Tanaka, Shigetoshi

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RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Absorption at Electron CyclotronResonance in Slightly Ionized Gases(II)Tanaka, ShigetoshiTanaka, Shigetoshi. Absorption at Electron Cyclotron Resonance in Slightly Ionized Gases(II). Memoirs of the College of Science, University of Kyoto. Series A 1961, 29(3): 371-3801961-09http://hdl.handle.net/2433/257442Memoirs of the Cellege of Science, University of Kyoto, Series A,            Vol. XXIX, Ne. 3, Article 8, 1961.   ABSORPTION AT ELECTRON CYCLOTRON RESONANCE                IN SLIGHTLY IONIZED GASES (II)'}:                                                    '                                   BY                           Shigetoshi TANAKA                          (Received March 17, X961)                                ABSTRACT        An expression of the absorption at cyclotron resonance was derived for the micro-     wavepropagating through a $light]y ionized gas, transversally to the static rnagnetic     field. It was shown that the resonance occurs at ca,= Vth+2--cap2-y7t instead of cocpt-co    and the absorption curve is not Lorentzian and the half-width depends not only on     collision frequency but also on electron density. Therefore, one can obtain both the    electron defisity and the collision freqttency from one measured resenance absorption    curve. The same expression was derived by solving the Boltzmann transport equation    for the distribution of eiectrons under the same circumstances.1. Intreductien    In the first part:i`;i: dealing with the microwave propagating along the staticmagnetic field (ko//B), it was shown that the cyclotron resonance occurs attu==tuc, the absorption curve is Lorentzian and the haif-width gives a directmeasure of the collision frequency.    In this part, we coRsider the electron cyclotron resonance in a slightly ionizedgas, for the microwave propagating tansversaliy to B (ke j-B). Since the wavebecomes a so-called hybrid wave with transversal and longitudinal electric vectorsin the case of ko Å}B, the dispersion relation is more complicated in comparisonwith the purely transversal wave (ke//B) (1)•    In a similar way to the case of purely transversal wave, approximate expres-sions for the attenuation factor and the relative phase shift near the cyclotronresonance were derived from the specific inductive capacity for the hybrid wave.The same absorption expression was obtained by solving the Boltzmann transportequation for the distribution function of electrons and it was shown that thereis no power dissipation for the Iongitudinal electrical component of the hybridwave. * Reported in Japanese in "Kakuyugo Kenkyu (Nuclear Fusion Research)" 4 (June, 1960), 550,** The preceding paper, hereafter referred to as I,372 S. TANAI<A2. Dispersion o'f the hybrid wave    We consider a microwave propagating through a siightly ionized gas in thedirection of ko transversal to the static magnetic fieid B.    The electron motion parailel to B is unaffected by B, so the wave with theelectric vector initially parallel to B can remain as purely transversal wave, andthe dispersion relation is the same as in the case of no magnetic field. Such acase is left out of consideration. The interesting case to be considered is thatin which E is initially perpendicular to B. Taking the g-axis parallel to B andthe x-axis to ko, the wave, incident with Ey component only on the plasma slab,comes to have both E. and Ey componentsinside the plasma. Sucha wave isso-called hybrid transversal and long2tudinal wave (1, 2). ,    For the hybrid wave, the effective specific inductive capacity K-L is givenby (3, 4):                      K-L ==: 1- ci-,,--nwwn.,(,1wwwwBT,)Lia-iB•B)(2-,)-, (1)us!ng the notations given in I. We proceed in a similar way.    