Effect of Electric Fields on Electron Thermal Transport in Laser-Produced Plasmas

Transcription

1 FFC OF LCRIC FILDS ON LCRON HRMAL RANSPOR IN LASR-PRODUCD PLASMAS ffct of lctric Filds o lctro hrmal rasport i Lasr-Producd Plasmas Itroductio hrmal trasport plays a importat rol i dirct-dri irtial cofimt fusio. h Spitzr Härm hat flu q SH = κ SH has b cotioally usd i th dirctdri irtial cofimt fusio (ICF) hydrocods. Hr κ SH is th Spitzr hat coductiity ad is th lctro tmpratur. I th rgios of th stp tmpratur gradits whr q SH cds a fractio f of th fr-stram limit q FS = th Spitzr flu is rplacd by fq FS whr is th lctro dsity = mis th lctro thrmal locity ad f =.5. is th flu limitr. It has b kow for mor tha two dcads 5 that i additio to th trms proportioal to th tmpratur gradits (thrmal trms) th hat flu i lasrproducd plasmas cotais podromoti trms that ar du to th gradits i th lasr lctric fild. o our bst kowldg o systmatic aalysis has b prformd to addrss th ffct of such trms o th hydrodyamic flow i ICF plasmas. As show latr th ratio of th podromoti trms to th thrmal trms is proportioal to R= ( ) ( L L) whr = mω L is th lctro quir locity is th lctric charg is th amplitud of th lctric fild m is th lctro mass ω L is th lasr frqucy L ad L ar th tmpratur ad th lctric fild scal lgth ad is a costat. h ratio of th lctro quir locity to th thrmal locity is small for typical plasma paramtrs. Idd ( ). I5λ µ m kv whr I 5 is th lasr itsity i 5 W/cm λ µm is th lasr walgth i micros ad kv is th lctro tmpratur i kv. Usig I 5 ~ ad ~ kv w obtai ( ) ~. for λ µm =.5 µm. h ratio R howr ca b of th ordr of uity du to a larg ratio L L. Idd as th lasr rachs th turig poit whr th lctro dsity quals c cos θ th lctric fild dcays toward th ords portio of th shll as 6 ~ ma p ( ζ ) whr c = mωl is th critical dsity θ is th lasr icidc agl ζ = ( ω LL c) zl L ~ L is th lctro-dsity scal lgth ad z is th coordiat alog th dsity gradit. hrfor th lctric-fild scal lgth ar th turig poit bcoms L ~ L ( ω LL c). Substitutig this stimat to th ratio R ad usig L ~ µm ad ( ) ~. gis R~ ωll c ~ 6.. As will b show latr th cofficit is umrically larg ad proportioal to th io charg ; this maks R largr tha. his simpl stimat shows that th podromoti trms bcom comparabl to th thrmal trms i th lctro thrmal flu ar th turig poit. I additio th p-polarizatio of th lctric fild (polarizatio that has a fild compot dirctd alog th dsity gradit) tuls through th ords portio of th shll ad gis a rsoac lctric fild at th critical surfac. 6 h gradit of such a fild is proportioal to th ratio ωl i whr i is th lctro io collisio frqucy at th critical surfac. Substitutig typical dirct-dri primt paramtrs ito a prssio for th lctro io collisio frqucy at th critical surfac i ωl 5. kv shows a sigificat cotributio of th podromoti trms to th hat flu ar th critical surfac. I this articl th podromoti trasport cofficits ar drid. Such cofficits ha b cosidrd priously Rfrc 7 dlopd a mthod of solig th kitic quatio by sparatio of th lctro distributio fuctio o th high-frqucy compot du to th lasr fild ad th low-frqucy compot of th tim-aragd plasma rspos. Usig such a mthod th lasr filds cotributio to th lctro strss tsor was obtaid. A similar mthod was usd i Rf. whr th importac of th podromoti ffcts o th lctro thrmal coductio was mphasizd. P. Mora ad R. Pllat ad I. P. Shkarofsky 5 ha aluatd th cotributios of th lasr filds ito th hat ad momtum flus. As was poitd out i Rf. 8 by ot howr cosisttly takig ito accout th cotributio of th lctro lctro collisios th trasport cofficits i thir rsults cotai wrog umrical factors. A cosistt aalysis was prformd i Rf. 8 whr rsults wr obtaid i th limit of larg io charg. Such a limit was rlad i Rf. 9. h lattr rfrc howr cotais umrous typographical rrors so th rsults will thrfor b rdrid i this articl. h ffct of podromoti trms o th hydrodyamic flow i dirct-dri ICF primts will b discussd i dtail i a forthcomig publicatio. 5 LL Riw Volum 98

