Relativistic correction of (v/c) 2 to the collective Thomson scattering for high-temperature high-density plasma

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1 Chin. Phys. B Vol., No Rlativistic corrction of v/c to th collctiv Thomson scattring for high-tmpratur high-dnsity plasma Jiang Chn-Fan-Fu, Zhng Jian, and Zhao Bin CAS Ky Laboratory of Basic Plasma Physics, Dpartmnt of Modrn Physics, Univrsity of Scinc and Tchnology of China, Hfi 36, China Rcivd 9 Dcmbr ; rvisd manuscript rcivd 9 March Collctiv Thomson scattring is thortically invstigatd with th inclusion of th rlativistic corrction of v/c. Th corrction is rathr small for th plasma paramtrs infrrd from th spctra of th thrmal lctron plasma wavs in th plasma. Sinc th full formula of th corrctd rsult is rathr complicatd, a simplifid on is drivd for practical us, which is shown to b in good agrmnt with th un-simplifid on. Kywords: Thomson scattring, rlativistic ffct PACS: 5.5.Os, 5.5.Gj, 5.7.Kz DOI:.88/674-56//9/95. Introduction Thomson scattring,, which can provid highly rliabl tim- and spac-rsolvd masurmnts of plasma paramtrs, including lctron dnsity, tmpratur and plasma flow, is now widly utilizd to masur th lasr-producd plasma rlvant to inrtial confinmnt fusion ICF. 3 Thomson scattring is usually opratd in th collctiv rgim in th fild of ICF. In most of thos xprimnts, only th ion-acoustic faturs of th Thomson scattring spctra ar dtctd and th following quation is usually adoptd to infr plasma paramtrs from xprimntal data,, d P dω s dω = r I V s n n s Sk, ω, whr P is th powr of th scattrd light, r is th classical lctron radius, I is th intnsity of th prob light, V s is th scattring volum, n is th lctron dnsity, is th polarization of th prob light, n s is th scattring dirction and Sk, ω is th so-calld dynamic form factor. Th ω and k ar th diffrntial frquncy and th wav vctor, rspctivly, givn by ω = ω s ω, k = k s k, a b whr ω,s is th frquncy of th prob/scattring light and k,s is th wav vctor of th prob/scattring light. Th wav numbr of th lctromagntic wav dpnds on th plasma dnsity, k,s = ω,s /c ωp/ω,s / whr ω p is th Langmuir frquncy of th plasma. Whn th plasma dnsity is dilut and th diffrntial frquncy is small, th diffrntial wav vctor is usually approximatd by, k = ω c n s k. 3 In prvious xprimnts, it is Eq. 3 that is usd to calculat th diffrntial wav vctor instad of th mor accurat Eq. b. For collisionlss nonrlativistic plasma in th quasi-quilibrium stat, th dynamic form factor is givn by Sk, ω = χ n ɛ l F + Z χ ɛ l F i, 4 whr ɛ l is th non-rlativistic longitudinal prmittivity of th plasma, χ is th non-rlativistic lctron suscptibility, Z is th charg stat of th ions and F,i ar givn by F,i = f,i pδω k vd 3 p. = n,i πkv,i xp ω k v,i, 5 Projct supportd by th National Natural Scinc Foundation of China Grant Nos and 5 and th Innovativ Projct of Chins Acadmy of Scincs Grant No. KJCX-YW-N36. Corrsponding author. jzhng@ustc.du.cn c Chins Physical Socity and IOP Publishing Ltd

