Merlino Department of Physics and Astronomy The University of Iowa Iowa City Iowa 52242 Received 26 February 2007 accepted 14 July 2007 I give several simple examples of model Langmuir probe currentvoltage IV characteristics that help students learn ID: 23471
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UnderstandingLangmuirprobecurrent-voltagecharacteristicsRobertL.MerlinoDepartmentofPhysicsandAstronomyTheUniversityofIowa,IowaCity,Iowa52242Received26February2007;accepted14July2007IgiveseveralsimpleexamplesofmodelLangmuirprobecurrent-voltagecharacteristicsthathelpstudentslearnhowtointerpretrealI-Vcharacteristicsobtainedinaplasma.StudentscanalsocreatetheirownLangmuirprobeI-Vcharacteristicsusingaprogramwiththeplasmadensity,plasmapotential,electrontemperature,iontemperature,andprobeareaasinputparameters.SomeexamplesofLangmuirprobeI-Vcharacteristicsobtainedinlaboratoryplasmasarepresentedandanalyzed.Afewcommentsaremadeadvocatingtheinclusionofplasmaexperimentsintheadvancedundergraduatelaboratory.© asetofmodelplasmaparameters.Alinkisprovidedtoaprogramthatallowstheusertoinputtheplasmaandprobedensity,temperature,plasmapotential,andprobedimensionandplottheresultingLangmuirI-Vchar-acteristic.SectionIIIprovidestwoexamplesofrealLang-muirprobeI-Vcharacteristicsobtainedinlaboratoryplas-mas.IcloseinSec.IVwithcommentsadvocatingtheinclusionofplasmaexperimentsintheundergraduatead-vancedlaboratorycourse.II.MODELLANGMUIRPROBECURRENT-VOLTAGECHARACTERISTICSInthissectionIdiscusssomeofthebasicaspectsofLang-muirprobetheorythatareneededtoconstructmodelprobeI-Vcharacteristics.TwoexamplesofidealprobeI-Vcharac-teristicsarethengiven.Finally,adiscussionofhowtheidealcharacteristicsmustbemodiÞedtoaccountforrealprobeeffectsispresented.A.IonandelectroncurrentstoaLangmuirprobe1.TheioncurrentWhenthebiasvoltage,ontheprobeissufÞcientlynegativewithrespecttotheplasmapotential,theprobecollectstheionsaturationcurrent.Positiveionscontinuetobecollectedbytheprobeuntilthebiasvoltagereachesatwhichpointionsbegintoberepelledbytheprobe.For,allpositiveionsarerepelled,andtheioncurrenttotheprobevanishes,=0.ForaMaxwellianiondistributionatthetemperature,thedependenceoftheioncurrentusuallytakentobethenegativecurrentgivenbyistheelectronÕscharge,andistheBoltzmannconstant.Wheniscomparabletotheelectrontemperature,theionsaturationcurrent,isgivenby istheiondensity, istheionther-malspeed,istheionmass,andistheprobecollect-ingarea.Whentheionsaturationcurrentisnotdeterminedbytheionthermalspeed,butratherisgivenbytheBohmioncurrent=0.6 Thefactthattheioncurrentisdeterminedbytheelectrontemperaturewheniscounterintuitiveandrequiressomeexplanation.Thephysicalreasonforthedependencehastodowiththeformationofasheatharoundanegativelybiasedprobe.Ifanelectrodeinaplasmahasapotentialdifferentfromthelocalplasmapoten-tial,theelectronsandionsdistributethemselvesspatiallyaroundtheelectrodeinordertolimit,orshield,theeffectofthispotentialonthebulkplasma.Apositivelybiasedelec-trodeacquiresanelectronshieldingcloudsurroundingit,whileanegativelybiasedelectrodeacquiresapositivespacechargecloud.Foranegativelybiasedelectrode,thecharac-teristicshieldingdistanceofthepotentialdisturbanceistheelectronDebyelength