentsinthedrivesolenoids. 65

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3.2ImpulseMotorThedesignfortheimpulsemotorissimilartothepassingmotorbutmultiplecoilsarereplacedwithmultiplesuperconductingcylinders[ 74 ].ThemotoroperatesbycontrollingthecurrentinasinglesolenoidasmultipleNbcylindersarepassedthroughitscenter.ThisisinoppositiontothepreviousdesignwhereasingleNbcylinderispassedthroughmultiplesolenoids.Theimpulsemotorthatwasconstructeduses2Nbcylinders.Thesmallnumberofcylinderslimitstherange,butlessenstheprecisetimingrequirements.Additionally,byswitchingtoonecoilandmultiplesuperconductorsthedifferencesbetweenseparatecoilsisnolongerafactor.ThisnewdesignisinasensetheLorentztransformofthepreviousmethod.Thisdesignisalsoextensivelyexplainedintheliterature[ 75 ].Theoperatingprocedureforthismotorisasfollows:acurrentisappliedtothedrivesolenoidwhiletherstoftwoNbcylindersisinsidethesolenoid.ThisproducesanimpulseinoppositiontogravityandcarriestheNbcylinderoutofthesolenoid.ThesecondNbcylinderisattachedtothesameactuatingrodastherstandisinitiallylocatedbelowthedrivesolenoidandwelloutsideofitsfringeeld.ThearmaturetravelsupwardsuntilthesecondattachedNbcylinderiscarriedintothesolenoideld.BecauseoftheinitialpositionofthesecondNbcylinderbelowthecoil,theforceitexperiencesopposesthemotionandactsasabrakewhichstopsthearmature.Themomentumofthearmatureissetbytheinitialcurrentpulsesothesecond,orbreaking,cylinderwillslowthearmatureandtheattachedgridataspeciedlocationwherethesystemcanbeheld.Thisdistanceissetbytheseparationofthetwocans,2.1cm.Thismotorsystemwasdesignedtoproduceveryfastgridmotionsoveraveryshorttimescale.Theoriginalmotorwasdesignedtoproduce1m/sgridspeedsoveramotorpulllengthof22ofmm.Toproducethisacceleration,thesystemisdesignedwithshortsolenoidsandsteepeldgradientstheacceleration.Thetheory,motioncalculations, 66

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designs,andearlyresultsforthismotorarereportedinthedissertationbyShu-ChenLiu[ 75 ].TherealsoresultsproducedfromthissystemarepresentedinChapter 4 .Fortheseearlymotorexperiments,thepositionofthearmaturewasmeasuredbyacapacitivepositionsensor.Laterthiswillchangeasthegeometryofthemotorandexperimentalcelldictate.Thispositionsensorisdesignedsothatthearmaturetravelsaxiallybetweentwosemicircularconductors.Asthesuperconductormovesbetweentheplatesthedielectricpropertiesofthespacechange.Whenthesuperconductingtubefullyllsthespacebetweentheplates,thecapacitanceisatamaximum.Whenthespacebetweentheplatesisempty,thecapacitanceisataminimum.Thecapacitivemeasurementsarecalibratedat1Kandassumedtobeindependentoftemperaturebelowthispoint.Thepositionisalock-inamplierdetectorandcomparedtothecalibrationafterthemotormotionhasceased.Figure 3-2 showsthemannerinwhichthissensorisrunandFigure 3-3 showsthecalibrationdataforthesensor.Inthegure,theredcoloredmaterialisthecapacitivesensorandthebluecoloredmaterialistheNbarmatureinsidethesensitiveregion.Thearmatureisguidedbyphenolicknifeedgebearings.Theresultsofusingtheseringbearings,andtheheatingtheymayhaveproduced,werenevermadeclearduetouncontrolledheatingfromothersourcesintheexperimentalcell.Inthefuturemotordesignsitwasomittedinfavorofmagneticbearings.ThelargecurrentchangeovershorttimesfortheaccelerationinducesastrongbackEMFvoltage.AnunderstandingoftheEMFproducedbythedrivecoil,aswellasitsinteractionwiththecurrentamplicationhardware,arerequiredforaquantitativeunderstandingoftheentiremotormotion.Thiseffectwasnotaccountedforintheoriginalanalysis.Inaddition,theaccelerationproducedwiththismotorrequiredhighmagneticeldsrelativetothecriticaleld,Hc,forNb.AndthesharpcornersoftheNbcylindercan 67

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experiencepenetrationofthemagneticeld.Thereissomeevidenceforthis,whichispresentedinChapter 4 .PartialpenetrationcanoccurbecauseniobiumisatypeIIsuperconductor.Thismeansthatsomemagneticuxpenetratesthesuperconductor,butthesuperconductorremainsinthesuperconductingstate.Partialpenetrationoccursatthelowercriticaleldofasuperconductorandthesuperconductorgoesnormalattheuppercriticaleld.ThelowercriticalmagneticeldofNbisexceptionallyhighrelativetootherelementaltypeIIsuperconductors.Thismeansthatuptohighmagneticeldvaluesof0.2T)NbwillbehaveasthoughitwereatypeIsuperconductor[ 76 ].LowtemperaturetypeIandIIsuperconductorsareidentical,inthephysicalsensethatelectronsinbothformcooperpairsandfollowBCStheory.Thedifferencebetweenthetwotypesistheratioofpenetrationdepth,,andcoherencelength,.ForatypeI, >1,whilefortypeII, <1.Thedifferingratiosleadtoachangeinbehavioratthenormaltosuperconductingtransition.Thecriticalmagneticeldforasuperconductorisafunctionoftemperature.Thecriticaleldforsuperconductorscanbeestimatedbythefollowingrelations[ 76 ], Hc1'0 2(3) Hc2'0 2.(3)Hc1isthelowercriticalmagneticeldwherethenucleationofasingleuxoidisallowed,andHc2istheuppercriticaleldwherethesuperconductingstatecannolongerbemaintainedandthematerialrevertsbackintoitstypicalnormalstate.0isauxoidorasinglequantaofux.FortypeI,Hc1
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~Aisthemagneticvectorpotential,~Bisthemagneticeldandisthemagneticuxinthematerial.FollowingfromAshcroftandMermin[ 77 ],forasuperconductorthisturnsouttobeanintegermultipleof2~=qinasuperconductor,ormoreprecisely o=2~=2e'2.067810)]TJ /F6 7.97 Tf 6.59 0 Td[(7gauss)]TJ /F3 11.955 Tf 11.96 0 Td[(cm2.(3)TheuxoidisanalogoustoaquantaofcirculationinsuperuidHe4asdiscussedinChapter 1 .MeasurementsonNbatzerotemperatureplacethepenetrationdepthat(0)=475nm[ 78 ]andthesuperconductingcoherencelength'13nm[ 79 ].InsertingthesevaluesintoEquations 3 and 3 ,wecanexpectvaluesofHC1=2,979.8gaussandHC2=38,949gauss.TheexperimentalvaluesforHC1andHC2areheavilydependentoftheorientationandgeometryonthesuperconductorrelativetothemagneticeldbutthevaluesabovearecalculatedforanidealgeometry.Thegeometryusedtoachievethemaximumcriticaleldisalongthinwiresuperconductorinaxialsymmetrywithanappliedmagneticeld.Thisisnotthegeometryusedinthemotordesign,sodiminishedvaluesofcriticaleldareexpected.AmorereasonableestimateforHC1isthenumbercommonlyquotedintextbooks[ 76 77 ]of1,980gaussatT=0.Thiseldiscalculatedforawideatgeometry. 69

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Figure3-2. Thisgureshowsacartoonofthecapacitivepositionsensor.Inthisgure,theblueistheniobiumtravelingthroughtwocopper(red)plateschangingthecapacitanceacrossthecircuitasafunctionoftheniobiumposition. Figure3-3. Thisplotshowsthecalibrationofthecapacitivepositionsensorasafunctionofniobiumposition.Themanytracesaredifferentexperimentalruns. 70

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3.3InertialMotorTheinertialmotorisashiftinphilosophyfrompreviousdesigns.Ratherthanusingashortsolenoidtomaximizeeldgradients,thisdesignhasonelongsolenoid.Theshortcoildesignessentiallyproducesanimpulsetothrowthearmatureandasecondarypulsetocatchit,whilethelongcoilusesamoresustainedforceintime.ThisnewtechniquemaintainsmagneticcontactwiththeNbcylinderforitsentiremotion,sothearmaturedoesnotneedtobepassedfromcoiltocoilasitmoves.TheFaradayvoltage,orbackEMF,whichcomplicatedthemotioncalculationsinthepreviouswork,isintegratedintothedrivemechanismofthisnewmotor.Infact,thebackEMFwhichpreviouslywasahinderance,isinstrumentalintheoperationofthisnewdesign.Theidealmagneticeldproleforthisactuationdesignisoneinwhichthemagneticeldprolewithinthedrivesolenoidisentirelyconstantanddirectlyoutsideiszero.Foramagneticprolesuchasthisandasuperconductorasanactuatortheonlydissipativeforceinthesystemisanon-lineardragforce.Thisdragforcedependsonthevelocityandisdifculttomodelthroughoutthemotion.However,ifalargerdissipativeforceisintroducedtothesystem,whichisdirectlyproportionaltothevelocityoftheNbtube,thedissipationisbetterdenedandcontrolledmotormotionispossible.AconvenientmethodtointroducethisfrictionalforceisthroughtheFaraday'slaw.Thislawcanbesummedupas,Achangingmagneticeldinducesanelectriceld"[ 80 ],andisdenedbytheintegralanddifferentialequations. I~Ed~`=)]TJ /F13 11.955 Tf 11.29 16.27 Td[(Zd~B dtd~a(3) r~E=)]TJ /F5 11.955 Tf 10.5 8.09 Td[(@~B @t(3)Togethertheserelationsleadtothewellknowngenerationofelectromotiveforce(EMF) "=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(d dt=d(IL) dt(3)whereistheux,Listheinductance,andIisthecurrentinthecircuit. 71

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Forthemotorproposedinthissectionaresistorisintroducedintothedrivecircuitparalleltothedrivesolenoid,creatingdissipation.Specically,astheEMFisgeneratedbythechangingofILasafunctionoftime,adissipativecurrentwillowthroughtheshuntresistor.AbackEMFisgeneratedfromtwodistinctsources;changingIintimeandachangingL.ThedL dttermdeterminedbyanalyzingthemotionofthemotor,whiledI dtwillneedtobeexplicitlyobtainedbyanalyzingthecurrentowinthedrivecircuit.Thistypeofelectricaldesigniscommontodemagnetizationstagesforultracoldcryostatswherethecurrentneedstobeslowlybledoutofthemagnet.Itiswellknownthatwhenasuperconductorisinthepresenceofamagneticeld,itsmagnetizationincreaseslinearlywithmagneticeld(untilHC1).HerethisimpliesthattheNbactsasaperfectdiamagnetwithamagnetization/B.Asaresult,asthearmaturemovesitwillcreateaFaradayvoltageacrossthedrivecoilandparallelresistor.Thisprocessisidenticaltotheeffectapermanentmagnetexperiencesasitisdroppeddownasolenoid.TheNbwillexperiencethesametypeofretardingforceasittravelsthroughthedrivesolenoid.TheEMFisdirectlyrelatedtothemotionofthearmature,sothevoltagecreatedbythemotionoftheNbcylinderproducesadissipativecurrentintheshunt.Thisdampingforce(Faradaydrag)iscalculatedandoptimizedinthefollowingpages.TheapparatusforthisexperimentisshowninFigure 3-4 .TheNbcylindersareburgundy,theinsulatingG-10spacerispurple,thedrivesolenoidandthepositionsensoraregreen.Thepositionsensorislocatedbelowthebottombearingmagnet(coloredgrey)andthedrivesolenoidisinthecenteroftheimage.Alsoshowninthegureisasketchoftheelectronicdrivecircuitry.ThebottomNbisusedasbothabearingandapositionsensorwhilethetopNbcylinderisusedtoactuatethemotorandasabearing.Inthisactuationmethod,thedrivecoilideallycreatesamagneticeldinitsinteriorwhichquicklydecaysoutsidethesolenoid.Withthissystem,ifthesuperconductoris 72

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insidethesolenoidasitisinFigure 3-4 anupwardsforceisestablished.Atthestartofthemotormotionthecurrentinthedriveissetsothearmatureisonthebrinkofmotion.AtthiscurrenttheMeissnerforceexactlycancelsoutgravity(seeEquation 3 )butprovidesnomoreforce.Afterthisisestablishedthecurrentinthesolenoidisraised,producingaliftingforceforthearmature.ByvirtueofthemagnetdesignthearmaturewillaccelerateuntiltheFaradaydragexactlycancelsouttheappliedforce.Fig 3-5 showsacartoonofthecurrentproleinsidethedrivecoilasafunctionoftime,withthemotoroperationovertheentirecourseofmotiondescribedhere.Initially,thecurrentinthedrivesolenoidissettothemaximumvaluethatdoesnotproducemotion.Therefore,anyadditionalcurrentwillproduceaforceinthezdirection.Theexactvalueofthecurrentinthemotoratthistimeiseithercalculatedorcalibratedbymonitoringthepositionsensorasthecurrentinthedriveisslowlyincreased.Whenanadditionalcurrentstepisadded,allofitowsthroughtheshuntresistor,becauseforthisstepwisecurrentincreasethereactanceofthedriveis1.Afterashorttime,,currentwillbegintoowthroughthedrivecoilaswell.Thisincreaseindrivecurrentproducesanaccelerationofthearmature.ThearmaturewillaccelerateuntilitapproachesaterminalvelocitythatisdenedbyamatchingoftheinducedEMFfromthecoilandthevoltageacrosstheshuntresistor.Thecurrentinthedrivecoilatthispointexactlymatchstheinitialconditionwherethemotorwasheldonthebrinkofmotionandthereforethearmatureandgridaretravelinginanearlyconstantspeed.Atthispointofthemotion,alloftheadditionalcurrentaddedtothecircuitbypassesthecoilandowsthroughtheshuntresistor.DuringthisconstantvelocitymotionsaconstantEMFisgeneratedbytheFaradaydragofthearmature.Thebrakingprocedureattheendofthemotionproceedssimilarly,butwiththeappliedcurrentbeingloweredratherthanraised.Withthismethod,linearmotionsatvariousspeedsareobtainable,andcontrolledbythesizeoftheinstantaneouscurrentstepattimet=0. 73

