ENSE:'iA:"ZA REVISTA MEXICANA DE FíSICA 45 (3) 322-325 JUNIO 1999 The ponderomotive force of a high frequency field acting on an isotropic plasma C. Gutiérrez- Tapia Instituto Nacional de !lIrestigaciones Nucleares. Departamento de F(sica Apartado posta//B-/027, //BO/ México, D.F., Mexico Recibido el 20 de marzo de 1998; aceptado el 12 de diciembre de 1998 Using the hyJrodynamic approach lO describe a plasma, an expression for the ponderomotivc. force ~f.a high frequency field acti~g on an isotropic plasma is obtaillcd and shown in a simple way. Non stationary (crms and the cffcctlvc COl1l51011 [requency belween partlcles are lakcn ioto account. Thc collisionless approximation is analyzed. Kl'.Y\\'ordc Pondcromo!ivc force; isotropic plasma Usando el modelo hidrodinámico para describir un plasma, se muestra en forma simple la obtención de una exprcsión para la fu~rza po~dero. motriz de un campo de alla frecuencia que actúa sobre un plasma isótropo incluyendo lérminos no estacionarios y una frecuencia efectiva de colisioncs. Se analiza e; límile no colisiona!. Descril'fores: Fuerza pondcromolriz; plasma isótropo rAes: 52.35.Mw I. Introduction Two non.linear mechanisms, excluding non-linear wave interaetions, rcsponsihle 01' high frequency (IIF) field mornenlum and energy transfer lo lhe plasma particles can be clas. sificJ. rhe ¡¡rsl one consisls in lhe transfer of momenlum p = f¡k and cnergy n" = n\.JJ by a mechanism rclated ro lhe inhomogcneily 01' lhe HF ficld amplitud~. Here, k, w and ti are lhe wavc vector. Ihe frcquency, and the Planck conslant diviJed hy 217. rcspcclively. J( is known lhat a time averaged 1<1I'cc1I.2J ., (fon) e- 'J = -,---"'--¡- vi Eo 1- , '1J11nW (I) ac!s on plasma particles. where eo ami mo are the dlarge anJ Ill.:J.SS nf lhe particles. respectively. Ea is Ihe HF field ampli[ude. Al Ihis point i[ is important to spccify I\\'o concepls used in Ihis \\'ork: the valid frequcncy mnge 01' lhe melhod used and Ih\.' validilY of Ihe.:assurned linear rclation bCI\\'ccn the currenl and lhe HF clcctrie field. The appro.xirnation of a HF ficld assumed in our (heory is: Ihe oscillation frequency -,-,'01" lhe elcctromagnctic field is largc compared wilh Ihe Laflnor (q:clotron) frequcncy O cJJ/lne.c ami wilh the reciproi.:al transit time t'/ L, where v is the particle vcloc. ity anJ L is the characteristic length 01' Ihe de"ice 13J. In (hc Cl.lllisinnal case, il can be imposed Ihe additional condirion ..•.' » /1, whcre 1I is rhe effeclive eollision frequency hetwcL'n paniclcs. There cxisl several applicalions, whcre lhe study 01 lile motion of charged particles in a HP fields is importanl as found in focusing and Irapping 01' particles. I"or L'\,-llllplc. These processes mean Ihat particles are elaslically hound lO an axis or a coordinate in spacc ir a binJing force acls on thcl11 increasing Iinearly with dislance r. = i.e., F = -er, where e is a constant (4]. In practice, 1'0cusing is ohtained using a paraholic potential 01' the forrn ,p '" (el]:' + (3y' + 'yz') , in eartesian coordinates [3,4). The components, along Ihe x, y and z directions, of lhe equation of motion of lhe partieles moving in a parabolic potenlial as abo ve can be written in tcrms of the Malhieu equalions [lA]. The analysis 01' the cocfficients 01' the Mathieu equations determines hoth the stahle regions (focusing and trapping) and the instahle regions (acceleration) of partiele motion respective1y [51. A particular case of particle motion in n paraholic potential is the 2-dimensional quadrupole speclroscopy where the focusing of particles is rcalized using a HF e1eelric field [4 J. In these de\'ices, the rnost rele\'ant frequeney region ocellrs when -.! > (l/ro) J2e ¡'/m, where V is the HF \'oltage applted nnd ro is the half-distnnee hetween elcclrodcs. To achieve a stl'Ong I"ocusing (trapping) of partieles (3-dirnensional quadrupole field), is tecommended (6) lo use rOlating magnetic flclds 13. l. 8] whcre. the frequency 01' rotalion (..¡.. is comparahle Wilh rhe Larrnor frcquency. An. olhcr areJ of application of HF fields is found in wave guides alle! n:son<ltors theory [5J. If .•: « Le - \I/here :..JLe = -1i711¡(,:.? /1I1e is Ihe electron plasma frequency, within Ihe framcwork of Ihe linear theory-the HF tleld can be introduccd as far as the skin penctration depth amilhe approxima. lion 01' [he ponderomorive force is no longer va:id. rhe assumption 01' a linear relation berwecn the HF field and Ihe curren! is valid when rhe influence ol' lhe HF field 0/1 lhe rlasmJ is slll,-lll. In ;'.n isolropic plasma, this condilion is e.xpressed as Eo « E,. [9J. where "",".t Ec Ep = -;:¡::- :.!:l/2 . Herc k = :-,.J/I.: and E1I -= 31~11If'..•.. ,l/e'2. Te is Ihe elcctron rcmpcraturc and 11I,. is the clcctron mass. In Ihe opposilc THE PONDEROMOTlVE FORCE OF A HIGH FREQUENCY FIELO ACTING O;>;AN ISOTROPIC PLASMA case. for a large-amplitude HF field (Ea» E,). we eross inlO Ihe region of non linear interaetion of a HF field with a plasma where lhe interaction between wavcs beco mes impor(ao1. One of lhe most intcrcsting processes of lhe non-linear interaction of wa\'cs is lhe parametric resonance, This proces s is related to the aetion of a strong (Iarge-amplitude) HF field inducing a strong periodie lime variation of the plasma pararnetcrs. These nuctuations, in conjunction with lhe externally imposed HF field can resonantly couple (parametric resanance). When the plasma parameters oscillations beeome sufficiently strong, sorne plasma mode is driven unstable by the externally imposed HF field at a different frequency (parametrie instability). Parametrie instabilities of this kind have been investigatcd intensivcly in relation to lhe 1a5erplasma inleraelions [la]. To show the influence of the force on a plasma. it will be considered lhe force (1) per unit volume of plasma (praetieally on eleetrons). =_ 2 wLo 16rrw2 vlEol2• (2) where Xo. and ;",;L} = 4rrXo.e~/mo. are lhe dcnsity and Ihe plasma frequency of the speeies a respeetively. In a cold isotropie plasma the dielectric permittivity has the fmm [9. 11] W2 [-l--!&. - w2 (fo) = -- 16rr (e - ., 1) vIEol-. In a more general form lhe cxpression (fo) = E:ij - 16rr óij lasers and masers capable of delivering peak powers [13. 14]. exceeding the tera-walt level the coneept of lhe ponderomo(¡ve force has aequircd a renewed interest. Reeently, (he problem for ohtaining an cxprcssion for the non s(ationary ponderomotive force of a HF field has been diseussed exlensively [15J. Among the most imponant applieations are the following: the beat-wave and weak field partiele aceeleration methods [16-18J. the ion eyelotron isolope separation [19-22J. lhe generation of intense magnetie fields using the in\'erse Faraday effeet [23-25J. lhe generalion of driven currenlS by non induelive methods [26-28] and recently a melhod has been pro po sed using the ponderomotive force. to produce a plasma cdge rolation in tokamaks in order lo supprcss, by a forming thermal harrier, the turbulence resulting [ram the anomalous transport and incrcasing the energy confinement time in lhese machines [29-31]. An express ion for the pondcromotivc force can be realized using single parlicle [32-34]. hydrodynamie [15. 