A Unified Theory of rf Plasma Heating. J.e. Sprott. July 1968

Transcription

1 A Unifid Thry f rf Plasma Hating by J.. Sprtt July 968 PLP 3 Plasma Studis Univrsity f iscnsin

2 INTRODUCfION In this papr, th majr rsults f PLP's 86 and 07 will b drivd in a mr cncis and rigrus way, and th tratmnt f lctrn cycltrn rsnanc hating will b xtndd t highr dnsitis. Th basic mthd is t calculat th pwr dp absrbd by a vlum f plasma dv in th prsnc f an scillating lctric fild E = E iwt in trms f a ral cnductivity, cr: dp <IV = whr j3 is th man squar lctric fild: and cr is fund frm a cmplx prmittivity E by cr = -w m E. Fr an azimuthally symmtric plidal magntic fild, th pwr can b writtn as () Sinc cr is gnrally frquncy dpndnt, it is usful t cnsidr thr sparat cass which span th ntir frquncy spctrum: ) w $ w (in cycltrn rsnanc hating) ci ) - w (lctrn cycltrn rsnanc hating) c 3 ) w» w (rsistiv hating) c ION CYCLOTRON RESONANCE HEATING Fr <. w xpct lins t b quiptntials: = ( ). If, - Cl in additin, w apply th lctric fild in an azimuthally symmtric way

3 such that E = 0, th lctric fild in th plasma can b writtn as dip nrb CI;jj' and Eq. () bcms = 4n ( r 4 (jrbd. _ Th prmittivity apprpriat [ pi t th prscribd cnditins is w - iv. - = (w. - v.) C and th crrspnding cnductivity is (j =.V. p w +. + v. C Nt that th cllisin frquncy v. shifts th rsnanc ff th in cycltrn frquncy. Fr th usual cas f v «w th cnductivity bcms (j = til. V p (' and th pwr absrbd is givn by - w +. C.) C + whr B = M B w ' ci 0 M = w, and b.b M = v i Fr b.b sufficintly small, mst f th cntributin t th intrgtal cms frm B B ' and B can b xpandd in a Taylr sris abut B ' B( ) B + I B

4 3 Th intgral can thn b valuatd xplicitly: dp_ (d J O Q.iji - 7Tn f -00 d5. () Th sum is takn vr vry intrsctin f th jj lin with th cnstant B surfac B = B ' Th in hating prducd by this pwr can b calculatd frm whr V ' = dv d5. = B' Th in tmpratur incras is thn In a ral xprimnt n wuld prbably masur (jj). Hwvr, an quatin fr (jj) can b drivd frm Pissn's quatin writtn in th frm Substitutin f th apprpriat prmittivity (with vi = 0) lads t th diffrntial quatin (s PLP 07):

5 4 - cj RB a B (B_B ) aljj. ELECTRON CYCLOTRON RESONANCE HEATING Fr - w c w cannt xpct ljj lins t b quiptntials and th lctric fild E must b rtaind xplicitly. cmpnnts prpndicular ffid paralll t B: E can b dcmpsd int E = Elt Th prpndicular prmittivity is similar t that [ w E = E l - - V O (w - c - _ \! ) fr th in cas: ] ZiON. Th paralll prmittivity ls smwhat simplr: Fr v «w c th rspctiv cnductivitis ar and

6 5 Th pwr absrbd can thn b writtn as d.q, B Th bundary cnditins n 3 rquir that Ell = cnst. and S.J. 3.. = cnst. can thrfr bring E I ut f th intgral and writ El as S Sinc * sj..s.l S whr a = w p /w = nin ' th magntic fild can b xpandd in a Taylr sris as in th in cas t giv prvidd w assum that a «. Th pwr intgral can nw b writtn as ] dp CliP = Z n + 6BEJ d.q, B B 0

