Tanaka, Shigetoshi

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RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Microwave Conductivity ofBounded PlasmaTanaka, ShigetoshiTanaka, Shigetoshi. Microwave Conductivity of Bounded Plasma.Memoirs of the College of Science, University of Kyoto. Series A 1961,29(3): 395-4061961-09http://hdl.handle.net/2433/257444Memoirs of the Cellege of Science, University of Kyoto, Series A,           Voi. XXIX, No. 3, Article 10, 1961.    MICROWAVE CONDUCTMTY OF BOUNDED PLASMA*                                   BY                           Shlgetoshi TANAKA                          (Received March V, 1961)                                ABSTRACT        An expression for microwave conductivity of bounded plasma was derived in    the absence of magnetic field and then another simiiar expression in the presence of it,    introducing the depolarization effect in each case. It wa$ shown that this effect is     to replace w in specific inductive capacity for unbounded plasma by co'=L-cv{1-L(cop/ua)2}.    The results were as follows: (i) plasma resonance occurs at ce= v'IZ cop, (ii) magneto-    plasma resonance oÅëcurs at two positions of electron density at each magnetic field,    both for the transversal wave (ko//B) and for the hybrid wave (kej-B).1. Introduetion    Microwave conductivities of plasma in the absence of or in the presence ofmagnetic field have been studied by various workers (1, 2), since Margenau's firstreport was published (3). The expressions for conductivity have so far beenderived by solving the Boltzmann transport equation for distribqtion function ofelectrons moving under the Lorentz force. The experimental results on the inter-action between plasma and microwave have been analysed in the theories (4, 5).    In these theories, the plasma was considered to extend through infinite space(unbounded plasma). It is not only theoretically interesting but also experi-mentally important to study the interaction of bounded plasma with microwave,since tke experimental plasma generated an our laboratories is actuaJly bounded.   Investigation on bounded plasmas with no magnetic field was made first byTonks (6) and he reported "plasma resonance" which depends on both electrondensity and geometrical configuration of plasma. Thereafter, the plasmaresonance for the cyHndrical plasma has been studied theoretically and experi-mentally by many researchers (7.vll). Moreover, the plasma resonance for plasmaconfined between parallel plates was theoretically investigated in detail by Wolff(12), by solving the Boltzmann transport equation for spatial distribution of elec-tron density.   In studies on cyclotron resonance of electron and hole in semiconductor,Kittel et al. (13) considered the infittence of bouRdaries, and introduc•ing the * Reported in Japanese in "Kakuyugo Kenkyu (Nuclear Fusion Research)" 5 (1960), 604.396 S. TANAKAdepolarization effect they discussed "magnetoplasma resonance" which dependson both electron density and geometrical configuration.    In this report, the rnicrowave coRductivity of bounded' plasma in the absenceof as well as in the presence of magBetic field was studied by introducing thedepolarization effect, similarly to the research on semiconductor by C. Kittel et al.2, Depelarization effect oÅí bounded plasma    For simplicity we consider the plasma of uniforin electron density withaxial symmetry whose axis, when it is in the presence of magnetic field (B),should coincide with the direction of B. Thus, a spherical or a circular cylindricalplasma is considered (see Fig. 1), in which the electric vector of electromagneticwave lies and is uniform ln transversal plane to B, aRd the frequency is sohigh that ions can be assumed to be at rest. Then, the electron gas is subject                                         ------                                       .': s               /+- + + + S+Nx                                   i tXx                                   /x                                  