RL-331 Novelmber 1965 T' 1 ': i;EXCI'I'ATIOI(N ()OF SURFACE CURRIEN'TS ()N A 'PLASMA-IAMMIERtSED CYLINI)DER BY INCID' EN 1,1: C 'I t 1 N IC AND ELECTROKINETIC WAVES by Edrmund Kenneth Miller A dissertation sulbmitted in partial fulfillmeint of the requirements for the degree of Doctor of Philosophy in The University of Michigan 1965 Doctoral Committee: Associate Professor Andrejs Olte, Chairman Professor Chiao-Min Chu Doctor Raymond F. Goodrich Professor John S. King Associate Professor Herschel Well RL-331 = RL-331

TN L -A' ON A PAM-MESDCLNE <7 VDIFK< E ES IR_~ w__i fNIxtC A'_,D E L EC IROKIIK1\EYI7C WAAITES w k1P(5 4fti slvcC Is toc eltar sur-face I rrv C1 K I 012 'n B) rieet ' noE 'or- iir. -nr-m o V. U <1- P213 ii i,~l.~ Z1O 2 cui0 i u c h cl 11-l tsm: S(- cc-oi_~ - I Pr - ' cvc 'i'- T-ln-.~ J -V ) ' -~, -.re i WIICt1 - Us Si Oxit- oSsc acvI:lc to 1 KintiCtica ninclaoc-" i ol>eooilin i-s, revel', ed by `uibcirc 11cc eol I, > i-)I ' 1'( '2S,- m1012i 10135.-t L-... ~l an eJll'c 1i-1OSS 01 meat izaio 1)mleads10 tolL siets ofr a_-1cq `~ V.1(V102 0ecc sI- ci' 1er o toer m1: dj 4]21: X l i 10'acuts~ C"Hii'tolt le1-'L `~f~o lca 01oasc DC 1-c202 I es 0071i4 00 shwT>:ViC 10 l(1211010~mIi -sI1Vece uriunjlui1. Vrias n i h sai' ls: 1o e'is -~il 2e 0Ii3Unn 0 t I1 -I t lriSar ccore fr h eLI on desert~(, 'I( c)i t 21c 2' iel 'variation I'- the ico ogro ussOf'? shoot]S require c (-n'rialslii ` i' 2(16ap iit:i the uriorr Dlam arb sove aa ti lv TueI I 1eIic tar ii 1(1 ID oiem foOvn s 'd r t crc tite cyi'- icr ato -1113,1 LI a-'','l ~1- escIn lo Cis, t1 is S s o:Lcj b { _li It bond ry condtion- is to,ethed

field iqutantities r at the cyl i iCder sur ace and the inhom ogeneous sheath-Li nifor n-i plasima inlt-rf:-,ee. An outline for the,procedure used in obtaining a numerical s tlI t ion to the boundary value problem is given. A second approach to the problem inv7olves replacinog the actual irhoAl oueneous sheath by a free —space laye r, which is called the \ acuumll sheath. Thle boundary value problem associatec with this mode] is also set up and an analytic solution for the various fields is given. Extensive numerical resutlts for the slIufac.e currents are presented. The 'a:c.lullum sl-ath resullts q-re given for:;ihbitl':-i'' ngle of incidence while the inlomogeneo;Lus:slhea-tlh results are restricted to normial wave incidence. It is shown that the surface c-lturrents for EK wave incidence are e\xpt;onlltially attenuated by the vacuum -.1,.,11, with the attenurltion increasing as the angle of incidence measured from the cylinder axis beco]nes smaller. Further it is found that the EK wave is not as efficient in producing surface currents as the EM wave, even \\hl'n the sheai;h attenutation is not fal;en into account. It is also'found that the sheath and pll:tslla compressil)ility havre little effect on the currents due to the El[ w ave for c ylinider-s with radii small compared with'the EM swavelengtth. Finally, the results of the iihinogeneous sheath and vacuum sheath imodels are found to be in:tlls.l:i'ltial agreement for normnal incidence. We can conclude from the results of this study that: 1) the..l I I compi e;ssil)ility and sheath can be neglected v. lihen considering the currents excited by EM waves on a plasmaLs-immersed cylinder of small Ira;diLus ccllnll:Lr l with the EM wavelerngt'l; 2) the sheath appears to quite ef:ectively screen the EK wave from the cylinder; 3) it appears that since the EK wave is less E rrit it. -, that the EM wave in exciting surface currents on the cylinder, it would be difficult to detect the EK wave in a. background of EM radiation, by a measulement of the surface currentll'ls which it produces.

A CKNOWILE DGMENT The author wishes to express his appreciation to the nembers of his,committee for their help andl gui dance during this study. Special thanks are euspecially (due to AnlIrejs Olte, the committee.l,'i:i il:,ll who suggested the problem. He ga\e i nvalL. li' support throlughoutt the c.'sll.et. of this work and furnished maIny helpful suggestions for carrying it through to completion. This research was supported in 1: 1 by the National Aeronautics and Sp1ace Alminiistration under Grant NsG-472 with the Langley Research Center at Ha nptoln Virginia. The n'lmu-rical computations were suppLi.,lted by The University of Michigan Co, 1.ti 11, Center. Special thanks are also due to the author's wife for her patience during the course of this work and for rewriting the original manuscript. ii

TABLE OF' CONTENTS ii i i Ai \(C N OW' L E L LIST O) F T( C -1A PT E R )(O1ENT T I1? FVq INTRODUCTION 1 1. 1 Review of Previous Work 3 1.1.1 The Scattering Problem 3 11. 1. a Plane Bclundary1.1. b Spiherical Blounarl.y 7 1.1.2 The Radiation Problelm 8 1.1.2a 'r 1 i ti'n from Isolated Sources 9 1. 1.2b RT.idi:ii in from Bodies 10 1.1.3 The Static Sheath Problem 10 1.1.4 Experimenlt il Work 15 1.2 Problem Areas Remiaining to be Solved 21 1.:3 Problem to he In\' stigalted 24 11 FORMULATION 27 2.1 The Boltzm atnn Equation 27 2.2 Development of the MaIcroscopic Elalt:lil ns 30 2. 3 Linearization of the Macroscopic Equations 36 2.4 The Sit:lll Plasmal; Sheathl 39 2.5 The Dn:ll c Sheath Equaltin 51 2. 5. 1 Wave Pr'l I;pagzat ioll in a Uniform Plas mla 5 2.5. 2 Elimiination of the Static Electron Velocity from Dynamic Sheath Equation 55 2.5. 3 Ordered Power Flow in Pl:isii: 519 2.5.4 Cot pled Wave P '1',.' " 6. 2.5.5 Specii: ic ti, of Inhomogeneous She a li Boundarl y Value Problem 65 2.6 The Vacuum Sheath Model 82 CHA PTEP'jR iii

CHAPTER 13 CHAPTER I APPENDIX APPENDIX APPENDIX APPENDIX APPENDIX APPENDIX NOTATION BIB LIOGRA PH I RESULTS 3. ] Vaniti.i, SheatlI 3. 1. 1 incident EK W\V-ve:3. 1.2 liidceit EM Wave 3. 1. 3 Comp,:aison oJf EM 'ii EK Induced Currents 3.2 Inhomogeneous Sheath 3.2. 1 Inciicdent EK Wave 3.2. 2 A Closer C, 'n.i1,ration of the Dvynamic Sheath Behavior 3. 2. 3 Lineari ation Criteria in Inho:lmogeneous She at]l 3. 2. 4 Coupled Field Variation in Sheath 3.2.5 An E\a',inaltikinl of the Possibility of Ordered Energy Loss in the Sheath. [V CONCLUSIONS AND RECOYMMENDATIONS FOR FURTHER STUDY 4.1 Summary and Cnclusions 4.2 Recomicm-indations for Further Study A Analysis of Electron Number Density and Velocity in Static Sheath B E',. lIt lion of the Static E1-t'Ll n Velocity Terms in the Dvn:limic Sheath Eq.luattions C Details of the Inhomrogeneous Sheath Analysis D A Suggested Experimnent on hl- Elect rokinetic Wave E Development of Vacuu-im Sheath Formulation and Approximate Solutions F Details of the Cylindriical Function Ev.1ii i;il 100 100 103 156 163 170 171 183 186 188 206 215 215 219 221 235 238 248 252 270 276 287 LY iv

LIST OF FIGURES Page CEAPTER FIG. FIG. CH APT ER FIIG. FIG. FIG. FIG. lii- I(,T. FIG. FIG. F IG. II 2. 1 2. 2 III 3. 1:3. 2 3. 3 3.4 3. 5 3.6 3. 7 3. 8 CYLINDER AND COORDINATE SYSTEM NORMAL CROSS-SLE('CTI(,N OF CYLINDER (zi MAGZNITUDE OF K) vs. AZIMUTHAL AJNGLE P 0 FOR NOMINAL PARAMETER VALUES PHASE OF K vs. AZIMUTHAL ANGLE 0 p FOR NOMINAL PARAMETER VALUES MAGNITUDE OF K vs. AZIMUTHAL ANGLE p 0 FOR NOMINAL PARAMETER VALUES PHA.SE OF 1] vs. AZIMUTHAL ANGLE 0,P FOR NOMINiIAL PARAMETER VALUES (2) MAGNITUDE OF K vs. A ZIMUTHAL ANGLE p FOR NOMINAL PARAMETER VALUES PHASE OF iK vs. AZIMUTHAL ANGLE. 0 FOR NOM INA L P A RA ME TER VALUES (0) MAGNITUDE OF K vs. AZIMUTHAL ANGLE P FOR NOMINAL PARAMETER VALUES PHASE OF K ()vs. AZIMUTHAL ANGLE FOR NO L P IETER FOR NON INAL PARAMETER VALUES 40 43 104 105 106 107 109 110 111 112 v

F[G. 3. 9 FIG. 3. ] 0 F]IG. 3. I 1 FIG. 3.12 FIG, 3.13 FIG. 3. L4 FIG. 3.15 FIG. 3. 16 FIG. 3.17 (z) VIMAGNITUDE OF K vs. AZIMUTHAL ANGLE 0 P WITH CYLINDER RADIUS c A PARAMETER PHASE OF Kz vs. A ZIMUTHAL ANGLE 0 FOR CYLINDElR RADfLTS c-0. 1 cm MAGNITUDE OF K( vs. AZIMUTHAL ANGLE 0 P WITH CYLINDER RADIUS c A PARAMETER (() PHASE OF K() vs. AZIMVUTHAL ANGLE 0 FOR CYLINDER RADIUS c=0, 1 cm MAGNITUDE OF K) vs. AZIMUTHAL ANGLE 0 P WITH CYLINDER RADIUS c A PARAMETER, (z). MAGNITUDE OF MAXIMUM VALUE OFA K vs. P CYLINDER RADIUS cQ (z) MAGNITUDE OF K vs. AZIMUTHAL ANGLE P FOR N- 0. 8 A ND NOMnINAL VALUES OF OTHER PARAMETE RSc MAGNITUDE OF K) vs. AZIMUTHAL ANGLE P FOR N-0. 9 AND NOMINAL VALUES OF OTHER PARAMETERS MAGNITUDE OF K() vs AZIMUTHAL ANGLE 0 P FOR N=0. 99 AND NOMIINAL VALUES OF OTHER PARAMETEPRS 114 115 116 117 118 119 121 122 123 vi

FIG. 3.18 FIG(T..19 FIG. 3, 2 0 FIC-. 3.21 MAGNITUDE OF K vs. AZIMUTHAL ANGLE 0 P FOR N-0. S \ND NOMIINAL VALUES OF OTHER PARAMETERS IiMAGNITUDE OF K() vs. AZIMUTHAL ANGLE 0 P FOR N=0. 9 AND NOMINAL VALUES OF OTHER PARAMETERS MAGNITUDE OF K vs. AZIMUTHAL ANGLE 0 P FOR N- 0. 99 AND NOMINAL VALUES OF OTHER PARAMETERS (z) MAGNITUDE OF K vs. AZIMUTHAL A-NGLE 0 i p FOR 0 =0. 05?r AND NOMINAL VALUES OF OTHER PARAMETERS 124 125 126 128 FIG. 3.22 (z) MAGNITUDE OF K vs. AZIMUTHAL i FOR 0 - 0. 15 AND NOMINAL VALUES PARAMETERS ANGLE 0 OF OTHER 129 FIG. 3.23 FIG. 3.24 FIG. 3. 25 FIG. 3.2 6 (z) lMA GNITUDE OF K vs. AZIMUTHAL ANGLE 0 i P FOR 0 -0. 357r AND NOMINAL VALUES OF OTHER PARAMIETERS (z) MAGNITUDE OF K vs. AZIMUTHAL ANGLE 0 i P FOR 0 o0. 45z AND NOMINAL VALUES OF OTHER PARAMETERS MAGNITUDE OF K vs. AZIMUTHAL ANGLE 0 i P FOR 0 =0. 05 AND NOMINAL VALUES OF OTHER PARAMETERS MAGNITUDE OF K vs AZIMUTHAL ANGLE 0 i P FOR 0 =0. 15 AND NOMINAL VALUES OF OTHER PARA METERS 130 131 132 133 vii

FIG. 3.27 FIG., 3.2 8 FIG. 3.29 FIG. 3.30a FIG. 3. 301b FIG. 3. 31 a FIG. 3.31b FIG. 3.32 F[G. 3. 33a MAGNITUDE OF K vs. AZIMUTHAL ANGLE 0 FOR 0 -0. 327r AND NOXMINAL VALUES OF OTHER PARAMETERS MAGCNITUDE ()F K vs. AZIMUTHAL ANGLE 0 i p FOR 0 S0. 457T AND NONIINAL VALUES OF OTHER PARAMETERS MAGNITUDE OF MAXIMUM VALUE OFtK AS A FUNCTION OF ANGLE OF INCIDENCE AND NOMINAL VALUES OF OTHER PARAMETERS MAGNITUDE OF K() x s. AZIMUTHAL ANGLE 0 i FOR 0 =0. 057r AND SHEATH THICKNESS X A PARAMETER MAGNITUDE OF K x s. AZIMUTHAL ANGLE 0 i ) FOR 0 =0. 05ir AND SHEATH THICKNESS X A PARAMETER MAGNITUDE OF K vs. AZIMUTHAL ANGLE 0 i P FOR 0 =0. 05wr AND SHEATH THICKNESS X A PARAMETER MAGNITUDE OF K ( vs. AZIMUTHAL ANGLE 0 i p FOR 0 =0. 05w AND SHEATH THICKNESS X A PARAMETER MAGNITUDE OF K xvs. AZIMUTHAL ANGLE 0 i p FOR 0 -0. 25wr AND SHEATH THICKNESS X A PARAMETER MAGNITUDE OF K 0) vs. AZIMUTHAL ANGLE p p FOR 0 =0.257r AND SHEATH THICKNESS X A PARAMETER 134 135 136 137 138 139 140 141 142 v-ii

FIG,. 3. 331 FIG. 3. 34 FIG 3.35 [(:. 3.36 FIG.- 3.37 IF. r. 3. 37 FIG. 3.38 FIG. 3.39 FIG. 3.40 FIG. 3. 40 FIG. 3.41 FIG. 3.42 MAGNITUDE OF K s. AZIMUTHAL ANGLE 0 p FOR () -. 25 2r AND SHEATH THICKNESS X A PARA ME TER (z) MAGNITtUDF OF K) s. AZIMUTHAL ANGLE FOR 0 -89.'9 DEGREES AND SHEATH THICKNESS X A PARAMETER (0 ) MAGNITUDE K vs. AZIMUTHAL ANGLE 0 For P i p 0 -89. 91 DEG'REES AND SHEATH THICKNESS X A PARAMETER (z), MAGNITUDE OF MAXIMUM VALUE OF K vs. i SHEATH THICKNESS X WITH 0 A PARAMETER MAGNITUDE OF MAXIMUMI VALUE OF'K vs. i P SHEATH THICKNESS X WITH 0 A PARAMETER (z) MAGNITUDE OF MAXIMUM VALUES OF K, and (0) i p K vs. ANGLE OF INCIDENCE 0 P (0) MAGN IT D OF MAXIMUM ALUES OK and K vs. RATI(-) OF PLASMA FREQUENCY TO p INCIDENT WAVE FREQUENCY, N (z). MAGNITUDE OF MAXIMUM VALUES OFK( ) and (0)p iK ) vs. CYLINDER RADIUS c p P MAGNITUDE OF CURRENTS EXCITED BY EM WAVE vs. AZIMUTHAL ANGLE 0 FOR NOMINAL PARAMETER VALUES MAGNITUDE OF MAXEi UM CURRENT AMPLITUI)ES EXCITED BY EM WAVrJES vs. CYLINDER RADIUS c 143 144 145 147 148 152 154 155 157 159 ix

FIG. 3.43; FIG. 3.44 FIG. 3.45 FICG. 3.46 FIG 3.47 MIAGNITUDE OF MAXEIMUM CURRENT AMPLITUDES EXCITED BY EM WAVE vs. RATIO OF PLASMA FREQUENCY TO INCIDENT WAVE FREQUENCY, N MAGNITUDE OF MAXIMUM CLiRlRENT AMPLITUDES 1 EXCITED BY EM WAVE vs. ANGLE OF INCIDENCE 0' MAGNITUDE OF MAXIMUM CURRENT AMPLITUDES EXCITED B' EM WAVE vs. THE SHEATH THICKNESS X MAGNITUDE OF CURRENTS EXCITED BY EM AND EK WAVES OF EQUAL POWER FLOW DENSITY vs. AZIMUTHAL ANGLE 0 FOR 0 i=/4 AND NOMINAL VALUES OF OTHER PARAMETERS MAGNITUDE OF CURRENTS EXCITED BY EM AND EK WAVES OF -EQUAL POWER FLOW DENSITY vs. AZIMUTHAL ANGLE 0 FOR 0 ir/2 AND NOMINAL VALUES OF OTHER PARAMETERS MAGNITUDE OF K vs. AZIMUTHAL ANGLE 0 FORt P INHOMOGENEOUS SHEATH MODEL WITH M=2 and -5. 34 VOLTS c (() MAGNITUDE OF K() vs. AZIMUTHAL ANGLE 0 FOR P INHOMOGENE1OUS SIHEATH MODEL WITH M=4 and =-5. 34 VOLTS c MAGNITUDE OF K 0 vs. AZIMUTHAL ANGLE 0 P FOR INHOMOGENECUS SHEATH MODEL, AND HARD BOUNDARY CONDITION ONLY, with 3 - 3.06 VOLTS MAGNITUDE: OF K() vs. AZIMUTHAL ANGLE 0 P FOR INHOMOGENE CUS SHEATH MODEL WITH M=10 atd =-3. 06 VOLTS AND HARD BOUNDARY c CONDITION 162 160 166 161 167 FIG. 3.48 172 FIG. 3.49 178 FIG. 3.50 179 FIG. 3. 51 181 x

F'IC. 3. 52 F IC. 3. 5 FIG. 3.54 FIG. 3.55 FIG. 3. 56 MAGNITUDE OF K( vs. AZIMUTHAL ANGLE 0 FOR INHOMOGENEOUS SHEATH MODEL WITH X=5 THE NORMALIZED ENERGY DENSITIES N and e P N vs. RADIAPL DISTANCE FOR EM WAVrE (e POLARIZATION) INC IDENCE AND SHEATHLESS CASE P THE NORMALIZED ENERGY DENSI IES N and Nh vs. RADIAL DISTANCE FOR EK WAVE INh CIDENCE ANlD SHEATHLESS CASE 182 192 193 e THE NORMALIZED ENERGY DENSITIES N and e P Nh vs. RADIAL DISTANCE IN SHEATH FOR EM WAVE h (e POLARIZATION) INCIDENCE AND INHOMOGENEOUS SHEATH WITH- HARI) BOUNDARY CONDITION 195 THE NORLI\,LIZED ENERGY DENSITIES Np rlln p p Nh vs. RADIAL DISTANCE IN SHEATH FOR EK WAVE h INCIDENCE AND INI-OMOGENEOUS SHEATH WITH HARD BOUNDARY CONDITION 196 FIG. 3.57 FIG. 3. 58 THE NORMALIZED ENERGY DENSITIES Ne and e P Nh vs. RADIAL DISTANCE IN SHEATH FOR EM WAVE h (e POLARIZATION) INCIDENCE AND INHOMOGENEOUS SHEATH FOR THE SOFT BOUNDARY CONDI I1( )N 198 THE NORMALIIZED ENERGY DENSITIES NP and p Nh vs. RADIAL DISTANCE IN SHEATH FOR EK WAVE h INCIDENCE AND INHtOMOGENEOUS SHEATH FOR 'HE SOFT BOUNDARY CONDITION 199 xi

FIG(. 3.59 THE NORMALIZED ENERGY DENSITIES N AND p p Np vs. RADIAL DISTANCE IN SHEATH FOR EK WAVE h INCIDENCE AND INHOMOGENEOUS SHEATH WITH X=5 AND HARD ]BOUNDARY CONDITION THE NORMALIZED ENERGY DENSITIES NP AND p Nh vs. RADIAL DISTANCE IN SHEATH FOR EK WAVE INCIDENCE AND INHOMOGENEOUS SHEATH WITH X-5 AND SOFT BOUNDARY CONDITION 202.( I(. 3. 60 203 APPENDIX FIG. FIG. FIG. FIG. FIG. A PPEN DIX A Al A2 A 3 A 4 A5 STATIC ELECTRON DENSITY AND VELOCITY vs. RADIAL DISTANCE IN SHEATH FOR M=2 AND — 3. 06 VOLTS c STATIC ELE(CTRON DENSITY AND VELOCITY vs. RADIAL DISTANCE IN SHEATH FOR M-=2 AND — 5. 34 VOLTS STATIC ELECTRON DENSITY AND VELOCITY vs. RADIAL DISTANCE IN SHEATH FOR M=4 AND =-3. 06 VOLTS c STATIC ELECTRON DENSITY AND VELOCITY vs. RADIAL DISTANCE ]N SHEATH FOR M-=4 AND =-5.34 VOLTS c STATIC POTENTIAL VARIATION IN SHEATH vs. RADIAL DISTANCE 229 230 231 232 234, FIG. B1 THE RATIOS R AND R2 vs. RADIAL DISTANCE IN SHEATH 236 xii

APPENDIX C FI(. CL MAGNITUDE OF K \s. AZIMUTHAL ANGLE 9 FOR SHEATTILESS CASE CALCULATED FROM INHOM1OGENEOUS SHEATH COMPUTER PROGRAM 247 APPENDIX IF FIG. Fl THE DIVISION OF ARGUMENT - ORDER SPACE AMONG VARIOUS METHODS OF COMPUTING THE CYLINDRICAL FUNCTIONS 2 72 x:L 11ii

CHAPTER I INTRODUCTION 'In,:. dynamic response of a plasman-i to high frequency electric fields has receive(l clonsiderable attention in recent years. By plasma we mean a partially ionized g',as that on the average is electrically neutral. The analysis of the p]asma may proceed from the BoltlzmantL equation and Maxwell's equations together vith appropriate boundary conditions and sources. The ultimate solution involves finding the electric and maognetic fields in the pl.l-llm i as well as the distributionl.inctions for the various plasma. species. Betca:Lus of the great matlhetmatical difficulty, this is seldom possible. Instead, the problem is converted to one involving,only imaL ro-se.( opic vCariables by taking velocity momenlets of the Boltzmann equation. In many pr)I Lctical problems it is physically reaslliblteL to neglect all moments higher than sco —c:nd order in velocity. This means we introduce as new macro-scopic val'iaZble the scalar number density, tlie vector velocity, and the tensor pressure for each plasma specie:. The moment equations obtained are nonlinear in these variables. They are linearized by re(lquiring"' the time varying perturbations of these ) a'riables to be smal.ll compared with the static parts. Under cer'tain conditions the tensor pressullrt reduces to scalarl' 1p - -me. The perturbed pressure is rtelatted by an e(luation of state to the perturbed number density. The problem at this poin't i<s one of finding the perturbed numberIT density and velocity of each Especi', in addition to the electric and n1gli'tli.: fields. Studying the plasma equations thus obtained one observes tha t for a uniform plasmanL free of;static magnetic fields the electric field can be decomposed iill. a 1

2 sol enoidaIl andll an irrotational part h ich ican be shown independently to satisfy ditttl renlt vx-ctor wax e equations. The tolt.l inlt. —varying mallgnetic field is then related to the solenoidal electric field and together they form an elec t romlagnetic (EMI) %\ L\ e The irrotational time varying electric field is associated with the pe.'tulbed:charge density.lccultulaltion (space charge). These two qulanttities form a \ave thaI h is been referred to as the plasma wxave, electro-acoustic, wave, and el ctrolkinltic (EK):' wave. We will use the 1. - term. In the EM wave the energy is shared between the solenoidal electric field and the mag ll. li: field. In the EK wa-ve the e'ler-gy is shared between the timne vary lilng irriotational electric field arn-d the ordered hinetiic energy of the charged particles. In the study of p-l1.tl1, one frequently findl,-t that to a reasonable approximation the j:tlusnma iun-der consideration may 1be divided into unifornm phasllma regions connected by 1onuLnliform plasmal transitions. Or we may find a uniformL plasma termi n-LaLted(c via a nonuniform plas1;ma (the sheath) into either dielectric, conductor, or free sp.ce. In the nonuniform plasma transition regions the EM;nd EK waves areC co.upledl due to the static electron density \aL iattion. This means that an EK wave (for (e-ample) propagating into such a region in addition to reflections will suffer a loss of energy into the EM wave and vice versa. A similar energy conversion between the two waves:.(tloccurs at sharp triansition regions such as the botundary between plasma sheath and dielectric (or metal). This is due to the bolncdary conditions on the tantgential electric avnid magnetic fields and the ordered electron velocity at the interface. The txwo waves will be significantly coupled 'Thi-s narne was suggested by G. Hok (1959).

,C) even in tuniform platsmna if Kth linearti;-&tion requirement on the velocity momllle ecquttion is re.l.i,_d. The same tlhing is true i if an external static 'agllnetic field is presc lt We are excluding the 1.- t two forins of coupling from this 'l1i.rli,:'i., Tlhite energy conversion between the wNtaves is of interest in;tapplic'attionts as diverse as ast-i ophysics and radiat ion or reception by pllaslma-immersed antennas. Some of the radio emissions froml stellar bodies ': ',e\plalined on this basis. The effect of the wave conversion on transmission and reception properties of the plals]ma-inmbedded antenna is of considerable imp.i tance. Tle latter is our field of i ntterest. Before we present our problem in detail, it is of value to review briefly some of the pertinent work that 11has been done in this general area. For an oircelty discussion it is convenient to arrarge the papers dealing vith the conversion problleml into two cateo-ries: the scattering problem and the r.adiation problem. Since in our area of interest the high freqlency responsel of plasma. presumes a ];l mi tedg of the static p1i isma sheath sollutionl, the work on tl is topic will be revxiewxed. 1. 1 IReview of Previous Work 1. 1. 1] The Se.!tt-. ill Problem A discussion of the scattering problem cLan be carried out conveniently accordirg to the geometry of the transition region. A further cl:tl.i fic iat ilo of this probleml can be made of the 1 I of whether or not the th ickness of the tr~ansition region is taken to be small com1pare d with the pertinent wavelengths so that it may be considered as an abrupt distr l'iutillnitj of the jplasIunaL properties. When the transition region is assulmed thin, the problem is one of considering

4 pLropagai on in a uniform lz:ilL;L togcthlr Ii Ljb appropriate boundary c(' nit inn':at the discontintiity. The use of - to1., a representation is an approximation sitnce a real plasma canU1' eanot.-.ill)ol't abrtupt c'l tcnges in its properties, such as those which occur at the walls confining la.horatory )lasma for example, without a L OIt 9' P)lC-iong traLnsition region where the plasma properties change in some contint.olis nlzta.nner. The validity of stuchl an approximation would be dependenlm in part on the wavelength of the waves incidenlt on such a tLransition region compa.Sred \\ith its hiLkness. The coupling between the EM and EK waves due to the nonunif)'ormity is thlus neglected with this La)proacl. When the transition region is thick the probic m requires consideration of the wave propagation in both the uniform plasma and the nonuniforlm ' ri i,,, region, together with appropriate )oundary conditions. This latter problem is obviously muLch more complex than the former. The boundary conditions x\\hitLh are taken to apply at an a'bitr-at'5 boundary are the usual ones from elL.olllagnetic theory for the tangential electric and maLgnetic fi. Ids. In addition, conditions analogous to those encountered in lacoustics are uslually employed for the pressure and velocity, that is, contin iuity of electron p1ressure and normal electiion velocity across a permeable boundary, or v;alnislilg of nor-emal electron velocity at a rigid bollundary. There is no unanimi.ty regarding the latter boundary condition however, and the use of a different boundary condition will be noted. 1.i1 a Plane Boundary Thin T[ri '.i ion Region. In the first general treatmentt of EM and EK wave propaga;tion in a plasma, Field (1956) considered the conversion of EK plan e waves

5 to EM p.ane waves at a sharp plasmna - vacuum interface. Field's formulation beganll i tho ut tak Lling into.Lcounllt the sta ttic lt-t -ic field which is present in reg'ion, s of el'c't ol density variation in the 1 la.L-iat, so his discussion of this as — pect of the prloblem is not correct. His requiremnent that the normal electron velocity vanish at the interface was also incor_-ect in not allowing for the possibility of a surfaice charge. However, Field's work was significant in that his tr-atlment e stablished the basic approach followed later by other investigators in iL ulnpi ~silng plane waves in uniform plIasma into EM and EK components. A pr-oblem similar to that considered by Field was investigated by Kritz and Mintzer (19'60), the difference being tlhat their problem involved a sharp plasmlapla1;- i1 L boundariy. They took the norimal electron velocity across the interface to be continuous. Analtic.al expressions for the trLansmitted and reflected elect t-ic fields for arbitrary angles of incidence were obtained. Some numerical results showing the reflection and transmission coefficients of the EK wave for an incident EM wave and thle reflection aind trlLlansmissionl coefficients of the EM wave for an incident EK wave as a function of the angle of lncidence are presented. The ratio of the electron density on the two sides of the interface was taken to be 2. The results showl that the EM waves are rmore efficiently converted to EK waves than are EK waves converted to EM waves. It is iriteresting that the EK wave excites. '),l,;Ll i)', EM waves for only a small ang1ular interval around normal inceidcrnce. Other-wise, EM surface waves, declay: ng exponenitially normal to the surface, are produced, due to the large difference in the propaga.tion constants of the E.M and EK waves.

6 Ti(lmanll1 (1960) found the conlversion efficiency of EK to EM elnergy for the case of a plane ]EPK wave at nori;mal incidence on a plasma-plaslma discontinutity. A St L1..I I form \7was used for the static pt-essure while the dynamic pressure w\.-,s taken to be a tenisor. His boundary con((iition on the normal electron)I velocity was different from that of Kritz and Miintze i in that the total current (displacemenit plus conduction) was taken to be c inti inu(.s cross the interface. A numerical res11b; for tlhe conversion efficiency was obtained for one particular disil.-tltil wiy where the static electron density changes by a factor of 2 at the inte-lrfce. Cohen (1';-62a) obtained expressions for the reflection coefficients of EK alnd EM waves incident on a plasma-metal interface. In formulating the boundary condlitiOl on thut normal electron velocity, he introduced a bilinear adnmittance relatioln between the velocity and the perturbed electron density and electric field. The re flection c;out'iitients which are given involve these adtcmittanc-es but no e splicit form is gixen for them, nor is there any discussion of a possible theoretical mnethod for obta Iing them. Tidnman and Boyd (1962) extended Tidman's (1960) earlier work on the pla:. Il.L-plasma interface. They used the same boundary conditions as those previously used by Tidman except for his continuity 1i i r I ii on the dynamic pressure. An integration) through the thin transition region was performed on the elec tron equatiion of motion, which gave the discontinuity in the dylnamic pll" r- -ll'V in tern-ms of the -tlttic eltr.itic field in the tra'11litionl region. E\pI.'-.-.i.ls for the tr.,,-, u i i, ll and reflection coefficients for an EK wave incident at an arbitrarv angle on such a. boundary are obtainlec

7 Fecdo-rchl nko (1962) C'examineid the refluction of EK and EM waves from a lplitsma-;:-dielec tric intetlface, ass nn~ - elastic reflection of the electrions fror the boundatry. The tiransmission coefficient of the EK wave due to an EM wave in.ident on such a dis-conltillnuity from the dielectr" Sidsit is found as a function of theangle of iincICildence for various ratios of the )Ilasma frequency to the radio frequency. These res.lts are plotted, I1.1 i, Illy and indicate transmission coefficients on the order of 10 to 50 times greater thaLn those obtained by Kritz and Mintzer for the pl.1 1n.11i-pla snma intel-rface. Thiclk TraIInsition Region. Tidman (1960) also investigated the EM radiation produced by an ElK wave propagating through a transition region thick comparled with the EK wavelength. He calculated the conversion.,Il'iei t ly of EK to EM energy as a function of the ratio of the length of the static El r-i' I density valIiRLtion to the EK wavelength He was able to show that the conversion efficiency becomes exponentially small for EK wavelenglths less than a certain scale length of the electron density variation. 1. i..b. _Sperical Boundary Thin Transition Region. Cohen (1962a) examined the scattering characteristics of a plasma bubble of constant t;latic electron density which is different from that of the surrouCnding plas1ma. Formulas were obtained for the scattering coefficientts of such a bubble when illulll ilt by incident EK and EM radiationl,,LSsLuinllt)in the bubble dimensions are much less than the EK wavelength. The results show that the cross secoion for scatterinto the EK wave are onhe order of (v/ v ) greater than the correspolndi ng EMN[ scattering cross sections, where

8 \ is ihl\ electi'on rnms velocity and vi is the velocity of light in free space. Tlie scattering of t:la.neB- EK and I'M waves by a spherically shaped discont inuity in plasllmL,II il> was t rC'e;te'd 1h Yildiz (1963). He considered the sitiiLliu i \v1here the -sphere and the surTrounding medium were plasmas of different static cl-ct l oii density, and also where the static electron density of either was zero. Expressions for the scattered fields and scattering cross section are obtained Tlhick Tlransition Region The 1probluci of the scattering of plane EK waLves Iy spherically shaped blobs of smI1all anpt1itude flu(ictlatitiis in plasnma density was il ig i lt,,d by Tidlman and Weiss (1961) The EM energy radiated was deterlminted fot the case where the electron density variation is Gaussian, decreasing frolm the center of the blob. It was found that the scatttered energy is exponentially dependent *n the squar.e of the ratio of the electroln density scale length to the EK wavell ength, dtecreasi ng as tlhe ratio becomes lar geIr. 1.1.2 The ]-Radi.Ltion Problem TlIl radialtion problemn may be conveniently discussed according to the kind of souL Lrc involved In physical probllems, the i ali.l ii source almost always consists of a physical stLucture or body wvhich is connected by a transmission line to the generator. The generator impresses voltage across parts of the raldi.l i 11 g body and producl:es current on it. These induced sources produce fields for which. solultion is desired. For mathematic.al convenience in many cases a problem is solved by ignoring the body, and finding the radiation from an,l '11 iil nt disttribution of isolated soul-rces. Thus, for example, if the far-zone fields of a thin Linear

9 dipole antutuiat a]re desired, one niav solve the corresponding problem for a filatmentary current souL'ce A solution of the radiattion from a body is difficult -ince pro)]priate bounLLdary conditions must bie satisfied on the surface of the body. The followlit; di(scussion of the raLdiaLtion problem is divided therefore into two parts: the first dealds wvith radiation fromn isolated sources and the second w\ith radiation from li tutie' 1.1.2a tRadiiation from Isolated Sources Cohen (16B2a) considlered the fields ldue to various kinds of surface distribution of soulrces in a uniform plasma11L. His analysis was limited to sources distributed uniformly over a plane surface infinite in extent, with plasma on both sides. In another paper the fields due to an oscillating current filament in a uniform plasmna wNelre found by Cohen (19f;2b). His analysis showed that the EM field is maximum in a p-laLIne noral to the filameni ui\\ile the EK field is LLmaximum along the filament axis. The radiation resistence of the current filamlent was found to be dominated by the effect of the EK wave Unfortunately, no definite conclusior can be reached about the behavior of an actual linear dipole in a plasma since the sheath eLfect arid the boundar(:Ly conditions which Lmust be satisfied on the real antenna are not taken into account by the currel t filament analysis. liessel et al (1962) found both types of fields excited within a uniform plasma half-space due to a magnetic line current source located in free space parallel to the plaslzL —free space interface. In another paper Hessel and Shmoys (1962) considered the exclitation of EM and EK waves by a short oscillating current filament in a uniform plasma. Like Cohen (1962b), they conclude thallt most of the

I ( radiated 1power is in the form of EK waves. 1. 1.2b Radiation frul, Bodies In the samLe pa)per in which he exl nined the radiation from an oscillating ( crrent filament in a pl)sma, Cohen (1962b) also set up the problem for radiation f, omn a wire dipole.antenna. He obtainl;ed two linear integral equations for the radiated Fields, but made no attempt to solve them. Hessel and Shcmoys (1962) also contsidered the fields due to a l,'.-lcb.,.'t ctlrrent disti i bution on the surface of a rigid sphere. They obtained some expressions for ll. -'i fields and com-pared them with the results of the short current filament 'Wait (1964a, b) solved for the fields produced by a slotted spherical antenna immn..4d in a uniform plasma He considered two cases, one when the plasma extends to the surfLfce of the sphere, and the other when the sphere is s. ep tt' ated from the plasma by a dielectric layer. The ]ati lec model is an attempt to account for the sheat.h. No numerical iresults are given, but expressions for the impedance and radialled power are olbtalined. He co,, 1.ides that the EK waves will be excited by stuch al1 antenna. 1.1.3 The Static Slhet;th Problem Before the dyna1mic plasma behavior can be theoretically analyzed insofar as wave proptaga-:ion is concerned, a knowledge of the static plasma description is required. Th:Is is not a problem where the plasma can be reasonably taken to be uniform, as in regions far from perturbing influences such as confining walls, or bodies immersed in the J1 li In..i1 In the regions close to such perturbing

11 influences, the plas-ma is not unifornm, however. Such regions are referred to as the plaLswia sheath. A cgreat deal of effort has been devoted to finding the static t11..- i i behavior in such sheaths Included below are some of the more pertinent analyses; which have been made of the static shteath pi-oblem. A solutiorn to the static plasma s;l t-.ith Lproblem involves finding the electron and ion 11iullber t 'iii ii L- s and the pote nl ial as Ca function of the space coordinates in the sheath. The ion and electron number densities are related to the potential via Poissoi' s equltion. Expressions for these number densities are obtained by Lapp r' ialtely integrating tflhir respective \velocity distribution functions olver the velocity. Upon putting these expresisions (which are functions of the potential) into P.Ai:;son's:equatLtion, and integrating subject to the appropriate boundaryiL conditions, a slution for the potenltial is obtained. It is apparent that in order to carry out the procedure outlined, the electron and ion velocity distribution functions miust be known. It is usual 1.o tC.-.-lull the electrons have a Maxwellian velocity dli- I Iiln'ilt ll function, but the ions on the other hand may be taken to p.. some other di:-f iliultion. The choice used depends to a large extent on the parI-ticular problem under consideration, as will be discussed in the followxing. Various other simnplifying assumptions may also be made, such as taking the shelath tc) be s11ILiL.ly defined from the uniform plasma and ignoring collisions in the sheailh. It should be observed that since the random electron velocities are muchl greater than the random ion velocities in the same plasma, due to their difference in mass, the sheath electric field about an insulated boundary will be such as to attract the ions and repel the electrons. This sheath is thus a region

