Numerical Simulation for Magnetic Mirror Effect on Electron Movement in a Hall Thruster

Transcription

1 Numrial Simulation for Magnti Mirror Efft on Eltron Momnt in a Hall hrustr IEPC-7-37 Prsntd at th 3 th Intrnational Eltri Propulsion Confrn, Florn, Italy Sptmbr 7-, 7 Darn Yu *, Hui Liu and Haiyang Fu HI Plasma Propulsion Laboratory, Harbin, Hilongjiang, 5, China Abstrat: h fft of magnti mirror on nar wall ondutiity (NWC) is studid in th hannl of a Hall thrustr. Eltron dynamis pross in th plasma is dsribd by tst partil mthod that ltrons ar randomly mittd towards th innr wall from th ntrlin of th hannl. Mont Carlo mthod is applid to sol th modl. h numrial rsults show that mor than half of th ltrons ar rfltd bak to ntral rgion by radial magnti fild for, whih ar unlikly to ollid with th wall. h urrnt dnsity pak du to NWC inrass linarly with th inras of magnti mirror ratio. Espially whn onsidring th anisotropy of ltron loity distribution in hall thrustrs, th magnti mirror fft would b mor obious. W also obtain th two dimnsional distribution of ltron urrnt du to NWC. Finally, w ddu th analytial rsults of magnti mirror fft whih aords with th simulation rsults. Nomnlatur Eltron tmpratur in th dirtion prpndiular to magnti lins Eltron tmpratur in th dirtion paralll to magnti lins Eltron loity in th dirtion prpndiular to magnti lins Eltron loity in th dirtion paralll to magnti lins w Wall tmpratur φ Ψ E N n j μ Eltri potntial Magnti flux Eltri fild Magnti fild Numbr of mittd ltrons Eltron numbr dnsity Eltron urrnt dnsity Eltron mobility α Effti ollision offiint * Profssor, yudarn@hms.hit.du.n PhD studnt, liuhui@hms.hit.du.n Graduat studnt, fuhaiyang@hms.hit.du.n h 3 th Intrnational Eltri Propulsion Confrn, Florn, Italy Sptmbr 7-, 7

2 ν w Frquny of ltron-wall ollisions β k m Magnti mirror ratio Magnti mirror ratio oltmann onstant Eltron mass Anisotropy offiint Γ Eltron flux to th wall w I. Introdution In hall thrustrs, ollisions btwn ltrons and nutral atoms ar not suffiint to xplain ltron transport aross th magnti fild lins insid th hannl, spially in th xhaust rgion whr th magnti fild is maximal and th atom dnsity is rathr small du to intns ioniation. Various thoris ha bn proposd to xplain this anomalous ltron transport. A. I. Moroo spulatd nar wall ondutiity (NWC) thory that whn an ltron ollids with th wall, and thus th stady hall drift of th ltron would b dstroyd, th ltron would obtain nrgy from ltri fild and transport aross th fild lins. h NWC thory sussfully xplains th anomalous ltron transport. Exprimntal rsults also show that NWC plays a signifiant rol in hall thrustrs,3, spially in xhaust rgion. h prious studis show that NWC is inflund by many fators, suh as rosion surfa, magnti fild gradint in radial dirtion, plasma wall rombination, osillating shath t 4-9. Prious studis of NWC only onsidrd th radial omponnt of th magnti fild. Howr, magnti fild in hall thrustrs has a ona shap spially dsignd to fous ion. his magnti topology lads to an additional fft of inrasd magnti fild magnitud nar th hannl walls in omparison with th hannl ntrlin, th so-alld magnti mirror fft. Magnti mirror fft forms a nt for to prnt rtain groups of ltrons to rah th hannl walls, and thrfor dminishs ltron flux to th hannl walls. NWC urrnt drass aordingly with th rdution of ltron flux to th hannl walls. h magnti fild intnsity nar th hannl wall is gnrally two tims strongr than that nar th ntrlin of th hannl in xhaust rgion, whih lads obious magnti mirror fft. In Rf., Sullian usd kinti (PIC) simulations, and found that whn thr xists magnti mirror fft, th ltron flux to th walls is lss than xptd, and th shaths ar wakr