The values of the critical current density of niobium slabs, measured with the help of trapezoidal ac magnetic fields are not in agreement with the critical state model. Extensions of this model are suggested to explain the observed deviation

Gijsbertse, E.A.

Klundert, L.J.M. van de

English

van de Klundert, L.J.M.

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Deviations from the critical state model observed in niobium

University of Twente Research Information

Volume 77A, number5 PHYSICSLETTERS 9 June1980DEVIATIONS FROM THE CRITICAL STATE MODEL OBSERVED IN NIOBIUME.A. GIJSBERTSEandLJ.M. vande KLUNDERTTwenteUniversityof Technology,DepartmentofAppliedPhysics,7500AeEnschede,TheNetherlandsReceived10 March1980Thevaluesof the critical currentdensity of niobiumslabs,measuredwith thehelp of trapezoidalac magneticfields arenot in agreementwith the critical statemodeLExtensionsof thismodel aresuggestedto explainthe observeddeviation.Introduction.In a generalizedcritical statemodel E/E = (d — ~ (2)(CSM) [1,2] the pinningof flux linesin type II super- 1 tconductorsis describedby a critical current density whereE1 is the inducedvoltage at thesameexternal/c’ which may dependon magneticinductionB and field, whenthe flux fronthasreachedthe centreplaneon position.Additional screeningof flux by surface of the slab(x~= 0); E1 is independentof theexternalcurrentsis expressedin termsof aBen(Ba)relation field at which the field changestartedanddependsin the caseof an increasingexternalinductionandof only on Ba. Thelocal magneticinductionat thisposi-Bex(Ba)in the decreasingcase.Ben(Ba)andB~(Ba) tionis givenby:give thevalueofB just insidea narrowsurfacelayer B’ B ~ — ~F~ ‘B b ~+B ‘Basa function of externalfield. Finally,underac con- ‘~~‘ a) 2 L ex’~ 0 — 0) en’~ aditionsalso flux flow effectshaveto be considered; — 1theywill be describedin thesameway asnormalcur- Bex(Bo — b0) + ~(Ba — BaO)dBen/dBalBao (3)rentsin termsof a flux flow conductivity (Yff. Bao is definedby Ben(Bao)= Bex(B0— b0). With theAn inductivetechniquemakinguseof a trapezoidal help of theseequationstheflux distribution at theacfield (amplitudeb0) superposedon a static field momentat which the externalfield changestartedB0 hasprovento be a powerful instrumentfor the canbe determined.An analogousderivationcanbestudyof the variousparametersof the CSM [1]. A given for a decreasingfield.carefulanalysisof the waveformof the inducedvolt-ageleadsto a separationof thevariouscontributions Experimentalresults. The experimentalmethodto the aclosses* . In fact, it is possibleto determine describedabovehasbeenusedto studythecriticaltheflux distributioninsidea superconductor,provid- statein a Nb slab.The sample(20 X 5 X 0.3 mm3)ed that flux flow effectsmay beneglectedandthat w s takenfrom commercialNb plateandsubsequent-‘c is independentof B within the appliedac field ly annealed.Experimentsweredonewith trapezoidalrange.In this casethevoltageinducedin a pick-up ac fields of variousamplitudesand~a/dt valuescoil arounda superconductingslab (thickness2d) in superposedon a staticfield. A moredetaileddescrip.a parallelincreasingfield is givenby: tion of theexperimentalequipmentis givenelsewhere*E = (d — xt)(dBen/dBa)(dBa/dt), (1) A numberof normalizedEversusBa curvesfor in-wherex is the positionto which theflux changehas *i .A moredetaileddiscussionof theexperimentalmethodentered.The position of this flux front canbe deter- andtheequipmentwhichis usedfor thesemeasurementsminedfrom: wifi bepublishedin a subsequentpaper.335Volume 77A, number5 PHYSICSLETTERS 9 June19801o5[Acm_2] o increasing fielddecreasing field3020 io40000 00oOoo~~~:,10 ~ I ~ 0JJill~!’ 102 I I0 100 200 300 400 0 100 200 300B~ [ml] B0~ [ml]20 Fig. 2. The critical currentdensity/c’ measuredin increasinganddecreasingfield, plottedversusthe averageinternal induc-1.5 b tionBay.10 I tures.Experimentsperformedin thesameway on aNb—50%Tisampledid not showthis phenomenon[3].0 Discussion.The apparentdifferencein/~values0 100 200 300 4(X)B. [~T] cannotbeexplamedby assummgthat ‘c is actuallyFig. 