超导物理学课件

MartinWilsonLecture1slide1SuperconductingmagnetsforAcceleratorsWhoneedssuperconductivityanyway?

AbolishOhm’sLaw!nopowerconsumption (althoughdoneedrefrigerationpower)highcurrentdensityampereturnsarecheap,sowedon’tneediron (althoughoftenuseitforshielding)Consequenceslowerpowerbillshighermagneticfieldsmeanreducedbendradius

smallerrings

reducedcapitalcost

newtechnicalpossibilities (egmuoncollider)higherquadrupolegradients

higherluminosityMartinWilsonLecture1slide2SuperconductingmagnetsforAccelerators

PlanoftheCourse

MartinNWilson(RutherfordLab

OxfordInstruments

consultant)

1Introductionwheretofindmoreinformationpropertiesofsuperconductors,criticalfield,criticaltemperature&criticalcurrentdensityhightemperaturesuperconductorsHTSmagneticfieldsandhowtocreatethemloadlinesandtemperaturemargindegradationandtraining2Controllingtraining&finefilamentscausesoftrainingminimumpropagatingzonesMPZandminimumquenchenergyMQEscreeningcurrentsandthecriticalstatemodelfluxjumpingmagnetizationandfielderrors3Cables&quenchingwhycables?couplingincablesfielderrorscausedbycablemagnetizationthequenchprocess,internalandexternalvoltagesdecaytimesandtemperaturerisepropagationofthenormalzone,quenchprotectionschemes,protectionofLHC4Manufacturingandtestingconductorandcablemanufacturemagnetmanufacture

measurementofcriticalcurrentandmagnetizationcurrentleadsandpersistentcurrentswitchingsomeexamplesofsuperconductingacceleratorsMartinWilsonLecture1slide3SomeusefulreferencesSuperconductingMagnetsSuperconductingAcceleratorMagnets:KHMess,PSchmuser,SWolf.,pubWorldScientific,(1996)ISBN981-02-2790-6HighFieldSuperconductingMagnets:FMAsner,pubOxfordUniversityPress(1999)ISBN0198517645CaseStudiesinSuperconductingMagnets:YIwasa,pubPlenumPress,NewYork(1994),ISBN0-306-44881-5.SuperconductingMagnets:MNWilson,pubOxfordUniversityPress(1983)ISBN0-019-854805-2ProcAppliedSuperconductivityConference:pubasIEEETransAppliedSuperconductivity,Mar93to99,andasIEEETransMagneticsMar75to91HandbookofAppliedSuperconductivityedBSeeber,pubUKInstitutePhysics1998CryogenicsHeliumCryogenicsVanSciverSW,pubPlenum86ISBN0-0306-42335-9CryogenicEngineering,HandsBA,pubAcademicPress86ISBN0-012-322991-XCryogenics:publishedmonthlybyButterworthsCryogenie:SesApplicationsenSupraconductivite,pubIIR177BoulevardMalesherbesF5017ParisFranceMaterialsMechanicalMaterialsatLowTemperature:EdRPReed&AFClark,pubAm.Soc.Metals1983.ISBN0-87170-146-4HandbookonMaterialsforSuperconductingMachinerypubBatelleColumbusLaboratories1977.Nonmetallicmaterialsandcompositesatlowtemperatures:EdAFClark,RPReed,GHartwigpubPlenumNonmetallicmaterialsandcompositesatlowtemperatures2,EdGHartwig,DEvans,pubPlenum1982AusteniticSteelsatlowtemperaturesEditorsR.P.ReedandT.Horiuchi,pubPlenum1983SuperconductingMaterialsSuperconductorScienceandTechnology,publishedmonthlybyInstituteofPhysics(UK).SuperconductivityofmetalsandCuprates,JRWaldram,InstituteofPhysicsPublishing(1996)ISBN0852743378HighTemperatureSuperconductors:ProcessingandScience,ABourdillonandNXTanBourdillon,AcademicPress,ISBN0121176800MartinWilsonLecture1slide4MaterialsdatawebsitesCryodataSoftwareProducts

GASPAK propertiesofpurefluidsfromthetriplepointtohightemperatures.

