Ando Masahiro

Enomoto Yoshio

Kawasaki Sunao

Yagi Teiichiro

川崎 温

八木 禎一郎

Kanazawa University Repository for Academic Resources

ScL Rotl.広島naz思waUniv. Vo!. 21. No. 1， p. 9-18 June 1976 Measurements of the Eled:ron Density of a Plasma in a Magnetic Field Y oshio ENOMOTO， Masahiro 閃， Sunao KA W ASAKI， and T eiichiro Y AGI ザPhysics，Faculか01Scie刀ce，Kanaza1ω Un抑制か(R邑ceivedApril 30， 1976) Abstract We compar日 th忠告lectrond邑nsitiesof a plasma in a magnetic fie!d measured by the disk， plane， spherical， and cylindrical prob記withthos氾 measur芭dby th巴microwavecavity method回 Whenw芭 comparethe el巴ctrondensiti関 inthe wid記r昭ionor Helium gas pressure， only the spherica! probξagrees with tle cavity method Thus， itb邑comesevident that the theory of sph邑ricalprob百ina magnetic field giv邑sus th居mostconfidence L Int]'oducHon. We are studying now the interaction between a Helium plasma and a spira1ling electron bεam. In this process， we need to measure t加 electrondensity of a plasma in CMA diagram more accurately. One of the fundamental method by which we measure the electron density of a plasma is an electric probe， that is， Langmuir probe. But the theorγof probεis extremely complicated. In many cases， such as in a magnetic a probe current depends on the transport coefficients of a plasma and the absolute magn・itude of a1 electron density can not be calculated with certainty. In spite of the method of probe is an important one for a local measurement. On the contrary， inthe presence of a magnetic field， a microwave cavity method measuring the electron density of a cold plasma has been suggested. In conception， itis very simple and has fairly high precision. But it is inefficient for the spatial distribution of a density which varies with position， so that is given an avεrage value alone. The prob吃 methodis effective， but it is doubtful wether it will act rightly or not 80 far as we do not try a cross check. In many cases， itdoes not give a convincing proof. So that we tried a cross check by the method of microwave cavity. 2. Tl:問。ryThe plasma which we are going to research now is introduced into a cavity reso-nator， the medium having a complex plasma conductivity in a magnetic field， as a result 一世10 Yoshio ENOMOTO， Masahiro ANOO， Sunao KAWASAKL and Teiichiro YAGI the shift in the resonant frequency of a cavity and the change in the Q-value of a cavity are mea8ured together. When an electron concentration is low (η ne2 / mεo w2<<1)， the first order perturbation i8 good， because the plasma does not appreciably disturb the electric field. When a density is increased， the electric field E in the pre-sence of a plasma becomes appreciably different from Eo in the absence of a plasma. Then the foHowing three effcts must be considered. 凶 driftof a plasma by Lorentz force in the presence of a magetic field. (到 acspace charge (commonly called収plasmaresonance" makes itself feel when η approaches unity， 80 caHed when the probing frequency approaches and it is equal to the plasma frequency.) (c) when both ηand the pressure are enough so that the Q-value 1S lowered， the overlapping of higher mode also causesE to be different fromEo. By proper method of design， the effects and (B) may be eliminated. (A) As the magnetic field in thεz-dire氾tionexists in our experimental device， ifthe distri-bution of the electric field in a cavity has the electric field Ez in the direction of a the p!asma i8 not caused to drift by Lorentz force. (B) This can be shown th台-oretically as foUows， combining Maxwell eq. with continuity eq. and assuming such a harmonic time variations as exp (jωt). ~乙= div i (l) at From (1)， -iwp = divσE div E =ρ/ε。(2 ) From ( 2 )， div E = divσE/( -jωε。)= div (l-K) E ; K =1十σ/iωε。-grad K. E + (1 - div E ¥div E = -grad K 0 E / K ; K =f(η)圃The eq. states thatρwill be zero when the applied electric field E is normal to the density gradient. If therefore， by a proper experimental arrangement， this condition is satisfied， the space charge effect will not limit the microwave cavity method from measuring a high electron 1n our experimental device， an electron density varies with radiaUy directional position from a axis (grad r nヰ 0).We selected TMo10 mode from the above reasons. A cylindrical cavity that oscillates in the TMo10 mode with a cylindrical plasma column placed along the axis of a resonatorフ satisIIesthe required condition. Fur咽thermore，TMo10 mode has large separation with the higher modesl。