Maxwellian Collisions

Transcription

1 Maxwllian Collisions Maxwll ralizd arly on that th particular typ of collision in which th cross-sction varis at Q rs 1/g offrs drastic siplifications. Intrstingly, this bhavior is physically corrct for any charg-nutral collisions and odrat nrgis: Th charg q polarizs th nutral in proportion to th fild (α q/r 2 ) and th dipol α attracts th particl with a forc F α/r 3 q/r 5. Fro our work on powr-law potntials, this is th intraction typ that lads to Q 1/g. Th siplification sts fro th fact that th group gq (g) appars in th intgrals for rs M rs and E rs, and can now b ovd outsid as a constant. W put, Q rs(g) = g 0 g Q rs(g 0 ) thn M rs = µ rs g 0 Q (g 0 ) f r f s (wm r wm s )d 3 w r d 3 w s = rs ws wr [ ] = µ f s d 3 w s Mw r f r d 3 w r f r d 3 w r wm s f s d 3 rs g 0 Q rs(g 0 ) w s w s " - " n s w r " - " n r u r w r " - " n r w s " - n s u s " M rs = µ rs g 0 Q rs(g 0 )n r n s (Mu r Mu s ) Dfin ν sr as th collision frquncy of on s-particl will all r-particls, ν sr = n r gq rs(g) constant for Maxwllian collisions (1) Siilarly, ν rs = n s gq rs(g) (Not: ν sr /n r = ν rs /n s ) M rs = µ rs n s ν sr (Mu r Mu s ) = µ rs n r ν rs (Mu r Mu s ) (2) For othr typs of collisions th valuation is uch lss straightforward, as it rquirs prior solution for f r and f s. Howvr, th for M rs = µ rs n r ν rs (Mu r Mu s ) can always b rcovrd, only th collision frquncy ν rs is gnrally not a constant, but a function of th lctron tpratur, and is calculatd fro so of th xisting odls for f r and f s. 1

2 For nrgy transfr, w will dal dirctly with th intrnal nrgy transfr rat, M E rs = E rs Mu s M rs (3) Fro th dfinitions, E rs = µ rs f r1 f s gq (g)(g M Mu s ) Mgd 3 wd 3 rs w 1 (4) w w 1 and for Maxwllian collisions, th group gq (g) is a constant and ovs outsid th inrs tgration. Th vlocity cobination insid can b anipulatd nxt. Dfin th rando vlocitis Mc s = wm Mu s, Mc r = wm 1 Mu r : s(u s+c s)+ r (u r +c r ) r + s (G M Mu s ) Mg = (Mu r + Mc r Mu s Mc s ) Mu s (Mu r + Mc r Mu s Mc s ) ( ) s Mu s + r Mu r r 2 s 2 = Mu s (Mu r Mu s ) + c c + (Trs linar in Mc r or Mc s ) r + s r r + s s r + s " {{ - " r (u r +s r u s) Calling for short r + s =, and ignoring th linar trs, bcaus thy intgrat to zro (notic (Mc s ) s = 0, (Mc r ) r = 0), r r cr 2 s c 2 r (G M Mu s ) Mg = (Mu r Mu s ) (Linar trs) Substitut into (4): E rs [ = µ rs (gq rs ) f r1 f s d 3 w 1 d 3 r (Mu r Mu s ) 2 w +... w w r c r 2 f d 3 w 1 d 3 r1 f s w s c 2 sf d 3 w 1 d 3 r1 f s w (5) w w 1 w w 1 2

3 Th first of th intgrals is siply n r n s. Th scond can b rorganizd into d 3 wf s f r1 r c 2 r d 3 w 1, w w 1 of which th innr intgral yilds 3kT r n r, whil th outr on givs n s. With a siilar argunt for th third intgral, w obtain E = µ rs (gq )[ r (M M ) 2 rs n r n s rs u r u s + 3k(Tr T s )] (6) This has an intrsting structur. Th r (Mu r Mu s ) 2 tr rprsnts an irrvrsibl intrnal nrgy addition (hat) to spcis s fro collisions with r, providd th two spcis drift at diffrnt an vlocitis. Th scond tr, in (T r T s ) is th transfr of hat fro r to s whn th two spcis hav diffrnt tpraturs. It is rvrsibl, dpnding on th sign of T r T s. For copltnss, w can now calculat th transfr of full kintic nrgy, E rs = E +Mu s rs M rs, with th rsult [ ] r Mu r + s Mu s 3k E rs = µ rs n r n s (gq rs) (Mu r Mu s ) + (T r T s ) (7) So sipl applications of th Montu Equations Elctrons Oh s Law - Excpt for high-frquncy ffcts (of th ordr of th Plasa Frquncy) or for vry strong gradints (lik in doubl layrs), th inrtia of lctrons can b nglctd in thir ontu balanc. Assu collisions of lctrons happn with on spcis of ions and on of nutrals only: 0 + P = n (E M + Mu B M ) + n [ν i (Mu i Mu ) + ν n (Mu n Mu )] (8) whr w usd µ i =, µ n =. In any cass, u i «u, u n «u, and w can siplify th quation by introducing th lctron currnt dnsity, Mj = n Mu (9) Divid by (ν i + ν n ) and dfin, n E M + P = Mj B M + 2 n (ν i + ν n )Mj σ = (Elctrical conductivity) (10) (ν i + ν n ) B β = (Hall paratr) (11) (ν i + ν n ) 3