7 6 Fr an istrpic lctric fild, E! and E ar f th sam rdr. In particular, Fr B «B and a «, th paralll cmpnnt f th intgral can b nglctd and th pwr bcms Cfiij = dp B 3" n E 3 TT n E = 3" B 0 L / db d _ = TT n E 3" B I / 0 db dj/, 7T - = 3 n E d V aj3a"iij (4) B 0 This rsult is idntical t that drivd in PLP 07 undr th mr stringnt cnditin f w p «wv. Th prsnt rsult is valid fr w p «r dv n «n c ' Th quantity lnds itslf t a simpl physical intrprtatin Slnc it rprsnts th diffrntial rsnanc vlum in th flux shll As in th in cas, th tmpratur incras can b writtn as mt / (5) prvidd diffusin and nrgy lsss ar nglctd. Th quantity d V. V' S plttd vs. fr varius valus f B in PLP 4. If inizing cllisins ar takn int accunt, th pwr balanc can b writtn as

8 7 whr U i is th inizatin nrgy. But can b writtn in tnus f th inizatin tim L i as Substitutin f this rlatin and Eq. (4) int th pwr balanc quatin lads t th fllwing diffrntial quatin fr kt : kt +U. d (kt ) + = E at L..J (6) If th duratin f th rf puls is shrt cmpard with L ' th scnd i tnu can b nglctd and th tmpratur incrass linarly with tim accrding t Eq. (5) until th rf is turnd ff whrupn th tmpratur dcays accrding t th quatin t (kt ) + U l.. kt + U. L i = O. If th rf puls duratin is lng cmpard with L i ' th tmpratur initially incrass linarly, but aftr a tim f -L i it lvls ff at a valu givn by Th dnsity thn incrass accrding t th quatin dn _ Tf E n dv at - 3V ' (kt f + U.) <maiji :

9 8 which has an xpnntially grwing slutin: Th inizatin tim T is unfrtunatly a cmplicatd functin nt nly f th lctrn tmpratur but als f th xact frm f th lctrn vlcity distributin. It can b frmally writtn as -.l =n nf T i 0 cr (v)vf(v) dv whr cr (v) is th inizatin crss sctin and fy) is th lctrn distributin functin. Th inizatin tim has bn calculatd fr th spcial cas f a Maxwllian distributin in PLP 4. Th rsult is sufficintly cmplicatd that an analytic xprssin cannt b writtn, but a graphical slutin is prsntd in PLP 4. Th fficincy f pwr transfr fr lctrn cycltrn rsnanc hating can b writtn as whr P is th input pwr t th cavity. Th man squar lctric fild can b xprssd in trms f P and th prturbd Q f th cavity as = - P Q E 'C.. w V Fr n( ) = cnst., th fficincy thrfr bcms n = Q n n c B v : B Hnc fr fficint hating, w want a high Q, a high dnsity, and an rf. B frquncy such that V : is larg. B

10 9 RESISTIVE HEATING Fr» w c ' hating taks plac nly thrugh cllisinal rsistivity, and th magntic fild srvs nly t inhibit th flw f plasma nrgy t th walls f th cnfinmnt dvic. Th prmittivity is istrpic and has th frm Th crrspnding cnductivity is = [ _ + i 'X> ] w + V (J = 0 OY V w + V As in th prvius sctin, th lctric fild can b dcmpsd int prpndicular and paralll cmpnnts which ar subjct t diffrnt bundary cnditins, and Eq. () taks th frm Sinc * ( - ) + V w (w + V ), th pwr can b wri ttn as Fr th usual cas f v «w, th abv quatin bcms

11 [ + v a a J ( - a ) J V I 0 whr It is clar frm th abv xprssin that th absrptin dcrass mnatnically with dcrasing dnsity and in th limit n «n c bcms (7) Th mass dpndnc indicats that th pwr S absrbd almst ntirly by lctrns rathr than ins. Fllwing th usual prcdur, th tmpratur ris can b calculatd: (8) Th tmpratur ris is indpndnt f if v is cnstant in spac. Inizatin can b includd by a mthd xactly analagus t that in th prvius sctin lading t th diffrntial quatin kt + u. T = - V E m w (9) Fr n) = cnst., th fficincy f rsistiv hating is givn by f dp n v n = p d d = Q n w c Hnc, rsistiv hating is lss fficint than lctrn cycltrn rsnanc hating by th factr