plasma physics lecture 1 ian hutchinson

8/3/2019 Plasma Physics Lecture 1 Ian Hutchinson

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Chapter1Introduction1.1 What isaPlasma?1.1.1 An ionizedgasAplasmaisagasinwhichanimportantfractionoftheatomsisionized,sothattheelectronsandionsareseparatelyfree.Whendoesthisionizationoccur? Whenthetemperatureishotenough.Balancebetweencollisionalionizationandrecombination:

Figure1.1: IonizationandRecombination

Ionizationhas

athreshold

energy.

Recombination

has

not

but

is

much

less

probable.

Thresholdisionizationenergy (13.6eV,H).iIntegral over Maxwellian distribution gives rate coefficients (reaction rates). Because ofthe tail of the Maxwellian distribution, the ionization rate extends below T =i. And inequilibrium,when

nions < iv >= , (1.1)

nneutrals < rv >

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Figure1.2: Ionizationandradiativerecombinationratecoefficientsforatomichydrogen>thepercentageofionsislarge(100%)ifelectrontemperature: Te i/10.e.g. Hydrogen

>isionizedforTe 1eV (11,600k). Atroomtemprionizationisnegligible.Fordissociationandionizationbalancefigureseee.g.DelcroixPlasmaPhysicsWiley(1965)figure1A.5,page25.1.1.2 PlasmasareQuasi-NeutralIfagasofelectronsandions(singlycharged)hasunequalnumbers,therewillbeanetchargedensity,.

=ne(e)+ni(+e)=e(nine) (1.2)Thiswillgiverisetoanelectricfieldvia

e.E= = (nine) (1.3)

0

0

Example: Slab.dE dx = 0 (1.4)

xE =

0 (1.5)7


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Figure1.3: ChargedslabThisresults inaforceonthechargestendingtoexpelwhicheverspecies is inexcess. Thatis,ifni >ne,theE fieldcausesni todecrease,ne toincreasetendingtoreducethecharge.Thisrestoringforce isenormous!ExampleConsiderTe =1eV,ne =1019m3 (amodestplasma;c.f. densityofatmospherenmolecules31025m3). Supposethereisasmalldifferenceinionandelectrondensitiesn=(nine)

so =ne (1.6)Thentheforceperunitvolumeatdistancexis

xFe =E=2 x =(ne)2 (1.7)0 0

Taken/ne =1%,x=0.10m.Fe =(10171.61019)2 0.1/8.81012 =3106N.m3 (1.8)

Comparewiththisthepressureforceperunitvolumep/x:pneTe(+niTi)Fp10191.61019/0.1=16Nm3 (1.9)

Electrostaticforce>>KineticPressureForce.Thisisoneaspectofthefactthat,becauseofbeingionized,plasmasexhibitallsortsofcol-lectivebehavior,differentfromneutralgases,mediatedbythelongdistanceelectromagneticforcesE,B.Anotherexample(related) isthatof longitudinalwaves. Inanormalgas,soundwavesarepropagated via the intermolecular action of collisions. In a plasma, waves can propagatewhencollisionsarenegligiblebecauseofthecoulombinteractionoftheparticles.

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1.2 PlasmaShielding1.2.1 ElementaryDerivationoftheBoltzmannDistributionBasicprincipleofStatisticalMechanics:Thermal Equilibrium Most Probable State i.e. State with large number of possible ar-rangementsofmicro-states.

Figure1.4: StatisticalSystemsinThermalContactConsidertwoweaklycoupledsystemsS1,S2 withenergiesE1,E2. Letg1,g2 bethenumberofmicroscopicstateswhichgiverisetotheseenergies,foreachsystem. Thenthetotalnumberofmicro-statesofthecombined systemis(assumingstatesareindependent)

g=g1g2 (1.10)Ifthetotalenergyofcombinedsystem isfixedE1 +E2 =Et thenthiscanbewrittenasafunctionofE1:

andg

dgdE1

==

g1(E1)g2(EtE1)dg1dEg2g1

dg2dE .

(1.11)(1.12)

Themostprobablestateisthatforwhich dgdE1 =0i.e.

1g1

dg1dE =

1g2

dg2dE or

ddElng1 =

ddElng2 (1.13)

Thus,inequilibrium,statesinthermalcontacthaveequalvaluesof ddE lng.

