M, We: S EjESELB-EWmME‘H ETH Mia L‘U E'E183E3f3r hi: agree 3? DE; H1} E‘EEE ESEE‘E SzETE UEW EEESE ;Y SEW .‘D " STEESE‘E‘EE ”T- u Michigan 5 £356 «.3 University This is to certify that the thesis entitled RADIATION IN A COMPRESSIVE PLASMA BY A SMALL SOURCE SURROUNDED BY A PLASMA SHEATH presented by David F. Strawe has been accepted towards fulfillment of the requirements for Ph D. degree in Elect. Engr. 13% MS Lfik Major professor Date fi/J‘Vfiffi - / /75] L/ / f 0-169 AN ABSTRACT RADIATION IN A COMPRESSIVE PLASMA BY A SMALL SOURCE SURROUNDED BY A PLASMA SHEATH by David F. Strawe The purpose of this investigation is to determine the effect of the nonuniform plasma sheath region surrounding a small dipole antenna,immersed in an otherwise uniform compressive plasma, upon the driving point admittance and the radiated electroacoustic and electromagnetic energy. Most previous analyses have ignored the sheath assuming an entirely uniform surrounding plasma. These analyses generally predict relatively large amounts of power radiated in the form of longitudinal plasma waves or electroacoustic waves as they are called; because of this effect the driving point admittance of the antenna is modified greatly from the value it would assume in a free space environment. It is then a goal of this study to check the validity of these predictions by including in the analysis the effect of a plasma sheath using a geometry previously analyzed assuming a completely uniform surrounding plasma. A simple geometry has been chosen consisting of two perfectly conducting spher'es connected by an extremely thin straight feed wire; the system is driven symmetrically at the center of the feed wire with a sufficiently small excitation voltage that the perturbations in electron plasma density are linearly related to the excitation. Around each sphere spherically symmetric sheath regions are assumed to form DAVID F. STRAWE whose thicknesses are sufficiently small that they do not overlap. Also,the electron density perturbations are assumed spherically symmetric. These symmetry conditions can be obtained approxi- mately if the spheres are sufficiently separated. Further, the dimensions are limited so that the fields surrounding the antenna are quasi-static. The input admittance is determined approximately from the quasi-static solution and corrected by a radiation conductance determined by the usual Poynting vector approach. The plasma is considered collisionless,and Landau damping is neglected; as a result,the plasma is lossless and the quasi-static input conductance can be related to the radiated electroacoustic power. The symmetry assumed allows the perturbation in electron density around each of the spheres to be described by spherically symmetric wave equations which are determined from the first and second moment equations derived from Boltzmann's equation. Solution in the uniform plasma is particularly simple, but within the sheath regions a pair of wave equations must be solved simultaneously; each equation involves the electron density perturbation and the perturbation in electron drift velocity. These wave equations are extremely complicated and must be solved numerically. The method chosen here for solution is an iterative one involving solution of a simplified wave equation in the density perturbation only as a first step. This approximate solution can be substituted into the remaining wave equationallowing approximate solution for the drift velocity perturbation; this solution is reintroduced DAVID F. STRAWE into the first wave equation to obtain a first order iterative solution for the electron density perturbation. Once this solution is determined the input admittance is derived from it. When the EM radiation con- ductance correction is made the total input admittance is determined. The relative amount of power radiated in each radiation mode is determined by the ratio of the quasi-static input conductance to the EM radiation conductance. Plots from which the quasi-static input admittance and the radiation conductance can be determined are given as are those of other parameters of interest. Comparison is made with the results obtained for the same geometry without the plasma sheath. Although there are many similarities in the two cases there are significant differences. The most salient of these is the general tendency for the magnitude of all solutions to be reduced when the sheath is considered. Although the trends of the solution for the quasi-static input conductance