Plasma Sheaths and Langmuir probes

Transcription

1 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 Last wk w ookd at msson of ght fom a pasma. In ffct ght msson s th smpst dagnostc too fo pasmas f th s ght bng mttd, th s a pasma. If t s bght th s mo pasma, tc. Cunt fom a pasma s th scond smpst dagnostc too fo xamnng a pasma. Howv, to undstand what that cunt mans, w nd to an mo about how cunt s tansfd to and fom a pasma. Ths mans that w nd to know about shaths. Pasma Shaths and Langmu pobs Th basc thoy bhnd pasma shaths can b dvd dcty fom th fundamnta fud thoy quatons. W know that th ctons and ons must av a pasma at th sam tm avagd and poston avagd at. (Ths mps that w can hav th ctons avng at on ocaton and th ons avng at anoth ocaton. It aso mps that th ons mght av at a ctan tm wh th ctons av at a dffnt tm povdd th tm dffnc s not gat.) W want to xamn pasmas n tms of ths qumnt of chag nutaty. Ou fundamnta fud quatons a as bfo: f ( n n )+ c v t v mn t + v v M m v f P+ qn E+ v B c c momntum ost va cosons momntum chang va patc gan/ oss Lt us xamn what ach of ths quatons man. Fst ntgatng ov th contnuty quaton n f dτ t d τ n d τ c ( v ) Vo Vo Vo Vo Vo n v Γ d s ds wh Γ s th fux of th patcs. ( ) Lkws w can wt of ngy quaton, h ntgatng aong th E fd to gv v m m + m d f d d q d t c c n vv M v + ( + ) n P n E v B Ths a n gna vy dffcut pobms to sov. Thus w w mak som appoxmatons. Fst, w w t th coson tms go to zo. Thn w w t th pssu dffnta go to zo. Both of ths appoxmatons a asonab fo many pasma systms. Typcay thy Pag

2 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 hav vy thn shaths compad to th man-f path and a n ow-pssu systms. Thus ou quatons smpfy to: n t d τ Γ d s Vo Vo v m + m vv d ( + ) t q E v B d W can futh smpfy ou mod by assumng tm ndpndnc and no magntc fd. Ths mnats a ot of a shaths that w mght un acoss but t maks th mod undstandab. Thus, w a dang wth th foowng st of quatons Γ d s Vo Γ nv const m vv d qe d q φ d m v q φ W w us ths quatons to gv us th on dnsty and vocty at dffnt ponts n th shath. (Th abov quatons a gna and w had not appd thm to any spcs.) Th ony thng that w nd to know s what s th potnta as a functon of poston. St, w can dtmn th vocty and th dnsty as a functon of th potnta. Thus w hav vs n v n s mv mv s+ ( φs φ) wh s stands fo th vau at th shath dg. Combnng th quatons gvs m n n v s mv s s + ( φs φ) mn v n mv + φ φ n s ( ( )) s s s s n + ( φs φ) mv s / / o n ns + ( φs φ) mv s W now hav on quaton and two unknowns, th potnta n th shath and vocty of th ons at th shath dg. W can gt on of ths unknowns by usng Posson s quaton to gt th potnta. Pag

3 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 ρ φ ( n n ) ε ε Agan w a mssng a pc of th puzz, but now t s th cton dnsty. Th a a coup of ady usfu optons. ) Th a no ctons n th shath. ) Th cton dnsty s st by Botzmann s quaton. Th scond opton s th on usd most commony, wh th fst s appopat ony fo vy spca stuatons. Intay, t us assum that th ctons foow Botzmann s quaton. Thus, φ / kt n n wh n s th cton dnsty at th pont at whch th potnta s zo. O w can wt ths n tms of th potnta at th shath dg and gt ( φs φ )/ kt n n s Rvw: At ths pont, w hav fou fundamnta quatons whch dscb what gos on nsd th shath, Engy consvaton: mv mv s + ( φs φ) v n s Fux consvaton: v n s ρ Posson s Equaton: φ ( n n ) ε ε ( φs φ )/ kt Botzmann s Dstbuton: n n s Not that a vaus a atv to th vaus at th shath dg. W w now us ths quatons to dtmn what xacty s th qumnt fo a shath dg! Takng ou quaton fo th on dnsty and puggng nto Posson s quaton to gv / ( φ φ) φ n s / kt s n s + ( φ ) s φ ε mv s. / n / kt s n s ( ) ε Ε s wh φ φ s < and Ε s mv s. Ths quaton can b ntgatd onc anaytcay. Ths s don n th foowng way. Fst mutpy by. Ths gvs / ( ) n / kt n ( ) Ε s s ε s Now ntgat ov d Pag 3

