Derivation of Moment Equations for the Theoretical Description of Electrons in Nonthermal Plasmas

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1 Advancs Pur Mahmaics,,, 4-5 hp://dx.doi.org/.46/apm..49 Publishd Onl May (hp:// Drivaion of Momn Equaions for h Thorical Dscripion of Elcrons Nonhrmal Plasmas Markus M. Bckr, Dlf Loffhagn Libniz Insiu for Plasma Scc and Tchnology, Grifswald, Grmany markus.bckr@p-grifswald.d Rcivd Dcmbr, ; rvisd Fbruary 7, ; accpd March 9, Copyrigh Markus M. Bckr, Dlf Loffhagn. This is an opn accss aricl disribud undr h Craiv Commons Aribuion Licns, which prmis unrsricd us, disribuion, and rproducion any mdium, providd h origal work is proprly cid. ABSTRACT Th drivaion of momn quaions for h horical dscripion of lcrons is of rs for modllg of gas discharg plasmas and smiconducor dvics. Usually, cra arificial closur assumpions ar applid ordr o driv a closd sysm of momn quaions from h lcron Bolzmann quaion. Hr, a novl four-momn modl for h dscripion of lcrons nonhrmal plasmas is drivd by an xpansion of h lcron vlociy disribuion funcion Lgndr polynomials. Th proposd sysm of parial diffrnial quaions is consisnly closd by dfiion of ranspor coffics ha ar drmd by solvg h lcron Bolzmann quaion and ar hn usd h fluid calculaions as funcion of h man lcron nrgy. I is shown ha h four-momn modl can b simplifid o a nw drif-diffusion approximaion for lcrons wihou loss of accuracy, if h characrisic frquncy of h lcric fild alraion h discharg is small comparison wih h momnum dissipaion frquncy of h lcrons. Rsuls obad by h proposd fluid modls ar compard o hos of a convnional drif-diffusion approximaion as wll as o kic rsuls usg h xampl of low prssur argon plasmas. I is shown ha h rsuls providd by h nw approachs ar good agrmn wih kic rsuls and srongly improv h accuracy of fluid dscripions of gas dischargs. Kywords: Momn Equaions; Plasma Modllg; Elcron Transpor. Inroducion Nonhrmal plasmas ar widly usd many chnical applicaions cludg plasma display panls, nrgy savg lamps, dvics for microbial dconamaion and ozonizrs [-4]. Thy ar characrizd by low gas mpraurs Tg h rang from o K and comparaivly high man lcron nrgis bwn and V, whr V corrsponds o mpraur of 65 K. Compur simulaions of lcric gas dischargs producg nonhrmal plasmas ar usd sc many yars o g a dpr undrsandg of fundamnal procsss and o improv chnical dvics [5-]. In ordr o dscrib all phnomna akg plac h discharg mchanism, prcipl, a mahmaical modl comprisg h kic Bolzmann quaion [] fs r fs q rv,, v rv,, Er, v Br, vfsr, v, s ms fs col () for h disribuion funcion f s of ach gas spcis s wih charg q s and mass m s svn-dimnsional spac of r xyz,,, vlociy v vx, vy, vz and im, nd,nd has o b solvd combaion wih Maxwll s quaions for h lcric fild E and h magnic fild B. Th righ-hand sid () accouns symbolically for h chang h disribuion funcion du o collision procsss. Howvr, such sysm is no solvabl rasonabl compug im and svral simplifyg assumpions hav o b akn o accoun. For h nonhrmal plasmas undr considraion, magnic filds ar ngligibl, and, sad of h hol sysm of Maxwll s quaions, h Poisson quaion N s r, r, () q n s s s for h lcric ponial is solvd for drmaion of h lcric fild E, whr N s is h numbr of gas spcis wih dnsiis n s and dnos h Copyrigh SciRs.