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Withanavekineticdissipationmodelinhandwecandevelopquantitativeequationstomatchthequalitativedescriptionofourmotion.Thismodelcontainsthefollowingassumptions:themagneticeldinsidealongsolenoidisat;immediatelyoutsidethesolenoidthesameeldfallstozero(describedbyEquation 3 );thedrivecoilisaperfectsuperconductorwithzeroresistance,andleadresistanceofthecircuitisnegligible.Thersttworequirementsstipulatethatthesolenoidmagneticeldisperfectlyboxshaped,andaperfectsuperconductorimpliesthatthecoilwillactasapureinductorwithnorealresistivecomponent.Figure 3-6 showstheidealmagneticeldinthisdesignasafunctionofaxiallocation,z.Physicallyspeaking,thismeansthecurrentwillneverhop"outoftheNbwireandintoitscoppercladconstructioncausingohmicdissipation.Expandingontheideathatasuperconductoractsasaperfectdiamagnet,wecancalculatetheexactvoltageproducedbytheNbcylindertravelingthroughourcoil.Itiswellknown(forexampleinTilleyandTilley[ 81 ])thatthemagnetization,M,ofasuperconductorisproportionaltoitssurroundingmagneticeld,suchthatM=)]TJ /F5 11.955 Tf 9.3 0 Td[(oH.Ineffect,theNbinsideofthedriveeldactslikeaperfectbarmagnetofmagnetizationM=)]TJ /F3 11.955 Tf 9.3 0 Td[(B,whereBoisthesolenoideld.ItiseasytoseethatastheNbcylindermovesoutofthesolenoid,theendofthecylinderstillinsidethecoilwillpassthroughthemanysuccessiveloopsofthedrivesolenoid.AseachloopispassedbythemovingNbcylinderanEMFwillbegeneratedinthesuperconductingsolenoidequaltod=dtintheoppositedirectionoftheappliedvoltage.Ifaresistoriselectricallyinparallelwiththedrive,acurrentwillbepushedthroughthe(dissipative)resistor.Letusspecifythatthemagneticdrivecoilhasalengthbandtheeldinthecoreofthecoilisexpressedbytheinnitesolenoidapproximation, B=onIInsidesolenoid (3a)=0.Outsidesolenoid (3b) 74

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whereBisthemagneticeld,oisthepermittivityoffreespace,Iisthecurrentinsideofthecoil,andnistheturndensity.Themovementofthesuperconductorleadstoachangeintheuxinsideindividualwireloopsofthesolenoid.Thechangeofux,d=dt,insidethesolenoidcanbemeasuredbyachangeininductanceorcalculatedasafunctionoftime,(Ndx=L)(A)(B) dt.IListhecurrentinthesolenoid,Listhesolenoidinductance,Nisthetotalnumberofturns,andAisthearmaturecrosssection.Inthisrelation,Ndx=Lisequaltothenumberofcurrentloopsinagivenlengthdx.Thisequationcanberearrangedtoproduce, ILdL dt=nABU,(3)whereUisthearmaturespeed.TheinducedFaradayvoltageofthissystemis "=RIR=d(LI) dt.(3)Thisincludesboththegeometriceffect,dL=dt,andthereactancecomponent,dI=dt.Here"=EMF,Ristheresistanceoftheshunt,andIRisthecurrentthroughtheshunt.Combiningequations 3 and 3 wearriveat, "=RIR=nABoU+LdI dt=nABoU.(3)Theseequationsarewrittenforarbitrarymagneticeldsandmotorspeeds.Combiningthesethreeequations, 3 3 ,and 3 ,andapplyingthemtoparticularsituations,wecansolveforthekinematicresponseforspeciccurrentinputs.Figure 3-7 showstheelectricaldiagramandthelabelsforthecurrentsineachpartofthecircuit.Thenetforceonthearmatureisthesumofthegravitationalforce,pointinginthe)]TJ /F4 11.955 Tf 9.45 0 Td[(^zdirection,andtheMeissnerforce,pointinginthe+^zdirection.TheMeissnerforceisdenedastheforceresultingfromthebulkexpulsionofmagneticuxfromasuperconductor.Itcanbecalculatedbynotingthatmagneticenergydisplacedbya 75

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superconductorisequalto U=)]TJ /F5 11.955 Tf 9.3 0 Td[(B28 2o(3)whereUistheenergy,8isthevolume,isthemagneticsusceptibility,Bisthemagneticeld,andoisthepermeabilityofspace.Givenforceisthegradientofenergy,rU=~F,acylindricalsymmetry,and=)]TJ /F4 11.955 Tf 9.3 0 Td[(1,theMeissnerforceonthearmatureequals, Fz=Adz 2o(2BdB dz).(3)Integratingthisequationalongthelengthofthesuperconductingcylinderandcombiningwithalocalgravitationalforcewearriveat, F=m.U=A 2o(B2B)]TJ /F3 11.955 Tf 11.96 0 Td[(B2T))]TJ /F3 11.955 Tf 11.96 0 Td[(mg(3)BBistheeldatthebottomofthesuperconductingcylinder,andBTistheeldatthetop.NowweassumetheparticularmagneticeldproleofB=onIfromEquation 3 .ThiseldproducesaforceontheNbarmatureequaltoF=A 2o(2on2I2))]TJ /F3 11.955 Tf 12.07 0 Td[(mg.Fornowweignorethepossibleeffectsfromresistanceinthecoilandsolvethesimplieddynamics.Thiswillbeincludedinthisderivationoncesomeofthemathematicalmachineryhasbeendeveloped.Areadditionalforcepresentformotionthrougharealuidistheuiddynamicdrag1 2AgridU2,whichwillalsobeignoredforthetimebeingaswell.Intheend,thistermissmallcomparedtotheenergydissipationintheshuntresistorandcanbeignoredinthenalresult,buthereitisanassumption.AddingEquation 3 tothelistofrelations,establishedinEquations 3 3 ,allowssimulationoftherealappliedmagneticeldrequiredforthedesiredmotion.ThemaximumcurrentappliedtothecoilthesuperconductorisI=Io.ThecurrentstepdeliveredtothemotorisIs,thereforethetotalappliedcurrenttothemotorcircuitisIa=Io+Is.Theshuntresistor(R)hasacurrentIr.ArrangingthevariouscurrentsIL,Ir, 76

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Ia,Is,andIorepresentthecurrentpassingthroughthedrivesolenoid,shuntresistor,thecombinedcircuit,currentstepincreaseandthecurrentrequiredtoexactlycancelgravityrespectively.ThediagramshowingthecircuitcomponentsandtheircorrespondinglabelsisshowninFigure 3-7 .Inthisgure,Aisthecurrentsource,aKepcobipolaroperationalamplierdrivenbyadataacquisitioncard(DAQ)andlabviewsoftware.TheresistorlabeledRistheshuntandtheinductorListhedrivecoil.Theresistanceofthelinesisassumedtobezeroandtheimpedanceoftheamplierisinnite.Thecurrentthroughthesolenoidandshuntforalltimesafterthestepcurrenthasbeenapplied,t>0,isdenedinequations 3 and 3 IL=Io+I(t)(3)isthedrivecoilcurrentandthecurrentintheshuntresistoris IR=Is)]TJ /F5 11.955 Tf 11.95 0 Td[(I(t).(3)I(t)isafunctionalcurrentsuchthatI(0)=IsandI(1)=0,iftheinitialreactanceofthecircuitisignored.I(t)isformallydenedasI(t)=Ia)]TJ /F3 11.955 Tf 12.68 0 Td[(Io)]TJ /F5 11.955 Tf 12.68 0 Td[("=R.SubstitutingEquation 3 and 3 intoEquation 3 andassumingtheeldprolefromEquation 3 ,theequationofmotioniswrittenas m.U=Aon2I2o 2+oAn2I(t)Io+oAn2(I(t))2 2)]TJ /F3 11.955 Tf 11.96 0 Td[(mg(3)andwithEquation 3 "=RIR=on2A(Io+I(t))U+Ld(Io+I(t)) dt.(3)Equation 3 canbefurthersimpliedbynotingthatAon2I2o 2)]TJ /F3 11.955 Tf 12.13 0 Td[(mg=0(Equation 3 )andbyassumingthatsecondordertermsarenegligible,oAn2(I)2=0.Linearizing,theequationreducesto m.U=oAn2I(t)Io.(3) 77

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SolvingforI(t)gives I(t)=m.U oAn2Io.(3)PluggingthisintoEquation 3 wearriveatthecompletekinematicequation "=R(Ia)]TJ /F3 11.955 Tf 23.91 8.09 Td[(m.U oAn2Io)=on2A(m.U oAn2Io+Io)U+Lm.U=oAn2Io+Io dt.(3)Rearrangingthetermsleadsto RIoAn2o mLIa=..U+on2A LIoU.U+R L.U+I2on4A22o mLU.(3)Itisknownthatiftimeistakentot!1,theFaradaydragwillimposeacriticalvelocityontheNbandUwillbecomeaconstant.Theexistenceofasteadycriticalvelocityatt!1impliesthat,U=U)]TJ /F3 11.955 Tf 11.96 0 Td[(UsteadywithUsteadyequalingtheconstantvelocityatt=1.SolvingEquation 3 fortheconstantvelocityconditionatt=1andsubstitutingBo=oIon, Usteady=R ABonIs.(3)ThisnewdenitioncannowbeinsertedintoEquation 3 tocalculatethevelocityatanygiventime, RIoAn2o mLIs=..(U)+on2A LIo(Usteady(.U)+1 2.(U)2)+R L.(U)+I2on4A22o mL((U)+Usteady).(3)Keepingonlythelineartermsofandrearrangingtheequationweareleftwith A=..U+B.U+CU.(3)HereA=RIoAn2o mL(Ia)]TJ /F3 11.955 Tf 12.51 0 Td[(IoAn2oUsteady)=0,B=n2Ao LIoUsteady+R L,andC=2on4AI2o mL=2gA bAo.TheconstantCissimpliedbyincorporatingEquation 3 andobservingthatL=Aoon2b,wherebisthelengthofthedrivecoilandAoisitsinnerarea.Thesolutionforthissimpledifferentialequationisrevealedbyassumingasolutionoftheform, 78

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Table3-1. InertialMotorParameters ParameterValue TurnDensity5072Turns=mArea1.52E-3m2Permeability4E-7N A2Current,Io1.5AShuntResistor0.1Inductance0.383H U=Uoe)]TJ /F10 7.97 Tf 6.58 0 Td[(t,whereisthesolutionofthequadraticequation 2)]TJ /F12 7.97 Tf 13.26 10.7 Td[(B+C=0.(3)Solving=)]TJ /F12 7.97 Tf 10.6 10.69 Td[(B=2q b2)]TJ /F4 11.955 Tf 11.95 0 Td[(4C,andinsertingreasonablenumbersfortheparametersfromTable 3-1 =0.257.6i.(3)ThissolutionproducesadampedoscillationtakingthemotoruptoaconstantspeedofUsteadywithatimeconstantof0.25soravelocityof U'Usteady+e)]TJ /F11 7.97 Tf 6.59 0 Td[(t=4sin(7.6t).(3)Withtheequationofmotionforthismodelsystemsolved,theexperimentaldatafromthismotordesignarepresentedinSecs. 3.3.1 and 3.3.2 .However,beforethisisdonethevalidityoftheassumptionthattheFaradaydragisamuchlargerforcethanthenon-linearhydrodynamicdragischecked.Todothisnumbersareinsertedintotheirequations, FFaraday=nABoIs(3)and FHydro=1 2AGU2.(3) 79

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WiththeparametersofoursystemFfaraday FHydro=212460.Thesecondorderterms.U2and.I2werealsoignored.Tocheckthis,theratioofIo=Isistaken.ThisisequaltoAU bAoorwithreasonablenumbersis0.0,showingthatthesecondordertermsareindeedsmallcomparedtotheleadingorder.ThenalassumptionisthatB2T<
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Figure3-4. Schematicformotorscontainingquadruplemagnets.Fortheinertialmotorthedrivesolenoidisasinglesolenoidasisshowninthisgure.Forthecontrolmotordesign,thedrivecoilissurroundedbyananti-Helmholtzsetofthreecoils. 81

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Figure3-5. Thisdiagramshowsthevaluesofcurrentineachcomponentoftheinertialmotorasafunctionoftime.Theappliedcurrentinthepurplegureisthesumofthecurrenttravelingthroughthedrivesolenoidandshuntresistorbranches. 82

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Figure3-6. Thisisthemagneticeldthatwasusedtodevelopthetheoryfortheinertialmotor.Withthisassumptionthecurrentisniteandperfectlyatinsidethedrivesolenoidandfallstozerodirectlyoutsideofit. Figure3-7. Thisdiagramshowsthenamesofthecurrentsusedforthemotormotiontheories. 83

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3.3.1InitialResultsat4KThetheoreticalframeworkdiscussedabovewasputtothetest.Adrivemagnet,similartotheoneinthetheory,waswoundonamandrel.A1=4"diameterhollowNbcylinderwasepoxiedtoaG-10tubeandinsertedintothecenterofthesolenoid.AphotoofthisapparatusisshowninFigure 3-8 .Theserstexperimentswereconductedina`suckstick'apparatus[ 82 ].Thesuckstickisadoublewalledstainlesssteelcryostat,whichcanbeeasilyinsertedintoaheliumtransportdewar.Withanappropriateexternalpump,thecryostatisabletoachieveabasetemperatureofjustabove1.3K.Thedrivecoilforthisapparatuswasa55mmlongsolenoidwoundwith4layersofcoppercladNB-Tiwire.ThissolenoidisshownattachedtotheopticaldewarinsertinFigure 3-9 andthecalculatedmagneticeldforthiscoilisshowninFigure 3-10 asafunctionofposition.TheNbusedintheseexperimentsiscommercialannealedtubingNbwithapurityof99.9%.ForthisinitialexperimentweconstructedapositionsensorsimilartothatusedontheimpulsemotordiscussedinSec. 3.2 .Whilethepositionsensingtechniqueremainedthesameasinthepreviouswork,thegeometrywasheavilyalteredtohaveamuchlargergapbetweenthemovingNbcylinderandtheoutercapacitiveplates.Thisincreaseddistancewasrequirednecessarytoremoveallnon-superconductingelectroniccomponentsfrominsidetheheliumcell.Theresultswiththisexperimentalsetupwereinconclusive.ThedatasuggestedthatthepositiondetectionoftheNbarmaturewasnotsufcientlysensitiveandthemotionproleasafunctionofinputcurrentwasnotdetermined.Toproduceabetterpositionsensor,aconsiderableamountofeffortwasexpended,whileavoidinganymetalorsemiconductingcomponentsintheheliumsample.Thisprocessledtothecreationofseveraldifferentpositionsensors,ofdifferentdesigns,andmadefromdifferentmaterials.AsmallsamplingofthesecanbeseeninFigure 3-11 .ThebestcalibrationfromthesetestsisshowninFigure 3-3 .Capacitivepositionsensorsarenotwellsuited 84