OnedefineslngastheEntropy.And[ d

dE lng]1 =T theTemperature.Nowsupposethatwewanttoknowtherelativeprobabilityof2micro-statesofsystem1inequilibrium. Thereare, inall,g1 ofthesestates, foreachspecificE1 butwewanttoknowhowmanystatesofthecombined systemcorrespondtoasingle microstateofS1.Obviouslythatisjustequaltothenumberofstatesofsystem2. So,denotingthetwovaluesof the energies of S1 for the two microstates we are comparing by EA,EB the ratio of thenumberofcombinedsystemstatesforS1A andS1B is

g2(EtEA)=exp[(EtEA)(EtEB)] (1.14)

g2(EtEB)9


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NowwesupposethatsystemS2 islargecomparedwithS1 sothatEA andEB representverysmallchangesinS2senergy,andwecanTaylorexpand

d dg2(EtEA)exp EA +EB (1.15)

g2(EtEA) dE dEThus we have shown that the ratio of the probability of a system (S1) being in any twomicro-statesA,B issimply

exp (EAEB) , (1.16)T

wheninequilibriumwitha(large)thermalreservoir. Thisisthewell-knownBoltzmannfactor.YoumaynoticethatBoltzmannsconstant isabsentfromthis formula. That isbecauseofusingnaturalthermodynamicunitsforentropy(dimensionless)andtemperature(energy).Boltzmannsconstantissimplyaconversionfactorbetweenthenaturalunitsoftemperature(energy, e.g. Joules)and (e.g.) degrees Kelvin. Kelvins are based on Cwhich arbitrarilychoosemeltingandboilingpointsofwateranddivideinto100.Plasma physics is done almost always using energy units for temperature. Because Joulesareverylarge,usuallyelectron-volts(eV)areused.

1eV =11600K=1.61019Joules. (1.17)OneconsequenceofourBotzmann factor isthatagasofmovingparticleswhoseenergy is1 2 2mv adoptstheMaxwell-Boltzmann(Maxwellian)distributionofvelocitiesexp[mv ].2 2T

1.2.2Plasma

Density

in

Electrostatic

Potential

Whenthereisavaryingpotential,,thedensitiesofelectrons(andions)isaffectedbyit.Ifelectronsareinthermalequilibrium,theywilladoptaBoltzmanndistributionofdensity

eneexp( ) . (1.18)

TeThisisbecauseeachelectron,regardlessofvelocitypossessesapotentialenergye.Consequenceisthat(fig1.5) aself-consistentloopofdependenciesoccurs.Thisisoneelementaryexampleofthegeneralprincipleofplasmasrequiringaself-consistentsolutionofMaxwellsequationsofelectrodynamicsplustheparticledynamicsoftheplasma.1.2.3 DebyeShieldingAslightlydifferentapproachtodiscussingquasi-neutrality leadstothe importantquantitycalledtheDebyeLength.Supposeweputaplanegridintoaplasma,heldatacertainpotential,g.

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Figure1.5: Self-consistentloopofdependencies

Figure1.6: Shieldingoffieldsfroma1-Dgrid.Then,unlikethevacuumcase,theperturbationtothepotentialfallsoffratherrapidlyintotheplasma. Wecanshowthisasfollows. Theimportantequationsare:

d2PoissonsEquation = (nine) (1.19)2=

dx2 e

0ElectronDensity ne =nexp(e/Te). (1.20)

[This is a Boltzmann factor; it assumes that electrons are in thermal equilibrium. n isdensityfarfromthegrid(wherewetake=0).]

IonDensity ni =n . (1.21)[Applies far from grid by quasineutrality; wejust assume, for the sake of this illustrativecalculationthationdensityisnotperturbedby-perturbation.]Substitute:

d2=

enexp

e. (1.22)dx2 0 Te 1

Thisisanastynonlinearequation,butfarfromthegrid |e/Te


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12

Solutions: =0exp(|x /D)where|0Te

(1.24)D2e n

ThisiscalledtheDebyeLengthPerturbationstothechargedensityandpotential inaplasmatendto falloffwithcharac-teristiclengthD.InFusionplasmasD istypicallysmall. [e.g. ne =1020m3Te =1keV D =2105m=20m]UsuallyweincludeaspartofthedefinitionofaplasmathatD