are the same in each case, with the sheath it is greatly reduced indicating greatly reduced radiated electroacoustic power. With the sheath there is a very significant increase in input susceptance (i. e. , toward the value corresponding to a free space environment). The radiation conductance is zero for each case at and below the plasma frequency, and each conductance increases monotonically with frequency to attain the same limiting forms at high frequencies although the uniform plasma case shows somewhat greater conductance and hence,greater EM power in the intermediate I DAVID F. STRAWE frequency range. It is generally concluded that previous analyses have unjustifiably neglected the sheath and have thereby predicted excessively large effects due to electroacoustic radiation. RADIATION IN A COMPR ESSIVE PLASMA BY A SMALL SOURCE SURROUNDED BY A PLASMA SHEATH BY . , 1.0.” Dav1d F13 Strawe A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1967 ACKNOWLEDGEMENT The author is grateful to his major professor Dr. K. M. Chen for his guidance and encouragement in the course of this research. He also wishes to thank the members of his guidance committee, Dr. J. A. Strelzoff, Dr. P. M. Parker, Dr. M. J. Harrison and Dr. T. L. McCoy for reading the thesis. The research reported in this thesis was supported by the National Science Foundation under Grant GK-1026. TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES LIST OF SYMBOLS I. IV. IN TR ODU C TI ON 1. l. The Problem and its Motivation 1. Z. The Geometry l. 3. The General Approach BASIC E QUA TI ONS Basic Equations The Collision Term The Stress Tensor The Moment Equations THE PLASMA SHEA TH Sheath Formation Analyses of Low Density Plasmas Conductor Potential Selection of a Sheath Model Wave Equation Form Solution from the M_oment Equation Elimination of the E1 Term The Drift Velocity Wave Equation The Drift Correction to the Density Wave Equation Final Forms B OUNDAR Y CONDITIONS 5.1. 5. 2. 5. 3. 5. 4. 5. 5. Introduction Sheath-Plasma Boundary Conditions Boundary Conditions at the Surface of the Metal Sphere Hybrid Boundary Conditions for the Density Solution Boundary Condition for the Drift Equation ii Page iv 53 53 55 60 62 64 68 71 71 72 73 77 79 TABLE OF CONTENTS (concluded) Page VI. ANALYTICAL FORMULATION 83 6. 1. Introduction 83 6. 2. The Hydrodynamical Wave Equations 84 6. 3. Transformation of the Density Wave Equation 88 6. 4. Matching Solutions at r = d 91 6. 5. Matching at r = a 95 6. 6. Matching at r = (11 97 6. 7. The Far Zone Field 100 6. 8. Radiation Conductance 104 6. 9. Quasi-Static Input Admittance 105 6.10. EA-EM Power Ratio 108 6. 11. Application to Resonance Probes 109 VII. FORMULATION FOR NUMERICAL SOLUTION 111 7. l. The Mathematical Sheath Model 111 7. 2. Transformation of the Wave Equations 112 7. 3. The Iterative Solution 117 7. 4. Boundary Conditions for the Drift Equations 119 7. 5. Reduction of the K Forms 120 7. 6. Comparison of Degenerate Forms with Fejer's Results 125 VIII. NUMERICAL RESULTS 130 IX. CONCLUSION 1 58 LIST OF REFERENCES 161 LIST OF FIGURES Title Antenna-Plasma Geometry The Plasma Model Maxwellian Distribution at r = d Self's Plasma Geometry Comparison of Sheath Potential and Density Profiles Sheath Potential and Electric Field Physical and Mathematical Models for n and p Physical and Mathematical Models for fr Zeroth Order Ky Zeroth Order Kz Zeroth Order lN(a)Kal Zeroth Order Ka Zeroth Order KC Zeroth Order Ke Zeroth Order Kp Preliminary Solutions for y1 Preliminary Solutions for y2 iv Page 12 13 50 50 51 52 128 129 137 138 139 140 141 142 143 144 145 Table Numbe r 3.1. 8.1. 8. 2. 8. 3. 8. 4. 8. 5. 8. 6. 8. 7. 8. 8. 8. 9. 8.10. 8.11. 8.12. LIST OF TABLES Title Self's Values for I(nw) Zeroth Order Kz for 6 = O. 01 Zeroth Order Ka for 6 = 0. 01 Zeroth Order |N(a)Kal for 6 = O. 01 Zeroth Order K for 6 = 0. 01 Zeroth Order Y and Ge for 6 = 0. 01 Zeroth Order P /Pe and CT for 6 = O. 01 Zeroth Order K for 6 = 0. 5 Zeroth Order Kz for 6 = 0.1 First Order Kz for 6 = 0. 01 First Order Ka for 6 = 0. 01 First Order |N(a)Kal for 6 = 0. 01 First Order Kp for 6 = 0. 