In the presence 01 a static Ma釘leticfield， a plasma becomes an anisotropic medium. As thθresult， a plasma conductivity is a tensor function of position and a dielectric coεfficient of a plasma is defined by the foHowing tensor2). Me，ωuremenお01the Elecfron Densi，か01a Plasma in a Magnetic Field 11 50cm Lーー」ー~ー'-'Plasma Souce Electron Gun Fig. 1 Schematic diagram of experimental device G 30 ~戸νヘ(C， C， 3A 1. 11'1:1:=調"・'.c，-c， 8A 1ρ .国 民属高... c" C， 3A c，-c， lA 1. C，-C， 8A C，-C， lA 。c3'c" Cs Ce10α ¥ c， C， C， ca zC2 Csc. C5C6 C7 C8 _C9 50 10‘ 150 佃50 150 αn -100 Cb C， 3A 官 民首，'..‘ C.-C， 8A C， C， 3A I. 2目。 C，-C. -2A C， C， 8A Ip c，-c， -2A 1. -300 -3001 、./Fig. 2 Magnetic field intensity (theory) Fig. 3 Magnetic field intensity (experimental value) [Krr Kro 0 [ K=l+σjjωeo =1 K併 Koo 0 1 100K"， 1 K is given by K"z = 1 +σZZ !jWeo 1一一 r1一(1 -j分in the similar fashion ????????? ???? ? ????ァ???????? ??????where a2 ω~IωZ ， β=ωblω'， y IIclω， and ω主 ne2 1mε0， here n is a electron density，ω eBI m is a electron cyclotron frequency and Ic is a electron collision fre-quency for momentum transfer. For the TMomo mode in the Absence of a plasmal)， Er = Eo = Hγ= 0 12 Yoshio ENOMOTO， Masahiro AN以)，Sunao KAWASAKI， and Teiichiro YAGI Ez = 10(k1r) = 10 (Xom r/a) He = -10 (k1r) = -10 (Xom1グa)，where Xo拙 =mth root of 10 (x) = 0 and a is the radius of a cavity. With the plasma present， the axial electric field is given by solving Maxwell eq. div E = 0， d2Ez 1 dE 一一~ + k2 Kzz Ez 0 1子f-+ ~crr 0 三 r 三 R d2Ez 1 dE --.，-..-十一一一寸三一 +k2 Ez = 0 ar“ r ar ;R 三~ r三;a，where R is the radius of a plasma and k =ω/c ;ωis a resonant angular frequency. The solution of above Maxwell eq. is given by [AJ a2 =マ<l， y=OEz = Eo 10 (k K泣r)+ E' No (k K1J:;r) 0三 r三REz = A1o(kr) + BNo (kr) ; R :Sr三 α(3) (4 ) as In order to have the finite solution (3)， E' becomes zero. The electric field Ez and its derivatives must be continuous at r R and Ez must vanish at r a. From abo. ve-mentioned relations2) c=一品i担L10(初)α。(統)+ No(統)C]1(kR) + N1(kR) 10(kK泣)K1J:U1 (kKljlR) Eq. (6) gives the equation for a resonant frequency. [BJ a =マー~ 1， Y = 0 In the similar way as [AJ， Ez = Eo 10 (k / a -1) 0孟 r孟 REz = A10 (kr) + BNo (kr) ; R 壬 r五αα。(版)+ N，。(統)C]1 (版)+ N1 (百訂=I。(k/ a -1 R) -/a2-111 (R/a-1 R) So that eq. (9) gives the equation for a resonant frequency， too. ( 5 ) (6 ) ( 7 ) (8 ) (9 ) In the limit of vanishing small plasma densities， Kzz 1 and eq. (6) properly reduces to 10 (初 0，which is the characteristic equation for a empty cvity. Eq. (6)， (9) posses an infinite number of modes of TMo刷 class.It was solved numerically for its lowest root in terms of a dimensionless quantity初. The vesonant frequency is also given according to the electron density of a plasma. As a result， the relation between the shift in the resonant frequency of a cavity and the electron density is given from eq. (6) and (9). Meo-surements 01 the Eledron Density 01 a Plasma in a Magnetic Field 13 ka I ka R=l.Ocm 42 0.9 ---2.5ト0.8 〆/〆〆〆 4 グ 0.9。7 ぷ予一三 0.80.6 0.7 3.0 0.5 委主主三三二;:2.4 2.4 5η Fig.4 The r邑lationbetween Ka and甲 (1) Fig. 5 The relation between Ka and叩 (2)ka I 忌ニム年旦0.9 。50 100 Fig. 6 The relation beween K昌and早(3)CM-' 1011 Ne 1010 10' L一一一一ー~ーーー」10-1 1013 Ne 1012 lQll l K=O.lcm ムfFig. 7 The relation between Ne andムf(1) '---..l. 10 ムf」一ー」10' MHZ Fig. 8 The relation betw巴enNe andムf(2) 1.0cm 14 Yoshio ENOMOTO， Masahiro ANDO， Sunao KAWASAKi， and Teiichiro YAGI Then， the relation and are plotted in FigA~6 and the radius of a plasma is put as a parameter， so-called plotted the resonant frequency of a TMo!o mode cylin-drical cavity as the function of the electron density of an axially located plasma， and in Fig. (7)， is plotted the electron density as the function of the shift in MHz unit of the resonant frequency of a TMo!o mode， where the radius of a cylindrical cavity is 2.98 cm. 3. Probe theo:ry in the pre愚容nceof a magnetic field The electron concentration of a plasma is got from the electron currεnt near the space potential. In the case of a plane it can bεshown that the number of electrons StI北 ing1 cm2 per second of a solid plane parallel to a magnetic field B