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tothistypeofmotor,becausethecapacitanceofthesensorisinverselyproportionaltothegapdistancebetweenagivensideofthecapacitorandtheNbrod,whileitisdirectlyproportionaltothedistancethearmaturemoves.Thisgeometrydictatesthatthepositionsensorissensitivetoradialandaxialchangesinposition.Inaddition,theincreasedgapnecessaryforthenewcellmakethisanunsuitabledesign.But,toseeifthetheoreticalmotioncalculationsareaccurate,themotorwasmountedinaglassdewarwherethemotioncouldbedirectlyobserved. 85

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Figure3-8. Thisgureshowsthetestcellusedinmanyoftheexperimentsinthesuckstick.Thedrivesolenoidisseeninthegure,aswellasthesharpedgeusedtobearthearmatureduringitsmotion.Thearmatureusedinthiscellisshownlyinginfrontoftheapparatus. Figure3-9. ThisisanimageoftheapparatusshowninFigure 3-4 .Seeninthisgurearethetwoquadruplebearingmagnetsandthe65mmdrivesolenoid. 86

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Figure3-10. Thisgureshowsthecalculatedmagneticeldforthe55mmdrivecoilusedtotesttheinertialmotortheory. Figure3-11. Thisisaphotographofthevariouspositionsensorsconstructed.The1inchinductivesensorisontheleftnextto4differentcapacitivesensorsofslightlydifferentdesignandsophistication. 87

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3.3.1.1OpticalCryostatResultsThemotorwasreassembledandattachedtoanewmountwhichisshowninFigure A-3 .ThemotorassemblyusedforthisworkisshowninFigure 3-4 ,exceptheretherewasnopositionsensormounted.ThepertinentdimensionsforthisapparatusaregiveninFigure 3-9 .TheglassdewarapparatusinsertisshowninFigure 3-9 .Unfortunately,therstcooldownofthedewarrevealedacrackontheheliumsideoftheglassdewar.Duetothis,theliquidheliumfromthetransferseepedintothevacuumspaceandwewereunabletoproceed.However,adifferentdewarwasavailableandthecryostatinsertwaschangedtomatchthenewdesign.Thisismentionedbecausethereplacementdewarusedinplaceoftheoriginalwaslonger;thishadthepracticaleffectofraisingourbaseexperimentaltemperatureto4.2Kandeliminatingourabilitytosubmergethismotorinsuperuid.Thereforethersttestsonthemotorwhenthemotionwasveriedwereconductedinaopticalglasscryostatat4Kandthemotionwasobservedvisuallyandrecordedonastandarddigitalcamera.Thepurposeofthisexperimentwastoverifythatthemotorbehavedqualitativelyasweexpectedsincenoaccuratepositionsensorhadbeendeveloped.Figure 3-12 showsaseriesofstillframesofthemotormovingandFigure 3-13 showsselectmotormotionstakenfromanalyzingthevideosofthedigitalcamera.Thepositionaxisofthevideoisundeterminedintheplots,butthetotalmotionwas25mmandthescalemeasuredinthevideosisassumedtobelinear.Becauseoftheexploratorynatureofthisexperiment,theshuntresistorwassituatedoutsideofthecryostatinstedofdirectlyatthecoil.Hencetheleadwirestothedrivecoilhaveatotalseriesresistanceofabout1.5,ratherthantheinsignicantresistanceofasuperconductingcoil.Thisinturnrequiredalargerpowersourcethanavailabletorunthemotorasdesigned.Theminimumshuntresistanceparalleltothemotorcoilwas3,or30timeshigherthantheoptimumcalculatedvalue. 88

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ThetestsshowninFigure 3-13 showveryslowchangesinthemotorposition.Thedrivecurrentwaschangedveryslowlyintime,ratherthaninstantaneouslyaswasproposedinthetheorysection.Thesetestsservedtwopurposes;theymapped"outthemagneticeldofthedrivesolenoidasafunctionofposition,andshowedthestabilityofthearmatureateachpositionalongthetraverse.ThemagneticproleofthedrivecoilisobservedintheshapeofthemotortracesinFigure 3-13 ,becausethecurrentwasrampedlinearlyintime.Therefore,theequilibriumpositionofthemotorforagivencurrentcanbecalculated.Foreachrunofthemotor,atearlytimesthereisaquickrisetothetopofthedrivesolenoid,followedbyaslowcontrolleddescentofthearmaturetothemiddleofthedrivecoilwherethearmaturebecomesunstableandfallsoutoflevitation.Thisbehaviorisindicativeofthechangingmagneticeldgradient.Whenitisinthebottomhalfofthedrivecoilthederivativeofthemagneticeldispositive,andwhentheNbissituatedabovethecenterofthecoilthebottomofthecoilsitsinstableequilibriumpositions,exactlyasexpected.Becauseofthisfact,onlythesolenoidmagneticeldforforpositionsabovethecenterofthecoilcanbemapped.Onenalcuriousfeaturefromthisdatasetistheviolentoscillationsonsomeofthemotorruns.Thesewereidentiedasphysicaloscillationsinitiatedbynitrogenintheshieldevaporatingandareignoredastheyareirrelevanttotheoperationofthemotorinadilutionrefrigerator. 89

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Figure3-12. Thisgureshowsstillframesextractedfromthevideosofthemotormotion.Thedarkcylindermovingdownwardsacrossthethreeguresisthetopoftheniobiumbeingloweredbythemagneticeld.Thesevideoswereanalyzedtoproducetherstquantitativedescriptionofthemotormotion. Figure3-13. Themotiontracesinthisgurewereextractedfromthevideoofthearmaturemotionshowninthegureabove.Thesethreeguresshowaloweringofthearmatureintheexperimentalcellatvaryingspeeds. 90

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3.3.1.2ResultswithinductivepositionsensorAfterthemixedresultswiththemotortestsintheopticaldewar,anaccuratepositionsensorandamorestableworkingenvironment.Forthelatter,anon-opticalheliumdewarwhichdoesnotcontainaevaporatingnitrogenjacketwasused.Inthisdewar,alloftheheliumbubblescanbesuppressedbyapplyinganexternalpressure(aboveT)orpumpingbelowthesuperuidtransitionwherecavitationisnolongerpresent.AninductivepositionsensorwasdesignedandwoundtotaroundtheactuatingNb.TheinductivetechniquehastheadvantagethatitdoesnotdependontheradiallocationoftheNbandhasasensitivityoftheorderofthediameterofthewiresusedinitsconstruction.AphotographofthepositionsensorisshownalongsidethecapacitivesensorsinFigure 3-11 .Theinductivepositionsensorworksbychangingthemagneticsusceptibilityofthespaceinsidethesolenoidasthesuperconductormoves.Theinductance,L,ofasolenoidcanbecalculatedbytheformula, L=(1+)AuoN2 `.(3)Asthesuperconductormoves,itllsmore(less)ofthesolenoidcausingadecrease(increase)ininductance.Inthisequationisthesusceptibilityofthematerialinsidethepositionsensor,Aisthecrosssectionalareaofthecoil,oisthepermeabilityoffreespace,Nisthenumberofturnsinthesolenoid,and`isthesolenoidlength.Thersttestsofthisapparatuswereconductedat4K.Forthese,thequadruplebearingmagnetsweretemporarilyabandonedduetotheconnedgeometryofourcryostat,astheseexperimentswereconductedinthesuckstickapparatus.TheexperimentalapparatusisshowninFigure 3-14 .Theseexperimentsweredesignedtoproceedwiththesameexperimentalprocedureastheprevioustestsintheopticaldewarwithouttheinterferingmechanicaleffectsofaboilingnitrogenjacket.Themotorapparatuswasthesamedesignaspreviouslydiscussed,withtheexceptionthatthedrivecoillengthwasincreased.Thenewdrivesolenoidis10layersofsuperconducting 91

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wire,ratherthan4,andthelengthofthesolenoidwasincreasedfrom55mmto65mm.Thelongercoillimitedthemotortraverseto20mmwhileneverallowingthebottomoftheNbtoenterthepositiveeldgradientofthemagneticeld.Inotherwords,themagnetcoilwasmade10mmlongerandthestartinglocationfortheNbwasmovedtothecenterofthesolenoid.Theadditionallayerswoundonthedrivemagnetallowedittooperateatlowercurrentsandachievethesamemagneticeld.Withasmallercurrent,asmallershuntresistorcouldbeused,ttingbettertheparameters.Forthesetests,thecurrenttothedrivewasdiscreetly,ratherthancontinuously,adjustedtomovethearmatureasetdistance.Aftereachchangeinpositionthestabilityofthemotorwasexamined.Thismethodofmotionproducesastepwisechangeinthearmaturepositionwithtime,whichcanbeusedasacalibrationfortheequilibriumcurrentofthemotorsystemasafunctionofposition.AtypicaltraceofthisdataisshowninFigure 3-15 .Thesedataweretakenwithoutthepresenceofashuntresistorandthereforewithoutasourceofdissipation.Afteranewcurrentlevelisdeliveredtothedrive,thearmatureapproachestheequilibriumposition,overshootsit,andoscillatesaroundthatpositionwithaveryweaklydampedsinusoidaldecay.ThiscanbeseenintheinsetFigure 3-15 .RecordingthecurrentrequiredtoholdthearmatureineachlocationrevealedinformationaboutthemagnitudeandderivativeoftheMeissnerforceonthelevitatedNbrod.ThecalculatedfrequenciesarederivedbytreatingtheNbasamassonaspringwherethespringconstantk=dB dzandthefrequencyis!=p k=m.ThemeasuredequilibriumpositionasafunctionofdrivecurrentisshowninFigure 3-16 andcomparedtotheprediction.Aftermappingoutthesolenoideldandforcesonthearmature,themotorwasreadytobetestedintheoperationmodedescribedinthetheory.Uponbeginningthiswork,acuriouseffectbecameevident;thearmaturenolongeroscillatedafteritmoved,andchangesincurrentnolongerproducedpredictablechangesinarmaturelocation. 92

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ThesedataareshowninFigure 3-17 .Unlikewiththepreviousmotormotions,Figure 3-15 ,thesizeofthestepsinthisplotappeartohavearandomandunpredictabledistributionofsizeandarestronglyhysteretic.Sowhathaschanged?TheNbmusthavebeenexposedtoaeldaboveBC1.ThereforeinFigure 3-17 uxonsmusthavepenetratedthesuperconductingcylinder.Thepenetrationofuxintheseexperimentsappearstobeagradualeffect.Figure 3-18 shows3consecutivemotormotionswithidenticalcurrentinputs.Inthisgure,therstmotion,whentheNbisstillvirginandnotyetexposedtoaeldisshowninblackandontopoftheothertwocurves.Theredandbluecurvesbelowaresubsequentrunsanditisobservedthattheamplitudeofoscillationsontheserunsdecease.Eachtimethemotorisrun,themotionappearsmorestable.Thisisattributedtotheprogressiveincreaseoftrappeduxinthearmatureandthedissipationassociatedwithmagneticvorticesmovingthroughamagneticeld.Alsoshowninthisplotisthedrivecurrentasafunctionoftime.Inthisexperiment,themagneticeldaroundthearmaturewasneverraisedabovethecalculatedzerotemperatureeld,soitispostulatedthatthemagneticpenetrationoccurreddueinparttothegeometryoftheendsoftheNbcylinders.TheNbcylindersusedinthisworkwerecappedwithmachinedNbstopperssothatthesuperconductorwouldbeoneclosedcylinderwithnolocationfortheeldtopenetratetheinteriorofthecylinder,andthereforemaximizingtheMeissnerforce.However,itseemslikelythatthemachinedNbcapsandtheirsharpfeaturesarenucleationsitesformagneticuxinthebulksuperconductor.Arecentpaperontype-2superconductors[ 83 ]showsminimaleldleakageintoahollowsuperconductorofsimilaraspectratiototheoneusedhere.ThisistrueevenforDCelds.Therefore,theremovaloftheendcapsfromoursystemshouldnotintroducealargeamountofdegradationofourMeissnerforcebecausethemagneticeldwillnotsagintotheinteriorofthesuperconductor.Thisstatementimpliesthatthesuperconductorwillexpel 93

