DRAFT 1 Collisions btwn lctrons and ions Flix I. Parra Rudolf Pirls Cntr for Thortical Physics, Unirsity of Oxford, Oxford OX1 NP, UK This rsion is of 8 May 217 1. Introduction Th Fokkr-Planck collision oprator can b simplifid significantly whn w considr collisions btwn lctrons and ions. Th simplification is a rsult of th larg mass diffrnc. Assuming that th ion and lctron tmpraturs ar of th sam ordr, T T i, 1.1 w find that th thrmal spd of th lctrons t is much largr than th thrmal spd of th ions ti, T Ti m t ti t, 1.2 whr and ar th mass of lctrons and ions, rspctily. W will xploit this diffrnc in charactristic locitis to study collisions btwn lctrons and ions. 2. Simpl stimats Bfor attmpting a rigorous driation, w gi som simpl stimats. W considr lctrons of charg and mass colliding with ions with charg Z and mass. Naily, to ha a collision, th distanc b i btwn th lctron and th ion must b such that th kintic nrgy of th lctron is of th ordr of th potntial nrgy du to th Coulomb forc btwn th particls, Thus, only lctrons at a distanc 1 2 2 t T Z2 4πɛ b i. 2.1 b i Z2 4πɛ T 2.2 of ions ha significant collisions with thos ions. Th charactristic collision frquncy th inrs of th tim btwn collisions can thn b stimatd using b i. An lctron mos a man fr path λ i bfor it ncountrs an ion. Sinc th lctron only notics ions at a distanc b i, it sampls a olum πb 2 i λ i. In this olum th probability of finding an ion is unity, so it must satisfy n i πb 2 i λ i 1. From this stimat w obtain th man fr path With th man fr path, w obtain th collision frquncy λ i 1 n i πb 2 4πɛ 2 T 2 i Z 2 4. 2. n i ν i t Z 2 4 n i. 2.4 λ i 4πɛ 2 m 1/2 T /2
2 Flix I. Parra This stimat ignors th fact that wak collisions btwn particls sparatd by th Dby lngth λ D dominat. To includ th ffct of ths wak collisions, w only nd to rcall that w stimatd th ffct of wak collisions to b largr than th ffct of collisions btwn particls at a distanc b i by a Coulomb logarithm ln Λ i 1. A largr ffct is roughly quialnt to mor collisions, that is, it is quialnt to a largr collision frquncy. Thn, w nd to multiply 2.4 by a factor of ln Λ i to obtain ν i Z2 4 n i ln Λ i. 2.5 4πɛ 2 m 1/2 T /2 Similar stimats gi us th typical collision frquncy of lctron-lctron collisions and ion-ion collisions, and 4 n ln Λ ν 4πɛ 2 m 1/2 T /2 2.6 ν ii Z4 4 n i ln Λ ii. 2.7 4πɛ 2 m 1/2 i T /2 i Th stimat that ld to 2.5 is not alid for th ffct of ion-lctron collisions on ions. It is tru that lctrons and ions collid oftn, but th ffct of a singl collision on an ion is small. Du to consration of momntum, if th chang of th lctron locity in a collision is, th chang to th ion locity is i = m m t ti ti. 2.8 Thrfor, a singl lctron-ion collision only modifis th ion locity by a small amount of th ordr of / 1/2 ti ti. This typ of collision that only changs th locity by a small amount can b thought of as a random walk in locity spac. To achi a total chang in th ion locity of th ordr of ti, w nd a numbr of lctron-ion collisions 2 ti N c 1. 2.9 i Thn, th ffcti collision frquncy of ion-lctron collisions is 1/N c smallr than th lctron-ion collision frquncy, ν i ν i ν i Z2 4 n i m 1/2 ln Λ i. 2.1 N c 4πɛ 2 T /2 Combining quations 2.5, 2.6, 2.7 and 2.1, and assuming T T i, Z 1 and n n i, w obtain m ν ν i ν ii ν i ν i ν i. 2.11 Th lctrons collid with lctrons as oftn as thy collid with ions. Ions collid with ions much mor rarly, and ions ar affctd by thir collisions with th light lctrons only aftr a tim much longr than th tim btwn collisions with othr ions.