HEPAKpropertiesofheliumincludingsuperfluidabove0.8K,upto1500K.

STEAMPAK

propertiesofwaterfromthetriplepointto2000Kand200MPa.

METALPAK,CPPACK,EXPAK referencepropertiesofmetalsandothersolids,1-300K.

CRYOCOMP propertiesandthermaldesigncalculationsforsolidmaterials,1-300K.

SUPERMAGNET fouruniqueengineeringdesigncodesforsuperconductingmagnetsystems.

KRYOMnumericalmodellingcalculationsonradiation-shieldedcryogenicenclosures.Cryogenicproperties(1-300K)ofmanysolids,includingthermalconductivity,specificheat,andthermalexpansion,havebeenempiricallyfittedandtheequationparametersareavailablefreeonthewebat.Plotsandautomateddata-look-upusingtheNISTequationsareavailableonthewebforafeefrom.Otherfeewebsitesthatusetheirownfittingequationsforanumberofcryogenicmaterialpropertiesinclude:(cryogenicpropertiesofabout100materials), and(temperaturedependentpropertiesofabout1000materials,manyatcryogenictemperatures).Commerciallysuppliedroom-temperaturedataareavailablefreeonlineforabout10to20propertiesofabout24,000materialsat.thankstoJackEkinofNISTforthisinformationMartinWilsonLecture1slide5ThecriticalsurfaceofniobiumtitaniumNiobiumtitaniumNbTiisthestandard‘workhorse’ofthesuperconductingmagnetbusinessitisaductilealloypictureshowsthecriticalsurface,whichistheboundarybetweensuperconductivityandnormalresistivityin3dimensionalspacesuperconductivityprevailseverywherebelowthesurface,resistanceeverywhereaboveitwedefineanuppercriticalfield

Bc2

(atzerotemperatureandcurrent)

andcriticaltemperatureqc(atzerofieldandcurrent)whicharecharacteristicofthealloycompositioncriticalcurrentdensitydependsonprocessingField(Tesla)Temperature(K)Currentdensity(kA.mm-2)MartinWilsonLecture1slide6Thecriticallineat4.2Kbecausemagnetsusuallyworkinboilingliquidhelium,thecriticalsurfaceisoftenrepresentedbyacurveofcurrentversusfieldat4.2KniobiumtinNb3SnhasamuchhigherperformanceintermsofcriticalcurrentfieldandtemperaturethanNbTibutitisbrittleintermetalliccompoundwithpoormechanicalpropertiesnotethatboththefieldandcurrentdensityofbothsuperconductorsarewayabovethecapabilityofconventionalelectromagnetsCriticalcurrentdensityA.mm-210102103104Magneticfield(Tesla)Nb3SnNbTiConventionalironyokeelectromagnetsMartinWilsonLecture1slide7Filamentarycompositewiresforreasonsthatwillbedescribedlater,superconductingmaterialsarealwaysusedincombinationwithagoodnormalconductorsuchascoppertoensureintimatemixingbetweenthetwo,thesuperconductorismadeintheformoffinefilamentsembeddedinamatrixofcoppertypicaldimensionsare:wirediameter0.3-1.0mmfilamentdiameter10-60mmforelectromagneticreasons,thecompositewiresaretwistedsothatthefilamentslooklikearope(seeLecture3onfilamentaryconductorsandcables)MartinWilsonLecture1slide8Criticalproperties:temperatureandfield1wherekBisBoltzmann'sconstantandD(0)istheenergygap(bindingenergyofCooperpairs)ofatq=0CriticalField

Bc:Type1superconductorsshowtheMeissnereffect.FieldisexpelledwhensamplebecomessuperconductingItcostsenergytokeepthefieldout.Criticalfieldhappenswhenthecondensationenergyofthesuperconductingstateisjustequaltotheenergypenaltyofkeepingthefieldout.CriticalTemperature

qcwhereGistheGibbsfreeenergyofthenormal/superconductingstate.BCStheorysayswhereNF

isthedensityofstatesattheFermisurfaceofmetalinnormalstate-calculateitfrom:wheregisSommerfeldcoefficientofelectronicspecificheatMartinWilsonLecture1slide9Criticalproperties:temperatureandfield2combiningthepreviousequations:'thermodynamiccriticalfield'