2 FFC OF LCRIC FILDS ON LCRON HRMAL RANSPOR IN LASR-PRODUCD PLASMAS Modl W cosidr a fully ioizd plasma i a high-frqucy lctromagtic fild: ω ω ε = rt i L t rt i L t () ω ω β = B rt i L t B rt i L t () whr ad B ar slowly aryig (with rspct to i ω L t ) lctric ad magtic filds ad * ad B * ar th compl cougat (c.c.) of ad B. h lctro distributio fuctio f obys th Boltzma quatio β t f f ε r pf c = J[ f f] Ji[ f] () whr is th low-frqucy lctric fild. Hr Λ J [ f f]= m δk k d k f ( ) ( ) ( ) f. (7) Nt followig Rf. 7 w sparat th lctro distributio fuctio o th slowly aryig part f ad th high-frqucy compot f : iω t i t f = f f L ω f L. (8) Substitutig qs. () () ad (8) ito q. () ad collctig th trms with qual powrs of i ω L t w obtai J f i= i δk k 8 k is th io lctro collisio oprator i = Λ m is th lctro io collisio frqucy f () (5) f iω f f f t L r p i pf [ ( ) ] f ω p L = J[ f f] J[ f f] Ji[ f] (9) f f f J f J f f t r p i i i ii i i = is th arag io charg i is th io umbr dsity is th lctro dsity i is th io charg m is th lctro mass Λ is th Coulomb logarithm = m ad is th lctro tmpratur. h sum i is tak or all io spcis i th plasma. h lctro lctro collisio itgral is tak i Ladau form (6) i = pf [ ( ) ] pf ω L J f f c.c.. () h to rlat f with f w assum that th lasr frqucy is high ough so f ca b padd i sris of LL Riw Volum 98 55

3 FFC OF LCRIC FILDS ON LCRON HRMAL RANSPOR IN LASR-PRODUCD PLASMAS () ω L : f = f f ( whr f ) ( f ) ~ i ωl <<. Substitutig th lattr pasio ito q. (9) gis () i f = p f () ω L { } f = t f J f ω r p i p. () L is th mass locity ρ = m i m i i m i is th mass dsity m i is th io mass i is th io locity ad f i is th io distributio fuctio. Wh k = th dscribd procdur yilds th mass cosratio quatio; k = ad k = gi th momtum ad rgy cosratio quatios rspctily. Omittig lgthy algbraic maipulatios w rport th fial rsult: 8 t ( ) ( V)= (6) o limiat f from q. () for th low-frqucy compot of th distributio fuctio w substitut qs. () ad () ito q. (). h rsult taks th form 8 f f f J f J f f t r p i ρ ρ ( ) = ( p p ) σ ρ t k r k rk i r k k r k r k c.c. mω L { } (7) = ω f f p L m pip f ( i c.c. ) t r m rip ( ) c.c. t r i f p p p J p f J p f p f i i i. () quatio () is sold assumig a small diatio of th lctro distributio fuctio f from Mawllia f M : f fm ψ p t () = whr ψ <<. h kitic quatio for ψ is obtaid by substitutig th pasio () ito q. () ad rplacig th tim driatis t f M usig th trasport cosratio quatios. hs quatios accordig to th stadard procdur ar obtaid by multiplyig th kitic quatios by ( ) k with k =... ad itgratig th lattr i th locity spac. Hr = ρ dmf d m f i i i i i (5) p t ( r ) ( V) q 6mω L m m = i V i ( i ). (8) m Hr w us th stadard dfiitios σ i = df V= d f = (9) m q= d ( ) ( ) f () = m d f p δ () k k k whr i is th io tmpratur is th lctro dsity is th currt dsity q is th hat flu σ k is th strss tsor p ad p i ar th lctro ad io prssurs = mω L ad ρ = i i is th charg dsity. o simplify th driatio of th trasport cofficits w assum = ad glct trms of th ordr of mm i. Nt th quatio for th corrctio ψ to th Mawllia distributio fuctio is drid by substitutig q. () ito q. () ad usig th cosratio quatios (6) (8). h rsultig quatio taks th form 8 56 LL Riw Volum 98

