Negative Electron Emitting Planar Collector Immersed in a Plasma that Contains a Mono-energetic Electron Beam

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Jemima Heath
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Intrnational onfrnc Nuccllarr Enrrgy fforr Nw Eurrop 9 Bld / lonia / ptbr 14-17 Ngati Elctron Eitting Planar ollctor Irsd in a Plasa that ontains a Mono-nrgtic Elctron Ba ABTRAT Toaž Gyrgyk 1,, Milan

1 Intrnational onfrnc Nuccllarr Enrrgy fforr Nw Eurrop 9 Bld / lonia / ptbr Ngati Elctron Eitting Planar ollctor Irsd in a Plasa that ontains a Mono-nrgtic Elctron Ba ABTRAT Toaž Gyrgyk 1,, Milan Črčk,, Jrnj Koačič 1 1 Unirsity of Ljubljana, Faculty of Elctrical Enginring Tržaška 5, 1 Ljubljana, lonia Jožf tfan Institut Jaoa csta 9, I-1 Ljubljana, lonia Unirsity of Maribor, Faculty of iil Enginring, tanoa 17, Maribor, lonia Toazgyrgyk@funi-ljsi, ilancrck@ijssi A on-dinsional fluid odl for th analysis of th potntial foration in front of a ngatily biasd or floating lctron itting lctrod (collctor irsd in a twolctron tpratur plasa, that has bn dlopd and prsntd rcntly [T Gyrgyk, B Jurčič-Zlobc, M Črčk, J Koačič, Plasa ourcs ci Tchnol, 18, (9, 51], is odifid Th high tpratur lctron population is rplacd by a ono-nrgtic lctron ba o prliinary rsults of th odl ar rportd 1 INTRODUTION In dg plasas of plasa fusion dics occurrnc of nrgtic lctrons is rathr coon uch lctron populations appar du to strong radio frquncy filds during ion cyclotron and lowr hybrid wa hating and radio frquncy currnt dri Th prsnc of th nrgtic lctrons has a rarkabl ffct on potntial foration in th plasa In our rcnt work [1,] nrgtic lctrons wr odlld by a two-tpratur Maxwllian locity distribution In this work th hot lctrons fro th prious work ar rplacd by a ono-nrgtic lctrons, which ha th dirctions of thir locitis uniforly distributd MODEL W us a on dinsional fluid odl Th potntial profil in th shath is dtrind by a on-dinsional Poisson quation: d Φ = ( ni ( x n1( x n( x n( x (1 dx ε Hr n i (x is th dnsity of singly chargd positi ions, n 1 (x is th dnsity of th cool lctrons, n (x is th dnsity of th hot lctrons, n (x is th dnsity of th ittd lctrons, is th lntary charg and Φ is th potntial In th asyptotic two scal liit [] usd in this work th boundary conditions at th shath dg (at x = d ar: dφ Φ ( x= d =Φ, ( x = d = ( dx 811

2 81 Th potntial of th lctrod (collctor locatd at x = is Φ and at a ry larg distanc fro th collctor th potntial is zro Φ(xØ = At a ry larg distanc fro th collctor th ions ar assud to b all at rst o at a distanc x fro th collctor th nrgy of th ions is: 1 Φ( x ( x Φ ( x =, ( x = ( i i i i Hr i is th ion ass and i (x is th locity of ions at th distanc x fro th collctor in th dirction towards th collctor Th ion currnt dnsity to th collctor is j i Fro th assuption of th flux consration th ion dnsity n i (x at a distanc x fro th collctor can b xprssd in th following way: ji ji ni ( x = = (4 ( x Φ( x Th ittd lctrons ar assud to b ononrgtic and thy all la th collctor surfac with th sa initial locity c, which is allowd to b largr than zro W assu that th flux of th ittd lctrons fro th collctor is j and that this flux is ondinsional in th dirction prpndicular to th collctor It is also assud that this flux is consrd o fro nrgy consration i and flux consration on finds 1 1 Φ = ( x Φ ( x, (5 n j = n( x ( x, (6 ( x = j ( Φ Φ x ( (7 Th bulk lctron population has Maxwllian locity distribution and thir dnsity obys th Boltzann law: Φ( x n1( x = n1xp (8 kt Hr k is th Boltzann constant and T is th lctron tpratur Th nrgtic or th ba lctrons ar assud to b ononrgtic Far away fro th collctor, whr th potntial is zro, all th lctrons ha th sa locity, whil th dirctions of thir locitis ar uniforly distributd Thir dnsity far away fro th collctor is n Thir locity distribution function is gin by: n f( = ( 4π δ (9 uch a distribution is calld watr bag distribution and it is usually a good approxiation for th dscription of th priary lctrons in low prssur dischargs [4] Th dnsity n (x of th ba lctrons at th distanc x fro th collctor is found by intgration of th distribution function: Procdings of th Intrnational onfrnc Nuclar Enrgy for Nw Europ, Bld, lonia, pt 14-17, 9