XN                                                   1                         ffrf SllbSss'                                  XNP z                                    -xZ                                     X --                                      X-                                      -s- --                                        .g--"-                     Fig. 1. Depolarizatien of bounded plasma.to rigid-body displacement in transversal plane from equilibrium position andsurface charges appear oR boundary surfaces of bounded plasma. The inducedfield opposite to the external fieid exerts the force upon the electroR gas, tendingto return it to its originai position. The effect of this restoring force of plasmamay be tal<en into account by introducing a depolarizing field, (•-LP/eo), whichis uniforrn in transversal plane. Tlten, the !ntez'nal electric field is:                               Ei -rm Erm LP/ee, (1)where L ls the depo}arizing factor associated wlth the plasma shape and givenby:                   . -ww- { l/,g Igi 2:.,2e.EggaL,zzS,M,IE,} ,},,-a.The polarization P due to displacement of electrons is :                       P:= eAix =- eNSvdt == ieNv/a], (2)               MICROWAVE CONDUCTIVITY OF BOUNDED PLASMA 397where x, v are displacement and velocity of respective electron aRd N is electrondensity, taking the time dependence as exp(-dat). Substituting (2) into (1),the internal field is:                +                                '                             E,ww-Ewwd-LewwA-r.. (3)                                      eodi '    The equation of motion for an electron in unbounded infinite plasma underthe influence of constant magnetic field and rf electric field is given by :                        m(-dtu+y)v=- e(E+vxB), (4)where v is the effective collision frequency. On the other hand, for the boundedplasma the equation of motion is :                    m{- j'tu(1-L toopt: )-Y y}v =e(E+vx B), (5)where tup is the plasma frequency. Thus, the effect of depolarization is to replaceto by of, which is:         '                     of x to {1-L(tup/di)2} =: di(1-Lr/). (6)    It is noted that the uniformity of the depolarizing electric field in the trans-versal plane is necessary for the above conclusion, as it is satisfied in this case.3. Plasma resonance in absence of magnetiÅë field    By substituting tu' for tu in the specific inductive capacity K for the unboundedplasma, K for the bounded plasma is given by :         , K"" 4ww {1-L'(lco,7'Ebn)-i'//' ff-Li,',,co,"b//,,`O))22} -y ju/., (7)or using Brown's notations (5),                        I< =: 1-l hl Lrp (1 wu Loptp) {n rs. (s)(a) Rofractive index n   From (s), n at zero comsion limit is giveh by:                            n2 =: 1- V L.                                                                    (9)                                   (1-Lop)2•   In Fig. 2 n2 is plotted against op for L=-Ll/2. In this figure, the dotted curvepresents the dispersioR curve for unbounded plasma (L::'LO).398               LO               0,5            pa             q                 0              -O.5             •- !•0(b) Conductivity   From (8), thewhere, ae ='=- e2N/mv 'occurs atas shown in Fig. 3.               1           b"           x           6ct 0,5 -s. TANAKA      '//!IXrestli}',k kX f.' x F x"  Si(>t   'XN'    lx,    ,N    LN,nv--+4 6Xxeq!/v8  ?o1  Fig. 2. Dispersion curves for bounded plasma.real conductivity (on) is given by:            aie - 32            ao (1-Lv)2 -g- B2 'is the static conductivity. According to (10),        1-Lrp == O or co == •J/rr o)p,   ""''' -T-wwwwNwwP                      1                      1                      1                      l                      1                      1                      i                      1                      l'                      1                      1                      1                      l                      l                      I                      I                      i                      lplasma     (10)resonance     (11)0Fig. 3. Resonance       (f/Labsorption curve)rof bounded   7plasma.               