4 FFC OF LCRIC FILDS ON LCRON HRMAL RANSPOR IN LASR-PRODUCD PLASMAS ψ ψ δj ψ δj ψ t r i ad = i q 5 ( V) 5 l l k ( ) k i 8 r k d ( ) whr kl r 8 l ( ) k = ( ) δ J i δj k () [ ψ]= i k δ k k ψ () 8 ψ i = f d f f k M M M { k k k k ω Lm δ. (6) Nt w sol q. () assumig that th lctro quir locity is much smallr tha th lctro thrmal locity << ad ordrig ~ i. h fuctio ψ is padd as ψ = ψ ψ whr ψ << ψ. h first approimatio ψ is obtaid by kpig oly th trms of th ordr of ( i ). h scod-ordr corrctio ψ is drid by rtaiig th first driati of th lctric fild ad th scod driati of th lctro tmpratur ad dsity. First-Ordr Approimatio Rtaiig th first spatial driatis i tmpratur ad dsity ad also trms proportioal to q. () yilds δji ψ δj ψ i i 5 i i l l 8 = d. (7) ( ) δ ( ) k k ( ) δ k [ ] ψ( ) ψ () k k ( ) δ δ δ kl 5 k l kl k l l k (5) W look for a solutio of q. (7) i th form ψ Φ i i Φ = Φ Φ l l. (8) i Usig dfiitios (9) () th currt dsity hat flu ad strss tsor i th first approimatio bcom i LL Riw Volum 98 57

5 FFC OF LCRIC FILDS ON LCRON HRMAL RANSPOR IN LASR-PRODUCD PLASMAS () = λ l l (9) q () = λ q l q l () σ k σ m () 5 k () = whr λ = i is th lctro ma-fr path. h umrical cofficits i qs. (9) () ha th forms ( ) = d Φ () 5 ( q ) = d Φ () q σ 5 = d Φ. () quatios (9) () show that th lctric currt ad th hat flu i th first approimatio ar proportioal to th gradits i tmpratur ad prssur. h strss tsor o th othr had dpds o th lasr lctric fild. 7 though th fuctios Φ ad Φ do ot tr ito th first-ordr hat flu thy cotribut to th hat flu i th scod approimatio. hus w d to fid all four fuctios Φ. h gral form of th solutio ψ [q. (8)] ca b sparatd o th followig thr typs of fuctios: typ I dpds oly o () I th locity modulus ψ = Φ( ); typ II is proportioal to th locity ctor ad locity modulus ψ ( II ) = A Φ ; ad typ III dpds o th locity tsor ad locity modulus ψ ( III ) = ( ) ( ) i i Φ whr A i is th ctor proportioal to th tmpratur prssur gradits or th lctric fild. Accordig to such a classificatio th gorig quatios for th fuctios of ach typ bcom yp III: ( ) Φ δji i i = δj Φ i φ (7) i i whr φ() is dfid by th right-had sid of q. (7). Sic th io lctro collisio oprator has a ry simpl form it is straightforward to calculat Ji[ ] ad Ji[ ( ) i ] usig q. (): δ J δ J i [ ]= i (8) i 9 i i = i. (9) h lctro lctro oprator is mor complicatd ad th aluatio of δj [ Φ( )] δj[ Φ( ) ] ad δj[ ( ) i Φ] rquirs lgthy algbra. Blow is a dtaild calculatio of δj [ Φ( )]. h itgral part i th lctro lctro collisio oprator ca b rwritt as ( ) ( ) k k d fm( ) δ Φ Φ ( ) k = ( ) () whr fuctio Σ() is foud by multiplyig q. () by k. his yilds d f M Φ Φ ( ) [ Φ]= (5) yp I: δj φ { [ ]}= yp II: A Φ δ Ji δ J Φ φ A (6) dy y y = () 58 LL Riw Volum 98