12 \\h1er the ion ldensity exceeds the election density. T"onlks and Langmuir (192)a) analyzed the low pressure d:i.seh.tlge for plane, eylindrhitll and spherical geonmetries. Thley assumed a Maxwellian distr'ibution function for the electron velocities. Ion generation was also taken into accour-t, assuminl the ions to be genelrtted at rest, so tha1t their velocities depended only on the potential difference through which they L11l after generation. Ion collisions were ignored. Poisson's lequation was the used together with kllnuledgt of the ion;In1 electron velocity distribution function to obtain an integral equation fo:r the poteiti;ll, which was called the comlplete plasma-sheath llquation This ecuation exhibits a (:el)pendence on the k-ind of mode]:Lssulled for the ion generation. It was simplified by assuming that the ion and electron densities are equal to a high degree of accuracy (Tonks and LallngmuirnL called this the plasnlul equation) and it was solved for two ion-generation models. A solution to the simplified equation is good for the body of the j1plasma, but it is not, of course, adequate for the sheath region where the densities of the ions and electrons may be very different. Tonks and Langnm-ir- tried to remedy this by im-atchingli ) an approximate sheath solution to the pl;sma solution. This was only partially successful in that the sheath potential dr op is correctly obtiined but the potential profile is not accurate. Othele investig ations of various aspects of the sheath problem to improve on the works of' Tonks and Langnmuir have been carried out by Allen et al (l197 rj), Harrio-.unl and Thompson (1959), Auer (1961) and Ca-Lruso and C(.llilere (1962) among others. Solutions of the sheath problern, such as these, have characterist ic.lly been restricted to special cases or subject to limiting approximations

13 due to the c-oriplexity of the platsma sheatll equation. Recently, however, with the a;vailLaility of higll speed computers, t'11r.i. ' numerical analyses, valid thr-loughout the plasnia, have been carried out. Numierical solutions to the plasma sheath equaLtion hi.\ve been given for plane geometry by Self (1963) and for cylindrical geom-etry by Parker (1964), using it-s.ettidill y the analllysis of Tonks and La, —ngmui. Curves for Ilit shealtll potential and charge density are given for various -.iJ, e.., of the disclharge and ion genleiration models. A -,illiat\ltt different approach from that of Tonks and Langmuir has been followed by Bernstein and PALbinlowitz (1959). They alnalyzed the problem of cylindrical and spherical probes in a plasma on the basis that in the absence of collisons,.he general solution of Boltzmann's equation for a;particle distribution function is an arbitrary function of the colnsttalts of the motion. The constants of the nmotion \which they used are the energy and mIl.agnitude of the angular momentumll'! The elect ro(n velocity distribution function was asilIlsmed to be Maxwellian while aC imono-energetic velocity di..lt 'i Ilo ll flunc 'tion was used for the ions. After integral:ting the distribution functions over the constzants of the motion to get the respective particle number densities, a numerical solution of Poisson's equation was obtained for the potential. The essenltial difference between the analysis of Bernstein and FRabinowitz and that of Tonks and Langmuir is that the former consider the problem of a probe placed in an infinite plasma medium (the exterior problem) while the latter investigated the 1plasnia confined by walls of a given geometry (the interior problenm). (An earlier paper by Mott-Sniith and Langmuir (1962) developed the theory of praobes in a plasma, but did not discuss

L 4 'he pole,tti' v; nistion in tilt sheath. ) The exterior problem is fundanmentally liI f ei11 from the inlerior problem (besides the obvious difference in geometry) an the lt ot.ion (,f t lti particles which are attracted to the boundary. In the ' xlerior pi blc hn only those iOns \itli 1he proper comnbination of velocity and 1; IIIr moLen 1etuminL will strike the probe, while in the interior problemn, all ions wiill et 'entu:illy reach the boLundary. This requires a somewhat more elaborate tre.tientl 1o calculate the ion number dLensity in the exterior problem. An additional li-['f i-enc(e between the two developments is that Bernstein arid lRabinowitz aid not at [,1ow far ion Dgeneration. Finally, both an;alyses neglect the exist ae.c,f a net e-lectron.'currenlt to the boundary. This is an approximation whose validity Is reason.ible for boundarCLy potentia1ls equal to or less than the potential an insulaLted body would aLsstnum-e when imlmerse-d in the plasma. An e teniiioni of the work of Bernstein and Rabinowitz has been carried out -ecently )y La F]?lrambois (I'l,;1). He also considers the probes of spherical and (,ylindric al geotmetry but takes the ion velocity distribution function to be Maxwellian r.tihrt iim mono-energetic, as did Bernstein and Rabinowitz. Lami (.l64) has also analyzed the cyli jdrical and spherical probes following Bernstein an0d Rabi riowitz for the asymptotic limit of a very large probe radius to Debyelengthl ratio. Wasserstrom et al (1964) also treated the case of a spherical probe in a 1 iasi7nra I I, i r- mthod was to assume MIaxxwellian velocity distribution for both tI e ion:s andc! electrons. These distribution functions were divided into two parts r f veloc-ily:.space according to whether or not the particle velocity vectors in

1 I5 hs;ical,, were directed along a straight line intersecting the sphere. The Pon r part-s; f tle distrib ution fLncltions were tal un to be different unknown functions )f rtij radi us v.ari;ble. The problem las then to find these unknown radial m:nlctiont; for b tKh the ions and electrons by using the zeroth and first order -elocity mncioents of the Boltzlmann eqiations, the results of which could be Ised.vith Poi;- scons equation to find the potllential. The effect of collisions was taklen into account in this.aInalysis..1. ] Ex.. je trimenltal Work In addition to the theoretical stu]dies wlhich have been made concerning wave prc:?pagation in a plasmna, a considerable amount of c\per- 'inlll('iIal work has been perfotrmed in this area. Some of the more pertinent published results, particularly those (dealic.nig with the excitation and detection I EK waves, are discussedre belIow. Thhe first conm-prehensive experimelntal inm.-. tig.it io into the subject of plasima t)sc illa.li)ns is that of Tonks.lld LLlangmnii (1 i"'I). The purpose of thleir experiInent was; to find the electron plasmta.:.ill.illis which Dittmer (1926) llLha -uggCesteld wv,~ere jresponlsible for Itn11-'lllui.'iing the velocity of mono-energetic olecu-;Ao)n l)e,llms whlen interacting with a plasma. Arrangements for detecting plasma oscillations were made by connecting a crystal detector and galvanometer between two- electrodes in the plasma. The oscillation frequency was determined by m]e;astlriiI the \wavelength h-r a pair of Lecher wires. n, ill.ii;is were found in the i algit. of 1 to 1000 Me in a spherically sl'aped mercuryl plasma. A theo — Iretical e;pression which was obtained, relating the oscillation firequenticy to the electron d-ensity, was found to satisfactorily account for the oscillations of fre

16.jOt;y> 2,-, t't'; than 1. 5 Mc. (The expression which was derived i it il duced the ' 4"-l'on tp).1smit L't'ji'-1. ) Oscillations of freqluency less than 1. 5 Mc were.tt:'itIttd 1to) thle iLons..An expression for the.on oscillation frequency was also,oiC.tl nid 'l'e existcnce of ion sounLl ast\V''S was postulated, but none were io nd. M riii Lrnc n- Wetbb (1939) also miade a careful study of the iitteractionl of ditrect ci'1l tIlut electron beams with a plaLsnma. They used a movable thin wire -probe,: fc:- inaeasu-ling the electlront velocity distribution function. The same probe was Lalso Li -ed, in conjunction with a Lecher wire system, to measure the fre — cquenc:y of [plas-nva osc illlttions. The results showed that the beam electrons were sc ittered in narrow, well-defined regions, and that oscillationls at or near thie ele:ec' '(1 1 pla'sma frequency also occurred in these regions. No such oscillations w7e re founld loutside these scattering regions. It was concluded that since he o —,-i I'itu ions occur only where the beam electrons undergo scatterinLg, the Jct.' el'ectr'ons are the source of energy for exciting the oscillations. A -imila:t experriment was performed by Wehner (1951). He also found that 'I:.<'ilitilon occur at the plasma frequency when the beam elections'; undergo scatterinIg. Wllen the discharge conditions were such that no abrupt scattering of the )bean- electrons occurred, no oscillations were detectable. By correlating the tpr-ob e in'._ —I Ient.-nts with visual observation, Wehner was able to conclude that llie sc;ttelg tlook iplace in a thin layer at the dge of the ion sheath which ( Lrrounded the beeaim control grid. Wehner -II no -l td that it is reasonable to I elieve that the excit tion of plasma oscillations by an electron beam is always

17 associateld withlbe layers near a plaslnma boundary, i.e., near the edge of an ion sheath. Further evidence to support Wehlner's resullts was obtained by Looney and Brownl (1955). Their experimenlt diffei ed from illht of Merrill and Webb and Wehner mentioned above in one important respect: whereas the former utilized Onfly one cathode to form the discharge and to provide the electrons for exciting plaml. I-fSillations, Looney and Brown used separate cathodes for each function so that the exciting beam and the pll;LslI.a could be independently controlled. A Imovable wire prole, \\hic l was capacitatively coupled to a superheterodyine receivler, was used to detect the preseince of plasma oscillations. It was found that lti:u lil wave patterns in plasmla oscillations were set up in the region between the e\citing beam electrodes. The oscillations occir'IFr'c] near the electron plasmia f:-equency. The presence of sheathsls or the exciting beam electrodes ',a.' found to be,c 1..,, 'y for the excitation of the oscillations. By varying the excitin beam electrode sheath thickness, it was observed that the standng wave palttern always adjusted so as to keep Ca node of the pattern near the visible sheath edge. A calculation of the phase velocity of 11( standing wave components indicated that the oscillations are not EM in origin. The authors suggest that it is a longitudinal pressure wave (EK wave) set up in the plasma electrons by the exciting electron beam. An experiiment was performed by Gabor et al (1955) in order to clarify the plaslma oscillation question, particularly as related to the shke-%tht The experiment was motivated by a desire to tccoutlt for Maxwellian electron velocity distributions

18 n a p1 as mu even vwhen its inil-, iilions we many times less than the electron rmean tree path length. It was suspe(cted by the JI.uthurs that the explanllaLtion of this vht.,normenon would be found to be at:-,.:i.tt<d w.th plasma oscillations. Contrary,o the pi iviously mentioned experim-e its where an electron beam was used to *xcite oscillations in a plasma, an electron beam was utilized by C;iLoin to serve is a preobe or detector. A direct current tltutlun beam of 1 to 20 KV energy was c.oll initted and arranged to pass perpendicularly through the edge of the cylinldri,.al mercury plasma1, and the deflection of the beamn was observed by Ltarious arrangements. It was found ti tt when the beam was passed through Ilie ila:sma uUtside the sheath region, it suffered little deflection. When, howeverr, t wa;s passed tlhrough the sheath region near the wall, the beam was deflected in a direction per-penlicular to the wall at frecquencies on the order of 120 Mc. It \,as alsoc determined from the experimental data that the static sheath potential Awas nearly parabolic and hence the reflection time for all electrons is the same. ThRe Lauthors, thus conclude that electrons which reflect from the sheath can gain or lose energy depending on their phase relation tothe sheath osc illatil)Is at the time of (cnt:y. The energy which could be gaired in this way could be on the order of several eklc tronl volts, thus providing an explanation for the existance of high temperalure Max\vellian electron velocity distributions in such plasmLts. It is 3nelevant to note 1LLt the electron plasma frequerrncy outside the sheath region was Jound to be about 500 Me. No consideration was given to the possibility that the electron be nim may have excited the oscillations. Subsequent experiments which involved probing the static plasma sheath with

9 tiI eelle ct on betamI were reported by Gierke et al (1961) and Harp and Kino (19163). Nco evidence ol oscillations within the sheath, as reported by G(ab.i- et al, was:(oind. [Earp alld Kino also perfl.' ll-d the electron beam pr'.obillg experiment \xlLhe 1 1 slhealtih was -,lject to an rf electric field directed normally to the shei.tlh. T leiri mi.i:- i ' ments on the electric field variation in the sheath were' tonliI( to be in grood agreenment with a theory developed by Pavkovich and Kino (1lf;;[). [oirmultlted from the collisionless B. it(/niia. I 1 I1'i il;iI..Aiotlher aspect of plasmLa which has recr.i\c.l considerable aLttention is the (cattering of p].ane electromagnletic waves by aL cylindrical pll.usma column1. Tonks (19>) sbhowed that a bounded plasnma would oscillate at a single frequency proportii Ial I to the plasima fr eqlency, wilth the proportionality constant determined )by the si ze and shape of the discharge, the nature of the mode of oscillation, and..he d-ilectl'ic constant of the material sturrounding the plasma. This result was )btaineid by regardlling the plasma as a uniform dielectric with a dielectric constant 2 2 C,: —.J Wto ) where u and t are the electron plasma frequency and the lradcio o p p -'requenc'y respectively. Subel.(qilelit t-' \ler1 ImllCltal results obtained by Romell.3951) anld [Iattner (1!957, 1963) have shown tha.t there is a series of resonances in Itle scKlttering cross sectionl of a plasLma column, rather than the single resonance preiclted by Tonks. These additional 'resonancesL could not be explained successully L.si:ig a dielectric theory of cold?plasma. GoAuld (19:59).itllte tcledl to account for them by talkin into account the random Lt'er'l.l mrotions(- of electrons in a ullifolrm hot plasma 1111111 and allowing for:ia dial Els wave motion based on a scalar electron pressure. His theoretical

2 N]lcS] i1n tor l.i re'sollnance frequencies showed that the -r s 'I;aie.s would be t mot r!e closely >ua).,ceed, by labou)lt an order of magndlitude, than those measured e> x-:e]im-lll-ental]y. I[n a recent paper, Parker et al (1964) have extended Gould's:ea't' ltt to the case w\\here the pl1asn-ma columln- electron density varies in the a(hial direection. The model for the linol'luliiformity of the electron density was )atcsed on Palrker's (1964) resullts which were lenltioed previously. The analysis:)1(occede(d fl rolv the linearized /:e r'th and first order velocity mnoments of the olliotsionl ]e ss Boltzman etquationl and Ihe Maxwell's equations, together with a scalarlr eectron prt - -l, *. The quasi-static atl. p'lximation was invoked for the EM' field (this means the electric field. is represented by the gradient of a potential) alnd a foI trtlb order differential equation obtained for the potential. This equation w\-as solved subject to the boundary cll ml'ition that the radial electron current ',tanistl at the ' 1 i i I I i (.,1 pl'lalsm boundary. The resonance spectruml for exci — tlation cof 1he radial waves in the cylindrickt1 plasma columnL by multipole devices (e. g., a sp'lit cyllinder capacitor) was then calculated. The frequenlcy spectrum i;as found. to- depend on the square of the ratio of the plasma columnn radius to the i oot — ina n-square electron Debye length. Experiments were perfot'med using dipole and cluadrupole excitation devices rather than performing the free space ptl:tlt' \'ewa, e scati.ering experiment, since the desired multipole mode of the cylindc(rical columnl can be preferentially ctx-ited bv the former a; rrangeltent., whereas tie la:tter experiment does not have this advalnta:;ge. Good.Ltg l*i11(. L. was obtained between the experimental and theoretical spectra. This is a significant result

21 n that i p ovis a reasonably conclusive (ILlonlstl'ra;tion for the couplling of lM.niid-l EIK \\%wves at a plasma discontinuity as well as in the inhornogeneous Lisma-t itself. The existence of thi rLadLial EK wave is, however, only indirectly I hs e i lI throu(ghi the agreement in the results, and is not indepenldently estal)lished,y (tl-etr e:xperimental servations which would be l..i'..l.tle.. It is also obvious t-h.t lUany th0eor -etical treatmnent intendedt to.l-c.( Utlll for experimental observations of a Itboratory plasnma must take into consideration the ollllnuniformi electron ltensity,fi such aI pltasIma. Thus Gould's (1959) original treatment of a uniform plasma columLn,'a.-; not ale to explain experimn-ental observations that were successfully accouinted A for by the Parker et al (1964) treatment of the nonuniform coluimn. 1.2 Problenms Renmaining) To Be Solved TU to this point, the generall subject of wave propagation in a plasma has t-eel clowiis-dered, and a Iather extensive suLlrvey of previous work on this topicias beeL given. It should now be apparent which areas have been the more c.-,nm-:letely inlvestiglated and also where the major unsolved problems lie. We tuntr Mlarize brir efly then the previous coni tributions, particularly as related to the lield of pl.- 1l.l —ilmmersetd antennas, \which is of primary interest to us. Or)t the theloreticbal side, the pla;.smal;l-plas ma interface scattering p'i'bllc nl hlas beenl so.lved for a variety of boundary models and boundary conditions for plane., 1 1 spherical boundaries. Tichlllan and Boyd (1962) and Tichldan and Weiss * iL61) ha;ve probably given the most c.L-arefutl trleatment of the problem in the ')OL1llkda.ry co.-nditions and boundary models used. Problems of this type are of it:eest primarily to astrophysicists. Solutions to the elect:ro.dynaL nI ic plasma

22 s] t'at1h problem \lnn tlhe bClloundary is a solid material such as a dielectric or I-m tA;l 1hVLa(2 been confined to the case of a pltane 1)ounCLidary where even then the p-1 tLsm; shecalh is neglected, except for l-e cylindrical plasma column. The ics ' lated sio.urce problem has been ct.is idlered for various types of current st iurces iii a uniform plasma with the indickation that more power can be radiatecd a:- EK V.V\\as than as EM waves from filament currents. The radiation from a bo(dy 1i is ben confined to a consideration of the spherical antenna with the sheaLth b<. ing neglected or replaced by a layer of uniform permittivity. On the experimental side, the capability of exciting and possibly also of d( tectinog plasma oscillations (which are atpparently EK waves) by means of diriect culrrent elc't con beams has been demonstrated. The experimental results of variou.0is authors are in general agreement thatt it is the sheath region which is in porl tant in providing an excitation ct'l 1ianis n for electron beamns to produce suc'1h o)scillat ions. There is expt.iinlt il evidence that thin wire probes can also be used to detect pl:sma oscillations excited by electron beams. Indirect experimental evidence i'as been obtained which indicates that EK waves can be excited in a cylindrical pla;smat columnl by EM waves. Thus;, apart from the cylindrical plasma column, no solutioll has been obtained for either the scattering problem or radiation problem which includes tl etfiect coef the nonuniforml plasma sheath at the boundary between the plasma and the sc ttt{ring l or radiating body. No solid theoretical understanding of wave pr )ptpagatiotll in a plasalL can be achievecd until the sheath effect is taken into ac)count, rIF the indications are, from the results of the study on the plasma column

23 rt sojiunc u's xs wl1 as other expt.-init iLA evidence, that the nonLuniformn sheath i nrgiot may be the most important single factor in these problems. There are a il m'ber oi areas that one might consider investigating then in an effort to c la rify thl. sheath influelnce on wave propagation in a plasma. The.-i.ilit i ni., characteristics of plane, cylindrical and spherica.l body-plasma boundaries are of particular imtl rest. These are boundary shaLLpes which are readily handled and wxhlich are conmi only el-ncountered in dealing with laboratory-generated plasmas. The radiation from spherical and cylindrical dip:)ie antennas is another problem a1(:t-a whici is of great interest, particularly since the results of isolated source studies indicate that more power can be radiated in the EK wave than in the EM wave. Botl of these are problem areas where theoretical work in connection with carefully performned experiments could make important contributions to the state of our ptresent knowledge. There is an.rladitit1onal problem atr. which is a potentially rewarding one on both a theoretical and experimental baisis, and which encompasses aspects of both the scattering and radiation problems. It is the subject of the Sptcc ific excitation and, more importantly perhaps, the detection of EK waves. All of the theoretical cons it-rations devoted to p)lLas iLa wave pI )Lopagalt ion to the present timne have ultimately been concerned primarily with EM waves, either with the aLoun)t of IEM radiation converted from EK waves at some plasma transition region, or il,11i,i perturbing influence of the EK wave on the radiation charalct: r — istics of an antenna structure or source distribution which should be essentially a source of EM waves. No consideration has been given to the topic of plasma

24 nt r:intrssUcd;lltennlls from the viewpoint of detecting EK waves. An,ana.lysis ->ould 1< (;Lrr iec out to determine if it is possible to detect with an antenna the Wss's1nc1C of EKl x;ves in a plasma. This is partic-l.lllaly important if any experiillts atre to be performed in this area, for the purpose of testing the theory.. Presently, the exist of the EK wave can only be inferred indirectly, as, for xamp.Ln e, from its theoretical effect on the impedance characteristics of an..ntijtl;iL, ori the resonance of the cylirdrical plasma column. It would be nmore 1desiraLble to achieve this end directly by some appropriate detection device..3 P 'r)ll-im To Be Investigated The problemn is to devise and analIIyze promising schemes that may allow the detection oF EK waves by a plasma —immersed antenna. Ideally, a device I1i.Lt woulid measure the divergence of the electric field at radio frequencies is ecquired. For example, the analog for measuring the curl of the electric field is L, small loop autenna. In addition, it should be a passive device that does not tundul7y c(Jsturb either the static or the dynamici plasma behavior. Finally, it should t1ie. il to accomplish its task in a strong backgrounlLUig of EM radiation. A Langumuir probe such as a thin wire suggests itself as one possibility. Thin wire probes a,\x e indleed been successfully used in previous experiments to detect pIlasma oseillat.ton-s and to measure static electron densities. However, the oscillat- 'jn \whin2ich wez re observed by using such probes we:re the result of high energy beams cdriving Lhe pla:t.-_ ll into oscillations. Whether or not such a probe could detect EK wave oscillations of such small aLlllitude that the linearized theory applies would have to tbe carefully investigated. A capacitor with grid-like wire meshes

25 Co i p-Iai)s mi 'ht also be considered as a j1) i il.ly useful device for thlis applicaj on i, I tis wold have the draL'\\1);bac' that in order to determine the capacitan-l,_e, \L iic: \\ ()d b(h a ftunction of the chargce between the plates, a voltage would haveLlz (o hW Ijllied atcross th12 plateCs, with a resulting disturbing influence on the sl '.'t AnIot!ler method which was previously employed for detecting plasma osc il,'lti )s tit i,,ed the electron betam. Unforltunlately, the probing beam could also at h'ie same time excite oscillations. The Ilueslion arises now about what other lmeasurelment conceivably may be miaLde oil the EK walVve, if a direct mei sLurlemeelt of the divergence of its electric tield clo,:s.iot appear practical. If it is recalled that an EK wave which is scattered from La pllllasmaL diSCon1tinullity -may have some of its energy converted into EM L 1Aliation, we are provided with one possible mechanisll for such a tea. surtIrmnt. Wlhile the incid.nt EK wave has no magnetic field, the -.i'tl,.f ' Ei' wla\ does 1":s;ess a;magnetic field and so there will be surface currents induced (cn a co(nducting scattering obstacle. If these surface currents could be e-ncasurdt in-d their wavelenlgths ascertained, then, because EK tand EM waves at i'e same ft Illl qLncy have wavelengthus which differ on the order of the ratio of the velocity onf l ui in a vacuum to the plasma electron thermal velocity, an incidlent EK wave may, theoretica-,_-at least, be detected even in the presence of EM tadiatlti: ' The problem would be to determine whether from theoretical coni;idl rai tm s:.cl i nduced current imeasurements could be carried out on a practical 1, - is. A cylindrical geometry appears to be an attractive one from both a theoetlic(al anct e.xp.Sriimental standpoint..On the theoretical side, a fo'rmulation of such

3 ''t._ tn! ) '' n cli (':cal go ometry is relatively straightfor~ward, and static it l, STUt ( 5 t'I c crtd Lo this geomet(rN are available. An experimental measulreitL (o tt 't' fe currents i11nduced )byN i wident wavaes on a hollow metal cylinder (Hi1ci em i d(1 bv terminaliting transmission lines, which run inside the cylinder. n: sl, t aplIj opria.' ()t'iltely on the cVylinder surface. The specific problem stldli,'d theoelicall isr th i iwvestigation is the excitation by incident plane EK and EM waCves:f surfa.e carreots on an infinitely long, metallic circular cylinder immersed in it l.ts.L, in otder- to determine the practicality of making surface current -1(easI'urcntiIts tior the purpose of deteceing the EK wave. The (1mAillCnder of this thesis is divided into two main sections. In the lollowting, Chapter II, tlln. theoretical fi niulation of the problem is given. th ipler II presents 1he results of the Lnumerical analysis and conItains a hdiscussion,i their signifi,:cance. The main feature, of this study and conclusioins reached e om it, -.re given in ('1t.ipler IV. Several appendices follow which contain detailed ais-lyses of various aspeCts of the problem which are not necessary for an understanding of the ma'Lin part of the thesis. The rationalized meter-kilogram-second (Gliorgi) s -y;teni is; used th t1ough out, unless ot i r\\ise indicated.

CHAPTER II FORMULATION 2. 1 1 ne B3 11. i,,. Eluli,,n Cne starting point in the t'rltlltmnlt of an assembly of various kinds of charoged iin1l Iticli rged particles such as is foundtll in a partially ionized gas or l)lasmi:l, is the Iitzmrnn equation. This equation aIccounLts for the force effects of both conlact collisions between particles as well as the macroscopic forces (tue to eldclric, magnetic and gravitztl, i-nll fields. Other forlmllattions, such as the Foklk;er-P'lalnch equation, have 1 1 d dleveloped in order to overcome so rne difficulties in accounting for force effe cts which are not contact collisions but rather Lblog-range Coulomb collisions in highly ionized gases. A typical labora — tory plasnmah however, in which exper' i,Ills might be performed to check on the results of the following development, will be only slightly ionized, so that the Boltzman:.equation approach, which is used here, is a reasonable one. A completely general treatment of the N-comiponent plasma would now involve N equations of the form (2. 1) where f. f.(r, u, t) is the tdistrlibt ltiul, function of the jth component, which 3 j gives the density of particles in ordinary space per unit volume of velocity spa2ce at space point r., velocity u and time t. mj is the mass of the jth species, F. is the force ac-ting on it, and V and V are gradient operators in physical space r v and velocity sptace. The N-equations which result may be coupled through the 27

28 collisi,)n term:andl the force term. A solution of the problem would reLquire l indli ng the N distribution functions I:'pji>:aing ia the N-equations. Once the distributionl fulLti ills have been obtaintied, all the macroscopic physical observ - tbles of interest lc'l as density, velocity, pressure, etc. could be generated from moments of the dist1ribulti-on fullnc i.ns. This is a forl l1 l., 11 p1roblem andl,:,ne whiclh can seldom be solved, even when simplifying assiumnptions are emploved. 'The usual approach is to employ a linearity condition, i.e.,, the timer arying components of the variables are assumed(l to be small compared to their stattic parts. Sluch phenomenon as I,1" 1,, dlamping can be developed following this anrl ilsis, which is called the kinetic theory approach. When the problem under consideration involves in addition, electric and miagnletic fields in the plasma, there is required in addition to the system of e(quations in (2. 1) the Maxwell eLquatiotns (2, 2 a) (2. 2b) (2 2 c) — 1r where E;inc H are the electric and magnetic fields in the plasma. u and e - o o are the permeability and permittivity of free space. The complexity of the problem is increased still further then, and it is obvious that the kinetic theory treatment is unworkable for all lbt the.ir-lllps probl))ems.

29 An atIernat:lvl e ap:lproach t to the soluti)on of (2. 1) and (2. 2) is the hydrortviui:]:c treatlent. This in olves takling velocity moments over the distrilotiln uLIleti ll:'i to generate an inllfinite set of nc i 1ll equationll in which the I'i c'ro-,iclopic ~\arial)les of Ilnimler cle-tsity, velocity, pressure, etc. are the uinlnxns. A solution to the problem requires finding the va'riti n, in time and:-lpacce of ihese qLuantities for all the plasma (components. It is usual to invoke linLerii. when using the lh dlrod,:, n:'nli approach also. There is no essential philosophical difference between an exact so] ution obtained from the kinetic appro(:.ach and that obta:iiu.-d from the hydrod3ynllamic approach, since a knowledge of the( distribultion functions means that all the macroscopic var-iables can be f',til]. If, onL the other ]lh1nd, all the infinite set of IlMalci.'l-'.e,.,it variables is Ilnown. the dist'ilbution function can be c.onst tructed. There is, however, a I;,' rai practical difference between the two imethods when one considers the comnpglexitv of the Il;l lilt:ll li s whllich is 11ncounte red, even in solving a problem \lhere r!1'i1n sjilplif\ ing assumlptions can be made. Oster (1960) points out that in order to avoid excessixely complicated maLthematics in the kinetic theory app rl)oac.h, it is nectss:rii to make such assumptions that it is more reasonlable to use Ithe hydru i,, n'l:llic equlations. The h3 dll'.lnT:lllic approach is the one to be used here. Before further dleveloping the formi-1ulation, it is appropriate to discuss the a-sLlsmptions which will be made a)bout the plasma. The plasma will be con.siderecl to be o-f infinite extent and of uniform temperature throughlout, and to consist of electrons, positive ions anci neutral particles. The electrons and the ions arj taken to have the same numlber dens ities on the average so that when

30 the pllasnma is un1,iform, it is electrically neutr-al. Collisions of the electrons,. ith io.,ns; andc nelutralls are ignored, in so far as their effect on cllltclix. plaslma oscillations is concerned. This seems jlstifilable since the electron collision Ilreil'lu'- icy in a tvypical labol.iltlr plasiia;t may be a factor of 100 or so less than the elec tllon plasinla frequency. Eleectcron collisions are important however, tco the p)rodcllction of a MaT:xwellian electron velocity di:>- I il ui l iI1 which will be used here. This alsco seems justifiablle, in that Maxwellian electron velocity distri — 1butions have: been found in laboratory plasnias, even when the electron mean free path is longer than the plasma ldinetnsions., The way in which this distrililition is produced in such a plasma is referred to as Langmuirts paradox and has been discussed by Gabor (1955). JF'illly, the plasma is lil;i to be quiescent, i.e., no.li:lrg' is being crltealttd or destroyed. This iSSlllupti)on primarily affects tlhe static plasma behavior in sheath regions, and simpllifies a 'liscmmsi' ' of that part of the problem. 2.2 Development of the Macroscopic Equations The proc-dull're to be followed now is to generate macroscropic plasma variables by taking, velocity moments of the cl I' i.inlli..s Boltzman equations for the ions and electrons, which are (2.3) (2.4)

31 I nd(. ire 'i electrn a nd ion distiribution fUlllltiins, u and u are the elec-: i — e ~-~L t, r'];td I I n I vel ocities, 'and m and mi are their respective masses. The force e i (2.5a) (2.5b),.here — e is the electron charge, and E is the electric field in the plasma. With this choice for the force term, the subsequent dCiscuLssion is liminted to the cacse ',here _,: \ itll i ntal and magnetic forces in the plasima may be neglected in cornlpa ris' ci ilh the electric forces. This is justified in a plsml.as,'t which is not hot CnoULih for relativistic effects to become importnant. In addition, no external iL.,-lne ic field is considlered to be present. TIt first four moiiit-vts over the di.sl ri.iiiin function produce, in Cartesian coo]'(linatces, particle density: (2. 6a) particle current: (2. 6b) pressure. (2.6c) heat flux: (2. 6 d)

32 114 V I,(,(Cl itx is A~- a uiate 11u 1ciiA crof CMI ipted mioment equal 1ion)Is, '\hich contain only the Ia, cur acI C ic plasman war ialhles suich asGien liv (2. 6) mayT be generated from (2. -)amiI (2. 19. If it a ejve necessary- to deal with this entire set of eqluations), I I-: x'voiilId JIme no inhderen-Wat advaintage of the- hi N rod~ynnamiic a-pproa'tch over the, Fia L trea ~_jtmen1(1t. Howev~ _er, it is pvs-sicall~ n me an ingful in mnany practical lprci)]etns. to term-inate the infinite sin.t of rnmnt equations by taking the nth C)rder,_ ~'e~lacitv~ mom1"LCent o f the? dli strI ibutiton function to be zero. The p 'mt at wifich the ni.momert equaitions can be terminall-tedl is discus —sed 6wPa rh-Le r et Cal. (1 964). \Vheneosiein the propaga,,tio-n of longituidina-l wNaves in ai plitsumL it -: ci be shu% a that reta iningno the hio igt. r, order veloc-ity mioments a n gemrait i g the momen(_nt equations is equiv-alent to in11C'ILudCing 0 the higher term's in 1* Xp"Ins ion of (rain hio frequeney/ C l- C ectron plasmia frequenIcy) 2in term-s of (era+pw /a v speed) in the dilspe~rsion relation for 1 'ngmmidiiimal waves p Onpgaia in a plasmia. Park-er et al.( (( Incude that minc'mtnts up to the pres — s'ure_ ten's a)r a IId Possibly thle hea~L't flux tensor are required, but none of the hi~ghertmromentrts thanu the(:se wvill e mn~t1ibu~te: sufficiently- to the accura-cy of the results to o:iitbeing uIseCd. Accepting as rea)sonable the termiinatio-n of the moment equations with the hatii fitK ix,nsorl, the problemi has be:m mmtml 1filed a great deal but, is unfortunately siall N, CIf-ifficult. If no fuirther si m,6 ificaticns were possible, one would be, requitred to rc-la] with Can equiv-alent 19th o-rder differential ItquaI'Itio~n. The choice

i- i(i o. t(I so se(,t i lIt h(tit lilux telnso' r ' (l al to zero, an:lapproxilnatioiric"' is 1' it (tlx ', d see lor ex.npf-Il TidCinan and Boyd (1962), Parker et: L. (1;4). K ritz i Land M ilntzer (19(60). 7] iis 'et( ices the o},rder olt thu., diflerential eqlualti,n to 10th order, and ii ill.t e tlialt 'lhe electron and ion velo.:ity (lisl i, illt ius are isotropic Rose and (lark (Ilh1) p. 119 A f.:nAl assuinpt,<n, consistent with setting the heat fluI' tens-, r e.,j'il to zero, and which reduces the differential t.iuatlit)onl to 8th order, is thaLt of replacingl thei pressurle tensor Lx a scalar. The view which is thus taketin lir:1) is that the electrons behavelNiv as a conlltinUous fluid and the effect of all the ele.tic )n interaciccons is represented by a scalar pressure term. The use of a sc(alar pressure is ex.act for a uniform plasma. In the preseIce o1 a pl;asMla: inhom)- w l( ity, which is to be considered shortly, it is an Ipproximation. The representati, rl of the electron tbehaxvior in the plasma by an isotrp.:ic,scailar ],Lprssure' is (),le \hich tlis lt:t.n widely used Cohen (1962a), Fedorchclnkio (14il,;2). Field (.1955), Parker et a1. (1964), Kritz and Mintzer (1960) Good agreement 3- t'\-etel theoretical ald experimnental results has been obtalined [)- Parkl.er et:l. (lo-,~4), where the theoretical results were based upon the use of a sc:lac: ell tl 'il pressure. Hence the indications are ill.i this is not an unreas on:I 1) app I)r;i ioion. The m )ment equat i-ons which a:-e to be used then are uobtaintll from the zeroth tanC first order velocity moments of (2. 3) and (2. 4) a; i are (2. 7a)

34 (2. 71)) (2. 8a) (2.8b) Vwhell re hi sul)script e and i denote lquantities associated with electrons and ions ifspI:ix\ ely. The Maxwell equations can now be written (2.9a) (2.9b) (2.9c) (2.9d) This sl.et (of e(LLuations is completed by an tlqultiion of state which relates the press;1'. to the nu mber densities of the charged particles. The non-time varying or sitilc p) reC ssulre P, from the assumption of Maaxwellian velocity (list ribullti.li, is. (2. 10) ',llere k is B ilxzllnil's,-,;nstajnt and T is the; temperature. It should be noted thit thc) is thel, first use to be Inade of the Maxwtlli:mln velocity distribution.

35 All of lt1 development to follow rests on this I liii. *. For the dynamic I)rcssire va' iiilions Pl it follows that (2. 11) 1 1 'a\h.re fo:rc one dimensional, adiabatic pressure \ ariations, y = 3, (Cohen, 1.955). The temperature for a Maxwclliani distribution of velocities is given in tern's of the ro)t-I-Jear-square (rms) vel)ocit, v as r r (2. 12) The set of Eqs. (2. 7) to (2.12) constitutes the starting point from which all of the developments to follow are derived. They contain within them the complete -I't'c and dynamic plasma desc.riptio)n with which this treatment is concerned, subject to the approximati(o1s and Ztassumptions which have been pointed o).1 above. These assumptions are again: that the plasma is ] ii. —ul:ttivistic, of unifo(rm tetmpi."ttalti. throughoutt, infinite in extent, and only slightly ionized; the i. -i'ged.iparticles are electrons.,,il positive ions baving Maxwellian velocity dCist ributi Lns; they suffer no collisions and their behavior can be represented by a scalar pressure. The use of these assumptions on the physical nature of the plasma has been discussed above and no further mention will be made of them e)oCvt/ inl thle suhsI.L'.l11t_'tlut theoretical development. There are however, some simpliLfvin; app i mations which are basically mathematical in nature to be ntrocduced in the following.

36 2.3 Linearz,I;.ii of the Macroscopic Equations While a great silmplification has been effected in going from the Boltzmann equations to the system of Eqs. (2. 7) to (2. 12), so that the problem is inucL more m.nl':^I*, ':11, it is to be noted lthat these equations exhibit still a ii:tjur difficulty. That is, of,,L u'Sc, the fact that some of these Eqs., (2. 7), (2. 8) and (2. 9c) are non-linear, thus posing a still very complex mathematical problem. This difficulty may be side-stepped by the artifice of lineiaization, a very coimmonl3 used technili'.lt in hydrodynamics. This linearization is accomplished by assuming that variations of the plasma variables from equilibriumn are small enough that products of these \ari:atifnls can be neglected. The reasonableness of this assumption is dependent upon whether or not the effects produced by perturbing influences which satisfy the linearity requirements are large enough to be seen experimentally. In other words, there is no doubt that linearization can be valid, but the question is, are perturbing influences which do produce llmeasurL able effects small enough so that a linearized theory applies. This cannot be.i.-., ' r.,1 until some theoretical answers are obtained. The indications are, however, that good success can be obtained with a linearized theory, which is to be employed here, as shown by the results of Parker et. al. (1964) and Pavkovich and Kino (19614). Before the linearized equations are written, one further observation is made. Since the ratio of the mass of the positive ion to the electron may vary from about 1800 in a hydrogen plasma to 360, 000 in a mercury plasma, thern the ion plasma frequency may be from 1/45 to 1/600 that of the electrons. This means that in the radio frequency range to be considered here, where the electron plasma frequency is always less than the radio frequency, that the ions will

37 contri}ibute ncgligib)ly to collecti\'e 11i..i uI.i oscillations. Thus for practical purposes the ion motion can be neglected in so far as the time varying behavior of the plasma is con(cerned. The linearization of (2. 7) to (2. 12) is now accomplished by the following substitutioTns: (2. 13a) (2. 13b) II - I i (2. 13c) (2. 13d) (2. 13e) (2. 13f) The subscript o denotes a quantity which does not change with time (these will silbJ.-.cLltLcltly be referred to as static q(ulntities) and the subscript 1 denotes a time-varying quantity (this will be referred to as a dynamic term). With these substitution, Eqs. (2. 7) to (2.12) can be written f - (2. 14a) (2. 14b)

MISSING PAGE

39 of ec ua t iions cdiffe,rs considerably from the incorrect approach used by Field (1955) \\here the staLtic electron velocity and electric field terms were neglected. It also differs from the forml ulation used by Parker et. al. (1964) because o:F their neglect of the static electron velocity. The reason for the applearance of static velocity terms will become tlapparent in the following discussion. A static magnetic field term was incIludetld in (2. 14) for generality; it will be shown to be zero below. An obsr-rv f ii, of the linearized equations shows that eight of them contain only static quantities, whereas the six remaining equations may contain. both dynamic and static terms, exhiblitingl: the influence of the static plasma characteristics on its dynamic behavior. Thus it is necessary that an understanding of the static behavior of the plasma be reached before its dynamic response can be investigLaed quant it lti tly. It is for this 1e:, sun that in the following section, the st-t tic p] a sma (chlracteristics are in estioLgated for the particular geometry Vwith which the re ma inder of this study will be concerned. 2. 4 The Static Plasma SIht:1ht As was mentioned in the introduction, the purpose of this study is to investigatue the surface currents which are excited on an itnfinitely long plasma immersed circular metallic c3liillLr by plane electromagnetic (EM) and electrokinetic (EK) \\.i\t. s. The plasma surrounding such a cylinder is changed by its priesence, forining an inhomogeneous region called a sheath. Because of the cylinder geometry, the plasma \variation in the sheath is a function only of the radius, 'iriblt, p. Figure 2.1 shows the cylinder in relation to the coordinate system.