If op <<1 and B2<<1, by expan(ling (1) with respect to op, the refractive index(n.J.:-=--:nl-l-1'nrt.') can be approximated by :     S n .1. r-= i-m --?27--- (i iSilllimu-II3.o Stii3-, )'72i. "rB22)(ll'l+wwi3 1>)2                      o72 {(1- 77)(1- )7-r2) +B2}2- l?2{(1- o7)2 -- -r2+l92}2 (2 )                      8 [(1-of)-7'2-B2)2-F32(2-o7)2}2 'and              nfuse2va(i-,(-irh,Z)B2,")-,r.2`B'/•(2-2-,),                   "i[i-Fga=(z'1{-wwopi/im(iB-,)V,F,"/'2,2(g22,),]• (3)    If the last terms in (2) and (3) can be neglected, the relative phase shiftdell- and the attenttation ev]. per one free space wavelength 2Lo are :                   demL =t rrvaÅí1rpwwmrp.r),(ww1ff,)op,iA/'2,)(g-B2rp), (rad/R,), (4)and                                (1-v)2+r2-I-B2                                                 ,(nep/2o) . ( 5 )         ev-L == nrpP                           (1- o7 - "r2- E?2)2+ 32(2- of))3. Resonance position and maaximura absorption    The cyclotron resonance occurs near the value at which the first term ofthe denominator of (5) is equal to zero. In the neighbourhood of resonance, aibecofnes approximateiy :            ELECTRON CYCLOTRON RESONANCE IN IONIZED GASES (II) 373                                    i92(2-T)2                       alL == CUTLm(?.,t,-?.,),-i-3,(2-mp),, (6)where                               azt.m=-=-e=um-Lij,-rrBO, (7)and                              )]2o di- 1 - v - B2 , ( 8 )or caZ =: to2-to,2,-yZ. (8a)Thus, frorn (6) in the above approximation, the cyclotron resonance occurs at7'==7]o which is on the lew magnetic field side where 7,::t-w./o) is smalier than unityand the maximum attenuation is a-L.,(nep/2,). This is characteristic of the hybridwave.    Next, we investigate to what extent the approximation of (6) is permissible.The exact resonance position (rft) of (5) can be determined by (7cri./6r==-O, fromwhich follows :                                                                  '                     ?'?t ::-=- (1- "o){1 -t- -if2-- (IiE op)2} a- o(B`), (g)where O(3`) is of the order of magnitude of 3`. Therefore,                        ?•?i-7•g -- -SL'iun' rprp+o(B4) >o, .(lo)which ifnplies that we can regard the positioR of cyclotron resonance as ?'o,corresponding to the neglect of B2 against unity.    Substituting (9) into (5), we can obtain the following expression for the exactattenuation at cyclotron resonance (evÅ}R) :                               f.o  crcr-irS. t" i"-Z-(Er,Ey)2 N""".,.,                                           "         +O(3`)>1, (11) Q8 N" ..,                                                 "tf'hO,M..W.\•ACII.l)lett,C.a.n,trie.g.ar,d, pa'"Osia6 X"itN:}Å~{/.tsog..,.,                                                                Rxcrj-,n , in accordance with theneglect of B2 against unitY• o.4   In Fig. 1, rg is plottedagainst if. In Figs. 2 and 3cr.L,n is plotted against if andP respectively.    o.i o.2 5.3 o.4 os            7--(ccip/w )2Fig. 1. Resonance position (7o2) verstes ?i.374 S. TANAKA2ioliii'2lo-i Doh.ilO'   oxk   o•p& op?o>o6 tO-tilo-5 2 46 lo- 2 4  T =t (CUp/w )2 2Fig. 2. cU.1.)n versus ".2'j'lk2g2IO-' '2ok  9 os<oIPxxo.Oeq.tlo-5 46 iO-Fig. 3. ila+Lnl2 46 io-'fiversus P,2            ELECTRON CYCLOTRON RESONANCE IN IONIZED GASES (II) 3754. Absorption eurve and halÅí-width    It will be seen from (6) that, although the,absorption curve plotted against 72is symmetrical with respect to the line r2mr2o, the curve plotted against 7' is notsymmetrical and somewhat steeper on the side of ;'>ro.    