MICROWAVE CONDUCTIVITY OF BOUNDED PLASMA 399                                 '    Especially, the plasma resonance of circular cylindrical plasma (L==1/2) occurs                              tu == tup/}/2un, (11a)which was called the transverse resonance of plasma column by Goald (i4). Thisplasma resonance has been studied by various worl<ers (6•vll). Solving theelectric fields inside and outside the circular cylindrical plasma, Herlofsonobtained the result (11a) (11).4. Magnetoplasma resonance for transversal wave (ko//B)   A Iinearly polarized wave incident on an infinite slab of plasma in a parallelmagnetic field is split up into two circularly polarized waves, the right-handed andthe Ieft-handed waves with different phase velocities (15, 16). We consider awave propagating through a circular cylindrical plasma, aiong the magnetic field,paralle} to the cylinder axis.(a) Right-handed circularly Polariged wave   The specific inductive capacity K- for the bounded plasma is :                     Knv =""1 (1-ILrp)a-LopLt r)+i's' (12)Accordingly, nrm at zero collision limit is given by :                        "1 ='"ma 1ww (1 um Lop)('IV um Ln-r)' (13)                                l ix                                     ixeN'ol'E-j.-.---Fig. 4.NNx NxxNanoisrep1'SD,      t  --rt-        b (1-L?)N x  x  x   !N i,C X .l curves forNx xc  x  Å~    Å~      ss        ---hi.a ----L?) l             J  a: L7</  b: Lrg >1  c: L :0right-handed wave..s./400 S. TANAKAng.is plotted againstr with 'o--const. in Fig. 4, which shows that ne increases toinfinity at "r--1-L"o for the case of Lrp<1. As n& increases to infinity at 7'-=1for infinite plasma (L =O), as is represented by a dotted curve, the effect ofdepolarization results in the displacement of na-axis by (-Lv).    On the other hand, oR is given by:                            tiR- B2                                                                        (14)                             oe ww (1-Lv-"l)2-t-B2' •-From (14), the cyclotron resonance for the bounded plasma (oR..x), which iscalled magnetoplasma resonance, occurs at                                ?'o ==: 1-Lop• (15)The resonance position "ro versus rp is shown in Fig. 5, which implies that themagnetic field ro at resonance decreases with increasing electron density op. For rp>1/L, the resonance does not occur, as ro becomes negative. The resonanceabsorption curve represented by (14) has the halÅí-width dr-th2B. It is to be notedthat for the unbounded plasma, "ro==1 independently of "o.                     I•                   1                ),sq                   0                                     //L NXx. )2                                              SNXx                                                   N'ssNx                     Fig. 5. 7e versus ol for right-handed wave.(b) Left-handed circularly Polarieed tuave   Similarly KÅÄ of the left-handed wave is :                       K"" "lm a.ILmp) (1 ww Lopl{J -r) -v'3' (16)Accordingly, n+ at zero collision Iimit is given by :                         'n'd' :== inv (Lv-o(rpLop-i-r)' (i7)                         'n2-. versus r with op==const. is plotted in Fig. 6, which shows that nft increases toinfinity at r=Lop-1 for the case of Lif>1, but it monotonously increases to unkyfor the case of L"o<.'1, in contrast to ehe infinite plasma (L=O), in which n?:.monotonousiy  MiCROWAVE CONDUCTIVITYincreases to unity as showh                        !Nllo/OF BOUNDED PLASMAby a dotted curve incr  -/Fig. 6.4e!                     -                    -                   l                  IC                                 a                  Fig. 6. ])ispersion curves    Next, aR is given by:                           ale                           aewhich implies that the magnetoplasrna                               7'o =:Therefore, althoughthe magnetoplasma resonance occurs at 7'oLv>1. In Fig. 7 the resonance position ?'o              XiS"i                 +f              (Lrp-1-r)2"B2                 resonance                 Lrp-1the cyclotron resonance does noto/riCL7-/) c)-      a: Lv >l     b: L7 <1i C: L=0Ifor left-handed waxre.B2       , (18)      occurs at  . (19)        ccuy for the infinite plasma, for the bounded plasma in the range is plotted against T.           /        /       -Fig. 7. 7o versttsL//---dednaht'fe1rofny wave.7402 - S. TANAKA                         '(c) Linearly polariged zvave    Combining the right-handed and the left-handed waves, ore for the linearlypolarized wave is given by :                    a.R, "= B2{(1-Lop)(21wwFi•2L-lf)322?2"Cl$BE2(1-Lv)2' (20)Accordingiy, the magnetoplasma resonance occurs approximately at ro given by :                              r2o == (1-Ltp)2-32. (21)In Fig.8 the resonance position 7'o is plotted against v. It is evident from thisfigure that the resonance occurs at two positions of v, when the electron densityv is varied with the magnetic field fixed. This is characteristic of the boundedplasma. Ie is to be noted that the resonance at the smailer rp is due to theright-handed wave and that at the larger op to the ieft-handed.               ts"                i                o                  Fig. 8. 7e versus 71 for transversal wave (ko.!B).    The absorption ,curve(aR versus 7') is asymmetric and its half-width (dAr) is :                        ils Y-E':; == I-i-s- A/i F• 2Bii ;.gLop • (22)For B2<<1, dr is approximated by :                                  d"r '-w 2B, (22a)with the exception of r2o=xO.    McGrath et al. (17) derived (20) in a similar way, but they made error intaking L:=1. Frorn the experimental studies on the cyclotron resonance of theafterglow, they reported that there may be two resonance positions for eachraagnetic field actually.,-9rr77N 7777P--XNxxN -l/L 7ttt5.theAccordingly, nÅ} atThe dispersion[A-I] 1-Lrp=l=O(a) rg==a-Lrp)   n2Å} increases against r2 with             eqs               /              o              -l           MICROWAVE CONDUCTIVITY OF BOUNDED PLASMAMagnetoplasma resonance for kybrid wave (keÅ}B)We consider a wave propagating perpendicularly to the cylinderhybrid wave is given by :     K.i--1- op (1-L•4)2-lfÅÄne(1-kop)  1- Lrp (1- Lop) {(1- Lmp)2- rp- 7•2-B2} A- 7'3 {2(1- Lop)2-   zero collision Iimit is given by :      n?L =" i-op (ind L,)2vaLgeXiiij-T-r2} '    'occurs as follows :2- rp>O to infinity at r2==rg, as shown in Fig. 9, in which n?L op w= const. ..sixa.t'KLta 403 for(23)(24)is plottedb                          a                              ,                              l                Fig. 9. Dispersion(b) ?'2,--(1-Lrp)2-T<O   nl increases monotonously toto that for the infinite plasma, forhowever, that there are twothere is one v-region where rg>O[A-Il] 1-Lop==O      curves     unity as r2rp-regions where     fordo2 d2         '               ?     a: (1-Z7)-rg >0     b: (1-L7S?-rg <0    for hybrid wave (konv1-.B).      increases. The dispersion is similar which r3'::=1-v is taken. It is to be noted,      r3> O for the bounded plasrna, while the unbounded plasma (as discussed later).404 S. TANAKAwasAccording to (24),discussed in ?Rs3.Next, aR is given by:        aR sonft=:-oo for any "r, and{(1-Lop)2-this is theop} 2 a- (1 - Lop)2(r2 -}- B2)same condition            tio (1-Lv)2{(1-Ltp)2-rp-r2-B2}2-yB2{2(1-Lop)2-op}2'Therefore, the magnetoplasma resonance occurs approximately at 7'e obtainedthe following equation:                     (1-Lop)2{(1-Lop)2-rp-r,2-32}2 .. o.[B-I] 1-LT=l=O   The resonance position re is given approximately by            r'S =•:=-- (1- Lop)2- rp -- B2 :•==-- {Lrp -- (1 + Ii}l2-:-)}2- (tt-- -i- 41L2)-B2 ,which is plotted against rp in Fig. 10.that(25)from(26)(27)64%'N2o-- 2-4 xxw-"ktbx L7x x     N   Å~  xg <x4 6x<-CPXN"  XX,:X)NOÅé, 8 0E   SincefollowingthetwoFig. 10. 