4 M P so that σ E + = M j + M M j β (12) n Notic: (a) Elctron prssur gradints can driv lctron currnt. This is sotis calld a diaagntic currnt. vp = vp n n (b) As a liit, if boundary conditions forbid currnts, j = 0, thn E M + = 0, E M, which ans dnsity gradints can st up a fild th Abipolar fild. If T = const. kt n kt n φ = φ = φ 0 + ln (13) n n 0 Which strongly rsbls th kintic Boltzann rlationship (xcpt this ti w only look at avrags). (c) Th Hall paratr is th ratio β = ωc of lction gyro frquncy to lctron collision ν frquncy. It can b larg in low-dnsity plasas, vn with odrat B filds. (d) Th currnt is not alignd with th driving filds. Additional dviations fro th lctric fild rsult fro E M = E M + vp n () Eq. (12) can b solvd for Mj in trs of E M = E M + v n. Start by ultiplying (cross products) tis β M : P σβ M E M = β M Mj + β M (Mj β M ) " -{{ " β 2 j β (β j ) considr only th currnt prpndicular to B M, so that B M Mj = 0: Mj β M = β M 2Mj σβ M E M and substitut this into (12): σe M = Mj + β 2Mj σβ M E M, or M σ E M j E = + β M M plus Mj = σe M 1 + β2 (14) 4

5 This is sotis organizd as a tnsor quation. With z takn along B M : j x j y j z 1 β 0 1+β 2 1+β2 = σ β β 1+β E x E y E z (15) which aks th anisotropy of th situation or clar. In Ionosphric Physics, this is also put as a conductivity tnsor σ P σ H 0 M σe M j = M Mσ = σ H σ P 0 (16) 0 0 σ σ P = Pdrsn conductivity (vry sall in th ionosphr, β» 1) σ H = Hall conductivity (intrdiat) σ = σ Paralll conductivity (vry larg in th ionosphr) Abipolar Diffusion Considr a sipl cas with B = 0, ngligibl inrtia. Writ both, lctron and ion ontu quations: DMu i i n + P M i = En + n [ ν i (Mu Mu i ) + µ in ν in (Mu n Mu i )] Dt P = En M + n [ν i (Mu i M u ) + ν n (Mu n M u )] Add togthr, not n ν i = n i ν i (and also n = n i ), DMu i i n + (P i + P ) = n i µ in ν in (Mu n Mu i ) + n ν n (Mu n Mu ) Dt " {{ - " usually sallr Also, norally T /T «n /n. In addition lt us assu that ion inrtia can b also nglctd in coparison with th othr trs in th ontu balanc (although kping th tr would b or gnral), i or i or, k(t + T i ) n = n µ in ν in (Mu i Mu n ) k(t + T i ) n (Mu i Mu n ) = µ in ν in n 5

6 Sotis nutrals rturn fro rcobination of ions, so, n n n Mu i = n n Mu n thn n (Mu i Mu n ) = Mu i + Mu i n = (n n + n i ) Mu i n n n n n n k(t + T i ) ρ i n Mu i = n ν in = n n g in Q in n i + n n = ; µ in = n i + n n µ in ν in i 2 n n k(t + T i ) n Mu i = i ρ n 2 i n n Q in g in n Mu i = D a n D a = 2k(T i + T i ) ρq in g in Abipolar diffusivity Back to th lctron quation, if w nglct both collision forcs, n P = En M kt = φ n (φ kt ln n ) = 0 φ φ 0 = kt ln n n 0 Equivalnt Boltzann quilibriu 6

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