2 Chin. Phys. B Vol., No with f,i bing th Maxwllian momntum distribution function of th lctrons or th ions, δx th δ function, v,i = T,i /m,i th lctron/ion thrmal spd, T,i th lctron/ion tmpratur and m,i th lctron/ion mass. Th thoris of Thomson scattring ar thus dvlopd to includ various ffcts that may hav important ffcts on th dynamic form factor Sk, ω, such as th lctron positron plasma, th supr- Gaussian lctron vlocity distribution du to th strong invrs brmsstrahlung absorption, 3 4 th frqunt Coulomb collisions 5,6 and th plasma inhomognity. 7,8 Th scattring spctra from th thrmal lctron plasma wavs, which allow th dirct masurmnt of th lctron dnsity and th tmpratur, wr also succssfully dtctd by Glnzr t al. 6 Noticing that th plasma tmpratur was rathr high in that xprimnt and that th diffrntial frquncy ω was no longr ngligibl in comparison with th prob frquncy ω, w rvisitd th first ordr corrction of v/c whr v and c ar th lctron and th light spd, rspctivly to th scattring powr spctrum 9 and obtaind th sam rsult as that prsntd by Shffild, d P dω s dω = r I V s n n s Sk, ω + ω. 6 ω Th Sk, ω in Eq. 6 is th sam as th on in Eq. and th nw trm ω/ω coms from th first ordr corrction of v/c. Anothr corrction to Eq. is implid in th xact diffrntial wav vctor, k = ω s c n s k. 7 Equations 3 and 7 ar diffrnt from ach othr by a trm proportional to ω/ω. With th inclusion of th first ordr of v/c, w find that not only th intnsity but also th spctral profil of th scattring light of th lctron plasma wavs changs significantly. 9 With Eq. 6, th infrrd lctron tmpratur from th scattring spctra in Rf. 6 is about 3% highr than that obtaind with Eq.. 9 With th inclusion of th dpndnc of wav numbr on th plasma dnsity, th diffrntial wav vctor bcoms a littl smallr, lading to a smallr wavlngth of th rsonanc pak of th scattring spctrum. 9 As indicatd in Eq. 6, th corrction trm of ω/ω can bcom of ordr whn th light scattring from th lctron plasma wavs is dtctd. This fact mans that th first ordr corrction may not b nough for th fitting of th Thomson scattring spctra obtaind from th lctron plasma wavs in high-tmpratur high-dnsity plasma. Howvr, th lctron tmpratur in th coronal rgion of th lasr-producd plasma may rach 5 kv and th plasma bcoms mildly rlativistic in th following ignition xprimnt. Thrfor, a thory of collctiv Thomson scattring including th rlativistic corrction, i.., v/c, may b ncssary for such hightmpratur plasma. In fact, rlativistic corrctions to Thomson scattring wr addrssd many yars ago. 6 Howvr, ths prvious studis concntratd on th rlativistic ffct for incohrnt Thomson scattring. In this papr, w calculat th powr spctrum of collctiv Thomson scattring with th corrction up to v/c. Whn th ratio of th lctron tmpratur to th lctron rst nrgy is ovr a fw prcnt or whn th phas vlocity of th dtctd fluctuation is largr than.c, th rlativistic ffct should b includd in th thory of Thomson scattring. Th lctron tmpratur obtaind with th corrctd thory is a littl highr than that obtaind with th thory only including trms of v/c ordr. Sinc th full formula for th corrctd spctrum is rathr lngthy and complicatd, a simplifid on is drivd for routin applications. Th rsults may mak collctiv Thomson scattring a mor accurat mthod for diagnosing plasma paramtrs in lasr-producd high-tmpratur plasma.. Basic quations In th wav zon, th spctral nrgy mittd from N acclratd lctrons into th solid angl dω is approximatly givn by 9 d E ns,ω s = N 8π c 3 i= iωsτ iωs/cns riτ dτ d dτ v i τ n s n s v i τ/c, 8 whr is th lctron charg, v i and r i ar th vlocity and th coordinat of th i-th lctron, rspctivly, R is th obsrvation coordinat, τ is th rtardation tim dfind as t = τ + R/c n s r i τ/c and ω s dnots th frquncy of th mittd lctromagntic wav. Th incidnt wav is assumd to b plan, monochromatic and linarly polarizd, E = E cosk r ω t, 9 95-

