Sssio : Plasmas i Equilibrium Ioizatio ad Coductio i a High-prssur Plasma A ormal gas at T < 3000 K is a good lctrical isulator, bcaus thr ar almost o fr lctros i it. For prssurs > 0.1 atm, collisio amog molcul ad othr particls ar frqut ough that w ca assum Local Thrmodyamic Equilibrium (LTE), ad i particular, ioizatio-rcombiatio ractios ar govrd by th law of mass actio. Cosidr utral atoms () which ioiz sigly to ios (i) ad lctros (): + i (1) O form of th law of mass actio (i trms of umbr dsitis j = p j /kt, whr T is th sam for all spcis) is, i = S(T ) () Whr th Saha fuctio S is giv (accordig to Statistical Mchaics) as, 3/ q i πm kt V i S(T ) = xp (3) q h kt Groud stat dgracy of io (q i = 1 for H+) Groud stat dgracy of utral (q = for H) Mass of lctro (m = 9.11 10 31 kg) Boltzma costat (k = 1.38 10 3 J/K) Plak s costat (h = 6.6 10 34 Js) Ioizatio pottial of th atom (V i = 13.6 V for H) Excpt for vry arrow shaths ar walls, plasmas ar quasi-utral: So that, ca b usd. Th ioizatio fractio is dfid as α =. + = i (4) = S(T ) (5) Giv T, this rlats to. A scod rlatio is dd ad vry oft it is a spcificatio of th ovrall prssur, 1
Combiig (5) ad (6), p = ( + i + )kt = ( + )kt (6) ( p ) = S(T ) = S(T ) ( ) kt Whr = p/kt is th total umbr dsity of all particls. W th hav, ad, + S S = 0 = S + S + S = _ (7) 1 + 1 + S Sic S icrass vry rapidly with T, th limits of (7) ar, (T 0) S Wak ioizatio (T ) Full ioizatio Oc alctro populatioxists, alctric fild E will driv a currt dsity Ej through th plasma. To udrstad this quatitativly, cosidr th momtum balac of a typical lctro. It ss alctrostatic forc, EF = E (8) It also ss a frictioal forc du to trasfr of momtum ach tim it collids with som othr particl (utral or io). Collisios with othr lctros ar ot coutd, bcaus th momtum trasfr is i that cas itral to th lctro spcis. Th ios ad utrals ar almost at rst compard to th fast-movig lctros, ad w dfi affctiv collisio as o i which th lctro s dirctd momtum is fully giv up. Suppos thr ar ν of ths collisios pr scod (ν is th collisio frqucy pr lctro). Th lctro loss momtum at a rat m Euν, whr Eu is th ma dirctd vlocity of lctros, ad so, O avrag, or m Euν = E, so that, EF frictio = m Eu ν (9) E + E F F frictio = 0
( ) u = E (10) Th group µ = is th lctro mobility [(m/s)/(volt/m)]. Th currt dsity is th flux of charg du to motio of all chargs. If oly th lctro motio is coutd (it domiats i this cas), Th group, j = u = ( ) E (11) is th lctrical coductivity of th plasma (Si/m). σ = (1) Lt us cosidr th collisio frqucy. Suppos a utral is rgardd as a sphr with a cross-sctio ara Q. Elctros movig at radom with (thrmal) vlocity c itrcpt th ara Q at a rat qual to thir flux c Q. Sic a whol rag of spds c xists, w us th avrag valu c for all lctros. But this is for all lctros collidig with o utral. W ar itrstd i th rvrs (all utrals, o lctro), so th part of ν du to utrals should b c Q. Addig th part du to ios, ν = c Q + i c Q i (13) c is vry diffrt (usually much largr) tha u. Most of th thrmal motio is fast, but i radom dirctios, so that o avrag it arly cacls out. Th o-caclig rmaidr is u. Thik of a swarm of bs movig furiously back ad forth, but movig (as a swarm) slowly. Th umbr of lctros pr uit volum that hav a vlocity vctor c dig i a box, d 3 c = dc x dc y dc z i vlocity spac is dfid as f ( c, x)d 3 c, whr f ( c, x) is th distributio fuctio of th lctros, which dpds (for a giv locatio x ad tim t) o th thr compots of c. I aquilibrium situatio all dirctios ar qually likly (isotropic), so f = f (c ) oly, ad o ca show that th form is Maxwllia, ( / m ) 3 ( ) m f = xp c πkt kt whr c = c x + c y + c z (14) With th ormalizatio, f d 3 c = Th ma vlocity is th, 3
c = 1 8k c f d 3 T c = πm (15) For lctros, c = 610 T (m/s with T i K) (16) Not that if thr is currt, th distributio caot b strictly Maxwllia (or v isotropic). But sic u c, th ma thrmal vlocity is vry clos to Eq. (15) ayway. Rgardig th cross sctios Q ad Q i, thy dpd o th collisio vlocity c, spcially Q i. This is bcaus th i coulombic itractio is soft, so a vry fast lctro ca pass arly udflctd ar a io, whras a slow o will b strogly dflctd. Th complt thory yilds axprssio, whr l Λ Q i.95 10 10 (m with T i K) (17) T 1 l Λ 11.35 + l T l p (T i K, p i atm) (18) so that l Λ is usually aroud 6 1, ad cav b tak as a costat ( 8) i rough calculatios. For th utral hydrog atoms, th collisios ar fairly hard, ad o ca us th approximatio, Q H 10 19 m (19) 4
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