. Elctron-ion collision oprator Collisions btwn lctrons and ions W start with th ffct on lctrons of collisions with ions. W prform an xpansion in / 1. W first considr th lowst ordr in / 1, and latr w kp highr ordr trms..1. Elctron-ion collision to lowst ordr in / 1 Th lctron-ion Fokkr-Planck collision oprator is { [ C i [f, f i = γ i f i g g g f f if / T } f f i d..1 m } i {{} f if / T i To find th ordr of magnitud stimat, w ha usd f f / t and f i f i / ti. Th trm m 1 i f f i is thn ngligibl. Moror, w find lading to g =,.2 t ti g g g = 2 I.. With this rsult, and using f i d = n i, quation.1 bcoms to lowst ordr in m / 1 C i [f, f i L i [f = γ in i 2 I m 2 f..4 This approximat oprator is known as Lorntz collision oprator or pitch-angl scattring collision oprator. To undrstand th pitch-angl scattring oprator, w rwrit it using th sphrical coordinats {, α, β} in locity spac, shown in figur 1. In th orthonormal basis {ˆ = /, ˆα = α/ α, ˆβ = β/ β }, th gradint and dirgnc with rspct to th locity of gnral functions f and Γ ar and f = f + α f α + β f β = f ˆ + 1 f α ˆα + 1 sin α Γ = 1 [ J J Γ + α J Γ α + = 1 2 2 Γ ˆ + 1 1 sin α Γ ˆα + sin α α sin α β β J Γ β Γ ˆβ f β ˆβ.5,.6 whr J = dt[/, α, β = [ α β 1 = 2 sin α is th dtrminant of th Jacobian of th transformation, α, β. Using.5 and.6, quation.4 bcoms L i [f = γ [ in i 1 m 2 sin α f + 1 2 f sin α α α sin 2 α β 2..7 Th Lorntz oprator diffuss th distribution function in α and β, but las its structur in unchangd. Th rason for this lack of diffusion in is that lctrons do not chang th magnitud of its locity whn thy collid with hay ions. According to 2.8, th locity of th ion barly changs in a collision with an lctron. Thn, th ion kintic
4 Flix I. Parra ẑ ˆ ˆ ˆ ŷ ˆx Figur 1. Sphrical coordinats {, α, β} in locity spac. Th orthonormal basis {ˆ = /, ˆα = α/ α, ˆβ = β/ β } is also sktchd. nrgy dos not chang, and sinc th total kintic nrgy of both th lctron and th ion is consrd, th kintic nrgy of th lctron is th sam bfor and aftr th collision. Thus, only th dirction of th lctron locity changs aftr a collision, lading to th diffusion in α and β sn in.7. Th Lorntz oprator tnds to mak th lctron distribution function isotropic, that is, it tnds to gi a function f that is only a function of and not of α or β. To show this proprty, w pro that th Lorntz oprator satisfis its own H-thorm. Th ntropy production du to th Lorntz oprator is σ i L = ln f L i [f..8 Using.4 and intgrating by parts, w obtain σ i L = γ in i m 2 f ln f 2 I ln f d = γ in i f m 2 ln f 2 ln f 2 d = γ [ f 2 in i f m 2 + 1 2 f α sin 2 d..9 α β Thus, th ntropy grows until σ i L =. Th ntropy production σl i anishs only whn ln f is proportional to, that is, whn f is only a function of th locity magnitud. Intrstingly, w did not nd to add th ntropy production of th ions du to lctron-ion collisions to show that th ntropy incrass. Th isotropization procss is thn indpndnt of th ion distribution function bcaus to this ordr in / 1, th ions sm just stationary particls compard to th fast lctrons..2. Elctron-ion collision to first ordr in / 1 W ha argud in 2.11 that th lctron-ion collisions ar much mor frqunt than othr typs of collisions. Thus, it is usual to ha an lctron distribution function that