From (6), the half-width (dr) of the absorption curve against ?' is:                                    2-T                                          +O(fi3), (12)                            dr -- B                                  (1nvrp)i12and the measure of asymmetry is given by :                           d7•pm dT+ - B2 (2-rp)2                            2'2 unT(1-v)3i2' (13)which implies that the curve is more asymmetric with increasing B. If Bo andAB denote the magnetic field intensity at which cyclotron resonance occurs andthe half-width respectively,                      d.B, :== 4'g' - B2iii,rp-yo(pe) :::-iieL2--T,, (i4)which is useful for determining vc from the experimental data, since we candetermine Be and dB by experiments.2oL8Q6Q .4oili2Qofi # o.os        /       /      -     ---  - !Z-X           IZ #: O.2          /x         !x         /N        1       Itx              x       '       1     /f f)<    l   /1 // /l // /f11/        IZ       /l    // IL !/ .ijn"ixsl  1  /lx/lx  N    C.O5    r-, O.Ol      N   xNx    kX         N     xx     Xx Xx      xx       xx        xx        xx Xx.. Nxx    Xx      xx          s-s-sfull curves caiculoted   from (6)dotted curves calculated   from (5a)    X:N            -tg;.I . N  lYY/l                      O,80 O.90 l.OO LtO                                           .di = (Wc /`a )                  Fig. 4. Absorption curves versus 7 for various ny.    In Figs. 4 and 5 several absorption curves (evi/ath,.) are plotted against r. Inthese figures, full curves are calculated by (6), and dotted curves by the followlngexpression, which is derived from (3) and (6):376 S. TANAKA                            '                      CU.1. - B2(2rm•o)2                     CUJJm im (rb'-r2)2-f"r/f2(2mu .ij)2(2nv 'n•fL) ,with nl approximated from (2) by:                      nft-:-:s-i---Z-S-/8te-I-i-i:rp--'i-S-//il•i-:'k---i-J<-522---;ir,E-?•       l.2 thl                             it!                                   t"` t               t? =o.t      t.o ..•,r"[i lxl                               i                              llx    XV-' - i. 1 111 'iliXXxxlxxNXNx g                        -N                            s...s                              / S-"`-L.       O,ilfuti curve•s catculated    from (6)dotted curves catcuiated   from (3aj .x Å~Å~    xx       xx          sNXNxs t`s ..<3a)(2a)                .80 Q90 LOO LtO                                         f = Ccac/ca j                 Fig. 5. Absorption curves versus 7 for various B.It is to be noted that the dotted curves are more exact than the full curves andthat the resonance positions of the former are on the slightly smaller side ofr and the maximum absorptions are slightly Iarger than those of the latter.   In Fig. 6, (nl-1) calculated by (2a) is plotted against T.              i.2             Q8                                               tL"x                                              lx/gQ4 o-O.4-Q8.t t,2 '     T in7-- Ql, fi =o,os --N             s.   7= o,2       ,P;O,05    Fig. 6.V=O,05,fi=OO508 i O.91/xDispersionxes'vurc    ---i  i  t L 1' ?7tQt,/eao,o2    V=O,f,P=QOi verstts 7. N/t- mmlL:  1.o d' = rCac/cu 1           ELECTRON CYCLOTRON RESONANCE IN IONIZED GASES (II)5. Cemparisen of cyclotroR resonaRces between yarious waves   The behaviours of cyclotron resonances are quke different according towave types. The characteristics are tabulated in the following table:                   Table. Characteristics of cyclotron resonance.773ehtRight-handed   wave Transversalwave (ko//B)Hybrid wave  (hoiB)                            7==1 i 7o2=1wwB2                                                      7o2"=1-n-B2       Resonance position independent independent dependent                            Of lVlc,Yc Of Ne On Ne,Vc                                                      1 - ny 7v"                            rr" 1rrny       Maximum attenuation                                                      2-ny B                            B 2S                                                      (2-n)S   Half-width                            2B 2p       Absorption curve                                                non-LQrentziansyrnmetriÅëal Lore tzian       Approximate expression (8) inl (20) inl i S?gsg:ttg:rt      --nvN--tt ' '6. Derivation by Boltzmann equatiop    We can derive the same expression for the absorption curve at cyclotronresonance by solving the Boltzmann equation.   