Bc

solikethecriticaltemperature,Bcisdefinedbythe'chemistry'typicallyforNbTig~103Jm-3K-1

soifq=10KBc=0.24TConclusion:

Type1superconductorsareuselessformagnets!Notehowever:Meissnereffectisnottotal,themagneticfieldactuallypenetratesasmalldistanceltheLondonPenetrationDepth.Anothercharacteristicdistanceisthecoherencelengthz-theminimumdistanceoverwhichtheelectronicstatecanchangefromsuperconductingtonormalMartinWilsonLecture1slide10Criticalproperties:type2superconductorsTheoryofGinsburg,Landau,AbrikosovandGorkovGLAG definestheratiok=l/xIfk>1/

2themagneticfieldcanpenetrateintheformofdiscretefluxoids-Type2asinglefluxoidenclosesfluxwhereh=Planck'sconstant, e=electronicchargeuppercriticalfieldinthe‘dirtylimit'wherernisthenormalstateresistivitythustheuppercriticalfieldforNbTi:g~900Jm-3K-2

rn~65x10-8Wmqc=9.KhenceBc2=18.5TMartinWilsonLecture1slide11Criticalproperties:currentdensityFluxoidsconsistofresistivecoreswithsuper-currentscirculatingroundthem.spacingbetweenthefluxoidsauniformdistributionoffluxoidsgivesnonetcurrentJc=0,butagradientproducesanetcurrentdensitygradientsareproducedbyinhomogeneitiesinthematerial,egdislocationsorprecipitatesprecipitatesofaTiinNbTifluxoidlatticeat5TonthesamescaleMartinWilsonLecture1slide12Criticalproperties:summaryCriticaltemperature:choosetherightmaterialtohavealargeenergygapor'depairingenergy'Criticalfield:chooseaType2superconductorwithahighcriticaltemperatureandahighnormalstateresistivityCriticalcurrentdensity:messupthemicrostructurebycoldworkingandprecipitationheattreatments -thisistheonlyonewherewehaveanycontrolSimilareffectsinhightemperaturesuperconductingmaterials:fluxoidlatticeinBSCCOMartinWilsonLecture1slide13Uppercriticalfieldsofmetallicsuperconductors

Note:ofallthemetallicsuperconductors,onlyNbTiisductile.AlltherestarebrittleintermetalliccompoundsMartinWilsonLecture1slide14Hightemperaturesuperconductorsmanysuperconductorswithcriticaltemperatureabove90K-BSCCOandYBCOoperateinliquidnitrogen?YBCOYBa2Cu3O7MartinWilsonLecture1slide15HightemperaturesuperconductorsYBCOstructureConductionlayersconsistoftwoCuO2layersseparatedbyyttriumatoms.Thechargelayerconsistsofcopper,bariumandoxygenatomsNote:thisstructuremakesthepropertieshighlyanisotropicMartinWilsonLecture1slide16Irreversibilityline-abigdisappointmentUnlikethemetallicsuperconductors,HTSdonothaveasharplydefinedcriticalcurrent.Athighertemperaturesandfields,thereisan'fluxflow'region,wherethematerialisresistive-althoughstillsuperconductingTheboundarybetweenfluxpinningandfluxflowiscalledtheirreversibilityline

metallicHTSMartinWilsonLecture1slide17Fieldsandwaystocreatethem:(1)IronConventionalelectromagnetsironyokereducesmagneticreluctance