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uxintheabsenceofaclosedgeometryandtheendcapsusedinthisexperimentaresuperuous.Toexaminethisproposition,theendcapswereremovedfromthesystemandthemotorwasonceagaincooledto4K.Figure 3-19 showsmeasurementsofthecurrentvs.positionofseveralmotionpathsofanuncappedarmature.SimilartoFigure 3-17 and 3-18 thereisevidenceforpresenceoftrappedux.Thisimpliesthatthesourceoftrappeduxwasnotentirelythemachinedendcaps.ThissetoftestsshowthatthemotorcanoperatewithoutthemassiveNbcapseventhoughthetrappeduxdoesnotallowpropermotortests.Thesedata,Figure 3-19 ,arepresentedinadifferentway.Theyshowthepositionchangesvs.inputcurrenttothedrivecoilforseveraldifferentmotormotionsratherthanjustmotionvs.time.Thisallowsforamoreindepthunderstandingofthesystem.Foreachsubsequentmotion,themotorrequiresalargercurrenttobothinitiatemovementandtotravelacrosstheexperimentalspace.Thisisseenbylookingattheblack,red,thengreenlines,respectively.Inthisplottheblackmotionisrst,followedbyredandgreen.Eachshifttotherightontheplotrepresentsalargerinputcurrentforthearmaturetomovetotheequilibriumposition.Thedrivecurrentisippedandthemotioninitiatesatalowercurrent,seeninthecyancoloredtrace.Thecurrentinthedriveisthenippedagain(backtotheoriginal)andthebluedataarecollected,showingsmallermotionsforthesamecurrent.Thereductionofmotionforagivencurrentandanobviousmagneticpolarityeffectstronglysuggestthetheproblemswiththismotorareduetotrappedmagneticux.Thisexperimentalsoshowedthat,evenwithouttheobviousnucleationsitesonthemachinedendcapthismotorcannotbereliablyoperatedatatmosphericpressureheliumtemperatures(4K).Thisplotalsoshowsthatintheeventthatmagneticuxdoespenetratethesuperconductor,itgrowswithrepeatedmotionthroughamagneticeld.However,thecriticaleldofNbrisessubstantiallybetween4and1K(Figure 3-20 ),sothetemperatureofthisapparatuscanbeloweredtoavoid 94

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trappedux.Thesetestsat4Kgiveagooddescriptionofwhatmotionwilllooklikewithtrappedux,sothatafterlowertemperatureteststhiseffectcanbelookedfor. 95

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Figure3-14. Thisisaphotographoftheapparatususedtotestthemotorinthesuckstickcryostat.Thepositionofthearmatureintheexperimentswiththismotorweremeasuredwiththeinductivesensoratthebottomofthegure. Figure3-15. Thisgureshowsthemotionofthearmatureasafunctionoftimeforequalstepwisedecreasesincurrentasafunctionoftime.Aftereachdiscretechangeinpositionthearmatureisseentofollowadampedringdown.Thisisshownoronesuchstepintheinsetofthegure. 96

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Figure3-16. Thisgurecomparesthecalculatedandmeasuredarmatureheightasafunctionofmotordrivecurrent.TheexperimentaldatainthisgurewereobtainedbylinearlyrampingthecurrentasafunctionoftimeandthecalculateddataaretheresultsofsettingthesumofthegravitationalandMeissnerforcestozero. Figure3-17. Thisdatatraceshowsthearmatureresponsefordiscretechangesindrivecurrent(similartoFigure 3-15 )asafunctionoftime.Inthisplot,unlikethepreviousone,thereisnoringdownbehaviororpredictablearmaturemotion. 97

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Figure3-18. Thedatainthisgureshowthemotionfromthreeconsecutivemotormotionsgeneratedbyidenticalrampsintheappliedcurrent.Theseplotsclearlyshowanoticeabledecreaseintheoscillatorymotionofthearmatureforidenticalappliedcurrentproles. 98

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Figure3-19. Thisgureshowsevidenceoftrappeduxintheniobiumcylinderswithoutendcaps.Thedatawascollectedbyrepeatedlyapplyingidenticalcurrentramps(exceptforthemagneticpolaritywhichwasswitchedtwice).Thedataintheblacktraceweretherstcollectedwithavirginniobiumrod.Afterafewcurrentramps,anoticeabledecreaseinarmaturepositioncomparedtodrivecurrentisobserved(redcurve).Uponrepeatedcurrentinputsthecurrentproledropstothegreencurve.Afterthedrivepolarityisswitched,thedatafollowthebrightbluecurve,andonceitisshiftedbackthedatafollowthedarkbluecurve. Figure3-20. Thisplotshowsthecriticalmagneticeldofniobiumasafunctionoftemperature. 99

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3.3.2InitialMotorTestsat1KThemotortestsat4Kmadeitclearthatuxpenetrationintothesuperconductorisaproblemandmustbeavoidedtoproduceamotorthatfunctionsinapredictableway.Thecriticalmagneticeldofsuperconductorsistemperaturedependent[ 81 ], Hcb=Ho[1)]TJ /F4 11.955 Tf 11.95 0 Td[((T Tc)2].(3)InthisequationTcisthemaximumcriticaltemperature,Hoisthezerotemperaturecriticaleld,andHcbisthespeciccriticaleldatthetemperatureT.Therefore,loweringthetemperaturewillallowustousehighermagnitudeeldswithoutproducingtrappedux.Ifthetemperatureisloweredfrom4to1K,thecriticaleldisraisedbynearly20%forNb.Todothis,theNbcylinderwascycledabovethesuperconductingtransitionclearinganytrappedux;thenthetemperaturewasloweredto1Kbyevaporativecoolingoftheheliuminthesuckstick.Atthistemperature,thesameexperimentswereperformedasinFigure 3-18 .TheresultsareshowninFigure 3-21 .Thereproducibilityofthesemotionsisobvious,especiallywhencomparedtothe4Kdata.Theoscillationsinthisgurewerereducedbyrampingthecurrentataslowerrate(noadditionalinternaldamping).Thesedataarefromover20differentmotorrunsandthereisnoindicationoftrappedux.Tocheckthis,thepolarityofthedrivecoilwasreversedtolookforashiftofthecalibrationaswasseeninFigure 3-19 .InFigure 3-23 ,nearlyhalfofthedataareofonemagnetpolarity,whiletheotherhalfareoftheother.Unlikeat4K,themotionat1Kisindependentofbothmotorhistoryanddrivepolarityultimatelysuggestingthatthereisnoornegligibletrappedux.NotethatwiththeMeissnereffecttheforceis/B2,sothepolarityofthedriveshouldbeindependentofthearmaturedynamics.However,ifthesuperconductorhasbeenpenetrated,thetrappeduxdoeshaveapolarityandisdependentonelddirection.Atthisreducedtemperature,itisclearthatthenecessarymagneticeldstoactuateaNbcylinderareproducedandmaintainedwithoutchangingthesuperconductingstate. 100

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Withthispreliminaryworkaccomplished,themotorwasthentestedtoproducevariouslinearmotionpatterns.Thisexperimentwasdesignedtocharacterizethemotorsystem,andverifythatfutureworkwillbepossiblewiththesemotordimensions.Therefore,thereisnoshuntresistorpresentandtheoperationprocedureoutlinedearlierinChapter 3 isunapplicable.Whilethesystemremainedcold,ratherthanwarmingupandstartingagain,linearvelocityproleswereattemptedwithouttheresistorusingadifferenttechnique.Thistechniqueusesacurrentrampintimesuchthattheequilibriumpositionofthearmaturemoveswithaconstantvelocity,ratherthaninstitutingastepwisechangetothedrivecurrent.ThecurrentinputisshownasanarbitraryfunctionoftimeinFigure 3-21 .DatafromtheseexperimentsareshowninFigure 3-22 .Inthisgure,z=0isthecenterofthedrivesolenoidandincreasingpositioncorrelatetotheNbmovingupwardsandoutofthedrive.Thesedataarequiteencouragingfordrivingthemotor.Overadistanceofnearly0.5"alinearmotormotionisobserved.Thismotionisconcentratedonthepositionswherethemagneticeldgradientisrelativelylarge.Therefore,runningthemotorinthisoperationmodehastheoppositeidealspecicationsasthemethoddescribedinthetheory.Forthistypeofmotormotionalargemagneticeldgradientisusedtoholdthearmatureinamovingpotentialwell,asopposedtotheoriginaldesignwherethemotorisprovidedanimpulsetoincreaseitsmomentumbeforeitmovesthroughaforcefreeregion.Intheoperationprocedureheretheeldgradientinthecenterofthecoil,)]TJ /F11 7.97 Tf 10.49 4.71 Td[(dB dz0hasverypoorcontroloverthemotormotionandatthetopwhen)]TJ /F11 7.97 Tf 10.49 4.71 Td[(dB dzismaximalthemotionisverywellcontrolledandlinear.Thisissimplyvisualizedbyobservingthatatsmallvaluesofz,inthecoil,verysmallchangesincurrentmovetheequilibriumpositionagreatdistancewherenearthetop,arelativelylargercurrentchangeisneededtomovethearmatureashortdistance.SeveraldifferentmotormotionsareproducedwiththisnewactuationprincipleandareshowninFigure 3-23 101

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Afterfullyexploringthismotorwithoutashuntresistorpresentandthisnewactuationmethod,ashuntresistorwassolderedparalleltothedrivetoexploreitseffectandthemotoroperationasdesigned.Figure 3-24 depictsthreedifferentmotormotions,withthreedifferentresistiveshunts.Thevelocitiesofthemotorinthisgurerangefrom80mm=sto27mm=s,thefasterofthesespeedsapproachthegoalof100mm=s.However,ifthespeedofthemotorisincreasedasinFigure 3-25 ,alargeoscillationisroutinelyobserved.Forreference,thecurrentstepusedforFigure 3-25 isshowninFigure 3-26 .TheoscillationsatopFigure 3-25 werefoundattheendofallinertialmotormotionswithspeedsabove80mm=s,regardlessofshuntresistor,currentstepsize,orlength.Inaddition,foragoodbrakeafteramotortraverse,theinertialmotordesignisnotwellsuitedformotormotionsmuchslowerthan10mm=s,becauseoftheinherentinstabilitiesofaveryweakpotentialwell.Intheend,therequirementsofthiselegantmethodaretoogreatforitspracticaluseandtheforceholdingthearmatureafterthemotionistooweak.Thereforeanewdesignwasdeveloped. 102

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Figure3-21. Currentinputtothedrivecoilrequiredtoproducelinearmovementsintimefortheinertialmotorrunwithoutashuntresistor.Thetimeforthiscurrentrampdependsontherateatwhichthemotormoves. Figure3-22. Onetypical`linear'motionforthemotorat1K.Thegridvelocityforthisparticularmotionis0.61mm=s 103

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Figure3-23. Thisgureshowsthereproductivityofequilibriummotorpositionsoverthecourseofmanymotormotionsat1K.Roughlyhalfofthemotionsinthisgureareforapositivecurrentdrive,whiletheotherthedrivecurrentisnegative. Figure3-24. Thisgureshowsthepositionoftheniobiumasafunctionoftimefordifferentshuntresistorswhenthemotorisrunaccordingtotheinertialmotiontheory. 104

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Figure3-25. Thisplotshowsthemotionoftheinertialmotorwitha1shuntresistorwhenthearmatureistravelingintheoppositedirectionasgravity. Figure3-26. Thisplotshowsthecurrentstepusedtocreatethemotionintheinertialmotor. 105

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3.4ControlMotorAfterspendingaconsiderableamountoftimeworkingouttheoperatingprocedureoftheinertialmotor,thelimitationsofthedesigninstigatedanewdrivesystem.Thisdesignissimilartotheinertialmotor,inthatlongsolenoidsandcontrolledmagneticeldsareused,butwiththecontrolmotorthemagneticeldgradientwasspecicallychosenratherthantheBelditself.Thischangeintheeldstructureallowsmoreexibilityandcontinuityintheoperationofthedrivesystem.Thisdesignusedasimilarmathematicalformalismtotheoneoutlinedinsec 3.3 ,butitiscomplicatedbyallowingaconstantchangeincurrentasafunctionoftimeandanon-constanteldgradient.Achangeinthecoildesignwasrequiredtocreatethisnewmagneticprole.Thedesignandconstructionofthiscoilisoutlinedinsubsection 3.4.1 ,themathematicalmodelin 3.4.2 ,andresultsin 3.4.3 .Theprincipleofthecontrolmotoristhatduringthelinearmotionofthearmature,boththemagneticeldatthebottomofthearmatureandtheeldgradientinthesamelocationareconstantintime.Thiswillleadtoaconstantmechanicalresonantfrequency(!=p k=m)ofthearmatureinthedrivesolenoid,whichcanbematchedtotheresonantfrequencyoftheRLcircuit. 3.4.1ControlMotorDesignThenewmagnetarrangementwasneededtocreatetheeldproledictatedbythe`control'design.Therequirementsofthismotordesignare;thederivativeoftheforceeldneedstobeconstantinz,theinputcurrenttothemotorshouldhaveamathematicallysimpleform(eitherlinear,constant,oraHeavisideprole)inordertoprovideaconstantforceonthearmatureasitmoves,andthedesignshouldbesimplewithasfewdifferentcircuitsaspossible.Therewereseveraltheoreticalmodelscreatedforsuchacoil.Twoarereviewedbelow.Therstattemptatcreatingaeldproleofthistypewasbywindingasteppatternontopofauniformeldsolenoid.Theeldgradientisdecidedbythenumberofwindinglayersalongthedrivesolenoid.Thisprocessarrangesseveralsolenoidsalongaline, 106

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eachwithadifferenteldandsuperimposesthoseeldstogether.AcartoonschematicofsuchasystemisshowninFigure 3-27 ;acalculationofthemagneticeldisshowninFigure 3-28 ;andthecorrespondingMeissnerforceinFigure 3-29 .Theadjustableparametersforadesignsuchasthisarethenumberofsteps,theirsize,bothinnumberoflayersandwidth,andthesteplocation.Fromthesegures,itisclearthatwithproperlocationsofstepsandthenumberofwindings,acustomcoilcanbecreatedthatproducesalinearvariationmagneticeldasafunctionofposition.ThederivativeoftheforceeldasafunctionofpositionisshowninFigure 3-30 .Thiscoildesignispromising,exceptthatchangingthecurrentduringoperationtoactuatethearmaturealsochangesthederivativeoftheeld.Thisisbecausewhenthecurrentisincreased(decreased)thelowzmagneticeldisraised(lowered)disproportionatelytothelow(high)zmagneticeld,leadingtoanon-constantdF=dzasafunctionofcurrent.Becauseofthis,thestepmotorcoildesignisnotidealandwasnotpursuedfurther.Thefailingofthesteppedsolenoiddesignledtoalooseningofthesimplecircuitrequirementandtwoseparatecoilswithindividualcurrentsourcesareused.Thissystemrequiresoneextrasetofleadwires,butstillonlyneedsonevaryingcurrentinput.Thismotorwascreatedbyplacingananti-Helmholtzcoilaroundthe65mmdrivecoilusedinthepreviousdesign.Withthisdoublelayerdesign,theanti-Helmholtzcurrentismaintainedataconstantvalue,providingaconstantdB=dz,whilethelongsolenoidhasitscurrentraisedandloweredtochangethemagnitudeoftheBeld.AschematicofthemagneticcoilsisshowninFigure 3-31 andaphotooftheapparatusisshowninFigure 3-32 andthemachineschematicsareshowninFigure A-8 .Themagneticeldsofeachofthecoilsintheanti-HelmholtzpairandtheirrelativepositionsareshowninFigure 3-31 ,wherethetwolargehumpsarethetwocoilsoftheanti-Helmholtzpairandthesmallerhumpisacentralcompensationcoilforthe65mmsolenoid.Figure 3-33 showsthesumoftheindividualanti-Helmholtzeldsalongwiththedrivecoil,andFigure 3-34 showstheforceproleofthiscoil.Inthesegures,the 107