40 z +00 Cylinder I i I 10 I 0 -r:-, — QT-: r"13P: jiili-.-, - Y I," w — =;-, 17 `r i:1 I-1 Cc s-, ~ Ir 1~ '' r d 1 9r" k, F rc_-e a r 4 - r`` r ' iC'* 5; '" ~I -~2~ a;-P= ~- ' 1CI —r,: a,~ ~ 41~; 'iS = riJ b r -— r 91'.~ _ Yi- 'iQ~,.? i; ~~~. Ya Ir? e -e __F1-iC i 4 s F "; , = z~ _- 00 FIG-: 2.-1: CYLINDER AND COORDINATE SYSTEM:-i -= -F r

41 The mechanism for the formation of the inhomogeneous sheath is r -- fact that, since the electrons are lighter than the ions, they have a larger rms velocity, even when they are both at the same temperature. In a typical laboratory plasnma the electron temperature miay exceed the ion temperature by a factor of 10 to 100, thus, further increasing the velocity discrepancy. Due to this velocity difference, more electrons (ian ions tend to strike the cylinder per unit timrne. If the cylinder is insulated, that is allowed to acquire a potential which is determined by the plasma paramieters, as will be assumed here, it acquires an excess of electrons which give the cylinder a negative potential. This negative potential repels all but the fastest incoming electrons and it adjusts to the point where the ion and electron currents to the cylinder are equal. i e' equilibrium is established. This particular sheath is thus a region where there is an excess of positive ions. The origin of the static ion and electron velocities, which for the cylin — drical sheath have radial components only, is thus apparent. Note, however, that there is no net current to the cylinder, and that since as shown by (2. 14), the static ion and electron flows are divergenceless, the static electron and ion currents are then everywhere equal. As a result, there is no static magnetic field in the sheath due to these currents, and H = 0. -o Tlhis inhomogeneous sheath region extends to infinity on a theoretical basis, since the static currents cannot go to zero, except at an infinite distance from the cylinder, according to the plasm1a nmodel which is being used here. However, the contribution of the static currents to the plasma inhomogeneity decreases on the order of 1/ radius as the (listance from the cylinder increases, so that

42 from a ]j1l:l i -il viewpoint the pla.sma becomes homogeneous in a finite distance. In addition the collisions \\hich occur between elect'rons and neutrals in a real p1:i-l'.l provide a source for l ill iihlil ii to the electrons and ions which flow to the cylinder, thus further limiting the extent of the inhomogeneous region around it. The picture of the sheath to be used here is that it is of finite extent, the thickness of ewhich is to be established below, and that the plasma outsicle the sheath can be taken as ulifi 'ri 'l, * 1.. Figure 2.2 shows the sheath in relation to the!iniforml'l plasma"n' al the cylinder. An lamilysis of the pl:isml:i sheath for a planar geometry which was recentl) carried out by Self (1964) and in which the generation of electrons was taken into account, indicates that the electron and ion densities are within 5 per cent of each other at a distance of 10 to 20 electron Debye lengths (Do) from the bounding plane wall. (We note that D =. vr is the electron rins er velocity.) In another ana:lysis by LaFT.rambois (1964) for spherical and cylindrical geometrics, but in which electron generation w\'as not included, the sheath thickness defined by the same criterion was also found to be on the order of 10 el.ectron Deby'e lengths. The electron Debye length for a typical laboratory plasma 4 o with an electron temperature of 10 K and a pliasmi:l frequency of 700 Me is -3 8.851 x 10 cm, so that the sheath thickness is on the order of 1 mm. It i's thus very much less than the 'a\ eleiingth of EM waves near the plasma frequency but may be twice the EK wavelength in thickness, as will be verified below. The fact that there is a static component of electron and ion velocity indicates that the electron and ion velocity distribution functions are not exactly Maxwellian. The departure from a Maxwellian diistribution is dependent upon -. 'L";~ ' ', - s ' L..r' c.:" -."ct e I S.~' L

43 ii.~ -i:.t,f-,: *'' t~ ~~:.' W z s s r b ff -, X'~ '~ -~- - ~~ ' c~.. / ~!!?i ~.. ~.i:: -~ r Uniform Plasma,: -..... ~... ~ — _- IIC. ~~ a X I Direction of Incident Pla-ne Wav FIG. 2..- NORMAL CROSS-SECTION OF CYINDE c_, ' 'tsW'

44 the dis:ta.nce from the cylinder. At distances from the cylinder greater than -: electron mean free patlr(MFP), the velocity distribution for the electrons will be Maxwellianl with a superimposed radial drift velocity die to the static velocity. At distances less than the electron MFP, the electron di-luijutLion function will become truncated due to a deficit of fast electrons in the outward direction. This deficit is brought about by the recombination of the electrons and ions \\hich reach the cylinder. ITmm diately at the cylinder wall, there will be a minimum in the number of outward travelling electrons. The ion velocity distribution becomies almost entirely one-sided as the cylinder is approached since the ions are attracted by the cylinder. It is observed then that the static radial velocities are real drift velocities at large distances from the cylinder. As the observation point is brought closer to the cylinder, the static velocity is primarily caused by the deficit of electrons and ions with outward radially directed velocities. A qualitative picture of the sheath region has been developed, as a region where there is an excess of positive ions, extending roughly 10 to 20 electron Debye lengths from the cylinder and where the electron and ion velocity distribution functions may differ from Maxwellian. A rigorous quantitative analysis of the sheath requires finding the solution of the Boltzmann eciuations for the ions and electrons top-ptblpr with Posiqson's equation. This is a very complex problem. The usual method is to find expressions for the number densitieions ofin terms of then potential by integrating the Boltzmnrn eLquation. The resulting expressions are then put into Poisson'E

45 eqll (,i on, ifrom whlichil the poteuntiat can be found. The complexity of the equation to b.e solved for the potential varies widely depending upon the rigor of the t-reLt.mennt. In a report by Chen et. al. (1963) a closed form solution for the potential is obtained for a planar geometry als.suming Maxwelli>an velocity distributions and equal telmper atures for both the ions and electrons. On the other hand, La Frambois (1964) when considering spherical and cylindrical probes, had to treat a system of non-linear integral equal ions in using the orbital anlsis of Berstein and RaLbinowitz (1959). In our case, the static she..ath solution could be obtained from equations (2. 14). The purpose of this study is, however, to examine the dynamic sheath behavior while the static sheath description serves only as a 'ei c.L]IS to that end. Since there are more rigorous static analyses such as th.i.t of LaFrambois available in the literature, it is preferable to use some of their results here. Some of the static sheath parameters can be varied whllen perfo.rming the dynamic analysis so that the results whlich ares obtainedd can rea- 'iuLltly be expected to contain those closest to physical reality. At the same time they will exhibit the sensitivity of the dynamic sheath behavior to the static parameters. An examination of the theoretical results presented by LaFrambois (1964), and Self (1964), and the experimlental findings of Gabor et. al. (1955), Gierkc et. al. (1961) and Harp ancd Kino (1962) shows that the static potential varlition in the sheath may be closely represented by

46 (2. 16) \.'., I is the potential at the bolundin1g' wall, s is the sheath radius, c c the cylinder radius, and p is the radial co-orditnate. M is an adjustable parameter, the best fit value of M for the experimental results quoted being about 2 and for the theoretical results about 4. Expression (2. 16) is, of course, only an approximation since the potential cannot generally be solved for in a closed form. It does, however, give a reasonably accu-rate fit for the potential variation in the sheath in a form conmentient for numerical Ca'.L 'utl t i (tAIS. The cylinder potential c will be calculated from (2. 17) This form is due to Self (1964) for the planar geometry. An expression due to Chen et. al. (1961), derived for planar geometry, but with assumptions very different from those used by Self yields nearly identical numerical results. Unfortunately, no similar closed form has been found in the literature for the wall potential in the cylindrical geometry. However, values for ' obtained from (2.17) and some graphical results given by c LaFrambois (1964) for a cylindrical probe 20 electron Debye lengths in diameter, are in agreement to within 10 percent where LaFrambois graphs may be, t. iI,.i, ly read. In addtitioni, Parker et. al. (1964) obtained some

47 numerical values for the wall potenitial of a cylindrical envelope encloae wig; a plasm.a which agree %\ ith (2. 17) to within 10 percent for envelopes mnore than a few electron Debye lengths in diameter. Further, their I I~,lii show the wall potential to be relatively insensitive to the envelope diameter. The evidence illdicaLtes that the wall potential is almost independent of crosssecti;nitld geometry so long as the dimensions of the probe or envelope sufficititilly exceed the Debye length, the situation in which we shall be interested, thulls justif itng the use of (2. 17). will be treated as an adjustable c parameter through varying m. so that the effect of a reasonable variation in its magnitude can be observed. Once the sheath potenltial vari. iit il is known, then it can be observed fromll equation (2. 14) that all the other static sheath parameters can be derived on the basis of the known sheath potential. There is thus obtained (2. 18a). c (2.18b) Cc (2. 18c) (2. 18d) (2. 18d) (2. 18e)

43 where is the electron and ion density in the uniform plasma, er and are the elucti,.un and ion rms velocities and I is the electron ir and ion particle flux density in the sheath. This is an awkward set of equations to hallndle, being transcendental in nature. In addition, it does not provide an accurate pictLul'e of the ion density and velocity near the cylinder since the use of a scal ar pressure for the attracted particles (the ions in this case) beconmes increasingly less valid as the attracting surface is approached. Some conclusions about the sheath patralleters can be reached from it however. First of all, the electron density can be seen to decrease in the sheath as was previously surmised, since the cylinder potential is negtire. A natural question which arises is whether this decrease is due primarily to either or. eo or whether both together have relatively the samle influence. Now we cbCrITe fl.. m ep'"Ctio^ (.2r, ) tht ti i -"cr"l'e re — rt inn to th+ e r o-' Cf r: 'Ycn 'rc ',th.n -e,n - decrease in r-.ni t.e aS te errincelr is n r*cfCo -t.c..r rcnc ' t '- tit n t must be correspondingly increased with decreasing radius. Thus eo should have its largest value at the cylinder's cylinder surface. If all the incident electrons are absorbed at the jr Il!,r surface, then V/ ~/u /2 eo er there. But from (2. 17), the cvlinder potential. is such that - c C varies between 3 and 6, depending upon the ion mass, so that Since. decreases approximately eo as. when crossing the sheath away

49 froml the I1 i.il('dr, which is a faster variation than that of, then everywhere in the sheath. Consequently the Boltzmnrr.: distributi,,Lt if - I appears to be a good approximation for the electron density variation in the sheath. This conclusion is vtrlified by the more detailed analysis of Appendix A. When the ions are considered, the potential and velocity are seen to have opposing effects on the density variation in the sheath, the p'IKt in i.l] tending to increase it and the velocity to decrease it. Consequently, nothing: further can be concluded about the ion density variation in the sheath unless a specific numerical case is considered. While it seems that it may be a good:tppl)u\imatiol, to thus neglect the static electron velocity in analyzing the static sheath problem, this may not be justifiable when the 'iy.tniir behavior of the sheath is concerned, since the dynamic response is coupled to the static variations. The ion I ariation in the static sheath is of course not coupled to the dynamic behavior by reason of the large mass of the ions in comparison with the electron mass. In order to give fuller consideration to the static sheath variation, especially from the viewpoint of determining the static electron velocity effect on the dynamic behavior, a semi-quantitative analysis of the static sheath, dealing with those variables related to the electrons only, is carried out in Appendix A. The electron number density, and velocity are obtained there by integrating

50 a tlillilL' ted ]\laxwellian electron velocity distribution, the tlruncation point being Ildetermined by the deficit in outward travelling fast electrons. Expression (2. 18b) is also employed to get the static electron density. As a further check, the method of Bernstein and Rabinowitz (1959) is also used to get the electron number density and velocity. Figures Al to A4 show the electron number density and velocity i]n the..li.th for some typical plasma parialmetel values, obtained as outlined in App:enldix A. As can be seen from these results, the use of the Boltzman distribution for the static electron density variation in the sheath departs from the other analyses which take loss of ele.ctrons to the cylinder into account by less than 5 percent over 85 percent of the sheath. The maximum discrepancy between the various llappr oachlies occurs at the cylinder, but the difference is no more than 50 percent. Since the -variationl between the most rigorous analysis, following Bernstein and Rabinowitz, the semi-quantitative analysis of Appendix A, the expression given by (2. 18b), and finally the simplest approach which yields the Boltznri distribution is negtligible over almost all of the sheath, the tentative conclusion reached above, that it is reason;able to use the Boltzm-Ln distri)butionl for the static electron density variation is justified. This is, of course, equivalent to neglecting the loss of electrons to the cylinder, or in other words, regarding the static electron velocity as being equal to zero. Self (1963) arrived at a similar conclusion in analyzing the sheath.

51 Still unanswered at this point is the question of whether the static electron velocity can be nieglrcted in relation to the dynaimic sheath response. This clannot be resolved until some further consideration is given to the drynamic problem. iThe- results of Appendix A for the static electron velocity are used in Appendix B, i.,,a il. with some results to be obtatinedl in section (2. 5) to answer this (luestion. In s.imllllylll'i-, the static sheath description is based on a representation of the sheath potential by equation (2. 16). This potential variation is then used to calculate the static electron density according to equation (2. 13b) in which the flow of electrons to the ~ 1;"1l- is neglected. The static electric field is then obtained from the niegative gradient of the potential. All of the quantities are functions of the radius variable only, due to the axial and azimlllthal symmnetry of the sheath. 2. 5 The Dynamic Sheath Equations The e quations which contain the description of the dynamic sheath behavior are given by (2. 15) and are rewritten here with an harmonic eW time dependence, as (2. 19a) (2. 191))

52 (2. 19c) (2. 19d) (2. 19e) (2. 19f) In order to simplify the notation, and since no ambiguity can now arise, the subscript e is omitted from the quantities dealing with the electrons and the 1 is omitted from the dynamic terms. Equationl (2. 19a) is the continuity equation for the dynamic component of electron motion, and (2. 19b) is the force equation, while (2.19c) to (2. 19f) are the usual Maxwell equations. Equations (2. 19) exhibit. an explicit dependence upon the static sheath vaLriables which, as will be shown below, couple the magnetic field and the dynamic electron density in the sheath. 2. 5. 1 Uniform Plasma Equations When the plasma is homogeneous, then these equations take the simpler form (2. 20a) v (2.20b)

53 (2. '2 c) (2. 2Cd) (2.20e) Following Field (1955), the total electric field can be broken up into two parts, one part denoted by EE having zero divergence and the th ther part E whose curl is zero. Then the set of equations (2. 20). exhibits two independently propagating waves, the first having electric field E t) O -E corres;londing to the usual electromagnetic (EM) wave, and the other with electric field Ep being the electrokinetic (EK) wave. The results of thi s -. 1.1 p,i. ion of the total electric field are summarized by the following equations: (2. 21aa) (2. 21b),.(2. 21c). (2. 21d) (2. 21) P P

54 from which (2. 22a) where (2.22b) (2. 23a) (2. 23b) (2.23c) A (2. 23d) (2. 23s) (2. 230 r3 and is the velocity of light in free space. The wave equations (2. 22) can alte L naltivXely be written as (2. 24a) V - (2. 24b) This form for the wave equations is more instiructive than the first since it emphasizes the independence of the magnetic field and the dynamic electron density

55 when EM and EK waves propagate in a uniform plasma medium. These are quan1tities which belong separately to either of the two waves, while each wave possesses an electric field. Equation (2. 21) to (2. 24) thus serve to describe wave propagation in the uniform medium outside the sheath reg-ion. It should be pointed out that the frequency interval in which the EK wave can propalgate unattenuated is limited to For becomes imaginary so that the waves is exponentially. I.!i i i. t1, and for l., the - -c Pl: r:- -subject to Landau damping. The EM on the other hand is unattenuated so long as. When both waves together are considered then we require that 2.5.2 Elimination of the Static Electron Velocity from Dynamic Sheath Equation We leave this aspect of the problem now and return to the more com.plex question of the non-uniform medium, as described by the set of equations (2. 19). Since a great simplification Xlould be accumplished if the static electron velocity terms were not present in Lqu:lt luns (2. 19), it is appropriate to consider here whether the magnitude of these terms is such that they might reasonably be neglected in the analysis. A,i, i ltl Ft ward way of determining the relative importance of these terms is to compare them in magnitude with the other terms in the equation which involve the same dynamic variables. This has the advantage that the dynamic part, of these terms

56 may be factored out, so that the explicit dynamic solution, which is not of course available at this point, is not required. It is thus of interest to compare the ratio of (2. 25a) (2. 25b) and (2. 25c) All of these ratios involve radial compolinents only. Now the dynamic electroin density cancllels in (2. 25a) so that we can obtain the ratio (2.26) which involves static quantities only the Ltmagnitudes of which are given in Appendix A. In order to similarly factor the dynamic electron velocity from (2.25b), and obtain an expression in the static variables only, the radial derivarc-eof ' must be approximated in terms of < itself. For plane wave propagation in a homogeneous medium, rt fn mryle G;mLtnh res-cot to the z:2;i cr s te

57 where K is the.-:- (-*,, c w, 've. It seems reasonable in the present case to replace the t —, t er- ' \ by, so that 'r -- e t- e ts C- Cr a ch ft s (2. 27) Similarly, ira arm v(r b^'rlr on t.I-: dI-r: r-r n ft t, r etics In(?25c cAm (2.27b) The requirement on R1, R and R3 in order that terms in can be dropped from (2. 19) is that they be much less than unity. This is discussed in detail in Appendix B, where it is shown that R1 and R2 are less than 0. 1 over 85 percent of the sheath, a:,d thast it is a reasonable ili,'s i i rl. in to drop terms in from equation (2. 19b). It follows as a result, that the terms in _ can be dropped from (2. 19a) and (2. 19e) with the same degree of approximation as for equation (2. 19b), c - - _~ It should be noted here that the approximations which have been made by omitting the static electron velocity altogether from both the static and dynamic analyses are based on the LIsunplllition0 that the cylinder is insulated and drawing no net current from the plasma. When this assumption is not met, as for example when the cylinder potential is altered by connection to an external source of emf, the argument above may no longer hold. In particular

58 when the cylinder potential is i auiked to the plasma potential, there is no sheath, but there is a l.aldizal static electron velocity. This velocity provides the only couplilng mecll;aisinl between the EM and EK waves in this case and thus cannot readily be dropped from equations (2. 19). It follows that the relative contlibutions of the -taltic electric field and static electron density gradient may be conipa rable al some cylinder potential, so that all the coupling mechanism;s must then be considered. At the same time, the static sheath picture becomes more comnplih'..itt.lt with a variable cylinder potential. The problem of a potenltifl pos:itive with respect to the insulated cylinder potential is not considerecl in this study. The dynamic equations can now be written (2. 28a) (2.28b) (2.28c) (2. 28:) - (2.28e) while the static lshealth variables are simply (2. 29,)

59 (2.29b) - (2. 29c) This is the final form of the equations which are the basis of the development to follow. No further simplifications or aLpplUltxillatiollS will be made in them. Note that one of the simplifying effects of dropping the static electron velocity terms from (2. 19) has been to reduce the order of the set of differential equations. The reason for this is that equations (2. 19a) and (2. ld) are redundant, either equation together with (2. 19e) serving to derive the other. As a result, either of these two equations are used with the remaining three to form a complete and,- 1V i t.-li set of differential equations for the dynamic sheath behavior. Since a dciilrivati - of appears only in (2. 19a) it is natural to use the other of the two redundant equations, so that becomes a dependent parameter which is determined by the other dependent variables H, E and. If on the other hland, terms in the static electron velocity are retained in the equations, then a derivative of the dynamic velocity appears in (2. 19b) as well, so that is then a dependent variable of the differential equations. 2. 5. 3 Ordered Power Flow in Sheath A generalization of Poynting's theorem can be obtained from (2. 28) in the usual way. We take E- the conljLig;Ltte of (2. 28e) and Hi (2. 28c) and

60 upon snblijt acting obtain Equation (2. 28b) can be used to u ~-h-:: The term in, and then upon using (2. 28a) to evaluate we get (2. 31) where the factor of V/Z of needed to represent avela'giing with respect to time. Field (1955) obtained a similar expression x\ithout the. term. In a uniform plasma, the terms on the left are the average power flow dchiis.itivs in the EM and EK waves respectively, while the first four terms on the righoit hand side ax te tic -r rc cf, magnetic and electric energy densities in the EM wave and the kinetic and potential energies in the EK wave. The last term is rather interesting and is considered here in greater detail. A volume integral of (2. 31) leads to a Poynting's theorem for the plasma as (2.32)

i6 where WH, WE WK and Wp are real quantities which represent the time average magnetic, electric, kinetic and potential energy, stored in the vclume of integration. The real part of the left hand side gives the average power flow a'cross the c-lsu.d1 surface bounlding the integration volume and the imaginarI-,ly part is proportional to the difference between the energy stored in the form of magnetic and kinetic energy compared with that in electric.ini.1 potential energy, within the volume. In a lossless, passive mledliluml, the real part of the surface integral is zero. Since the last term in (2. 32) which involves E can apparently be complex, there is the possibility then that the 0 sheath may be lossy. We define loss to mean a net decrease in the ordered energy content of the plasma, varying with the frequency of the incident wave. Gain is defined as negative loss. Now nv is a current flow in the sheath due to charge accumulat lllltion, and depending 1upon11 the direction of this current flow with lre spect to E, there can be a gain of energy or loss of energy by the -O wave in the sheath. This can be accounted for in another way by observing that in the static case, the sheath electrons have a potential energy due to the negative sheath potential. In the 1)p sL.incrc of the dynamic electron motion, the total electron energy with respect to the static energy, may be increased or deciresed, cepending upon the l-al. l. 1' ni of the electron from its static position by the wave. If we rewrite the term in question as

62 (2.33) where, and x is the electron.i.l-j'J.Li. 1I,, Il from equtilil) iumt this viewpoint becomes clear. Thus the term in (2.3 ) which shows the effect of the static sheath on the ynanl;Illllic electron motion in the sheath can account for both a reactive energy st oratge as well as a gain or loss of power in the sheath. Whletlher or not there is a net gain or loss due to this effect can be determined from the integra;iADn of the real part of either the right or left hand side of (2. 32). 2.5.4 The Coupled Wave Equations The:. Lmposil iti(nl of the electric field into solenoidal and irrotational partes:;- was done for the homogeneous plasma medium is not meaningful here, in terms of se)]paratilng the EK and EM waves. This is illustrated by using (2. 2Db) l -id (2. 28e) to obtain an equation for the electric field, which is. (2.34) If as before, we attempt to use (2. 35a) it is a:pp.arent that there now is no simple relationship between p and: X p

63 and between E and H, due to the E term and the spatial variation of e,, -E - - -o A somewhat more reasonable way to attempt decomposing the total electric field rnmay be using (2. 35b) (2. 35c) since an EM wave alone propagating in a medium with a variable dielectric constant, satisfies (2. 35b) rather than (2. 35a). There is thus obtained from (2.35) (2. 36a)... - (2. 36b) which does succeed in giving E in terms of H and E in terms of n. However now now r- - i L (2. 36c) which shows that Ep is not irrotational, thus coupling to the magnetic field. It is of interest however, to obtain wave equations involving the magnetic field and dynamic electron density, as was done for the 1'LIogLc.v' us;

64 medium. This is most simply done by successively ta.ldkilg the curl and divergence of (2. 34), and with the new variables (2. 37a) (2. 37b) there is obtained (2. 38;l) (2. 38b) These equations clearly exhibit the coupling between the magnetic field and the dynamic electron density due to the sheath inhoiogenAiety. This is in contrast to the case in the uniform plasma where the magnetic field and dynamic electron density satisfy inllepenldelt wave equations and are quantities.l-ct i.tL. (. with only the EM and EK waves respectively. Equations (2. 38) reduce to (2. 24) when the plasma is uniform, since then P and E become zero.

65 2.5.5 Specification of the Inhonmogeneous Sllhetllh Bou..nllldarly Value Problemn Since analytical solutions cannot be obtained to equations 2,'?," so that resort must be made to IIIni-,'i.- L1 t' computtll titiions, it is prefe-:'..)ile to deal with the first order differential equations. For this reason, we Ict'l(i.11 to (2. 28) which are rearranged as follows. (2. 39a) - ' ~. l; (2. 39b) (2.39c). (2. 39d) A separation of the angular dependence of these variables for the non-uniform plasma follows from the requirement for single valued solutions as varies in incxtei'nc-ll, t of 2 radians, in the same way as for the uniform plasma. The variation may be deduced by dividing the sheath region into a number of cylindrical homogeneous shells whose

66 proj )e 'ties represent an average over that section of the inhlomogeneous sheath which I:he3y i't [)pLcct. The resultant wave quantities must be periodic in the direction within each shell due to the boundary conditions applied at each shell interface, with the periodic ity determined by tihe incident wave. As the t-liclkness of the sicll1. is taken to be zero in the limit, the v'ariatio:,i is seen to be constant through the inhomogeneous sheath. Thus with the aIzilmuthal separation variable or mode number m, which is an integer, and the direction wave number, there is obtained r; t (2.40a) 2 ^ < (2.40b) (2.40c) (2. 40d) The summlnation is understood to be from " to i zt -, and is summied over the 3 cylindrical co-ordinates and

67 Upon intl-oducing (2. 40) into (2. 39) and utilizing the orthogonality of the lequaltion with rt.spelct to, there is obtained the following set of diff e 4- nti lI equaiItions ~ -- (2. 41 L) (2. 41b) (2.41c) -, (2.41d) (2.41e) L: '(2. 41f) where the prime indictates differentiation with respect to p. It should be noted tliht no derivatives of or appear in these equations. is given by

68 (2. 41g) and is obtained from ' (2. 41h) (2. 41i) (2.41j) The dynamic sheath behavior is thus specified by a system of 6 first order ordinary diffterl-eltial eq(luations. Before a solution of this system of equations is undertaken, some further consideration mlust be given to the equations which describe the waves propagaL;ti1ng in the uniform plasma, since the waves which are incident on the sheath from the uniform plasma serve as a source for the excitation of the dynamic sheath variation. Now a plane elect l:onmtgnetic wave of ai Jit arLy polarization, when referred to the plane of incidence in a particular co-ordinate system, can be broken up into two waves of specific polarization, the transverse magnetic (h) and transverse electric (e) waves. In the case of the cy-lindrictal co-ordinate system, the plane of

69 incidence is taken to be the plane formed by the - axis of the co-ordinate system and the propagation vector of the incident plane wave. The h and e waves are those waves for which the magnetic and electric vectors respectively, are perpendicular to the plane of incidence. An equally valid criterion which can be used for specifying' the pola;rization is on the basis of the components of the fields. The h wave has no c- omponent of magnetic field and the e wave has no component of electric field. The fields scattered from the cylindrical sheath can thus also be specified as to their state of polarization as transverse magnetic or tranlsverse electric modes on the basis of a - component of electric or magnetic field. (The scattered fields when menlltioned specifically will be referred to as modes since they are funletionls of the mode number m, and this serves to distinguish them from the incident waves. ) We will thus use h and e to refer to the state of polarization of both the incident EM waves and scaLtle'Ced EM modes. The incident EK wave and scattered EK mode will be indicated by p. As a re sult of there being three kinds of waves or modes propagating in the uniform plasma, three wave equations are required for that region. Two of the kind given by (2. 24a) are required for the EM fields, one each for the h and e polarizations, and the other as given by (2. 2hb) for the EK wave. It can be seen then, that the complete mathematical descriiption of the dynamic response of the homogeneous plasma and inhomogeneous sheath involves solving a twelfth order system of linear differential

70 equations, six first order equations for the sheath as given in (2.41) and three second or'der wave equations as given by (2. 24). There are then twelve constants of integration to be 'il- r iljied by the 1oit lIIr.t ') conditions. However, one constant of integration fr~ lin each of the wave equations for the homogeneous medium will be associated with the inc'hvent waves and is therefore an adjustable parameter. This leaves nine constants of integration to be determined, and t-u^ Sr. ';- scI r boundary conditions are ileq uired. The 1)boundary conditions to be imposed at the interface between the sheath and uniform plasma, aLssuming no discontinuity in the static plasma variable-s in crossing the inter face, are the usual ones from electromagnetics and acoustics, which can be derived from (2. 28). They are:c.l'tiiiii of the taliglt ilul electric and magnetic fields, 1nd continuity of the normal dynamic electron velocity and dynamic electron density. The other three boundary conditions are specified at the cylindeir surface. Since the cylindeer is taken to be infinitely conducting, then the and components of the electric field are zero there. The final boundary condition must involve in some way, the hydrodynamic aspects of the electron flow. Tihe usual approach is to assume elastic reflt-crtlion of the incident electrons from the metal surface, so that the norpal comlponent of the dynamic electron velocity is set equal to zero. Cohen (1962) d eiscusses this boundary conditio in some detail, and concludes that a more realistic way of accounting for the effect of the boundary would be to use

71 -- ' (2.42 ) Y and Y are called bi-linear aldmiittance relations between the velocity, A B and the electric field and electron l)ressure. Cohen does not justify this bound'rl ]lLy condition except on heuristic grounds, nor suiggest any theoretical way flor calculating YA and Y.B There is however, from ordinary Lcoustic thecry,l' some justification for the term in YB since the surface effect in acoustics is represented by a surface admittance which relates the normal velocity to the pressure, as in (2.42). A rigid, impenetrable boundary in,acousLti(s is represented by zero surface;hadmlitttanc-e, while a completely po)()rous boundary has an infinite value of surface admittance. It would seen to be reasonable then to include these two extremes by using as the last boundary condition at the cylinder surface, either zero velocity or zeropressure, corresponding to YB equal to zero or infinity respectively, with YA equal to zero. Due to the lengthy numerical calculations which will be required to solve (2. 41), it does not seem practical to perform a parametric study on the effects of using other valuts of YA and YB. The nine bolunldary., i.lIii i,,,*s to be used are written below, with the subscripts 1 and 2 used to denote field quantities in the inhomogeneous sheath and uniformn plasma respectively. At the sheath interface (p = s ):

72 (2. 43a) (2. 43b) (2. 430c ) (2. 43d) and at lHii cylinder (p = c): (2.43e) (2.43f) This set of bolundarl.y ' firli.ions comnpletes the specifications of the problem. We wish to find the -,i)lliutiill to equations (2. 41) which apply to the inhlomogeneous sheath, oalid of (2. 24) which apply) to the uniform plasma, subject to the boundary c-,llditiI 1.i (2.43). The boundary conditions (2.43a) to (2.43d) serve as the link between the sO.llutionsl; for the two regionls.

73 Before proceeding to the numerical solutions of (2. 41), we first write [Lhe analytic solutions for the Llnifolrm plasma.. It is well known (Stratton (194 ) p. 393) that solutions to the vector wave eqLtutioil can be generated fromn solutions to the scalar wave equation by the method of potertials. This is due to the fact that, in cylindrical co-ordinates, the compornent of the vector wave equation is identical to the scalar wave equation. The three possible vector solutions are given by (2. 44c.) (2.441b). (2.44c) where C and are solutions to (2. 446) for and - i- for - - The cI1vl lic anid imagnetic fields in terms of the potentials, are given by i *

74 (2.45) (2. 46a) (2. 46b) (2.47a, (2. 47b) is the radial direction separation constant, related to - and - by (2. 48a) (2.48b)

75 where is the direction separation constant, given by (2.49) i is the angle of incidence measured from the - the lr',I. Ig.L i' I, constant of the i cid..it. plane wave. iml'1pedance for the EM wave, given by * ~ ' r - - i axis, and is rn is the plane wave (2. 50) with the free space impedance. The potential of the incident plane wave solution of (2. 44d) may be expres>sed,:t in tellms of an infinite series of Bessel functions, as tx- - '(2.51a) where,and X are subscripted according to the kind of incident plane wave, and is obtaiied froml (2. 49). The plane wave is taken to be incident on the cylinder from the direction. The scattered fields have as their potential (2. 51b)

76 where a.gain and A have -.ll-ciits corr -et t)(, ('ii to the kind of scattered mode. A is the Fourier scattering 1 1i, a,1,i. where the second subscript m — denotes rhe kind of incident wave and the thlird the type of -clatllt'led mode. (2) H( is Ihe Hankel function of the second kind, given by with J and N the Bessel and Neumann functions. im m Note that and vary in the ratio of *i. so that A -and.X are quite different in value. It should be observed that X- becomes imaginary when an EK wave is incident at an angle such that or when 0 is different from normal incidence by approximately V\J., / radians. When this occurs the Hankel function of:l..tL olid kind with argument AEP becomes a modified Bessel function. The negative imaginary root for XE is, required so that the fields do not grow as the radial argument increases to infinity, since the medium is passive. The'refore:' 7 "' is replaced by J) z, which decreases exponentially with radius for large..t'rgunun'ltS. This does not occur for EM wave incidence. Upon u-ling (2. 41), (2.43) and (2. 51), and eliminating the Fourier scatteiringl coefficients, the boundary conditions at the sheath can be reduced to

77 (2. 52a) r. - (2.52'b) ~... (2.52 (2. 52) (2.52e),' - -- I;, (2.- 52 )..(2.52f) In writing these equations, it is assumed that only one wave type is incident, so that two of the three nmust be zero. The boundary conditicr at the cylinder may be written as (2. 52;g)

78 (2. 52h) and (2. 52i) or, - I (2. 52j) r" Q being zero is the condition that the dynamic electron pressure be zero at the cylinder wall, while the equation (2. 531 is the condition that the radial dynamic electron velocity is zero at the cylinder. Only one of these two equations is used. Now cylindrical functions of real argument satisfy (2. 53a) and those of ilmaginary argument satisfy (2. 53b) Then by using (2. 52) and (2. 53), it can be verified that for p and h wave:; IlI;. I.. II,.