6 FFC OF LCRIC FILDS ON LCRON HRMAL RANSPOR IN LASR-PRODUCD PLASMAS whr y = cos θ ad θ is th agl btw ad. Itgratio or th agls gis y dy y = [ ( ) ] if < =. () ( ) if > Substitutig q. () ito q. () yilds = ( ) = d Φ Φ ( ) () d Φ Φ ( ). () whr fuctio φ is rlatd to φ by itgratig q. (6) φ = d φ( ). (8) o sol q. (7) w tak th driati of both sids of q. (7). his gis d φ = γ Φ Φ ( ) (9) i whr γ( )= d is th icomplt gamma fuctio. Itroducig a w fuctio g = d ( ) Φ q. (9) bcoms C g = d ( ) ( φ (5) i γ ) γ ( ) whr C is th itgratio costat. hus th fuctio Φ() ca b prssd as a multipl itgral of φ: Φ = C C d d g. (5) hus th lctro lctro collisio itgral rducs to δj i [ Φ]= f f k k M. (5) M h t stp is to substitut q. (5) ito q. (5) ad sol th lattr for Φ. o simplify th itgratio th right-had sid of q. (5) ca b rwritt i th form φ= φ f k f k M. (6) M h itgratig q. (5) oc th followig itgro-diffrtial quatio is obtaid: φ = i ( ) (7) Nt w rport th quatios corrspodig to th fuctio of th scod ad third typs [qs. (6) ad (7) rspctily]. quatio (6) rducs to φ i Φ i γ Φ = ( ) γ γ Φ Φ d Φ 5 ( ) 5 d Φ. (5) LL Riw Volum 98 59

7 FFC OF LCRIC FILDS ON LCRON HRMAL RANSPOR IN LASR-PRODUCD PLASMAS quatio (7) for th fuctio of th third typ bcoms φ i = Φ iφ 5 γ γ Φ ( ) γ Φ 5 ( d Φ ) d Φ. (5) h hat trasport cofficits i th first approimatio dpd o fuctios Φ ad Φ which blog to th fuctio of th scod typ ad ca b foud by solig q. (5) with φ 5 = for Φ = Φ i (5) φ= Φ= Φ. (55) for i o sol th itgro-diffrtal quatio (5) fuctio Φ() is traditioally padd i Lagurr polyomials Φ= A( ) L. As proposd i Rf. 9 it is mor coit to us a mor-gralizd pasio i trms of Lagurr polyomials L ( ). h choic of ths polyomials coms from thir orthogoal proprtis L L d m if m = Γ ( )! if m= > =.... (56) aluatio of th itgrals i qs. () ad () bcoms particularly simpl if = β β Φ ( ) A L. (57) Id β is dtrmid by matchig th polyomial pasio (57) with th act solutio of Φ i th limit of. Calculatios show that such matchig spds up th corgc of th trasport cofficits with th umbr of polyomials i pasio (57). akig th limit i q. (5) yilds Φ = φ. (58) i h th choic β = / k with k = will satisfy th rquirmt of matchig q. (57) with th act solutio (58). h paramtr k is dtrmid by miimizig th umbr of trms i th polyomial pasio () to match th act solutio for ad () to rach th dsird accuracy of th trasport cofficits for ~. Calculatios show that for th cas of fuctios Φ ad Φ β = / satisfis such a miimizatio critria [it taks fi trms i q. (57) to obtai th trasport cofficits with % accuracy]. hrfor th pasio bcoms = Φ ( ) A ( ) L. (59) Multiplyig q. (5) by L s with s = N [whr N is th umbr of polyomials i th pasio (57)] ad itgratig th lattr i from util w obtai th systm of N algbraic quatios. Figur 98. shows a dpdc of th cofficits ad q o th umbr of polyomials i th pasio (57) with β = / β = / ad β = / rspctily. Obsr that th cofficits corg fastr with β = /. Nt w dri th umrical cofficit σ of th strss () tsor σ i. his rquirs that q. (5) b sold with Φ() = Φ ad φ = i γ γ 5. (6) 8 6 LL Riw Volum 98