3 Chin. Phys. B Vol., No B = n E cosk r ω t, whr ω and k ar th frquncy and th wav vctor of th incidnt wav, rspctivly, and n is th propagation dirction of th incidnt wav. Hr w do not discuss th ffct of th plasma polarization on th lctromagntic filds, which has bn justifid in our prvious articl. 9 Undr th action of th incidnt wav, th acclration of an lctron is givn by vτ = E m v c + c v n c v v cosk rτ ω τ + Ov/c 3. Hr w nglct thos trms of ordrs highr than v/c. For simplicity, w assum that is prpndicular to scattring dirction n s in th following calculations, i.., n s =. This condition can asily b achivd by a suitabl xprimntal stup. Substituting Eq. into Eq. 8 and prforming th nsmbl avrag, w can obtain th spctral powr of th Thomson scattring via th procdur prsntd in our prvious articl, 9 th only diffrnc bing that th Klimontovich distribution function in this papr is dfind as F r, p, τ = N δr r i τδp p i τ i= instad of F r, v, τ in Rf. 9. Hr p dnots th momntum. Th spctral powr is hnc givn by d P ns,ω s = I r πn V s n d 3 pd 3 p + c Ξ + c Ξ δf rl k,ω,p,p. 3 Hr Ξ and Ξ ar th first and th scond ordr corrctions, rspctivly, givn by Ξ = n s n v + v, 4 Ξ = A: vv + v v + B: vv, 5 whr vlocitis v and v ar functions of p and p, rspctivly, and A and B ar two tnsors dfind as A = I + n sn s n n n s, 6 B = n s n n s n + n n s, 7 with I bing th unit tnsor of rank two. Th δf rl k,ω,p,p in th intgrand of Eq. 3 is th spctral dnsity of th corrlation function of th Klimontovich lctron distribution functions, which has alrady bn obtaind. 7 It should b pointd out that with th inclusion of th rlativistic ffcts, δf rl k,ω,p,p is diffrnt from th non-rlativistic on δf non k,ω,p,p prsntd in Rf. 8, bcaus both fluctuating lctric and magntic filds hav an ffct on th particl fluctuation. Introducing th notations, δf rl k,ω,p,p πn d3 pd 3 p = Ak, ω, 8a vδf rl k,ω,p,p πn d3 pd 3 p = Bk, ω k k, 8b vv + v v δf non k,ω,p,p πn d3 pd 3 p = C k, ω kk k + D k, ω I kk k, 8c vv δf non k,ω,p,p πn d3 pd 3 p = C k, ω kk k + D k, ω I kk k, 8d Eq. 3 can now b writtn as whr d P ns,ω s = I r N s σ + σ + σ 3 + σ 4, 9 σ = Ak, ω σ = c n s n k Bk, ω, k b σ 3 = { k ns k n s n c k C D + n s n D C + D σ 4 = c { k ns n k C D a }, c + n s n 3 + n s n D }. d It should b pointd out that Eqs. 8a 8d ar valid only if th fluctuation in th plasma is statistically isotropic. 3. Rlativistic corrctions Th intgral Ak, ω in Eq. 8a is actually th auto-corrlation function of th lctron dnsity, 95-3