4 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 / ( ) d n kt s n s ε Εs / kt s ( ) d + ε Ε n kt n s s / d Ε s / d wh s th shath dg and s a poston n th shath. Now, d so that w can us th d fst pncp of cacuus and gt d d d d d n kt / kt n s / Ε ( ) s + s d ε Εs o kt n / kt n Ε s ( ) s + s ε ( () ) ( ) n Ε s / + () } } } / kt () () Ε () n / kt / kt s s ns ε Εs Ε s () / kt ( () ) s ε ( () ) n / kt () s ε / kt Ε () [ ]+ n s s Εs ( ) + () } } } / kt () n () / kt () / kt Εs n s s ε Εs Εs / kt () () Ε [ ]+ n s s Εs / / Bohm Pshath Rqumnt Th s no known way to futh anaytcay ntgat ths quaton. Howv w can st t that th tm nsd th squa oot cannot b ngatv o s th potnta s magnay. W can an somthng fom ths qumnt. n kt / kt n Ε () s () / s [ ]+ s Εs Fo sma potntas, w can xpand th xponnta tm to gt Pag 4

5 n Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 s kt / () Ε s () + + ns kt Εs / Ε s () ns()+ ns Εs Futh w can xpand th squa oot tm to gt Ε s () ns()+ ns Ε s () Ε s ns() ns Ε s ( ns() ns() ) ns ns mmb that ()<! Ths mans that th cton dnsty must b ss than th on dnsty but not by much. Ths s not supsng as w hav aady sad that th shath occus bcaus th ctons woud othws av fast than th ons. Ths maks t such that cton dnsty s sghty sma than th on dnsty. Now, f w assum that th cton and on dnsts a appoxmaty th sam, so that ns ns ns as w typcay assum n th pasma, w can now go back and xamn th ght hand sd of ou nquaty agan. n kt / kt n Ε () s () / s [ ]+ s Εs / kt kt Ε () s () / [ ]+ Εs Now w w xpand th quaton to th scond od so that kt () () + + Ε s kt k T Ε s 8Ε s ()+ () () ()+ kt 4Ε s + () () () () kt 4Ε s kt Ε s kt vs v B / on acoustc / Bohm vocty M Ths s known as th Bohm Shath cta. In ffct, t says that th ons hav to b tavng at th on acoustc vocty bfo thy nt th shath. Ths maks sns as th f th vocts w zo at th dg of th shath, thn th fux of th ons nto th shath woud b zo (unss Pag 5

6 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 th dnsty was nfnt!). Ths mans that w hav to hav a gon n whch th ons a accatd fom to th acoustc vocty. Ths gon s known as th pshath. What dos ths man? Fst to accat th ons to th Bohm vocty w nd to hav a potnta dop such that th kntc ngy ncass. At fst w w assum that ngy s consvd,.g gno cosons. Thus, w fnd that m v q φ 678 MvB vpasma q( φb φpasma) M kt qb M q B kt Lkws w know that w can dtmn th cton dnsty at th dg of th shath fom Botzmann s quaton. q B kt n n s pasma npasma 6. n pasma Th abov dscusson mps that th has to b som sot of pshath gon whch accats th ons fom zo to th Bohm vocty. W can gt som hand on what th pshath ooks k f w assum a fw smp thngs. Fst, that th on and cton dnsts a qua thoughout th pshath. Scond, that th cton dnsty foows Botzmann s quaton. Thd and fna, that th ons nt th pshath at zo vocty and av on th oth sd at th Bohm vocty. Thus w hav th foowng pctu. Pag 6