than xptd as wll. Furthrmor, h drid onditions for an ltron outsid th shath to rah th walls. Kidar usd D and D fluid modl to simulat potntial distribution in a Hall thrustr, taking into aount th magnti mirror fft on th ltron dynamis. h rsults show that magnti fild gradints in radial dirtions afft th prshath strutur and potntial distribution. h lngth of th radial prshath rgion drass in th prsn of a magnti fild gradint. h two-dimnsional potntial shap is afftd by magnti mirror ratio 3. In this papr, w alulat th urrnt dnsity du to NWC in urd magnti fild. Sin it is diffiult to us th analyti mthod in Rf. du to th fild nonuniformity, th ltron dynamis pross in th plasma is dsribd in th tst partil mthod in urrnt thoris that ltrons ar randomly mittd towards th innr wall from th ntrlin in th hannl. Mont Carlo mthod is mployd to numrially alulat th momnt of ltrons, thrmally mittd from th hannl wall. hn th ltron mobility du to NWC an b alulatd by statistially araging th axial ltron loity. h numrial rsults show that magnti mirror fft influns ltron mobility gratly, whih rdus with th inras of magnti mirror ratio. Espially whn onsidring th anisotropy of ltron loity distribution funtion (EVDF) in hall thrustrs, i.. th tmpratur of ltrons prpndiularly to th magnti fild is gratr than that of ltrons paralll to th magnti fild, ltron mobility du to NWC drass furthr. hus w onlud that th anisotropy an magnify magnti mirror fft. sids, w simulatd th r- distribution of ltron mobility and ltron dnsity. Finally, w ddud th analytial rsults of magnti mirror fft on NWC whih aords with th simulatd rsults. h organiation of th papr is as follows. In S. II, w brifly dsrib th modl and basi quations usd to alulat th urrnt dnsity and ltron mobility du to NWC onsidring th magnti mirror fft. In S. III, w numrially sol th modl quations in diffrnt ass, and analy th physial mhanism. h papr is thn summarid in S. IV with onlusions. II. HE MODEL In hall thrustrs, th magnitud of magnti fild inrass from th anod to th xit plan, and th maximal h 3 th Intrnational Eltri Propulsion Confrn, Florn, Italy Sptmbr 7-, 7

3 alu loats in xhaust rgion. Considr that th ltron man fr path for all typs of ollisions ( λ m ) is muh gratr than th hannl lngth (L 5 m) in xhaust rgion whr ollisions btwn ltrons and hay partils an b ngltd, ollisions btwn ltrons and wall would play a mor important rol in this rgion 4. For ths rasons, th modl inluds th only xhaust rgion. Exprimntal rsults in Rf. 5 showd that th lngth of xhaust rgion is about mm. In our work, imposing that ltrons ar mittd from hannl ntrlin to th innr wall, w adopt th lapfrog mthod of oris 6 for disrtiing th quations of motion. W dfin and ltron tmpratur in th dirtion prpndiular and paralll to th magnti fild lins rsptily, and assum th initial EVDF is Maxwllian and anisotropi. h EVDF in ths two dirtions an b writtn as: m m f ( ) n xp( ) () πk k m m f ( ) n xp( ), < () πk k In abo quations, dnots initial stat. Eltrons olliding with th innr wall ar dfinitly aommodatd and rmittd aording to a half Maxwllian distribution of wall tmpratur w. W us th polar form of th ox Mullr transformation to sampl loity. h magnti fild is gnratd by a magnti modl using th Finit Elmnt Mthod Magntis (FEMM) softwar pakag 7. Supposing th magnti fild is axisymmtri, w only onsidr th axial and radial omponnts of magnti fild. Undr th old plasma assumption, quipotntial ontours tnd to lin up with th magnti fild lins. his is th as in onntional Hall thrustr hannls. h rlation btwn th ltri potntial and th magnti flux an tak a form of φ αψ 8, whr, φ and Ψ ar ltri potntial and magnti flux in th hannl, rsptily, and α is a onstant. Finally, th