4 Chin. Phys. B Vol., No which ssntially has th sam form as that for th non-rlativistic plasma, Ak, ω S rl k, ω = n χ ɛ l F + Z χ ɛ l F i, whr ɛ l is th rlativistic longitudinal prmittivity of th plasma, χ is th rlativistic lctron suscptibility and F,i ar givn by F,i = f,i pδω k vd 3 p, with f,i bing th momntum distribution function of th lctrons or th ions. Th rlativistic dynamic forming factor Eq. is diffrnt from th nonrlativistic on givn in Eq. 4, as th rlativistic momntum distribution functions ar usd instad of th non-rlativistic ons. For rlativistic plasma, th momntum distribution function of th lctrons is givn by th Maxwll Jüttnr function, n α f p = 4πm 3 c 3 K α xp α + p /m c /, 3 whr α = T /m c and K x is th modifid Bssl function. With th rlativistic distribution function givn in Eq. 3, th function F can b writtn as is usually lss than.. Hnc th rlativistic corrction to th ral part of χ is small. Th ral part of χ can b approximatly writtn as s Appndix A R χ = R χ + α 5 k λ D 4 3 ξ ξ ξ 3 ξ4 ξ R Zξ, 5 whr λ D is th Dby lngth of th plasma, R χ = kλ D + ξ Rξ is th ral part of th nonrlativistic lctron suscptibility and R Zx is th ral part of th plasma disprsion function. Whn ξ, th asymptotic xpansion of R χ is givn by R χ = kλ D + 8α 3 4ξ 4 5 α ξ +. 6 With Eq. 6, w can obtain th rlativistic corrction to th disprsion rlation of th lctron plasma wav as ω = ωp 5 α + 3k v, 7 whr ω p = 4πn /m / is th Langmiur frquncy. Equation 7 is th sam as that obtaind by Buti. 3 As indicatd by Brgman and Eliasson, 9 Eq. 6 is accurat nough whn α.. F = n α / kv K α xp αξ + α + αξ α α αξ, 4 whr ξ = ω/ kv. Th function F bcoms zro whn αξ, as no particl can mov fastr than light. In Fig., w compar function F with th nonrlativistic function F givn in Eq. 5 in th cas of α =.. As sn in Fig., th two functions ar diffrnt from ach othr only whn ω/k bcoms closr to th spd of light. Th two curvs ar not sparabl whn ξ <. Th ral part of th full rlativistic lctron suscptibility χ is vry complicatd and no analytical form is availabl at prsnt. Dtaild discussion on this function can b found in rcnt articls on th linar wav disprsion rlation of th rlativistic plasma. 9,3 For lasr-producd plasma, paramtr α Fig.. Comparison btwn F and F in th cas with paramtr α =., i.., T = 5. kv. Th imaginary part of χ can b analytically calculatd and is givn by Im χ = α k λ D K α ξ αξ + α + αξ α xp α αξ

5 Chin. Phys. B Vol., No Whn αξ, i.., ω/k c, Im χ =. In this cas, th Landau damping of th lctron plasma wav disappars, as no particl can catch up with a wav with a phas vlocity fastr than light. Sinc th rlativistic corrctions ar proportional to v/c in th cas of v/c, th intgrals of C, k, ω and D, k, ω, which appar in th trms proportional to /c, can b calculatd with th nonrlativistic spctral dnsity δf non k,ω,p,p. Th calculations ar straight forward and similar to thos prsntd in th Appndix of Rf. 9. Sinc th first corrction trm is of ordr, th intgral Bk, ω in th first corrction trm should b calculatd with th rlativistic spctral dnsity δf rl k,ω,p,p. Its calculation is also similar to that prsntd in Rf. 9. Ths intgrals ar givn by B = ω k Srl k, ω, 9 C = ω k Sk, ω + ω p χ + χ k n ɛ l α /F α ɛ l + ɛ l α=,i F, 3 D = T m Sk, ω, 3 C = ω Sk, ω, 3 k D = T m F, 33 whr α is th charg of th α-lik particl and F α ar th non-rlativistic limits of functions F α dfind by Eq.. In Eq. b, th unit vctor k/k should b approximatd by k k = n n s n n s + ω n + n s. ω 34 Thn th