is isotropic to lowst ordr in / 1, Collisions btwn lctrons and ions 5 f = f + f 1 +....1 isotropic m m f i If this is th cas, w nd to continu th xpansion of th lctron-ion collision oprator to nxt ordr in / 1. For g g g, instad of th lowst ordr approximation in., w kp th nxt ordr corrction to find g g g Mg = M M M =..11 Substituting this rsult and th xpansion in.1 into.1, w obtain { [ C i [f, f i γ i f i du to isotropy f + f 1 f f f i d }..12 Using and f i d = n i, f = 1 f,.1 f i d = n i u i, f i d =,.14 th lctron-ion collision oprator in.12 can b rwrittn as C i [f, f i γ in i m 2 On furthr usful manipulation is f 1 1 f u i..15 = I =..16 With this rsult quation.15 finally bcoms [ C i [f, f i γ in i m 2 f 1 + u i = γ in i m 2 f [ f 1 + u i f = L i [ f 1 + u i f..17 Th lctron-lctron collisions ar usually as frqunt as th lctron-ion collisions s 2.11, and as a rsult, it is usually th cas that th lowst ordr lctron distribution function is not only isotropic, but also Maxwllian, f = f M n m 2πT /2 xp 2 If this is th cas, quation.17 bcoms [ C i [f, f i L i f 1 u i T f M 2T..18..19
6 Flix I. Parra W ha sn that th Lorntz oprator tnds to mak th distribution function isotropic. Thus, th collision oprator in.17 will gi f 1 = g 1 u i f..2 whr g 1 is isotropic. Whn calculating th total distribution function f f + f 1, w can absorb th isotropic corrction g 1 into th lowst ordr isotropic distribution function f, lading to f f u i f f + u i f f u i..21 Thn, th lctron-ion collisions tnd to gi an lctron distribution function that is isotropic around th arag locity of th ions u i. W procd to calculat th collisional friction forc and th collisional nrgy xchang using.17..2.1. Elctron-ion collisional friction forc Th collisional forc on th lctrons is F i = C i [f, f i d..22 Substituting quation.17 into this xprssion, and intgrating by parts, w find F i = γ in i m I f 1 + u i f d = γ in i f 1 + u i f d..2 Intgrating th first trn th intgral by parts again, w find F i = γ in i f 1 2 m 2 I f 4 u i d..24 To simplify th intgral furthr, w us that f is isotropic, and w tak th intgral in th sphrical coordinats sktchd in figur 1. Sinc = [sin αcos β ˆx+sin β ŷ+cos α ẑ, th intgral or th angls α and β gis 1 4π π dα 2π dβ sin α = 2 2 ˆxˆx + ŷŷ + ẑẑ = I..25 Thn, 2 I f 4 d = 8π I f d = 8πf I..26 Using this xprssion, and mploying 2 = [ = [ = 2 = 2 = = 2,.27 quation.24 bcoms F i = γ in i 8πf u i 2 f 1 d..28
Collisions btwn lctrons and ions 7 Th friction forc dpnds on th arag ion locity and on a momnt of th corrction to th lctron distribution function f 1. Not that only th alu of f at = ntrs in th xprssion, and that th intgral or f 1 is wighd towards smallr du to th factor. Th alu of th lctron distribution function at low nrgis is mor important bcaus slow lctrons ar mor likly to ha rlant intractions with ions. Exprssion.28 bcoms mor transparnt if w assum that th lctron distribution function is a Maxwllian with arag locity u u i t, /2 m f = n xp u 2 = f M u 2πT 2T f M u f M = f M + u f M,.29 T f f 1 whr f M is th stationary Maxwllian dfind in.18. In this simpl cas, /2 m f = f M = n,. 2πT and using.25, w find that f 1 d m u = f M d T = 2n 5/2 u m xp 2 2π T 2T With ths rsults, quation.28 bcoms d = whr th lctron-ion collision frquncy is dfind to b ν i = 4 2π 2n 2π m T /2 u..1 F i = n ν i u i u,.2 γ i n i m 1/2 T /2 = 4 2π.2.2. Elctron-ion collisional nrgy xchang Z 2 4 n i ln Λ i.. 4πɛ 2 m 1/2 T /2 Th collisional nrgy gaind or lost by th lctrons is 1 W i = 2 2 C i [f, f i d..4 Substituting quation.17 into this xprssion, and intgrating by parts, w find W i = γ in i 2 f 1 + u i f d 2 = γ in i m f 1 + u i f d =..5 To this ordr in th xpansion in / 1, thr is no xchang of nrgy. Th magnitud of th locity of th lctron barly changs in on collision, and as a rsult, th transfr of nrgy is minimal. To calculat th nrgy transfr, it is bttr to us th ion-lctron collision oprator than to xpand th lctron-ion collision oprator to nxt ordr.