79 (2. 54a) and (2. 54b) For e wave incidence there is obtained (2. 55,L) and (2. 55b) It '[he dynamic sheath problems has now been reduced to solving the sixth orZlder system of diffetretial equations (2. 41) subject to the six boundary conditilons (2. 52a) to (2. 52c) and (2. 52g) to (2. 52j). It is interesting to observe

80 that the lboundary colndition equations at the sheath edge do not involve both EM and EK qaiLllItities t ),ttller in the inl q tl ioliln, as a result of eliminating the Fourier c:tttc'lL'ing coefficients. The numerical solution of a set of ciffetrential i lIt Lii il-, with bounda ry condihtions specified at two boundaries is discussed in detail in Appendix C. The basis problem is to obtain the starting values for the numerical integration. This is done by setting, at one boundary, all but one of the modal variables ia (2. 41) eLqual to zero, the one retlllaiin being set equal to unity. The boundary conditions are then used to find the values of the derivatives and other variables which may be relat tc.d to the non-zero input. A numerical integration through the sheath is carried out which yields the output values of the variables and their derivatives at the second boundary. This process is repeated with a new input variable set equal to unity, as many times as there are boundary condition equations at the second boundary. A linear combination of the output values, with a coeff i lent corresponding to each input, is put into the boundary condition equations at the second boundary, to obtain a matrix, which upon inversion, yields the coefficients of the linear combination. These are the desired starting values for the final numerical integration, which when carried out will produce a solution th:at satisfies the bountdary conditions at both boundaries. The surface currents are then obtained from the tangential copoents of the magnetic field at the cylinder as (2. 56a)

81 (2. 56b) A subsli ipt in place of the dash on the current symbol will indicate the kind of illci:lI-It wave. An dclditional complication to the numerical solutinM of (2. 41) is due to the.C[act that these equations are complex. There are thus effectively twelve tLeal eq-uations, rather than six, to be solved. The computer time requirc-d to carry out the numerical calculations involved can become prohlibitiv., especially when the results are desired to be accurate to three or four silgnificarit figures. This degree of accuracy is required since the final resuLlts, are obtained from summintIg the Fourier series, where the errors involved in the individual terms in the series are additive. An interesting feature of equations (2. 41) and the boundary condition equations (2. 52) is that when, corresponding to normal incidence, these equationsl break up into two sets. The and components of the velocity and elec tric field, nnd the z component of the magnetic field and the electron pressur e are in one set, with the - component of velocity and electric field and the and components of magnetic field in the other. In other words, the Land modes are coupled but independent of the mode equations which stand alone. The coupled and mode equations consist then of four complex diffe ential equations and boundary ~.,,iili i.,.s, with the result that the inumerIi'-al comlputration time can be dct-cleaLsed by more than one half. There

82 is an;tadicitional feature in the fact that there are fewer terms in these differential equations with a resultant decrease in the generation of erirors. For these reasons the numerical computations for the inhomogeneo)us sheath are performed for the case of normal incidence only. It was felt that the problem is an extremely difficult one even for the case of normial incidence and thlLat there would be an illncreased poslsilbility of better understandingo the physics involved, since for a given amollun.llt of computer time, more numerical results could be obtained. There is, in addition as discussed in Appendix I), a potentially significant experiment that could be carried out for the case of normal im-ithl,,,r., whereas the case of oblique incidence may not lend itself so easily to experimental invl.-.l igation. The results for the ili,,,l,,,.,,,.,u. sheath solution are contained in (Clllptci' III. 2.6 The Vacuum Sleatll Model It was pointed out above that analytic solutions can be olbtailned in the homogeneous region outside the slleathl. A natural question to consider now is whether the inhonmogeneous sheath may be reasonably app;lo\inated by rleplacilng it with a homogeneous region with properties which may be different from thmeof the external uniform plasma. In this way analytic solutions could be obtained everywhere. The results obtained from this model could serve as a measure of the relative ilortance of the inhaotih omgeneous sheath;1and the boundary conditions to the coupling of EM and EK waves, when coLpared- with the inhomogeneous sheath solutions. Also, the uanalytic solutions are easily obtained for aribiti raryl angles of incidence, in contrast to the case

83 for the inhoimogeneons shea. th. A co'tmparitison of the results obtained from the t\\ i s;heath models for normal injcidence would serve as an indication of how i iii-,i -il are hui results obtainedfor oblique incidence from the homnogeneous, sheath model. If the hlonmgeneous sheath region is taken to be a plasma with different p op-l, 1s than the external homogeneous plasmla, then in the case of oblique incidence, there will be nine boundary conditions to be satisfied, as was the case for the inkiomnogeneous sheath. This results in the requirement that a 9 x 9 matrix be ilnverted fo.r the solutilin of the Fourier coefficienits of the various modes, a task of such complexity that it would have to be carried out numerically. It is felt that there is little loss of generality by using as a sheath model however, a homogenweoCus region void of plasmna, and having the electrical properties of free s:.i(ce, which will be called the vacuLum sheath. This reduces the number of bcuLn,-l1jry conditions from nine to seven, since the p mode will not be tralls.;1lted thr]:ough the vacuum sheath. As a result, the matrix inversion becomes much more manageable, and can be easily cat''it, d out analytically, so that tlhe numericcal computation become considerably less involved. In the following discLt!-.:ion the fields scattered from the sheath-plasma interface will be called scattered modes, those \\hieli pass through the sheath to the cylinder are denoted as tr'aLs:Lslitted modes, and those scattered at thei cylinder wall are called reflected modes. The latters S. T and R will be used as siLpeigescriipts on the Fourier coeffecients for the potentials from which the

84 various field ILIquanltities are derived to indicate them as the scatter1ed. transmitted o-,r 1 t fl-cted modes. As before, the subscripts will iil,,,.lri, mode nulmber t-he incident wave type and till mode type, in that order. Six of the boundary conditions to be used are the same as those discussed for the inhomogeneous sheath. They are continuity of the tangential electric and magnetic fields at the sheath edge, and vanishing of the tangential elect-ic field on the cylinder. The last boundary condition for the vacuum sheath should be so chosen as to be consistent with the remaining two conditions at the shlllath-plasma interface which were used for the inhomogeneous sheath. There are two possibilities then, corresponding to the continuity of prfts 'e and continuityof normal dynamic electron velocity. Since there is no plasmna within the vacuum sheath, these conditions would lead to the vanishingo of either the electron pressure or normal electron velocity, at the sheath edge, -\which are the same conditions mentioned previously for use at the cylinder wall, in connection with the inhomogeneous sheath. The consistency of the boun)ldary coI.di t is used for the two sheath models in thus maintained when:the vacuum sheath tliclklness is taken to be vanishingly small, so that the sheathless case can also be studied with this formulation. The boundary conditions mnay be stated then, with the sub scripts 1 and 2 used to refer to the vacuunm zsheath and plasma quantities respectively, at the sheath edge ( ) (2. 57a)

85 (2. 57b) (2. 57c) and at the cylinder ( pc ) - (2. 57d) The field qucltlltities are again generated from their respective potentials, as was done for the analytic solutions in the uniform plLasiml in connection with the inhomogeneous I.-Lt.1 ll model. Upon writing the boundary cunoililtils expressed by (2. 57), and utilizing the orthogonality of the resultlng equations with respect to m, the matrix equation from which the modal Fourier coefficients are obtained can be written;. -` ' ) - ~ r s ofZ e (2.58)

86 where the prime indicates a,IL ix Lti x. with respect to. The middle subscript on the Fourier coeffi ients, indicated -' - - -:, will be p, h or e, corr'-esponding to the kind of incident wave determined by the source S. The sou tce terms are given by: (2.59a) i - (2. 5'1b):\ (2.59c) where the only which is now zero is tiaiL[ of the particular incident wave. We obtain Ap, XSE and XE (which appears in ) from (2. 60a) (2. 60)) (2. 60c)

87 where the is given by (2. 60d) when the wave is the incident wave, or (2. 60e) when the or waves are incident. O the tangle of incidence is measured from the positive axis. is given by (2. 61) (see insert next page), where (1) and H) are T-T;IIll.c-l functions of the first and second kind. m m The bl.)tcl''Ttitol made albout the radial separation constant i, possibly becoming inmginclary with an incident EK wave in discussing the inhromogeneous sheath model applies here also to 'E and E. When this occurs the minus sign must be used on the imaginary roots so that the Hankel funlctions of the second kind,,whdich represent the scattered fields in the p..liaL, do not increase in magnitLide with increasing radius. Hafinkel functions of a negative imaginary targumlenlt. are given by the modified Bessel functions r and as

OH(2) (-S m P 2 XE (2) xS -H (A s) KEm U A.Eu I (1) A S (x o s)E 2 2 0 X E (2) K om Eo U -imH (2) (A s) @.QLH (2) (x S) s m P K~sm F H(2)l LS m ~ PM H(1 (A. s) H (x) A.E S) K sm -Eu m E F:-, Pm H(1) ) (2)' A- S Kos m Eu m Eu 0 0 0 1o (2)' -~H (A. S) KAm E KE 2 xE (2) KY 7Hm (E S E [3m H(2) (-S K EsnT m ~E im (2) S s Hm('E 0 2 xo (1) XEH (A. S) Ko U 0m Eu 0 1 H(1)'IL S U~ m Euo u P0m H( ) (X s) I4 H(2)'0 Eo S) K sr s Tm Eu U0m E [S3m (2) H (A- s) KEo u ~me 00) co 2 - o (2) X S Ko ~)oH (Aos 1H (2)' XS N2m P 0 0 0 0 0 0 0 2 XE o(1) - H (A.E c) Eu 0 2 XEo (2) (' C Ko m Eu 0 0 0 0 [3m (1 (1)' K_ c~m( Eu C Hm( Eu tO [3m H(2) xC K- c m Eu tu (2)' -H (A. E C) (2. 61)

89 (2. 62a) -j, -~.), (2.62b) and vary approxinmately as and for large ~. Since the fields transmitted from the sheath interface towards the cylinder vary as there exists the p ssibility of a large attellnuation in the fields reaching the A-linder, Ejc mTintg a scthe curr'I'ents excited. This is a significant outcome which is discLussed more fully in the next chapter when the numerical results are presented. The angle at which AE changes from real to imaLginary will be denoted as If the boundary condition rather than - is used at, then the fifth row of. 'nlld the fifth row of the source vector on the right hand side of (2. 58) are replaced by A;... '- -,^ (2. 63a) (2. 63b) It is evident with (2. 63) put into (2. 58) and (2. 61), that there is no coupling between the EM and EK waves due to the boundary conditions. But the only

90 possibility for coupling between the EM and EK waves in a homogeneous - _ ' ': '' boundary. Thus we can conclude that the EM and EK waves are not coupled when prupL.a^iting in a uniform plasmaL terminallted by boundacries at which the electron pressure is zero. This suggests the intriguing possibility of being able to:-cpl.LILt( the relative contribution of the inhomogeneity and boundary coupIliing effects in the analysis of the illnhomnlogeneous sheath, since any coLupling which is observed when the electron pressure at the cylinder wall is takenlu to be zero should be due entirely to the sheath inhomogeneity coupling. This point will be discussed further in the presentation of the ILnumerical results which are given in the next chlapte'r. The solution of (2. 58) is straightf ---^ '-trt but tedious. The procedure followed was to eliminate four of the coeff cients. - and by solving for them in te rms of the other three. A 3 x 3 matrix was gent'rated as a result, from which the desired coefficients were obtained. Results are presented here for the three coefficients found directly from the matrix. Solutions for the other coefficients and a brief outline of the above procedure are given in Appendix E. Before writing the expressions obtained for the coefficients, some shorthand notation which was used to fae ilitate nwriting these rather lengthy expressions is given here. Due to the requirement for boundary conditions to be satisfied at two boundaries, the cylinder wall and the sheath edge, a number of Wronskian-type relations appear in the coefficients, but with Hankel functions

91 of ai t',un iits eval]L ti d at two boulilrlies. As a consequence, the Wronskian relation- cannot be evalulhated analytically, as when all the.Li't,1I-1 li,- appeal ing in the *..r\t,,.ssion are the same. In order to avoid writing these numerous relations out in full, they are abbre x-iated by using. (2.64a) (2. 641b)? (2.64e) ' (2. 64d) The prime now indicates differentiation with respect to argument after which the argument is fixed at the indicated value. Note that /: Appendix F contains some approximate evaluations of these Wronskians. We will also use

92 (2. 65a) (2. 65b) (2.;i;,c') (2. 65d) The determinant D of the matrix m can then be written -,t (2.66)

93 S Trhe Fo-.urier coefficients A - rn-p A~ and A InI-h rn-e are: Incident p-wave - /tSR?- C i IFI -1I - -, I I I I I -:. -I -1 I I tI - -- .7 - I -- I I! I,- -1 I \ - Ii I - - I ( f-, 4 1 9 - I I (K (7C)

94 I necident h-w~ave - I.I - ) p~ / Incident e-wave -, 7, I I- - I- I t — I Cs~- a)

95 Upon using (2. 53), it can be shown that.- is a property of the Fourier coefficients for the scattered, transmitted and reflected fields of the same type as the incident wave. This is also a property of the coefficient c.oupling the h and p fields. The coefficients of the other modes dissimilar to the incident wave type have the property A - (-1) A -m m The surface currents excited. on the cylinder can be written in terms R ].q of A and A as m-h m-e -- (2. 70a)

96 (2. 70b) where the dash subscript on the and the will indicate the kind of illci'.nllt wave. Due to the behavior with change in the sign of m of the various n1oclcl coefficients, we observe that -: (2.71a) (2.71b) These expressions are rathler lengthy, and conseqtuently present relatively little interpretable information in their present form. There is however, one interesting fi-I all 'iC of the coLupling i pi.l of the problem which can be seen from a consideration of the Fourier coefficients. The coefficients show that the results of the coupling between the incident wave and the various mnodes can be summarized quriali tatLi ely by a matrix C whose elements are proportional to the respective Fourier coefficients as

97 (2. 72a) where (2. 72b) There is no dListinction made between the scattering, transmission and reflection ooefficients for a given mode since all exhibit the same proportionality to the kitnd of incident wave, denoted by the vector. It is evident for no z variation in the iincident field, i. e. for nornimal incidence, that the h mode 'It-X&uL)les from the other two, in agreemenet with the conclusion reached in the (liis-;I'Iussion of the inhlllgencous sheath. There is also a decoupling of the e mode from the p and h modes when the azimuthal variation is zero, which 'o]'r-esponds to the cylinder, in the limit of its radius approaching infinity, being- replaced by a flat surface. Cohen (1962) and others have pointed this out. for scattering from a plane surface. One Additional aspect of this problem can be studied using the results given above. This is the effect of including the EK wave in the formulation upon the statturiIlg properties of the cylinder when illumiILited by EM waves.

98 Or in other wlords, the results obtained by using a compressible model for the plasmal can be compared with those which are gotten from a t eltlllellt in which the plaslma is iIl('oilllpL'Lc' -ibl(.- and cha:L racterized only by a dielectric coinstanlt. The only change in the present F,-. 1ml Ii iii to obtain the incompressible plasma results would be to set A = 0 and to eliminate the boulndary condim-p tion on the normal electlron. velocity at the sheath edge. A solution of the resulting equations for the new Fourier coefficients is not required since the new coefficients can be obtained directly from (2. 68) and (2. 69) by setting the terms containing H in these expressions equal to zero. This comparison of the compr essible and incompressible plasma results should be informative in showing whether it is necessary to allow for comnpressive plasma effects in the problem of the scattering of EM waves from plasma immersed obstacles. The czalculLtions which are IIc-'t'ssulry for obtaining numerical results require the evaluation of Hankel rlllle liolns of both real and imaginary argumenell t, over a wide rantge of order and 1mag;Ltllitude of arglument. This presented a difficult conLmpulational problem, and the approach used to solve it is presented in Appendix F1. There is.Lso contained in Appendix F a discussion of the limitation on the accuracy of the final results for the surface currents. The limitations on the accuracy of the iiihomogeneous sheath results are discussed in Ap.pendix C. This concludes the formulation of the problem of finding the surface currents excited on an infinitely long, insulaLted, plasma-immersed circular

99 t-tet.ll cylinder by incident plane EM: and EK waves. To summlarize, the p L.. IiLL l'as been taken to be of nlliflorm temperature thlroughout. The L-(.Lit ions which govern its behavior have been obtained from the Boltzmann equa.l;tion and Maxwell's equationts using a linearized theory and a scalar electron pressure. Magnetic force effects a.re neglected, as are the radio fr'eque ncy i o n motion and the static e 1 e c t r o n velocity. The i nhomogeneous sheath3L region is assumed to be confined to a layer on the order of several electron Debye lengths thick next to the cylinder surface. Two models are used for the sheath, one taking the lsheathl inhomogeneity into aLc c' ountt, and the other x\lhich L;pprox. i llitc s the sheath with a,tvacLuun layer. Finally, in the bounda ry co-diltion lwhich relates the electron pressure to the normal electron velocity at a cylinder surface, the surIface admittance is taken to have either the v:IluL zero or infinity. The nunmerical r..-slls which are obtained from the fcolrmulatioln;lc)(ve follow in Chapte i III.

CHAPTER III RESULTS Thle tesults presented here are given first for the vacuum sheath model, an(- folli)wig,, those obtained from the inhomogeneous sheath. 3.1. V;jl.c'u M! Sihea.tht Due to the compleKity, and hence the time consuming nature of the numerical call-la.ti(:'l:- required for the evaluation of the expressions which give the Fourier 'oufficiullt:.s for the \ a'i. ui>. modes arising in the vacuum sheath formulation, it was not practical to carry out a coiimplete paranmetric analysis involving all the pai aellters and x'ariables appearing in these expressions. Instea;d, the following results ar e obtaained as f:unctions of only certain of the parameters and variables, which are;iv n below. The (lqantitieis which are independenll parameters in this analysis are T = electron tem perature e w = 27rf = radian radio frequency i V - voltage 1magnitude of incident wave u =27 f =radian plasma frequency P P c = cylinder 1 ad;ius s = sheath radius 0 - angle of incidence of plane wave 'Thle surface current results are obtained as functions of the parameters 0, c and s. Variations in s are presented as variations in s-c, in units of D. ThI sheath lhickee ss measured in D will be denoled by X. The radian plasma -C 100

101 fr(. luencv ' is varied in terms of N, the ratio o[ wp to tu. The other three par — P P an etetJ]:s ai'e fixed at constant values throughout the calculations. V of course, is simply tai s,'ale [aLt,,,'r for the incident wave, and is set equal to 1. 0., w is set at 2-r x 10 / sec., CrL't''sponding to a frequency of 1 G c, and T is fixed at e 10 000 K. lThis value for T correspocnds to a typical electron temperature for e a, old cathode laloratory li.r.'l ge. Since the velocity v is proportional to the squarle root of thl ttumpera ture, a range of temperature variation typical of the cold cathiode disclharge produL1ces a much Esmaller change in v., so it seems reasonable to fix 'T at this nominal value. e In aLcdlition there are the independent variables of the co-ordinate system, p, z and 3. But since the z variation of all the field comlponents is e, z can be fixedL at 1a vIalue of zero in the numerical analysis with no loss of generality. Also, since thie surface currents, which are of the primary interest, are evaluated at p-c, while the field variaLtion in the sheath can be deduced to some extent from the sulrfaLce curLrent and from observing the asymptotic forms of the field expressions;, p is fixed at the value of the cylinder radius c. The 0 variation on the othler h:.nd is of importance, and the surface currents will be plotted as a function of j. Some preliminary c.i,i ii1 i;,. were performed to determine what would be the most-)s reasonable nominal values around which the paramneter variations could be nu.ide. The cylinder radius c, was set at 1 cm for these calculations. It -aas found that, while the computation time required to obtain a convergent so lution for c: 1 cmi and EK wave incidence was not excessive (on the order of 0. 4.* -]t A t 1 o e, bt re st.... -I

102 seconds per term in the Fourier series), the X;\aittion of the suLrface current with p was so finely structured, as to make plott ing the results on 8 1/2 x 11 inch graph paper iil)p., lcl i,l. Because of this, the nlominal radius was decreased to 0. 2 cm. The noninal values for the p..ll..ilerters and the range of variation which was investigated is sunmmarized below. Parameter Nominal Value Variation t 27r x 10 /sec None T 10 K None e V 1.0 None N = f /jf 0.7 0. 7 to 0. 99 0 0. 257 0.05r to 0.57 c 0. 2 cm 0. 005 to 1. 0 cm X= (s-c)/Dg 0; 10 0 to 20 (s-c=0 to 0. 177 cm) Generally, only one parameter at a time was varied from this nominal set of values, except at end points of the range of variation. The situation where the vacuunl sheath is taken to be of zero thictkess will be referred to in what follows as the sheath]ess case. In plotting the surface currents, since they are complex quantities, we can prisent either their real and imaginary parts, or their mllag'nitude and phase. The latter choice is the one used here, since this is more physically meaningful and laboratory measurements are performed on this basis. The phase of the current is not presented for all the results which follow, however, since the phase behavior can be visualized easily fromlll the few represenltative curves which are

103 pilsented. The surface currlents excited by the EK wave are given below, followed by those due to the EM wave in section 3.1.2. The salient features of these rvutults will be commented upon as the various sets of curves are presented, but a d(etailed disc ussion of them will be deferred until all the graphs have been given, as there are various features shared in common by the res ults generated fro)ml the different parameter variations. 3.1.1 Incident ElK Wave Figures< 3.1 and 3.3 show the magnitude of the z and 0 directed surface curr nts K() and K, excited by the EK wave, for the nominal parameter values. Ace P The ordina.te on these t.LI'I-. as in all the graphs to follow which show the surface currents, is amperes/cm and the Jl.i-, -.L is the azimuthal angle measured (in degrees) fl rom the front of the cylinder. Results are plotted from 0 to 180 only, since the curves are symmetric in the 0 variable. The maximum value of the current will be frequently mentioned; it is the value of the current maximum over 0~ 0 the. \,Lri,lti it) from 0 to 180.In figures 3.2 and 3.4 are plotted the phase of these cuLrrents re stricted to the interval - r to w. There are two rather striking features exhibited by these curves which compare the cylinder surface currents of the vacuum sheath 10 De thick, with the sheathless case. The first is the attenuation of the currents by the vacuum sheath, a decrease on the order of 100 - 200 times compclred with the sheathless currents. The other p[J |i(._ ij I effect of the vacuum sheath has been to reduce the fluctuations of the currents as a function of the azimuthal:angle }. These effects are produced by a sheath of thickness 0. 0885 cm, while the EEK wave length is 0. 0945 cm. The

% I 104 I.I f i I II i I 11-1 II;I k f I i I I I i I I II t I I II I I I I i I t I I i j i I II t I I I (Z) p. ft I""> TX I J I ' I! it '4 1 ii I 1/ 11 Ii r jo (Aminperes/ cm) Iis ''1. I 4.F1' 444: -44 —. Mil44 -r, lXjQ N>;i~%~ i~tr4rtt ' A~ r-M Ide r = ]. 4ck-S li: 0.7 -7-1 10 f,L- c — O, 2 c i m, 91.= 7 /4 V' Vacuuam IS heath M oc 45 ''90 135 -I i14 FIG. 3. 1: MAGNITUDE OF K v&AZIMUTHAL ANGEO p NOMINAL PARAMETER VA-L~IJ&S

1 05 C'u tit iPhIase 11 RalL di ans ---) I. 5 I f= I Go 4 o - T =il0 Ke N= 0. 7 c=0.2 cm 1 o urff/4 Vacuu~m Sheath Model 1.0 0. 5 ~ 7> 01 I I iI 'IS — - A&M - -1 -1 t II i I f i I t I tI Ir I I i I.1 I -V I ii II I"11 -0. 5 t, I -- t I I I I -1. ai I 411 9 A ,,I;, 4 L 111*," i i t -1. 51-1 — I k Vi Ii 45 90 FIG1C. 3. 2; P:HASE OF K(Z vs. AZIMUTHAL ANGLE 0 p 1FOR NOMINAL PARAMETER VALUES 135 180 I so-amm"WbAft --- - - -- "Nomma ~.. 0 (Degrees) X=: -- -; ~-0 x = 10 I - -

MISIN PAGE

107 Current Phase 1.5 K (Radians) 44 I ~~~N 77~ri c 0-2cm 0.5L; I I I Vacuum Sheath / ModelI __ _ _ _ _ _ _ _ _ _ -0.N 2 W -1. 0, ssc 4 Al' r t.t4 FIG. 3.4: PHASE OF K VS. A ZIMUTHAL ANGLE,4 OR N~OMIWNAL~L PARAMETEY VALUE1,S XcU

108 [ I i L t i J ti1 ns of the current with 0 for the sheathless case can thus be attributed to interference effects of the incident and scattered fields since the cylinder is approxinmately 13 EK wave lengths in c ilcumfelrenlce. The aLttenluation of the sheath is ALt i.',l t-,l for by noting that the angle of incidence of the EK wave,?7/4, is less than wr/2 - v /vf. the approximate angle at which the scattered EM fields become evanescent. The angle. 7r/2-v Jv, will be referred to as 0. As a result the fields which reach the cylinder throuLgh the sheath are attenuated, exciting currents of smaller magnitude than those produced when there is no sheath. This effect will be discussed in greater detail at the coInclusionr of this section, after presenting the other results for the incident EK wave. A comparison of the phase of the cuLjrents anid the magnitudes reveals that abrupt changes in amplitude are accompanlied by corresponding rapid cli.Lagc.s in phase. The sheath has the effect of decrea sing the phase variation as a function of. Figures 3.,5 and 3.7 show the current magnitudes, and 3.6 and 3.8 the current phase, ag'ain for the nominal parameter values, except that the angle of incidence 0 is now 89. 91 degrees = 0, so chosen as to produce a propagating scattered s EM mode 'with a scattering angle equal to 7r/4. (See the discussion of Appendix E i -(z) on the scUtte ring angle). 0 is not set enurl to 7r/2 since K would then be zero. It -is inteiesting that now K is relatively unaffected by the vacuum sheath and is also insensitive to 0. This behavior can be verified by the approximate (z) expression for K given in (E13a), Appendix E, which shows that in this case K( is delermined primarily by the m = 0 term in the Fourier series. The K( P p current on the ofli- 1 hand, is very similar to that for 0i = 7r/4, for the sheathless case. HoEceer, the sheath has considerably less attenuating effect now,

109 ki I I II I -4 10 0 I 1 5',, - I ) C-:" rl I- j'':.,,, I I4 4 II ilw -t I v isV FF~1 -ic f =IGec 1' 10 oK e N 0.7 c 0. 2c cm aL 17 2hr/ 89. 9l'Degrees (Os 7r/4) Vacuum Sheath Model XO0. ~0 (Degrees) 45 90 135.....- IA FIG. 3. 5: MAGNITUDE OF Kp vs. AZIMUTHAL ANGLE 0 for NOMINNAL PARAzMETER VALUES

11l0 -fa Curretnt Pliasc, (HR ctci,IIs ) if -I Ge, if z 10 oKE N z90.7 1. t., - ez1:0.2 cm 0'89. '91 deg~rees "Vu-cuim1 Sheath Mode-l..a f i.,, -, -V, -, - Z' 0. K ~:.;............. U -u.,- _ Al -oil0 WMUUSlfX 1! jttr I~z~c~rAp fI.- 1. i I I -.1. 5 - 45 90 FiG. 3, 6: PIHASE: OF K vs. AZ][MUTHAL ANGLE $FOR p NOMINAL PARAME TER VA LUE"S 1a& z7

I-I f;. I t, — l I I 11 I f'-,. I- I 4,!" t I "I I I" -.11 %,t I I t I 111 i1 U - ^ ' -^ - -r " -- - -. t^, lltA^ - a9 15 ''I 1C;I: j I I" III j -o 6.. I* i I j<(o p I (Ampr:eres cm) 4A Apt~ - - 7 -'- gI-....-. ^. X = 10 - Go 4 o T - 10 K: e N = 0.7 1-7 c = 0.2 cm,L ( I 0 = 89. 91 Degrees - Vacuum Sheath Model r } i I f -8 10 f |trFv$ i 1" 4 4j 45 I* - - - - - 90 - -1 - 1, > AZIMUTHAL ANLIE 4FOlE >TO-NAL,'' ', ' '4 FIG. 3.7: MAGNITUDI)E OF K vs. PARAMETER VALUES

1 12 C-uiienu- Phase (RPtLcifaLns) eI ~N:=0. 7 cr=0. 2 cm 6' 89.9 1 Degrees XVacuunm Sheath Model It 0.5:,k s,I II i t I 0 I 7 11 I I II I -0 5 I --- - It II 31 I "- e -i*, i i i II - I O,f f 11 I 11 li ji I -1 3 I,- -- I II t TI I I 45 I FI G. 3. 8: PHASE OF vs ZIMUr FOR NOMINAL PAHAMETE] rHAL ANGLE R VALUES XC > - - - t-;

113 i and has reduced the p fluctuation compared with that for 0 =r/4. As was the case with 0 =7T/4, the current phase is observed to fluctuate more rapidly when there are rapid changes in amplitude. This reduction in the 0 variation of the current by the vacuuml sheath is somewhat LuIexpNected. However, the decreased attenuation caused by the sheath is evidently due to the excitation by the EK wave of plropLagating EM modes which do not decay in the radial direction. The above graphs serve as a measure of comparison for the curves which follow below and which show the results of the various parameter variations about the nominal values. The results which are given have been selccted as providing a representative condensed picture of the computer cliculat ionI s which were carried out. Some additional graphs will be presented which are derived from the basic curves of the current as a function of angle 0, some of which are not included below, as a means of illustrating the more interesting aspects of the results without presenting an excessive amount of detail. Figures 3.9, 3.11 and 3.13 show the nagnitude of the surface currents K() P and K( as the cylinder radius is varied from 0. 01 cm to 0. 1 cm, for the angle p 0 =7r/4, and the laLgittlde of K' for the angle 0 =, and the nominal values of p s (z) the other parameters. K for the latter angle is shown in figure 3.14 as a P function of cylinder radius directly, since it is a straight almost horizontal line as a function of 9. Graphs of the phase for c=0.1 only are given in figures 3.10 and 3.12 for i0 =r/4. The most obvious change which occurs when the cylinder radius decreases is the decreasing number of fluctuations in the current mLagllitude as a function of $.

10 11 114 I / 4 f x L i4 1 10 - (Ampere s/cm) -Ki -, - ra". lo,_, I - I-,, - - - -5 - - -r - KY 10 8 7,-. -1. t i i I i i i i i 1 1 ~ c= 0.01 cm ~ C= 0. 02 cm c=0. 05 cm e N=0.7 Vacuum Sheath- Model 4 - - I, — "I --- - - --- -I -.- --- - llll ' - I I ***ft --- V7 Vm-, FI.t:MAGNITUDE OF K vs. A ZI MU TR-IAL ANt p S~ &- -- -WITH CYLINDER RADIUS CAP

115 i II II 1. 5 'I-. i C~urrentPhs (Radians) f = I Go T: 10 4 K e 1. 0 0. 5 0 A I r.-Il ---m ~t -0. 5 -1. 0 -455 V V-Z 2ta 4; z 1 -3-7- — I=- - - - -. -, -,, ",- -- &Mi*AtWhlkWi, - WI PHASE OF Kvs. AZIMUTHAL AN-GLE -PO p CYLINDER RADHEB 0=0. 1 cm0 - t-t - -- - f --- -'f, -- -,, - t N I - - " - I - - - - - - -- - -t - - On — r, -- i j- 1 2, -, I

(Amperes4/cm)/ -I 0 Wc -. A-1 cV %t!o2 I40"Aeot f". c 0 0 m Ft V 136 10 II aiia tvR% -60. JI Vacuum Sheath Model\ VI +:1: jjA FIG. 3. 11: -MAGNITUDE OF $1vs. AZIMUTHALANGLRWT p 1 CYLINDER RAISc A PARAMETE'R": J*Ot'*,

117 f= 1 Gc T = 1040K e N=0.7 c = 0.2 cm Current Phase, i (Radians) 0 /4 1.5 f_ Vacuum Sheath Model i "' I \ A -0.5 1 0 45 90 135 FIG. 3.12: PHASE OF K vs. AZIMUTHAL ANGLE - FOR CYLINDER RADIUS c =0.1 cm X0 -;X ' I -;.~ -F 5 -rg1 -Lr r 2-C *C,- ~ `" TI P -1 3 -? I 'r - r, b

K-~ P,~ " -~e~zL~ ~etk~-/ (~n lt~re;th~2~~ilr "rru( /:' le,,z- - "s llit wmat~-,sSB Fr eP- -npi la.A "Wt-,- 47.. d" VI x=o w = 1 A -mrt- a c 0.01 cm IM" %Iona* - - c0.Olcm02 cm -6 c 0. 05 cm 1 0 YI~I1. ~ ~PIc = 1 ~I1a cm.f = IGc 4 o T17=10 K a N =0.7 0 89. 91 Degrees VaumSheat Model -8 -Deg-i -e e I 0c FIG. 3.13: MAGNITUDE OF K vs. AZIMUTHAL ANGLE9'IT p CYLINDER RADIUS c A PARAMETER

119 10- I Vt XI (Amperes/rcm) -10-2 I \ 1-1 I x 1k;t., 1% X =-O X = 10 _-pppp-ppgnwm. 1f = 1 G e N=0.7 0= 89. 91-Degrees I I p - -A -':; -, -, ---.,4 - -1; - " -. 2:- - _- t - - -- 2. - Vacuutm Sheath Model -I rc(cm) \\j). t- -2 1-0 '"I L I I - -- - - -I -- - I - I I f - ->1, - I I ---. -- 1, F. "I -W,- - -" & limp 'o - - - - --- A - U, - -- -- FI.31:MAGNITUDE OF MAXIMUM VALUE OFI lsCYINDERp RADIUS C I k - - 4]-~! -1 - -s" -4

120 This is reasonable silce the cylinder circumference measured in EK wave lengtlhs is 1k. i-t.. ing. The attenuation caused by the vacuum sheath however, does not seem to be very senlsitive to the cylinder radius. In..l. l. 1 il the magnitude of the 1a1;L;xi1iun11 current also seems rather insensitive to the cylinder radius, except for the z component of current when 0i=0. K( in that case, as shown by figure s p 3.14 exhibits an inverse re& dlil!il.lj. to the cylinder radius, varying with the -1. to -3/2 power of the radius, a result which can be deduced from equation (E13a.) in Appendix E. This is aninteresting thouLg perhaps not too practically useful result, since a cylinder radius less than 0.1 cm could not likely be used for the experimental measurements sluggested in the introduction. In.addiionl, this behavior occurs for only a very narroNw angular interval, on the order of 5 to 10 -3 x 10 raLdianls (0.2$: to 0.57 degrees). The surface currents are shown as a function of the paramlllete r N in figures 3.15 to 3.20, for the angle of incidence 0e=T/4 only. The currents are characterized by decreasing fluctuation as a function of 9 and decreasing magnitude as N increases. There is also less attenuation due to the vacuum sheath with increa.sing N, a result to be expected, since as N increases, the EK wave length also increases and the sheath thickness to wave length ratio becomes smaller. It is (z) interesting to note that K for N larger than 0.85, exhibits an approximate P exponential attenuation in the 0 variable, characteristic of a surface wave. This occurs with or without the presence of the sheath. The same behavior will be observed as the angle of incidence is varied. Figures 3.21 to 3.28 present the currents as the angle of incidence 0 is

,l~n~j'trie-s ent N-K N CG T KC ( 0, 21 c(i V Cu~ SeatModel0(eres 4510 F IG 53,5, M>~AG TNTJTUI-DE OF Kz)vs ZIMIUTHAL -— ANGLE C) FOR K 0 8 p I —D NOMINs,L V>,-LUES OF OTHER PA'RAMETERS

u i:- - (z Kp ' 122 (Amperes fcm) 1 1 0 '... ~ i " '; 104:e 4 - -9. -. IG to _ -- - 100 K{ I -7 -r 0X 2 cm 1V \ T 10.- 0 K *\ 0 ' -m |~ Vacuum Sheath Model -+ j (Degrees): -: 10-8 5 90 15- - -..i - -- FIG. 3. 16: MAGNITUDE OF K. AZIMUTHAL ANGIE 0 FOR N=-0.9 AND NOMINAT, VAT.TCRSOT-TR. PAO R.HAAr. TR.,S —

- c 10 ', } Is K'Z) p I (Amperes/ cm) 123 10-4 10 L -,10-5 -6 1 0,'s 10 4 - - - J10 T b '"*^ ''.. S-. EN E N - N A,, V, -N - - x = 10 f= 1 Gc T=10 K e N = 0.99 c=0.:02 cm- - a = 7/4 Vacuum Sheath Model 5 r - j +(Degrees) - 45 90 135 180 1-8! 10 T; |- — ___- ___-_ FIG. 3.17: MAGNITUDE OF K vs. AZIMUTHAL ANGLE i FOR'N -0.99 - - P AND NOMINAL VALUES OF OTHER PARAMETERS

II (A\mperes / cm) 124 5 1 i 11 i -7 4 - - i,, I I I II i; i 4 f I, I t " I ~ I I I V ' H II ' V I k I 4 i fI I i I I t i I I s I1 I t I f i f 41 f t i 6, I itI v tif I.t- x = 0 x= 10 t if 1- Ge e N= 0.8 c. = 0. 2 cm i 0 = w/ 14 Vacuum Sheath Model ~ Dges., lf, - ~ 0 (Do- ia 45 90 135 II vs. AZIMUtHAL ANGLE 0 FOR N-: 0-.-8 189 -I -. FIG. 3.18: MAGNITUDE OF p AND NOMINAL VALUES OF OTHER PARAMETE-RS

10~ K i p e (Amperes,/cm) 125 -4 10 s-5 i P C 6 -t I ' -r — - t A. r;C" -rr.4 A4 ' P-i, Y., -. I — r-vv I ~,:, A~,;~ist'A t k n - - r f \ j t \ s \ yi | i 7 I, 3 -M-S X= = 1 0 f 1 Gc T 10o K -e N=0.9 c = 0.2 cm = r /4 Vacuum Sheath Model 10 - s.1 i I f — I t f f' I" I f - i I (Degrees) 10-8 i - 45 90 135 8 FIG. 3.19: MAGNITUDE OF K' vs. AZIMUTHAL ANGLE FOR N=0.-9 AND NOMINAL VALUES OF OTHER PARAMETERS; - 0 -

10 KpFl P (Amperes/cm) 126,I. 10-5 I4 Fr / 10-7 lo p1 -', ' - - y x A -_. X=O X-10 f 1 Gc T = 10 K e N -0. 99 c= 0 2 cm i /J4 0 7r/4 Vacuum Sheath Model N.t "IN 4114 N1 ~ 9K riK \ 1B 'I t - e. + (Degrees) 45 90 135 FIG. 3.20: MAGNITUDE OF K( vs. AZIMUTHAL ANGLE 0 FOR N =0.99 P AND NOMINAL VALUES OF OTHER PARAMETERS 0

127 varied in itepl of 0.17r from 0.057r to 0.457. There is generally an increase ir the fluctuation of the current 1Imagnitudce as a function of 0 with increasing angle of incidence. Also, the attenuation of the maximum current value caused by the.1cuLiLi sheath is observed to Geucttase with incr.- wsing angle of incidence. An (z) i exponential decrease in K as a function of 0 is observed, for 0 =. 457r, again exhibiting a behavior characteristic of a surfatce wave. K(0) for this angle of P incidence also has an exponential aitteILnuati(.on with 0 after the maximnum value is reached at about p 2(fto 25~. There is an additional graLph given in figure 3.29 which shows the maximum value of (z) for the sheathless case as a function of the angle of incidence on a p logarithmic scale, measured from normal incidence. The Km.tiun i | Ki occurred generally near 0=0. There is a narrow spike in this curve centered about the angle where 0 =0. This is caused by the leading term in the Fourier C series for K(z). No similar peaking occurs for K. P p The results f r the final parameter variation, that of the sheath thickness expressed in D, are shown in figures 3.30 to 3.35 for 0 =7r/20, 7r/4 and 0. The curves for the first two angles of incidence generally show a reduction in amplitude and 0 variation with increasing sheath thickness. For sheaths of 2 and 5 Df in thickness, and 0 -z/20, there is however increased 0 variation in the currents,, land at some angles they exceed the sheathless values. The currents for 0 =0 s are reduced much less in alltijIitiide with increasing sheath thickness, though there is a decrease in the 0 variation of K(0) at this angle of incidence also. p

p (Amperes/cm) 128 1-4 10 z z N / A ItII:, i-~ --- I i ' - -6 1 10 I-4 - ~ f-lGe i T X10 1 K e -7 -10- N= 0.7. c 0.2cm:: 0. 05r | Vacuum Sheath Model \ +t0 (Degrees) -8 | 45 90 18 10 FIG. 3.21: MAGNITUDE OF Kz vs. AZIMUTHAL ANGLE 0 FOR 0 0. 05 5 p AND NOMINAL VALUES OF OTHER PARAMETERS 0

10 - p (Amperes 'cm) 1299 tI I i I I r f, i I s, 4 t i I I I t i 1 1 1? i t i, I X -, I -Ft i --, I 1 -- I " I, I - I I iI I I r i I i I t I i, t i " I I 1: 7 ti1 i, u i zi11 4i I.I iC Fwi I 7 — i f4 1 IV — IlkN ltl- -- - A%. 1-1 iL IZ %41r — `7 t tA Af i: I- all, f t f 1 k I 1 4 -I r I e - I1; lvII II I tI -- i i -7 -1 I I 0 1 4t --- L II s Ii it ill I i, i I t 8 w I 0 I FIG. If =I G N= 0.7 c=0. 2 cm i 0 =0. 1157w Vacuum Sheath Model t 0 (De grees) 45 90 135 8 3.22: MAGNITUDE OF K(z VS. AZIMUTHAL ANGLE 0 FOR 6Lb 0. 15wrp AND NOMINAL VALUES OF OTHER PARAMETERS 0 I

10 K - p (Amperesicm) -I 130 7 f II l i 10 f~ I It -- I io6 - 1 1 - N 11 I i 14; 14 I j i k, i 0 I i I i I t t z 11 I t f f i z i -..-. X = 0 I, 1 X= 10 F' f-lGc I _ f - 40...7 i Al - iu K 'l e I. - - - - i N -0.7 7- Vacuum Sheath Model + (fDegrees) L - 145 90 135 180 FIG. 3.23: MAGNITUDE OF K vs. AZIMUTHAL ANGLE 9 FOR 0 0. 357r p AND NOMINAL VALUES OF OTHER PARAMETERS

IJ U 1A p (Ampeies \ in) 131.-4 I! 0! i-i t tt il I, " i - k 1~1 X t$~ 10-6 i0 X-= X — 10 i'= 1 GC r = I4~K oK e N - 0. 7 c = 0.2 cm 0r= 0. 457 VacuLum Sheath Model 1-7 10 + 6 (Degrees) 45 90 135 180 FIG.:3. 24: MAGNITUDE OF K vs. AZIMUTHAL ANGLE B FOR0 =e 0.45 AND NOMINAL VALUES OF OTHER PARAMETERS

C..... < LO If a. K 1c4 00 tOtm c0q c I

p 133 (A mpere s em) r4 ii I Ii 4 I 1010 K e~ ICIN \if: 'N 10140 N = 0. 7 =0.2 cm 0.15w +i(Degrees) 10-s f 'Vacuum Sheath Model 10 i45 90 FIG. 3.26: MAGNITUDE OF K vs. AZIMUTHAL ANGLE 0OFOR p fl;0 7c A Mfl NTflM/TMVAT. XTA T TTrC nr nmrsrn t 1 DAnVAxffVrmrW!>Q

k,i 1i i I:f I-1 -4 10 (.Amperes -em) 10 -1 0 x=0 f =1 Gc e N=0. 7 c 0. 2 cm i 0 =0. 35wr Vacuum Sheath Model 45 - I i I,- It Ip It r f, i, YI i i, 19 it LI I 90t 9 ig y 1 'X it lj 4 —"I, ~d (Degrees) 135 180 (0)i FIG. 3.27: MAGNITUDE OF K' VS. AZIMUTHAL ANGLE -0 FOR 0i- =0. 3Sir p AND NOMINAL VALUES OF OTHER PARAMETERS