01 'O’U’O N Page 41 146 147 148 149 150 151 152 153 154 155 156 157 LIST OF SYMBOLS I. Subscripts Except where noted by the inclusion of exceptional terms in the nonstandard term section the following subscripts indicate: e Electron related quantities i Ion related quantities 0 (or eo) Unperturbed quantities 1 Perturbation quantities 1e EM mode perturbation quantities 1p EA mode perturbation quantities 0° Limiting values at infinity 3 Sheath edge values r Radial components 6 Angular components (1) Angular components w Values at the sphere's surface 11. Vector Quantities Quantities with bars are vector quantities while the same symbols without bars are vector magnitudes of the same quantities, i.e., El versus El for example. III. Standard Symb oli s m E Electric field B Magnetic flux density ITI- Magnetic field intensity vi av av '9' x CD 3 9| 4| 8 5| Time averaged Poynting vector Time averaged radiated EM power Feed current Surface charge density Applied voltage Dielectric permitivity Permitivity of free space Volume current density Permeability of free space Electronic charge Boltzmann's constant Scalar potential Normalized potential (fii) Gas temperature 6 Radial coordinate Debye shielding length (Xi) = __e_z_o_) nm e 2 2 1'1co e Radian plasma frequency (w = 1 p m E e 0 Velocity Acceleration Average drift velocity RMS electron thermal velocity Gas number density Radian frequency Time Differential volume in configuration space vii .iilrlxl II..I[3|.|\' .111! I, A(r) m|| Differential volume in velocity space Probability distribution function in phase-time space Unit normal in the direction of increasing r Del operator Del operator in velocity space IV. Nonstandard Symbolism Ga 5 pres sure Potential c omponent Differ ential linear momentum Collision term in Boltzmann's equation Critical electron velocity below which electrons have insufficient kinetic energy to reach the sphere Velocity at r of an ion generated at r' 3 - E The i—th component of L3 The counterpart of d?3 in 07 space Sphere radius Sheath edge radius in sheath model Differential of i-th component of .17 Differential of wi Differential of A(r) Area Average of ; over ; space Boundary between regions 11 and III The largest plasma dimension viii 1(7)) Mean free path -- ions through neutrals Normalized distance variable in Self's analysis (8 = r/L) Wave length of Bohm's plasma standing wave pattern r-a Collision term in second moment equation Plasma stress tensor Collision frequency - (b r 431 Electron mass Ionic mass Ion generation function (ions/vol/time) Ensemble of generated ions r(s) ) Normalized generation function ( O Balmain's absorption coefficient Velocity of sound in plasma Normalized ion current density] 2 2 " w - (1) EA propagation constant ((3 = TR ) o [3 l6! ix N1(r) N(r) CO’CI’ CAB 2’ 9 0.4, (15, (16 E,F,H 1,2,3,4 E field contribution to complete plasma wave (3 quation Drift velocity contribution to complete plasma wave equation Effective normalized local plasma frequency Cohen's surface admittance coefficients Coefficients for hybrid boundary condition Linear ordinary second order differential operators r n 1 Normalized spherical density solution A Nl(r) Constants determining solutions for nl Constants determining solutions for ¢1 Plasma parameters involved in solutions for major plasma variables Normalized version of C2 Normalized quasi—static input admittance Normalized solution from which n1 is determined Normalized version of Ge Normalized EA - EM radiated power ratio Propagation factor analog in complete density wave equation Jw2_ 2f b.) J 1' o v Radiation conductance Quasi-static input admittance Re[Yp] “1*U'TJ "D NH(I‘) Np(r) Radiated EA power Radiated EM power Source volume current density Source volume charge density Plasma volume current density at r = a 211. 8x1 Dipole model feed current Dipole model charge Singular point in Fejer's solution for nl(a) Dummy variable Linearly independent solutions to density wave equation Integration c onstants Geometry index Generation factor ZKT e Specific heat ratio Number of degrees of freedom in perturbation system Homogeneous solution for N(r) Particular solution for N(r) xi CHAPTER I INTRODUCTION 1.1. The Problem and Its Motivation The purpose of this investigation is to determine the effect of the nonuniform plasma sheath region surrounding a small dipole antenna,immersed in an otherwise uniform plasma,upon the driving point admittance and the radiated electroacoustic and electromagnetic waves. It is hOped that certain generalizations can be made as a result of this study which will be of general applicability to antenna problems where the effect of the plasma sheath is important. Generally,when antennas in plasmas are analyzed the plasma sheath is ignored; this usually leads to the prediction of a relatively large amount of power radiated from the antenna in the form of electro- acoustic waves and of a considerable modification of