decreases as B rises， and the electron current near the space potential 1S given by the following equation3)ヲ1 沌v 8+ (ωτ)2 {1 -exp 2π/ωτ1 e 4 2 ¥4+ (ωτ12 f where j is a current density near the space potential， e 1Sa charage ofτis a mean collision time， v is a mean :radon / 3 i(T e/m (m is an electron mass and Te is an electron temperature)， and ω1S a angular frequency. The magnitude of a electron current near the space potential is given by Bohm， Burhop， and Massey for a probe of arbirary shape4)， {L21主ー (K+ 一一生一 )~14 16π;-αCD where K 1S a constant (K = 1)， S isa pobe surface area，α=Dょ/D = 1/( 1 +ω21)， D =λ ，λis a mean fr問 pathalong a magnetic field， C 1S a capacity of probe， and D， Dょisa difusion coefficient and across the field. In the case of a spherical probe， b=α トーな~αd=〆すい +λ)句 J京:where αis the radius of a spherical probe and な isLamor radius. And C= d (1-ρ2)112 tanh~l (1 -ρ2)1/2 ;ド十壬 1d (ρ2 _ 1 )12 tan~l(戸2 _ 1 )12 C= ; t = す~ 1 These formulas give the saturation electron current at a small positive potential under the assumption that く< T e， that orbital motions can be neglected in the free -faH region， and that quasi-neutrality obtains elewhere. Measurements 0/ the Eledron I加'si，か01a P!asma in a M.向gneticField 15 4. Expe:rimeniaI apparatus The experimental apparatus used now is schematically shown in Fig.1. A plasma generated by the plasma source placed at the left end flows into a vacuum chambεr through the hollow anodεof the plasma source and the vacuum chamber is made of Pyrex glass. Solenoidal coils placed outside the vacuum chamb信rproduce d.c. magnetic field. The intensity of the magnetic field is varied from 0 to 600 Gauss. Helium gas is supplied into the central region of the vacuum chamber from a gas bombe through a leak valve in the left end. The pressure of the arrangement is varied in the range from 10-5 to 10一1 Torr and the lowest attainable pressure is in the range of 10-6 Torr. The magnetic field intensity as the function of a position along the linear machine is shown in Fig.2 (theoretical value)， where C1 ~~ C9 denote the positions of coils. And Fig.3 denotes the experimεntal value of the magnetic field intensity in our experimental device. The cylindrical cavity is made of brass. We made holes on the axis of it and covered these holes with wire net. The wir・enet arrests the lowering of the Q-value. We put the cavity at P in the vacuum clamber， then the beam plasma passes through the cavity. The shift of the resonent frequency of cavity is measm:・edby a spectrum analyser. 5. Exp凹 imenialr記sultsThe radial density variation in a plasma column is measured by a radially movable probe at P in the vacuum chamber， whose width of a half value is about 2 cm in spite of the kind of probe used in the measurement. The probes used are disk， plane， spherical and cylindrical probe. Each probe is 1，(mA) J， 10 -40 ん=0.5Aι=8AP=5X10-'Torr 7.;=1)， ]，=3A B=200Gauss .-"'?~堵~tan α=e/xTe>く2.303xTe=10.1eV 。心 40 V V" Fig.9 Probe char呂cteristic(spherical prob邑)16 Yoshio ENOMOTO， Masahiro AN以)， Sunao KAWASAKI. and Teiichiro YAGI consisted of a tungsten electrode and a tungsten wire (0.5 mm diameter) and the tungsten wires are al sheathed in glass. The radius of the disk probe is 1 mm and the thickness of it is 0.5 mm. The plane probe is sheathed by a glass except for the circular part of the disk probe and the radius of it is 1 mm. The radius of the spherical probe is 1 mm. The cylindrical probe is made of turgsten wire and the radius of the elec-trode is 0.5 mm and the length of it is 1 mm. We neasured probe characteristics by using a X -Y recorder or a synchroscope， but we mainly used a X-Y recorder. One example of a probe characteristic is shown in Fig.9. 