scond trm in Eq. 9 can b approximatly writtn as c Bk, ωn s n k ω k = ω ω ω S rl k, ω. In Eqs. c and d, th unit vctor k/k can b approximatd by Thus w hav k k = n n s n n s. 35 k n s k n s n k = n s n, k n s n k = n s n. Finally, w obtain th spctral powr of th Thomson scattring with th corrctions of v/c, d P ns,ω s { = ri V s n S rl + ω S rl ω ω ω + c C + C D sin θ s C + D S rl + D sin θ s }, 36 whr θ s is th scattring angl givn by cos θ s = n s n. In Fig. a, w plot th thr spctral powrs rspctivly givn by Eqs., 6, and 36. Th paramtrs that w tak in th calculations ar th sam as thos usd by Glnzr t al, 6 i.., T = kv, n =. cm 3, λ =.566 µm, and θ s = 4. Equation 7 is usd in th computations of Eqs. 6 and 36, whil Eq. b is usd in th computation of Eq.. It is shown in Fig. a that th maximum intnsity givn by th corrctd formula Eq. 36 is highr than that givn by Eq. 6, and that th pak position prdictd by Eq. 36 is slightly blu shiftd in comparison with that prdictd by Eq. 6. Th physical rason is that both th damping rat and th frquncy of th lctron plasma wavs bcom a littl smallr whn th rlativistic ffct is includd. 9,3 As sn in Eq. 7, whn th rlativistic ffct is takn into account, th frquncy of th lctron plasma wav is a littl downshiftd through th factor 5α/, making th pak of th lctron plasma wav blushift as shown in Fig.. Th damping of th lctron plasma wav also bcoms lightr bcaus of th rlativistic ffct. As sn in Fig., th imaginary part of lctron suscptibility Im χ, which is proportional to F, is smallr with th inclusion of th rlativistic ffct, lading to slightly highr pak of th lctron plasma wav. Th profil of Eq. 36 is rathr clos to that of Eq. 6. This rsult mans that th rlativistic corrctions ar not significant for this xampl. In ordr to show th rlativistic ffct mor clarly, w plot in Fig. b th profils of th thr quations in th cas of T = 3 kv, n = 4 cm 3, λ =.566 µm and θ s = 4. In Fig. b, w can clarly s th diffrnc btwn th first and th scond ordr thoris. Th rlativistic ffct could bcom mor notabl whn th lctron tmpratur bcoms vn highr, say 5 kv, that may b achivabl at th National Ig- 95-5

6 Chin. Phys. B Vol., No nition Facility or whn th phas vlocity of th masurd lctron plasma wav is closr to th spd of light. and for th scond ordr fitting curv solid lin ar T = 4.48 kv and n = 4. cm 3. From ths rsults, w conclud that th first ordr corrctd thory can vry slightly undrstimat th lctron tmpratur. Fig.. Profils of th scattring spctral powr. Th paramtrs usd in calculations ar a T = kv, n =. cm 3, λ =.566 µm, and θ s = 4, and b T = 3 kv, n = 4 cm 3, λ =.566 µm, and θ s = 4. W pointd out in our prvious articl 9 that in comparison with th fitting rsults obtaind with th uncorrctd thory, th infrrd lctron tmpratur could b significantly undrstimatd and th infrrd lctron dnsity is narly unchangd whn th corrctions of v/c to Thomson scattring ar includd. W now invstigat th influnc of th rlativistic corrction on th infrnc of th plasma paramtrs. W still us th uncorrctd thory to gnrat an artificial scattring spctrum and fit it with Eqs. 6 and 36, rspctivly. On of th comparisons is shown in Fig. 3a, whr th plasma paramtrs for th dashd lin ar T = kv and n =. cm 3, whil