1i 5 t 1E? f i E 4 r a P"t f g i P.S i I:I: I I r r; i r i 3 i i Lb J i ij i: i. i r i r I t a r I;: ih: I s: i f " I (0) Ke) p (Amperes fcm) I i I I I 10 l -5 -10 10.. i lI r5 il X=0O.... X = 10 f= 1 Gc 40 T = 10 K e N= 0. 7 c=0.2 cm 0i= 0.457' Vacuum Sheath Model + A (Degrees) 10-8 45 90 135 -18 FIG. 3.28: MAGNITUDE OF K vs. AZIMUTHAL ANGLE j FOR = 0.-45 - AND NOMINAL VALUES OF OTHER PARAMETERS AND NOMINAL VAL UE S O F OTHER PARAMETERS - ~-; i0

10 P 136 (A]nperes 'cm) f= 1 Gc 40 T = 10 K e N- 0I7: -2 c 0. 2 cm X- 0 Vacuum Sheath Model -3 10 II e 10-3 10 i-4 -2 - _4: 10 10 L0 _ 4 - G, i,! - -. FIG. 3.29: MAGNITUDE OF MAXIMUM VALUE OF (K AS A FU-NCTION F ANGLE OF INCIDENCE AND NOMINAL VALUES OF OTHER PARAMETERS

10 FI\ (Amperes,, cm) 137 io-4 II ti I II I -5 k 10 I I zj I 41 i I - t — I i N i k" f I i i f I i j I-1 1I I14 II ii t I I II I 1., i, %-.aI I i i 1-1-11' i I F I t - I 7-I N, I~ 10-6 X i i i I I I I I I i I i I 4 I i I " I i;1I I I I I 1 1 11 1; ji I I f =-I Ge - 40o T 10 K e N=O.7 c 0. 2 cm Vacuum Sheath Model +ji (Degrees) 108 90 135_ I 180 I -A -- - - & - -,, I *-k- - I FIG. 3. 30a: MAGNITUDE OF K VS. AZIMUTHAL ANGLE P FOR0hO0r AND SHEATH THIEKNESS X A PARAMETER

IU 138 Kz) p (Ampe re s cm) \' —" 10-8 10 a~ l -, t t \ \ / V *14~~ -10 10 X= 15 X= 20 f= 1 Gc 4o T =10 K e N=0.7 li I i -d - 10-11 c = 0.2 cm 0 = 0. 05r Vacuum Sheath Model + 0 (Degrees) 135 I 0-12 10 I 45 4f 90 180 * + FIG. 3.30b: MAGNITUDE OF K() vs. AZIMUTHAL ANGLE f FOR 0 = 0.-057 AND SHEATH THICKNESS X A PARAMETER

U K' 139 P (Amperes 'cm) -6 I. 1 0 - - " /, 3 X \ I -!, 10 _ K i. q b j 7, i i i- '. /.j: - I; ' - f X - 5: —! I 40 -T1 — K ": ':',,-8, _-_ X O \ T =104~K, j " N=0.7 0. 7 I c =0.2 cm oi o0. 05: r- + (Degrees) _-9 45 90 135. 180 10 9__ FIG. 3.31a: MAGNITUDE OF K vs. AZIMUTHAL ANGLE FOR - 0. - OS AND SHEATH THICKNESS X A PARAMETER

T0 - - z - - I - f i I I f, 4bi i -0 r I i (Amperes, cm) i It x r Ok -Alp' 1 O't -10 10 -5v Ni It I I -I I 1 ~ II W-0 X = 20 I I i i - - t I I I t I f - 121 -10 i i — II I I 1, I t - i i i t I f = 1 Go T 04oK e o 0.2 cm I 0 = 0. 057r Vacuum Sheath Model A j. j (Degrees) 135 45 i -------;___EI 90 180 - FIG. 3. 31b: MAGNITUDE OF K~9 VS. AZIMUTHAL ANGLE fi -FOR Ot:-0.057rAND SHEATH THIeKNESS X A PARAMETIER —

p (Amperes3 cm) 141 I e f f y -, j I - 1 3 I II I I i i I 2:I iI I I I II I I I I K, I II I I - I f I -5 10 10V V0 - I jt Its 4* 44 A4 N It N U Vt I ~ I / I I 9 -I I 41- 1 t, '111 I I i I II I I I I I I I i. 4r 14 - -A — fl 1 j, .4 %, -V x -i-I N 4AM - e N - t, -N" - x = 0 - --- - - - - X =2, — " —X =5 eVacuum Sheath Model It %%000 Awdip 1' N If 'tL X= 1 V ~ ~ nX=20 ~ (Degrees) 4590 -135 1 FIG. 3.32: MAGNITUDE OF K(z) vs. AZIMUTHAL ANGLE y FOR- 0i.5 AND SHEATH THJ&KNESS X A PARAMETER p 10-8

-LU (ArpereSI, cm) 1 42 II tI 11 i I i f I I, I i I I I I, t,11I It i I a 1 i t11 i I kI I I I,I I t "I 'I; Ii I, I f I:v i 1 2 % I II i I s; i 5. i i I i i; I I z, - L I i I I% I 11,L 4 I I I i i -; 1, I I,,It;, I,,,: t - li if -, I, e -,Jl.1I I II z i I i I9 i I 1 4 I II I t t t tI II 1 I I I , I 1 c I I .f k I II I> V - 5 10 x=0 -- - I- -- - X =2 X=5 - I! ~ U I I V 0 —, I t i I j i i I I i I i t 4 f I i i 1. I I I i I i f - I i I I k I I 1 4 1 11 i I i I I I I i It I\9 ' I — 1? i i1 4' -6 ~ 10 IJ I,-qoe~#,:X = 10.9;*AF,,- X = 15 -7 -Vv,? 0 I -N I t a I VI 14 -UTe= 104 F4 NzO. 7 a- 01 0. 25 j4 acuum __ 10 I1 4 44,~ V 0 K cm N ' 'I )T7 Sheath Model 1v A. I " I ~,I I 99 'I1 0(Degrees) I 135G 45 FI..33:MAGNITUDE OF vs. AZIMUTHAL-ANGLE ~4FOR- Ct 0 25 AND SHEATH THtKNESS X A PARAMETER

I-0- ' p (Amiperes "cm) I,1' 1 '~0) i0o8 I, I I T, I? II- 7 1 4 I 1 r-I f 11 i i i I L - '1 I I - -- off i i I _9 I I lo II I j I t 7 - 11 - i i 4 / I/ vi k I I. rI 11 1 Vl I I 10 1 I'i t -11 10 I f = I GoC T =1040 K e N =0.7 c=0. 2 cm i 0 = 0. 2571* X = 20 Vacuum Sheath Model VI -I i&12 FIG.33b + 0i (Degrees) 45 90 135 180 MAGNITUDE OF Kvs. AZIMUTHAL ANGLE fiFOR 0k -. 257rAND SHEATH THIEKNESS X A PARA'METER

1U -4 10 I i I ii f i i:r } I i I 144 Kz) p (Amperes /cm)....... X= 0 X= 10 f =lGe 40 T =10O K e N= 0.7 c = 0.2 cm i= 89.91 Degrees Vacuum Sheath Model I iI I i i-. I f - 10 + 4 (Degrees) 45 1 90 l 135 11 180: I L r ---- r ---- -----------— ^ --—,,~.L^...~_ I i FIG. 3.34: MAGNITUDE OF K() vs. AZIMUTHAL ANGLE 0 FOR 0i. 39.91 DEGREES AND SHEATH THICKNESS X A PARAMETER -"

I U I 1 45 I 1 ".,I i, f I I I I I I IIT Ii 'I - 5 j%, 10 4 -1 I., I —, I I. N - i " I 11 l f)i 9, -f - I t - t. 4 7 t" I i i i II e,, - t t (0) K-. (Amperes/7cm) iI t I I X = 0 ----- - -- X z=2 --- - -- - X z=S X= 10 X = 20 i II-TI10 F f t-"I - II I e 11 I -f 1 Go e N = 0. 7 o 0. 2 cm 01= 89.91 Degrees Vacuum Sheath Model ~. 0 (Degrees) 135 8i' 45 90 iqo - (0) FIG. 3.35: MAGNITUDE K, vs. AZIMUTHAL ANGLE9 FOR 6~k 89.81cl DEGREES ANII SH4EATH TH11ICKNESS X A PARAMETER I -

146 There are several features of the results presented above which are interesting in their own right, but there are two which are of the greatest significanCrte from the viewpoint of measuring these currents in order to detect an incident EK \wave. Thile most important is the shieldingl effect of the vacuum.ltlI sheath, which was Found to produce as much as 60 db attenuation in the surface currents for sheath tllicl'a.'.c.'.cws up to 20D~. The implication of this is obvious, in that the sheath may act to screen the EK wave from the cylinder and thus prevent its detection. Figures 3.36 and 3.37 summarize the sheath effect, showing the maximumL current values for changing sheath thiclkness, at three angles of incidence. The attenuation due to the sheath is seen to be nearly exponential, with a scale factor oO0 ), determined by the angle of incidenc{., and may be approximately represented by A(0O X)exp -a(01) X (3.1) X is the sheath thickness in D~ and should exceed 2 for (3.1) to be valid. This is a useful result since an L)approximate magnitude of the current for an arbitrary angle of incidence and sheath thickness can be obtained from the current value cat a reference point, once a(0 ) is known. This exponential dependence of the attenuation on the sheath thickness was deduced from the approximations to the exact solutions, given in Appendix E. There it is shown that the sheath attenuation may be approximately accounted for by a factor given by 1 1 cosh X (s-c) cosh /(s-c) (3.2a) eo

147 1'z) P -3 10 (Amperes cm) 5 -4 o1 5; - 6 10 -75 10 5 I 5 0 0. 05 -5 1 -89.1 Degrees io i 5 ' -9i = 10X 10 e N= =0.7 "*V._ Vacuum Sheath Model } 'K O 5i' I10 -0. -=. 0 57r, 0i= e 89. 91 Degrees - -: X f=lGc -9! T = 10 K c = 0.2 cm Vacuum Sheath Model X\ 4 8 12 16 20 FIG. 3.36: MAGNITUDE OF MAXIMUM VALUE OFjK( vs. SHEATH THICKNESS X WITH 0 A PARAMETER

148 10-3 5 "I" 101 I i (d) K l P... (Amperes,cm). -"P, C~3 10 1 5 I — i 5 5 1i lo 1 - - -6 10 5 5 i I I1 -I i — I II i, II i i v "\ 0,S \, N,_,,,, __. 0. 057.. i 0= 0. 257 - - 0i= 89. 91 Degrees 5 -9 10 f= 1 Gc 40 T-: O^K e N= 0.7 c = 0.2 cm Vaculum Sheath Model 5 i x X 4 8 12 16 20 FIG. 3. 37: MAGNITUDE OF MAXIMUM VALUE OF IKpf lvs. SHEATH THICKNESS X WITH 0 A PARAMETER

149 when cos0 v. When /(s-c) 1 this becomes 1 --- 2 exp -~(s-c) (3 2b) cosh.(sb-c) Thus we should have - -. (3.3) As a check O3D 0. 418 and 0.:-[, for 0 0. 257r and. 057r respectively. Using ihe graphs, we obtain the corresponding values for a as 0. 46 and 0. 636 from K () and (z) 0. 417 and 0. 575 from K. The agreement between the maximum current values P from the exact solutions and the approximations is thus quite good. Note that c, can be written in a more general way as a 1-N2 coso/N (3.4) The physical explanation for the attenuation of the currents by the sheath is straightforward. It has been pointed out previously that the radial separal i i constant of the EM fields becomes imLaginary when 0of the incident EK wave differs sufficiently from normal incidence. Further attention was directed to the fact that the modified cylindrical f iIIct ions serve then to describe the radial dependence of the EM fields. These modified cylindrical functions exhibit an exponential dependence on p for large arguments. Consequently the fields in the plasma outside the sheath far from the cylinder decay exponentially, and are known as evanescent waves. They carry no energy in the outward radial direction, but instead form surface waves along the sheath-plasma interlface (p = s ). The fields within the sheath also decay with the radius moving away from the -sheth-n ~interfce (c ss so that the fields at the cylinder may be significantly less thea

1 50 those at the sheath interface. If we consider the sheathless case, the currents on the cylinder orfi determined by the magnetic field at the cylinder surface, which is also then the slheath-plasma interf:tcet. As the sheath interface is gradually moved away from the cylinder surfa ce, it may be deduced that the mnagnetic field at the sheath ilterface is not very sensitive to the cylinder radius, as shown by the graphs of figures 3.9 and 3.11. The magnetic field at the cylinder however is decreased in value from that at the sheath interface by the amount of attenuation caused by the sheath, so that the sheath attenuation of the nimagnetic field is proportionately reflected in a decrease of the vacuum sheath current compared with the sheathless current. A second feature which is of considerable interest is the 0 variation of the EK induced currents (note also the z variation e which is not plotted graphically here). Since the variation is on the order of the EK wave length, which is less by the ratio v/ive than the EM wave length at the same frequency, then this variation may possibly be used to discriminate between EM and EK induced currents. Note in this regard however, that with increasing sheath thickness, the 0 variation is correspondingly decreased. Some of the other aspects of the preceding results deserve mention. One of these is the peaking of the z component of current at the angle where the scattered EM modes change from propagating to evanescent in the radial direction. This occurs for 0 =0 when there is no sheath, at which point the EM mode is a pure surface wave, propagating unattenuated in the z direction. A similar peaking of radiated power was observed by Balmain (1965) for the radiation from a slot in

151 an: iufinite pl.lne. He conclut.d: tlht while the maximum power in the peak is very lare. the \vidth of thl- peak is so narroNw that its integrated power in conmparison 'v ilh the total power radiated by the slot, is negligible. The current peak in the present case is also very narrow, on the order of 0.1 degrees at the point \he{re the current falls to 0.1 its nmaximumn value. As a consequence of the e'xtremie nlarrovw.'ness of this current peak as a function of the angle of incidence, it does not seem promlising in a practical sense for detecting an incident EK wave. A similLar observation was made above concerning the increase in the z:ireecte 2 current with decreasing cylinder radius. Another striking property of some of the current graphs is the exponential attenuation of the current as a function of 0. This is particularly noticeable for an angle of incidence of 0.45r. It is characteristic of a surface wave in the f direction, as was previously;menltioned. While this may be accounted for as being a consequence of the energy reachingl the cylinder's surface primarily from the front part off 1ti sheath, it is difficult to find the combination of circumstances between cy]inder radius, angle of incidence and sheath thickness which leads 0;o this situation. Three final graphs of this sequence are presented in figures 3.38 to 3.4( which summll arize the maximum current value for the sheathless case and a sheath 10 D thick, as a function of 0, N and c respectively, where 0 for the latter two curves is wr/4. In the first, where 0 is the independent variable, it can be seen that the currents for the sheathless case are not very sensitive to 0, (except for (z) the peak in K discussed above which is not shown here). However, when the p

U s \Ai-iipres cmj t i D/ - /-- X x -4 T 10 ~ I rI } I t o-5 L 10o / / / -Al / / X= 0 X ---K| ij XP --- f= 1 Gc T - 1040 K N= 0.7 Vacuii Sheath Model 0 (Radians)/(7r/2) 0.2 i 0.4 I1 0. 6 0.8 I 1.0 i --._-. - - - FIG. 3. 38: MAGNITUDE OF MAXIMUM VALUES OFJKp jandK Jpvs. ANGLE OF INCIDENCE 0. - - --

153 case where the sheath is 10 D tlick is considered, a strong dependence upon 0 is found. Tllis can be explainied by the fact that the effective sheath thickness increases with increasing obliquelness of the angle of incidence, thereby producing greater attenuation. The approxinmate solution derived in Appendix E correctly predicts this behavior, since a, the sheath attenuation scale factor, as given by (3.4) above, increases proportionally to cosO. An examination of figure 3.39 reveals that the sheath attenuation decreases with increasing N, also in agreement with (3.4). There is in addition more than an order of magnitude decrease in the sheathless currents as N increases from 0.7 to 0.99. This is to be expected, since if N were exactly unity, there would be no propagaLtionL of energy and no currents would be excited on the cylinder. Finally, in figure 3.40 where the cylinder radius varies from 0. 005 cm to 1 cm, it can be seen that there is a fairly regular increase of K() and a decrease of K() with increasing cylinder radius for a 10 D~ thick sheath. For the sheathP less case, there does not appear to be any systematic current variation with changing cylinder radius. According to (3.4), the sheath attenuation is not a function of the cylinder radius, so that the sheathless and the 10 D~ thick sheath currents might be expected to exhibit similar variations with changing cylinder radius. But (3.4) was based on a large argument approximation to the cylindrical functions, which becomes invalid for a radius less than 0.1 cm, so that (3.4) is not applicable then. It should also be noted in this regard that the sheath attenuation as expressed by (3.4) depends on the order of the cylindrical function also being less than the

ILU.... 154 (A.mperes cm) -4 10 /'- - 1 0 X_. X 10 (z), 4o 1e 0. 25. iV J — I- i-c iff 10 [., X= O c=.2 cm c1 O. 2 cm 0.25~ t Vacuum Sheath Model 0i - N -8 0.75 0.80 0.85 0.90 0.95 10. FIG. 3.39: MAGNITUDE OF MAXIMUM VALUES OF!KP ANDIK() s. RATD OF PLASMA ~[REQENY TO ItCL-ENT WAVE FRRUg CY, N

MISIN PAGE

156 argumeint. Since, as discussed in Appendix E, the Fourier series for the current,lS 1not converge before this 'rlliriellent is violated, then the sheath attenualtion may differ from (3.4), resulting in the curves of the five previous graphs being at variance from the behavior predicted by the approximate formula. 3.1.2 Incident EM Wave The surface currents excited by the e and h waves for the nominal values of the parameters are shown as a function of 0 in figure 3.41. This graph is distinctly clifferent from those for the incident EK wave in two important points. First the y variation here is quite regular, being approximately sin$ for Kh (z) (z) (0) and K and almost independent of ) for Kh and K. The second difference is in the fact that the vacuum sheath has considerably less effect upon the -.,,,,illt:-.- of the currents excited by the incident EM wave compared with its effect in the case of the EK wave. It is of interest to observe however, that K shows an increase in magnitude in the presence of the vacuum sheath. It should be noted in this regard, that the h wave when illumlinating a perfectly conducting cylinder in free space, does not excite a, component of current. In the present case, the comipressible plasma and the sheath both act as a means for causing its excitation. It however is small in magnitude compared with the z component of current excited by the h wave. Becatuse the surface currents due to the h and e waves have a consistent 0 variation, it is unnecessary to present them as a function of 0 for each parameter value used. Instead the currents are shown directly as a function of the various parameters. The current values given in these curves will be the maximum values KTThle magnitude of the currents only is presented since the phase is practically ii,'.v I At Xj l f 0.

I'Ampeles ' cm) 157 V _IW F --,", I z -w - — 4- r 7 i+ne. p-~TSrr j*Lj9lif;5js.k---=__,. asbsi re-i p-~ZS ya - 4p lrYa95~a-li t. 9 ~ — -;,,8- z'f;%gir g r S.4iD~BIF?'PB`PrBqdiebLh L (r C L3,'is-lgPD~7'3B:~EBiLDiC5. 9 i sP ~iVlirbSa?Bsgedi$$ i Ai;Tlw.B$aSn,cc- a i bt.$LFJSfcSbBB.10 N Ni X =0 10-5 10 X= 10 X= 10; X= 10; IK(Z) I.e 0>avsgvK~sae * ~ Zs ffssiS f -6 io ' < f l Gc T =10 K e N= 0.7 c = 0. 2 cm - -- 01 7T /4 Vacuum SheatW-'" Model,.N i A? I - / t t i 10-7I FIG. _ (Degrees) 90 135 45 180 3.41: MAGNITUDE OF CURRENTS EXCITED BY EM WAVE vs. AZIMUTHAL ANGLE P FOR NOMINAL PARAMETER. VALUES

158 from the 0 variation of the current. Figures 3.42 to 3.45 show these results as a function of the cylinder radius, the rattio of w to u, the angle of incidence and tle P sheath tliclness respectively. Two curves are presented for each current, one for the sheathless case and lthe other for a sheath 10 D~ Lllilck, when the difference between the currents for the two cases is large enough to show graphically. The most striking feature to be observed from these graphs is that, apart from K(P, Kh these currents are relatively unaffected by tlhe vacuum sheath. Another impor(0) tant point is that these currents, excepting again K, have magnitudes on the order of, or larger than, tlllhe produced by the incident EK wave. This will be taken up in greater detail in the next section. It was mentioned in the preceding chapter that the incompressible plasma would also be investigated in connection with the incident EM wave. Taking the p1isntal to be inlcole.lll'-ibIe means that the EK wave does not appear in the analysis and the plasma is treated simply as a dielectric as far as the EM wave is concelrned. These calculations were also carried out,and the currents were changed less than 5 percent from the compressible plasma case except of course for K ). This component of the current becomes zero when the plasma is incompressible and the sheath thickness is zero. It was found to depend primarily on i11- existence of the sheath for its excitation, and with a sheath of nonzero thick — ness, the plasmaLL compressibility affected its magnitude less than 5 percent also. -— Y. --- -----— ~ --- --- An important consequence oi these results is that when considering the scattering of EM waves from plasma-immersed obstacles (small compared with the EM wave length), the plasma can, for piractic'al purposes be replaced by an equivalent dielectric whose permittivity is c (1-NT ) This statement o

k) 159 (Amperes cm) 1-9 10 I-A - - -Z- - f- ,_ - -- - -i.. AlNx4w, 1 -,F " -1 - - -el-I -'. —t4 1 -' X 10 ()K z) 2"(0 ) l 0 -10 2xK(0 f-= 1 Gc T =l0 4K e N =0.7 i /4 Vacuum Sheath Model c (cm) 0. 01 0.02 106 i 0 - 0.05 0.1 0.2 0.5 1.0 -..... - -... __ FIG. 3.42: MAGNITUDE OF MAXIMUM CURRENT AMPLITUDES EXCITED BY EM WAVE vs. CYLINDER RADIUS c.

MISIN PAGE

MISIN PAGE

1Anmpevres encii )16 1 (32 -C3 - -- - v — - - -.- ---- -. -o- - r, t " i 0:=: 0. 25w7 i o 0. 05w, ( Z)~ KKI e. I f =1I Go e N= 0.7 c = 0. 2 cmn Vacuum Sheath Model loG6~' I"' — -- - - -., U~ X-1 ~ -. — - - 4 8 12 16 20 -7 1.0 4'FIG. - 1I- III --- - - ~ ~ ~ =.. I -- — A - 3. 45: MAGINITUDE OF MALXIMUM CURRENT AMPLITUDES EXCITED BY EM WAVE vs. THE SHEATH THICKNESS X.

163,tll- not 10e I!!l; for obstacles of arbitrary size since the present analysis has t-eein r-strJctec l o a cylinlder no larger than 1/15 EM wave lengths in diametelr. 3.1,3 Comparison of EM and EK Inldul.-.ed Currents We c, nit now to the very imnportant question of deciding the feasibility of mialkin; surface current ItHllsl ur ements for the purpose of detecting an EK wave t;cident onI the cylinder. Since suchl a measurement would conceivably have tb be performli ed in a strong l1.;lck1grc)lAlound of EM radiation, it is pertinent to compare ille malgnitl des of the currents excited by both kinds of waves. It was observed previously tl[hat thie EM waves generally induced currents of equal or greater lman itude than those due to the EK wave for unit V. This may be misleading hel,\\ ever since there are two considerations to be taken into account belore the co nlp:l'ison is meaningful. These are: (1) the power flow density in fa;clh \\':V1:111n (2) the satisfying of the linearity requirement upon which this tnalysis llins been based. The former point is important to the relative magnittLdes of the induced currents, which the latter places a limit on their absolute magnitudes in connection with a linearized analysis. Tile only reasonable criterion for a coit)l'parison of the induced currents catused by Ilie two kinds of waves would seem to be that when the incident power flowo dens ilt is arel equall A Poyntings vector for the power flow was given in Chlapter II: fronm which it can be shown that the equal incident power flow reqtiremel- nl c lan be expressed as i i (3.5) V-. V P

1i64 where V represents both V and V. For an electron temperature of 10, 000 K, e L (3.5) reduces to V = 4.74x10- N V (3.6) P which shows that in order to compare the curves of sections (3.1) and (3.2), the scale for the EK induced c-.i ret.lil.s must be reduced by a factor of 0. 0332 to 0. 0470. This changes the picture considerably, since instead of having currents on the same order of magnitude induced by the two;inds of waves, the currents are a minimum of 15 to 20 db apart. The linearization requires that the dynamic electron number density be small compared with the static number density, and in addition that the dynamic electron velocity be small compared with v so that the plasma is not heated by the dynamic r fields. These requirements can c:-.^. 4o T is 10 K in the numerical expressions. L is a parameter which is the ratio e of the static to dynamic electron number density or the rms to dynamic electron

165 velocit-, and is thus a measure of the validity of the lineairizationl. It is interesting to note that in the case of the EK wave, the linearity conditions on both the dynam; c dcensity and velocity reduce to the same form given by (3. 7 a). If L - 10 and N= 0.7 then Vi 0-. 178 Volts (3. 8a) p V 1.59x102 Volts (3. 8b) so that results obtained using V-= 1 and V determined by (3. 6) will satisfy both -- p the linearity requirement and equal power flow criterion discussed above. Note however, that equations (3. 7),l.ply to the incident plane wave only. It is possible that the inttraction of this plane wave with the cylinder will give rise to fields in the vicinity of the cylinder which are so large that the linearization is invalid even though (3. 7) are satisfied. These equations thus represent an upper limit for i i Vi and V. It is of interest to observe that whlen (3. 8) holds the power flow P density in the plane EM and EK waves respectively are 5. 5x 103 and 6. 1 watts / (meter). When V is set at 0. 0332, corresponding to N = 0. 7, and for the sheathless p case only, then we find the EK and EM induced currents for 0 = 7T/4 and 0 = 7/2 given in figures 3. 46 and 3. 47. It can be seen that the z component of the current excited by the EK wave is about 0. 003 that due to the h wave and 0. 05 that due to the e wave for 0 = 7/4. The 0 component of the EK induced current is about 0.1 that excited by the e wave and about 100 times that produced by the h wave for 0 = r/4. Tle same relationship holds for the EK and e induced 0 components of current for 0 =7r/2. Thus apart from the K) current, the currents excited by the

(amiperes / cm) 166 f~ 1 G 4 o r = 10 K e N-::0,7 II (7z), eK 4 e I (0) - i K I 4 - I 10-3 ] () c = 0.2 cm i X - 0 i i -V = 1. OVolts e n V- 0. 0332 Volts -1 __ ^ — r ^ ^^ hK e 'K'f --- -- I Kp Vacuum Sheath Model 1-4 I" I\ ' 4 f t it i f i - i I I t u t II I i r I f t if I:. r - i i I I It f I, if. I i I k il.: 'I 1 -: 1 " - ) Z I. - i II I;4 II i I I -1 I? t il I I " I -x-, t I I -1I I, I I 'L I. i t I 4 i I I i i f I 1 7 I t, I i. II 1.- I f i I i i f i I II I i I I 4 I i!\ $4 4. 1; \ c- 'I -6 10 t o —7 10 45 rII 90 135 4 - 3.80 N |, \ - FIG. 3.46: MAGNITUDE OF CURRENTS EXCITED BY EM AND EK WAVES OF EQUAL POWER FLOW DENSITY vs. AZIMUTHAL ANGLE p FOR qfl- - A d Mf XfTTTT A T IT A T TT r f' l D r T A D & 'r - rr rflc-t

l Amneres cm) i67 )-4 -4 0 r 'V At I ~S 'I ii if I i I-6 l 0 Vacuum Sheath Model (z)....-.K II KhI - -I — - -. ---/K-K I f= 1 Gc T = 1040K e N -- 0. 7 " ii F r i t 1 i ' 5 I 1 ri ' f % r i i r 5 t- 5! f:: r 5 "'k i Vi 1 eV h1. eh Volts Vi= 0. 0332 Volts p c = 0. 2 cm 1 = 7T/2 X= 0 45 + 0 (Degrees) 90 135 180 I FIG. 3. 47: MAGk(; NI TUDE OF CURRENTS EXCITED BY EM AND EK WAVES OF EQUAL POWER FLOW DENSITY vs. AZIMUTHAL ANGLE f FOR 0L T' 2 AND NOMINAL VALUES OF OTHER PARAMETERS.

168 EK wrave for the sheathless case are 0.1 to 0. 003 the corresponding currents causea by the, EM wave. When the attenuation due to the vac'uunl sheath is lal.:. ilt).coiintl, the EK induced currents are reduced by a further factor of (). 01 to 0. 001 excepti for ineatL normal incidence, where the attenuation factor may be 0. 5 to 0. 1. It canl thus be.,,-.1.ll 1; that the EK wave is considerably less effective than the EM wave in the excitation of currents on the plasma-immersed cylinder. Th re is: an iddiitional factor to be taken into consideraLtion here, from the viewpoint of p'Llacti(aclly measuring these surface currents. A slot which is very thin in the cdirection of the current flow has been previously mentioned as a means of car-'ryinog out these measurements. The effect of the slot is to perform a line integration of the magnetic field parallel to it, which is the current flow normal to 11.: slot. If[ the current is of constant phase and amplitude over the surface of the cylinder, then the longer the slot the larger the voltage which can be BILLmeasured across it, as long as the slot length does not -rcach the smallest length at which it will resonate. One of the distinct differences between the EK and ENM NwTave induced currents is the much greater fluctuation with the azimuthal angle > of the magnitude and phase of the EK currents. This limits the practical effective size of the slot and thus the magnitude of the signal to be obtained in (,) measuring 1K', while the z component of the EM current is not subject to this limitation. A similar argument follows for the 0 directed currents due to the t iltrX. nct in the z variation behavior of the EK and EM currents. (Z) (~ ) The cur t'\ves of K and K on figure 3.46 show that the average sEenrr.P P tion of the c.tllrrent minima are on the order of 12~to 15~. Since these minima are

169 (z) 's' )'ciate' i t:h rll Iid phase 'lt,ljes, thl slot for me suring K should thus P slU Itl 13 l00 i,. t(r Liit- tint that encil'is-) d by tN.\wo succ'essive miinima, which le-ds to L. slo-' lenglth o abhout 0. 0535cmr. Assumling that the slot is centered on a ciir't'l, m-:imllL te t t '1 compuient of the c.trrent available for nmeasurement -5 i i'lt to thil E:K wAv\\ would be no greater than 1.5 x 10 V amper-es, for the P sh.l:;thiess c lIse. Wirth V equal to its maximum value for which the linearizatior P -6 is val thle curl-nt is Tbout 3x 10 amperes. If this current could be introced to a 50 'hi- transmisn-sion line without couplling loss, then a power on the orrcer of 5x 10 wa)tts would be available at the output of the line. An rf power lev\el of this lmagInitude is well within the capability of the more sensitive laboratory tc chnlliut s conmmonly in use..n incident h wave with Vh related to V by h p -5 (3.6) vould proluce C n output power from the line on the order of 3x 10 watts for the z coillpozl-_t-' l of f11i curr'ent. A consideration of the 0 components of the surface curreint shows that due to the EK wave to be one tenth the current caused -10 by 1?n e wave, t 'ith res pective otput powers on the order of 5 x10 and 5x 10 x at. t. It shouldc be kept in mind that the calculations outlined above are for the sheathlass case, so tlhat the output power figures mentioned for the EK wave may -6 be a factor of up to 10 lt —s thanl those calculated above when the vacuum sheath is allowed. for. Note also that the larger of the two current components excited by the h and e wave.s has beein used in the comparison, since the incident EM \wave must be.tssuilted of arbitrary polarization. The prospect of detecting the EK \;\ave in a background of EM radiation would thus appear to be a very difficult

170 one T'l variationZ of the EK induced surface cu'rre-nts in the p and z directions toes howit(ever,, suggest a means for observing 1,. ll in the presence of the EM i'j''rents.t sincet a relative motion between the slot and source would produce a -- iiiation in tlhe output which is dependent upon the EK wave length. It would apear tlhit the ]nost favorable circullstances for texperimientaLly detecting the 1EK current would be for noiLalll incidence, since the sheath attenuation is a ininimutmn there. Th _t-er is one additional observation which can be made concerning the P componlentt of the. —: f K-r crr:'. It was pointed out that this current is not excited in the absence of the sheath when the plasma is incompressible. Further, I-\ cept in the sheathless case, the (coulI rssibility of the plasma has little effect on its Ima Ognitude while it is fairly sensitive to sheath thickness. T'herefore, a meisu-rement of K) would be informative in determining the effective sheath thickness. 3.2 Inhono) eneous Sheath The tmnerical solution of the inhomogeneous sheath equations given in Clhaptej I1 take 20 times more compulll)tter time than the corresponding vacuum sheath cal.'ulatio1is. Because of this, the range of parameter variations which could be inlavestigated for the inhomog-eneous sheath is much more limited. Furtl-lheore, tihere atre additional parametric quantities associated with the inhomogeneo'DUS sheath whiti are not encountered in the vacuum sheath analysis. Theyr are the cylinder potential and the potential variation in the sheath, as well as the

171 boundary condition thait the dynamic election nulimber d.ensity be zero at the cyli '! I.-. This b,uIll -rl'y condition does not need to be considered in the numerical calculations for the vacuum sheath since the EK lurrent is then zero. Consequently in order to obtain the lmaximumi IIU benefit from the computer time available, a judiciOUs choice had to be made for the values of the parameters to be used. Due to these considerations, the only vacuum sheath parameter varied here is the sheath thicklness. N, c, a, T and V have the nominal values given e on page 10', while the angle of incidence is of c ourse fixed at 7t/2. A limited numlber of values for the cylinder potential and for the exponent of the potentia.lvariation in the sheath are employed, for both boundary conditions, the vanishing of the dynamic electron number density, and the normal dynamic electron velocity at the cylinder surface. These will be referred to as the soft and hard boundary conditions for obvious reasons. While the number of cases which can be investigated numerically for the inhomlogeneous sheath is small compared with the vacuum sheath, for the same computer time, there is a great deal of very useful and interesting information in addition to the surface currents which carn be obtained from the inhomogeneous sheath results relating to power flow, energy density, etc., and which is not obtained in the vacuum sheath analysis. This mate - rial is presented after the surface current curves are given and discussed 3.2.1 Incident EK Wave Figure 3.48 is a graph of the surface current (note that there is only the s component for normal incidence of the EK and e waves) for both the hard and soft

10 - K() ' Ampeesc (Amperes/c m) 172 N N, f i I - $J I05 I,I t I9 -5i t i i t IIc I '\ I, / ', ~t, \; \ I I. I i I { i" r0,E, s j. h Hard Boundary. Soft Boundary f= 1Gco T=10 K e N= 0.7 c = 0.2 cm I - — Pf = <2 - X = 20 M o -- c.s-c - - 0 C- C.S-C 2 M=2 - = -5, 34 volts.^C + 0 (Degrees) -8 45 90 135 180 FIG. 3.48: MAGNITUDE OF K Vs. AZIMUTHAL ANGLE $ FOR INHOMOGENEOUS SHEATH MODEL WTH M-2 AND C = -5.34 VOLTS -C C

173 bountdaries. Til sheath tilicl:i ss is 20 D A. A parabolic sheath potential varication is usea (M-2) anid the cylinder potential is obtained from (2.17) for a mercury plasmla (my200 atomic mass units), which yields a -value of -5.1 volts for C -n im.port.int feature of figu r e 3.48 is that the soft boundary condition leads to currLlents a factor of 5 to 20 times less than those obtained from the hard bound try. It was pointed out in Chltapter II that the use of the soft boundary condition will give coupling between the EK and EM waves due only to the sheath inhomogeneity, and thus serves to separate the relative contributions of the inhomogeneity and boundary coupling for the ii.homoll gei,-l{.us sheath. The tentative conclusion might thus be reached that inhomogeneity coupling is less effective by a factor of at least 5 than the hard boundary coupling. A comparison of figures 3.48 and 3.7 is now very informative. It can be seen that the current from the sheathless case in figure 3.7 exceeds that for the inhomogen1.leous sheath and the hard boundary condition in figure 3.48 by 3 to 10 times, while the current for a vacuum sheath 10 De thick is on the same order less than llis. The two curves from the vacuum sheath model thus bracket the inhomogeneous sheath current, which is a desil rable result from the viewpoint of establishing the validity of the vacuum sheath model. Before drawing any conclusions from this outcome, some further observations should be made. First the VLLCuLIuII sheath model, it should be remembered, has as its only coupling mechanism, that due to the hard boundary condition. The soft boundary. condition produces no coupling for the vacuum sheath model. The inhomogeneous sheath on the other hand, has the additional coupling mechanism of the sheath

I i / l-,mogee ity e \ ill-hit expec(t then that tee inlholmogeneous llheaL}, with bo:;h f K, ',-i.:_. i-!i; t'- cnlcl rS liss in effect. could pi oduce larger surface currents hL1an those ol'btairel w 'ith the vacuuml' sheath, which has the single coupling iecha - ismi. Tl'is Cdues 'ot t turn out to be th- case howevEer, except when the vacuumn -I ath J:I mo e1 th.n 5 D tlhic1. The implication would seem to be thatt tle inihomog- eIwouas sheaith serves more as an attenuator for the incident EK wave th-an1 it cult t ibutes to the coupling pictulre. The effective atte tuation appjears to he fairly well approximated by the vaLcuuml sheath model with a sheath about 5 L, in thilckness. A second obserxation of importance concerns ili uncer'tainty of the inhojnogeneouls sheath ctIllurtllt results due to lack of lnowledge about Y We mer.tion ag.il t.iat Y ilntroduced by Cohen in the bounldar-y condition on the normal electron velocity at thel boundiill surface (equation 2.42), is set at C' or oo in this Canalysis. corn1 espondling respectively to comnplete reflection ar complete absorption of the electrons). It is likely that the correct value for YB lies somewhere between thlese two extremes. Now it is shown in Appendix E, equation (E5), tiat for ihe shlil.thless case and EK nwave incidence, Y appears only in the denominator for Ili e and h Cmode reflection coefficients. The surface currents induced bv the EK wxave:]ius show the sal me dependence on YB. _'I.\ ilsly then, when Yb is large enough, the boundary contrlibution to tli EK surface currents become, inversely proportionCal to YB going to zero when YBis infinity. The b1,uL1ncli' contribuhtil)on to the surface current for the inhomogeneous sheath and vacuum sheath mode-ls mCavy be concluded to e-xhibit a similar deipendence on YB. In the W5

it te' ('::e. Y) v ' uld correspond to a surface.,LJittai.e'e at the sheat)h-uinifo-rm tpLi 'Sm1 i tr' 'tfaeC, ~-e fuithE l nF-ote t}h.Vt an implicit -:, lption in the preceding dis('ussion hjas 1bee-n tlli:t thie contributions to the imagnetic field on the cylinder surtface by cl. sheatll in]oirn,,:th itv and the boundary are additive so that the two effects together, arising from11 use of the hard boundary condiition, would lead to a larger surface current than wvould the sheath inhomogeneity alone as is the case for the sF 1,ui, it!. rv c, ].iition. An examinaltion of the real and imaginlaLr, parts of the surface cLrrents showed that the current )phase for the hard bounidary was generally,,ithin 45 of that for the soft boundary. This r'esLult, together with the increasel iiruellt malgnitude obtained from the hard boundary condition indicates ]le apprx'imate ":dliditv of this assu-'lllption. CotnseIquenC 1l.-, N \e are justified in conjecturing ll:tt, when the matgnitude of the EK curreneits. il t;liued from the inhomogeneous sheath model is considered as a function of YB' a minimum is obtained for YB= oo. Uiifl'tunilaely, we cannot ('conclude that the ('curlt'rt hlias cor'tespotndingly its -.aximum value when Y =0. It may be seen in equation (E5) thiat the e aind h reflection coefficiint s can be increased over those results obtained for Y B=0 by a value of YB which makes the cenonminator D small]er. AS a matter of fact, (E5) shows that if YB is complex m B and of the pr)ol)er value, D can be zero, leading to surLface currents of infinite m magnitude. This is unacce,_-ptable on plhysical grolunds, indiczatingc that the boundary condition the dynamLic norimal electron velocity to the dynamic electrcn ldensity throughll Y is an over-simiplificationi or this value for Y is not realistic. t B B

I 7Ti it niicht l:he ccu,- cl:ldl fl',]i! this th:'t Y should not be set equal to zero in equati-, n (242), We &% s c thI, th ttlth qlue stion of the surface impedamnce of the boundary ifot thie CiYnlAi1ic e']tectr)ll Ilmotion there is a conmplicatedi one which has a strong inllutence,on the co.littlril 01on1 of ltic boundary to the EK induced surface current. As; a relsullt of this n(llOertaltv in the co 11il iuLtioul of the boundary to the surface curl-rent as a luncltion of YB we cannot establish an upper limit to the EK induced cu'rrent in t11he saume way as a lower limit has been established, for the inhomoge neouLs sheallth. iA finll obse:rvation about these graphs concerns again the vacuumi sheath results in c o npari son with those from the il' 11 io>gneou I s sheath. It is apparent that, the va-cuum sheath curlrents for sheath thicknesses of up to 20 D~ exceed the cut rent obltained fti-ol the i11,,, 'l'ogeneLous sheath with the soft boundary condition. On the othlier hand, the vacuumn- sheath curr- ents are in reasonlably3 good agreement with those fIl toi-1 tl-le ilnhomoeneous sheath for the hard boundary condition. It thus aLppears that the \xtlmll11 sheath model using Y 30, provides a good approxiIl,.liion to the ilnhoi>llg Lrels slheath also using YB=0. It naturally fails, however, when Y-( -L a coi;nsequenL'ce of the fact that the boundary coupling becomes B inversely proportional to YB alone, and the current goes to zero. Thus we see -B that the range of uncertainty in the EK induced current from the vacuum sheath model due to,tarial;ion with Y is much greater than that for the;,.1...,. I 1, 111, B sheath wlhere a lower limit to the current has been,..1l.lli.ll..i. Also the validity of the vaUt.oilnll sheath model is thls much more dependent upon the correctness of the hard boMnnl0ary condition than is the inhomongeneous sheath model.