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[Thisequationcomesfromelementarygas-kinetictheory. Seeproblemsifnotfamiliar.]Themeanspeed 8T T .v=

m mBecauseofmassdifferenceelectronsmove mi fasterandhencewoulddrainoutofplasma

mefaster. Hence,plasmachargesupenoughthatanelectricfieldopposeselectronescapeandreducestotalelectriccurrenttozero.Estimateofpotential:

1Ionescapeflux4nivi1Electronescapeflux4n evi

Primedenotesvaluesatsolidsurface.Boltzmannfactorappliedtoelectrons:

n =nexp[es/Te] (1.26)ewheres issolidpotentialrelativetodistant()plasma.Since ions are being dragged out by potential assume ni n (Zi = 1). [This is onlyapproximatelycorrect.]Hencetotalcurrentdensityoutofplasmais

j = qi14nivi +qe

14 neve (1.27)

= en4 {viexp

esTe ve} (1.28)

Thismustbezero sos =

=Tee

Tee

ln|vive|=

Tee

12ln

memi

12ln

TiTe

memi

[if Te =Ti.](1.29)(1.30)

Forhydrogen mime =1800so 12 lnmemi =3.75.

Thepotentialofthesurfacerelativetoplasmaisapproximately4 Te.e

[Note Tee isjusttheelectrontemprinelectron-voltsexpressedasavoltage.]

1.2.5 ThicknessofthesheathCrudeestimatesofsheaththicknesscanbeobtainedbyassumingthationdensityisuniform.Thenequationofpotentialis,asbefore,

d2= en exp e (1.31)

dx2 0 Te 1

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Weknowtheroughscale-lengthofsolutionsofthisequationis

120Te

D =e2n theDebyeLength. (1.32)

Actuallyourprevioussolutionwasvalidonlyfor|e/Te 1 (Collectiveeffectsdominateovercollisions) (1.35)1.4 SummaryPlasmaisanionized gasinwhichcollectiveeffectsdominateovercollisions.

[D 1 .] (1.36)14


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1.5 OccurrenceofPlasmasGasDischarges: FluorescentLights,Sparkgaps,arcs,welding,lightingControlledFusionIonosphere:

Ionized

belt

surrounding

earth

InterplanetaryMedium: Magnetospheresofplanetsandstarts. SolarWind.StellarAstrophysics: Stars. Pulsars. Radiation-processes.IonPropulsion: Advancedspacedrives,etc.&SpaceTechnology InteractionofSpacecraftwithenvironmentGasLasers: Plasmadischargepumpedlasers: CO2,He,Ne,HCN.MaterialsProcessing: Surfacetreatmentforhardening. CrystalGrowing.SemiconductorProcessing: Ionbeamdoping,plasmaetching&sputtering.SolidStatePlasmas: Behaviorofsemiconductors.ForafigurelocatingdifferenttypesofplasmaintheplaneofdensityversustemperatureseeforexampleGoldstonandRutherfordIntroductiontoPlasmaPhysicsIOPPublishing,1995,figure1.3page9. Anotherisathttp://www.plasmas.org/basics.htm

1.6 DifferentDescriptionsofPlasma1. SingleParticleApproach. (Incompleteinitself). Eq. ofMotion.2. KineticTheory. BoltzmannEquation.

ft +v.x+a.v f = t

col.(1.37)

3. FluidDescription. Moments,Velocity,Pressure,Currents,etc.Usesofthese.

SingleParticleSolutionsOrbitsKineticTheorySolutionsTransportCoefs.FluidTheoryMacroscopicDescription

Alldescriptions

should

be

consistent.

Sometimes

they

are

different

ways

of

looking

at

the

samething.

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http://www.plasmas.org/basics.htmhttp://www.plasmas.org/basics.htm


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1.6.1 EquationsofPlasmaPhysics

.E = .B = 00

B 1E (1.38)= = 0j+E t B c2 t

F =q(E+vB)1.6.2 SelfConsistencyInsolvingplasmaproblemsoneusuallyhasacircularsystem:Theproblemissolvedonlywhenwehaveamodelinwhichallpartsareselfconsistent. Weneedabootstrapprocedure.Generallywehavetodoitinstages: CalculatePlasmaResponse(togivenE,B) Getcurrents&chargedensities CalculateE&Bforj,p.

Thenputitalltogether.

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plasma physics lecture 1 ian hutchinson

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