8 FFC OF LCRIC FILDS ON LCRON HRMAL RANSPOR IN LASR-PRODUCD PLASMAS (a) 7. (b). b = / 7.5 a. / a q / C655 N N Figur 98. Cofficits ad q as fuctios of th umbr of polyomials i th pasio (57). h rsults corrspod to β = / (dashd li) β = / (solid li) ad β = / (dots). Similar to th priously cosidrd cas w pad fuctio Φ i Lagurr polyomials. o prss σ through ust o cofficit i such a pasio w tak β 5 Φ β = B( ) L. (6) h choic of th powr id β coms from th coditio of matchig pasio (6) with th act solutio i th limit of. Nglctig trms proportioal to i qs. (5) ad (6) gis = Φ 8. (6) It is asy to s that th alus β = satisfy our rquirmt. Calculatios show that pasio (6) with β = has th fastst corgc with th umbr of polyomials. abl 98.II shows a summary of cofficits σ for a diffrt io charg. Obsr that th strss tsor has a ry wak dpdc o (% ariatio i σ from = to = ). O mor fuctio rmais to b dtrmid i th first approimatio: th corrctio Φ to th symmtric part of th distributio fuctio. his fuctio blogs to th first typ ad ca b foud i th itgral form usig q. (5) with φ= i (6) abl 98.II: rasport cofficits i th first approimatio q q σ LL Riw Volum 98 6

9 FFC OF LCRIC FILDS ON LCRON HRMAL RANSPOR IN LASR-PRODUCD PLASMAS ad Φ = Φ. h itgratio gis 8 = C C Φ γ ( t) dt t ( t) (6) whr C ad C ar dtrmid from th coditio of zro cotributio of Φ to th lctro dsity ad tmpratur d Φ d Φ =. (65) = Coditios (65) yild C =.7 ad C =.5. Not two misprits i Φ rportd i Rf. 8 [th diffrt sig i frot of th itgral ad ( t) istad of ( t) isid th itgral]. h corrctio to th symmtric part i th distributio fuctio coms maily from balacig th irs brmsstrahlug hatig i ( ) with th lctro lctro collisios δj. Sic δj ~ i fuctio Φ bcoms proportioal to th arag io charg as show i q. (6). As mphasizd i Rf. 8 th symmtric corrctio Φ gis th domiat cotributio to th hat flu i th scodordr approimatio. Scod-Ordr Approimatio Corrctio ψ to th distributio fuctio i th scod approimatio satisfis th followig quatio: 8 () () 5 q Φ l i Φ l Φ ( ) Φ Φ 6 ir k ik ( 8Φ ) t Φ i i l Φ i rir Φr i r l ( ) ik ( ) rk i Φ = δji ψ δj ψ 8. (66) A gral solutio of q. (66) ca b writt as ψ = Φ Φ i ( ) ( ) ik r k i r i Φ ( ) k Φ i Φ r l r k Φ5 t i i i i l δ rk r k l δ k l 6 Φ l Φ 7 l. i (67) h lctric currt ad th hat flu i th scod ordr tak th form i r k ik r i λ = λ ti (68) ik qi q rk q ri = λ q λ ti. (69) 6 LL Riw Volum 98