177 Figur'e 3.49 shows the i lnhomogleneous sheath.ll'Hi..c currents for the saime conditions ris figure 3.48 except that M =4. The principle effect of this change is to increac(:se the current fluctuation with p and to increase the magnitude of the cuLrve r-e-.-ltiing orl the soft boundary condition by Jlabout 2 times while having little effect on the maIl,,itudlle' of that for the hard boundary. This increase in tihe inholmogentc ity cou Lpluing is somewhat sup L'i-i sing, since the static electron density gradienlt and the static potential gradient for M =4, the quantities which produce the coup1ling, are less than the M=2 valLCues over more than half the outer part of the sheath, as can be seen in the graphs A2, A4 and A5 of Appendix A. Evidently the,greater degree of sheath inlhomoogeneity near the cylinder surface more than compensates for the decreased inhomogeneity in the outer part of the sheath. In order to determine the effect on the surface currents of varying the cylinder potential, the above calculations were repeated for the hard boundary condition only, for a hydrogen plasma, giving a cylinder potential of -3.06 vol'ts. lThe resulls are shown in figure 3.50 \w\here it canl be observed that the currents are just slightly larger, by a factor of 1.5 to 2, than the Ctu'rnlpon l)iii curves at the higoher potential. It can also be seen that the curve for the hard boundary. i., ii 1 with -3.06 volts and M=4 resembles that of figure 3.7, for the sheathles:; case i,lre closely than any of the other curves presented. This is reasonable, since the lower cylinder potential and larger M decrease the sheath inhomnogene ty, so that in effect this case is now closer to resembling the sheathless mrodel than the others for the lower cylinder potential and smaller v. lue of CLE. Proceeding further with this line of reasoning, these computations

K p (S'%Iltp-iles cm) 1 78 <C t(1, Ii Vk I I 12 " r,!- I I II! ' t 'I-, i, I I 11 i I i " I i 4 q iI I I II i I I I0 -E HardI Bounduary Soft Boundary 40o Ir I10 K e N0. 7 c 0. 2 crn IC X z:20 0 C S-C -r. 34 Volts 10- 7 10- 8 + 0 (Degree~s) 45 90 135- 180 FIG. 3. 49; MAGNITUjl'DE OF K vs. AZIMUTHAL ANGLE O O NIMOGENROU p SHE-lATH MODEL WITH M 4 AND 2=-5.34 VOLTS

MISIN

180 for the hard boundary condition and the hydrogo-en plasma were repeated for Me 10, the expectation being that with the sheath i nhumogeneity thus further confined to a nar.row region close to the cylinder surface, the surface current obtained from the inhomogeneous sheath should even more closely approach that for the sheathless case. That this is the case can be readily seen from a comparison of figure 3.51 which presents this result with the curve for zero sheath thickness of figure 3.7. There is a sl il;ig] similarity between them. VWhile the use of sheath thicknless on the order of 20 De seems to be the most reasonable choice in lilght of l11, the theoretical and experimental informnation which is available, it is desirable to investigate at least one other sheath thickness to see if the results obtained are very sensitive to this parameter. A sheath thickness of 5 De was used for the mercury plasma with M=2 for both the hard and soft boundary conditions. The currents are shown in figure 3.52. Comparing these results with those for a 20 Df sheath given in figure 3.48 reveals that the currents from both boundary conditions exhibit an increased magnitude and fl-UIiattioii as a function of 5. The soft boundary current shows the largest increase in mnagnlitude which is consistent with the increased values of the coupling quantities, the static electric field and the static electron density gradient, in the - thinner sheath. This would suggest that by making the sheath sufficiently thin, the inhomogeneity coupling would eventually exceed that caused by the hard boundary. Since a sheath thilnnier the 5 D~ does not seem physically realistic, however, for the insulated plasma-immersed cylinder, this result is of more mathematical than physical interest. It is interesting to note in addition that the

J -' 4 o T: ',0 Kh I si i I z v I I I yI I I I i II I 1;I I " 1 -I I e -N>T 0). 7' cd.2 '-cm i 9 M4 S -b o C S-C cr- <-3.06 Volts M i CO X= - 20 -1p3 (Degrees) 45) 90 135 180 FIG. 3. 51: MAGNI'tTUDE OF K~ V.AZIMUTHAL ANGLE 0 FOR INHOMOVGENLEOUS p SH-rEA-~TH MODEL WITH M10AD2=-36VOLTS AND HAR.PDBOUITNDARY CONBITT)TA~NT

0d CW zl

183 curves of figure 3.52 are very similar to those obtained from the vacuum shea th analysis given in figure 3.35. This concludes the presenLtatilnn of the surface currents excited by the incident EK wave coming from the inhomogeneous sheath analysis. Since the currents obtained for an incident EM wave from both the vacuum sheath and inhomogeneous sheath analyses were found to be the same for the first three significant figures, no graphs are included here for the latter case. This outcome strengthens the conclusion reached from the vacuum sheath analysis, thlat the sheath has a negligible effect on the scattering of EM waves from plasmaimmersed obstacles. Since the current values produced by the inhomogeneous sheath analysis are in substantial agreement with those arising from the vacuum sheath model, the comparison which was previously made concerning the problem of measuring the EK and EM induced surface currents is essentially the same. There are however interesting results to be obtained from both EM and EK wave incidence and the inhomogeneous sheath model pertaining to power flow and energy density in the sheath which are given in the next section. 3.2.2 A Closer Examination of the Dynamic Sheath Behavior There are some aspects of the inhomogeneous sheath solutions which are of considerable significance to better understanding the physical processes occurring within the sheath that are not revealed by the surface current results, which necessarily give a kind of spatial average of the IT)mrt tsath rspns?- to tcincident waves. Such quantities as for example the dynamic electron velocity and number density, and the electric and magnetic fields, as they vary with position in the

184 sheath, mnay provide a moire det iled picture of the sheath behavior. In particular the inhomogeneity coupling is one phenomenon that canll be more closely examined. There is also the question of whether the requirements on the incident w-ave amplitude which were necessary for the lineacrization to be valid for the vacuum sheath are also adequate for the inhomnogeneous sheath, since the static electron numbl)er density decreases by two or more orders of magnitude as the cylinder is approached through the iilnhologgeileous sheath. It is obvious that with the many quantities which vary with both the angular and radial coordinate in the sheath, it is impractical to find the complete spatial variation of each or of even one for a given situation. Rather, it is necessary to limit the examination to some selected paths in the sheath, the results of which may serve to form an overall picture of the sheath behavior. The decision was made to look at the radial variation for some particular azimuthal angles, since the dependence on the radius is of more interest than the 0 variation in connection with the static sheath w-hich has a functional dependence on radius ialine. In addition, the angular variation of the surface currents provides some indication of the dependence of the other dynamic quantities on 0. The dynamic electron number density, and the dynamic electric and magnetic fields are the dependent variables of the differential equations for the inhomogeneous sheath, and consequently are of possible significance as quantities to be investigated as discussed above. While these are quantities which can provide an answer to the question concerning the linearization requirement, they are otherwise of limited usefulness in themselves in arriving at a fuller understanding

05 '1 'h( (th nl\ i, 1 allh behavior, towev'er, they can be used to derive some physitally intelisting gli nti i.ics which are perhaps more meaningful and more open to interp etatio n. in particular as related to the coupling of energy from one wave tVp), to another.11 rro >1pw el'ciric. we return to the expression for the inhonmogelneous shea-'h given by eclattjio (2.31) of Chapter II, which is reproduced here as (3.9) As expltain-led in ltChapter II, the terms on the left hand side of this equation give the time aver tage power flow density in the( plasma while those on the right hand side )yield the tiille av\eragoe density of the various type: of energy stored in the plasma. [n addition the last termn on the right za.lho allows the possibility of a gain or loss of energyo' in the.11heath. A comparison of the various energy storage terms should provide sorme insight into the conversion of the purely EM or EK energy of Lhe incidcn. wave into energy associ.ite;d with the other type of wave. Nowv it wasv observed in Chapter II, that while both the EM and EK waves in a homrn,'erlc(ous plasma have an electric field and also a dynamic electron velocity field, the EM wave alone has a magnetic field while the EK wave alone hlas a charli.;e ac liMullittio in the form. of a dynamlic electron numlber density. Thus the relat ive Lnag nitucles of the manetll i and the potential energy density terms of

186 ^qtl tion (3.9) should 1 c espc, cially inforl'mative.about establishiingo the con' ' sion oI t he incidlent wave energ'. The c't' iiinicg I's '' may provide ad iLo0n: l i-nfLilat ion a'bout the relative i tlortance of the various energy;ftliUogt ne'lhalnismis 1and their d:epednict e upon the kind of inciden. wave. Th.e power flow terms on the other hand. since they do not inlcorporate field quantities Dassociated with either wntave alone when propagating in a nrr-i-: -. plaslma. nr not suited for investigatilng the conversion prCt-li>- It is of interest however, to know the power flow in -li sheLath in comparison with that of the incident \la-t to determline what effect llt sheath has on the flow of energy to-wards tile,-y5 er'. An i itegration of the power flow across the sheath-uniform plasma i I itt r; L t'C also serves to establish the net vpoweri flow to the cylinder, which should he zero if 1i1e sheath is passive and lossless, and the cylinder itself albsorbes ino energy. The topics of the liniearization requirement, conversion.- ';and the possible loss of energy in the sheath are considered separately in turn below. 3.2.3 Linea:riza-tion Cliter ia in Inhomogeneous Sheath T e linearlization of the Boltzmann nmoment equations requires that the dynamllic quanltities be smnall compared with their static counterparts so that products -: i '.t.i:- te:rms may be neglected in the equations. In order to check this the -Jlaln ic qulalities obtained in the inhollogeneorus sheath solution were calculated as a fulnction of radius in the sheath, at angular intervals in the 0 rvariable of - /8 fr'om the front to the back of the cylinder. As before ve regard the linearization condition to be that the amplitude of

187 tle incider-it \iWI\'oc lie QLchl that the dyn.r Alic variables do not exceed a fraction, L, )f the eoi''spo:b;gng staLtic (quLnt11ity. 'JL)on observing the matgnitudes of these,;uiatitites. it \\,;E ifotund that the lineariza tion criterion established for the oi)oim(i'?'1n'-' p.lAsf J'see equation (3.7) did not hold for the inhomogueneous sheath. For a fixe.d ielue of L — 10, it \\is found (Ii.0 l1:. Imlaxilmumln allowable anmplitude, for particularly i:he incident EK wave, was sensitive to the sheath model and tPe lboundatry condit.ion on the dyZnamic electron number density at the cylinder wall. The use o(8f the soft boundary condition, i. e. complete absorption of the incident electrons, genera tlly resulted in smallller all\\ Lable values for V than the hard P -4 -3 I oundary. on the order of 10 for the former compared with 10 for the latter. Co'rrespon:lintg values for V1 are 0. 25 and 0. 5 respectively. e Theste values are considerably smaller than tlhose obtained for the homogeneous plasma, which were Vi= 0.178 d,-t Vi= 159. They are also less than the P e values us.ll: for tlie cormnparison of the EK and EM induced currents and the disc(ussion on. the po. sibil iy of measuring the currents in section 3.1. 3, the values used therlc. beino V = 0. 0332 and V = 1. The importance of pointing this out is p e that the surface curr'ents which can be expected to be produced when the power in the incident wave does not exceed the maximunm for which the linearized theory applies, may be an order of magnitude or so less than those discussed in section 3.1.3. Ti is has.ie effect of rendering the measurement of the current produced by the EK wave, as predicted by the linearized theory, somewhat more marginal The erity itthoe eo eane r The linearity condition for the homogeneous plasma was based on plane

188 wave prt,).ation. It is interesting to see what effect the diverging waves in the vi(Linity of-' tlhe c7linde' my have on it, when there is no sheath. When the cylinder potenlt til was f - at zero. to check the sheathless case, it was found th:It with L- t10 V;'nd V were required to be about 50 percent of Ihose values p e given in (3.7). which vwas derived for the homogeneous plasma. It may be recalled th it this p)ssihility wvas mentioned in the dLiscuLSsi(ol citfcceiiing the derivation of (3.7' While the lc estion of the linearity condition being satisfied is a necess -' r~ne in a dj.Iscuss.Ion of the Jabsolute magnitudes of the various quantities involvad, 1? '- - i. i., as in the subsequent.discussion1, the relative magnitudes of the effects cL.l sed by the EK and EM wave are under consideration. What is more important then is 11 i I the incident power flow in the two waves be the same. Since the followino discussion is coincerned with quatdities which are normalized with respect to i11 incident wave, the only restriction on V and Ve in the following is that (3.6) )e satisfied. 3.2.4 C!oupled Field Variation in Sheath Thil. conver)-;ion or coupling of EM to EK waves and vice versa in the inhomogeneous sheath and at p)lasmna boundaries is one of the most interesting theoretical aspects of this problem. It is also of practical interest since without the coupli1ng, there would be no txcitation of surface currents due to an incident EK AN-ave, and1 no mechanism for detecting it, in the context of the present study. Before prese nting- some of the results dealing with this phase of the problem, it is infovrmative to consider the magnetic energy! density of the incident plane EM

189 \\N e C1 r-i tlic ))tt'tial energy il-nsity of the EK N wave. Having some know\ledge 'iI it h1(- tc!si\ u.-1Sities cssoscated xith 'he incident waves Nwhich mayn< be comnt roe iit tLheL COt energy:-nSllr is ities in the reflected waves may pro-^ide a )atlsi fior, sta)lisl'lihii' the relative coupling betNween the two kinds of waves. If,,we use -I w1i formlulas given in tiliations (2.45) to (2.47), (2.51), and (3.9), jt is easy to show lhit the time average magnetic energy density, wh, for a plane EM wave is given by (3.10) Similarly, the time average potential energy density w of a plane EK wave is (3.11) i i VF and V are thlie ImLagnitudes of the potentials from which the wave fields are E p o)tal'ned. Upon fob'ming the ratio wh to w, we find h p (3.12) i i This lelads lto the situation, when V = V that w exceeds w by about p E' p h (v /v ), which is also the square of the ratio of the EM to EK wave length. This is simply.1 illustlr.tion of the fact that, for a wave of a given amplitude, the energy de;ns;ity is ilversely proportional to the square of the wave length. Now i i if we relate 'V a ld V by equation (3.5) so that the incident waves have equal F p

190 po\v, er flow de tnsity, then (3.13) Thuls, e;ven ith t'le eqlul power flow requiremlelnt, which is used here, the maiIltII tic ( anlcd potential energy densities of the plane EM and EK waves respectively tvary by the ] rtio of iheir propagation velocities. Bec.ause, of rhis large iffereince between wh and w, a direct presentation of their lmagnlituLdes for the purpose of exhibitiing the relative coupling between the EM an d EK xwaves is not very. - '. ': Instead we generate normalized energy dens;ities Nh and N defined as the ratio of wh ailnd w in the sheath to p p 1l respective valL es of wh and w in EM and EK plane waves of equal power flow den!sity. The superlscitilt denotes the kind of incident wave, so that we have. (3.14a) - (3.14b) An exLanlinatiLn of the variation of the -normalized energy densities about unity will servel two puirposes. First, the normalized energy densities Nh and N h P provide an illndication of the perturbing effects of the cylinder and sheath on the propagation of the incident wave near the cylinder. Second, NP and N give a h P me-Lsure of the energy converted to the other wave form. For convenience in the distllussion-, we refer to the former as incident energy densities and the latter as coupled energy densities. The r; phs tIo be presented show the normalized magnetic and potential erLergy densities Nh nd N as a function of in the sheath at an angle 90 fromthe h P

191 i ou>t, I l 1 cy]:r This angle was chosen since at the front of the cylinder. >onlv the fieil \' a i;.ales associated \\ith the incident wave are non-zero. A check -, - of N:an I N at )i lher values of zit' i utihalil.iL Igl- revealed no significant Ii I l i l.-i. h P in their -,1 i;iation L;s i fulnction of p. Thrp>ee sets of y3raptlhs are shown. for the sheath molldls that have been used in the cur.lrent presentation and a c5 lii:lcer radius of 0.2 cm. There is one graph in eaclh set for Tlhe incident c wave and EK wave. Figures 3.53 and 3.54 show Nh and N for- the she thlless case a:di the hard boundary. No cort'espoldill c'uves aere shown for the soft bo111undary since then there is no coupling. Figures 3.55 to 3.,5817 pre sent simillar results for the i homoge1 tneousllt LI sheath, with M=2 and m. 200 atomic mass unlits (: -5.34 volts), for both the hard and soft 1 c )otndarties, and a sheath thiclkess of 20 Di. Obse -rationl of figur'es 3.53 and 3.54 shows that for the sheathless case, tlere is some sil ilarity between the results of e and EK wave incidence. In 1)oth cases, the incident energy densities are on the order of unity, while the -3 c ouplped e. nergy densities are about 5x10 3antl. of very nearly the same magnitude at the cyl;inder's surface. Both the incident and coupled energy densities exhibit a different dependence on p however. e It is;_t r,.,.sting to note that in the case of the incident e wave, N i. un- ity independent of p. This is simply an illustration of the Born approxij;tion, v\llich is that the total field in the vicinity of a scattering obstacle small compared,'ith the wave l]thllig1 is m^'c-:cnsre~l ccual tothe- intc The ld. Thi~; L;"' r ' _...:-.>",i - v-; x.... r:'' - r te in....ien io O', -'"s s te t

II~ p l'To) t'.insfo im n' ma li d energy densities to same scale, e Multiply N by v 'v p.~- r 450.Lt / - ^, -I- ^.^ I-R, -, I ^ I - I J _-. I ii i -9) 0D *-4 j -1 i d potential energy density d magnetic energy density i - on 1 -3. ) N ' Normalize P ---..... Nh' Normalize f-= Gc T = 10 40K e N: 0. 7 c - 0 2 cm 0 - 712 X - V,, =0 " c 4 j (p-c)/'s-c) 0.6 0.7 0. 1 0. 2 0.3 0.4 0,5 J 0.8 I 0.9 I1 I t FIG. 3. 53: THE NORMALIZED ENERGY DENSITIES Ne AND Nh vs. RADIAL DISTANCE FOR EM WAVE (e POLARIZATFON) INCIDENCE AND SHEATHLESS CASE

MISSING PAGE

194 -catet' ElK e-nergy. A standing wave of Np is produced by interference bep A\Vt 't;-: ihK i.ciaent and SL-iltkred EK wax-es on the other hand since the cylinder.iam;eterl is aifl)o3t 4 timies the EK wave length a;nd the scattered fields are not,leg igi})le. WNXhitn the coupled energy density terms are considered for the shea.tiess (ease it is seen — hat N falls off more rapidly than N with increasing p. This h p is catlsed by the -'ifftei ct' in the cylitlnder radcius to wave length ratio of the two wa\-es, Cs iilc(Ce the lar.ger1 this ratio, the more nearly the scattered fields resemble those produced by a planar obstacle. Conlsider 1now0 figures 3.55 and 3.56 which present results corresponding to fllose julst giTven but for the inh(omogeneoLus sheath and the hard boundary. Again, the Boi'n.Ipproxitnation is seen to be satisfied, for e wave incidence, as shown by figure 3.55 --- — unity throughout the sheath. The coupled energy density N 1now e.:hibits 1an- oscillaltory variation with p in the sheath, whereas in the P sheathl1..-: case it w1as monotonically decreasing from the cylinder. This ill ustr;ies that there is now coupling within the inhomogeneous sheath, since the standing wave palttern requires energy to be propagating in both directions in the sheath. Apparently as a result of the inhomogeneity coupling, the coupled energy density NI is abo-it 4 timles larger than it was for the sheathless situation. lWhe ni we losk at the results for the incident EK wave in figure 3.56, there are obsl, red to be standing wave patterns in both Nh and N, again showing the effect of i ihomogueneity coupling. It is interesting to see that variations in Nh and NP co-inccide quite closely in the outer part of the sheath, but that on approachp

I:, ' I-.:". -r ' rigy - ensities to same scale, multiply I-x v * 45) 1i) ( I ' —A I4 I I i I Z-p}wr-QSw f 1 Gc T - 10 K e N 0. 7 c -- 0 2 cm i = 7 X- 20 = -5.34 Volts - c M = 2 Ne p e N -2 - -9 0 - ^\ \ - \ I Ii -i* 0. 6 0.,7 o.8 0.9 > 10-, 0. I FIG. 3. 55: 0 2 0-3 0.4 I THE ]NOPRMA[LIZED ENERGY DENSITIES N AND N vs. RADIAL DISTANCE IN SHEATH FOR EM WAVE (e P POLAFIZATION) INCIDENCE AND rNHOIOMOGENEOUS SHEATH WITH HARD BOUNDARY CONDITION

MISSING PAGEN

19'7 ing the cylinder surface, Nh increases in nearly the same way as for the sheathless case. T1his ntdlicates thlat the magnetic field near the cylinder is principally determin2ed by the contribution from the boundary coupling, while farther out in the sheath, the ',1,.,,, iiy coupling effect is predominate. The increase in Np upon Hearing the c(5lill tK' \\ utild seenin to be clue to the fact that the potential p energy density is inversely proportional to the static electron density, which of course becomes smaller with decreasing p. Since the increase in NP is on the order of the decrease in static electron density, the implication is that the dynamic electron density is relatively unaffected by the sheath. This is a question which is considered further in the following. Finally we direct our attention to figures 3.57 and 3.58 which show the results firom the illllil:' 'i, 'LI- sheath and soft boundary condition. It is strikingly evident that the soft boundary condition plrduces field variations in the sheath quite different from those obtained with the hard boundary especially for the EK wave incidence. In the case of the incident e wave, the difft'ttrence is not so great, and there is again - L.b i tia i ilonl of the Born approximation. (Actually Ne may become less than 1.0, on the order of 0.99, a difference too small to show on the graph). The coupled energy density N however, is now about 400 times p larger than that for the hard boundary. It is nearly the same at the cylinder as that produced in the sheathless case with the incident EK wave, but decays more quickly away from the cylinder, with more cycles of amplitude variations. The situation where the EK wave is incident is characterized by extremely large increases of Nh and NP with increasing radius contrary to that shown by the Z1h p

IO - '., NT'7, N "7 P-_ h N NCTo trainsform- energry densities to same scale, IIL inhiply N0 byVv, r p ~ 450 10 0 N 10- I / Ir I I ", - I I11-4 " I 4 Ii I Ii i I i i -1 3 i i I t 10-2 I --- I 4 II I I I 9 4 1 1 4 P III II I I i I" I i i # N e p e h flGc 10 4T lo4K X=20 NO0.7 = -5. 34 Volts c 0.2 cm M 2 0. 5 0.1 0. 2 0. 3 0. 4 0. 6 0. 7 0. 8 0. 9 FIG. 3. 57: THEP NORMALIZED ENERGY DENSITIES SNN v.RDADis IN SHEATH FOR EM WAVE (e POLARIZA ON) INCIDENCE ANDINHOMOGENEOUS SHEAT1H FOR THE SOFT BOUNDARY CONDITION -1 I4 -1 *1 -I I I1 I IAN

06 Cf) Cf) z 0

2 00 1;. I'd boutlndarty results. Both of the Ii.,rimllized energy density telrms now exceed unity at t He sheath -uniforlm plasma interface, indicating a large accumulation of energy in the outer x.'prt of the sheath. Again, as for the hard boundary and inhomogeneous sheath, NP varies in the same way through the sheath as N. Nowv \with the use of the soft boundary condihtion, there is only the inhomogeneity coupling mleclhanisml in the sheath to produce the coupled energy densities observed. And yet, as is shown in the precedinlg graphs, the coupled energy densities can exceed tho(.e which are obtained when the additional coupling nmechani:sm due to the boundary is included by the use of the hard boundary condition. It is further apparent that the soft boundary has much more effect on w than wh, which is reasonable since the dn-.r-c &ftrcsi F:rE- 't' more intimately p h i*sociati l with the form-ler, its effect on the laiter occurring only through the coup1ling. T his conclusion is based on the fact that for e wave incidence, wh is unaffected by the chlange in boundary conlditions, while for EK wave incidence, wh follows the variation in w p Tlhlis for the particular sheath model that has been studied, we find that complete abtsorfption of the electrons at the cylinder wall can produce effects which are substantially different from those obtained when the electrons are reflected, particularly in the sheath region, but which may also extend out into the uniform plasma [n order to investigate the latter, some calculations were performed using a potential variation with M=10. Thus the sheath inhomogeneity is confined more closely to the cylinder, and the numerical integration over the outer part of the sheath is effectively carried oat in the uniform plasma. It was

201 found for EK wave incihlt Flu that the soft boundary then produced Np values close p to those of figure 3.58 near the cylinder, but \\hich decreased in tilt. outer part -4 of the sheath to about 10 of the max il la near the cylinder. Some furl thIer calculations were carried out ewhich differed from the sheath model usled for figures 3.53 to 3.58 by incorporating a sheath 5 rather than 20 De thick. The results of the EK wave incidence on such a sheath are sho\\ for both the hard and soft boundaries in figures 3.59 a —. 3.60. These graphs exhibit variations of N. and Np in the sheath quite different from those for the 20 De h p I thick sheath. In particular, there is not the large increase in the energy densities that were found for the thicker sheath and the soft boundary. The overall picture provided by the results above is one which shows both the illhoollloeneous sheath and the boundary to strongly influence the behavior near the cylinder of the fields which are produc-ed by an incident EK wave. Of these two influences, the ^-<. crtrr^-i aominates the behavior of the dynamic electron number density in the sheath. The soft boundary condition leads to increases in the dynamic electron density \\ ithl increasing radius, while the hard boundary hals little effect on the dynamic electron density in the sheath. Significantly less effCect is found to be exerted by the sheath and boundary on the fields arising from e wave incidence. This is particularly true of the magnetic energy density, which was ~fnid 7<.r- > s:; 3 - that of the incident wave, indicating relatively little energy traLnsfe r to a scattered EK mode. The conclusion that the sheath ald t the copressibility of the plasma have negligible effect on the scattering by the (ylinder of incident s plane waves, which was previously

L U 202 P p To transform energy densities to same scale, multiply NP by v v -450 P -r N h 1 — j; 1 -4 I { i — i I I 101 100 F i ---, I-T~ i I E^ f= I Gc 4o T = 10 K e N=0. 7 \ N N~C K,: =0.2 cm I - 7 611= w 2 X = 5 "N N M = 2 N. "I., = -5. 34 Volts 0C 1 iz ^ I t I iI I I - t 10 3 fl3 1 11 In p NP ^ ^ _ _ ^ _ _ r i II I i rI i (p-c)/(s-c) -2 10 II i C0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0,9 L__ ~L__ L.___Li___LL _LL t FIG. 3.59: THE NORMALIZED ENERGY DENSITIES NP AND Nvs. RADIAL DISTANCE IN SHEATH FOR EK WAVE INCIJENCE hAND INHOMOGENEOUS SHEATH WITH X = 5 AND HARD BOUNDARY CONDITION

I p 10 Nl p To transform energy densities to same scale, multiply Np b v v -450 p -, -~ r — 1 I --- 1 10 -i 10 t Pi i tI i 4 1-2 10 -_ N p P p Nh h f = 1 Gc T -= 10 K e N= 0. 7 c=0.2 cm 0=wr 2 X= 5 M=: 2 = -5. 34 Volts c (p-c)/(s-c) 0.2 0.3 0.4 5 0.6 0.7 0.8 - 0. 9 10 FIG. 3.60, THE NORMALIZED ENERGY DENSITIES N AND NP vs. RADIAL. - DISTANCE IN SHEATH FOR EK WAVE - "INCIDENCE AND — INHOMOGENEOUS SHEATH WITH X=5 AND SOFT BOUNDARY-CNDITIrC

204 reached on 11(i basis of the surfa lce currents, is thus further reinforced from lh1 energy density results. A physical interpretation of the - - — 'r T cumlulation in energy density in the sheath is that the cylinder and sheath have a focusing effect on the incident EK plane wave, causing enerogy corieilt rations appreciably larger than those found in the incident wave. Another way of looking at this is in terms of the effective area of the cylinder and w.alulh, which may become much greater than the physical area, as evidenced by the energy accumulation. This picture is llso true oC - rc ftc — uwL, as the effect of the E-ylinder on the plasma may extend beyond its physical bounc' r-, We have as yet made no quantitative statement about the relative conversion of EK to EM energy and vice versa. It is difficult to give a definite answer to this question on the basis of the previous results because of the energy density,- effect. This is especially true of the incident e wave. Thus there e is seen in figure 3.57 a situation where N exceeds unity while N is nearly unity, p h indicating that a relatively small portion of the total energy has been removed from the incident e wav-. It appears obvious however, that a small proportion, certainly no more than 1 percent of the incident e energy is converted to scazt-re'e ETL energy, since the magnetic energy density is in all cases within 99 percent of that in the incident wave. On the other hand, when the incident EK wave is considered, it seems reasonable that since it is more.11lk. t.' Il by the focusing action of the sheath and

205 p p cylinder tlhan the EMI wave, a direct comparison of Nh and N should give a h P llenit;LllnIgfl m-,leasure of the conversion efficiency. Pr;, c.t li,, on this premise, it may be \erific 1! fron- the graphs of figures 3.54, 3.56 and 3.58 that in no case does the normali et'i I;glOi.-tic energoy density represent niore than 1 percent of the normalized putt tial energy density. What is especially interesting to observe is that,:.t the surface of the cylinder, the largest value of Nh is obtained in the shejAthless case, with the i-mallest value occuring for the soft boundary a.nd. the inii1omogeneous sheal;th. This has of course been previously commented on in the discussion of the surface currents. By way of contrast, at the sheath-plasma interface the situation is reversed, and Nh is a maximum for h the soft boundary condition, as a result of the rapid increase of N h with increasing radius. This situation is somewhat perplexing, but may perhaps be explained by the following reasoning. A pronient feature of NP in figure 3.58 is that only one maxima occurs as a function of radius and that is located near the cylinder. As a consequence, it seems logical to conclude that the incident and reflected waves of potential energy are of about the samne amplitude only near the cylinder. Since N increases on h approaching 1he sheath interface, a further conclusion is that it is the outward traveling conmponent of NP which is predominate in the outer part of the sheath. This is based on.issu:mlinig that the energy converted from potential to magnetic propagates in the same direction as the original energy, so that if the inward travelling part of N were larger, Nh would increase on approaching the cylinder. If the proceding logic is correct, then there must be regions in the sheath

2 06 where the energy flow is predominaniitly towards the cylinder, to supply the flow away Fl'-,i the cylinder in other areas. A complete picture of this phenomenon \\u tl(l require mlapping the power flow over the entire sheath region. The calcut i O.-; '.hich i ould be required to acc u 1mpl)j ish this are beyond the scope of the present investiga t ion. 3.2.5 An Examilntion of Pussibility of Ordered Entergy Absorption by Sheath A highly significant feature that appears in this analysis is the implication in the Poynting"s theorem given in equation (2.32), that there exists a possibility for a gain or loss of energy in the sheath, by the incident wave. As was exaplained in connection with the derivation of this equation, we use the term loss to mean a net decrease in the ordered energy content of the plasma, varying at the frequency of the incident wave. Gain indicates 11i,- converse, a net increase of the ordered energy content. We shall in the following use loss to cover both eventualities, since gain can be described then as a negative loss. Used in this context then, there is no loss or gain of energy associated with the production of scattered EK radiation as, a result of an ilncidentl EM wave, but rather a conversion of the ordered energy from one mode to another. The implication for a loss of energy by the incident wave in equation (2.32) lies in the interpretation of the imaginary parts of the equation to represent stored or reactive energy and the real parts to represent real power flow. Thus a non-zero real part of the left hand side of (2.32) shows there to be a loss of power in the shelath since the net power flow across the sheath-uniform plasma interface is not zero. We are thus interested in examining

2 0T (3.15) There is no -,i r.,,1 integral over the cylinder surface since the boundary cui'ti-.tions there make the power flow into the cylinder eqlual to zero. Since (3.15) has been obtained usig an lloutward pointing surface normal vector, we interpret a 1minus result fl0rom1 the integration to indicate a loss of power to the sheath from the incident wave. It should be noted that the right hand side of (3.15) provides the loss mnechanisml for extracting energy from the inc.dent wave, while the left hand side merely provi ides an inflow of orderied energy into the sheath to balance the losses which occur in the sheath. The term on the right hand side of (3.15) is thus a cause and Ihat on the left ain effect in the cause-effect relationship between energy loss and power flow. For this reason, while (3.15) requires both integrals to be numerically equal, we refer to the right hand side as the loss term. The loss term was investigated by numerically integrating the power flow term on the left- side and the loss term on the right side of (31$) respectLvelv ovr t- te" st-c.'>. vi 4 olume. It is unnecessary to integrate both sides of the equation, since the same answer should be obtained from each. This however provides a check on the consistency of the numerical results. We e;xpress the shc-. -h IrS: -er -:t c' C r.: — r length by forming the ratio

2 08 of the net power flow across the sheath interface to the power flow density in the inciclent plane wave multiplied by the cross sectional sheath area per unit (3.16) where either V or V - 0. less thall one indicates that the sheath's p e equivalent cross sectional area for absorbing the incident wave energy is less than its physical area, while greater than one indicates the converse. These calculations were performled for several sheath models for both, aLd e wave incidence. The results;s;lho that for e incidence regardless of the sheath model. Simnilarly there is found for EK wave incidence, and the hard boundary conidition also. These numbers are too small to be of sigHnificance and most likely represent the limitation on the computer accuracy. They are useful in that they indicate the sheath to be lossless for practical purposes in these palrticular cases. On the other hand, when the soft boundary condition is used for EK wave incidence, there is a very significant.Lit'ffer' ice in the results, and is found to hae vues which range from 10-3 to 0. was found to vary found to have values which range from 10 to 10. was found to vary

2 09 with she;th thick'ness. cylinder radius and the value used for the exponent of the potential distrilbution in the sheath. The signific:ltllc.- of this result is discussed below. Sonme ilnte.esting 'it.- 1 t s were,l i otined by clculllatilng both the real and imal;,l..:, l parts of the volume integral appearing in (3.15), which is (3.17) -6 The real part, to which the sheath loss is proportional, was found to be 10 or so, of the imnagi nary part for EK wave iHncidence and the hard boundary condition. Use of the soft )bou1ndary condition made little change in the imaginary part but increased the real part by a i i. 6x ordelrs of l tignlitude. This means that in the case of the hardt boundary the power flow across the sheath interface is almost wholly re- Lactive, and the sheath volume serves primarily to store the energy from the i nc ident wave. The imlportant point here is that the ratio of the real to the imaginary parts of (3.17) depend:sll on the boundary condition used. Since the real and imaginary 1 il u of (3.17) are obtlainied by simriy rearranging the real and imaginary parts of the product nv, the change in this ratio with boundary condition appears to be numerically significant. This is particularly so because the change is larger in olders of ma-gnitude than the number of significant places which are accuLrate in the calculations. The possibility for a loss of energy in the static sheath appears thenl to be a real one, rather than merely resulting from uncertainties in the cal

210 ullations. A moiL' thorough invtestigation of this pic.'olem would have to be undertalken belor-e) cdiefiniti (e answers could be obtained, an analysis which was beSyond the intendled scope of this study. Soime iresults obtained by Pa- xovich and ]ino (1964) are relevant to this disc zssi)on. A I heoretical formulation of the variation of rf fields in a plasma sheath for a plane boundary was carried out based on an integration of the collisionless Boltzir ann equation. An integral equation was developed for the rf electric field in the sheath which is related to the perturbed electron distribution function and the static sheath potential. The numerical results obtained from this analysis.1.Io indlicate the possibility for a gain or loss of rf energy in the sheath. No^v, there is no question of a net gain or loss of energy by the system as a whole, since the total energy must be conserved. This being so, it is necessaryr to look into this question nmore cartefully, to determine the possible sources or - i i. 1. of energy that may be associated with the cylinder and sheath. This is especially important since the present formulation does not show the other endc of the energy exchange mechanislm. There -may- seem to be some inconsistency in a system of equations which admits of a loss of energy -\ i lli out indicaLtingc thel new form into which this energy is converted. That this is not the case is illustrated by the interpretation of the loss term. in connection with EM wave propagto.n_ in a medium with a complex prl.1'itti\ ity'. A term appears in the derivation of Poynting's theorem which is related to the imaginary part of the permittivity, C, of the medium. This term

21 le:. ds to;i net flow of EM en lergy into the volllmit proportional to '. This is intt Apiet-led to mean that energy is lost fromi the EM wave propagating in such a e-,tlium, thr'oulgh the mec,:,isnism of the complex permittivity. It is informative to explain the loss of the ordered energy fromn the EM wave on the basis of there being in-phase loinponents of the electric field and current flow as a result of c. A farther analysis of the current flow in this situation leads to the conclusion that the ordered EM energy is dissipated as heat in the medium, and is thus randomized. Since the Poynting's theorem was derived to account only for the flow of ordered energy in the EM mode through the mediumn, then it is physically necessary that any transformation of energy to a form other than that in the given EM mode be manifested in a loss term. It is important to emphasize that the loss term cannot tell us what new form the energy lost from the EM wave will talke, although it, may provide a good i,'Iicut lii of this. That can only be established by formulating a Poynting's theorem which takes into account all the types of energy appearing in the system. It is instrluctive to examine more closely the loss term, given in (3 15), to see if some physical insight will provide a clue to the question of the ordered energy loss. We have previously obsc rved that this term may be regarded from two viewpoints. In the first, there is observed to be a current flow in the sheath, arising frora the ordered electron motion, nv, which is acted upon by the static sheath electric:field. Depending upon the relative directions of current and field, there may be a loss of energy either by the field or the ordered electron flow. The other viewpoint looks at the displacement of the sheath electrons from their

212 ejLiLibrium positi'.s is as a eLnge in their potential energy, again corresponding to a. gain or' 1,,.-. depending on the displacemlent relative to the field. In -itiler O:,f these two equivN7, [it viewpoints, there is seen to be a means of exchanging energy between the ordered electron motion and the static field. Noswe it has beenll an ass Limption all along thalt while the dynamic fields in tle sheath are depene lt0it on the static sheath, the converse is not true. It would seem that this asstumpltion may have to be re-examined. This is because the EK wave propagates in a homogeneous plasma by a process of energy exchange between oradered kinetic energy and potential energy I. lll 1 by charge accumulation. When the wave encounters an inhomogeneous egion such as tsheath, with a static electric field, the kinetic energy can tbe e:\chait ged with the static field as well as the potential energy of charge.tetumLulllti.o The wave therefore can upset the energy distrlibution of the static sheath. At first thought, it might seem that such an (effect would be inconsistent with the linearization of the original equations, but the same term can be obtained without recourse to linearization. There are two observations which can be made here. First, it is apparent that whetheri or not there is a net transfer of energy to the sheath is strongly influenced by the static sheath itself, since the overall effect is obtained by an integratlli in of (2.33) over the sheath volume. Second, assuming that there is such net energ.y transfer, then it must be concluded that the static sheath is indeed changed Dr perturbed by the incident mrave. The change involved depends on how efficiently the sheath is able to redistribute the energy it receives from the wave.