thos for th first ordr fitting curv dottd lin ar T =.7 kv, n =.5 cm 3, and for th scond ordr fitting curv solid lin ar T =.75 kv and n =.5 cm 3. Anothr comparison is shown in Fig. 3b, whr th plasma paramtrs for th dashd lin ar T = 3 kv and n = 4 cm 3, whil thos for th first ordr fitting curv dottd lin ar T = 4.44 kv and n = 4.8 cm 3, Fig. 3. Fitting of th first and th scond ordr corrctd thoris to th uncorrctd on. Th plasma paramtrs usd to calculat th uncorrctd spctrum ar a T = kv, n =. cm 3, λ =.566 µm, and θ s = 4 ; and b T = 3 kv, n = 4 cm 3, λ =.566 µm, and θ s = 4. Equation 36 is rathr lngthy and complicatd. For routin applications, w can furthr simplify it. Noticing that th rlativistic corrctions ar important only in th high-frquncy part of th Thomson scattring spctrum, w can nglct th low-frquncy rspons of th lctrons that scrn th ion motion in th plasma, i.., w approximatly hav χ i and F i whn ω/kc s, whr c s is th ion-acoustic spd of th plasma. Thn, th normal dynamic form factor can b writtn as Sk, ω ɛ l n F. Also, C and D can b simplifid into C k, ω ω k ω p ω Sk, ω, D k, ω T m ɛ l Sk, ω. 95-6

7 Chin. Phys. B Vol., No With th abov approximations, Eq. 36 can b writtn as d P ns,ω s { = ri V s n S rl k, ω + ω S rl k, ω ω ω + ω p ω 4 sin θ s / + α 4 sin θ s + ɛ l sin θ s ω } Sk, ω. 37 Hr w hav rplacd ω/ω S rl k, ω with ω/ω Sk, ω, sinc ω/ω is small in th scond ordr corrction. Th trm proportional to α can b nglctd, bcaus α for th lasr-producd plasma with tmpratur lowr than 5 kv. Th trm proportional to ωp/ω can also b nglctd, bcaus ω ω p around th maximum of th collctiv scattring spctrum. Thrfor, w approximatly hav d P ns,ω s = r I V s n + ω S rl k, ω. 38 ω In Fig. 4, w plot th profils of th scattring spctral powr sparatly givn by Eqs. 36 and 38. Th paramtrs usd in th calculation ar T = kv, n =. cm 3, λ =.566 µm and θ s = 4. As sn in th figur, th diffrnc btwn th two curvs is rally vry small, indicating that Eq. 38 is a good approximation to Eq. 36. Th accuracy of th approximation bcoms wors whn th plasma dnsity bcoms highr, bcaus th diffrnc btwn ω/ω S rl k, ω and ω/ω Sk, ω incrass with th plasma dnsity incrasing. Fig. 4. Comparison btwn th unsimplifid rsult from Eq. 36 and th simplifid rsult from Eq. 38. Th paramtrs usd in th calculation ar T = kv, n =. cm 3, λ =.566 µm and θ s = Conclusion W rvisit th thory of collctiv Thomson scattring for high-tmpratur high-dnsity plasma. Th spctral powr of th Thomson scattring with th inclusion of th rlativistic corrctions of v/c is drivd with th aid of th fluctuation thory. It is found that with th inclusion of th rlativistic corrctions, th infrrd lctron tmpratur is vn highr than that obtaind with th thory with th inclusion of only th first ordr corrctions of v/c. Th full formula Eq. 36 for th scattring spctrum with th inclusion of th rlativistic corrctions is rathr lngthy and complicatd. For practical us, a simplifid formula Eq. 38 is givn. A comparison btwn th two quations shows that th lattr accords wll with th formr. Acknowldgmnt Th authors ar gratful to Prof. A. Brs of Massachustts Institut of Tchnology for usful discussions during th prparation of this papr. Appndix A: th drivation of Eq. 5 Th ral part of th rlativistic lctron suscptibility can b writtn as 3 R χ = k λ + ω D kc α 4K α Aftr intgrating by part, w hav R χ = { k λ + ω D kc 4K α xp α cosh x ln α cosh x tanh x ω/kc tanh x + ω/kc sinh x cosh xdx. A d ln dx } tanh x ω/kc tanh x + ω/kc cosh x dx. A 95-7