Inthevicinityofanegativelybiasedprobe,boththeelectronandiondensitiesdecreaseastheparticlesap-proachtheprobe,butnotatthesamerate.Theelectrondensitydecreasesbecauseelectronsarerepelledbytheprobe.Incontrast,theionsareacceleratedtowardtheprobe,andduetothecontinuityofthecurrentdensity,theiondensitydecreases.Apositivespacechargesheathcanformonlyiftheiondensityexceedstheelectrondensityatthesheathedge,andfortheiondensitytodecreasemoreslowlythantheelectrondensity,theionsmustapproachthesheathwithaspeedexceedingtheelocityuToachievethisspeed,theionsmustac-quireanenergycorrespondingtoapotentialdropof,whichoccursoveralongdistanceintheplasma.Thefactorof0.6inEq.isduetothereductioninthedensityoftheionsinthepresheath,whichistheregionoverwhichtheionsareaccelerateduptotheBohm Fig.1.Schematicofbasicdevicesforproducingaplasma.Adischargetubeinwhichaplasmaisformedinalowpressuregas1Torrapplyingseveralhundredvoltsacrossthecathodeandanode.AcylindricalprobeisinsertedintothedischargetomeasurethepropertiesoftheSchematicofamultidipolehotÞlamentplasmadevicewithaLangmuirdiskprobe.TheplasmaisproducedbyelectronimpactionizationofargonatomsbyelectronsthatarethermionicallyemittedandacceleratedfromahottungstenÞlament.ToenhancetheionizationefÞciency,thewallsofthechamberarelinedwithrowsofpermanentmagneticofoppositepolarity.Thelowerdiagramisanendviewshowingthearrangementofmagnets.ThemagneticÞeldlinesaresketchedasthedottedcurves.InthismagneticcuspconÞguration,thebulkplasmaisessentiallymagneticAm.J.Phys.,Vol.75,No.12,December2007RobertL.Merlino rentshouldalsobeusedasthebaselinetodeterminetheelectroncurrentforthecaseshowninFig.,althoughinthatcasethepositiveioncontributionisnegligible,andthelinecanusuallybeusedtomeasure.WhentheplasmacontainsasigniÞcantfractionofhighenergytensofelec-tronvoltsionizingprimaryelectronsinadditiontothesec-ondaryelectronsresultingfromionization,itisessentialtoÞrstsubtracttheioncurrentfromthetotalcurrenttoobtainaccuratevaluesoftheelectroncurrent.C.EffectofsheathexpansiononprobecharacteristicsThesharpkneeattheplasmapotential=1.0VintheI-VcharacteristicandßatelectronandionsaturationcurrentsshowninFig.areidealprobefeaturesthatarerarelyseeninpractice.ForthisreasontheI-VcharacteristicinFig.ideal.RealLangmuirprobeI-Vcharacteristicshaveroundedkneesandsaturationcurrentsthatincreasegraduallywithincreasingvoltage.Thelackofsaturationisrelatedtothefactthatasheathisformedaroundtheprobe,andthissheathexpandswithincreasingbiasvoltage.Sheathsformaroundanyelectrodeinaplasmaifthebiasvoltagediffers.TheformationofasheathistheplasmaÕswayofmaintainingchargeneutralityinthebulkoftheplasma.Anelectrodewithapositivebiasrelativetoattractsanelec-troncloudtolimitthepenetrationoftheelectricÞeldintotheplasmatoadistanceapproximatelyequaltotheelectronDe-byelength,,deÞnedinEq..Iftheplasmadensityisgreaterthan2eV,thenthesheathwidthwillbe0.1mm.Inthiscasetheexpansionofthesheathwillproduceonlyanegligibleincreaseincurrentastheprobebiasisincreased.Forlowerplasmadensitiesandsmallprobesthesheathexpansionproducesanincreaseinthecol-lectedcurrentbecausetheeffectiveareaforparticlecollec-tionisthesheathareaandnotthegeometricprobearea.AnotherwaytothinkaboutthesheathexpansioneffectistorealizethatforaÞniteprobe,thecollectionofplasmapar-ticlesislimitedbyfactthatsomeparticlesthatenterthesheathwillorbitaroundtheprobeandnotbecollected.Asthepotentialontheprobeisincreased,theminimumimpactparameterforwhichparticlesarecollectedincreasesandthusmoreparticleswillbecollected.Sheathexpansionoccursforboththeionandelectroncur-rentsandmustbetakenintoaccountintheinterpretationoftheI-Vcharacteristics.ThesheathexpansioneffectcanbeincorporatedintheidealprobecharacteristicsothatwecanlearnhowtodealwithitwheninterpretingrealLangmuirprobecharacteristics.Anillustrationofthiseffectisshownin.TheparametersusedtoproducethisI-Vcharacteristic=4V,=100=4eV,and=0.1eV.Thesheathexpansionwasmodeledasalinearfunctionofthebiasvoltagewiththeions,and=0.7fortheelectrons.FigureshowsthefullI-Vcharacteristic.Realcharacteristicsrarelyshowthesharpkneeattheplasmapo-tential;ratherthekneetendstoberoundedasillustratedbythedottedcurveduetothepresenceofoscillationsoftheplasmapotentialoraveraginginthedataacquisitionoranalysisprocess.Theroundingofthekneecomplicatesthedeterminationoftheelectronsaturationcurrent,buttheloca-tionofthekneeismademoreevidentbyreplottingthecur-rentonasemilogscale,asshowninFig..BothcannowbeeasilydeterminedasthecoordinatesoftheintersectionoftwostraightlinesÑoneparalleltothecurveabovethekneeandtheotherparalleltotheslopingpart.TheslopeofthestraightlineÞttotheelectroncurrentinFig.isusedtodetermine,as/ln,where1and2refertoanytwopointsontheline.Theelectroncurrentbeginsfallingoffthestraightlineduetothecontributionoftheioncurrent.AnaccuratemeasurementoftheioncurrentinthiscaserequiresthatbeobtainedforasufÞcientlynegativeprobebiassothattheelectroncontributionisex-cluded.TheprocedureformeasuringisshowninFig..TheioncurrentisplottedonanexpandedscaleandastraightlineisÞttedthroughthepoints;istakenasthevalueof.MoreaccuratemethodsofdealingwiththenonsaturationoftheioncurrentarediscussedinRefs.III.EXAMPLESOFLANGMUIRPROBECHARACTERISTICSFROMLABORATORYInthissectionIprovidetwoexamplesofLangmuirprobeI-Vcharacteristicsobtainedinmorerealisticlaboratoryplas-mas.TheseexamplesdemonstratehowthebasicprinciplespresentedinSec.IIareappliedintheinterpretationofrealA.MultidipoleplasmaAmultidipoledeviceisarelativelysimplesetupforproducingaplasmathatcanbeusedforbasicplasmaphysicsexperiments.AschematicdiagramofatypicalmultidipoledeviceisshowninFig..Itisessentiallyalarge20lstainlesssteelsouppotsometimesliterallywhichispumpeddowntoabasepressureofTorr,andthenÞlledwithagassuchasargontoapressureofapproximatelyÐ10Torr.Theplasmaisproducedbyelectronemis-sionfromasetoftungstenÞlamentsthatarebiasedtoanegativepotentialofapproximately50V.Thethermioni-callyemittedprimaryelectronsthatareacceleratedfromtheÞlamentsionizethegasproducingtheplasma.Toenhancetheprobabilitythataprimaryelectronwillundergoanion-izingcollisionwithaneutralatom,thewallsofthedevicearelinedwithpermanentmagnetsinrowsofoppositepolar- Fig.3.LangmuirprobeI-Vcharacteristicforaplasmawithpositiveandnegativeionsofequalmassandtemperatures.Thepositiveionandnegativeioncurrentsarealsoshown.Am.J.Phys.,Vol.75,No.12,December2007RobertL.Merlino