ltron urrnt dnsity and mobility du to th NWC fft ar alulatd as follows: n j ( r) μ ( r) n Figur. h simulatd rgion, magnti fild topology and ltri potntial distribution in simulation N ( r) i ( r) 3 h 3 th Intrnational Eltri Propulsion Confrn, Florn, Italy Sptmbr 7-, 7 N jr E ( r) N N ( r) i ( r) E ( r) (3) (4) Whr n is th ltron dnsity, N is th amounts of mittd ltrons, N (r), (r), E (r) ar th amount, loitis and loal ltri intnsity of ltrons in diffrnt radial position. Paramtrs of simulations orrspond to th xprimntal rsults of th Hall thrustr oprating at 5.4A, 3V 3 in Rf.5.. n / m III. CALCULAION RESUL AND ANALYSIS A. Influn of magnti mirror ratio on NWC h simulatd rgion, magnti fild topology and ltri potntial distribution ar shown in Fig.. h innr and outr radii, ar rmm and r35mm, rsptily. h axial domain is mm long insid th thrustr from xhaust plan. Potntial drop in ntir rgion is V. Eah ltron is mittd in th ntrlin of xhaust plan, and

4 5 th mittd numbr is N. In hall thrustrs, th EVDF is anisotropi (Rf. 9,,). It an b xplaind that ltrons in hall thrustrs ar hatd in th dirtion prpndiular to magnti lins and sattr du to lasti ollisions in th dirtion paralll to magnti lins, but th nrgy rlaxation tim for ltrons ar longr than th sattring tim du to ollisions. h ratio of th ltron tmpraturs is typially two. In our modl, w us th simulatd rsults in Rf., V, V / rsptily, and nglt th shath as in Rf. 9. h ltrons ar rfltd isotropi aftr olliding with th wall, with tmpratur w V. (a) Figur. h initial distributions of ltrons ollid with th wall and rfltd by magnti mirror (b) Figur 3. h ltron mobility ariations alulatd with diffrnt magnti mirror ratios h simulation rsults ral that th mittd ltrons an b diidd into two parts apparntly du to magnti mirror fft, (s Fig. ). Most of th ltrons ar rfltd by magnti mirror. h ondition of ltrons olliding with th wall an b dsribd by loss on. Only thos ltrons in loss on an sap from magnti mirror and ollid with th innr wall. In a Hall thrustr, th ltron urrnt dnsity ausd by th ltron-wall ollisions an b xprssd as: ν w n E j, NWC E m n ν w (5) ω m Whr, ν w is th frquny of ltron-wall ollisions. It is known that th E onfiguration in hall thrustrs rsults in only an aimuthal drift of th ltrons without momnts in th axial dirtion unlss othr disturbans prturb th stady Hall drift. In our modl, only if an ltron ollids with th innr wall, th hall drift is dstroyd; th ltron aquirs nrgy from ltri fild and trars magnti lins towards anod. Howr, th ltrons rfltd by th magnti mirror an not produ NWC. Dnoting as ν rf th ltron-wall ollision frquny in th absn of a magnti mirror, and α th ffti ollision offiint rprsnting th ratio of th flux dnsity of wall-olliding ltrons to that of inidnt ltrons, ν w α ν rf. hrfor, NWC is proportional to α. w W dfin β as magnti mirror ratio. w and ar th magnti fild intnsity of th innr wall and th ntr of xhaust plan rsptily. h magnti mirror ratio was hangd by adjusting oil urrnt whil kping th magnti fild intnsity onstant in th ntr of xhaust plan. h ltron mobility du to NWC 4 h 3 th Intrnational Eltri Propulsion Confrn, Florn, Italy Sptmbr 7-, 7