1 eis electron current. Electron temperatures }{ Te are about 10e V in spite of the kind of probe from probe charateristics. Space potential Vρare about 5 V in these probes. The radius of a cylindrical cavity is a = 2.98 cm and the length of it is L = 5.00 cm， so that the resonant frequency of a cavity is 3.84 GHz. The microwave cavity method gives information averaged over the radius of a plasma R = 1.5 mm. We put in the input to a cylindrical cavity by a rod antenna and received the output by a rod antenna too. We set both of them symmetrically at r = 1 cm points from the cetral point. The effect of a magnetic field or a discharge current 1 d or a Helium gas pressure for measured electron densities is shown in Fig.l0， Fig.ll， and Fig.12. Here， a cylin-drical probe is plotted without taking account of the effect of a magnetic field. In the first place， an electron concentration versus a discharge current 1 dis shown in Fig.l0. The microwave cavity method and the probe method agree within the exper-imental errors in spite of the form of an electrode besides the cylindrical probe， where P = 5 X 10-2 Torr and B = 200 Ga附.In FiG.ll， an electron density versus a magnetic field is shown. The probes are in ?????? ?I.=0.5A P=5 X 10-' Torr 。20 'X resonant cavity o disk probe 0 a spherical 0 4plane • cylindrical ?????????P=5 X 10-' Torr )( measued by the resonant cavity ゐsphericalprobe odisk Ii plane • cylindrical 。x ，{ ???10 10 oX x o x d o ~ A X Il o ??????????????? ??????。???。???。 ???? ?. 1.5A 。g 企R，"o • I.A BG Fig. 10 Electron density versus discharge current Fig. 1 Electron density versus magnetic field Meωuremenお0/ihe Electron Density 01 a Plasma in a Magnetic Field 17 agreemεnt with the microwave cavity method in the region where B 孟 480 Gauss within the己xperimentalerrors besides the cylindrical probe， so that the probe theory in a magnetic field holds in thεregion of this measurement where P = 5 X 10-2 Torr and Id =O.5A. FigJ2 shows the relation of an electron dεnsity versus a Helium gas pressurεat a particular condition. The spherical probe is in agreement with the microwave cavity method within the experimental errors through the whole region of pressure. But the plane probe fairly differs from the microwave cavity m.ethod in the region from 2 x 10-4 to 10-2 Torr. On the contrary勺 inthe region from 10-2 to 10-1 Torr the probe theory in a magnetic field holds in spite of the sho.p台。fan electrode. The cylindrical probe gives a very smaller electron density than that measl1red by the microwave cavity method as a matter of course， becal1se the effect of a magnetic field is not taken @ Ne CM-3 〆@ X measured by the resonant cavJ.ty 'I} clisk probe " plane M spherical o cylinclrical @ ? ??ー???e' <1' ♂ @ @ ， @ 1012 u=O.5A IT=8A Ig=lA e' ⑮ 9' ⑮ 4ダ101lト 安 x x X x fi ~ ltefli d ベ!)(A .s A '"主 ムぎ " l'f 八A 金 6静X 自由 o 0 d⑩ O A む1:;，9' d 。O 。。。。O 108」 '70 L.:a 10-4 10-3 10← 2 P 10-1 Torr Fig. 12 Electron density versus pressure 18 Y oshio ENOlV!OTO， Masahiro ANDO， Sunao KAW ASAKI， and Teiichiro YAGI account oL As the result of trying a cross sheck by the microwave method， a spherical probe is in relatively agreement wIth the microwave cavity method and so the for a spherical probe in a magnetk field holds though this probe theory neglects electric fields and orbital motions and is valid for li Te > > )(T i' Also the theory of a probe in a field which Is derived for an infinite plane is not strictly 6， Condnsioi!s We compare the electron densities m悶 sured the spherical， and drical probe with that measured by the microwave cavity method， Cylindrical probe gives a very smaller electron density than that measured the microwave cavity method as is expectedフ becausethe eIfect of a magnetic field is not taken account ot The probes besides a cylindrical prob忠 agreewith the method in spite of the strength of a magnetic field (β 豆 480 in the ceration of the higher Helium gas pressureo But when we comparεthe electron densities in the wide of Helium gass pressure， only the spherical agrees with the cavity mεthod， There-fore， itbecomes evident that the theory of a probe in a field gives us the most confidence固Ref官renre51) CG， Momgomery， Techniqu己oIMicrowav巴Measurement，Massachusetts Institute of Technology， Radi ation Laboratory ser匂s1 (MacGraw-HiIl Book Company， Inc.， 1947) P，298-P，299 2) S，J Buchsbaum， L How巴r，and Soc. Brown， The Physics of Fluids 3 (1960) 806， 3) Rl Rickerton and A V on Engel， Pl'OC， Phys， Soc. (London) B 69 (195) 468， 4) F，F， Chen， Plasma Diagnostic T記chniquesed， by RH. Huddleston巴， and S.L. Leonard (Academic Press， New York， 1965) P，164， 

Measurements of the Electron Density of a Plasma in a Magnetic Field

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Measurements of the Electron Density of a Plasma in a Magnetic Field

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