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Table3-2. ControlMotorCoilParameters ObjectDimension DriveSolenoidRadius7.5mmAnti-Helmholtz(AH)Radius22mmDriveSolenoidCompensationCoilRadius22mmDriveSolenoidLength65mmDriveSolenoidWireLayers10LayersDriveSolenoidPosition0mmPositiveAHLength12mmPositiveAHWireLayers13LayersPositiveAHPosition0mmNegativeAHLength11mmNegativeAHWireLayers20LayersNegativeAHPosition48mmCompensationCoilLength17mmCompensationCoilWireLayers2LayersCompensationCoilPosition24mmWireDiameter0.19mm currentintheanti-Helmholtzis2.5Aandthecurrentis1.5Ainthelongdrivesolenoid.Duringoperationthecurrentinthedrivesolenoidisvariable,butoneparticularvaluewaschosenfortheseplots.ThevariousparametersofthedrivesystemaregiveninTable 3-2 andtheschematicisshowninFigure 3-31 .WiththeseparametersfromTable 3-2 ,thecurrentinputintothedrivesolenoidfordifferent,constant,anti-Helmholtzcurrentscanbecalculated.ThesearepresentedinFigure 3-35 .andtheeffectivespringconstantsproducedbythesameanti-Helmholtzcurrentscanalsobecalculatedasafunctionofposition.ThesearepresentedinFigure 3-37 .Thesecalculationsshowthattheeldprolerequiredforthistheorycanbecreated.Asinthetheoryoftheinertialmotor,theshuntresistorwillbeplacedacrossthelongdrivesolenoidtodamptheoscillationsofthearmature.Sincetheanti-Helmholtzsolenoidsmaintainaconstantcurrent,thedissipationthroughthemistrivialandsmall,duetotheirlargeradius.Themathematicsofthearmatureinteractionwiththedrivesolenoidandthedynamicsofthemotormotionareexpandedoninthenextsection, 3.4.2 108

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Figure3-27. Thisgureshowsacartoonoftherstcontrolmotordrivecoilconceived. Figure3-28. Thisgureshowsthecalculatedmagneticeldproducedbyasteppedmotordesign. 109

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Figure3-29. Thisgureshowsthecalculatedforceproleproducedbyasteppedmotordesign. Figure3-30. Thisgureshowsthenonconstantderivativeofthemagneticforceforasteppedcoildesignforachangingdrivecurrent. 110

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Figure3-31. Schematicdrawingsfortheanti-Helmholtzmagnets.Inthisguretheinnerredcoilisthevariablecurrentdrivesolenoidandtheouterthreebluecylindersarethesurroundinganti-Helmholtzcoils. Figure3-32. Photooftheanti-Helmholtzmagnets 111

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Figure3-33. Thisgureshowsthecalculatedmagneticeldproducedbyaanti-Helmholtzmotordesign. Figure3-34. Thisgureshowsthecalculatedforceproleproducedbyaanti-Helmholtzmotordesign. 112

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Figure3-35. Thecalculatedarmatureequilibriumcurrentsvs.armatureposition Figure3-36. Calculatedmagneticeldinsideeachofthethreesolenoidsoftheanti-Helmholtzcircuit 113

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Figure3-37. Calculatedeffectivespringconstantsofthecontrolmotorvs.armaturepositionforseveraldifferentanti-Helmholtzcurrents 114

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3.4.2ControlMotorTheoryThemagneticeldinsideanarbitrarycoilcanbedenedas, B(z)=onILP(z)(3)whereIListhecurrentthroughtheelectromagnet,P(z)isfunctionofzdescribingthestrengthofthemagneticeldalongthecenterofthesolenoid,andnistheturndensity.TheforceontheNbcoilsdenedbyEquation 3 ,anditisassumedthatB2T<
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whereIaistheappliedcurrent.Theforceequationnowbecomes m.U=1 2on2A(Io+t)2(a+bz))]TJ /F3 11.955 Tf 11.95 0 Td[(mg.(3)Becausethereisnoshuntresistor,Ia=ILwhereIListhecurrentintheinductor.TheschematicoftheelectronicsisthethesameaspreviouslyshowninFigure 3-4 .Byexpandingequation 3-4 wearriveat m.U=1 2on2A(I2o+2tIo+2t2)(a+bz))]TJ /F3 11.955 Tf 11.95 0 Td[(mg.(3)Thehigherordertermsaretakenout,reducingtotheformofawellknownpartialdifferentialequation, ..z+Az=Bt,(3)whereA=on2AI2o 2mandB=on2AaI2o m.Thesolutionforthisequationwiththeconditionsx(0)=0and.x(0)=0is z(t)=2a (t)]TJ /F3 11.955 Tf 13.15 8.09 Td[(Sin(!t) !),(3)wherethefrequency,!,isdenedby !=q A=r on2AbI2o 2m(3)andthelinearportionisdenedbythenon-homogenoussolution.ThedataforthismotordesignareshowntomatchexpectationsinSec. 3.4.3.1 .Fortheproductionofalinearlowtemperaturemotionandisotropic,homogenousquantumturbulence,ashuntresistorisintroducedintothecircuit,similartoSec. 3.3 .Withthepresenceofthisshunt,adirectsourceofstrongdampingiscreated.DeningtheinputcurrentasIa=Io+t,thecurrentinthedrivesolenoidis, IL(t)=Ia+IL.(3)ThelabelsfortheelectronicsindifferentpartsoftheapparatusarethesameasusedbeforeandshowninFigure 3-4 .Onceagain,wecancalculatethecurrentintheshunt 116

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resistorasaresultoftheEMFduetothetravelingsuperconductor: LIL dt=R(Io)]TJ /F3 11.955 Tf 11.96 0 Td[(IL)=R(t)]TJ /F5 11.955 Tf 11.96 0 Td[(IL) (3a)L.IL+(Io+IL).L=R(t)]TJ /F5 11.955 Tf 11.96 0 Td[(IL) (3b)andfromEquation 3 wecanextrapolatethattolowestorder .L=nAU(onILp ap 1+bz=a) IL'nAU(onILp a) IL.(3)Puttingtheseequationstogetheryields L.IL=R()]TJ /F5 11.955 Tf 11.96 0 Td[(IL))]TJ /F3 11.955 Tf 11.96 0 Td[(Aon2a1=2IoU.(3)TakingEquation 3 torstorder,itcanbereducedto m.U+1 2on2AbI2oz=on2AIoaIL.(3)UsingthestandardelectromagneticmachineryintroducedaboveandsolvingforILwearriveat IL=m on2AIoa.U+bIo 2az(3)whichhasaderivativeof .IL=m on2AIoa..U+bIo 2aU.(3)SubstitutinginEquations 3 and 3 intoEquation 3 ,asecondorderpartialdifferentialequationisobtained: Lm on2AIoa..U+LIob 2aU=Rt)]TJ /F3 11.955 Tf 27.55 8.09 Td[(Rm on2AIoa.U)]TJ /F3 11.955 Tf 13.15 8.09 Td[(RIob 2az)]TJ /F3 11.955 Tf 11.95 0 Td[(Aon2a1=2IoU,(3)DifferentiatingEquation 3 bytandrearrangedgives ... U+D1..U+D2.U+D3U=D4.(3) 117

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HereD1=R L,D2=on2AIoa Lm(LIob 2a+Aon2a1=2Io),D3=Rbon2AI2o 2Lm,andD4=on2AIoa m.ThehomogenoussolutionforthisequationisU(t)=C1e)]TJ /F10 7.97 Tf 6.59 0 Td[(1t+C2e)]TJ /F10 7.97 Tf 6.59 0 Td[(2t+C3e)]TJ /F10 7.97 Tf 6.59 0 Td[(3tandthecompletesolutionincludingtheadditionalconstantis U(t)=C1e)]TJ /F10 7.97 Tf 6.59 0 Td[(1t+C2e)]TJ /F10 7.97 Tf 6.59 0 Td[(2t+C2e)]TJ /F10 7.97 Tf 6.59 0 Td[(3t+2a bIo.(3)Theconstantsiarethesolutionstothecubicrootequationdenedinthedifferentialequation.Thesolutiontothisequationiscumbersomeandwillnotbereported,buttheleadingorderrealcomponentisequaltoR=3Lforeachiwithseveraladditionalrealandcomplextermsfollowing.Theleadingordertermgivesanideaoftherateofapproachtotheconstantvelocitysolutionastheoscillatorcomponentsdecay.Theaboveexercisequantitativelydescribesthesimpliedmotorsystemtohighprecision.However,duringoperation,ourmotorsystemisnotalwaysinsidetheboundsofourassumptions,particularlywhenthecurrentisrampedquickly.Asthesmalltermsincreaseandbecomeimportantintheelectro-magneticcalculations,thedifferentialequationsabovebecomenon-linearandunsolvable.Becauseofthis,itiseducationaltoremoveourselvesfromtherigorouselectro-magneticcalculationsaboveandexaminetheforcesonthearmatureatanygiventime,togainsomeintuitiononthemotoroperationforvaryingcurrentrampspeeds.Withaconstanteldgradientononeendofthearmatureatalltimes,boththeMeissnerforceandtheequilibriumoscillationfrequencyofthearmatureisunchangedthroughoutitstraversewhenthecurrentisrampedattheproperspeed.UsingEquation 3 tocalculateboththelevitationforceonthearmatureatanygiventime,andtheuseofmathematica,themotorcanbeeffectivelysimulatedasafunctionofagivencurrentprole.Withtheuseofasimplemodel,amovingmassonaspring,thehassleofformalcircuitcalculationsandthesimplifyingassumptionsrequiredmaybecircumvented.Toqualitativelyvisualizethisconcept,theNbarmaturesitsinthebottomofapotentialwelldenedbytheeldgradientandtheMeissnerforce.IfdFz=dzisconstant 118

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inz,thenthepotentialwellisaperfectquadraticwelldenedbythecoilgeometry.FromhereitissimpletovisualizeapotentialwelltravelingatsomegivenvelocityUalongwiththearmature.TheNbactuatorispushedalongbytheupwardslopingandtrailingedgeofthewell,alwaysmaintainingthedistancebehindthewellminimumrequiredtoproducethemotiveforce.TheNbwilloscillatewiththefrequencydenedbythewelluntilthemotionofthewellhasstopped.AtthistimetheNbwilldecayintothebottomofthepotential.Inthispicturetheoscillationfrequencyiscalculatedbytheforcesandmatchedasthecurrentinthedrivecoilisdrivenintotheshuntresistor.Therefore,iftheR/Ltimeconstantofthedissipationcircuitismatchedtotheoscillationfrequencyoftheactuator,theoscillationsaftertheruncanbecriticallydamped. 3.4.3ControlMotorExperimentTheseexperimentswereconductedusingthesameinsertastheglassdewarexperiments,Figure A-3 .Thedewarusedfortheseexperimentswasastandardsuperinsulatedheliumdewarwhichcanbeexternallypumped,reducingthetemperaturetoabout1.3K.Themajorityofthedatapresentedinthissectionweretakenwitha1"longpositionsensor,usedpreviously.However,forthelongermotionsanew1.5"copperwoundcoilwasusedtogivealongerrangeofsensitivity.AphotoofthiscoilisshowninFigure 3-38 anditscalibrationisshowning 3-39 .Fortheseexperiments,quadruplebearingsareusedandshowninFigure 3-40 .AphotoofthesystemisshowninFigure 3-32 .Severaldifferentexperimentswereconductedusingthisapparatus,includingmotoroperationswith,witha0.01,a0.1,anda1andinniteshuntresistor.Eachofthedatasetswascollectedbythesamemethodology.Initially,themotormotioniscreatedbyslowlyincreasingthedrivecurrentovertime.Fromthesedata,thecurrentvs.inductanceisextractedandcomparedtotheinductancepositioncalibration.Fromhere,adataleiscreatedwhichwillinputalinearchangeinequilibriumpositionsasafunctionoftime.ThesearecalculatedandplottedinFigure 3-35 forseveraldifferentanti-Helmholtzcoilcurrents.Withaninputleexpressingthecurrentsrequired 119

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forlinearmotion,therateatwhichthisleisreaddictatesthespeedofthemotortraverse.Forexceptionallyfastmotormotions20)]TJ /F4 11.955 Tf 12.25 0 Td[(30mm=stheinputcurrentproleissimilar,butnotexactly,thesameasdescribedintheinertialmotorsection, 3.3 ,asthemotionistoofasttoreadintheentireinputle.Thesedataarepresentedbelowanddividedbythesizeoftheshuntresistor. 120

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Figure3-38. Thisisaphotographshowinganinductivepositionsensor Figure3-39. Thisgureshowsthepositioncalibrationfora1.5inchsolenoid.RecordedinthegurearetheinductancesmeasuredwithaLCRbridgeat3differentfrequencies. 121

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Figure3-40. Photographofquadruplemagnetsandtheirmounts Figure3-41. Thisgureshowsthemeasureddataoftheequilibriumpositionsofthecontrolmotor,withoutshunt,asafunctionofdrivecurrent. 122