8 Chin. Phys. B Vol., No Introducing a nw variabl t = α cosh x, w can rwrit th scond trm of Eq. A as ω α kc K α xpα + αt + ln αt + α t / + αt ω/kc αt + α t / + αt + ω/kc { t ω/kc αt + α t ω/kc αt + α t / + αt } Du to th rapidly dcrasing factor xp t in th intgrand, th important contribution to th intgral coms from t <. Whn α, w can xpand th bracktd part in th intgrand in sris of α and obtain { R χ = αξ k λ + D K α xpα x x ξ + α x ξ x 4 x ξ + 3 x 4 } x ξ dx. Noticing that th ral part of th plasma disprsion function can b writtn as dt. w hav R χ = R χ + α k λ D R Zξ = ξ π x ξ x dx, ξ ξ ξ 3 ξ4 ξ R Zξ. A3 Rfrncs Evans D E and Katznstin J 969 Rp. Prog. Phys. 3 7 Shffild J 975 Plasma Scattring of Elctromagntic Radiation Nw York: Acadmic Prss 3 La Fontain B, Baldis H A, Villnuv D M, Dunn J, Enright G D, Kiffr J C, Pépin H, Rosn M D, Matthws D L and Maxon S 994 Phys. Plasmas 39 4 Glnzr S H, Back C A, Estabrook K G, Wallac R, Bakr K, MacGowan B J, Hamml B A, Cid R E and d Groot J S 996 Phys. Rv. Ltt Glnzr S H, Back C A, Sutr L J, Blain M A, Landn O L, Lindl J D, MacGowan B J, Ston G F, Turnr R E and Wild B H 997 Phys. Rv. Ltt Glnzr S H, Rozmus W, MacGowan B J, Estabrook K G, d Droot J S, Zimmrman G B, Baldis H A, Hart J A, L R W, Williams E A and Wilson B G 999 Phys. Rv. Ltt Bai B, Zhng J, Liu W D, Yu C X, Jiang X H, Yuan X D, Li W H and Zhng Z J Phys. Plasmas Wang Z B, Zhng J, Zhao B, Yu C X, Jiang X H, Li W H, Liu S Y, Ding Y K and Zhng Z J 5 Phys. Plasmas Yu Q Z, Zhang J, Li Y T, Lu X, Hawrliak J, Wark J, Chambrs D M, Wang Z B, Yu C X, Jiang X H, Li W H, Liu S Y and Zhng Z J 5 Phys. Rv. E Froula D H, Davis P, Divol L, Ross J S, Mzan N, Pric D, Glnzr S H and Roussaux C 5 Phys. Rv. Ltt Froula D H, Ross J S, Pollock B B, Davis P, Jams A N, Divol L, Edwards M J, Offnbrgr A A, Pric D, Town R P J, Tynan G R and Glnzr S H 7 Phys. Rv. Ltt Zhng J 6 Chin. Phys Zhng J, Yu C X and Zhng Z J 997 Phys. Plasmas Liu Z J, Zhng J and Yu C X Phys. Plasmas Myatt J F, Rozmus W, Bychnkov V Y and Tikhonchuk V T 998 Phys. Rv. E Zhng J, Yu C X and Zhng Z J 999 Phys. Plasmas Rozmus W, Glnzr S H, Estabrook K G, Baldis H A and MacGowan B J Astrophys. J. Suppl. Sr Wang Z B, Zhao B, Zhng J, Hu G Y, Liu W D, Yu C X, Jiang X H, Li W H, Liu S Y, Ding Y K and Zhng Z J 5 Acta Phys. Sin. 54 in Chins 9 Zhng J and Yu C X 9 Plasma Phys. Control. Fusion Lindl J 995 Phys. Plasmas 3933 Papprt R A 963 Phys. Fluids 6 45 Pchack R E and Trivlpic A W 967 Phys. Fluids Ward G and Pchack R E 97 Phys. Fluids 5 4 Kukushkin A B 98 Sov. J. Plasma Phys Chn S C and Marshall T C 984 Phys. Rv. Ltt Naito O, Yoshida H and Matoba T 993 Phys. Fluids B Klimontovich Y L 974 Kintic Equations for Nonidal Gas and Nonidal Plasma Oxford: Prgamon 8 Lifshitz E M and Pitavskii L P 98 Physical Kintics Oxford: Prgamon 9 Brgman J and Eliasson B Phys. Plasmas Brs A, Shkarofsky I P and Shourci M 9 Phys. Plasmas Buti B 96 Phys. Fluids