the driving point admittance over its free space value; many analyses of this form have appeared in the literature. Wait1 has analyzed the spherical aperture antenna in a uniform compressive plasma. He assumes the applicability of the linearized Maxwell and moment equations neglecting all loss mechanisms and replacing the stress tensor by the gradient of a scalar pressure as is, it seems, conventional in these analyses (this idealization neglects the existence of Landau dampingZ which is very significant for frequencies appreciably above the plasma frequency); his solution is a modal analysis exact within the framework of the basic equations which he uses. Wait's theory predicts a rather large effect due to the radiation of EA waves. On the other hand,Larsonf in his doctoral dissertation analyzed the symmetric version of the same model with : essentially the same basic equations and approach but retained the plasma sheath with its nonuniform electron density and static potential distribution (be neglected the electron drift current); this theory predicts a much smaller effect due to EA propagation. In the class of analyses in which the existence of the plasma sheath is ignored the common conclusion drawn is that the radiation of EA waves has a relatively great effect upon the antenna driving point admittance. Those analyses including the sheath show,as a general rule, a much reduced effect. Chen4 has analyzed the hertzian dipole as well as the thin cylindrical dipole assuming a uniform plasma and no plasma sheath; he obtains relatively large effects in each case. Chen5 also has analyzed a conducting cylinder in a uniform plasma with no sheath; he has found that the radar cross section due to the excitation of EA waves is orders of magnitude greater than that due to EM wave excitation, i. e. again a large effect due to EA wave propagation. Fejer6 has analyzed a short dipole antenna consisting of two metal spheres connected by a thin feed wire driven at its center and assuming a uniform plasma with no plasma sheath. He determined the quasi-static fields surrounding the dipole using Poisson's equation and the linearized moment equations derived from the collisionless Boltzmann equation; he calculated the driving point impedance based on this quasi-static solution and proceeded to correct this expression with a radiation resistance using the usual Poynting vector method in conjunction with the radiation zone fields which were matched to those in the quasi-static zone of the dipole. The model was simple,if somewhat unrealistic,but yields some particularly simple and interesting analytical results; his results also predict a substantial effect due to the excitation of EA waves. There are many criticisms which might be made of these various antenna -plasma analyses, but the most seemingly questionable assumption is that of a completely uniform surrounding plasma (i. e. , neglecting the plasma sheath); the vast difference in the results of Wait and of Larson,which represents essentially a comparison of the same geometry with and without the sheath,dramatizes this contention. The exact mechanism accounting for this difference is in question. This mechanism may be simply a matter of the depleted plasma sheath region, which forms a relative void near the antenna's surface, effectively isolating the antenna from the plasma for EA excitation. It is well known that there is a coupling between EA and EM waves in the presence of a density nonuniformity or discontinuity; this latter explanation seems more plausible since the sheath seems too thin for a void of that thickness to be an effective decoupling agent. The most rigorous of the analyses referenced here is that of Larson,whose approach does not depend upon a dipole approximation involving solution for only the quasi-static solution in the vicinity of the antenna but solves for the total fields and is,therefore,not limited in antenna dimension. Prior to this analysis,Schmitt7 analyzed acoustic plasma resonances in a cylindrical plasma column solving a modified form of the homogeneous Klein-Gordon equation with the plasma frequency written as a function of position to account for the nonuniformity in the electron density; the electric field of charge separation and the electron drift current were neglected. As mentioned before,Larson included the electric field but neglected the electron drift current. Inspection of the electron density wave equation developed in chapter IV indicates that the effect of the drift current is not entirely negligible, if small, so that it should be included in a rigorous solution of antenna -plasma problems. 