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3.4.3.1Shuntresistance=0Themotorapparatuswascooleddownto1.2Kwithnoshuntinthedrivecircuit.Thequadruplesolenoidsandtheanti-HelmholtzcoilsweredrivenbyaHewlettPackardHarrison6303B,0-7.5V,0-3ADCpowersupply,thedrivesolenoidisdrivenbythesameBOPaspreviouslymentionedandthedataiscollectedbyacombinationofanAgilentLCRmeter,Keithlyvoltmeters,andaNI6009DAQ.TheprocedureiscontrolledbyLabViewsoftware.TheslowrampingofthecurrentasafunctionoftimeisshowninFigure 3-42 .Theraisingandloweringofthearmaturewasdoneveryslowly,sothatthearmatureremainedinequilibriumwithgravityandthelevitatingforce.MotormotionsareshowninFigure 3-42 andFigure 3-43 withspeedsrangingfrom1mm=sto20mm=s.Thedataisarrangedsothatthemotionstarttimesforeachrunarecoincidentandthefastermotionsnishrstwhiletheslowertakemoretime.Themotionisshowntobelinearintime,superimposedwithaweaklydampedsinusoidwhichwaspredictedinsec 3.4.2 .Asthemotionbecomesmorerapid,theamplitudeoftheoscillationsgrows.Thisisintuitive,consideringthetheoryofmotionpresentedinChapter 3 andpredictedbythecurrentramprate,,intheamplitudeoftheoscillationinEquation 3 .Thefrequencygenerallyappearstobeindependentorweaklydependentonarmaturevelocityandlocationinthedrivecoil.ThereisaphaseshiftinFigure 3-44 ,duetothearticialshiftintusedtoensurethatthemotionshavethesameinitialstarttime.Figure 3-44 isolatesthreeseparatemotorrunsshowninFigure 3-43 .Twoofthesethreemotorrunshavethesameexactcurrentinputlesandinputratesappliedtothedrivecurrentsource.Themotionisnearlyidenticalinthetwoplots,showingahighlevelofreproducibilityinourmotion.Thedifferencebetweenthetwomotionsisshownintheplotinsetandappearstobeelectricalnoise.Thethirdplotinthisgureisfromaslightlydifferentsetofinputdata,wheretheinputleisinterpolatedsothereare0.5timesasmanypointsandthecomputerinputsthepointsat0.5timestherate.Comparingthis 123

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motormotiontotheothertwoprovesthattheslewrateoftheBOPandtheoutputrateoftheDAQarenotrelevanttothemotionofthearmatureatthesevelocities.Uponincreasingthecurrentrampratetoourmotor,andthereforeincreasingthemotorvelocity,astrangebehaviorappears.Atparticularspeedsthenaturaloscillationfrequencyofthemotorsystemandthetimescaleforstartingandstopingthemotionappeartoshowevidenceofaresonance.Whenthephasechangeofthenaturaloscillationsmatchesthetimescaleofthemotion,largeoscillationsatthestoppingtimearepresentandwhenthetwoareexactly180odifferentinphase,thereisnearlyanabsenceofmotoroscillationsattheendofamotion.ThisisshowninFigure 3-45 asatrainofseparatemotormotionsoperatedatslightlydifferentrampspeedsof40mm=sto71mm=s.Theseparationintimeisaddedtoeasetheeye.Itisclearlyobservedwhenlookingatthistrainofmotormotionsthat,dependingonthephaseofthemotion,rapiddecelerationsareeitherpossibleortheyinducelargescaleoscillatorymotionattheendofthemotion.Theinteractionbetweenthephaseoftheoscillationsandtheamountoftimethemotortravelscanbeusedtoproducelinearmotionproleswithfairlyhighvelocities.Fig 3-46 showsonesuchtunedmotion,wherethetraveldistanceforthearmaturehasbeenreducedtocoincidewiththenaturalharmonicsofthesystem.Byusingasystemlikethis,asetofmotormotionswereproducedforpossibleuseinthestudyofquantumturbulence. 124

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Figure3-42. Thisgureshowsslowmotormotionswithspeedsrangingfrom1-20mm=sforthecontrolmotorwithnoshuntresistance. Figure3-43. Thisgureshowsmotormotionswithspeedsrangingfrom15-45mm=sforthecontrolmotorwithnoshuntresistance. 125

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Figure3-44. Thisgureshowsthreedifferentmotormotions.Thetwomotortracesthatlieontopofoneanotheraremotionsfromidenticalcurrentinputs.Theirdifferenceisshowinthegureinset.Thethirdlineonthegureshowstheresultofasimilarcurrentinput,butwiththedriveproledeliveredtotheamplierathalfthespeed. Figure3-45. Thisplotexempliestheeffectobservedwhentheintegratedtimeofmotormotionisanintegermultipleoftheresonanceperiodofthearmatureataparticularanti-Helmholtzfrequencysetby!=p k=m. 126

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Figure3-46. Thesemotormotionsarecreatedbytakingadvantageofthemotorresonancesandactuatingthegridforaspeciedamountoftimetomaximizethearmaturespeed. 127

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3.4.3.2Witha0.01shuntresistorTherstshuntresistorusedintheapparatushadavalueof10m.Thisshuntisroughly10timessmallerthantheidealvaluecalculatedforthemotortheory.Thisreducedvalueofresistanceincreasestheinductivetimeconstantofthecircuit(L=R)anddoesnotallowforfastchangesinthedrivecurrent.Figure 3-47 showsaslowmotorrampasafunctionoftimeforthehighestanti-Helmholtzcurrentused(2.8A)andFigure 3-48 showstherelationshipofdrivecurrenttopositionforthedifferentanti-Helmholtzcurrents.Ifthisgureiscomparedtothesimilaronefornoshuntresistancethedifferenceischosen.Becauseofthelongtimescalesneededforthemotortocomeintoequilibriumwiththeappliedcurrent,thereisdelayinthemovementofthemotorwithachangeincurrent.Duetothelowshuntresistor,oscillationsonthemotorarestronglysuppressed,inparticularforthehigheranti-Helmholtzcurrents.Fig 3-49 showsseveraldifferentmotormotionswithnearlythesamevelocitybutwithdifferentanti-Helmholtzcurrents.Thefourierinformationfromthesedifferentmotionsarepresentedintheinsetofthegure.Toproducefastmotionsinthiscoildesign,changesincurrentofnearly1Aarerequired,wherethereactanceofthemotorcircuitandnotallow.ThiscanbeseeninFigure 3-50 wherestepwisechangesincurrentproduceinductivelyslowedmotions. 128

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Figure3-47. Slowmotorrampshowingcalibrationdataforthe10mshuntmotor.Thecurrentrecordedfromthismotionisusedtoproducetheinputlesforthismotordesign. Figure3-48. Currentvs.positionforthe10mshuntmotor.Thisgureshowsacuriouseffectwherethecalibrationdatafromthedifferentanti-Helmholtzcurrentsdonotdifferbymuch.ThisisattributedtothelongelectricalrelaxationtimeoftheLRcircuit. 129

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Figure3-49. Thisgureshowstwosimilarmotionprolesfordifferentanti-Helmholtzcurrents. Figure3-50. Thisgureshowsthelongequilibriumtimesforthemotorafteracurrentstepwiththe0.01shuntresistance. 130

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3.4.3.3Shuntresistor=0.1and1Inthemotortheorycalculations,ashuntresistanceofabout0.1matchedthenaturalharmonicofthesystem.Aswasseenbefore,alowervalueofshuntresistanceproducesunderdampedmotion.Thisisapparentbythefasterreactiontimeofthemotorcircuit.Themotormotionsforbotha1anda0.1shuntresistorarenowshownbelow. 131

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Figure3-51. Thisgureshowsa19.7mm=smotionforthe0.1shuntresistancecontrolmotor Figure3-52. Thisgureshowsa66mm=smotionforthe0.1shuntresistancecontrolmotor 132

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Figure3-53. Thisgureshowsa109)]TJ /F4 11.955 Tf 11.96 0 Td[(160mm=smotionforthe0.1shuntresistancecontrolmotor Figure3-54. Thisgureshowsa92)]TJ /F4 11.955 Tf 11.95 0 Td[(163mm=smotionforthe1shuntresistancecontrolmotor 133

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CHAPTER4MILLIKELVINEXPERIMENTSTherstlinearmotorbuiltforexperimentsatmKtemperatureswasconstructedbyS.C.Liu,G.LabbeandG.G.Ihas[ 74 ].InChapter 3 thismotorconstructionwasreferredtoastheimpulsemotor.ThisexperimentalapparatusandtheresultingworkaretherstincarnationsofisotropichomogeneousturbulenceproducedbymovingagridatmKtemperatures.Thedetailsoftheexperimentandapparatusarethoroughlyreviewedin[ 75 ]andtheoperationprocedureofthemotorisreviewedintheimpulsemotorsection,Section 3.2 .AschematicoftheexperimentalcellisshowninFigure A-12 andtherelevantparametersareinTable 4-1 .Theapparatushasbeencooledtomillikelvintemperaturesseveraltimes,eachtimewithaslightlydifferentoperatingprocedurewhichisreviewedinChapter 4 .Theinitialexperimentswereconductedwithaspringsteelgridproducingeddiesinaheliumbathandreportedin[ 84 ].Concernsaboutthemagnitudeandtypeofheatingobservedinthisworkledtoanewgriddesignandmaterial.Astainlesssteelmeshofidenticaldesignwasconstructedtoreplacethemagneticspringsteelgrid.Resultsfromthisarepublishedin[ 85 ].Withthisadditionalwork,theresultswerestillnotwellunderstoodandthegridwasremovedfromthesystemandthecellwasrunasacontroltomeasuretheheatingfromelectronicandfrictionaldissipationfromtheactuationprocedurealone.Afterthecontrolmeasurements,thestainlesssteelmeshwasre-installedandthe Table4-1. Relevantpropertiesoftheimpulsemotor CategoryResult SolenoidRadius5mmLengthofSolenoid10mmNbcanRadius3.175mmLengthof1stNbcan13mmLengthof2ndNbcan15mmGapbetweenNbcans21mmMassofarmature2.6g 134

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impulsemotorcellwascooledagain,toreproducethepreviousmeasurements.TheseexperimentsandthecomparisonsbetweenthemarereportedinChapter 4 .Thismotorwasdesignedtocreateamassiveamountofturbulenceinaveryshorttime.DatafromStalpetal.[ 86 ],suggestedthatat1.4Kgridvelocitiesofupwardsof1m=swererequiredtoestablishalongandwelldenedinertialregime.Atthetime,theorysuggestedthatwelldevelopedquantumturbulencerequiredfairlyhighgridvelocities,withthisinmindamKexperimentalcellwascreatedtomeetthesedemands.TheoperationofthismotorisexplainedinthemotordesignChapter 3 inthesectionontheImpulseMotor.Thecellwasmachinedfromoxygenfreecopperandheatsunktothemixingchamberofadilutionrefrigerator,withoutasinteredheatexchanger.Themotorwasdesignedtohaveaweakthermallinktothemixingchamber,sothatitcanbebothcooledbythemixingchamberandisolatedenoughtoobservequickchangesintheheliumtemperature.Themotorwasplatedwithsuperconductingleadtoformaninsulatinglayer,bothmagneticallyandthermally,betweentheheliumandthecopper.Theleadplatedcopperwasmountedonagoldplatedheatstagewhichwasattachedtothemixingchambervia41/4"copperrods.Thissetupisthesameaswasusedtocalibratethethermistors.Nosinterwasusedbecausetheexperimentwasdesignedtomeasurefasttemperaturechangesfromalargeamountofturbulencerapidlydecaying.Therefore,thethermometersaredesignedtomeasurethechangeintemperaturebeforethecellequilibrateswiththemixingchamber.Duetotherapidnatureoftheproposeddecayprocess,theKapitzaboundaryresistancebetweentheleadandheliumandleadandcopperwasusedtoinsulatethecell.Theleadisasuperconductorandthereforeunabletoholdmuchheat(duetotheabsenceofelectronmodes),butthetwolayersofboundaryresistanceandpoorthermalconductivityofleadinsulatetheheliumonthetimescaleofourmeasurement. 135

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Theexperimentalprocedurewastollacylinderwithapoolofhelium,stabilizethetemperature,dragatowedgridthroughthepool,andmeasuretheheliumresponsewithvarioussensors.Theturbulencecreatinggridispositionedatthebottomofthecylinder,suspendedjustabovethethermometers,heaters,andpressuresensors.Duringdifferentphasesoftheexperiment,differentgridsareusedincludinggridsmadefromstainlesssteel,springsteel,andG-10mesh.Thethermistorsweremountedonaplasticcircuitboardjustabove,butnotindirectthermalcontactwith,thebottomofthecell,whiletheotherthermometersweresuspendedintheliquidbytheirelectricalleads.Thegridistowedupwardswitharapidacceleration,average50m=s2,tospeedsof1m=sthendeceleratedattheendofthemotionbyasimilarrapidtechnique.Thegridispulledthroughtheuid,thenpulledthroughthesurfaceorleftintheliquid,dependingonexperiment.Oncethegridisoutoftheliquid,itcanbeheldabovethesurfaceonplungedbackthroughtheliquidbygravity.Throughouttheexperiment,thethermistorresistanceismonitoredwithawheatstonebridgecircuitdrivenat70kHzandanintegrationtimeconstantof300s.Thepositionofthegridismonitoredbythecapacitivepositionsensorinaseparatebridgecircuitdrivenat90kHz.Thepositionofthemotor,celltemperatureandheliumtemperatureareallcontrolledand/ormonitoredbyaNationalInstrumentsBoardandLabViewsoftware.TheDAQboardhasamaximumsamplingrateof50s.Therstmeasurementsusingthismotorsystemwerecompletedwiththespringsteelgridatatemperaturearound500mK.Forthedatapresentedhere,thegridwaspulled4mmthroughtheliquidheliumandheldabovethesurfacefor0.03sbeforeitwasallowedtofall.Aftertheinitialriseandfallofthemotor,thegridwasallowedtorelaxandsitinthelowestpositionofthecell.AnexampleofoneparticularmotormotionispresentedinFigure 4-1 [ 84 ].Inadditiontothemotorspeed,themeshReynoldsnumberforthemotionisgivenusingviscositiesfromtwodifferentsources[ 84 86 ].Inthegure,thepositivegridspeedscorrespondtomotionsopposinggravityandthenegative 136