10 FFC OF LCRIC FILDS ON LCRON HRMAL RANSPOR IN LASR-PRODUCD PLASMAS Cofficits q ar calculatd usig th followig rlatios: ( ) = d 5 Φ ( ) (7) 5 ( ) = d 5 Φ ( ). (7) q thus β taks th alus β = / k with k =. h fastst corgc of th cofficits ad q is obtaid with β = /. A summary of ad q for diffrt io charg is gi i abl 98.III. Nt w fid th fuctio Φ. h act matchig of th polyomial pasio (57) with th act solutio Φ for Φ = Φ (76) Nt w fid fuctios Φ Φ ad Φ 5. hs fuctios ar of th scod typ; thrfor to obtai thm w sol q. (5) with φ = ( ) 8 Φ for Φ= Φ i (7) φ = Φ ( Φ Φ) for Φ = Φ i (7) 6 φ = Φ for Φ = Φ5 i. (7) Followig th mthod dscribd i th prious sctio fuctios Φ Φ ad Φ 5 ar padd i sris (57). h act solutio for Φ as bcoms Φ = 5 ; (75) caot b do sic Φ dos ot ha a polyomial structur [s q. (6)]. It is asy to show howr that Φ ( )~ ad Φ 5 ( )~. hrfor th pasio of Φ with β = rproducs th asymptotic limits for << ad >>. akig β = ad kpig N = 5 trms i pasio (57) gis alus of ad q which ar rportd i abl 98.III. Obsr that ths cofficits bcom quit larg for >>. o fid th rmaiig cofficits i th hat flu ad lctric currt w sol th quatio for th fuctio Φ 5. I th limit of th fuctio Φ 5 bcoms Φ 5 6 = Φ = 9 ; (77) thus β = matchs th polyomial pasio (57) with th act solutio i th limit of. Calculatios show that β = rquirs a miimum umbr of polyomials i pasio (57) to achi th dsird accuracy. h alus of ad q ar summarizd i abl 98.III. Nt w abl 98.III: rasport cofficits i th scod approimatio q q q LL Riw Volum 98 6

11 FFC OF LCRIC FILDS ON LCRON HRMAL RANSPOR IN LASR-PRODUCD PLASMAS combi th lctric currt ad th thrmal flu i th first ad scod approimatios. h rsult is q q β = i r i r = λ l i l i ad q q β = r k ik r i λ t i (78) qi = λq r q i r l i l i q rk ik q r i q λ t i. (79) ca b rprstd with th followig fittig formulas: 8. β =. β. = (8) 9. β = 5.. (8) 7. Imposig a coditio of zro currt = ad also assumig t i << (whr t is th tim scal of ariatio) dfi th slowly aryig compot of th lctric fild I additio th cofficits i th lctric fild ca b fittd as follows: i r k = r ik i r i l l = r i. (8) =. 6.. (8) Substitutig from q. (8) ito q. (79) gis th hat flu i lasr-producd plasmas 85. =.. 8. ri q r k i = ik λ β r β l i β (8) whr β = q q Cofficits β ad agr with priously publishd rsults. Nt w discuss th alidity of th drid trasport cofficits. As show arlir th mai cotributio to th scod-ordr hat flu coms from th corrctio Φ to th symmtric part of th distributio fuctio. h fuctio Φ is gi i th itgral form by q. (6) ad has th followig asymptotic bhaior for small ad larg locitis: 6 LL Riw Volum 98

12 FFC OF LCRIC FILDS ON LCRON HRMAL RANSPOR IN LASR-PRODUCD PLASMAS Φ ( )= (85) it f f = M p Φ (9) = 5 5 Φ. (86) h alidity coditio of th Chapma skog mthod Φ << braks dow for is a good approimatio to th symmtric part of th distributio fuctio for. For such a purpos w fid th asymptotic bhaior of th fuctio that satisfis th followig quatio: t f J f f. (9) = ad < 6 (87) W look for a solutio of q. (9) i th form f = A Ψ whr Ψ = F g( ) ad A is a ormalizatio costat. h tmpratur dpdc is combid i fuctio F ad locity dpdc is i g; th th tim driati of f bcoms 5 >. 5 (88) Accordig to q. (7) th mai cotributio to th hat flu coms from th suprthrmal lctros [which corrspod to th maimum i th fuctio 5/ Φ()]. hrfor th limit (87) imposs o rstrictios o th applicability of th drid rsults. h lctro distributio fuctio for th subthrmal lctros rthlss is diffrt from th limit (85). As drid i Rfs. ad 5 th irs brmsstrahlug hatig modifis th distributio of th cold lctros to u du f( << )= p ( ) (89) u V L whr VL 8 is th Lagdo locity. 6 o chck th limitatios du to th scod coditio (88) w fid that th maimum of = 5 5 Φ ~ Φ ~ corrspods to ma /. his limits th applicability of th Chapma mthod to <.. though th modulus of Φ bcoms largr tha uity for larg [s q. (88)] w ca show that t t f = ff g = ffg = i (9) whr w substitutd t i du to th irs brmsstrahlug hatig. h lctro lctro collisio itgral rducs i this cas to J 6 Λ F f I (9) m = I( )= d f ( ) g g d f g g. (9) I th limit of I bcoms I g d f g ad q. (9) taks th form = F g g Fg = 5 F 8. (95) LL Riw Volum 98 65