reducesampereturnsrequired reducespowerconsumptionironguidesandshapesthefieldIIB100A/m-100A/m1.6TH-1.6TBIronelectromagnet–foraccelerator,HEPexperimenttransformer,motor,generator,etcBUTironsaturatesat~2TMartinWilsonLecture1slide18Fieldsandwaystocreatethem:(2)solenoidsnoiron–fieldshapeissetsolelybythewindingcylindricalwindingazimuthalcurrentflow -egwirewoundonbobbinaxialfieldBIIfieldlinescurveoutwardsattheendsthiscurvatureproducesnonuniformityoffieldverylongsolenoidshavelesscurvatureandmoreuniformfieldBIcanalsoreducefieldcurvaturebymakingthewindingthickerattheendsthismakesthefieldmoreuniformmorecomplicatedwindingshapescanbeusedtomakeveryuniformfieldsMartinWilsonLecture1slide19Fieldsandwaystocreatethem:(3)transversefieldssimplestwindingusesracetrackcoilssaddleshaped'dipole'coilsusedtomakemoreuniformfieldsknownasdipolemagnets(ironversionhas2poles)forgooduniformityneedspecialwindingcrosssections(LHCdipolewinding)someiron-butfieldshapeissetmainlybythewindingusedwhenthelongdimensionistransversetothefield,egacceleratormagnetsIIIBMartinWilsonLecture1slide20CurrentdensityIndesigningamagnet,whatreallymattersistheoverall'engineering'currentdensitytypically:forNbTimat=1.5to3.0ielmetal=0.4to0.25forNb3Snmat~3.0ielmetal~0.25forB2212mat=3.0to4.0ielmetal=0.25to0.2lwindingtakesaccountofspaceoccupiedbyinsulation,coolingchannels,mechanicalreinforcementetcandistypically0.7to0.8MartinWilsonLecture1slide21Importanceofcurrentdensity:(1)solenoidsthefieldproducedbyaninfinitelylongsolenoidisinsolenoidsoffinitelengththecentralfieldis

wherefisafactorlessthan1,typically~0.8sothethickness(volume,cost)ofasolenoidtoproduceagivenfieldisinverselyproportionaltotheengineeringcurrentdensityJe

MartinWilsonLecture1slide22SuperconductingsolenoidssmallsuperconductingsolenoidsforresearchapplicationsalargesolenoidinroutinecommercialoperationforthemagneticseparationofKaolin(chinaclay)MartinWilsonLecture1slide23Solenoidsarenotmuchusedinaccelerators.Theyarehoweverfrequentlyusedindetectors,wherethemagnetfieldprovidesmomentumanalysisofthereactionproducts.World'slargest:DelphisuperconductingsolenoidMartinWilsonLecture1slide24Importanceofcurrentdensity:(2)dipolesfieldproducedbyaperfectdipoleisJe=375Amm-2120mm

9.5x105Ampturns=1.9x106A.mpermJe=37.5Amm-2

9.5x106Ampturns=1.9x107A.mpermILHCdipole660mmIIBMartinWilsonLecture1slide25SomeengineeringcurrentdensitiesMartinWilsonLecture1slide26Criticallineandmagnetloadlines16weexpectthemagnettogoresistive'quench'wherethepeakfieldloadlinecrossesthecriticalcurrentline

*86422468101214FieldT1234567CurrentdensitykAmm-210

temperatureKMartinWilsonLecture1slide27TemperaturemargintoallowforpossibletemperatureexcursionsoperatemagnetbelowIcegrunningat1600Amm-2allowstemperaturetorise0.5Kbeforequench108642246810121416FieldT1234567temperatureK