2 44 M. M. BECKER, D. LOFFHAGEN prmiiviy of fr-spac. Furhrmor, havy paricls ar frqunly assumd o b hrmodynamic quilibrium and macroscopic fluid quaions wih consan mpraur Tg ar akn o accoun for racg h spaiomporal bhaviour of ions and nural paricls [-5]. In conras, h non-local kics of lcrons plays an imporan rol h discharg mchanisms and h applicaion rang of fluid modls which do no dscrib lcrons adqualy is vry limid [6-8]. Thrfor, hybrid modls ar frqunly usd which fluid quaions ar solvd for havy paricls and lcrons ar rad kically [9-]. Howvr, i has bn pod ou rcnly ha fluid modls ar abl o capur lcron kic ffcs, if h lcron nrgy flux is adqualy dscribd [,]. In h prsn papr, a high ordr fluid modl comprisg momn quaions for paricl dnsiy, paricl flux, nrgy dnsiy and nrgy flux of h lcron componn is consisnly drivd from h lcron Bolzmann quaion. In addiion a novl drif-diffusion approximaion for lcrons is proposd. Rsuls ar compard o hos of a convnional drif-diffusion modl frqunly usd [4,5] and o kically obad rsuls a h xampl of argon gas discharg plasmas.. Kic Dscripion of Elcrons In spi of h crasg spd of compurs, h soluion of h lcron Bolzmann quaion svn dimnsions is compuaionally no fasibl. A convnional approach for rducg compug im is o dcras dimnsionaliy by dcomposiion of h lcron vlociy disribuion funcion (vdf) f rms of sphrical harmonics vlociy spac [6,7]. In h planar sysm considrd h prsn sudis and dpicd Figur, whr all grads and h lcric fild ar assumd o b normal o h lcrods, h gnral sphrical harmonics xpansion rducs o h Lgndr polynomial xpansion [7,8] frv,, fx, v, vx v, () f lx, v, Pl cos. l In his cas h vlociy disribuion funcion bcoms symmric around h lcric fild and dpnds on h spac coorda x, h vlociy magniud v, h dircion cos cos vx v and im. Th subsiuion of h xpansion () o h lcron Bolzmann quaion frv,, vr frv,, (4) f Ev f r, v, m col Figur. Planar discharg gomry. wih lmnary charg and h ransformaion of h xpansion coffics o h spac of kic nrgy U m v accordg o f lxu,, π f l x, v,, (5) m fally yilds h fi sysm of parial diffrnial quaions [6] U U f x U f x U m x,,,, E x, U f xu,, f xu,, h m nh x, U Qh U f x, U, m U l Nr, h, r,, h h r U h Un x Q U f x U Nr,,,, n x U U h h r h r h r h r hr, hr, hr, hr, Q U U f x, U U U fl x, U, h h h, r l h r hr,, (6a) m l l U fl x, U, U fl x, U, l x l x l l E x, U fl xu,, fl xu,, l U l l E x, U fl xu,, fl xu,, l U N r l n x, Q U Q UUf x, U,, l (6b) l for h xpansion coffics fl, l,,. Hr, Qh and Qhr, ar h cross scions of lasic and lasic collisions of lcrons wih havy paricls wih dnsiy Copyrigh SciRs.

3 M. M. BECKER, D. LOFFHAGEN 45 n and mass and h m and h Nr dno h numbr of havy paricl spcis and racions, rspcivly. Th kic nrgy ha is los h corrspondg lasic lcron collision is dnod by U hr, and h paramr hr, dpnds on h diffrn kds of lasic lcron collision procsss. I is zro for dissociaiv aachmn of lcrons and on for xciaion, dissociaion and dxciaion procsss. Usg h assumpion ha h bdg nrgy is qually shard bwn h wo rlasd lcrons hr, quals wo for an ionizaion vn [6]. In ordr o solv sysm (), i has o b runcad afr a rasonabl fi numbr of quaions. Wih h common framwork of h wo-rm xpansion [9-], only h firs wo quaions for f and f ar akn o accoun and f l is s o zro for l. Usually, h rapidiy of h mporal chang of h anisoropic disribuion f is by som ordrs of magniud grar han ha of h isoropic disribuion f as long as h characrisic frquncy for h fild alraion is small compard o h powr dissipaion lasic and lasic collisions []. In his cas, h im drivaiv rm f (7) for l, which dscribs h sablishmn of f o h quasi-saionary sa f x, U, N r l nhx, Qh UQh, r U h r fx, U, fxue,, x, x 5 x fx, U, fxu,, fxu,,, U 5 U 5U (7) can b nglcd. If