5 alulatd with diffrnt β ar shown in Fig. 3. W an find from th rsults that whn inrasing magnti mirror ratio, th ratio of rfltd ltrons inrass, th alu of α and th ltron mobility paks dras aordingly.. Influn of anisotropi EVDF on NWC m h quantity th wall must satisfy: Eq.(6)an b writtn as: is known as an adiabati inariant. In th absn of ltri fild, ltrons olliding with < + W an s from Eq.(7)and Fig., only if th ratio > 5 h 3 th Intrnational Eltri Propulsion Confrn, Florn, Italy Sptmbr 7-, 7 β β of an ltron is gratr than β, th ltron in loss on an ollid with th wall. Whn w hang th ratio of initial loitis of ltrons, th ratio of ltrons olliding with th wall would altr. W dfin a paramtr to dnot th anisotropy of EVDF. hn is arid, kping onstant to hang paramtr. h ltron urrnts with diffrnt ar shown in Fig. 5. With th drasing of, i.. inras anisotropy of EVDF, th ratio of ltrons olliding with th wall and th NWC urrnt drass narly linarly. Whn th disharg oltag inrass in hall thrustr, as th rsults shown in Rf. 9, anisotropy of EVDF inrass furthrmor. So w an ddu that magnti mirror fft would bom mor signifiant, and NWC urrnt will b n smallr in as of high disharg oltag. On th othr hand, whn ltrons mo towards th wall, nrgy in th dirtion paralll to magnti fild will b partially onrtd to th dirtion prpndiular to magnti fild, so magnti mirror fft an aggraat anisotropy of EVDF. C. D distribution of NWC urrnt Prious studis of NWC in Rf. 4-9 only onsidr D NWC urrnt distribution along radial dirtion. Howr, NWC urrnt hangs along axis aordingly, as th magnti mirror ratio and magnti fild intnsity ar arid along axial dirtion. W analy th spa profil of ltron numbr dnsity and ltron mobility du to NWC, both in axial and radial dirtions. In our D modl, w initiali a uniform distribution of ltrons along hannl ntrlin. Ery Eltron urrnt (A) (6) (7) Figur 4. h ltron urrnts alulatd with diffrnt Figur 5. Eltron dnsity distribution whn magnti lins ar urd (normalid with ltron dnsity on th ntrlin)

6 ltron mos towards th innr wall as w did in th d simulation of NWC abo, and w also assum th initial EVDF is Maxwllian, V, V ; and alulat ltron mobility using Eq.(4). Eltron 3 dnsitis on th ntrlin qual to: n / m. Eltrons ar rfltd isotropi aftr olliding with th wall, with tmpratur w V. h total amount of simulatd ltrons is 6. (a) (b) Figur 6. Profils of ltron mobility (a) with only radial omponnt of magnti fild and (b) magnti lins is urd Fig. 5 shows th D ltron dnsity distribution in as of urd magnti lins. W an find that whn ltrons mo from th hannl ntrlin to innr wall, parts of ltrons ar rfltd by magnti mirror fors, and ltron dnsity drass gradually from th hannl ntrlin to innr wall. Whn thr xists only radial omponnt of magnti fild, magnti mirror fft anishs, and ltron numbr dnsity qual rywhr in th hannl. h ltron mobility distribution du to NWC in th hannl is shown in Fig. 6. W an obsr that th pak and width of ltron mobility inras from thrustr xhaust plan towards anod dirtion, no mattr whthr th magnti lins ar urd or not. h rason is that ithr th pak or th width of ltron mobility du to NWC is proportional to th inrs of magnti fild intnsity. Whil magnti fild intnsity drass from xhaust plan to anod dirtion, th pak and width of ltron mobility inrass aordingly. ut from Fig. 6 (b) w an find that if thr is magnti mirror fft ausd by urd magnti lins, th pak of ltron mobility is smallr in ry axial position ompard with th rsult of only radial omponnt of magnti fild. W onlud that, whn onsidring th magnti mirror fft, NWC is lss important than stimation bfor. h rdution of mobility in xhaust rgion du to th magnti mirror fft indiats th inrasing of potntial drop in this rgion, whih is profitabl for alrating ion. D. Analysis of physial mhanism Now lt us disuss th physial mhanism of th influn of magnti mirror fft and anisotropy of EVDF on NWC. If an ltron is rfltd by magnti mirror, th stady hall drift is not dstroyd. h rfltd ltron an not produ NWC urrnt and thrfor NWC urrnt is proportional to ltron flux to th wall. First, it is nssary to analy magnti mirror fft and anisotropy of EVDF on ltron flux to th walls. In Rf., Sullian ddud th rlation of magnti mirror ratio and ltron flux, but h did not onsidr anisotropy of EVDF in his modl. W disuss th physial mhanism basd on th modl of Sullian, onsidring anisotropi EVDF in hall thrustrs furthrmor. For onsration of magnti momnt, th total nrgy an b intgratd in th form: w m + m φ m + m φ (8) Sin th axial omponnt of magnti fild is small in xhaust rgion, w an assum that th magnti lins ar prpndiular to th walls approximatly. φ is th ltri potntial, is magnti fild intnsity. rprsnts 6 h 3 th Intrnational Eltri Propulsion Confrn, Florn, Italy Sptmbr 7-, 7