CurrentdensitykAmm-2temperaturerisemaybecausedby -suddeninternalenergyrelease -aclosses -poorjoints -etc,etcMartinWilsonLecture1slide28Temperaturemargin-dependsonfieldB(T)Jc(4.2)Amm-2Jc(4.7)Amm-2Jc(4.2)Jc(4.7)33881333586%62000158179%954522742%108642246810121416FieldT1234567CurrentdensitykAmm-2temperatureKMartinWilsonLecture1slide29Degradedperformanceand'training'anearlydisappointmentformagnetmakerswasthefactthatmagnetsdidnotgostraighttotheexpectedquenchpoint,asgivenbytheintersectionoftheloadlinewiththecriticalcurrentlineinsteadthemagnetswentresistive-quenched-atmuchlowercurrentsafteraquench,thestoredenergyofthemagnetisdissipatedinthemagnet,raisingitstemperaturewayabovecritical -youmustwaitforittocooldownand thentryagainthesecondtryusuallyquenchesathighercurrentandsoonwiththethird,knownastrainingaftermanytrainingquenchesastablewellconstructedmagnet(bluepoints)getsclosetoit'sexpectedcriticalcurrent,butapoorlyconstructedmagnet(pinkpoints)nevergetsthereMartinWilsonLecture2slide30recaptrainingpoorqualitywindinggoodqualitywindingMartinWilsonLecture2slide31TrainingofanLHCdipolemagnetMartinWilsonLecture2slide32Causesoftraining:(1)lowspecificheatthespecificheatofallsubstancesfallswithtemperatureat4.2K,itis~2,000timeslessthanatroomtemperatureagivenreleaseofenergywithinthewindingthusproduceatemperaturerise2,000timesgreaterthanatroomtemperaturethesmallestenergyreleasecanthereforeproducecatastrophiceffects4.2K300KMartinWilsonLecture2slide33Causesoftraining:(2)Jcdecreaseswithtemperaturebut,bychoosingtooperatethemagnetatacurrentlessthancritical,wecanallowatemperaturemarginatanygivenfield,thecriticalcurrentofNbTifallsalmostlinearlywithtemperature-soanytemperaturerisedrivesthe conductorintotheresistivestateMartinWilsonLecture2slide34Causesoftraining:(3)conductormotionConductorsinamagnetarepushedbytheelectromagneticforces.Sometimestheymovesuddenlyunderthisforce-themagnet'creaks'asthestresscomeson.AlargefractionoftheworkdonebythemagneticfieldinpushingtheconductorisreleasedasfrictionalheatingtypicalnumbersforNbTi: B=5TJ=5x108A.m-2

soifd=10mm thenQ=2.5x104J.m-3Startingfrom4.2K

qfinal

=7.5KworkdoneperunitlengthofconductorifitispushedadistancedzW=F.dz=B.I.dzfrictionalheatingperunitvolumeQ=B.J.dzcan

you

engineerawindingtobetterthan10mm?MartinWilsonLecture2slide35Causesoftraining:(4)resincrackingCalculatethestainenergyinducedinresinbydifferentialthermalcontractionlet: s=tensilestress Y=Young’smodulus

e=differentialstrain n=Poisson’sratiotypically:e=(11.5–3)x10-3Y=7x109Pan=1/3

Wetrytostopwiremovementbyimpregnatingthewindingwithepoxyresin.Unfortunatelytheresincontractsmuchmorethanthemetal,soitgoesintotension.Furthermore,almostallorganicmaterialsbecomebrittleatlowtemperature.