213. steady stte.-liould be attained where the redist1ributtiotn by the sheath of the {n er'gy receivecd firom the wave proceeds at the s-tInL ruIlte at which it is absorbed. Thi- v aty in which the sheath mighl redistribute this energy may be contcCtured to occur in two ways, one involving the ions and the other the electrons. We hjave discus-.:edi in connection with the static sheath the imotion of the ions under the influ:ence of the accelerating potential in the sheath. The ions are the recipients of the kinetic energy lost by the plasma electrons energetic enough to reaclh the cylinder. There is in other words, an energy exchange mechanism at \\ ok intthe static- sheath which leads to the kin tic energy of the electrons being tra -ns.fe' red to the ions. Ultimately, this eierg-y reappears in the form of higher te Inlperature neutral partticles which result from the recombination of ions nlld electlrons at the cylinder wall. Thus, an increase in cylinder potential ) brougtt about by the absorptionl of incident wave energy by the sheath could Eresult in raising the temperlllplatLure of the neutral gas particles. An alternative to the mechanism discussed above involves the interaction of tl:.e plaslma electlrons with the perturbed sheath. As the individual electrons move through the sheath they are subject to the local potential. Wi-hen they come lunder the influe nce of the perturbation in potential caused by the incident wave, their patlis are altered in a different way than would be the case in the unperturbed static sheath. If the electrons are moving faster than the fields of the -ir iden t,\ axe, t hey can, as a result, move into areas where their dynamic mnotion does not match that of the wave fields, thus leading to a decrease of:or-dere'd energy. The important point is that the sheath field provides a mechan

214 ism for redirecting the electron motion with respect to the incident w7ave, or in other woirds, it effectix ely brings elec.- ron1 coll isions into the picture. An inherent limitation of the hyd-r'odynamic a-tpp'roach is that tlt. electron nmotion is av eraged so the theo-ry c-iannllot show effects on a microscopic scale. It becomes apparent here also, that our neglect of the heat flux tensor in the plasma cannot be strictly correct when there are losses, for the ordered energy lost in the sheath by the wave must be removed from the sheath by a heat flow arising from a t.emnperature gradient.

CHAPTER TV CONCLL[S ONS AND REC OMiMlENDATIONS FOR FURTHER STUDY -. 1 SIm 1nar:ndo Cnlrni ( lsions A lhb oretical i!-nvstigation of the surface c.urrents excited on a plasmainers'I,d et:'l e; lilund by incident )lane elect ronmi1gonetic and electrokinetic x,:a vess h b-s leen carried out. The assoi:Ltt_,1 problem of the static plasma shi:etll ha-ts a1so been examnined. For lnormal incidence, the actual inhomo-;eneus plasma sheath hLas been accounted for, while for arbitrary incidence, the sheath ihas b1een replaced by a free space layer, called the vacuum sheath. It;i. -: been shown that the static electron, locity in the inhomogeneous sheath sIurroinding the insuhlted cylinder has a negligible effect on the static electron densit' variation in the sheath. It was "tso shown that the effect of the ti tic electron -velocity on wave wl'o)a:itio within the sheath is negligible in coun)xpr tison.'ith the effect of the sheath inh-l, mogeneity. 'T'h. mlost significant findingl of thle \;.ltiu;n sheath analysis is the large eDtten-la tion,.lich -imaxN be caused by the sheath of the surface currents excited on thle ( Uinder by:-n incident electrok.inetic wave. The attenuating effect of the sheath- con the c-urrents can be as great as 60 db compared with the currents;.hel the:re is no sheath, lltpendlii on the angle of incidence and sheath thicke},ss. 'I ir i'tlr'ents excited by the electromallgnletic wave on the other hand are found to b:,e unaffected, for practical purposes, by the vacuumi sheath. In additionl. the compressibility of the plasma iLls'l has little effect on the currents due to the electr)omagnetic axe. Thes esresults for EM wave incidence are based on cylinnde rs with radii small cocp. 1'edC with the EM wavelength, the case for v, hich theltl numerical calculations have been performed. 215

216 An analysis of the inhomllogeneous sheaith for normal wave iiicidtfice ilrther cdemom)strates the negligible effect of the sheath on the elect romagnetically indLuceed cut rrenlts. The currents cdle to the electrokinetic wave for the hard buIL<'rx condition iand an inhomogeneous she at 20 D thick are found to be in good agreeent A. ith the results of the vacuumL sheath analysis for a sheath 5 D thickl. This result ca:n be taken to in(licate the credibility of the vacuum sheath a.na,-lsis for oblique wave incidence. C0 nsider.-tli n was also given in both analyses to the effect on the surface currentsE of the boundary condition on the inormal electron motion at the cylinder surface, as represented by the surface admittance Y. It was observed that B when Y 0,. co_'rcsponding to a perfectly reflec'ting or hard boundary, there are contribuions to the surface current -excited by the electrokinetic wave from both the boundary and sheath inhomogeneity. The other extreme value of Y = o, correspsonging to the soft boundary, results in a surface current due to the sheath inijlmog,:-neitv alione, the boundary contrlibutionl then being zero. It was found that the hard boundary was more efficient in contributing to the surface current than the- sliL-alh ilnhomogeneity, for the inhomogeneous sheath models investigated. We may briefly summarize the results of the surface current calculations with the following remarks. (1) The sheath and plasma compressibil].ty can be neglected when the scattering of electromagnetic w/aves from a cylinder which is small in diameter compared with the electroi'magnetic wavelength is considered.:Calctjla.tions performed after the writing of this thesis show this result to hold for a cyli nder with a diameter of 4 (free space) EM wavelengths.

217 (2) Thei electrokinetic -..;\,- is screened from the cylinder by the sheath, Avith the screening effect inlcreasing as the angle of incidence mealsulred from the cvlinder axis. is decreased. (3) T'he elec troklinetic wave is less efficient than the electromagnetic wave in exciting surface currents even when the screening effect of the sheath is not taken into:;ce.lut, by a factor of 10 to 500. It thus apj)ears that detection of the elt-ctrokilnetic wave in a background of electroina:l'etic radiation b mieasuring the.>irrf:ilet currents would be difficult to accom plish. A further significant finding of the inhomogeneous sheath analysis is that the potential and nmagnetic energy densities in the sheath are strongly influenced by the value of Y". In particular, the soft I.ullrlndjary condition leads to values of B potential energy density in the sheath much larger than that in the incident electrokinetic wave. This indicates that the perturbing influence of the cylinder on electrokinetic \\ave propagation in the plasma extends far beyond the cylinder's physical boundary. In addition, there is found to be the possibility for the loss of energy from the incident electrokinetic wave in the she ath when the soft boundary condition is used. When we compare the results obtained in this study for the scattering of electroma-gnetic waives from a pIrllll imlllnrsed cy'linder with the results of Parker et. al. (1964) for the inverse pr'oblem, the scattering of electromagnetic waves from a plasma cylinder, an interesting observation can be made. For Parkers' problem, both the plasma compressibility and sheath effects are required

218 to t'ccoi'tlt for th(r resonances in tl:hi scattering; properties of the plaslma cylindc-r, h'.1er.eas th]eir effects are negligille i-lc nerning the scattering properties of a plasma it mmnersed cylinder \which is s aIl] compared with the t-1, (ctl*,laiil:netic:. velength. This implies that the tre'itrl-ment of electromagnetic wave sc attering fron-l obstacles in c* inttact vwith pIlasnmlas requires allovwino for inhomno-aeneity min(1 colnpressibility effects in the plasma only if the geometry is such that standing electrokinetic waves mayn be excited in the plasma by the electromagnetic vLave. It is interesting to observe that the production of evanescent electrolimagnetic wave's by electrokinetic waves incident on a plasma discontinuity has been previously pointed by Kritz and Mintzer (1960), Tidman and Boyd (1962), Cohen (1962b).1-' Fedorchenko (1962). The significance of this as far as screening the elcttrL. in-tiC wave from surfaces bounding the plasma has not been ment ioned howe er. Finially, we should inquire as to the possible significance of the results obtained in this study in relation to the problem of the plasma immersed antenna. Trit analysis of Cohen (1962b) predicts very large effects, due to the excitation of the elcectrokinetic wave, on the radiation re;sistlance of a linear filamentary current source in a plasma. The present results, as mentioned above, show h(ox ttver that the electrokinetic wave has little effect on the electromagnetic scattering propertiei s of the cylinder. We,can infer from this that the effect of the electrokinetic wx ave on the impedance characteristics of the plasma immersed cvlinderical antefnna would also be small. Wait (1965) arrived at this conclusion for an 1antenna large compared with the electrokinetic wave length.

219 4 2 e.cl;l i: id: tins for Furitll,. StLudy The results obtuinled in tlhis stud'- sugtgest some areas for furlthier- inv'esti — gati.on. 01(. of the imost vital points which needs additional effort is the boundary conditic-n on the. e lectron Io,)tion a:t thlu cylinderl sulrflace. This is a Lquestion the 1inr'tarlc of which ha:s been recognized for some tili-, but little clarification i ncr rnt i'i r, it h:as been accomplished. It appears that it would be worthwhile to carrv out a paramlletric iluv on the surface admittance for the electrons, YB B concerning its effect on the surface currents and the energy distribution in the she ath. It also would be valuable to extelnd the irdonmogeneous sheath calculations which havre been performed here to the case of oblique incidence. This would be especially illmpo)rtant to furL ther verify the screening effect of the sheath as obtained fromi the vacuum sheath analyisis. Oiit.iddii iorial signific. nllt subject that Should be studied is the radiation by a ctx lindrical antenna inmmersed in a plaslma. It has been conjectured here that, contrasted to a current filament in a plasma, a physical cylindrical antenna imimersed in a pl;lasma would not strongly couple to the electrokinetic wave. Before this can be accepted without reservation, a careful analysis of the radiation problem for the plasma immersed cylinder must be performed. Wait (1965) has recently begun an analysis of such a problem which does indicate the electrokinetic wave to exert only a minor influence on the cylindrical antenna. His treatment however neglects the inho trc,gunl.1.utlS sheath. The inhomogeneous sheath should be taketln into (account before any definite ncli uCLsioSkins are drawn from such an analysis.

220 Finally, a theoretical analysis of the exMleriment proposed in Appendix I) sholcd i:e perfri-ed, bzsed on the form.ulation of tlis study. Numerical results obtained from such an CanaNlysis should indica te the likelihood for this experiment to be carried to a successful con'clusionl. If the indications are indeed that this is the case. the performance of such an exper iMent would seem to offer an attractive -e: ns for checklil^ the validitv- of the theoretical formulation.

APPENDIX A ANAL'-SIS OF] ELECTRON NUMEER DENSITY IN STAT'IC SHEATH An ana-lysis of the static shea.th probltcm is basically one of finding the potential in thle sheath via Poisson's equat aion an(l integrals over the velocity of the electron:tnll ion distribution functions. The difficulty in accomplishing this is determ-ined primarily by \wThat:isl-t111p I il-;s one is willing to make about the ion and electron dis:ribution functions. Addlition l1 complications arise when electron generation is allowed for. Finally, the geometry of the plasma configuration, whether it is an internal or external problem which is being considered, also has a -trung influence on the complexity of the analysis. The intent here is not to solve the static sheath problem. Rather what is desired is to imake use of some of the results previously obtained in order to specialize them to the particular probleim of Ihe plasma-immersed cylinder. For this reason, l e potential varia iion in the sheath is not solved for here, but is assumlt.d to be given by (A. I) The experimental iwork of Gabor et al (1955) and Harp and Kino (1964) confirm the validity of (A. 1) with M 2, while the theoretical analysis due to Self (1964), La Frambois (1964) and Eernstein and Rabinowitz (1959) indicate that M is about 4. 221

2222 It is anr eZasy matter h1L'!_c to fi:',1_ the static sheath electron density froml (A. 1) and using a Maxxwellian electlron ve-locity distribution with a superimnposed sl:itli drift velocity given by -, (A.2) whiere ' (A.3a) (A.3b) (A. 3c) v is the static electron drift velocity toN-iart:s the cylinder caused by electron-o ion recombination at the cylinder wall. N (p) is the usual Boltzmann distribution e for particles in a potential well. The quantities which we want to find are the electron density and velocity in the sheath. An integration of (A. 2) over the velocity between the appropriate limits w7ill yield the number density, A hile a -.i ilhir integration of f u will give e — the velocity. The integration limits on u are -oo and u where u is the e m m truncation point of the distribution function, determined by a deficit of fast

223 t-l.-f r. 1I. refl-C-eting from the sheath. (This iLthlod will be referred to as tht i 1 ',': ii,, appro al ) If the.-:lI;,ttll is thin conlpared with the radius of curvature of the cylincler, so that at any p.iii-i, in the sheath all electrons which ihave sufficient rr;dii.l velocity to reach the cylinder do so, then u is given by (A.4) When tile integration which leads to the electron number density and velocity is carried out, there is obtained (A. 5) (A. 6) where (A. 7) and erf is the error function. By way of comparison, the result which was arrived at for n (p) from the integration of the Boltzmann equation for the electrons is, as shown by (2.18b)

224 (A.8) T^>-e sa lie velocity in (A. 8) '-.ulcll be obtained from sol\ ing the set of t-quall ionl:; (2. 18) \\hjere v is r _.la~-.d to n by (2.13d). eo eo We now have available two expressions for the electron density in the:i;:, '.,;'ii if the two approaches are consistenr, should produce the same:restt1. Lanw (1964) pointed out that (A.8) is an accurate representation for the electron number density in the sleath provided that few electrons are lost to the boutndiing;'ulrface, in this case the cylinder wall. Self (1964) on the other lt'tn1 a:rrived at (A. 5) from basically physical reasoning; however, he did not in1clnlI( the static electron velocity in (A. 7). Both expressions obviously are tI sare sa, when tiere is no electron flow to the wall. A different analysis based on the orbital approach used by Bernstein and RaNbim:-!tz may also be instrulctive for compalrinlg with the preceding results. Th1._a.is of I -ie for mllilation depends on the fact that when there are no collisions, thie Zgeneril S -,lLition of the Boltzmann equation is an arbitrary function of the constants,of the motion. By assuming the form of the distribution function at an infinite d stance from the body and with a lmnow]edge of the reflective and absorp — tihe pro(erties of th,_- body, the particle density and flux can be obtained in terms of an inLegration over the constants of the moLion. r'h,. es: et ial difference of the Btiernste,_in,,i'1 Rabinowitz approach from that of I oLiMU ir lies in eliminating the velocity co-ordinates and using instead

2 5 hoSe con.stants of hIie motion in the ingalatll;-Ls wliich give the particle nunlmber density and flux. In additioni there was no assumiption that the static electric field is: illf: cid to a well-defined llhcthi letgion, When the radius of the body is large te'L Ln1 Lh so as to preclude the L. -.- iI lli y of trapped ions, then the loss of panj' jitlS 1.0 tflh. body is accounted for by the appropriate integration path in energy- angular mnomentumrn (E. J) space. This integration path does essentially B1;' thI tri;L, iio n of the distribution function in the preceding calculation a.c:,-r-pl]ishedC. It exclud(es those electrons whlich reach the probe from being i clT iec,. in the integration of the outward-travelling electrons. Follo\x ing Lam 's(1964) development of the Bernstein and Rabinowitz theory ')r electrons in a spherical geometry, we have for the cylindrical case (A. 9) 4-C whe re E andl E are electron energy components associated with electron motion parallel aid perpendicular to the cylilder and J is the angular momentum. J and 2 are given by

9226~" (A, IIa) (A. 1lb) nu" p ti.cle flux is (A. 12) (A. 13) Al.ter p~e r-foxm11 kg-, the inruegrat,'ions over J and E,and with some si-mpliJitzriii01 flAiers is ob~tu-inced (A. 14) (A. 1 5)

227 (A. - a) (A. L f Sc-nte observations are required before the numerical solution of (A. 5), (A. A8. (A. 8). (A. 14) and (A. 15) are obtained. First, the same potential variation giv2-en by A,, 1) is to be used for all three methods of obtaining the electron nuavl)er cd.nsitcy Obviously if one could find the potential from eaetn method of tni;.tlysis. ihe r;.sullt obtained in each case would be expected to be different. tHoweve(r, in Iusing the.sai me potential for these calculations, the relative consistelncy of tih three analyses can be checkec. At the same time an indication is oiveni of the a 'l-lllenl. whichi might exist between the potential solutions whicdh could be obtained if one solved for the potential from each analysis ':1Il. ' than. as:Limu:ng its form. Second, in finding' the nmbler density from (A. 8), the static N veloc ity will bI e rusedl from (A. 6) rizt i Li than solving the transcendental equation (2.. 8d) for it. Tlhird, the static velocity (A. 6) contains a term which is the real elecwtron d(rift velocity in the plasnma and for which a n u. Me I- i c1 value is required. Now at ihl. - s-he-athl edge, and for that mlatter all through the sheath, the elect:ron anm: ion currents are equal. Since the densiti.es are also approximately the same at the e,cdge of the sihe. th., thlen the ion and elt ct iron drift velocities are equal there. If we use the Bolhm (1949) sheath criterion, tlIlu the edge of the sheath

228 s i-the point w\h- re the ion drift velocity equals the ion rms velocity, or where tile tranm ition from,!l subsonic to supersonic ion flow occurs. then:1, the:-, aLth edge. This term will be 1:,il I to be constant throughout the sheath,, The results of the numerical computations for a sheath thickness of 20D1, wiith rm. = 1:i i 200 atomic mass units and M = 2 and 4 are shown in figures (A-1) to (A-4) as a function of position in the sheath. It is in:nleiitfly evident thlat the electron number density obtained by the three methods outlined. above is in agrereemrent with the BoltzmaLrn diistribution to within 5 percent over alnmocs t e tilt-:-l aih. Further, it mnay be observed that the Bernstein and Rabinowit zn method and the trLtncat ion analysis are shown as a single curve since thley agree numerically to swNithin 1 percent and cannot be separated on the graph. Finally we note i lhat the curve showing the results from the Boltzmann equation integration lies between the Boltzman.m listribution and the Bernstein and Rabinowitz result:s. When we::observe the curves for the electron velocity, there is seen to be remarkably go od agreement between the two models with the difference never exceeding a Factor of 2. Surpri-; i'15 the velocities are practically identical at the cyolinder Surface for both models regardless of the potential variation. This is \lJll:.iiw1 l by noting that the electron distl'ibution fuInction is one-sided at the surface. so that the electron drift -velocity there is dependent only on the electron

I n1 eco, Co 212, 9 (!cm, see) 7100 -I - I (-AI 10 0, ---iI i I I t LQ11% lb %t, "I -- -I *%. — l -41 I — 1-. I-) - 1 4 - --- - - - T-%,:.- 4- - 41-. 4- Z - - 02 1 z -4 f i -11.1I I "M* "I 10o6 I n eo n eo eo ~eo v eo 4oo C, a 0, 2 cm X 20 M2 3.06 Volts -BoIt o znriul t-ls t riu13uti on A Integ-rat i on of Boltzima-nnEquation Truncation, and Bernstein and Rabinowitz Truncation A nalysis analysis Bernstei n and Rabinowitz Analysts i I -I II II I -4 i i t i i i 10 (p.-c)/(s-c) i 0.5 0.1 0. 0. 3 0. 4 0. 6 07 0. 8 0. 9 FIG. Al: STATIC- ELECTRON DENSITY AND, VELOCITY vs. RADIAL DISTANCE [N SHEATH FOR M-2 ANI) -3.06 VOLTS

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0I C, C,

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233 i- lpe 'e at t.irC and not upon tle potential variation or the cylinder potential. The drift velocity at the cylinder is 0. 468 that of the rms velocity in the uniform T.ie close agreement which is thus obta-ined by the three analyses outlined is grait'ilg'. It J l ti.lic-s the use of the scalar electron pressure in the 13oltznlanm eq uation formulation since the results obtained from its use differ little from the other two analyses where the anisotropy of the electron velocity di.;;i ibiition function is taken into account. It should be realized that scalar electron p1ri- * u., may not be generally valid however, as when the cylinder is lbiased to collect more electrons than ions. 'he scalar pressure depends for its validity oi few of the particles involved reaching the cylinder, so it is a good azpproximation for the repelled particles only, which in this case are the electrons. This is why a determi'nation of the deasity and velocity of the attracted par icli s requires a more rigorous manalysis than indicated by (2.18). Further, -i!c'(: the Boltzmann distribution is so little different from the densities due to the other methods, it appears that the flow of electrons to the cylinder can be neglected as far as the static sheath picture is concerned. Consequently, the Bohtzmann distribution will be used to calculate the static electron density in the slh-ath in the s-ulbseqluent formulation of the dynamic sheath problem. The results for tile slatic electron velocity in the sheath are used in Appendix B to examine the influe'nc-e of this velocity on the dfTr1riC sh cCth?nealsisE

2:34 M - - s-P 0 -~ C S-C 0~ C 1 o Pl 0, (3 I N 333 0. 6' 05,1 0 41 0:3 0. 1 M=2 N, II *3 Mz-4 It >It MZ i033I (p-c)/(srt) to0 0.1I 0. 2 0 FIG1(". A. 5: STrATlIC POTENTIAL VARIATION IN SHEATH vs. RADIAL DISTANCEK

APPENDEI B EL IMINATION OF THE STATIC ELECTRON VELOCITY TERMS IN THE DYNAMIC SHEATH EQUATIONS The lrc-stion, of the influence on the dynamic sheath solution of the static (fIectro)n velocity ';.is found, in Chapter II, Section 2.5.2, to reduce to an evaLuation of the ratios The reluie i'evMnent for the:ppxiniat ioni of omitting the static velocity from the,?-namic equations to be valid is that R and R be small compared to unity; 1L 2 the smallir their values, the better the approximation. 9 R and R2 are calculated using a frequency of 10 cps and the velocity 1. 2 obt)ained bhy the ]3ernstein and Rabinowitz analysis in Appendix A. Figure B. 1 shows Rt1:nd R,, as a fulihct in, of position in the sheath for M= 2 and M- 4 with pl.s-n: ions of 1 atomic mass unit. We see that the ratios are everywhere smaller for the larger value of M except for R1 near the sheath-uniform plasma in-iterfa:e. Tli[s occurs since R is inversely proportional to the gradient of the potcntiild, vwhic:h becc.,,. -, increasingly concentrated towards the cylinder surface for increasing values of M. 235

1, 0 'KitA _1,0 I1 ci 41o 10 K(Il 1', e N; - 0.7 o- -~ 0 2 cm 2 'l- -- - I - ---. - - 1 'r llit,W, (p -e)f(~ s -c) () 1 02? 0. 3 0. 4 0. 5 0.6(' io-4 0. 7 0- 8 0. 9 LO_ I 4 - I - - -. - 1 - 1- - 1 - 4 1. - -........ I FIGY. 131:. THUE R'ATrIOS H AND It vs. R.ADIAL DISTANCE IN SHEATH'1 I

23 The ratios are less than 0. 1 over 85 per cent of the sheath, with the a\ cl'rag vailue of illt largest 0. 058. Similar results are obtained for a plasma.,-ith an ion mn-ass of 200 atomic mass units. T'he omission of the static velocity term:s ifroal th. cdynamiic equations, in view of the fact that the contribution of these t( l is s i-mall in comparison with the terms which ar r etained thus serats to.o 1be a very reasonable appril il;lLiJI

A P'PE NDIX C DETAILS OF INHOMOGENEOUS SHEATH ANALYSIS The solutLion of the inhomnogeneous sheath] model would follow in a fashion similar to that of the vacuum sheath in t l if analytic solutions could be obtained to the eqalltionlls whlich describe..7- -- -'. inhoiCol ceoi.i.us sheath. The Atpplicat oon of the boundary conditions at tLu cylinder and the sheath-uniform plasma ilii tlcr, would lead to a Iii..tili\, which when inverted would yield the Fourier coefficients of the various modes prm 'li'-l.t i in in the sheath and the uniform plasina. Such a procedure cannot be followed when solutions must be obtained by,hlmerical computation. The reason for this is that the boundary conditions are exp ressed at two boundaries by certain of the dependent variables being zero, as in (2.52g) to (2.52i), or in terms of linear comblinations of the dependent variables and their derivatives, as in (2.52a) to (2.52c). Consequently, the boundary condtitions at one boundary alone are not sufficient to specify the values of all of the dependent variables at that boundary. Part of this information is contained in the boundary conditions at the other boundary. But before starting the numerical inttegra.tion of the differential equations at one boundary, the values of all the delen dent v ariables at that boundary are required. In contrast to the situation where analytic solutions are attainable and the functions can be evaluated at the bounlldry by using the appropriate argument, the numerical solution has yet to be obtained. The linearity of the differential equations allows the use of the superposition principle to -, ' -!!, -l this difficulty. One unknown dependent variable is set 238

239 at nivty, thus leading to values for the o1her variables related to it by the boundary conditions Land the differential equations, at one bountLdary. The:i'L.n;l i ig variables are then set equal to zero to start a numerical solution at this boundatry. An integration aci oss the, sheath to the second boundary is performed. This process is repeated for a different unknown variable equal to unity, and the process repeated as many times as there are boundary conditions Cat the second boundary. A linear combination of the solutions thus obtained is formed at the second boundary, and is required to satisfy the boundary conditions there, leading to a matrix. The required star-inog values for the dependent variabLes at the first boundary which were successively set at unity are proportional to the coefficients of the linear ct ibiiti o 't i which are obtained from inverting the matrix. At the same time the boundary c-inditions at the first boundary are satisfied by finding the starting values for.1 remaining variables through the boundary conditions there. To illuLstralte this procedure, we use the differential equations and boundary -)lotditiols given by (2.41) and (2.52). These equations cr^ be written for normal incidence as (Cia) (Clb) (Cic)

240 (Cid),rith the bouindLQ ary conditions (C2a) (C2b) tL p =C Lt tnd (C2c) (C2d) at p = s. N tt' tlaLt the prime above ilndicates derivatives with respect to p. S and S are ogi'en by (C3a) (C3b) Vie put the ecuation s into a more convenient form by transforming the real and imagina~ry3 parts of the dependent variables to Y1....Y as

2 41 Y 1 Y8 where thel subscripts r and i indicate the real of the depelidenl variables. Then with H-(Z) rnr (z) H (Z mi Q mr Q ml (C4) EF mr (p) nil mr (1) mi and imaginary parts respectively then (C6) where

I9 1) 2 t~, a,( 111 si ci i ffojvntia] eIcot ions to be so'llved.- The bip, condiltions are I]n ax Z t -c v, y3 4 0 t adc 0 (C7a) (Cmb) (C7c) 4 I 1 At p 5- sO 1)ZL axt with (2) H (KES) (2) C H (2)( s) m (2) H (K s) 1 -(2) S +is< S IT(2) (K- S) 4 - Y. - S. + H.. Y. (C~a) (C8b) n IId (C8c) (08d) (C8e) then i, jzl,... 4 where (C8f) R E - I~ R E 0 0 0 0 0 0 H - I 0 ID I -, R

2 439 If ii ' [e nurmerical integration is begun at p - c, some of the starting values i Y. a ndc Y are determined by the boundary conditions and the differential equai l oions (C6). But it can be seen that, which ever set of tle boundary conditions are ulsed,..espondi-, g to zero. ' -..: c" '.r.c -- t ve]ocity, not all dependent -variables Y. are specified at p = c. In the former e case Y 1 Y,, Y and Y are not known, while in the latter case Y1 through Y are 6 ' 4 not known. The information for determin'il those unlmowns is contained in the l,.uid-,ary condition at the other boundary, (C8), which cannot be used until the intt,!tL. tion has been perforlmed across the sheath. This then is the reason for the indetcl- rminancy of the starting values of the dependent variables. We overconme this obstacle to the numerical solution by setting all but one of the unknown Y. equal to zero at p = c, the one being set at unity. The numerical integration of ((6)1 is then perfolrmed \v itli this set of starting values, leading to a set. of Y. (p) and Y. (p) where a has the' value of the index of the particular unknown Y. with the value 1L at p c. This procedure is repeated for a new unknown Y. 1 1 equal to unity with the rest zero again at p = c, leading to a new set of Y. (p) and Y. (p). When the integration has been performed as many times as there are unll1;kowl Y. at p = c (or equivalently the number of boundary conditions at p =s ) a linear c ombination of these solutions is formed as Y.(s) = C Y (s) (C9a) (s) = C s) (C9b) Y'(s) = CaY (s) (C9b) 1 1

244 Then upoin using (C8e) there is obtained Ca Y. (s) -R.. Y (s) S. (C10) - 1 13 1 The coefficients C are found from()i inverting the matrix of equation (C10). The significallce of the C is that their nllumerical values give the correct starting values of the corllrespondiing dependent variables at p = c. When the ill,t l g i, I, of (C6) is then carried out for the final time, Jwith these starting values for the previousl lly unknown Y.(c) and the rest deternined by the boundary conditions at The actual numerical integration of (C6) was started by using a Taylor series expansion up to the third derivative in the dependent variables for the first step. The Rlunge-Kutta technique was used for the next four steps in the integration, in order to set up the va llIes for the Milne preIl (diclor-corr-ector technique which was employed forl the remainder. A Taylor series expansion was required for the first integration point since the first and second order derivatives of some of the dependent -ariablles was zero at p = c, while the third order derivative was not. The lurnble ' of inltegration points to be used was established after some experimentation. It was determined partly by comparison of the sheathless and ilhomlogeneous sheath results for the current when the cylinder potential was set equal to zero with an integration interval of varying magnitude. Also a determining fatclto here was the fact tht the Milne predictor - corrector routine provides infornm ation givzi ag the maximum lerrlo in each of the dependent variables at every integration point. A comparison of the error terms from the predictor-corrector routine and the results obtained from twice solving the same sheath model using 25

2 45 InI( th]en 50) integration points established the reliability of the errotr termns in C oI'r-ectly show\-in,_ the accuracy of the calculat: ons. As a result it was found that 25 intgttr-ation points provided an acceptable degree of accuracy, on the average ai u:inimun- of three significant figures. n otlher possible source of error in these results besides that arising fromn oe numeirical integoration comnes fromn the numerical evaluation of the cylindrical [unctions. The method used to calculate them and the relative accuracy obtainable is c'iscuss,ed in Appendix F. The accuracy of the inhomogeneous sheath.- omputations was not limited due to this consideration. F inall.. errors are generated by the matrix inversion required to find the C The inversion was carried (Jut in a doLuble precision routine on the computer, which provides an additional 8 significant places, for a total of 16. Consequently, the IUmaximum possible errors in the C are determined not by the matrix inversion. jw hich is accurate to about 8 figures, but by the uncertainty in the Y. due to the numerical integration. The overall accuracy of the solutioin was checked by calculating the ratio of the bn1i -:Ir y condcition (C8e) to the sumn of the absolute values of the terms ppearinllg in (C8e) at the conclusion of the final integration. If the fractional error in the largest termns appearing in the boundary condition is on the order of,. then this ratio should also be on the order of 6. It was found that this ratio -4 -7 was c(_]R istently less than 5 x 10 and generally averaged about 10, for the bouncary condition equations in which S. was not zero. In order to show the accuracy of the inhomogeneous sheath numerical integration,

246 ti.1 C1 is )- seintcd which gives ihe surface cLiurents.taillt-n. for a cylinder 1)otnt il,;l to zero. This reuLitL, —s the probl-1m then to the sheathless case. Acomprilson of figures Cl and s.3 7 show|s 1itt the currelnt from the!i i.' [it, i;:s seath numerical integration:ii that froml the vacuum s1healith analysis cannot be differet l i:, d to the limit of the graphical:t tcullacy. Actually, the currents bttintd from the two models in this case were found to agree to an average of three to four significant figures, after the summation of the Fourier series. This provides an excellenit verification of the at-cu CLacy which may be expected of thte numerical results obtained from both the vacuum sheath and the inhomogeneous sheath anal.ses. As a final check, the numerical integration of (C6) was performed for 0 0= and the soft boundary condition witii EK wave inlcicdence. This is equivalent to the sheathless case also, butl there is no coupling for this boundary condition and henlce no surface currents should be excited. The currents obtained from (C6) -6 were fouwnd to be( about 10 of those values obtained for the hard boundary. Consecque nt1ly, s ince the limitations on the computer accuracy keep the computer from obtainingl exactly zero, this result can be interpreted to mean that the overall accuracy of the calculations is on the order of 5 or 6 significant figures for this particular case.

b, - -- - =,A - -, - - Ca Q) b14M +1 C. I I- il -I, L6 -co, -, Hto.0 Oct2 > OM 0~ a) 0, U1) 0. 0. 0 0t"ztQ C% r P fl) tt -q k%. ' -.i4 -ft- 0) 6 4 — H Z q - N I I t 1 4 1 00 Co LO 0 — co I LI I

_APPE NDIX D A- SUGGESTED EXPE IMENT ONT THE ELECTROKINETIC WAVE It Aw,-ou d be desirabllle to perform an experiment for the purpose of iletermt'illi'g the vlidity of the thicoretical results; obtained in this study. Unfortunately, tie ipracltical problelms- assuciated N7ith conlductijng an experiment designed to coi)for),i- to the jflasmi l-cylinder geometry for which this analysis has been perf',rn'ed seem to preclude this possibility. This is pLrti, l.llly so because of t'le reqil'lement for an incident pltane EK wave. If we were to attempt generat — iig this \le by causinllg a p)LIne EM wave to impinge on a planar plasma slab it appears from this and other analyses: (Tidman and Boyd (1962), Kritz and Mintll.r 0l960) ) that relatively little energy could be expected to be converted to a plane E K:\ ivxt in the plasma. Consequently, the possibility of discrimi etillg betxteen the currents. -.'il d by the EM acndl EK waves on the plasmaiinmmersed cylindler wNould seemn to be unlikely, especially in view of the fact the thlieory iindicevates this to be a lmarlginal prospect even with equal power flow (densities in botlh waves. However, if we do nol require a plane wave incident on the Ipltsmasa-itinitersed cylinder at an arbitrary angle, but instead confine the e\p)ei lime ntal investigation to normal incidence, an experiment with more likelillood of success can be visutalized. We maty recall from the introduction il,.t Parka' et. al. (1964) recently performed aLn experiment i-nvolving the E scLttlt-ri ng of EM waves from a plasma crliinder. Their e xpe 'iment was motivated by the heretofore unexplained res:onutipncs in the scaltering properties of the plasma cylinder. The good 248

249' agreemlent \.hic- they 1 ta i led between their exper;imental observations and theoretical predictions indicates that an EK \7waN\e is excited in the plasma by the EM fieled with a 'l.illlm, iiamplitude at the resonance frequencies. In order to excite a single miode in the plasma. they arranged dipole and quadrapole excitation structures, parallel to and coincentric with the cylindrical glass envelope containing the plasma, thus effectively selecting only one mode at a time from the modal expansion of an incident plane wave. The;artn:llemellnt which Parker et. al. used would seem to be one which could be employed in our situation also. The only change to be required would be the addition of a metal cylinder along the axis of and inside the cylindrical glass envelope containing the plasma. By observing the resonances in the excitation circuit, one could determine the point of maximum EK wave amplitude so as to optimize thet possibility of detecting the surface currents which the EK wave would induce on the plasma-immersed cylinder. Similar measurement of the surface current in the absence of the plasma would provide a means of separating the EM induced current from that due to the EK disturbance. In addition, since the EM current would vary as cos mO while the EK current varies as sin mp, where m is the mode number and p the azimuthal angle, a determination of the angular variation of the currents would provide an additional means for separating out the two components. Finally, by usilng a metal cylinder whose diameter c is such that K;c <~1 and K c~?, then an increase in the ratio of EK to EM currents can be achieved by increasing the mode number m, since the EK current is relatively insensitive to m when Kp c > m while on the -P

250 other ha:!-J, the E M current decreaLses 'a.Lpp rox imntely as (KE c) This experiment would have two important aspects. First, it would provide a source for l[he EK wave in a well-defined geometry and the indications are fromn the present study that the currents would be largie enough to measure. The most critical point here v CMuld be the practicaLl one of how efficitfily the surface currents could be coupled to a transmission line which would carry the signal to an external receiver. Due to the magnetic field having a z -component only for normnal incidence of the EK wave, it does not appear that a coaxial line could be efficiently coupled to the surface current. However, by using a hollow metal tube cut with 2m axial slots as the plasma-immersed cylinder, where in. is the mode number of the exciting structure, one would have a circular wave guide that would be excited by the surface currents in the TE mode and which could carry the signal to the receiver. For the TEll ml 11 mode and a cutoff [requency of l.0Gc, a cylinder about 4.4cm in diameter would be required, but this size could be reduced by loading the waveguide with a dielectric. A diameter of this size is not unreasonable. A second feature of this experiment would be an extension of Parker et. al. (l1fi;4) work on the plasma resonances. It seems likely that the resonance spectrum of the plasma column would be modified by the presence of the metal cylinder, an effect which could be -_i.t ldied in the experiment outlined. There would be some further analysis required to that done in this study to predict the results of such an experiment. In this respect then, we are proposing to solve a problem different from that which we have already done. How

251 tver]. \\'lhat \\'e ciesire is a, cheek on the theoretical formulation which has been 'sed for tilu plresent study of the cylinder ill ersed in an infinite plasma lmediunz This awould be aLccoimp1lished by usinog Ihf- sanme forimulation for the problem which I>s >Llil':Gd here, for whichl the extperimenltall check could be made.