1. 2. The Geometry The geometry of the problem considered here consists of two perfectly conducting spheres, each of radius a , and separated by a distance D in a compressive plasma of infinite extent and of uniform density nm except in the vicinity of the spheres' surfaces. The spheres are connected by and symmetrically driven in antiphase from thin feed wires lying along the line between the sphere centers as indicated in Figure 1.1. The main body of the plasma consists of electrons and positive ions in equal numbers per unit volume assuring plasma neutrality; each constituent gas has the constant density nco there, and the electron gas is assumed to have a Maxwellian velocity distribution with a constant temperature Te so that the gas obeys the ideal gas laws. Due to the thermal motion of the electrons and ions a "plasma sheath" of nonuniform plasma is formed around each sphere. Because of their small mass relative to the ions, the electrons accumulate on the spheres in sufficient numbers to create a negative static potential on each of the spheres which accelerates ions to them and repells further ingress of electrons; this leads to an equilibrium situation in which electron and ion densities decrease to relatively small values and in which charge separation, with an accompanying electric field, occurs as the spheres are approached; in addition, there is a steady drift of electrons and ions toward the spheres in this sheath region. The densities and drift velocities as well as the electric field due to charge separation return to their uniform plasma values asymptotically with distance from the spheres; drift velocities and the electric field vanish in the uniform body of the plasma. Although the sheath variables return asymtotically to their uniform values, uniformity has been essentially reestablished within a distance of approximately ten debye shielding lengths so that, in order to facilitate analysis, the plasma may be considered uniform for distances from the spheres of the order of,or greater than,this figure. The plasma surrounding each sphere is to be modeled,then, by assuming a certain plasma density profile in a sheath region which returns to uniformity smoothly in a finite and arbitrarily specified distance from the sphere; the rest of the plasma is considered uniform so that analysis of the propagation of electroacoustic and electromagnetic waves is greatly simplified, in the absence of a constant magnetic field, by the mode separation technique developed by Cohen. 8 The feed wires are considered to be so thin that there is no appreciable charge accumulation upon them and hence,neg1igible sheath formation around them. The assumption is made that the sheath thickness is much smaller than the center to center sphere spacing D so that the sheath formation is essentially spherically symmetric about the spheres' centers; this requirement is not unreasonable since in most plasmas the sheath thickness will be quite small. Further,the sphere spacing D is assumed sufficiently greater than the sphere radius a so that when the system is driven the perturbation in the spheres surface charge density,as well as all perturbations in plasma variables within the sheath)have spherical symmetry about the center of the sphere in question. The dimension D is assumed to be much smaller than the electro- magnetic wave length in the uniform portion of the plasma so that the radiated electromagnetic fields may be determined considering the antenna to be an electric dipole in the uniform plasma consisting of two point charges (located at the sphere centers with charge equal to the sum of that on the sphere's surface and that in the sheath) separated by a distance D; for this assumption to be valid there must be no charge accumulation on the feed wires,a condition which requires that D not be extremely large relative to the sphere radius a (the antenna is an end loaded dipole and requires coupling between the spheres to guarantee uniform current in the feed wires and,hence,no charge accumulation upon them). There is obviously a conflict here with the symmetry requirement that D be much larger than a; even though there may be a compromise range of values for D/a in which each requirement is adequately met it seems that this range would be small at best. Despite this conflict, the assumptions made are necessary to make the problem