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speedsareformotionalignedwithgravity.ThemeshReynoldsnumberisdenedas, ReM=VgM=,(4)whereMisthemeshsize, =isthekinematicviscosity,andVgisthegridvelocity.Measurementsat600millikelvinweretakenwithoutthepresenceofliquidheliuminthecell.Thesemeasurementsareusedasacontrol,showingtheelectricalcouplingbetweenoursensorsandthedrive,aswellastheheatingproducedbythemotionofthegridwhenonlyaheliumlmispresent.Notethesmallriseintemperatureseeninboththenobulkheliumpresentandtheheliumpresentdata.Whenthereisnobulkheliuminthecell,thesecondlargertemperatureriseisabsent.DatatakenfromthisgridpullareshowninFigure 4-2 [ 84 ].Thesedatashowanimmediateriseintemperaturewhichquicklyplateausfollowingthecurrentdrivetothemotorandthecorrespondingmotionofthearmature.Thissetofdataiscomparedtoasimilarmotormotionat520millikelvinwhenacolumnofheliumispresentinthecellFigure 4-2 .Inthissecondexperiment,afteraninitialtimeperiodwherethemeasureddatafrombothrunsmatch,asecondlargertemperatureriseisobserved.Forcomparisontouiddynamicstudies,thechangeintemperatureshowninFigure 4-2 canbeconvertedintointernalthermalenergyoftheheliuminthecell.UsingtheheliumvolumeandthetablescompiledbyDonnellyandBarenghi[ 34 ],theheliumtemperaturecanbeconvertedintoanenthalpy.ThisisshowninFigure 4-3 alongwiththepositionofthegridasafunctionoftime.Plottingbothdatasetsonthesameaxisshowsthattheinitialriseintemperatureseeninboththeemptycellandheliumfulledcellcoincideswiththeinitialmotionorinitialapplicationofpotentialtothedrivecoil.Thissignalislikelyelectricalandindependentoftheuidmotion.Thesecondriseappearstobemoreinteresting.Itclearlyoccursafterthemotionofthegridhasceased.Becausethethermalconductivityofsuperuidheliumisextremelylarge,itisnotlikelythatthisheatingwascreatedintheheliumatthetimeofthemotormotion. 137

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Theexpectationsforheatingafterthedecayofquantumturbulencecanbecalculated,bothasafunctionoftimeandthemagnitudeofenergyreleased.Firstwecalculatethedragforceonthegridassumingheliumisaclassicaluidandthedragcoefcientisoforderone, F=CAV2g=2.(4)Aistheopaqueareaofthegrid(1.78cm2),Cisthecoefcientwhichistakentobeone,andisheliumdensityat0.52K.Integratingthisforcethroughthetraverseofthegridthroughtheheliumwecanobtainaninjectedenergyofabout30J.Therefore,themaximumexpectedkineticenergyoftheheliumisonly1/5oftheenergywemeasure!Therefore,basedonthisrudimentaryorderofmagnitudeestimation,somethingunexpectedishappeninginthesample.Theunexpectedmagnitudeofheatmeasuredinthisexperiment,pairedwithadifcultytoreproduceconsistentmotormotions,forcedustorethinkthecelldesignandsearchforalternative,otherthenturbulent,sourcesofheatingintheexperimentaswellasreasonsforanirreduciblemotormotion.Thecellwaswarmedupandtheoriginalmagneticspringsteelgridwasreplacedwithanon-magneticstainlessgridinanattempttomitigatethesmallmagneticcouplingbetweenthegridandthedrivemagnet.Inanefforttonotchangemorethanoneparameter,theexperimentalcellwasre-cooledwithonlythisdesignalteration.DatafrommotormotionswiththestainlesssteelgridareshowninFigure 4-4 [ 85 ].ThesedataweretakenaftersimilarmotormotionsasshowninFigure 4-3 .Themotormotionisalsoshownontheseplotsinred.Thesedataaretakenoveranexpandedtime-scaletoshowtheextentoftheheatingmeasuredbyourthermometry.ThetoptwoplotsofFigure 4-4 showashorttimeresponse,stillmorethan10timesaslongasthepreviousgure,wheretheinitialtemperatureriseisseen.TheshorttimescaledatacoverarangeoftimesignicantlylongerthanthegurespresentedinFigure 4-3 ,sotheshortelectronicheatingsignalisnotdetailed.Again,themotormotionisprovided 138

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togiveareferencetimescale.Thistemperaturechangeissimilarto,butsmallerthan,thedatapresentedforthespringsteelgrid.However,ifthesignalistrackedforlongerperiodsoftimeasecond,evenlargerheatsignalismeasured.Sincetheoriginaldatawasonlyrecordedoverashorttimescale,itisunknownwhetherthissignalwaspresentinourrstseriesofexperiments.Onceagaintheseheatingsignalsaretoolargetobedueentirelytoturbulentenergydecay.Withtheinabilitytoidentifythesourceofheatinginourpreviousexperiments,theturbulencecreatinggridwasremovedandthemotorsystemwasrunwithjustthearmature.Figure 4-5 showstheevolutionofthemeasuredheatvs.timeforamotorrunwithoutthegrid.Inthisplot,therearealsodatameasuredfromasquarevoltagepulseintoacarbonheater.Thetwoguresareverysimilarisshape,showingasharpinitialspikeintemperaturefollowedbytheabsorptionofthatheatbythecellwallsandmixingchamber.Theonlydifferencebetweenthetwoguresisthattheheatingfromthemotormotionappearstohaveabetterheatlinktothethermalbath.Theheatingpatternwithoutthegridpresentcanbecomparedtothemeasuredchangeinenergyforthepreviousrunswhenthemotorwasrunwithastainlesssteelmesh.ThesedataarepresentedinFigure 4-6 .Thesedatashowacleardistinctionbetweenthebehaviorofoursystemfromthedatatakenwiththeeddycreatinggridpresentandthedatacollectedwhenthegridwasremoved.Inthisgure,theshapeofthemeasuredtemperaturechangeiscompletelydifferentbetweenthetworuns,asistheabsolutemagnitudeoftheheatingproduced.Thenalmeasurementsconductedusingthisapparatuswerewiththestainlessgridreattached,checkingthesubstantialdifferencemeasuredpreviouslybetweentheheatingmeasuredwithagridandwithoutthegrid.DatafromthesemotorrunsarepresentedinFigure 4-7 .Eachdifferentcoloronthisplotrepresentsadifferentcurrentprolesenttothedrivemotor.Predictably,thelargertheinputcurrentandthefasterthemotormotion,thelargertheheatingmeasured.Sinceourpreviousworksuggestedthat 139

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theentireheatingsignalwasnotduetotheturbulentenergydecay,butacombinationofelectricalheating,frictionaldissipationandturbulentenergydecay,thedifferencesbetweentheenergyreleasedfromeachmotorrunwascomparedtothechangesinthemotionprole.Figure 4-8 showsdatafromtwosimilarmotormotions.Themainplotinthegureshowstheenergydifferencebetweenthemeasuredthermalresponseofbothruns,andtheinsetshowsthearmaturepositionasafunctionoftimeforeachmotormotion.Figure 4-9 showsthesametypeofdatabutfromtwoverydifferentmotions:oneinwhichthegridmovesthroughtheuidandtheotherinwhichthegriddoesnotmoveatall.Forbothgures 4-8 and 4-9 ,thedifferenceinappliedcurrenttothemotoristhesame,butinonethereisalargedifferenceinthearmaturemotionwhileintheother,thereisnot.Thechangeinenergyforbothplotsareverysimilartooneanother.Thisimpliestheheatingmeasuredisnotduetothemotormotion,butsomeothersource.BothoftheenergydifferencesareplottednexttooneanotherinFigure 4-10 demonstratingthiseffect.Fromthesedata,itappearsthattheenergyproducedbythemotorisindependentofthegridmotion,anddependsmoredirectlytothecurrentlevelappliedtothemotordrivecoil.Becauseoftheindependenceoftheheatingonthemeasuredmotionofthegridandtheenormoussizeoftheheatingmeasuredintheexperimentalcell,itisshownthatthemajorityofthesignalobservedwasnotduetoturbulentenergydecay.Inaddition,theheatinginthecellappearstobedirectlydependentonthecurrentsuppliedtothedrivecoil.Thisimpliesthatthereiseitheraresistancealongthemotorcurrentleadsinthecellwherepowerisdissipatedoramagneticcouplingtoanotherconductivepartofthecellwhichproduceseddycurrents.Therstattemptatreducingthelargeheatingsignalwastoreplacethebrasselectronicpinsconnectingthesuperconductingmotorleadstothecopperheatexchangeroutsidethecell,reducingohmicheatingfromthelargecurrentstraveling 140

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thoughthepins.Onceitwasshownthatthisdidnothaveanoticeableeffectontheheatsignalmeasured,attemptsweremadetoreducetheeddycurrentsproducedbythedrivemotor.Intheoriginalmotorconstruction,discussedintheliterature[ 74 ]andusedhere,thiswasdonebycoatingthecoppercellwithsuperconductingleadtoreducetheeddycurrentsinthecoppercellitself.Thisalsodidnotseemtobethesourceoftheheating.Currentsinthemetalgirdwerealsoconsideredandaplasticgridwasconstructedtoreplacethesteelone.ThisgridisshowninFigure 4-11 .Thevaluesoftheeddycurrentsinthespringandstainlesssteelgridsarecalculatedbythestandardformula[ 87 ]usingtheformula, _Qe=PV_B2=,(4)andPisthegeometricfactordenedbelow,Visthebodyvolume,_B2isthefunctionalchangeinmagneticeld,andistheelectricalresistance.P=d2 16(w=d)2 (1+w=d)2,wherewanddarethewidthandlengthofthemeshonthegrid.Sincethemagneticeldoutsideasolenoiddecaysrapidly,theeddycurrentsonthegridarenotlargeenoughtoproducetheheatingobserved.Afterthenalexperimentalrun(datashowninFigure 4-7 ),itbecameapparentthattheproblemwiththismotordesigndidnotstemfromanyofthesourceslistedabove,soanentirenewexperimentalsystemwascreatedwhichremovedallofthenon-superconductingmetalfromtheexperimentalvolume.ThisexperimentisoutlinedinChapter 5 .Whenthetroublesomemotorwasdisassembleditwasobservedthatthecapacitivepositionsensorwasextendedwiththeuseofconductivesilverpaint.Thispaintnotonlyformedalargeconductiveloopinthedrivemotor,italsocovereduppartsofthesuperconductingcylinders.Thepaintwaslikelyaddedtoincreasethesensitivityrangeofthepositionsensor,butithadtheeffectofprovidingalargeconductionpath 141

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foreddycurrents.ThesecurrentscanbeestimatedbyasimilarsetofequationstoEquation 4 ,butmodiedforthecaseofathinlm[ 88 ] _Qe=2_B2tV 6,(4)wheretisthelmthickness.Dependingontheresistivityusedforthesilverpaintniobiumcircuitcombination,thisheatingsourcecanrangefromWtomW.Thisismorethanenoughenergytoaccountforboththeheatingobservedintheexperimentsandtheirreproducibilityofthemotormotions. 142

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Figure4-1. ThisgureshowsthegridspeedfromtheinertialmotoratmillikelvintemperatureandtheassociatedmeshReynoldsnumber[ 84 ].Thereissomequestionastowhattheeffectiveviscosityoftheheliumisatthistemperaturesobothpossiblescalesaregiven[ 7 34 ]. 143

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Figure4-2. Thetemperaturerisemeasuredbythethermistorafterthegridispulledforthecaseswherethecellcontainsheliumandwhenitdoesnot[ 85 ].Thelargerriseisonlypresentwhenthereisliquidheliuminthecellwhiletheearlytimesmallriseispresentwhetherthereisbulkliquidheliuminthecellandjustaheliumlm. Figure4-3. Thisgraphshowsthetemporalrelationshipbetweenthemotormotionandthemeasuredtemperatureincreaseinthecell.Therstriseinenergydirectlycorrespondstotheinitializationofcurrentinthedrive,whilethesecondriseoccurssometimeafterthemotionhasstopped[ 84 ]. 144

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Figure4-4. Thisplotshowsfourdifferentinstancesofmotoractuationintheheliumandthesimilarheatprolesfollowingthemotion.Thedataarearrangedsuchthattheshorttimescalesareinthetoprowandthelongertimesscalesareonthebottomrow.Inadditiontothedifferenceintimescalesthelowertemperaturedataareontheleftandthehighertemperaturedataareontheright. 145

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Figure4-5. Plotshowingthedifferenceinheatgeneratedfromacarbonheateremersedinheliumandduetomotionofthearmaturesansgird.Theheatfromtheheaterwascalibratedtodeliverasimilaramountofenergyaswasseenbythemotioninthecellwithouttheturbulencecreatinggrid. Figure4-6. Thisgureshowsthedifferenceinthemeasuredheatsignalswhenthegridisattachedtothearmatureandwhenitisnot. 146

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Figure4-7. Theseheattracesaremeasuredafterdifferentgridpulls.Foreachtracemovingfromthetoptothebottom,lessencurrentisdriventhroughthedrivecoilofthemotor.Inthebottomfewtracesonthisplotthemotordidnotmoveatall. Figure4-8. Thisgureshowsthedifferenceinmeasuredenergybetweentwotraceswithsimilarmotionproles.Themotionofthegridisshownintheinsetofthegureandthedifferenceinthemeasuredheatisshowninthemaingure 147

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Figure4-9. Thisgureshowsthedifferenceinmeasuredenergybetweentwotraceswithdifferentmotionproles.Themotionofthegridisshownintheinsetofthegureandthedifferenceinthemeasuredheatisshowninthemaingure. Figure4-10. Thisgureshowstheprevioustwoenergydifferences( 4-8 and 4-9 )plottedonthesameaxis.Thisprovesthatthebulkofenergyweareseeingfromourmotorsystemisnotduetorealmotionofthegrid,andthereforenotduetoturbulentenergydecay. 148

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Figure4-11. ThisgureshowsaturbulencecreatinggridmachinedfromG-10 149