his cas f l is s o zro for l, h sysm (6) rducs o h sgl parabolic diffrnial Equaion (6a) usg h xprssion (7) wih f for h anisoropic conribuion f o h vdf []. In h pas, h sysm (6) has bn solvd wo-rm approximaion [9,] usg h xprssion (7) wih f as wll as mulirm approximaion considrg highr ordr conribuions o h vdf anisoropy [-5] o sudy h bhaviour of lcrons prscribd im-dpndn as wll as saionary lcric filds. Bu h coupld soluion (saionary or im-dpndn) of h kic Equaions (6) for lcrons, fluid quaions for havy paricls and Poisson s quaion for h lcric fild is sill an ambiious ask and has bn achivd for a fw discharg siuaions, only [,6-8]. In h followg a macroscopic sysm of momn quaions is consisnly drivd from h sysm (6), which srongly simplifis h dscripion of lcron ranspor.. Macroscopic Transpor Equaions for Elcrons.. Four-Momn Modl Th drivaion of a sysm of momn quaions for h dscripion of lcrons nonhrmal plasmas sars from h kic sysm (6). Muliplicaion of Equaion (6a) by facors m and U m, rspcivly, and subsqun graion ovr kic nrgy U dircly provids h wo momn quaions n x x S x x,,, wx, Qx, x E x x S x wih macroscopic quaniis,,, n x, U f x, U, du paricl dnsiy (8a) (8b) (9) w x, U f x, U, du nrgy dnsiy () x, UfxU,, du paricl flux m () m Q x, U f x, U, du nrgy flux.() Th sourc rms S and S Equaions (8a) and (8b) dscrib h ga and loss of paricls and nrgy du o collision procsss. For a spcific gas, hy ar givn as h sum of ras of all rlvan procsss wih ra coffics dpndg on h man lcron nrgy w n, s,.g., [5,6] for mor dails. In ordr o consisnly driv parial diffrnial quaions for h drmaion of h paricl flux () and h nrgy flux (), Equaion (6b) for l is mul- iplid by facors U m and U m, rspcivly. As bfor, subsqun graion ovr kic nrgy U yilds h wo momn quaions x, / w x, U f x, U, du x E x, n x, f x, U, du m m U m 5 U (a) Copyrigh SciRs.

4 46 M. M. BECKER, D. LOFFHAGEN Q x, xm 5 5 U f xu f xu,,,, 5 4 E x w x U f m 5m m /,, xu 5 U U f x, U, du wih h man fr pah of lcrons du,, du (b) N r l h(, ) h Qh, ru (4) h r Un x Q U. Th dfiion of h s of ranspor coffics 4 5m x, U f x U U,, d x, vx, 5 x, U f x,, d, m (5a) U U n x (5b) 4 5m x, 5 U f x U U Q,, d x, vx, ˆ 4, U U n x x U f x,, d, 5m U x, f m U x U U x,, d, (5c) (5d) (5) 5 U x, f m x U U Q x U,, d, (5f) wih h man vlociy of lcrons v n allows o wri h four-momn modl (4MM) for lcrons h form nx, x, Sx, (6a) x x, wx, x, x, vx, (6b) x m E x, n x, x, x, m wx, Qx, x E x,,, xs x (6c) Q x, x 5 E ˆ x w x x n x m,,. x, nx, x, Qx, v x,,,,, xqx (6d) Elcron ranspor coffics usd fluid calculaions ar commonly obad by solvg h kic sysm (6) mulirm approximaion [6] or worm approximaion [9] for givn valus of h lcric fild, nglcg spaial and mporal drivaivs. Th rsulg coffics ar hn pu o lookup abls as funcions of h man lcron nrgy for h usag fluid calculaions. Th sam procdur, known as localman-nrgy approximaion [6,4], is usd for h drmaion of h nw ranspor coffics (5)... Drif-Diffusion Approximaion As mniond Scion, h rapidiy of h mporal chang of h anisoropic disribuion f is by som ordrs of magniud grar han ha of h isoropic disribuion f and h im drivaiv Equaion (6b) for l can b nglcd many discharg siuaions. Wih his assumpion and dfiion of h coffic 4 (7) 5 x, U f x, U, d U n x, 5m h nw drif-diffusion approximaion (DDAn) x, ˆ x, x m E x, n x, m x, Q x, x, x, x, n x, x 5 ˆ m x, x, nx, x, x, Ex, n x, (8a) (8b) is obad for h paricl flux and nrgy flux Q of h lcrons. Subsiuion of (8a) and (8b) o Equaions (6a) and (6c), rspcivly, rducs h fourmomn modl o a sysm of wo parabolic diffrnial quaions for h paricl dnsiy n and h nrgy dnsiy w of lcrons. Th coffic (7) is drmd h sam way as h coffics (5). As an xampl, h ranspor coffics (5) usd Copyrigh SciRs.