tractable, and herein lies the justification for their use. This geometry has precedent for its use; with the exception of deletion of the sheath region the same geometry has been analyzed by Fejer6 who obtained some very interesting results in surprisingly simple analytical form; a major goal of the present study is to check out Fejer‘s results by inclusion of the effect of the sheath region. This geometry, while not representing a practical antenna structure, lends itself nicely to solution, and the results obtained should shed some light on the effect of the plasma sheath on antenna performance in a compressive plasma. The spherical symmetry of the perturbed quantities in the sheath allows the analysis to center around the solution for electromagnetic and hydrodynamic (plasma density, drift velocity, etc.) quantities radiated from a single sphere; the problem then reduces to a one dimensional one greatly simplifying the analysis. The plasma surrounding each sphere will be arbitrarily subdivided into three regions. First Region I, the sheath region, extending from r 2 a to r : d is conSidered to contain all of the nonuniform plasma. Second Region II, extending from r = d to r 2 dl , is conSidered to have. uniform plasma everywhere,joining that of Region I smoothly at r 2 d, and to be entirely within the quasi-static zone of the antenna so that the EM fields can be approximated nicely by solutions to Poisson/s equation. Region 111 extends from r : d1 to infinity, has uniform plasma everywhere, and encompasses the radiation zone as well as the transition zone between the radiation zone and the quasi—static zone. The profiles of pertinent plasma variables in the three regions are plotted in Figure 1.2. 1. 3. The General Approach The driving voltage applied to the antenna just described is limited to magnitudes much less than the static potential built up on each sphere by the thermal motion of electrons and ions in the plasma in order that the perturbations in the electromagnetic and hydrodynamic quantities in the plasma be linearly related to the applied voltage. This allows linearization of Maxwell’s equations and the moment equations which represents a major simplification of the analysis; the development of the linearized equations is carried out in chapter 11. This study ignores the possible existence of a static magnetic field; therefore, in the uniform plasma region (Regions II and III) Cohen‘s mode separation approach can be used to advantage. This approach separates the solutions for the electromagnetic and hydrodynamic perturbation terms from the linearized Maxwell and electron moment equations, developed in chapter II, into an EM mode (electromagnetic)- and an EA (electro- acoustic) mode for a uniform plasma in the absence of a static magnetic field and loss mechanisms. The equations show that the modes are decoupled,with the EM mode containing all of the perturbation magnetic field and no charge accumulation,behaving as if in a lossless dielectric medium of relative permitivity 2 w l - % (where (up is the plasma frequency). No loss mechanism (.0 is considered here for simplicity although there exist, in reality, two major sources of damping,namely’the electron-neutral particle collision and Landau damping. Although not always negligible, collision losses above the plasma frequency, where EA prOpagation is possible, should not be great in most laboratory plasmas and in the ionosphere. Landau damping results from the trapping of electrons of high thermal velocity in the potential troughs of the EA wave resulting in a net transfer of energy from the wave into the plasma in the form of heat; this effect greatly clamps EA waves for frequencies higher than about twice the plasma frequency so that the results of this analysis should be applied with care beyond this point. Regions I and II lie entirely in the quasi-static zone of the antenna so that the electromagnetic fields can be approximated by solutions to Poisson's equation. A one dimensional Poisson's equation will be solved in conjunction with a one dimensional wave equation in the electron density perturbation analogous to the inhomogeneous Klein-Gordon equation; in Region II,this equation is, in fact, the Klein-Cor don equation. In each of these regions the Poisson equation is solved in terms of the solution to the density wave equation. The density wave equation is independently solved-~exactly analytically in Region II and approximately by numerical means in Region I. The solutions in Regions I