APPENDIY E DEV ELOPMENT OF VAC UUI SHE ATH FORMULATION AND APPROXIMATE SOL-UTIONS The s.luiion of ltne boundary concdiJion equations arisi!lg fri'c-: the vacuum sheath model [or the Fourier Lc-Lcfflci;t l.t of the scattered. tiransmitted and reflected ill:-s requires the irl.\ -sion of the matrix given by ecquation (2. 61)., A strai-ghtforward approach to performing this task leads to expressions that are complicated and lengthy and xwhich as a result co-nveyr little information. A close exanmination of the matrix reveals that due to the number of zeros in it, some of the coefficients can be easily found in terms of the others, so that the orlder of tihe nmatrix can be reduced a'id(1 its inversion consequently simplified. S R Proceeding in this way, the matrix w as reldulced to 3x3. with A. A m-p' m-h R and A appearing in it as lhe coefficients to be determined. The adv:rtiLtI, e rtr_- e of using the coe-fficients of the reflected d;i\ t' — is that ttiey are requirltred for calctlcltilng the surface currents and are relatedl in a sinmple way to the transmiiiTed wave coefficients wuichl are also needed for this purpose. The remainder of the coefficients are expressed in terms of these three as (E.la) (E. 1 b) 252

2 5 (E-lc) ' (E- L d) 1 L / (E-2a) (E-2b) (E-2c) (E-2d) 2 (E-2e) The idash.-Ibcript represents the kind of incident wave, p, e or h and the prlim,:.lerotes differentiation with respect to argument. The resulting three

S R c j- IO-It -s I or c It-rnii-l:, AS A a- p rn - e ', - A are tlen mn-h (E-3) (E-4a) (E-4b) (E-4c) (E-4d) (E-4e)

255 (E-4f) (E-4g) (E-4h) (E-4i) The terms in the source vector are (E-4j) (E-4k) f /.... (E-4e) The solutions for A A and A are given in equations (2.66) m-p' m-h m-e to (2. 69) and the surface currents in (2.70). The correctness of these solutions wvas establishetd by verifying that the original boundary condition equations were satisfied xvihen these results were put into themn. In addition the sheathless

2 56 modtl was treated as a seplarate prollemn,;iI1 the solutioLns obtained there w'ere comCIparedl with llt-c of thle vacuumn sheath when the sheath thiclrness goes to zero. The results obtained from those two diffc- ren( approaches were identical. The results obtained for the sheaihless case and EK wave incidence for arUitrary- Y are of value. They are Thy a re (E- Sa) - (E- Sb) (E-5c) where (E- 5d) and (E-5e)

257 We that RA n A become inversely proportional to Y as -mp mph B ---- oo, so that lthe current excited by the EK wave also is inversely proporioljnal to B. The same Jesuli is found for the cross-coupling coefficients for B e: 1,, h wave incidence which are not i,_Alui(,l here, We repeat now, for emphasis, what was mentioned in the discussion of (Ch-.IterL II section 2.6 concerning the radial] variation of the scattered fields. j3. ihe z-direct ion separa;tioi constant, is determined by the incident wave, so thati, the radial separation constants are obtained from - (E-6a) (E-6b) (E-6c) whllere (E-6b);inl] K. is the propagation constant of the incident wave. If in particular, K. =K p then.1

258 (E-7a) (E-7b) (E-7c) Due to 1he fiac that vt is generally much less than v, then whenever (E-8a) (E-8b) XE and X,, rlspec.-iively, become imaginary. This leads to the possibility for larlge shiea.th atren-a in when the EK, wave is obliquely incident on the cylinder, as will be sho',:nl -;^elow. Note that a.s 0 is x aried from normal incidence, AE p B first becomes inlaginary, followed by XE. This means that there can exist E o trapped Ln-ttenltaited EM waves in the vacuum sheath. Since v in this study r 7 has; been fixed at 6.67x10 cm/sec, and the nominal value of N is 0.7, then tlhe;ngi-lar inlerval for 0 in which this lphl.nolenon could occur is about 0. 05. For ang:les greater lhan about 0.18 from normal, both XE and kE are imaginary. i nl ih EM wave becomes a surface wave propagating on the sheathunifolrm plasma interface inl,- the cylinder surface.

2 )59 Wh-en cos6 < v /v iilhe in a fashion.i-' il:,l to (E. 7) we can wr-ite p P -r (E. 9a) (E. 9b) where 0E -tii.1 0E are tine "scattering" angles of the EM waves in the plaslma and vact iil slieath. Physically. they represent the udirection of the vector sum of the power fLow in the scattered fields in the radial and the z-directions from the front of the cylinder, i.e., the baclkward scattering angle of the EM modie. It is convenienlt to specify Lhe angle of inciclde-ce of the EK Twave in this situation by giving the angle 0E whiclh varies from 7r/2 to 0 ',, ii 0 varies from 7T/2 E P to 7r/2-V /vQ, and is thus easier to visualize than 0. r' " p The siluation is very different when K. = KE, for then (E. l Oa) (E. 1 Ob) (E. 1Oc) Since v /Kv << 1t then 4p Kp, and the scattered EK fields propagate away F/vI << 1, t he n 1 C.L tD,

2,,60 ITPO { l< cIli"nc r in a nearly normal dire cior. T.,ere is no possibility for the cxcd.it.'K:,. uf stul;]u' w\'aves in this case. -r:i - to check the results obtaine d from the computer calculations O'Xlc'it me i-xt1,t exprtio for -Lihe current, given by (2.70), some forniuIta v c ]'Tr- Ter usingc approximate forms for thLe cylindrical functions appearinu ij' sio-t equations. Tie approximate expressions which are obtained in this Pay vidie. i acitional benefit of reducing thle complicated appearance of the,ua1m }io:ns sivinsl tne currents to a form more easily interpreted. T.se ra(tius of the cylinder chosen for this investigation was on the order of 1 cT.j a size (i:tid e partly by tlmeoretical considerations which required that a o ion1 b e.; J ained 1with a rL fsuiiui le- number of terms, and partly by practicaL ct. tLlra-ons associated with- the feasilbility of performing an experiment. These -'tendc to ~e- opposing restrainLts, the former leading to smaller radii and the hiti r to larger. In the end a third consideration resulted in decreasing the noI) in l r'adius to 0.2 cmr, due to the oscillation of the EK currents as a function of 0 -b inhLg so rapid tnat plotting the currents on 8-1/2 x 11 inch graph paper was road i ii: ractical. For the nominal choice of other parameters used, such as T. f, etc., KlE 0.14 cm and Kp 66cm. Thus KE s<<1 and K s ~> ]1, so ta lit small argument approximations can be used for the cylindrical P Luncti-..- of ai ui;.:en t proportional to K s while large argument approximations c.an L,:e.iscd for those of argument proportional to K s. The leading term in the ascenu-(g series for the cylindrical functions is used as the small argument approsimnIl icion anI u iankel's asymptotic expansion is used for the large argument;lp 1-)L;) 1Uii- iO 1,

261 TN.- (currents excited- v h ~%ot pes of EM waves are oiven bythe liv tubGtvb follow~ing appr)ioximiate expressions, where only 1ie leading term in thne -steriesz is included to be ct w Ii: — cut with theLup-jT-'roximai-Ions used-I. (E.l1la) (E.Il~b) (E.lIlc) -- I E\ (B 1ld) D0 and D Iare f r - -- -- I - I \ I - x N i " -,-, , - (E. lle)

'262 (E.llf) I) is _,to:,n i: a; f Lnc-ion of m since it is used also for the EK current expres-.siolns. m - 1 ihen D1 is used in the eqlltions above. Ti' t.-:- ll urre[nt do not show a strong dcependence on the sheath, in contrsast to the case for the EK wave discussed below. Numerical results obtained us il-u (EJ. [ 1)!w;ere foundc to be in good agreement with those calculated from the exacit etxpressions. In addition; since the exalct results showed lict currents to be tIl.j,'tJcted b)y tihe sheath and the compress bility of the plasma, it is instructive to re, rite (E. 1 1) for s = c and with tie terms resulting from the EK wave omitted[, as (E.12a). -~ - (E. 12b)

2 6:3 (E. 12o) (E 1 2z1I) (E. 12e) J Uizbha jar pred'l~ted by thiese e i'-iinis exhibited by the graphs of figur-e's 3-42to 345 wien how the currents aLs a f'undC ion o ~N n lated W.itilouLLapuii u.Nt ii KF is nt excited unless thiere is a s n~eai n or tnre pins mia comipre-,ssibility is al'mdfor. Whlen tn EK inl hiCed curnsare cosiJ ed,- a cittfeiwruni lm is en ic oui~in teCred in m-caking, th-e approximiations. Tni-re is, first of all, the fieit that s~omei of Ithe cylnricalI fuLn ct ions can ha ve in: iriiry argumeInItsl. SSecondly, the" CcoAnv(~ergece~ of t Le currenClt expre.-.- is w~ with m is controlled by thie ratio of the( E K wav elength to the cylIinder radiuIs, so that tire first term in the series no longYer Le'eu 'at ely represents the sum of the series. As a matter ol fact, if wewis-h to repi esen~t the entire series, ditfei-rent aprini.;would be requI~irl't1 iX en thie Lam, umntmlt >3 n twhen the argument m and -w!hen the ai rgumrent<c ax T Ihere is ntigto be D-ained from goin thoughDd His:: lengthy

26.4 procedurei, LsinCt the calculations are to be calrried out using the exact expressions anyway, but by deriving the first few tetrnls in the series froil the approximations, a chLeck can be made on the exact results, and we may also be able to deduce something about the dependence of the currents on the various parameters. Thus the equations which fullo,w are reasonable approxinations for the leading terms in the series, based on K s >> 1 and K s > m. The first set z P P of equations is for the case where thef EM nmodes produced by the incident EK wave have real arguments and the second for the case where their arguments are imaginary. Case 1: Real EM mode arguments With XE c, XEc << and rea: Xpe > 1 and m -. — " o E > - a -F (E. 13a) (- - 3b -- - - - 2 / (E.13b)

24i5 iD1 i.-: oiven by (E. lf). It is interesting to observe that when s =c,. then K is i,ien t-e m 0 terml only, a result wilich, incidentally, is dependent,on) ly '. c < 1 being true. i; hi- result explains the realatively little influEo (z) (z) cnC of i-he sneath on K alnd also the little variation shown by Ki as a P P functicn ot 0;hl, is exaibited by the graphs in Chapter III. The peaking of K( which changing 0 near n-.' llil incidence can also p t5,e showvi f]rom, (E. 13a). This peai.ilg occurs essentially because the last two.La'; ix in thie (llnolll:l I i of (E.13a) for the m = O0 term niay become small or equal to zero. Since the j2 term for these angles of incidence is much less than either oi the other two terms, until 0i approaches 7r/2 - v /V, then the mag(z) iu ol K) is inversely proportional to the terms in XE and XE. Conse-.P yuently. as XE an d XEo decrease in value with increasing obliqueness of 0, K( SlhowS a corTrespolnditng incr'ease in magniiude, which for the sheathless (z).a'Ise bcomnes L:;illltllu when XE = 0. When s +c, the peaking of K is not E P so aZ2eenLuaied, since XE;Itd XEo do not go to zero at the same time. (z) i' le dependence of K on c can also be observed from (E.13a). When P the s;ealles, s case is considered, we see that for c 1 cm, K( a -3/ P,i)'uxin ':l1tly. This dependence on c is shown by the graph of Fig. 3-14. Case 2: Imaginary EM mode arguments. Thle situation where cosi >> v /v, leads, by the use of equation (E. 7), to (E. 14a)

266 (E.:L4b) The minus sicgn is chosen for the imaginary root to ensure the proper behavior of the fields as op- o. Now with,si >v /v?, 3 can vary from 0- 0.01 K to K with decreasing 0. But over the range 0 <850, > 01 K so that -1 with the nominal value of K 66cm, ln]l with c 1 cm, /3c > 1 holds over most of this range in 0. We are thus interested in the large argument approxinmations for all tIle cylindriccal functions occurring in the exact formulas. If we require then where X = XE and XE XEo there is' obtained E P Po Po (E. 15a)

267 (E. 15b) 1) is given by 1 (E. 15c) I JL-i t\ a n d (E. 15d) These expressions are quite involved and would be of little use other tlan serving to check the numerical coii'pututitiois, if it were not for the fact that they:sow the effect of the sheath on tue first few terms in the series for the currents. We see that cosh '6 appears as a factor in these equations. When 6' >'1, then (E.16a) (E. 16b)

268 Tius the Icr';-s wiiicl ari proportional to ltallt 6 do not lalrc-uieo large in relation to the othler ternm-s, but the cosh6i factor exerts an overall influence, tending to decrease the current llag'l-itudes. The sheath influence can be represented by an atti.-niation factor (E.17a) With the sheath thickness ogiven in D, this becomes (E. 17b) where X is expressed by (E.17c) It is enlightening to give the sheath attenuation Ad, in db, as db' (E. 17d) i With N 0.7, X l= 10, and 0 7T/4, A — 15.1 d. db This attempt to predict the sheath 1i, li,;Iiiii),l is a very rough approxiniation of course, based on the first few terms in the Fourier series for the c ulr t.. l I..

269 rI ~ ^;..,' -.i resCllts i;l lts however. n copared i lg th ee attenpruationi ii i-.rI i'n me ic cure I+LS I.' ilated front tlie exact expressions.

APPENDIX F DETAILS OF THE CYLINDRICAL FUNCTION EVALUATI )N The accuracy of the umtl.-ricatl results presented in this study is determined to a great extent, and particularly in the case of the vRacuum sheath z;n:1lysis, almost wvholly by the accuracy. ith which the cylindrical flll(i, ts could be obtained. A wide range of order,itnd argunient values had to be considered for the cylindrical functions of both real and ihnllgin:lr3 argu(nents. As a re — sul t, no single method of evaluating these fLunctions could be used, but instead a number of various paths had to be available depending upon the area to be covered in argument-order space. This was due to some extent to limitations of the computer, which since it must work within the constraints of a finite memory, can handle only numbers of limited magnitude and numbers of significant places. ''litis for example, while the ascending series solution for the Bessel function of the first kind, J (z), is nicely convergent for large (m/ z):ratios, the numbers which are generated in the i1mlin niL'al terms in the series may be quite large and overflow the computer, i. e., exceed the computer capacity. It also happens that in a representation such as HankelRs asymptotic expansion, with an imaginary argument, the answer is much smaller than the individual terms of alternating sign which occur in the series, thus requiring their difference to be smaller than the last sigiliii':rt figures carried along by the computer in summing the series. Inaccurate or wholly incorrect results are then obtained. Another consideration, of course, is the inherent maximum accuracy which can be obtained in an asymptotic expransicn. 270

2 71 lThse considerations led to the use of three distinct methods for calculating the cylindricial functions. Thu formilulas used are the ascending series, Halnkele s as\:nptotic exp.ansion andll ()lver's uniform asymptotic r l'anli 'n, givKen )by Olver (1964) in Eqs. (9. 1.10) to (9. 1.14), (9.2. 5) to (9. 2.10), and (9.3. 35) to (9.3.41) respectively, for real aguLlments. When the arguments are imaginary, the coLrr spilldiig expressions are given in Eqs. (9. 6.10) to (9. 6.13), (9. 7.1) and (9. 7.2), acnd (9. 7. 7) to (9, 7. 11) respectively, in the same reference. The adlvantageto of using Olverts iiiir.rm zs\ mptotiet expansion in spite of its much greater colmplexity is due to 11i, fact that his formulas are valid for any argument to order ratio for large orders, so that it complements the other methods which are useful primarlily for small orders. The accuracf of the progorams.-.hiTch w\ere used to calculate the cylindrical functions was checked against the tables listed by Olver (1964). The various nmethods -,were also checked against each other to better establish the more desirablle formula to use for a given order and argument. The series appearing in these formullll's were calculated to the point where the last term ta;i-\en was no larger in absollute value than 10-6 of the first. If we denote the ascending series by AS, Hankel's asymptotic expansion by HE and Olver's lliform. asymptotic expansion by OE, then the region of orderargulm- l M nt space where each was used-I in the c('alcdul atiotIs can be depicted as shown in figure F. 1. The line dividing OE from HE is m= 10z, as it was found for m much closer to z than this, that HE began to lose accuracy, particularly for imnaginary ar gu lmellnts.

272 m. order AS: Ascending Series HE-= Hankel's Asymptotic Expansion OE= Olver's Asymptotic Expansion 40 30 OE i I f0 I *sw~sswwftBya-rtefeagatS i' _-a* -0-^ j0 HE AS 20 t i ve.->t^As-.w., i>er 30 I 40 z, argument 50 I I 0) —, A -., '.,a...... A h-.A. FIG. Fl: THE DIVISION OF ARGUMENT - ORDER SPACE AMONG VARIOUS METHODS OF COMPUTING THE CYLINDRICAL FUNCTIONS f-i --- Ih C,i LiJ_7 -rT -*z

The;l ili.-lnt I ove:"11 accuracy achiie\ ed for the cylindrical functions used in this study was tab.fut one digit in the fourth significant place. On the averague, the Iacc'uraco' was no less than one digit in the fifth place, and frequently the c lculated \ alslts checked those tabulated out to seven and eight places. This shIould not be taken to nean that better results could not be obtained if thfey were desired. That:vcwuld be a matter of taking more terms in the series from nwihich the function)s are obtained and possibly also of resorting to fancier coi,,,Ifi, r teclhniques such as double precision operation. In such a wav, one could achieve an arbitrary accuracyr up to the limit of the theoretical accuracy obtainable using the asymptotic expansion, at the expense of requiring more computer time. This was not done in the present study since the results which were obltained were felt to be adequate. It may be of interest to note that the ave rage time required for calculal ing a cylindrical function was about 0.05 seconds If we observe the expressions, which 1..:l to the vacuum sheath surface current (2. 70), it is apparent that the only source of random error in the calculations is that due to the cylindrical functions. A systematic inaccuracy is represented by the uncertainty in such parameters as the value of the velocity of light (or equivalently p and e ) Boltznmn's constant, the electronic charge, O 0 etc., over which we have no control. The values used for these parameters in the calculations are:

274 0 OU't81 41 A I I ctLa- ' LA 4i x 10 henrys/mn - 2. 997 x 1(18 m/sec k- 1. 8304 x t0-23 Joules/0 K - 1. e I1. 60207 x 10 coulombs -31 m -9.1072 x 10 kg e Due to the copll, 'ily of the expressions for the currents, a detailed etlror analysis is lri:' l. i. I Le iLtn l sti l,iii'g to tI c g' r tcst '. -acC occurs wvlhenr tVwo nearly c.A::1 nuInbers are subtracted since the errors are additive,, and can become larger than the result. Multiplication and division lead to rtloughly an N - fold increase in the errors when N nearly equal numbers n ith equal unceri'a nty are involved. Upon takinig into account the numblter of 1multiplications and divisions involvirag the cylindrical functions occurring in the current equaltins. it seems unrealistic to suppose that the inaccLuraLcy would increase by more than a factor of 10 due to this consideration. In order to chucl for errors arising from subtra;-l i ii, various colil.LItItt itnl points in the process of obtaining the solution were printed out. The. itt;i, i11 where nearly equal numllbers were being subtracted \\ s icientifil-ble Xi these chneck points consistently hixing zeros in the several last. signi[ic tlit figoures. A check of the results for the current in such a case revealed numbers which varied erratically from those otherwise obtained. The only time this was found to occur was for angles of incidence of the EK wave

2 7 5 near r/2 - v /. Numerical results for the culrrent in this case were obtained froml the:approximate expansions. Art overall ':I-c'i acy of at least three siognificant figures is felt to have been iimazintaiined in the illdix. idual terms in the Fo,.ii1r series for the ctlrr'..l. The convergence of the Fourier series; is not a Afictor here, since the terms were calculated out to the point \\here the last term ir no bigger in absolute Imagniitude than 10-6 times the m 1 term. This estin-iate of the a:ccuracy is probably somewhat conservative, as evicenced Lrv the fact that the inhomogeneous sheath and vacuum sheath results agreed to 3 and 4 places after the sullltilun of the Fourier series had been pertoried. An addlitional factor in.st:li] islhingi the credibility of the results is the toagreelii. ll of the calculated curves '. ih the expressions obtained from approximating the exact cLialtins. Finally, the regular \ ri:tiln of the currents with the azimuthal angle i indlicates the re'ative absence of random errors in the cal(uLlations. In this regard it should be noted th t current minima less than 10 of the tmaxima o;l uld be retgrded with caution since a three figure acclray in the Fourier coefficients for the c.ill iltts would indicate that the total error could then exceed the total of the Fourier series at the nminima.

NOTATION A Proportionality coefficient for electron velocity dcl -i- i i i, "I, function. i A(O X): Sheath attenuation function. AS: Abbreviation for ascending series method of calculating cylindrical functions.. A: Fourier coefficient for expansion of fields arising in the vacuum sheath analysis. Superscript indicates scattered, transmitted or reflected mode, and subscripts indicate mode number, incident wave type and mode type respectively. S Example: A p mph C: Dummy coordinate. a(0 ): Sheath attenuation scale factor. C( ) F:actor which shows the decrease in the static electron number density in the sheath due to collection of electrons by cylinder, arising from tlurc:cation analysis. B Proportionality factor appearing in exponential term for electron velocity distribu il tion function. 3: z direction separation constant; determined by angle of incidence with respect to cylinder axis of incident wave. C; Coefficients which determine the starting valuws of -te final integration in the inhomogeneous sheath analysis. C Matrix which shows the relation between the various Fourier coefficients obtained in the vacuum sheath analysis. C General cylindrical function of real argument and order m. m c: Cylinder radius. cm: Centimeter. cps: Cycles per second. 276

" t I ~J P t R ti 7f ct p( L< J ii i i. V or t i jron -as. '-;e — r > ' L- -,r th) -: tc:tin inant expansion of tL ' - I. I, d:n; t r;n'Nti l f xp: 1i(~1 f) n;r ater 1i'inn zero,,:, r'spownding to I)0. F: _; ct rc n )e)',-i o length Jor m:i,,rm plasma.. aD l e.xu:ct explrt-SSior(- 1 'tr t(i d( terrr inant -expansin of th. vac uu t she ith,. alls l s. d f lnhcremil l of \-)lu1me ini veloc. t spac:e. A (Coefficient matrix for deterir in ing the Fourier coeffic:ients (,f the vasCtnm shtaih an ilysis. C: oefficient matrix of i educed order. V, Gradient operator in physical space. V U Gradient operator in velocity space..i Sheath thickness tlll i )Itied bv 3. E Total electron energy E l Used as.>slsript to indicate [iatne wave pvopagat, ln constant and radial separationr constant of electroragnetic wave. E Component of electron energy parallel to the z axis. E C 11[,LIII ll n of electroa energy perpendicular to the z axis. E: Static pa.rt of electric field. O Ti me -varying part of electric field.

*.- -\ --:- s E ) Electric field of inhomogeneous sheath analysis. SApe - script indicates component relative to co-ordinate systlim-: - - and subscript shows mode number. - ^ EE: Electromagnetic component of total electric field- i. - uniform plasma. - - E Electrokinetic component of total electric field,.' - i uniform plasma.' EK Abbreviation for electrokinetic wave. EM Abbreviation for electromagnetic wave. E Total electric field. e Electronic charge. e Subscript which denotes quantities associated with trans -^. verse electric wave and abbreviation for transverse electric wave..e.: Charge of j'th plasma species. erf: Error function. c The flasia- p-ertdttivity.- i -. r. The dielectric constant of the plasma. E: Permittivity of free space. o cE Imaginary part of permittivity. F1: Integrand of integral for determining the electron nunP: -'density from the Bernstein and Rabinowitz. analysi4.- r.= -. F2 Integrand of integral for determining the electron velocity from the Bernstein and Rabinowitz analysis. F Electron force term. -e F. Ion force term., 1i ^, r F. Force on j'th plasma species appearing in the Boltzmann equation. -3 f: Frequency of incident wave.

279 f: Factor which relates the various Fourier coefficients of the electromagnetic wave in the vacuum sheath analysis. Superscripts indicate wave types as scattered, transmitted or reflected and subscripts indicate polarization. Example fSR eh. f; Static electron velocity distrib ution function. eo f Electron velocity distribution function. e f.: Ion velocity distribution function. f.: Distribution function of the j'th plasma species. f Electron plasma frequency. P 9 Gc 10 cps. H Static magnetic field. H Total magnetic field. H1 Time varying couinpulcnrt of the magnetic field. H: Abbreviation for H( 2)(X s). (1) H Abbreviation for H (LE c). Eo mn Eo ((1)2) H: Albbrevfiation for H (X ps). H (z) Hanlel function of first kind of order m and argument z. m (2) H (z) Hanklel function of second 1- ii] of order m and argument z. HE Abbreviation for ITlnkel's asymptotic Expansion method of evaluating cylindrical functions.

280 H M; iagnetic field of inullllt.lnugt l'.., lS sheath analysis. Superscript irl(dic alts comulpunlnt relative to co-ordinate system, and subscript indicatts mode number. H Total magnetic field. h Denotes transverse magnetic polarization of electromagnetic wave, and used as subscript to indicate quantities associated with transverse magnetic wave. Impedance for electromagnetic wave in plasma.: Free space impedance for electromrllagetic wave. I: Static electron and ion particle flux density in sheath. (2Y1 (2) IE Imaginary part of the ratio H (KE)/(H (K s) where indicates ditterenti: ltiun with respect to argument. (2)' (2) Ip Imaginary part of the ratio Hm (Kps)/H (Kp s) where t indicates differentiation with respect to argument. I General cylinldrical function of ilaginlary argument and order m. m i: Denotes imaginary quantities. J; Angular momentum of electron. J: Bessel functiOun1 of order m. m Jp Abbreviation for J (Xps). P mPm J Integration limit of electron angular momentum in Bernstein and Rabinowitz analysis. J2 Integration limit of electron angular momentum in Bernstein and Rabinowitz analysis. J Culrrent density. KE PI. l'.:l: l I 1l constant of electromagnetic waves in plasma. E K Propagation constant of electromagnetic waves in free space. Eo K. Pr1 1L:g;:i i, I' constant of incident wave. 1

-281 K: Propagation constant of electrokinetic waves. K: Cylindrical function of second kind and imaginary argument. K; Degrees Keix inl. K: Cylinder surface current. Superscript indicates component relative to co-ordinate system and subscript indicates incident wave type. k: Boltzmann's constant. L Parameter which relates the magnitudes of the time varying quantities to their static parts in the linearization of the plasma equations. n: Logarithm to the base e. X E Radial separation constant of electromagnetic wave in uniform plIasia. X E Radial separation constant of electromagnetic wave in \acuLum sheath. X: Radial separation constant of electrokinetic wave. M Exponent of static potential variation in sheath. MFP Electron Mean Free Path. m Azimuthal separation constant, an integer. m Electron mass. e m Electron mass. m. Ion mass. m. Mass of j'th species in Boltzmann equation. p: Permeability of free space. N Ratio of electron plasma frequency to radio frequency. N (p): Denotes Boltzrymnn distribution for static electron number e density variation in sheath.

282 N Neumann function of order m. m N Normalized energy density of kind indicated by subscript with incident wave ildicati. ed by sut1tpe rscript. n: Time varying electron density va-:riation. n Electron number density. e n Statict electron nflLu 1bel density. eo n. Ion number density. n. Static ion number density. 10 n Static particle density. 0 n1 Time varying electron number density. n Static electron number density. 0 n Electron and ion number density in uniform plasma. Go OE Abbreviation for Olver's asympl)totic Expansion method of evaluating the cylindrical functions. o: Angular frequency of incident wave. to Angular plasma frequency. P P Electron pressure. e P. Ion pressure. 1 P.. The ij component of the pressure tensor. 1] P Non time varying pressure. 0 P1 Time varying pressure. P Subscript to indicate electrokinetic plane wave propagation constant and radial separalti,)n constant.

283 P Measure of grdient of plasma inhomogeneity. p Denotes quantities.i. —.;.'i, 1 with incident electrokinetic wave or scattered electrokinetic mode.: Cylinder potential.: Static potential in sheath. Denotes potential of wave type indicated by suL)bscript ' - with superscript indicating incident, reflected, transmitted or f.iil i d potential.: Azimuthal co-ordinate. Q Quantity proportional to time varying electron number density. Qm Q of mode number m. Qijk The ijk component of the heat flux tensor. R Superscript denoting quantities associated with reflected mode. RE Real part of the ratio H2 (Ks)/H )(Ks), where inE Em m dicates differentiation with respect to argument. Rp: Real part of the ratio H()(Ks)/H (Ks), where' indiPm(K p)/H, where incates differentiati,,n with respect to argument. R Matrix used to write bcmd 1l'3 condition equations at sheath interface for inhomogeneous sheath analysis. RI 12, R3 Ratios to determine validity of dropping static electron velocity from equations. r Space point. rms Abbreviation for root-mean-square velocity. p: Radial co-ordinate in cylindrical co-ordinate system. p. Unit vector in radial direction. S Used as superscript to denote scattered mode.

284 S_ Source term due to incident wave type indicated by subscript in j:li..n, ii,,, 11 sheath analysis. S1 S, S4S: Real and imaginary parts of S and S, respectively. S_ Source term due to incident wave of type indicated by subscript for vacuum sheath analysis and boundary equation matrix of reduced order. S_ Source term due to incident wave of type indicated by subscript in vacuum sheath analysis. s: Sheath 'Irdius.: Summation symbol for Fourier series.: Normalized absorption cross-section for inhomogeneous sheath. T Superscript which denotes transmitted mode. T Particle temperature. T Electron temperature. e T. Ion temperature. 1 T Matrix of coefficients for the inhomogeneous sheath equations. t Time. i 0 Angle of incidence of incident wave. i 0 Angle of incidence of electrokinetic wave at which the'radial c separation constant of the electromagnetic wave becomes imaginary. 0 Angle of incidence of electrokinetic wave at which the backward scattering angle of the electromagnetic wave is 45 degrees from the z axis. u: Total electron velocity. e u. Total ion velocity. u. Velocity of j'th species in the Boltzmann equation. J

285 u m i V E V_ ( ) m v v eo v er v io V. ir: Velocity at which the integration is terminated in the truncation analysis. * Denotes amplitude of either polarization of incident electromnagnetic plane wave.: Amplitude of incident wave of type indicated by subscript.: Electron velocity of inhomogeneous sheath analysis. Superscript indicates component relative to co-ordinate system and subscript shows mode number.: Titme varying electron velocity. Static electron velocity.: Elect ii n root- mean -square velocity.: Static ion velocity.: Ion root-mean-square velocity.: Velocity of light in free space. * Electron root-mean-square velocity. * The i component of fluid velocity. Time varying electron velocity. * Static. 1. ct '. velocity.: Time average electric energy stored in sheath volume. (Lower case letters denote time average energy density.): Time average magnetic energy stored in sheath volume.: Time average electron kinetic energy stored in sheath volume.: Time average electron potential energy stored in sheath volume.: Abbreviation for Wlrnskian relation involving cylindrical functions of two different arguments. v r v. 1 v V 0 W WH WK wP Wp w(_, )

286 w The i component of particle random velocity. i X Sheath thickness 1measured in electron Debye lengths. x Electron displacement from,luili I i I, i nI t. YA: A d l i t t l rn 1:i1 -i between electron normal velocity and normal electric field at cylinder surface. YB Admittance relation between electron normal velocity and time varying electron number density at cylinder surface. Y': Quantity pr' 1l, ti' llal to Y. B B Y. Dependent variable of inhloml,,'netls sheath equations. 1 Y. Value for Y. obtained at sheath interface from ath integration 11 of inhomogeneous sheath equations. y Dummy variable of integratil in. Z Integration limit proportional to cylinder potential. c Z Variable proportional to static sheath potential. z. z xaria.bl of cylindrical co-ordinate system. z' Dummy argumlent of cylindrical functions.

REFERENCES Allen, J. E., R. L. F. Boyd and P. Reynolds (1957) "The Collection of Positive Ions by a Probe Immersed in a Plasma", Proc. Phys. Soc. of London 70, Pt. 3, 297-304. Auer, P. L. (1961) "The Space Charge Sheath in Low Pressure Arcs", Ionization Phenomena in Gases, Munich, Vol. 1, 297-305. Balmain, K. (;. (1965), "The hac.iation Resistance oi' a Slot Anten-:a in a Compressible Plasma,s" 1965 Spring USNC-URSI Mteting, Washington, D, C. Bernstein, I. B. and I. N. Rabinowitz (1959) "Theory of Electrostatic Probes in a Low-Density Plasma", Phys. of Fluids 2, 112-121. Bollm, D. (1949) The C.'I,.. I t..istics of Electrical Discharge in Magnetic Fields, Chapter 3, (Edited by A. Buthrie and R. K. Wakerling; McGraw Hill, New York). Caruso, A. and A. Cavaliere (1962) Nuovo Cimento 26., 1389. Chen, K. M. (1961) et. al., "Studies in Non-Linear Modeling III: On the Interaction of Electroimilagletic Fields with Plasma", Radiation Laboratory, Ann Arbor, Michigan, Report 4134-2-F. Cohen, M. H. (1962a) T"Radiationl in a Plasma II: Equivalent Sources", Phys. Rev. 126 389-397. Cohen, M. H. (1962b) "RadiaLtio n in a Plasma III: Metal Bounlidaries" Phys. Rev. 126,:3-8 —104. Dattner, A. (1957) Ericsson Technics 2, 309. Dattner, A. (1963a) Ericsson Technics 8, 1. Dattnei, A. (1963b) "Resonance Densities in a Cylindrical Plasma (Colullmnt", Phys. Rev. Letters 10, 205-208. Dittmer (19'26f) Phys. Rev. 28, 507. Fedorchenko, A. M. (1962) "Conversion of a Transverse Electromagnetic Wave into a,rilllotitli.inl Wave at a Dielectric-Plasma Boundary", Sov. Phys. Tech. Phys. 7, 428-430. Field, G. B. (1955) "Radiation by Plasma Osrcillations", Astrolphys. Journal 124, 555-570. 287

288 Gabor, D., E. A. Ash and D. Dracott (1955) "Llngmuiir's Paratdox", Nature 176, 916-919. Gierke, E., W. Ott and F. Schwirzke (1961) Proc. Fifth Internat. Conf. on Ionized Gases, Munich, Vol. 2, 1412. Gould, R. W., (1959) Proc. of the Linde Conf. on Plasma Oscillations, IndianapkaLs. ~Harp, R. and G. S. Kino (1963) "Experiments on the Plasma Sheath", VIth International Conference on Ionization Phenomelna in Gases, Paris, France. Harrison, E. R. and W. B. Thompson (1OY5J) "The Low Pressure Plane Symmetric Discharge", Proc. Phys. Soc. 74, Pt. 2, 145-152. Hessel, A., N. Markuvitz and J. Sl lhnlys (19CI62) "Scattering and Guided Waves at an Interface Between Air and a Compressitblet Plasma", IRE Trans. Antennas and Propag..):tgl.li, i. AP-10, 48-54. Hessel, A. and J. Shmoys (1962) "'rF\ci(lion of Plasma Waves by a Dipole in a Homogeneous Isoti opic Plasma", Proc. Symposiumn on Electromagnetic and Fluid Dynamics of Gaseous Plasma, Polytechnic Press of Polytechnic Institute of Brooklyn, New York, 173-183. Hok, G. (1958) "Electrokinetic and Electromagnetic Noise Waves in Electronic Walvcgitlcl-", Proc. of Symp. on Electronic Wave Guides, Polytechnic Press of the Polytechnic Institute of Booklyn. Kritz, A. H. and D. Mintzer (1960) "PIroplagaLtion of Plasma Waves Across a Density Discontinutiy", Phys. Rev. 117, 382-386. La Frambois, J. (1964) "Theory of Electrostatic Probes in a Collisionless Plasma at Rest", Fourth Internati ll'.t Symposium on Rarified Gases, Toronto Lam, S. H. (1964) "The Langmlluir Probe in a Collisionless Plasma," Aeronautical Eng. Dept., Princeton University Report No. 681. Looney, D. H. and S. C. Brown (1964) "The Excitation of Plasma Oscillations", Phys. Rev. 93, 9f;5 —(._;). Merrill, H. J. and H. W. Webb (1:'!-') "Electron ScaLttering and Plasma Oscillations", Phys. Rev. 55, 1191-1198. Olver, FW.S, (1964), Handbook of Mlathematical Functions, Chapter 9, "Bessel Functions of Integer Order, " NBS Applied lxthelmtics, Series 55.

289 Oster, L. (1960) "Linearized Theory of PhLsma OscillLtiotns", Rev. of Mod. Physics, Vol. 32, pp. 141-163. Parker, J. V. (1963) "Collisionless Plasma Sheath in Cylindrical Geometry", Phys. of Fluids 6, 1657-1658. Parker, J. V., and J. C. Nickel and R. W. Gould (1964) "Resonance Oscillations in a Hot Nonuniform Plasmat", Phys. of Fluids 7, 1489-1500. Pavkovich, J. and G. S. Kino (1963) "RF Behavior of the Plasma Sheath", Sixth Internat. Conf. on Ionization Phenomena in Gases, Paris. Romiell, D. (1951) Nature 167, 243. Rose, D. J. and M. Clark (1961) Plasmas and Controlled FUsion*', MIT Press; John Wiley and Sons, New York. Self, S. A. (1"3:) "Exact Solution of the Collisiconless Plasma Sheath Equation", Phys. of Fluids 6, 1762-1768. Spitzer, L. (1962) Physics of Fully Ionized Gases, Iite krscietcCL- Publishers. Stratton, J. A. (1941) ElectronlaLgnetic Theory, McGraw-Hill Book Co., Inc. Tidinan, D. A. (1960) "Radio Emission by Plasma Oscillations in a Nonuniform Plasma", Phys. Rev. 117, 366-374. Tidman, D. A. and G. H. Weiss (1961) "Radio Emission by Plasma Oscillations in Nonuniform Plasmas", Phys. of Fluids 4, 703-710. Tidman, D. A. and J. M. Boyd (1962) "Radiation by Plasma Oscillations Incident on a Density Discontinuity", Phys. of Fluids 5, 213-218. Tonks, L. (1931) Phys. Rev. 37, 1458: Phys. Rev. 38, p. 1219. Tonks, L. and I. Langmuir (1929a) "General Theory of the Plasma of an Arc", Phys. Rev. 34, 876. Tonks, L. and I. Langmuir (1929b) "Oscillations in Ionized Gases", Phys. Rev. 33, 954. Wait, J. R. (1964a) "Theory of a Slotted Sphere Antenna Immersed in a Compressible Plasma, Part I", Radio Science, 68D, 1127-1136. Wait, J. R. (1964b) "Thc1 ort) of a Slotted Sphere Antenna Immersed in a Compressible Plasma, Part II", Radio Science 68D, 1137-1143.

290 Wait, J. R. (1963), "On Radiation of Electromagnetic and Electrokir-lctiec Waves in a Pll,'sit, " -)Apl). Sci. Res., Scction B, Vol. 12, p~ 1. Wassv.Xsttoul, E., C. It. Su and R. F. Probstein (1964) "A Kinetic Theory 2Approach to Electrostaztic P-obes", Fluid Mechanics Report No. 64-5, Dept. of Mech. Eng., MIT. Wehner, G. (1951) 'Ele.ctron Plasma Oscillations", J. Appl. Phys. 22, 761-765. Yildiz, A. (1963) "S.'.tt. r'ing of Plane Pla:..'-lu Waves from a Plasma Sphere", Nuovo Cimento 30. 1182-1207.