itycreatingamagneticbarrier,whichinhibitstheionizingelectronsfromescaping.Duetothealternatingpolarityofthemagnetrows,themagneticÞeldhassubstantialstrengthonlyveryclosetothewalls,sothatthemainplasmaregionisessentiallymagneticÞeldfree.AtypicalLangmuirprobeI-VcharacteristicobtainedinthemultidipoledeviceintheUniversityofIowaÕsunder-graduateAdvancedPhysicsLabisshowninFig..Thischaracteristicwastakenwitha6mmdiameterplanardiskprobeinanargonplasmaatapressureof0.5mTorr.Thetopisthepositiveprobecurrentduetoelectroncollection.AsdiscussedinSec.IIseeFig.,theelectroncurrentcontinuestoincreaseslightlywithincreasingvoltageabovetheplasmapotentialduetothesheathexpansionef-fect.ThedeterminationoftheelectronsaturationcurrentandplasmapotentialisfacilitatedbyreplottingthecurrentonasemilogscaleasshowninFig..Thebreakpointoccurs4V,with100±5mA.Theslopeofthedown-wardportionofthelineonthesemilogplotgives1.5eV.Thenegativecurrentisshownonanex-pandedscaleinFig..Againweseethesheathexpansioneffectastheprobevoltageincreases.Theionsatu-rationcurrentisestimatedbyextrapolatingthelinearportionoftheioncurrenttotheplasmapotential,where0.85±0.05mA.Theßoatingpotentialisalsofoundfrom5V.TheionandelectrondensitiescannowbecalculatedusingEqs.=1.5eV.WeÞndthat8.3±0.5,and5.5±0.55.Eventakingintoaccounttheuncer-taintiesinvolvedinmeasuringthesaturationcurrentsfromtheplots,thereremainsa25%differencebetweentheplasmadensityobtainedfromtheionandelectroncurrents.ThisdifferenceisatypicaloccurrencewithLangmuirprobesmeasurements.Inamagnetizedplasma,thediscrepancyinthedensitiesobtainedfromtheelectronandionsaturation Fig.4.ModelLangmuirprobeI-Vcharacteristicincludingtheeffectofsheathexpansion,computedwith=4V,=4eV,=0.1eV,and=200.Totalcurrent.Thedottedcurvedepictstheroundingofthekneeduetoplasmanoiseoraveragingeffects..Theintersectionofthehorizontalandverticaldottedlinesoccursatthecoordi-.Theelectrontemperatureisobtainedfromtheslopeofthelinearpartofthedownwardslopingportionofthiscurve.Expandedviewoftheioncurrent.TheslopingdottedlineisalinearÞttotheioncurrent.Theionsaturationcurrentisfoundbyextrapolatingthislinetotheplasma Fig.5.LangmuirprobeI-Vcharacteristicobtainedinamultidipoleplasmainargonatapressureof0.5mTorr.Electroncurrent..Thesemilogplotoftheelectroncurrentprovidesacleardemarcationoftheplasmapotentialandelectronsaturationcurrent.isfoundfromtheslopeoftheexponentiallydecreasingportion.ExpandedscaleviewoftheioncurrentusedtoÞndAm.J.Phys.,Vol.75,No.12,December2007RobertL.Merlino currentsisconsiderablylarger,butexpected.Thegyroradiusoftheelectronsistypicallymuchsmallerthanthatoftheionssothatthecollectionofelectronsisaffectedmorethanthecollectionofions.Inthatcase,measurementsoftheplasmadensityusingtheionsaturationcurrentaremorere-Itisinterestingtocalculatethefractionoftheneutralar-gonatomsthatareactuallyionizedintheplasmausingthemeasuredvalueoftheplasmadensity.Thisfractionisknownasthepercentionizationorionizationfraction.Thedensityoftheneutralargonatomsis,whereisthetemperatureoftheneutralgas,andistheneutralgaspres-sure.ForTorrand300K,=1.65.With