1. A setof normalizedE versusBacurvesin increasing differentin bothdirectionsbecausethemethodfun-(a) andin decreasing(b) externalfield for a Nb slab(b0= damentallycannotdistinguishbetweenthetwo values.32 mT, dBa/dt= 0.67T/s). This is a consequenceof the fact that the inducedvoltageis determinedequallyby boththe flux profilecreasinganddecreasingfields are given in fig. 1. The in increasinganddecreasingfield. Additional confir.point at which an individual curvereachestheenve- mationof this point is given by the experimentalre-lope indicatesthepenetrationof the flux front to the suit that the/svaluesareindependentof field reversal.centreof the slab; theenvelopecurvesgive thevolt- Also the fact that/~dependsonB cannotgive any cx-agesEi(Ba) which are a measurefor dBen/dBaand planationof the observedphenomenon;it canbe(lBexI(lBa, respectively.A very remarkablefeatureis seeneasilythat in increasingfield a smallervalueofthat in decreasingfield thecentreplaneis reached j~shouldbeobtainedbecausetheaveragevalueof Bmuchearlierthan in increasingfield. Thissuggestsa inside thesampleis largerin this casethanin de-smallervalue Of/c in decreasingfield. This is confirm- creasingfield. With respectto thebehaviourof surfaceed by the resultsin fig. 2 wherethevalueof j~is screeningcurrentsno specialassumptionsweremade.plottedasa function of B. The/cvaluesare deter- A smallcorrectionto the resultscanbe obtainedbymined from the linearmiddlepart of theEversusBa taking into accountthe secondordertermin thecurves,accordingto eqs.(2) and (3). Herealsothe Taylorexpansionof eq.(3); this correction,however,differencebetween/ain increasinganddecreasing is notenoughto explain the largedifferencein/a val-fields is observed. ues.Althoughflux flow effectsmaynot be neglectedMost experimentswere doneat a temperatureof in this sample,theyare notresponsiblefor the observ-4.2 K butexperimentsat 2.0 K exhibit thesamefea- edphenomenon:the/~valuesareindependentof the336Volume 77A, number5 PHYSICSLETTERS 9 June1980applieddBa/dtin a wide rangeof d.Ba/dtvalues(0.67 atemperatureprofile with a high temperatureat theT/s to 27 T/s).A decreaseof the flux flow conduc- position of the flux front anda lowertemperatureattivity ~ with increasingB (which is commonlyob- the surface.The result of this temperaturegradientisserved)will leadto an apparentlylargervalueof j~in a forceon the flux linestowardsthesurface.In in-decreasingfield, whichis in contradictionwith the creasingfield this meansthat flux penetratesthe slabexperimentalresults, lesseasily,i.e.,a largercritical currentdensitywhileConclusivelyit canbesaid, that theobserveddif- in decreasingfield flux movesoutmoreeasily,i.e.,aferencein/a valuescannotbe explainedin termsof lower/avalue. Sothermo-magneticeffectsmight givethegeneralizedcritical statemodel,sincenoneof the an explanationof the experimentaldata.Finally whenparametersof this model nor their B-dependencescan thesetwo possibilitiescannotbeheldresponsibleforaccountfor this behaviour.So far a realisticexplana- the observedresults,a more fundamentalbreakdowntion cannotbe given. Severaldeviationsfrom the of the CSM in which alsovaluesof the supercurrentCSM maybe considered.First of all the modelmay be between±/~may existfor a finite timeinterval,mayextendedto a non-isothermalCSM. In this casethe beconsidered.A moreextensivepresentationof thetemperatureof the samplevariesduringoneperiod resultsobtainedon this samplewill be given in a sub-of the ac field due to dissipation;now also thetemper- sequentpaper.aturedependenceof the variousparametersmustbetakeninto account.A secondextensionconcernsthe Referencesinclusionof forceson flux lines due to temperaturegradientsinside thesample [4]. Thesethermo- [1] L.J.M. van de Klundert, E.A. Gijsbertse andH.P. vandemagneticforcesare directedtowardsthelower tern- Braak,Physica94B (1978)41.peratureregion.Numericalsolution of the heatdif- [2] J.R.Clem, J. Appi. Phys. 50(1979)3518.[3] E.A. Gijsbertse, Proc. Intern. Schoolon Superconductivi-fusionequationfor a slab with a dissipationdistribu- ty (Piechowice,Poland),to bepublished.tion similar to the one m a superconductingslab with [4] F.A. Otter andP.R. Solomon,Phys.Rev. Lett. 16 (1966)constantpinning force in anac magneticfield indicates 681.337

inclusion of forces on fluxlines due to temperature gradients inside the sample [4]. These thermo- [1] L.J.M. vande Klundert, E.A. Gijsbertse and H.P. vande magnetic forces are directed towards the lower tern- Braak,

Numerical solution ofthe heat dif- [2] J.R.Clem,

tion similar to the one m a superconducting slab with [4] F.A.

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Deviations from the critical state model observed in niobium

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