Desloge, Matthysse and Margenau (5) have shown that if one assumes thatthe effect of high frequency electric field upon an electron in a slightly ionized gasis on the average an addition, in the direction of tke field, of a velocity compo-nent, the magnltude of whlch is dependent on the electron velocity, then it ispossible to derive very simply tlte expression for conductivlty. Expanding theirinethod in the presence of the magnetic field, Dodo (6) has obtained a tensorspecific inductive capacity which is coincident with that derlved about uniformdistribution of electrons.   Following these methods, we derive the cyclotron resonance absorption curve.The Boltzmann transport equation for a plasma in a high frequency electric fieldE== Ee exp (--y'of) and a static magnetic field B may be written, using well-knownnotatlons :                                                                        tt                            +coaf.M-i-Fv,f--D-Dg/.f--, as)where                           F.., enEl -t- !Z (vx B), (16)                                mm               IDZCllf- -- Sf(v', t) W(v', v)dv'-f(v, t)S PV(v, v')dv', (17)                           W(v', v)-W(v', -v) (18)and                           S YV(v, v')dv' .-.- y(.).378 S. TANAKA   Next, if we assume that a}1 of the electrons which are in a given velocityrange in the equilibriurn distribution are shifted by the Lorentz force to a newrange in which the velocity differs from the original veiocity by an amount g,                f(v, t) =- fo(v-g)(i- 3.a,g.x)(i-ia3SIII't)(i-{3Sgi), ag)where fo(v) is the equilibrium distribution function and gi-wwgi(v, t) (i-mx, y, g).If f(v, t) of (19) is expanded in powers of gi and one assumes rapid convergenceof the resulting series,                           f(v, t) =fe-v•G, (20)where                            G = -,1- ro6s(fog), (21)which is even with respect to v, assuming g==g(v, t).   Substituting (20) and (16) into (15), we obtain:                .{ilti.-EXEtmlr3bSbo.S(GÅ~B)}+SEiti(G+,gib)                   ur vf(v, t)-!f(v', t) W(v', v)dv'. (22)Replacing v by -v in (22) and subtracting the resulting equation from (22),one finds                     tblt=-!iiE:L.;-liSto+ISi(GÅ~B)-vG, (23)which is coincldent with Eq. (14) in Kelly-Margenau-Brown's paper (7), inwhich f(v, t) is expanded in spherlcal harmonics and b is used such as:                              b-: -vG. (24)If G is obtained by (23), the current density J is given by:                  J" NeSvf(v, t) dv ==: -NeSv(v•G)dv (25)and the average power (P) absorbed by electrons per unit volume is                           P---S- Re (J•E'i`). (26)   Now we calculate for the two interesting cases.   (i) Transversal waveSince the wave is purely transversal, in the above-mentioned coordinates Jexists only in the xy plane. In quite the same way as Brown et al. we obtainfor the constant mean free time;            ELECTRON CYCLOTRON RESONANCE IN IONIZED GASES (II) 379                    ,is == I4ilS2o "voi"g'lf- {-(i- ?4)2, ms-o-t-a-y 73. )2,+si-}, (27) which is identica} with (22) in I.    (ii) Hybrid wave E and J have the x- and y- components. Solviitg (23), we obtain:               (g.x) -- -igi -g- i]/i 6e (d.-,1),ÅÄ .:,(IItu,,T:;v ]•.toLfi.)