5 M. M. BECKER, D. LOFFHAGEN 47 h four-momn modl (6) and h drif-diffusion approximaion (8) ar shown Figur for argon gas, whr h undrlyg cross scions ar daild rfrnc [5]. I bcoms obvious ha for man lcron nrgis V h disribuion anisoropy f bcoms imporan and should no b nglcd by usg h convnional wo-rm approximaion. Th drivd drif-diffusion approximaion (8) can b usd for dscripion of lcron ranspor sad of h convnional drif-diffusion approximaion DDAc x, Dx, nx, x b x, Ex, n x, Qx, D x, nx, x b x, E,,, x n x (9a) (9b) which is dducd from Equaion (7) wih [6] and wihou [8,4,4] considraion of h highr ordr conribuion f o h disribuion anisoropy. Th lcron diffusion coffics of paricl D and nrgy D ranspor as wll as h lcron mobiliis of paricls b and nrgy b ar drmd h sam way as h coffics of h four-momn modl (5) and (7) as funcions of h man lcron nrgy [6]... Convnional Thr-Momn Modl Th drivaion of h four-momn modl 4MM and h Figur. Argon ranspor coffics for lcron paricl flux drif-diffusion approximaion DDAn (a); nrgy flux drif-diffusion approximaion DDAn (b); paricl and nrgy fluxs four-momn modl 4MM (c); and dissipaion frquncis for paricl and nrgy fluxs (d). drif-diffusion modl DDAn is consisn h sns ha bsid h runcaion of h xpansion () no addiional assumpions ar ndd ordr o clos h sysm of macroscopic momn quaions. This is no h cas if momn quaions ar drivd dircly from h lcron Bolzmann Equaion (4) and no from h kic sysm (6), s,.g., [,4,4]. Th muliplicaion of Equaion (4) by facors, v k, k x, y, z and mv, rspcivly, and subsqun graion ovr vlociy spac yilds h sysm of hr momn quaions [44] nr, Γ r, Sr, (a) r, r, v r, p r,, k, k, k m n, E k,, k,, m r r r r (b) k x, y, z wr, Qr, n S r, Er, v r, r, (c) for h paricl dnsiy n, h paricl flux Γ and h nrgy dnsiy w of lcrons. In ordr o solv sysm () i has o b closd by cra xprssions for h lcron prssur nsor p r, m f r, v, v r, v r, d v () k, k, wih k x, y, z and h random lcron vlociy v v v as wll as for h lcron nrgy flux m Qr, f,, v d v. r v v () For ligh paricls such as lcrons, h prssur nsor p can b simplifid o h scalar lcron prssur [45,46] m pr,,,, d f r v v r v () m wr, nr, v r, and is hrfor drmd rms of h macroscopic quaniis n, w and v. Th drivaion of an adqua xprssion for h hird ordr momn Q rms of lowr ordr momns is a much mor difficul ask. Mos ofn, h lcron nrgy flux is rwr as [44] Q r, w r, p r, v r, q r, (4) wih h xac lcron ha flux m qr, f,,, v, d v, r v v r r (5) Copyrigh SciRs.