3 81 π Θc n δ ( ϕ 4π Θ= n ( x = f ( d = d d sinθd 1 1 Φ = n( 1cosΘ c = n 1 Hr Θ c is th critical angl btwn th noral to th collctor surfac and th locity ctor of a ba lctron gin by: ( x c c (1 1 Φ( x cos Θ =Φ( x, cos Θ = (11 Only thos lctrons for which th angl btwn thir locity ctor and th noral to th collctor is sallr than Θ c can o towards th collctor Th flux of th ba lctrons in th dirction towards th collctor is gin by: π Θc n δ ( ϕ 4π = j = cos Θ f ( d = d d cosθsin ΘdΘ (1 1 sin Θ 1 1 Φ ( x c = n = n ( 1cos Θ c = n Th currnt dnsity of th ba lctrons to th collctor dpnds on th collctor potntial Φ in th following way: 1 Φ Φ j = n1 H 1, (1 4 whr H is th Haisid unit stp function By such forulation of th flux j it is assurd that j is zro, if th rpulsi potntial Φ of th collctor is largr than th kintic nrgy of th ba lctrons obining quations (1, (4, (7, (8 and (1 th Poisson quation is writtn in th following for: d Φ ρ( x = = dx ε (14 j i / Φ( x 1 Φ ( x j/ = n1xp n 1 ε Φ( x kt ( Φ Φ( x i At th shath dg, whr th potntial is Φ, th plasa is quasi-nutral, so th th spac charg dnsity ρ(φ, dfind in (14 ust b zro: ji / Φ 1 Φ j/ n1xp n 1 kt = (15 Φ ( Φ Φ i Th ion flux j i is liinatd fro (15 and insrtd into (14 Th following quation is obtaind: Procdings of th Intrnational onfrnc Nuclar Enrgy for Nw Europ, Bld, lonia, pt 14-17, 9

4 814 d Φ ρ( x = = dx ε j Φ Φ 1 Φ n xp n 1 Φ( x 1 Φ( x j/ n1xp n 1 kt ( Φ Φ( x 1 Φ( x kt ( Φ Φ = ε (16 If a stabl shath is to b ford in front of a ngati collctor, th ions ust ntr into th shath with a crtain iniu locity This is a ry wll known Boh critrion [5] Th locity of ions at th shath dg ust fullfil th following: Φ * ( κti k T = = (17 i i x= d Hr κ is th polytropic cofficint and T * is th lctron scrning tpratur, dfind [] as: * n ( Φ T = (18 dn ( Φ k dφ Th lctron dnsity n can b rad fro (16: Φ( x 1 Φ( x j/ n ( Φ ( x = n1xp n 1 kt (19 ( Φ Φ( x Fro (17, (18 and (19 and with T i =, on gts: Φ Φ Φ β Φ j xp 1 1, kt kt = ( n1 ( Φ Φ W introduc th following dinsionlss ariabls: Φ( x Φ,, Φ, j, Ψ= Ψ = Ψ = μ = J =, kt kt kt i kt n 1 (1 n kt kt x ε kt β =, = N, =, z =, λ = n n D 1 λd 1 and th Boh critrion ( is thn writtn in th following for: β Ψ N Ψ xp( Ψ ( 1 Ψ 1 J = N Ψ Ψ Th total currnt dnsity j t to th collctor is gin by: ( ( Procdings of th Intrnational onfrnc Nuclar Enrgy for Nw Europ, Bld, lonia, pt 14-17, 9 (