35. 36] or kinclic approaehes [15.37]. As can be found in lhe lileraturc, the cxprcssions for the non.stationary ponderomotive force ohlained in lhese approaehes differ. This constilutes the origin of controversy between the expressions for lhe cnergy-l11olllentulll tensors proposed by }'1inkowski and Abraham [38.391. This paper is organized as follows: in Sect. 1 the basic cquations arc obw.incd. In Sect. 2 an cxpression for the non s(ationary pondcromotive force is obtained where the colli. sionlcss limit is analyzcd. . In this case {he exprcssion for lhe pondero motive force acting on a unir volume of plasma has lhe form 1 323 (3) (3) can be written as vEo,Eoj. 2. Basic equations Thc interaClion 01"a HF f1cld with plasma particles \ViII be dcseribed using hoth Ihe hydrodynamie plasma model proposeo in Ree 40. considcring a collisional tcrm and Maxwell cquations: (4 ) where E: ij and óij are the dielectric and unit tensors respectivel\'. As can be seen from (1). (3) m (4) this force has a pote~tial formo In general. the dieleetrie tensor slrongly depcnds 011 geometry 01' the electric and magnetic fields. Se\"eral rnethods to ohlain lhis tensor are shown in Refs, 9. 11, and 12. Thc second non linear mechanism is reIated to the time dependence of the HF field amplitude. In panicular. the force aeting on the plasma particles arises when the HF field amplitude increases \\'ith time. Duc to the (ime dependen ce of the amplitudc in lhe cxpression for the stationary pondcromotivc force (1). (3) or (4) new lerms will appear. In the lilerature. lhe expression for the ponderomotive force, which ineludes lime Jepcndcnt tcnns. is known as the non-stationary pond~rornotivc force. 1I is a faet that the magnitudc of the ponderomotive force is srnall hUI with the auvent of Bew radiation sourccs likc 1 DE 4rreo --D + --noVa, e t e 1 DB V x E= ---o e 81 v x TI = (7) (8) (9) whcrc flo ilOlhe cffcctivc eollision frequency between parti. eles. Assuming Ihal eleetrie E and magnetie B fields. lhe veloci(ics Vo and the densi(ies /lo have all a slow and rapid time dependcllccs. lhc)' can be v.'riuen in the form E = Rel'. Mex. Fú. 45 (3) (1999) 322-325 (E) + E. B = (B) + B, (J O) C. GUTIÉRREZ-TAPIA 324 whcrc (he anglc hrackcts denote an avcraging (.r(l))=- 1 [1+10 lo , over time [40] the following cquations for lhe fast variables are obtaincd fmm (5). (6) amI (8) 1'(1') <lt'. '0 Thc time inlcrval is largc comparcJ with Ihe characlcristic lime T 01' varialioll 01' lhe fast variables and is small comparcd with the charilCtcristic time of varialion of lhe slow variables. Il will he considcrcu lllal lile slnw ano fasl variables vary sllloolhly in spacc. i,e., suflkicntly \7 x SolUlions - E 01' lhese equations lOE ---o c al = within lhe approximation mcn. lioneJ ahove are the following: whcrc L and .