and II are matched at the interregional boundary (rzd) assuming the continuous differentiability of the potential perturbation and the electron density perturbation. In Region I,at the surface of the lO sphere, boundary conditions developed in chapter V are used to tie together the solutions for the density and potential perturbations as well as to tie the solutions to the driving potential V. In Region III,the mode separation technique of Cohen is employed to obtain solutions for the complete EM and EA terms. The EA terms are directly tied to those of Region II since the solutions are identical in the two regions. At the interregional boundary (r=dl) the scalar potential term valid in the radiation zone as well as in Region II (taken from modal solutions) is matched to the scalar potential derived in Region II from Poisson's equation. This matching process is surprisingly simple here; the scalar potential in each region consists of a term due to the electron density perturbation and an “EM" term; the EM term from Region III matches directly the degenerate EM term from Region II, their forms being essentially the same while the EA terms are identical. The antenna current I can be determined from the surface charge density perturbation on the sphere and the plasma current perturbation at r = a ; each is easily related to the driving potential V so that a driving point admittance (denoted YP) can be calculated. Since no loss mechanism was included in the analysis and since no EM radiation effect is included in this admittance, because only the quasi-static solution was obtained in Region I, the real part of this admittance accounts for the existence of electroacoustic waves. The effect upon the driving pomt admittance of the EM radiation can be included, because of the absence of all loss mechanisms, by the usual Poynting vector method; a radiation conductance term 11 will be calculated by this method and displayed, in addition to YP , in chapter VIII. The ratio of the power radiated in the form of electroacoustic waves to that radiated in electromagnetic form is also given versus frequency. Fejer describes the application of his theory to resonance probes and the modifications to the usual theory brought about by its application. He indicates that the usual theory can be grossly in error; this contention makes comparison of his theory with the present one on this particular point of interest. He argues that resonance indications take place at frequencies representing peaks (poles in his theory) of the electron density perturbation at the probes surface; the peaks in his theory occur appreciably below the plasma frequency whereas the conventional theory takes the resonance indications to occur at the plasma frequency. Plots of the electron density perturbation at the sphere’s surface versus frequency from the present theory are presented in chapter VIII for comparison with Fejer’s results. Admittedly,the model analyzed here is highly idealized as well as being quite specialized, and the boundary conditions to be used (developed in chapter V) are somewhat arbitrary and perhaps not entirely realistic; hopefully, however, the results of the analysis will yield some insight into the extremely difficult problem which antenna-plasma problems represent. 12 gmuflnofimom gudunm . m E commom\ \II, a flammom $305000 gndfimugoflzw J ._ 0.5th /. tram—05000 Manon—:4 “zoom .4 fl flowwom a :3me I. > ,0 ‘ v 13 ”HR .382 «Essa 2:. .N ._ 93th . H '0 ll 0.: u .505 E 8&th u coaom H aoawmm 0G H AHvOHH a I d i 4 a 8G u A: :1 _ — o> n A30> _ CHAPTER II BASIC EQUATIONS 2.1. Basic Equations The most basic description of the dynamical behavior of a plasmawould be a description of the position as a function of time of each constituent particle of that plasma. Obviously this kinetic or microscopic approach is impossible in general and recourse to the statistical hydrodynamic or macroscopic approach is required. A plasma being merely an ionized gas it is natural to apply the basic equations of gas dynamics to the description of the plasma. A basic set of equations applicable to gas dynamics consists of the Boltzmann equation a _ _ . 5fgw-VHEJL.vvf=(5?) (2.1) and its first and second moment equations v-(nfi) = - .31? (2.2) .ai — ’___:_ _1_... .s _.l.