(EE.x). (2s)Therefore, from (25) fx is given by:                    J. == g?iii;VS(dtu( 7;.Yww)g)x,l' to.c},Ept 4rrv33eo dv. (2g)lntegration is easily carried out for the constant mean free time and Jx, Jpt aregiven by:                 (f;) "rc 'lit?a{- f3i)2- ,;-2 ('I(}- i- IR EIB))(Z:)• (3o)    On the other hand, using the specific inductive capacity tensor (K) for thisconfiguration, the electric field vector should fulfi1 the following equation :                        v2E-vv•E---,i,(K)t-Dt. (3i)Therefore, the longitudinal component Ex can be expressed by Ey as :                Ex == bei f.' Ey == - (1-vww ?.2-i'Brp2r) +i3(2.rp) Ept• (32)    The average power absorbed from this hybrid wave (P) is the sum of thepowers absorbed from the longitudinal wave (Px) and from the transversal wave(P.). Substituting (30) and (32) into (26), we obtain:                 p-. -= g Re (f.E.*) -- o,and                 ,pl -- iillX2s2toa-B,21i,I(,i.,mh$2){'23-S(B,22,,), -= ,p, (33)which is coincident with (5). It is concluded from (33) that as there is no powerdissipation in the longitudinal compoment, the current component Jx is out-of-phasewith the fie!d Ex.   It has been shown by Watanabe (8) that the effecS of the longitudinal com-ponent of our hybrid wave can be neglected, and K is expressed by (1) to a goodapproximation under the following condition :                                7'ercTq                                     << 1, (34)                                -M,icme2''380 S. TANAKAwhere re is the ratio of specific heat at constant pressure to that at constantvolume of electron gas. The above condkion is fully permissible in the cold plasma.    If we express the ratio of the Iongitudinal field component to the transversalone as:                            E./E.-Aexp (7'e), (35)we obtain, by (32),                     A == opr/{(1-rp-T2 -B2)2A- 32(2-T)2} i12, (36)and                          e =--- tanmui{1'-k'('2umtl"2,,") B2}. (36a)Therefore, at cyclotron resonance,                          A:= rp7•,/B(2-op), e==O. (37)7. Conelusion    Using the specific inductive capacity, we have derived the expression for theab$orption curve at the electron cyclotron resonance of a slightly ionized gas forthe hybrid wave. It was shown Åírom "L'ne analyses that the behaviour of theresonance absorption by the hybrid wave is very different from that by thetransversal wave.    The maln features are as follows :(1) The resoRance occurs not at to,=-eJ but at to.:--]/to2-e)}Z,-yg;(2) The absorption curve is not Lorentzian;(3) The lta}f-width depends not oniy on colllsion frequency but also on electron     density.Although the expression for the absorption is more comp}icated, there is thefollowing advantage in this case. As shown above, one can caiculate both theelectron density and the collision frequency from the resonance position and thehalf-width, by measuring one resonance absorption curve. In addition an experi-mental apparatus becomes simpler. We shall report the experimental data andanalyses based on the above discussion in the next part.Acknowledgmerrits    The author wishes to express his sincere thanks to Prof. I. Takahashi forhis guidance throughout this work. The authoy's thanks are also due to ProÅíK. Mitani for discussions.                                REFERENCES1. J G. LINHART: Plastna Physies (North-Holland Publ. Co., 1960), p. 106.2. L. SplTzER: Plzysics of Fttlly Ionized Gases (Interscience Publ. Inc., 1956)T p. 55.3. S. K. MITRA: The UPPer AtomosPhere (the Asiatic Society, Calcutta, 1952). .4. M. A. HEALD: MATP17 (1959) Princeton University.5. E. A DEsLoGE, S. W. MAi"r}IyssF. & H. MARGENAv: Phys. Rev. 112 <1958), 1437.6. T. DoDo: (to be published).7• D• C. KF.LI.y, H. MARGENAti & S. C. BRowN: Phys. Rev. 108 <1957), 1367.8. K. WATANABF.: I<akuyugo Kenkyu (Nuclear Fusion Research) 5 (1960), 594 (in Japanese).

Absorption at Electron Cyclotron Resonance in Slightly Ionized Gases (II)