5 j = j j j j = t 1 i 815 kt Φ 1 Φ Φ n 1 xp β n 1 1 H 1 π kt 4 j Φ Φ 1 Φ n1xp n1 1 β i kt ( j Φ Φ Using th ariabls (1 it is writtn in th following for: 1 1 Ψ Ψ Jt = xp Ψ β1 1 H J π 4 β J (4 Ψ μψ xpψ 1 N ( Ψ Ψ If th collctor is floating, J t = Th Poisson quation (16 is also writtn with th ariabls dfind in (1: d Ψ β Ψ( z J = xp ( Ψ ( z 1 dz N ( Ψ Ψ( z (5 Ψ β Ψ J xp( Ψ 1 Ψ( z N ( Ψ Ψ Th boundary conditions ( ar transford into: d dψ d Ψ z = =Ψ, z = = (6 λd dz λd Th quation (5 is ultiplid by dψ/dz and intgratd or Ψ fro Ψ (at z = d/λ D to Ψ (at z < d/λ D : 1dΨ 1dΨ = dz dz β 4 = xp( Ψ xp( Ψ ( ( ΨΨ Ψ Ψ Ψ=Ψ Ψ=Ψ ( ( ( J N Ψ Ψ N Ψ Ψ β Ψ J Ψ xp( Ψ 1 ( ( g Ψ Ψ Ψ N ( Ψ Ψ According to (6 th scond tr on th lft hand sid of (7 is zro, so th function g(ψ, dfind on th right hand sid of (7 gis on half of th squar of lctric fild in th shath as th function of th potntial Ψ If th ission fro th collctor incrass, lctric fild at th collctor also incrass (bcos lss ngati and ntually th spac charg liit for th lctron ission is rachd In that cas th lctric fild at th collctor bcos zro Th condition for spac charg liitd (or critical ission is drid asily Th intgration boundaris of th abo ntiond intgral ar changd fro Ψ (at z = d/λ D to Ψ (at z = o th condition for th spac charg liitd ission rads: ( (7 Procdings of th Intrnational onfrnc Nuclar Enrgy for Nw Europ, Bld, lonia, pt 14-17, 9

6 816 g ( Ψ =Ψ = (8 If β, and N ar gin, th quations (, (4 (with J t = and (8 for a syst of quations for unknown quantitis: th shath dg potntial Ψ, th collctor floating potntial Ψ and critical ission currnt dnsity fro th collctor j REULT In this sction w show so prliinary rsults of th odl In Figur 1 th solutions of th syst of quations (, (4 (with J t = and (8 rsus β for th following paratrs: μ = 1/186, N = 1 and = 1 Figur 1: olutions Ψ, Ψ and J of th syst of quations (, (4 (with J t = and (8 rsus β In th scond row so dtails fro th rspcti plots of th top row ar shown on an xpandd scal In th botto row sa plots ar shown on a scal that is furthr xpandd Th othr paratrs ar gin in th txt In a wid rang of th dnsitis of th ba lctrons th syst (, (4 and (8 has solutions According to th absolut alu of th corrsponding shath dg potntial Ψ w call th th low, th iddl and th high Th iddl and th low solution xist in th rang fro β = 1 to β = 95, whr thy rg and dissappar Th high solution on th othr hand xists also for alus of β abo 115 and n highr For th high solution th potntials Ψ and Ψ ar constant for all alus of β >, whil th critical ission currnt dnsity J incras linary with β For μ = 1/186, N = 1 and = 1 th alus of th potntials ar Ψ = and Ψ = uch big alus ha no physical xplanation sinc ths potntials ar any tis largr than th nrgy of th ba lctrons Th Ψ of th iddl solution incrass, as β is incrasd This is a clar indication Procdings of th Intrnational onfrnc Nuclar Enrgy for Nw Europ, Bld, lonia, pt 14-17, 9