7 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383, n n n n, v n n n v v B kt M / kt pasma pshath shath X o To undstand th pshath, w nd to ook at th on fux though th pshath. Ths s smpy Γ n v. Takng th dvatv of both sds gvs n v Γ ( ) n ( v )+ v ( n ) now dvdng though by th fux, w gt Γ ( v )+ ( n ). Γ v n Puggng n Botzmann s quaton gvs Γ ( v)+ ( n ) Γ v n / kt ( v)+ / kt ( n ) v n q / kt ( v)+ q / kt n ( ) v n kt ( v)+ ( ) v kt What dos ths man? W hav th on fux n tms of th on vocty and th potnta. W woud k to dtmn th fux n tms of ony on paamt. If w mak th assumpton that ngy s consvd thn w can pac th vocty wth th potnta. Obvousy th a ways n whch ngy can aso b addd o subtactd! (Wh oth xamps a doab, th ony smp xamp s to assum consvaton of ngy.) If th ngy s consvd than m v φ. Ths mps fo ou cas that Pag 7

8 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 Mv o v M thus v M M M M v v v and at th shath dg M kt M kt Puggng ths nto ou fux quaton w fnd Γ ( v)+ Γ v kt + kt + at th shath dg kt kt Ths mps that Γ n th pshath f th ngy s consvd. Oth soutons typcay hav to b found va comput smuatons. Cassfd Shaths Th a a fw mpotant standad shaths. Ths a shaths to a foatng wa, shaths to a goundd wa and shaths to a dvn wa. W w stat wth th shath potnta at a foatng wa. Pag 8

9 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 Foatng shaths If a wa o oth objct s nstd nto a pasma and th s no way fo th chag to dan fom th sufac, w hav a stuaton n whch th wa s ctcay foatng. Ths gvs s to a foatng shath. In ths cas, th objct w chag up unt th fux of ctons and ons match. Thus to fnd foatng potnta of th objct w smpy nd to dtmn th potnta at whch th fuxs match. Th fux of ons to th wa s gvn by Γ const n s vb kt ns M Ths fux s matchd by th fux of ctons to th sufac. Th cton fux s thos cton hadd towad th sufac whch aso hav nough ngy to ovcom th potnta. Thus, Γ v x f() v dv x dv y dv z v mn wh v w mn m s st by ngy qumnts Thus m Γ n πkt 3 / m vx + vy + vz v w x xp m kt m n πkt n n kt πm kt πm / m( vx) vx xp dv w x m kt / m ( vx) x d mv xp w m kt kt / m w xp[ UdU ] ( ) ( ) dv dv dv x y z avag / spd kt } w w n xp πm n v xp kt 4 kt Now th fux of th ons and th ctons must com nto baanc so that Pag 9

10 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 kt n πm / xp kt w n w s n kt M kt M kt n πm M Goundd/dvn Shath In ths cas th ctod s ctcay connctd such that w can add o subtact any ncssay chag to povd baanc. As bfo th fux of ons to th sufac s Γ n ( x) v ( x) const (Not that wh I am dong ths fo fux, you ay want to do ths fo cunt. Th dstncton s that th ons mght b douby o gat chagd. Thus th on fux mght not b th sam as th cton cunt/cton chag. You ay nd to know how much chag you a cayng to th sufac.) To kp thngs as smp as possb, w w assum that th ngy s consvd. Thus M v. If w assum futh that th nta kntc ngy and potnta s zo hnc gno th pshath w fnd M( v vb) ( S) o ( S) v vb M but kt vb M kt S thus kt ( ) kt v M M M M and ( ) Pag

11 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 Γ n( x) v( x) const n( x) M M n( x) Γ ( x) Ths mans that n th shath, th on dnsty can b dtmnd f w know th potnta. W can fnd th potnta fom Posson s quaton ρ φ ( n n ) ε ε Puggng n ou on dnsty vaaton cton dnsty fom th Botzmann s quaton / kt n n w fnd ρ φ n / kt M Γ ε ε ( x) As w dd bfo, w can ntgat ths by mutpyng by φ and thn ntgatng. Thus + ( ) ( ) n / kt M Γ ε + ( ) ( ) / kt M d n Γ d ε n kt / kt ( )+ Γ ε ( ) d ( ) M d kt n / kt M ( ) ( )+ Γ ε kt n ()+ Γ ε M Now assum th foowng. ) ( ), ) ( ), AND 3) n (). Th fst two assumptons a asonab, wh th thd assumpton s an mpotant appoxmaton that aows us to obtan an appoxmat anaytc souton known as Chd s Law. Wth ths assumptons w fnd Pag