brittleness+tension

cracking

energyreleaseQ1

=2.5x105J.m-3

qfinal=16KQ3

=2.3x106J.m-3 qfinal=28Kuniaxialstraintriaxialstrainanunknown,butlarge,fractionofthisstoredenergywillbereleasedasheatduringacrackInterestingfact:magnetsimpregnatedwithparaffinwaxshowalmostnotrainingalthoughthewaxisfullofcracksaftercooldown.PresumablythewaxbreaksbeforeithashadchancetostoreupanystrainenergyMartinWilsonLecture2slide36Howtoreducetraining?makethewindingfittogetherexactlytoreducemovementofconductorsunderfieldforcespre-compressthewindingtoreducemovementunderfieldforcesifusingresin,minimizethevolumeandchooseacrackresistanttypematchthermalcontractions,egfillepoxywithmineralorglassfibreimpregnatewithwax-butpoormechanicalproperties1)Reducethedisturbancesoccurringinthemagnetwinding2)Maketheconductorabletowithstanddisturbanceswithoutquenchingincreasethetemperaturemargin -operateatlowercurrent -highercriticaltemperature-HTS?increasethecoolingincreasethespecificheatmostof2)maybecharacterizedbyasinglenumberMinimumQuenchEnergyMQE=energyinputatapointwhichisjustenoughtotriggeraquenchMartinWilsonLecture2slide37QuenchinitiationbyadisturbanceCERNpictureoftheinternalvoltageinanLHCdipolejustbeforeaquenchnotetheinitiatingspike-conductormotion?afterthespike,conductorgoesresistive,thenitalmostrecoversbutthengoesontoafullquenchcanwedesignconductorstoencouragethatrecoveryandavoidthequench?MartinWilsonLecture2slide38MinimumpropagatingzoneMPZThinkofaconductorwhereashortsectionhasbeenheated,sothatitisresistiveIfheatisconductedoutoftheresistivezonefasterthanitisgenerated,thezonewillshrink-viceversaitwillgrow.TheboundarybetweenthesetwoconditionsiscalledtheminimumpropagatingzoneMPZforbeststabilitymakeMPZaslargeaspossible where: k=thermalconductivity r=resistivity A=crosssectionalareaofconductor

h=heattransfercoefficienttocoolant–ifthereisanyincontact

P=cooledperimeterofconductorthebalancepointmaybefoundbyequatingheatgenerationtoheatremoved.Veryapproximately,wehave:lqcqohAJPEnergytosetupMPZiscalledtheMinimumQuenchEnergy

MQEMartinWilsonLecture2slide39HowtomakealargeMPZandMQEmakethermalconductivityklargemakeresistivityrsmallmakeheattransferhlarge(limitiscryostability)but

lowJemakeratiocooledperimeter/crosssectionareaP/AlargeMartinWilsonLecture2slide40LargeMPZlargeMQElesstrainingmakethermalconductivityklargemakeresistivityrsmallmakeheattransferhlargemakeratiocooledperimeter/crosssectionareaP/AlargeNbTihashighrandlowkcopperhaslowrandhighkmixcopperandNbTiinafilamentarycompositewiremakeNbTiinfinefilamentsforintimatemixingmaximumdiameteroffilaments~50mmmakethewindingsporoustoliquidhelium -superfluidisbestfinefilamentsalsoeliminatefluxjumping (seelaterslides)MartinWilsonLecture2slide41MeasurementofMQEmeasureMQEbyinjectingheatpulsesintoasinglewireofthecablegoodresultswhenspacesincablearefilledwithporousmetal -excellentheattransfertotheheliumMartinWilsonLecture2slide42Anothercauseoftraining:fluxjumpingusualmodelisasuperconductingslabinachangingmagneticfieldByassumeit'sinfinitelylonginthez

andydirections-simplifiestoa1dimproblemdB/dtinducesanelectricfieldEwhichcausesscreeningcurrentstoflowatcriticalcurrentdensityJcknownasthecriticalstatemodelor

Beanmodelinthe1diminfiniteslabgeometry,Maxwell'sequationsays

BJJxwhenasuperconductorissubjectedtoachangingmagneticfield,screeningcurrentsareinducedtoflowscreeningcurrentsareinadditiontothetransportcurrent,whichcomesfromthepowersupplytheyarelikeeddycurrentsbut,becausethereisnoresistance,theydon'tdecaysouniformJcmeansaconstantfieldgradientinsidethesuperconductorMartinWilsonLecture2slide43ThefluxpenetrationprocessBfieldincreasingfromzerofielddecreasingthroughzeroplotfieldprofileacrosstheslabfullypenetratedMartinWilsonLecture2slide44ThefluxpenetrationprocessBfieldincreasingfromzerofielddecreasingthroughzeroplotfieldprofileacrosstheslabfullypenetratedBeancriticalstatemodelcurrentdensityeverywhereisJcorzerochangecomesinfromtheoutersurfaceMartinWilsonLecture2slide45FluxpenetrationfromanotherviewpointsuperconductorvacuumThinkofthescreeningcurrents,intermsofagradientinfluxoiddensitywithinthesuperconductor.Pressurefromtheincreasingexternalfieldpushesthefluxoidsagainstthepinningforce,andcausesthemtopenetrate,withacharacteristicgradientinfluxoiddensityAtacertainleveloffield,thegradientoffluxoiddensitybecomesunstableandcollapses