7 an initial point. Only th ltrons in loss on an ollid with th wall du to th magnti mirror fft, and thos initial loitis of ltrons olliding with th walls must satisfy: w Δφ r > + (9) m Δφ φ φ w, dnots th potntial drop btwn mittd point and th walls. h flux of ltrons arriing th wall is: m Γ n πk 3 h innr intgral yilds: ( r ) mr Δ m w φ ( r ) /( ) k r m Δ d k φ xp r r xp r m mr Δφ w ( ) /( ) k k. m r, Eq.() rdus to k πk m So that with th substitution x Lt Δφ, 8 k r a k π m kr πm Γ n Δφ k,w an gt: Δφ w ( x ) /( ) k a β + β x β x dx π d () () n x Γ ( ) dx 4 a n a ( β ) a ( ) () 4 + ( β ) n a 4 + ( β ) Whn thr is no shath, ltron flux to th wall is n Γ (3) 4 + ( β ) Aording to Eq. (3) w an find both magnti mirror fft and anisotropi EVDF influn ltron flux n to th wall. Rgardlss of EVDF and magnti mirror fft, ltron flux is Γ. Whil aftr onsidring 4 ths two ffts, th ffti ollision offiint α rdus by th fator of ordr. Aording + ( β ) to Eq. ( 5 ), th ltron urrnt du to NWC is proportional to. NWC urrnt will also rdu α tims. So th alu of NWC urrnt an b signifiantly inflund by magnti mirror fft. h + ( β ) NWC urrnt alulatd using ths formulas agr wll with th alus obtaind in tst partil simulations (Fig. 4). 7 h 3 th Intrnational Eltri Propulsion Confrn, Florn, Italy Sptmbr 7-, 7

8 IV. CONCLUION h fft of th magnti mirror is dsribd in this papr. Numrial rsults show that th ltron mobility du to NWC drass gratly du to magnti mirror fft; th gratr th magnti mirror ratio, th smallr th NWC urrnt. h basi physial mhanism of magnti mirror fft on NWC is that whn ltrons mo toward th walls, partial of ltrons ar rfltd by magnti mirror fors, diminishing th ltron flux to th wall as wll as NWC urrnt onsquntly. aking into aount that anisotropy of EVDF inrass with th inrmnt of disharg potntial in a hall thrustr, ltron mobility du to NWC rdus; th magnti mirror fft would b mor idnt. sids, w show th two dimnsional distribution of NWC urrnt and ltron dnsity. Sin th magnti fild intnsity and magnti mirror ratio drass from xhaust rgion to anod dirtion, th pak and th width of ltron mobility inras towards anod. Nrthlss, th fft of th shath on NWC has bn ngltd in our modl. It should b notd that th xistns of th shath will dminish ltron flux to th walls furthr. h modl usd in this papr and prious work is quialnt to a singl tst partil alulation. h Mont Carlo mthod is atually an nsmbl arag of a randomly mittd ltron motion in fixd xtrnal ltri and magnti filds. Howr, suh ltron motions might at last prturb th ltri fild distribution. A slf-onsistnt fild tratmnt should b applid taking into aount th intration btwn plasma and th fild. W will onsidr ths ffts in futur work. Howr, w ha to point out that th magnti mirror fft disussd in this papr has alrady ontributd to th dsigns of Hall thrustrs. At last, w drid simplifid analytial formulas btwn magnti mirror fft and ltron flux to th wall n and NWC, Γ. Sin th magnti mirror fft diminishs th ltron flux to th wall, NWC 4 + ( β ) urrnt will b tims smallr onsidring th magnti mirror fft. + ( β ) o summari: In ordr to impro prforman and rais th hall thrustr disharg potntial, it dsrs furthr attntion to study how magnti mirror ratio would afft th prforman of Hall thrustrs. V. Aknowldgmnts Mr. Darn Yu would lik to thank Program for Changjiang Sholars and Innoati Rsarh am in Unirsity (PCSIR) No. IR5, Program for Nw Cntury Exllnt alnts in Unirsity (NCE) No. NCE- 4-38, and NSFC undr Grant No VI. Rfrns A. I. Moroo, Efft of nar-wall ondutiity in a strongly magntid plasma. So. Phys. Dokl,, 775, 966, (in Russian). A. I. ugroa., A.I. Moroo, and V. K. Kharhniko, Exprimntal study on Nar Wall Condutiity, Fi. Plamy,. 