6 48 M. M. BECKER, D. LOFFHAGEN and hn h ha flux is approximad by Fourir ha conducion accordg o [,] 5 q r, D r, nr, r,. (6) Howvr, his approximaion is known o b accura mos discharg siuaions [,4]. A mor sophisicad ha flux ansaz has bn drivd by Robson al. []. Unforunaly, hir ha flux xprssion dpnds on paramrs which ar no known for ral gass and is hrfor no applicabl wihou furhr bnchmark calculaions []. 4. Comparison of Macroscopic and Kic Modls In ordr o show ha h drivd sysms of parial diffrnial quaions 4MM and DDAn improv h accuracy of fluid modls for h dscripion of lcrons, numrical calculaions for wo diffrn discharg siuaions argon wr prformd. Firs, h lcron ranspor quaions wr solvd for prscribd puls-lik lcric fild (bnchmark modl). Scondly, an abnormal glow discharg low prssur argon was dscribd slfconsisnly h sns ha h lcron ranspor quaions wr solvd oghr wih ranspor quaions for havy paricls and Poisson s quaion for drmaion of h lcric fild. Th fi-diffrnc mhods usd o discriz h sysm of diffrnial quaions spac and im ar daild rfrnc [5]. 4.. Argon Bnchmark Modl Prob masurmns plasmas caus an abrup chang of h local lcric fild. This siuaion is considrd hr, and h four-momn modl 4MM (6) as wll as h drif-diffusion modl DDAn usg h nw flux rprsnaion (8) and h drif-diffusion modl DDAc usg h convnional flux rprsnaion (9) wr solvd for argon gas a a prssur of Pa and a gas mpraur of K usg h prscribd lcric fild profil shown Figur (a). In ordr o ra h rsuls of 4MM, DDAn and DDAc, h spac-dpndn lcron Bolzmann quaion was solvd kically accordg o Signgr al. [4] for h sam lcric fild, akg o accoun lasic and lasic lcron collision procsss. Figurs (b) and (c) xhibi h rsuls obad for h man vlociy and h man nrgy of h lcrons by mans of h diffrn modls. Bcaus h applid fild is im-dpndn and hrfor h mporal drivaivs of all quaniis ar zro, rsuls of 4MM and DDAn ar almos h sam. Th spaial profil prdicd by h modls 4MM and DDAn for h man vlociy and h man nrgy ar qualiaiv agrmn wih h kic rsuls. In conras, Figur. Prscribd lcric fild (a) and comparison of rsuls for h man vlociy (b) and nrgy (c) of h lcrons obad by h four-momn modl 4MM and drif-diffusion modls DDAn and DDAc wih kic rsuls. h rsuls of DDAc srongly diffr from hos of h kic soluion. Th rsuls show imprssivly ha h accuracy of fluid modls for h horical dscripion of lcrons is srongly crasd by h proposd mhods. 4.. Abnormal Glow Discharg Argon To dmonsra h pracical applicabiliy of h drivd momn quaions, h igniion of an abnormal glow discharg argon a a gas prssur of Pa and a gas mpraur of K was horically dscribd usg h discharg gomry dpicd Figur. A h powrd lcrod a x (cahod) a volag of 5 W was applid and h lcrod a x = cm (anod) was groundd. Th gnral procdur for solvg h coupld sysm of ranspor quaions for h spcis and Poisson s quaion has bn dscribd [5] and h daa usd for h lcron-aom collisions ar h sam as hos rpord his papr. Th rsuls obad by h modls 4MM, DDAn and DDAc for h man lcron vlociy, h man lcron nrgy and h slf-consisnly drmd lcric fild ar shown Figur 4 a hr diffrn sans of im. Obviously, all fluid modls undr considraion prdic qualiaivly h sam dynamic bhaviour. Shorly afr swichg on h discharg, a = μs, a quasi-saionary Townsnd phas is rachd which is characrizd by an almos consan lcric fild and small spaial variaions h man vlociy and man nrgy of lcrons. Copyrigh SciRs.