–afluxjumpMartinWilsonLecture2slide46Fluxjumping:whyithappensItarisesbecause:-magneticfieldinducesscreeningcurrents,flowingatcriticaldensityJcUnstablebehaviourisshownbyalltype2andHTsuperconductorswhensubjectedtoamagneticfieldBB*changeinscreeningcurrentsallowsfluxtomoveintothesuperconductorfluxmotiondissipatesenergythermaldiffusivityinsuperconductorsislow,soenergydissipationcauseslocaltemperaturerisecriticalcurrentdensityfallswithincreasingtemperaturegoto*DQDqDf

JcCurefluxjumpingbymakingsuperconductorintheformoffinefilaments–weakens

Df

DQMartinWilsonLecture2slide47Fluxjumping:thenumbersforNbTitypicalfiguresforNbTiat4.2Kand1TJccriticalcurrentdensity=7.5x109Am-2

gdensity=6.2x103kg.m3Cspecificheat=0.89J.kg-1K-1

q

ccriticaltemperature=9.0KNotes:leaststableatlowfieldbecauseJcishighestinstabilitygetsworsewithdecreasingtemperaturebecauseJcincreasesandCdecreasescriteriongivesthesizeatwhichfilamentisjuststableagainstinfinitelysmalldisturbances -stillsensitivetomoderatedisturbances,egmechanicalmovementbettertogosomewhatsmallerthanthelimitingsizeinpractice50mmdiameterseemstoworkOKFluxjumpingisasolvedproblem

soa=33mm,ie66mmdiameterfilamentscriterionforstabilityagainstfluxjumpinga=halfwidthoffilamentMartinWilsonLecture2slide48Magnetizationforcylindricalfilamentstheinnercurrentboundaryisroughlyelliptical(controversial)whenfullypenetrated,themagnetizationiswherea=filamentradiusNote:MisheredefinedperunitvolumeofNbTifilamentWhenviewedfromoutsidethesample,thepersistentcurrentsproduceamagneticmoment.ProblemforacceleratorsbecauseitspoilstheprecisefieldshapeWecandefineamagnetization(magneticmomentperunitvolume)NBunitsofHforafullypenetratedslabMartinWilsonLecture2slide49MagnetizationofasuperconductorTheinducedcurrentsproduceamagneticmomentandhenceamagnetization =magneticmomentperunitvolumeMBextMartinWilsonLecture2slide50Synchrotroninjectionsynchrotroninjectsatlowfield,rampstohighfieldandthenbackdownagainnotehowquicklythemagnetizationchangeswhenwestarttherampupMBBinjectionMartinWilsonLecture2slide51MeasurementofmagnetizationInfield,thesuperconductorbehavesjustlikeamagneticmaterial.Wecanplotthemagnetizationcurveusingamagnetometer.Itshowshysteresis-justlikeirononlyinthiscasethemagnetizationisbothdiamagneticandparamagnetic.

BMNotetheminorloops,wherefieldandthereforescreeningcurrentsarereversingThemagnetometer,comprising2balancedsearchcoils,isplacedwithintheboreofasuperconductingsolenoid.Thesecoilsareconnectedinseriesoppositionandtheangleofsmallbalancingcoilisadjustedsuchthat,withnothinginthecoils,thereisnosignalattheintegrator.Withasuperconductingsampleinonecoil,theintegratormeasuresmagnetizationwhenthesolenoidfieldissweptupanddownMartinWilsonLecture2slide52FinefilamentsrecapWecanreduceMbymakingthesuperconductorasfinefilaments.Foreaseofhandling,anarrayofmanyfilamentsisembeddedinacoppermatrixUnfortunately,inchangingfields,thefilamentarecoupledtogether;screeningcurrentsgouptheLHSfilamentsandreturndowntheRHSfilaments,crossingthecop