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CHAPTER5ONGOINGANDFUTUREWORKPresentedinthisdissertationare:anexperimentalimplementationofanexistinglinearmotortocreatetherststudiesofquantumturbulencecreatedbythismeansinthelowtemperaturelimit;identicationofthepossibleshortfallsofthispreliminarymotorsystemandthedevelopmentofanew,moresophisticated,driveapparatus;andthecreation,exploration,anddevelopmentofnewandexistingofsensorsformeasuringthepropertiesofquantumturbulence.Butalas,noworkisevercomplete.Thereisalwaysmoretodo,learn,andexplore.Afewoftheon-goingprojectsanddirectionsinlaboratorywillbeoverviewedandoutlinedinthisChapter.Specically,therepairandmodicationstotheimpulsemotorassemblyforanothermillikelvinexperiment,theimplementationofthecontrolmotorfortherstrunstoobtainnewinformationaboutquantumturbulentenergydecay,aswellassomeclosingthoughts. 5.1SecondSoundandtheControlmotorAsecondsoundcellhasbeenbuiltandcooledto1K.ThemachinedrawingsfortheexperimentareshowninFigure A-10 .ThisexperimentismountedonthesameinsertshowninFigure 3-9 andplacedabovethetopquadruplebearing.Threedifferentturbulence-creatinggridshavebeenmanufacturedandimagesofthemareshowninFigure 5-1 .Twoofthegridsareofthetypicaldesignwhereanarrayofsquareshavebeencutoutofstainlessstealstocktoformasquaremesh.Thematerialhasathicknessof0.001inandthegridsare0.8insquares.Thesegridswerecreatedtohavewelldenedlengthscalestocharacterizetheturbulence.Thethirdgridisofaverydifferent,modeledaftertheonlyothercryogenicheliumexperimentwherealinearlyactuatedgridisused[ 72 ].Thisistherstgridtobecooledinthisapparatus.Itwillbeusedinanattempttorecreatethedatapreviouslymeasured[ 72 ].ThecontrolmotoractuationdistanceismanytimesshorterthatthemotorusedinOregon,butstillmorethan10timesthe 150

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meshspacingofgridsthatwillbeused.Thereisreasontobelievethatthecontrolmotorchannelislongenoughtocreatehomogenousturbulenceforthismeshsize.Oncethisisestablished,thecontrolmotorwillbeusedtoinvestigatemanynewphenomenoninquantumturbulence.Ofspecicinterestistheapplicabilityofclassicaluiddynamiclikedescriptionsofgridturbulenceforvariousdifferentgridsizesanddesigns.Also,theearlytimeformationofturbulenceinthecontrolmotorchannel,specicallyifthemomentumstructureofthetangleobeysaSaffmanorBachelorexpansionforlowwavenumberswillbeinvestigated.Thecontrolmotor,grids,andchannelusedhaveaversatiledesign.Theymaybemountedbothontheapparatususedinmuchofthiswork,aswellasanopticalapparatuswhereheliumexcimermoleculescantrackthenormaluidmotion[ 58 ]andpotentiallytrackquantumvortexlinesinthesuperuidstate! 151

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Figure5-1. Thisgureshowsthreedifferentgridgeometriesbuilttoexploretheireffectonthecreationanddecayofturbulenceataround1K.Thegridswillbepulledthroughthechannelinthegureandthesecondsoundtransducersmountontheslitsshowninthechannel. 152

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5.2NonconductingimpulsemotorcellAnewnonconductingimpulsemotorcellhasbeencreatedwithsimilardimensionstotheoneusedinChapter 4 .ThemachinedrawingsandspecicationsofthecellarepresentedinFigure A-12 .Thecapacitivepositionsensorwascleanedandthesilverpaintwasreplacedbyalongerniobiumcylinder.ThestainlessstealgridwasreplacedbyagridmachinedfrompolycarbonateasshowninFigure 4-11 .ThisexperimentalcellwasmachinedfromGaroliteXX,whichisapaperepoxymixture.ThecellwasmachinedtothedimensionsspeciedandcoatedwithathinlayerofStycast1266epoxytoassureasuperuidheliumleaktightsystem.Onceatmillikelvintemperatures,theoperationofthismotorsystemwillproceedinamannersimilartothatdescribedinChapter 4 andsec. 1.6 .Themaindifferenceofthisexperimentalcellfromthecoppercellpreviouslyusedistheheatexchangewiththemixingchamber.TheoldcellreliedonthepoorthermalconductivityofsuperconductorsandKapitzaboundaryresistancewherethenewplasticcellismountedonathermalheatswitch.Withthisimprovementthecalorimetrycellcanbemademuchmoreindependentofthemixingchamber,andtheancillaryheatcapacitiesfromthecoppermountsarenolongeravailableasaheatsink.TheheatexchangerisalowmassmodicationofthestandardheliumheatswitchusedbytheIhasgroup.PhotographsoftheheatswitchareshowninFigure 5-2 .Theswitchoperatesontheprinciplethatindifferentstates,superconductingornormal,aindiumwirewilleitherconductheat(closed)ornot(open).TheendofthethermalswitchissolderedtoasetoftwosilversinteredheatexchangerswithexceptionallylargesurfaceareasuchasthosecommonlyusedattheUniversityofFlorida[ 89 ].Silversinterisawellestablishedmeansforheatcontacttoheliumatlowtemperatures[ 90 ].Thesinteredsilverisapressedpelletofmany70nmradiussilvergranulesaroundanesilverwire.Bycalculatingthenumberofgranulesfromtheassumedgrainvolumeandthemeasuredmassofthepopsicle,thesurfaceareaofthesensorcanbeinferredby 153

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multiplyingthenumberofgrainsbytheirindividualsurfacearea.Afteraccountingforsurfacesthattouchwithafactorof1=2thesurfaceareaoftheheatexchangerisshowntobe0.5m2.Thethermalconductivityoftheswitchcanbecalculatedwiththistableforboththe`open'and`closed'positionoftheindiumwire.Thethermalconductivityofthesliverwireconnectingtheswitchtothepopsiclesandthethermalconductivityofthepopsicleswiththeheliumarealsocalculatedbelow.TheKapitzaconductivityofthepopsiclecanbecalculatedat100millikelvin[ 66 ], _Q T=1 Rk'5,000cm2 15080(Kcm2=W)=0.33W=K.(5)Thethermalconductivityofthe1"longofsilverwireconnectingtheheatexchangertothepopsiclesis[ 87 ] _Q T=silverL A=110)]TJ /F6 7.97 Tf 6.59 0 Td[(2W=cmK1inch 0.012inch2=210)]TJ /F6 7.97 Tf 6.59 0 Td[(6W=K(5)andthethermalconductivityoftheindiuminitssuperconductingstate[ 91 ] _Q T=InsuperL A=1.5810)]TJ /F6 7.97 Tf 6.59 0 Td[(3W=cmK6inch 0.032inch2=6.210)]TJ /F6 7.97 Tf 6.59 0 Td[(9W=K(5)andthenormalstate _Q T=InnormalL A=1.020W=cmK6inch 0.032inch2=4.010)]TJ /F6 7.97 Tf 6.59 0 Td[(5W=K.(5)Theotherpropertyofnoteforthisheatexchangerisitslowthermalheatcapacity.Usingthestandardheatcapacityforametalatlowtemperaturec=T=T2+andasilvermassof0.6gthethermalmassofthesilveris[ 92 ] C100mK=0.174140.13+0.617170.15.5610)]TJ /F6 7.97 Tf 6.59 0 Td[(3=3.74410)]TJ /F6 7.97 Tf 6.59 0 Td[(4mJ=K(5)comparedtothe1.237610)]TJ /F6 7.97 Tf 6.59 0 Td[(1mJ=Kfor1.5molofhelium[ 34 ]. 154

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Figure5-2. Photographof0.5m2surfaceareasinterpopsicles 155

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APPENDIXMACHINEDRAWINGS 156

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FigureA-1. MachineDrawingofdrivesolenoid 157

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FigureA-2. MachineDrawingofinductivepositionsensor 158

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FigureA-3. Schematicdrawingsfortheglassdewarinsert 159

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FigureA-4. MachineDrawingoftopplatefromglassdewarinsert 160

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FigureA-5. MachineDrawingofbafefromglassdewarinsert 161

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FigureA-6. MachineDrawingofbottomplatefromglassdewarinsert 162

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FigureA-7. MachineDrawingofquadruplemagnetmounts 163

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FigureA-8. MachineDrawingoftophalfoftheanti-Helmholtzmandrel 164

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FigureA-9. MachineDrawingofbottomhalfoftheanti-Helmholtzmandrel 165

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FigureA-10. MachineDrawingoftophalfofthesecondsoundtransducermount 166

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FigureA-11. MachineDrawingoftophalfofthesecondsoundturbulencecell 167

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FigureA-12. MachineDrawingofbottomoftheplasticimpulsemotorcell 168

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FigureA-13. MachineDrawingofmiddleoftheplasticimpulsemotorcell 169

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FigureA-14. MachineDrawingoftopoftheplasticimpulsemotorcell 170

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[48] S.K.Nemirovskii(2010). [49] V.S.L'vov,S.V.Nazarenko,O.Rudenko(2007). [50] E.Kozik,B.Svistunov(2008). [51] D.C.Samuels,C.F.Barenghi(1998). [52] J.Maurer,P.Tabeling,Europhys.Lett.43(1999). [53] J.Salort,etal.,Phys.Fluids22,125102(2010). [54] A.M.Gunault,V.Keith,C.J.Kennedy,S.G.Mussett,G.R.Pickett,JournalofLowTemperaturePhysics62,511. [55] H.Yano,etal.,JournalofLowTemperaturePhysics156,132. [56] M.Bladkov,etal.,JournalofLowTemperaturePhysics150,525. [57] T.Zhang,S.W.vanSciver,NaturePhysics1,36(2005). [58] W.Guo,S.B.Cahn,J.A.Nikkel,W.F.Vinen,D.N.McKinsey(2010). [59] N.Giordano,JournalofLowTemperaturePhysics55,495. [60] G.Zimmermann,F.Pobell,JournalofLowTemperaturePhysics61,213. [61] R.A.Sherlock,D.O.Edwards,ReviewofScienticInstruments41,1603(1970). [62] V.Dugaev,etal.,Sensors,2002.ProceedingsofIEEE(2002). [63] Cryogenics47,474(2007). [64] R.A.Antonia,N.Phan-Thien,A.J.Chambers,JournalofFluidMechanics100,193(1980). [65] I.KHALATNIKOV,ZHURNALEKSPERIMENTALNOIITEORETICHESKOIFIZIKI22,687(1952). [66] G.Pollack,ReviewsofModernPhysics41,48(1969). [67] P.Keesom,G.Seidel,PhysicalReview113,33(1959). [68] InternationalJournalofHeatandFluidFlow8,82(1987). [69] R.K.Childers,J.T.Tough,Phys.Rev.B13(1976). [70] H.A.Nichol,L.Skrbek,P.C.Hendry,P.V.E.McClintock,Phys.Rev.Lett.92(2004). [71] D.I.,etal.,Phys.Rev.B85,224533(2012). [72] S.R.Stalp,L.Skrbek,R.J.Donnelly(1999). 173

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[73] M.Giltrow,D.Charalambous,P.V.E.McClintock,JournalofPhysics:ConferenceSeries150,012014. [74] S.-c.Liu,G.Labbe,G.Ihas,JournalofLowTemperaturePhysics145,165. [75] S.Liu,QuantumTurbulence:DecayofGridTurbulenceinaDissipationlessFluid,Ph.D.thesis(UniversityofFlorida,Gainesville,Fl,2007). [76] C.Kittel,IntroductiontoSolidStatePhysics,vol.2012(1995). [77] N.Ashcroft,N.Mermin,Solidstatephysics,Holt-SaundersInternationalEditions:Science:Physics(Holt,RinehartandWinston,1976). [78] B.W.Maxeld,W.L.McLean,Phys.Rev.139,A1515(1965). [79] Y.Asada,H.Nos,JournalofthePhysicalSocietyofJapan26,347(1969). [80] D.Grifths,Introductiontoelectrodynamics(PrenticeHall,1999). [81] D.Tilley,J.Tilley,SuperuidityandSuperconductivity,GraduateStudentSeriesinPhysics. [82] B.N.Engel,G.G.Ihas,E.D.Adams,C.Fombarlet,ReviewofScienticInstru-ments55,1489(1984). [83] S.Denis,M.Dirickx,P.Vanderbemden,M.Ausloos,B.Vanderheyden,Supercon-ductorScienceandTechnology20,418. [84] G.Ihas,G.Labbe,S.Liu,K.Thompson,JournalofLowTemperaturePhysics150,384. [85] K.Thompson,S.Liu,G.Labbe,G.Ihas,JournalofPhysics:ConferenceSeries150(2009). [86] S.R.Stalp,J.J.Niemela,W.F.Vinen,R.J.Donnelly,PhysicsofFluids14,1377(2002). [87] F.Pobell,MatterandMethodsatLowTemperatures. [88] F.Fiorillo,I.Mayergoyz,CharacterizationandMeasurementofMagneticMaterials,Elsevierseriesinelectromagnetism. [89] W.Pan,etal.,Phys.Rev.Lett.83,3530(1999). [90] Cryogenics24,477(1984). [91] P.-C.Ho,R.B.Hallock,JournalofLowTemperaturePhysics121,797. [92] K.DeConde,G.A.Williams,R.E.Packard(1974). 174

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BIOGRAPHICALSKETCH KyleJ.ThompsonwasraisedinthevalleyofAmherst,Massachusetts,lookingupatthefoothillsoftheBerkshires.Hegraduatedfromthesamehighschoolasbothofhisparents,AmherstRegionalHighSchool,in2002.AfterwardheimmediatelyenrolledatTheUniversityofMassachusettstopursuehisinterestsinmath,physicsandengineering.Duringhistimeincollegehegravitatedawayfromengineeringtothemorepurelyscienticpursuits.Earlyonheirtedwiththesoftsciences,workinginmicrobiologyforDr.Lovely'slab,butthensmarteneduptostudyfundamentalphysicsinDr.Hallock'slaboratory.Aftergetinghisfeetwetwithheliumphysicsandgraduatinginthefallof2006withhonorsdegreesinbothmathematicsandphysicsfromthecommonwealthcollegeheenrolledinaPh.D.programattheUniversityofFlorida.For5ofthenext6yearshewentontoworkinthelaboratoryoflowtemperaturephysicswithProf.Ihasproducingthedocumentinfrontofyoutoday.AfterhisPh.D.hehasagreedtotakeonapostdoctoralpositionattheUniversityofSaoPauloinSaoCarlosBrazilstudyingBose-Einsteincondensates. 175