7 M. M. BECKER, D. LOFFHAGEN 49 Figur 4. Comparison of rsuls obad by h fluid modls 4MM, DDAn and DDAc for an abnormal glow discharg argon a prssur of Pa and applid volag of 5 V. Charg carrirs ar producd maly fron of h groundd lcrod by ionizaion of argon aoms collisions wih lcrons and by scondary lcron mission a h powrd lcrod du o ion bombardmn. A μs nough charg carrirs ar producd o discly prurb h homognous lcric fild. Du o h cras of h lcric fild h cahod rgion srong charg carrir muliplicaion aks plac and fally h discharg ignis. Th discharg bcoms saionary afr approximaly μs. Elcrons mid a h cahod ga nrgy h srong lcric fild and ar hn slowd down lcron collisions. Th discharg is brighs h ngaiv glow rgion a approximaly.5 cm x.8 cm. Bcaus h characrisic frquncy for h fild alraion is small h discharg siuaion considrd hr, h rsuls of h four-momn modl 4MM and h nw drif-diffusion modl DDAn ar almos h sam. Small diffrncs occur fron of h boundaris du o h fac ha diffrn yps of boundary condiions hav o b applid for h sysm of firs-ordr diffrnial quaions 4MM and h parabolic sysm DDAn. Aga, h rsuls of DDAc diffr markdly. Paricularly h ransiion from h cahod rgion o h ngaiv glow a x. cm srong dviaions h rsuls for h man nrgy occur a sady sa. Th man nrgy mimum is srongly ovrsimad by DDAc and i has bn found ha his issu causs h occurrnc of a sgular po h mporal voluion of h discharg igniion if gas prssur, discharg chmisry or applid volag ar slighly changd. 5. Conclusions A nw sysm of momn quaions for h dscripion of lcrons nonhrmal plasmas was drivd by an xpansion of h lcron vlociy disribuion funcion Lgndr polynomials and h dfiion of ranspor coffics ha ar drmd by mans of h localman-nrgy approximaion. Th nw modl 4MM is consisn h sns ha no addiional assumpions ar ncssary o clos h sysm of momn quaions. I has bn shown ha h addiional rquirmn of a small characrisic frquncy for h fild alraion allows o rduc h sysm of four firs-ordr parial diffrnial quaions for paricl dnsiy, paricl flux, nrgy dnsiy and nrgy flux of lcrons o a parabolic drifdiffusion modl comprisg wo scond-ordr parial diffrnial quaions for h paricl dnsiy and nrgy Copyrigh SciRs.

8 5 M. M. BECKER, D. LOFFHAGEN dnsiy of lcrons. If his rquirmn is fulfilld, h rsuls providd by h nw drif-diffusion modl DDAn ar good agrmn wih hos of h high ordr fluid modl 4MM. Th comparison of rsuls obad by h modls 4MM and DDAn wih rsuls of h convnional drifdiffusion modl DDAc as wll as kically obad rsuls has pod ou ha h nw approach srongly crass h accuracy of fluid modls for h dscripion of lcron ranspor nonhrmal plasmas. Sc similar parial diffrnial quaions for lcrons aris h horical dscripion of smiconducors [47-49], i ponially improvs h horical dscripion of lcrons smiconducor dvics, oo. 6. Acknowldgmns This work was suppord by h Grman Rsarch Foundaion wih h Collaboraiv Rsarch Cnr Transrgio 4. REFERENCES [] B. Eliasson, M. Hirh and U. Koglschaz, Ozon Synhsis from Oxygn Dilcric Barrir Dischargs, Journal of Physics D: Applid Physics, Vol., No., 987, pp doi:.88/-77/// [] U. Koglschaz, Dilcric-Barrir Dischargs: Thir Hisory, Discharg Physics, and Indusrial Applicaions, Plasma Chmisry and Plasma Procssg, Vol., No.,, pp doi:./a:47985 [] G. Lisr, J. Lawlr, W. Lapaovich and V. Godyak, Th Physics of Discharg Lamps, Rviws of Modrn Physics, Vol. 76, No., 4, pp doi:./rvmodphys [4] J. Ehlbck, U. Schnabl, M. Polak, J. Wr, T. von Wodk, R. Brandnburg, T. von dm Hagn and K.