12 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 M ( ) Γ ε 4 / / Γ M( ) ε 4 ( ) 4 / / Γ M / ( ) ε Intgatng agan / 4 / ( ) Γ M 4 / d ( ) d ε ( ) ( ) / 4 / 4 3/ 4 Γ M d 3 d ε 4 3/ 4 ( ) ( ) / 4 / Γ M 3 ε / 4 / M 3/ 4 3 ( ) Γ ε 4 / 3 M Γ ε Ths s known as th Chd s Law shath n a fom ths s a bt mo gna than what s found n Lbman. It s th ony known anaytc dscpton, abt an appoxmat souton, to th shath quaton. Ion Matx Shath Th on matx shath occus whn th potnta on an objct changs vy apdy. In ths cas, th ctc fd chang s such that th ctons av th mmdat gon wh th ons man fxd fo a sma nstanc. (Th havy ons cannot spond as fast as th ght ctons.) Thn th s an ctc fd that occus though a unfom dstbuton of ons. How thck s such a shath and what s th potnta stuctu? Fom Posson s quaton w know: Pag

13 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 const n ρ } } n n ε ε n ε const ntgatng onc gvs n ε ( E) ntgatng agan gvs n ε nvtng ε n - Thus at th wa S wa λ Dby Wa ε n kt Wa, λ Dby εkt n Langmu Pobs At ths pont, w can bgn to ook at how w can us ths nfomaton to xamn th pasma. Langmu pobs w nvntd, aong wth much of th bass of pasma scnc, by Ivng Langmu s th 9 s. Langmu pobs a sma mta (o othws conductng) pobs that a nstd nto th pasma and whch can b st to a ang of bass. Fo pocssng and spac pasmas, ths pobs can gnay b postond any wh wthn th dschag. Fo thma o fuson pasmas, bcaus of th ntns hat oad, Langmu pobs can ony b usd at th dg of th dschag. Fotunaty ths s OK wth us as w a ony studyng n pocss and spac pasmas n ths cass. At fst bush, Langmu pobs a vy smp to constuct, vy smp to us and vy smp to undstand. Unfotunaty, non of ths a nty tu. Howv, w can an a sgnfcant amount about Langmu pobs by xamnng a smp mod of th pob. To do ths, w nd to fst ook at how a Langmu pob mght b st up fo a smp xpmnt. Pag 3

14 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 Vacuum Systm Pob Tp Vot mt Pow suppy Insuatng cov Pasma Ammt As can b sn fom th pctu, th pob tp s postond n th pasma, though a nsuatng cov so that th pob ntacts wth th pasma n ony th ocaton of th tp. (W w s at that wh ths s th ntnson, ths s not aways aty.) A vaab pow suppy s connctd to th pob such that vaous bass can b appd to th pob tp. In addton, a votmt and an ammt a connctd to th ccut so that th bas to th tp and th cunt though th tp can b masud. (Fo th vy smpst systm, dc gow dschag, such an aangmnt fo th Langmu pob s usuay suffcnt. Fo mo compx dschags,.g. pusd o f, th pob ccuty can b mo compx. Agan, w w dscuss ths bow.) To coct data wth a Langmu pob, on typcay swps th bas wth th pow suppy and masus th sutant cunt-votag swp. Typcay on w fnd a tac that ooks somthng k: Pag 4

15 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 Pasma fomaton though thmonc and sconday cton msson Ecton satuaton cunt Ecton and on cunt Pasma fomaton though sconday cton msson Ion satuaton cunt Foatng potnta Pasma potnta (Typcay on dos not gt nto th dashd aas on th cuv.) At ths pont w nd to dvop a mod of what s gong on. If th pob s basd bow th pasma potnta, thn w w daw ons to th pob tp. In addton, w w b ab to coct that poton of th cton dstbuton whch has nough kntc ngy to ovcom th potnta ngy ba poducd by th pob bas. W w fst da wth th ons. By th Bohm Shath cta, th ons w nt th shath at a vocty gvn by th potnta dop acoss th pshath gon. Namy kt v / M In addton, th ons w b at th sam dnsty as th ctons at th shath bounday, namy: n n 6. n Assumng that fux s consvd n th shath, w fnd that th on cunt (q*fux) dnsty s J nv / kt n 6. M (notng that ths assums that th ons a snguay chagd.) At ths pont, w nd to xamn th cton cunt to th pob tp. H th cunt s not so tva to cacuat but w hav don so n fo th cunt to a foatng pob. Cayng out th sam dvaton as abov but ths tm ncudng th chag, w fnd Pag 5