\ tJcnolc lhe charaClcristic distanccs over which thcsc quanlitics vary, rcspcclivcly. In this case, il can be showo 1.IOJ wilh respeel 10 lhe IIF field amplilude Eo. ¡hal in lhe cqualiolls for lhe slow variahlcs it is sufficicnt lO cnnsiLlcr lhe linear approximiltion ami in lile cqualions for lhe fast variables il is Ilcccssary lO use Ihe quatJratic approximation. SUhSliluting (10) inlo Eqs. (5)-(9) anu averaging in lime. the following equations for the slow quantities Vo = c'_' __ 21110 (1.10 + ((v,,), \7) (vo) = ~ \7 . (E) - ((v" . \7) vo} ( I 1) + \7. ((11 0) (vo) 1 \7x(E}=-~OI(I3}, \7 x (13) -111' '\' = -;:- L..(''' + (¡¡o v,,)) = le 13=--\7x O, ( 12) V ~ o = . (n ) r;l., • _ 1- I ., ("') J cOI ,- III~W is Ihe magnetizalion = O. v(t (14) ( 15) Eqllalions (11 )-(15) wcre ohtained using Ihe cuasi-neUlrality conditioll. Also. il can notkcd that the inlluence 01' the rapidly varying motion on the slow 1ll0tiOll is takell ¡nto acCOllnt in lenns 01' Ihe high frequency prcssure and the non. linear force in (1 1) (second and Ihird lerms on (he righl hand side) and in lerms 01' Ihe urag Ilow (¡¡o v,,) in ( 12) ami ( 14). oc - = ( 18) + e.e. exp (-iwl) ( 19) introduced in + r 0' l. -J eo r el - + w- (20) currenl • \7 x (Eo x Eo) , {II)(,;~ 4m2w (1/:!. + w:!.) o " x {1'(E v(; O' i I/n +w:!' Vl~ "')E' V o 1 o -:E ( \7. 75t DEI;) ó) + DEo (\7 . Eó)] [Eo (\7 . DE al. al 1 Z[Eo(r,t.)('xp(-iwl.)+c.c.), ( 16) D~to (\7 . Eó)] u (22) Inlroducing Ihe rclativc conuuclivity currenl where Eo (r, t.) is the slowly varying HF tielLl ampliludc. Considcring lhe tcnns to first orde!" in tcmporal derivalivcs, R('l'. Eo (\7 8~tó) + w2 [ U +c.c.} . as (21 ) anu w 3. Non stationary pOlldcromotive force E(r,l) (17) D~o] + e.e. (-iwt) - -) (I/.(lVn + e.e., whcrc + (¡¡ovo)) lhal lhe IIF field is descrihed iW) v (U) +~~(E), It \ViII be assumed Dt Using (17) amI (18) as \Vell as Ihe notation Rcf. 35. lhe drag t10w (ño n) lnkes the form " \7 . (E) + i OEo) (Eo - : 75t O ((110) (v,,) OEo} iw Va - [Eo - (~ X (,Xl' ~.:.u o 1 _ are oOlaincd: Illn 01 (11,,) {Eo x exp (-iwt) x~ 0 (v,,) 01 iw) - Mi'x. Fú. 45 (3) (1999) 322-325 vclocity 1I,,=(vo)+-( [35] which descrihes 1 "o }ro• the (23) THE PONDEROMOTIVE = O. the equalion and assurning tha! (E) [D;:" (n,,) 11I" FORCE OF A HIGH FREQUENCY FIElD ACTING ON AN ISOTROPIC PLASf\.-1A Iakes the fa/m (11) + ""U" + (u" . v) = 325 U,,] -1Itn (un) ((Vn . \7) Vn) + (~(:\(Un) (vn X ñ) +/1/. 0 D;~n + l/()IIl(\f n := fo. (24) This cxpression constitutes the general form ol"the ponderomolivc force. Suhstiluting (17)-( 19) in (24), it (akes the form (llo)e~ ., 2 ., ") vlEol 1 ( '11In v(; + w- + 1 '1Jlo (n,,)e~ (2 Vo + w 2) {i"" I - _ --"" +2iwEI) X ('"' v X DDEto) + iW)2 DE" i (u" w o -- --Eov'- DEo w DI -. /1'1 el iu" . D (Eov'E')-., + w2 Df o X DI W(I/~ +w'l) [1 1'" +w 1',,-iwE + Eo. vEol- --O' I/~ + w'l Eox (v x Ea) W w 11o + w,2. (v X Ea) DEo v- - Df. /In ., 11;:' + iw DEo +w 2 D t • . vEO + -i -D (E" . vEo) wDI ( Eov.---v.E' DEo DE" Df DI )] } +C.C .. o (25) = In Ihe collision!ess limit (vo O), the exprcssion (25) is retluccLl to that ohtained in Refs. 35, 36 (in the case of an isotropic plasma). Thc f¡rs1 1erm in (25) is (he known stationary ponderomotivc force (3). 1. A, V Gapollov anJ M.A. Mil1cr, SOl'. Phys . .IETP 7 (1958) 168. 20. J.M. Dawson. l'taJ../Jhy.\'. RCI'.Left. 37 (1976) 1547. 2. L.P. Pilaev~kii S(l\~PII."s.-JETP 12 (1961) J(X)8. 21. ES. WClhcl. Phy". /1",.. Le". 4~ (19XO) 377. 3. A.1. Morowv anJ L.S. 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