-D. Wlmann, Low Tmpraur Amosphric Prssur Plasma Sourcs for Microbial Dconamaion, Journal of Physics D: Applid Physics, Vol. 44, No.,, Aricl ID:. doi:.88/-77/44// [5] M. J. Kushnr, Mon-Carlo Simulaion of Elcron Propris rf Paralll Pla Capaciivly Coupld Dischargs, Journal of Applid Physics, Vol. 54, No. 9, 98, pp doi:.6/.76 [6] D. B. Gravs, Fluid Modl Simulaions of a.56-mhz rf Discharg: Tim and Spac Dpndnc of Ras of Elcron Impac Exciaion, Journal of Applid Physics, Vol. 6, No., 987, pp doi:.6/.9 [7] J.-P. Bouf, A Two-Dimnsional Modl of dc Glow Dischargs, Journal of Applid Physics, Vol. 6, No. 5, 988, pp doi:.6/.996 [8] G. J. M. Haglaar, G. M. W. Krosn, U. van Sloon and H. Schrudrs, Modlg of h Microdischargs Plasma Addrssd Liquid Crysal Displays, Journal of Applid Physics, Vol. 88, No. 5,, pp doi:.6/.8759 [9] R. Bussiahn, S. Gorchakov, H. Lang, D. Loffhagn and D. Uhrland, Ac Opraion of Low-Prssur H X Lamp Dischargs, Journal of Physics D: Applid Physics, Vol. 4, 7, Aricl ID: 88. doi:.88/-77/4//s7 [] M. Wnd, S. Prs, D. Loffhagn, A. Kloss and M. Kliz, Brakdown Characrisics of High Prssur Xnon lamps, Journal of Physics D: Applid Physics, Vol. 4, No. 8, 9, Aricl ID: 858. doi:.88/-77/4/8/858 [] S. Chapman and T. G. Cowlg, Th Mahmaical Thory of Non-Uniform Gass: An Accoun of h Kic Thory of Viscosiy, Thrmal Conducion and Diffusion Gass, rd Ediion, Cambridg Univrsiy Prss, Cambridg, 998. [] M. Wilcoxson and V. J. Manousiouhakis, Simulaion of a Thr-Momn Fluid Modl of a Two-Dimnsional Radio Frquncy Discharg, Chmical Engrg Scc, Vol. 5, No. 7, 996, pp [] H. C. Kim and V. I. Manousiouhakis, Dually Drivn Radio Frquncy Plasma Simulaion wih a Thr Momn Modl, Journal of Vacuum Scc & Tchnology A, Vol. 6, No. 4, 998, p. 6. doi:.6/.584 [4] S. Elaissi, M. Yousfi, H. Hlali, S. Kazziz, K. Charrada and M. Sassi, Radio-Frquncy Elcrongaiv Gas Discharg Bhaviour a Paralll-Pla Racor for Marial Procssg, Plasma Dvics and Opraions, Vol. 4, No., 6, pp doi:.8/ [5] M. M. Bckr, D. Loffhagn and W. Schmid, A Sabilizd Fi Elmn Mhod for Modlg of Gas Dischargs, Compur Physics Communicaions, Vol. 8, No. 8, 9, pp. -4. doi:.6/j.cpc.9.. [6] G. K. Grubr, M. M. Bckr and D. Loffhagn, Why h Local-Man-Enrgy Approximaion Should B Usd Hydrodynamic Plasma Dscripions Insad of h Local-Fild Approximaion, Physical Rviw E, Vol. 8, No., 9, Aricl ID: 645. doi:./physrve [7] A. Drzsi, P. Harmann, I. Korolov, J. Karácsony, G. Bánó, and Z. Donkó, On h accuracy and limiaions of fluid modls of h cahod rgion of dc glow dischargs, Journal of Physics D: Applid Physics, Vol. 4, No., 9, Aricl ID: 54. doi:.88/-77/4//54 [8] I. Rafaov, E. A. Bogdanov and A. A. Kudryavsv, On h Accuracy and Rliabiliy of Diffrn Fluid Modls of h Dirc Currn Glow Discharg, Physics of Plasmas, Vol. 9, No.,, Aricl ID: 5. [9] A. Bogars, R. Gijbls and W. J. Godhr, Hybrid Mon Carlo-Fluid Modl of a Dirc Currn Glow Discharg, Journal of Applid Physics, Vol. 78, No. 4, 995, pp. -4. doi:.6/.69 [] Z. Donkó, Hybrid Modl of a Rcangular Hollow Cahod Discharg, Physical Rviw E, Vol. 57, No. 6, 998, pp doi:./physrve [] D. Loffhagn and F. Signgr, Advancs Bolzmann Equaion Basd Modllg of Discharg Plasmas, Plas- Copyrigh SciRs.

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