16 J Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 Γ n n n n n / m m v v v + + v x πkt xp P kt 3 x y z m / ( x) m m v vx xp πkt P kt kt πm kt πm kt πm wh φ φ P pob pasma m m( vx) dv d mv kt kt / xp x P m / P m /. xp[ UdU ] ( ) ( ) P } xp n v kt xp 4 kt x avag spd dv dv dv P x y z Now th tota cunt to th pob, I P, s th sum of th two cunts tms th cocton aa, A P, IP AP J J ( ). Ths mod gvs a cuv that ooks k Pag 6

17 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 Ecton satuaton cunt Ecton and on cunt Ion satuaton cunt Foatng potnta Pasma potnta Th fst thng to not s that th on and cton satuaton gons hav constant cunts n compason to ou ncasng cunts n ou tu sgna. Ths s bcaus ou cocton aa s ncasng. W had mad th assumpton that th aa was th aa of th pob. Whn th pob bas s cos to th pasma potnta, th shath aound th pob s vy sma. Howv, whn th atv bas bcoms ag, th shath wdth bcoms ag, and bcaus of gomtc ffcts, th cocton aa gows. W can xamn ths b ookng at Chd s Law. Γ 3 / 3 / 8 M 43 / 6 ε 3 / 3 / 8 ( 6. ) ( ) ( ) 6 3 / 3 / 43 / n kt ε What w s fom ths s that th shath wdth gows appoxmaty nay wth th bas. Oftn fo 3 V bas, on mght hav shath wdths of ~ mm. (Ths of cous dpnds on th cton tmpatu and dnsty.) Assumng that th pob s a cynd wth a wdth of. mm, w fnd that th tu cocton aa of th pob s much ag than th physca aa of th pob. Ths cocton aa gows n both th cton and on satuaton gons thus ncasng th pob cunt. Undstandng ths smp phnomnon s th subjct of numous studs ncudng sva Ph.D. Dsstatons. (Th most famous s by La Fambous nd coct spng and tt tc.) Pag 7

18 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 To anayz Langmu pob data on must ook at th mod quatons, / kt J n 6. M / kt J n P xp πm kt W s that th on cunt s st pmay by th pasma dnsty. To gt to th coct dnsty, w nd to fst dtmn th cocton aa. Ths s don by fttng th on satuaton to cuv, fa fom th cton cunt, such as x I x ~ 5. to. Ths cuv s thn usd to dtmn th on cunt that woud b obsvd at th pasma potnta. By dong ths w hav st th cocton aa to th physca aa of th pob. Thn I J A n A kt / 6. M Now w nd to gt th cton tmpatu to gt th pasma dnsty. To gt th cton tmpatu, w nd to ook at ony th cton pat of ou masud cunt. To av at ths, w smpy mov th pat du to th ons. I Ipob I( ) / kt An P xp πm kt W s that w can av at th tmpatu by takng th og of th cunt and w fnd that th sop s popotona to th nvs of th tmpatu. Ln I P kt Ln An kt ( ) / πm Fnay, w fnd th pasma potnta fom th pont at whch th cuv os ov.g. th sop s not as stp. Oth ssus wth Langmu pobs. Pobs a subjct to o fo a ag numb of asons. Examps of ths asons ncud: ) Dty pob tps act k sstos and mt th tu cunt to th pob. Ths s usuay dat wth by can th pob though th cton o on bombadmnt. Fo cton bombadmnt to wok, th pob tp must daw nough cunt to bcom d hot (~ C), thus bunng off th dt. Fo on bombadmnt to wok, th bas must b sva hundd votag bow th pasma potnta, thus aowng sputtng to can th sufac. ) Tm vayng pasmas. Many pasmas mak us of an f pow souc o us a pus n th pocsss. Whn ths occus, a gat da of ca must b takn n th ccuty of th pob systm. Fo f dschags, choks must b addd to th ccut to mnat th f Pag 8

19 Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 fuctuatons. Fo pusd systms, RLC tm constants must b ookd at cafuy to match th pob ccuty to th qumnts of th masumnt. Pag 9