8 Flix I. Parra 4. Ion-lctron collision oprator W procd to calculat th ffct on ions of collisions with lctrons. W prform an xpansion in / 1. To simplify th problm, w assum that th lctron distribution function is almost isotropic and hnc it can b xpandd as in.1. Th ion-lctron Fokkr-Planck collision oprator is C i [f i, f = γ i { [ f g g g f i m } i {{} f if / T i f i f f if / T d }. 4.1 Th trm m 1 i f f i is thn small, and w can us th lowst ordr approximation f f in it. W also ha lading to g =, 4.2 ti t g g g Mg = M M + M =. 4. With ths rsults, quation 4.1 bcoms { [ C i [f i, f γ i f f i f i f } + f 1 f d. Using.1 and.16, w find 4.4 f = 1 f. 4.5 Employing.2, w obtain γ i f 1 d = With ths rsults, quation 4.4 bcoms C i [f i, f F i n i + F i n i f i + γ i { [ γ i 1 f u i d. f f i } d f i f u i 4.6. 4.7 W finish by taking th intgrals. Using.25 and = [ 2 I /, w find f d = 8π I f d. 4.8
Collisions btwn lctrons and ions 9 Using this rsult and.26, quation 4.7 finally bcoms [ C i [f i, f F i f i + 8πγ i f i f d + f u i f i. 4.9 n i If th lctron distribution function is a Maxwllian s.18, this oprator simplifis to [ C i [f i, f F i f i + n ν i T f i + u i f i, 4.1 n i n i whr ν i is dfind in.. W procd to calculat th collisional friction forc and th collisional nrgy xchang. 4.1. Ion-lctron collisional friction forc Th collisional forc on th ions is F i = C i [f i, f d. 4.11 Substituting quation 4.9 into this xprssion, and intgrating by parts, w find F i = F i f i d 8πγ [ i f i f d + f u i f i d. 4.12 n i Using.14, th collisional forc bcoms as xpctd. F i = F i, 4.1 4.2. Ion-lctron collisional nrgy xchang Th collisional nrgy gaind or lost by th ions is 1 W i = 2 2 C i [f i, f d. 4.14 Substituting quation 4.9 into this xprssion, and intgrating by parts, w find F i = F i n i f i d 8πγ i [ f i f d + f u i f i d. 4.15 Using that f i u i d =, w can writ f i u i d = f i u i 2 d. 4.16 By intgrating by parts, w obtain f i d = f i d = f i d = n i. 4.17
1 Flix I. Parra With ths rsults, quation 4.15 bcoms W i = F i u i + 8πγ i n i f d f f i u i 2 d. 4.18 For an lctron Maxwllian distribution function s.18 and an ion Maxwllian distribution function /2 mi f i = f Mi n i xp u i 2, 4.19 2πT i 2T i th collisional nrgy xchang bcoms W i = F i u i + n ν i T T i. 4.2 work don by friction forc Th first trn 4.2 is th work don by th collisional forc F i = F i on th ions. Th scond trs a collisional nrgy xchang proportional to th tmpratur diffrnc btwn lctrons and ions. This trm will tnd to mak th ion and lctron tmpraturs qual, but at th slow rat n n i ν i ν ii ν ν i. 4.21 Thn, th ions and lctrons can ha many collisions and thir distribution functions bcom Maxwllians without thir tmpraturs bcoming qual. For this rason, it is possibl to find plasmas with ry diffrnt lctron and ion tmpraturs. Du to nrgy consration, th lctron nrgy gain or loss is W i = W i = F i u i n ν i T T i = F i u work don by friction forc + F i u i u Joul hating + n ν i T i T. 4.22 This collisional nrgy gain has th work don by th friction forc F i on th lctrons, and th nrgy xchang du to th tmpratur diffrnc, but in addition to ths two trms, it contains Joul hating. This Joul hating trs du to th transfr of nrgy from th arag lctron flow to th lctron tmpratur.