6, 849, 99, (in Russian). 3 N. Gason, M. Dudk, and S. arral, Wall matrial ffts in stationary plasma thrustrs. I. Paramtri studis of an SP-, Phys. Plasmas, Vol., No., 3, pp Kolo AN, Modl of th nar-wall ondutiity in th iinity of a marosopially inhomognous mirror-rflting surfa, Plasma Phys Rp, Vol. 8, No.,, pp Zhi-Wn Wu, Da-Rn Yu, Xiao-Gang Wang, Effts of rosion surfa on nar wall ondutiity (NWC) in th Hall-typ stationary plasma thrustrs, Vauum, Vol. 8, 6, pp Da-Rn Yu, Zhi-Wn Wu, Xiao-Gang Wang, Numrial simulation for nar wall ondutiity fft on urrnt profils in th annular hannl of Hall-typ stationary plasma thrustrs, Phys. Plasmas, Vol., No. 4357, 5 7 A A Iano, A A Iano Jr, M aal, Efft of plasma wall rombination on th ondutiity in Hall thrustrs, Plasma Phys. Control. Fusion, Vol. 4,, pp Darn Yu, Hong Li, Zhiwn Wu, Wi Mao, Efft of osillating shath on nar wall ondutiity in Hall thrustrs, Phys. Plasmas, Vol. 4, No. 6455, 7 9 Moroo AI, Sal VV, hory of th nar-wall ondutiity, Plasma Phys Rp, Vol. 7, No.,, pp FF Chn, Introdution to Plasma Physis and Controlld Fusion, Vol. [M], Plnum Prss, Nw-York, 984, Chaps. R. Hofr and A. D. Gallimor, h Rol of Magnti Fild opography in Improing th Prforman of a High Voltag Hall hrustr, Prodings of 38th Joint Propulsion Confrn and Exhibit, July, Indianapolis, IN (Amrian Institut of 8 h 3 th Intrnational Eltri Propulsion Confrn, Florn, Italy Sptmbr 7-, 7

9 Aronautis and Astronautis, Rston, VA, ), AIAA Papr -4. K. Sullian, J. Fox, M. Martin-Sanh, and O. atishh, Kinti Study of Wall Effts in SP Hall hrustrs, Prodings of 4th AIAA/ASME/SAE/ASEE Joint Propulsion Confrn and Exhibit, July 4, Fort Laudrdal, Florida, (Amrian Institut of Aronautis and Astronautis, Rston, VA, 4), AIAA Papr No M. Kidar and I. D. oyd, On th magnti mirror fft in Hall thrustrs, Appl. Phys. Ltt, Vol.87, No.5, 5 4 Moroo A I, Sly V V. Fundamntals of stationary plasma thrustr thory, Vol. [M]. In: Kadomts, Shafrano, ds. Riws of Plasma Physis. Nw York: Consultants urau,, pp Jams M. Haas, Al D. Gallimor, An instigation of intrnal ion numbr dnsity and ltron tmpratur profils in a laboratory-modl Hall thrustr, Prodings of 36th AIAA/ASME/SAE/ASEE Joint Propulsion onfrn, July, Huntsill, AL, (Amrian Institut of Aronautis and Astronautis, Rston, VA, ), AIAA Papr No irdsall, C. K. and A.. Langdon, Plasma Physis ia Computr Simulation, Nw York:Adam Hilgr, 99, pp. 6 7 FEMM, Finit Elmnt Mthod Magntis, Softwar Pakag, Vr. 4., Fostr-Millr, In., oston, MA, 4 8 V. V. Zhurin, H. R. Kaufman and R. S. Robinson, Physis of th Closd Drift hrustrs, Plasma Sours Si. hnol, Vol. 8, 999, pp. R-R 9 D. Sydornko, A. Smolyako,I. Kaganoih, Y. Raitss, Kinti simulation of sondary ltron mission ffts in Hall thrustrs, Phys. Plasmas, Vol. 3, No. 45, 6 I. D. Kaganoih, Y. Raitss, D. Sydornko, A. Smolyako, Kinti ffts in a Hall thrustr disharg, Phys. Plasmas, Vol. 4, No. 574, 7 S. arral, K. Makowski, Z. Pradyn ski, N. Gason, M. Dudk, Wall matrial ffts in stationary plasma thrustrs. II. Nar-wall and in-wall ondutiity, Phys. Plasmas, Vol., No., 3, pp h 3 th Intrnational Eltri Propulsion Confrn, Florn, Italy Sptmbr 7-, 7