7 817 that also this solution has no physical xplanation In addition th corrsponding alus of J ar ngati Only th alus prdictd by th low solution ar physically possibl But n hr on should b carful in intrprting th rsults Whn β xcds th alu 755 th critical ission currnt dnsity J bcos ngati A or carful analysis shows that for μ = 1/186, N = 1 and = 1 and th alus of β btwn 11 β 115 th low solution splits inot parts For ths alus of β th syst (, (4 and (8 has 5 solutions all togthr Tripl floating potntials of lctrods that it lctrons and ar irsd in a plasa that contains an lctron ba wr obsrd xprintally [6] Th currnt oltag charactristicss of such an lctrod can b rproducd using quations (, (4 and (8 For th charactristics shown in Figur th following paratrs ar slctd: μ = 1/186, β = 11, N = 1, = 1 whil Ψ is gradually incrasd as an indpndnt paratr For ach Ψ quations ( and (8 ar sold for Ψ and J and thn J t is found fro (4 In th top lft figur th dpndnc of J t on Ψ is shown for a wid rang of alus of Ψ En alus of Ψ blow -5 (i blow / ar slctd in ordr to illustrat th ffct of th ba Figur : Th currnt oltag charactristics found fro quations (, (4 and (8 with following paratrs: μ = 1/186, β = 11, N = 1 and = 1 is shown in th top lft figur In th top right figur th part around th alus of th floating potntial is shown on an xpandd scal In th botto figurs dpndnc of Ψ and J on Ψ found fro th syst of quations ( and (8 is plottd In th top right figur th dpndnc of J t on Ψ is shown on an xpandd scal around th thr floating potntials In th botto figurs dpndnc of Ψ and J on Ψ found fro th syst of quations ( and (8 is plottd It can b sn clarly how th J t cur crosss th zro lin tis Th ain rason for this is th dpndnc of th spac charg liitd ission currnt J on Ψ shown in th botto right figur Procdings of th Intrnational onfrnc Nuclar Enrgy for Nw Europ, Bld, lonia, pt 14-17, 9

8 818 4 ONLUION W prsntd a sipl on-dinsional fluid odl of th shath foration in front of an lctron itting lctrod which is irsd in a plasa that contains a ono-nrgtic lctron ba with isotropic locity distribution Th odl prdicts ultipl solutions in a wid rang of ba dnsitis and nrgis, but only a sall part of thos solutions is physically accptabl Th odl rproducs tripl floating potntials of lctron itting lctrods obsrd in xprints [6] AKNOWLEDGMENT This work has bn carrid out within th Association EURATOM-MHET Th contnt of th publication is th sol rsponsibility of its authors and it dos not ncssarily rprsnt th iw of th oission or its srics This work was supportd by th grants J-916 and L-858 REFERENE [1] T Gyrgyk, M Črčk, A fluid odl of th currnt-oltag charactristic of an lctron itting lctrod irsd in a two lctron tpratur plasa, Eur Phys J D, 4, 7, pp [] T Gyrgyk, B Jurčič-Zlobc, M Črčk, J Koačič, "hath structur in front of an lctron itting lctrod irsd in a two-lctron tpratur plasa: a fluid odl and nurical solutions of th Poisson quation", Plasa ourcs ci Tchnol, 18, 9, 51 (18pp [] K-U Riann, "Thory of th plasa-shath transition", J Tch Phys, pcial Issu, 41,, pp [4] K N Lung, N Harshkowitz, K R MacKnzi, "Plasa confinnt by localizd cusps", Phys Fluids, 19, 1976, pp [5] D Boh, in A Guthri and R K Wakrling (Ed, Th haractristics of Elctrical dischargs in Magntic Filds, McGraw-Hill, Nw York, 1949, pp77 [6] hol-h Na, N Hshkowitz, M H ho, T Intrator and D Dibold, Multipl alud floating potntials of Languir probs, J Appl Phys, 6, 1988, pp Procdings of th Intrnational onfrnc Nuclar Enrgy for Nw Europ, Bld, lonia, pt 14-17, 9

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