,weobtain=0.005.Thus,only0.5%oftheneutralatomsareionized.Thefactthattheneutraldensityisroughly1000timestheplasmadensitymightleadonetowonderabouttheroleoftheseneutralatomsontheplasmaandprobemeasurements.Toaccessthepossibleeffectsofcollisionsoftheionsandelectronswithneutrals,weneedtoestimateafewtypicalcollisionmeanfreepaths,,whereisthecrossfortheparticularprocessconsidered.Forioniza-20for50eVelectronsonargon,so75cm.Thusisontheorderofthedimensionsofatypicallaboratoryplasmadevice.Therelativelylongion-izationmeanfreepathexplains,inpart,therelativelylowvalueoftheionizationpercentageÑelectronsthatareener-geticallycapableofionizingatomsaremorelikelytomakeittothewallbeforeionizinganatom.Thepurposeofthemagnetsonthewallsofthemultidi-poledeviceistoreßecttheionizingelectronsbackintotheplasma,thusincreasingtheirchancesofhavinganionizingcollision.Electronscanalsomakeelasticcollisionswithneu-tralatoms;atypicalcrosssectioninthiscaseis20,giving6m.Forcollisionsbetweentheionsandneutralatoms,themostimportantprocesstocon-siderischargeexchange,Ar+ArAr+Ar,inwhichanargonionexchangesanelectronwithanargonatom,result-ingintheproductionofveryslowargonionsandargonatomswithanenergypracticallyequaltotheinitialenergyoftheions.Thecrosssectionforthisprocessis19,giving12cm.Wenotethatforalloftheprocessesconsideredthemeanfreepathsaremuchgreaterthantheprobesizeandtheshieldingdistanceorsheathsize,sothateventhoughtheneutraldensityfarexceedstheplasmadensity,theneutralgasatomsproducenegligibleeffectsontheprobemeasure-B.Apositiveion/negativeionplasmainaQmachineisanexampleofaLangmuirprobeI-Vcharac-teristicinaplasmainwhichthepositiveandnegativepar-ticleshavethesamemass.Thisexamplemightappeartobeexotic,butitisnotdifÞculttoproduceaplasmahavingalmostequalnumbersofpositiveandnegativeionsofcom-parablemass.Wehaveproducedpositiveion/negativeionalsoknownaselectronegativeplasmasinadevicedaQmachine.InaQmachinetheplasmaisproducedbysurfaceionization,aneffectdiscoveredbyLangmuirandKingdonin1923.TheyfoundthatcesiumatomsthatcomeintocontactwithatungstenÞlamentheatedto1200Kemergeascesiumions.Thereasonisthattheionizingpoten-tialofcesiumis3.89eV,andtheworkfunctionfortungstenis4.52eV.SurfaceionizationisexploitedinaQmachinebydirectinganatomicbeamofcesiumorpotassiumatomsontoahot2000Ktungstenortantalumplate,usuallyseveralcentimetersindiameter.Bothpositiveionsandther-mionicelectronsemergefromtheplateforminganearlyfullyionizedplasmathatisconÞnedbyastrong0.1Ð0.5TlongitudinalmagneticÞeld.Therelativelygoodthermalcontactbetweentheplasmaandthehotplateresultsinaplasmainwhichboththeelectronsandpositiveionsareatroughlytheplatetemperature,typically0.2eV.TheQmachinehasbeenusedmainlyforstudyingthebasicpropertiesofmagnetizedplasmas,andinparticularplasmaTheQdesignationreferstotheexpectationthatathermallyproducedplasmawouldbequiescent,thatis,rela-tivelyfreeoflowfrequencyplasmainstabilities.NegativeionsarereadilyformedinaQmachineplasmabyleakingintothevacuumchambersulfurhexaßuorideSFatapressureÐ10Torr.ElectronsattachtoSFformingSFnegativeions.ThecrosssectionforelectronattachmenttoSFisenergydependentandpeaksintheen-ergyrangethatcoincidescloselywiththatoftheQmachineelectrons.Underthesecircumstancesitispossibletoproduceplasmasinwhichtheratioofelectrondensitytopositiveiondensityis.ALangmuirprobeI-Vcharacteristicobtainedinsucha/SF=3.7isshowninFig.Notethatthenegativeionpositivecurrentandpositiveionnegativecurrentsaturationcurrentsarecomparable.WithsuchacharacteristictheplasmapotentialismosteasilydeterminedasthevoltageatwhichtheÞrstderivativeofthecharacteristicisamaximum.Inthiscaseweseethat1V.Thecharacteristicisroughlysymmetricabout=0,withaßoatingpotential,aresultthatistobeexpectedinaplasmawith.Whenthenegativeionandpositiveiondensitiesarecomparable,itmayevenbepossibletoextractboththenegativeandposi-tiveiontemperaturesfromtheLangmuircharacteristic. Fig.6.LangmuirprobeI-Vcharacteristicobtainedinasinglyionizedpo-tassiumplasmaproducediachine.SFgaswasintroducedintotheplasmatoformanegativeionplasmabyelectronattachment.AsubstantialfractionoftheelectronsbecameattachedtotheheavySFmoleculesresult-inginanearlysymmetricprobecharacteristicwith.Thelowercurveisthederivativeoftheprobecurrent,.Theplasmapotentialisthevalueoftheforwhichisamaximum.Am.J.Phys.,Vol.75,No.12,December2007RobertL.Merlino beenergizedtoionizationenergiestomaintainthedischarge.EnergytransferbetweenthemassiveionsandlightelectronsisinefÞcient,sotheionsremainrelativelycold.Reference3,p.125;seealsothereviewarticlebyK.-U.Riemann,ÒTheBohnsheathcriterionandsheathformation,ÓJ.Phys.D,493Ð518,andtherecentarticlebyG.D.Severn,ÒAnoteontheplasmasheathandtheBohmcriterion,ÓAm.J.Phys.,92Ð94F.F.Chen,IntroductiontoPlasmaPhysicsandControlledFusion,2ndPlenum,NewYork,1984,Vol.1,p.290.Reference13,p.8.M.A.LiebermanandA.J.Lichtenberg,PrinciplesofPlasmaDischargesandMaterialsProcessing,2nded.Wiley,NewYork,2005,Chap.6.SeeEPAPSDocumentNo.E-AJPIAS-75-009710forathatcanbeusedtoproduceLangmuirI-Vcurves.ThisdocumentcanbereachedthroughadirectlinkintheonlinearticleÕsHTMLreferencesectionorviatheEPAPShomepagehttp://www.aip.org/pubservs/.Thisprogramcanalsobeaccessedfromwww.physics.uiowa.edu/~rmerlino/See,forexample,T.SunnPedersen,A.H.Boozer,W.Dorland,J.P.Kremer,andR.Schmitt,ÒProspectsforthecreationofpositron-electronplasmasinanon-neutralstellarator,ÓJ.Phys.B,1029Ð1039D.LeeandN.Hershkowitz,ÒIoncollectionbyplanarLangmuirprobes:SheridanÕsmodelanditsveriÞcation,ÓPhys.Plasmas,033507-1Ð4R.LimpaecherandK.R.MacKenzie,ÒMagneticmultipoleconÞnementoflargeuniformcollisionlessquiescentplasmas,ÓRev.Sci.Instrum.726Ð731A.LangandN.Hershkowitz,ÒMultidipoleplasmadensity,ÓJ.Appl.,4707Ð4710R.A.BoschandR.L.Merlino,ÒMultidipoleconÞnementofargonandpotassiumplasmas,ÓRev.Sci.Instrum.,2940Ð2950AgoodsourceforcrosssectionsrelevanttoplasmaphysicsisS.C.BasicDataofPlasmaPhysics,2nded.MITPress,Cambridge,MA,1967R.W.Motley,QMachinesAcademic,NewYork,1975I.LangmuirandK.H.Kingdon,ÒThermioniceffectscausedbyalkalivaporsinvacuumtubes,ÓScience,58Ð60BinSong,D.Suszcynsky,N.DÕAngelo,andR.L.Merlino,ÒElectrostaticion-cyclotronwavesinaplasmawithnegativeions,ÓPhys.FluidsB2316Ð2318S.B.Song,C.S.Chang,andDuk-InChoi,ÒEffectoftwo-temperatureelectrondistributionontheBohmsheathcriterion,ÓPhys.Rev.E Am.J.Phys.,Vol.75,No.12,December2007RobertL.Merlino