13 FFC OF LCRIC FILDS ON LCRON HRMAL RANSPOR IN LASR-PRODUCD PLASMAS Nt w mak a assumptio g << g which will b rifid a postriori. I this cas th solutio of q. (95) bcoms 5 F g = 5 5 F. (96) Obsr that th coditio g << g is satisfid i th limit of larg locity. h fuctio g by dfiitio dos ot dpd o tmpratur; this yilds for F F = C( ) F = F 5 C 7 (97) whr C ad C ar costats. h distributio fuctio f dpds o th product F( ) g( ) which accordig to qs. (96) ad (97) taks th form 7 F g = (98) Usig q. (98) th asymptotic limit of th symmtrical part of th distributio fuctio rducs to f 5 ( >> ) ~ p. 7. (99) it h lattr quatio must b compard to f i th limit [s q. (9)] it f 5 ( >> ) p ~. 5. () hus w ca coclud that th fuctio i th form (9) is a good approimatio to th distributio fuctio for thrmal ad suprthrmal lctros. ACKNOWLDGMN his work was supportd by th U.S. Dpartmt of rgy Offic (DO) of Irtial Cofimt Fusio udr Cooprati Agrmt No. D-FC- 9SF96 th Uirsity of Rochstr ad th Nw York Stat rgy Rsarch ad Dlopmt Authority. h support of DO dos ot costitut a dorsmt by DO of th iws prssd i this articl. RFRNCS. L. Spitzr Jr. ad R. Härm Phys. R (95).. R. C. Malo R. L. McCrory ad R. L. Mors Phys. R. Ltt. 7 (975).. I. B. Brsti C.. Ma ad J. J. homso Phys. Fluids 95 (978).. P. Mora ad R. Pllat Phys. Fluids 8 (979). 5. I. P. Shkarofsky Phys. Fluids 5 (98). 6. V. L. Gizburg Propagatio of lctromagtic Was i Plasmas ditd by W. L. Sadowski ad D. M. Gallik (Gordo ad Brach Nw York 96). 7. V. I. Prl ad Ya. M. Piskii So. Phys.-JP 7 (968). 8. A. V. Masimo V. P. Sili ad M. V. Chgoto So. J. Plasma Phys. 6 (99). 9. V. N. Gocharo ad V. P. Sili Plasma Phys. Rp. 8 (995).. S. Chapma ad. G. Cowlig h Mathmatical hory of No- Uiform Gass; A Accout of th Kitic hory of Viscosity hrmal Coductio ad Diffusio i Gass rd. d. (Cambridg Uirsity Prss Cambridg glad 97).. S. I. Bragiskii i Riws of Plasma Physics ditd by Acad. M. A. Lotoich (Cosultats Burau Nw York 965) Vol... N. N. Lbd ad R. A. Silrma Spcial Fuctios ad hir Applicatios r. glish d. (Dor Publicatios Nw York 97)... M. pprli ad M. G. Hais Phys. Fluids 9 9 (986).. A. V. Maksimo t al. JP 86 7 (998). 5. V. P. Sili Phys.-Usp (). 6. A. B. Lagdo Phys. R. Ltt. 575 (98). I coclusio w ha drid th trasport cofficits icludig th thrmal ad podromoti trms for a arbitrary io charg. h modificatio of th thrmal trasport du to th podromoti ffcts ar th critical surfac ad lasr turig poit will b discussd i a forthcomig publicatio. 66 LL Riw Volum 98