7,2inequalityop-reglons :rg>oversttsmustny for hybridbe sat!sfied,O<rp<opi andwave (kelB)•the resonance occursT>op2,only in the(28)where                MICROWAVE CONDUCTIVITY OF BOUNDED PLASMA 405                 z,;]...92.nvL..Å}--1.,.).mm-,.=i/24[L;,.g.tt-F.tli.:.i.3.:'.2..:!Ei-li., (,7,.c-(,7,). (2g)On the contrayy, for the uRbounded plasma the resonance position is given by :                                7'g -T- 1--fi-B2,which implies that the resonance occurs in the region op<1-B2 only, as shownin Fig. 10. Therefore, lt is characteristic to the bounded plasmas that the re-sonance occurs at two positions of rp at a fixed magnetic field 7'.    The absorption curve (aR versus r) is asymmetric and !ts half-width (dr) is :                  ;.ii, ASf; == -1+,VIT--Åí.],,{2(1-Lop)-1 EOLrp}• (30)For B2<<1, d7' is approximated by:                   iS/1'- =I;./7(:i rm BLT sep':-L-o {(i nv Lv)-H2a :Ii iii/ops}, (30a)with the exception of rg=y.O.[B-II] 1-Lrp-O    From (25), oR/oo=:1 is obtained for any r. The resonance is the same asdiscussed in bq 3. The plasma resonance appears independently of a magnetic field.According to Dattner's experiments (9), this conclusion may be true, althoughthe resonance absorptioll curve becomes complicated with increasing magnetic field.6. Cenclusion    Introducing the depolarization effect, the expressions for microwave conduc-tivity for the bounded plasma in the absence of as well as in the presence ofmagnetic field were derived. It was shown that this effecS is to replace tu in tkespecific inductive capacity for the unbounded plasma by to'==tu{1-L(tup/ca)2].    The characteristic properties of the bounded plasma are summerized asfollows :(i) Plasma resonance occurs at a)==i/]Z)-tup both in the absence of and in thepresence of magnetic field,(ii) Magnetoplasma resonance occur.s for both the left-handed wave and theright-handed and therefore, there are two resonance positions of electron density'o at each magnetic field 7' both for the transversal wave (ke//B) and for thehybrid wave (k,Å}B).    It has been assumed in these derivations that the electron density and thepolarization induced are spatially unifQrm in the transversal plane. For the406 S. TANAKAactual plasma, however, the electron density is not uniform in the plasma. Also,the surface charge lnduced, if the plasma is inserted within a waveguide, on theguide wall has been neglected above.Acknowledgments   The author wishes to express his sincere thanks to Prof. I. Takahashi forhis guidance throughout this work. The author's thanks are also due to Prof.K. Mitani, to Prof. S.'Takeda of Nagoya University and to Mr. H. Ikegami ofElectrical Communication Laboratory for their discussions.                              REFERENCES 1. H. MARGENAu et aL: Phys. Rev. Ie9 (1958), 6; 112 (1958), 1437. 2. I). C. KELLy: Phys. Rev. 119 (1960), 27. 3. H. MARGENAu: Phys. Rev. 69 (1946), 508. 4. for example, J. M. ANDERsoN & L. GoLDsTEIN: Phys. Rev. 100 (1955), 1037. 5. S. C. BRowN: Proc. 2nd. International Conf. on the Peaceful Uses of Atomic Energy,     Geneva. No. P/387 (l958). 6. L. ToNKs: Phys. Rev. 37 (1932), 1458;38 (1931), 1219. 7. N. HERLoFsoN: Arkiv. Fysik 3 (1951), 247. 8. D. RoMMEL: Nature 167 (1951), 243. 9. A. I)ATTNER: Ericsson Technics 2 (1957), 309.10. G. B. BoyD: Technical Report No. 11 (1959), California Institute of Technology.11. W. D. HERsHBERGER: J. Appl. Phys. 31 (1960), 417.12. P. A. WoLFF: Phys. Rev. 103 (1956), 845.13. G. DREssELHAus, A. F. Kip & C. KiTTEL: Phys. Rev. 98 (1955), 368.14. F. H. CLAusER : Sy}nPosium of Plasma Dynamics (Addison-Wesley Publ. Co. 1960), p. 111.15. for example, J. G, LINHART: Plasma Physics (North-Holland Publ. Co. 2960), p. 106.16. S. TANAi<A: Memoirs of the College of Science, University ef Kyoto (to be published).17. L. P. McGRATH, A. F. Cl{ATTERJEE & J. ?. MoNiER: Scientific Report No. 2 <1957),     University of Illinois,

Microwave Conductivity of Bounded Plasma

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Microwave Conductivity of Bounded Plasma

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