__ at+(u.V)u—a ~an w+nm PC (2 3) where f = f(-r_, V, t) is the statistical distribution function for the gas in question in phase space, _3 n = g f‘dv is the number density of the gas equal to the ' vs . .. integral of f over velocrty space, — -3 g‘ fv dv is the mean velocity of the gas particles, ' vs 5| II "I ll 5|“ 5|“ ' _ _3 S f a dv is the mean acceleration of the gas particles, ' vs 14 15 q; is the stress tensor related to the compressibility of the 8 _. -3 m 5361-8“ f wi w dw where VS gas and defined by (V ° Mi 2 E = V - E , PC is the mean momentum gain of the gas particle defined as ' _ _3 m 3 (g—E— v dv , vs coll m is the particle mass. The only other term requiring description is (3-5) 11 WhiCh co can be considered as the total derivative in time of the distribution ... _ ..3 _3 function f(r, v, t) due to collisions. Also (3%)c011dr dv is the net time rate of increase of particles in a differential volume dr3 d'v3 in phase space moving at velocity V and is due to collisions; this seems reasonable when it is recognized that the Boltzmann equation is just a continuity equation in phase space. In addition when the particles of a gas are ionized the dynamics of the plasma is also determined by Maxwell's equations. The plasma consists of electrons, positive ions, and neutral atoms so that Maxwell's equations take the form __ .3 _ V - E _ 6 (ni-I‘ie) (2.4) O — _ as VXE — - 5-?- (2.5) VXB— (" ‘)+ 53—37 (26) - IJ'oe niui-neue Ho oat ' v-Ez o (2.7) U3! where E and are the total electric field and the total magnetic flux density, respectively; e is the electronic charge; “‘0 and 60 are the permeability and the permitivity of free space, respectively. 16 The subscripts i and e on n and ii denote positive ion and electron quantities, respectively. Each of the three constituent gases in the plasma (electrons, positive ions, and neutrals) should be described in part by separate systems of equations of the form (2.1) through (2. 7); the neutrals, of course, are not affected by the E and B— fields so that only equations (2.1), (2. 2), and (2. 3) apply. The neutrals do not enter into the wave prOpagation picture and so are of little interest except in so far as they collide with electrons and ions thereby affecting their distributions. The ions are much too heavy relative to the electron mass to contribute to wave propagation directly although they do affect plasma sheath variations of electron density and the sheath potential distribution, and thus in this manner make their presence felt. Characterization of the positive ion gas by use of equations (2.1), (2. 2), and (2. 3) is found to be difficult so that different means must be found. The electron gas lends itself fairly well to description by means of these equations however, and use will be made of them as the basis of the following work. 2. 2. The Collision Term The collision term PC in equation (2. 3) is, in general, an extremely difficult quantity to evaluate; it can be approximated only under special conditions. In the case of the weakly ionized gases which are being considered the dominant collision effect is that corresponding to the collision of electrons and neutral atoms. One commonly used model in this case where the plasma is assumed 17 to be nearly in equilibrium assumes a constant collision frequency; then the collision term in the Boltzmann equation can be shown to be approximately 6 . (—-) = - fc (fe — fee) (2.8) where fC is the collision frequency between the electrons and the neutral atoms, fe is the distribution function for the electrons, and feo is the equilibrium distribution function for the electrons. The acceleration term 3 for the electrons is related to the E and E fields by e m e 3’: - [f+;x-§] if gravity and other external potentials are assumed absent. In this case equation (2.1) becomes Bfe _ e '5? ”'er-rn: (E + va)- vae = - fc (£8 -feo) (2.9) The collision term 5C then becomes _ __3 P =m§-f(f -f)vdv=-mf(nu-nu) c e'vs c e eo ec ee eeo sothat re _ _ n m 2 - fc (ue - ueo)' (2°10) e e Other models for this weakly ionized case are possible, such as a constant mean free path, but equation (2. 8) is likely to be as good of a model as any other yielding such a simple form for Tic . 18 2.3. The Stress Tensor The stress tensor term Hrln- V ° 4; is another term which defies description. It seems impossible to evaluate for the ion gas, but it can be approximated for the electron gas by assuming that the ideal adiabatic gas law applies. For the electron gas the divergence of the stress tensor can be replaced by the gradient of a scalar pressure derived on the basis of an ideal gas formulation. This can be shown by direct calculation of the i-th component of the divergence of the stress tensor 3 _ _3 (V-Lp)i=me fw.wdw . 8 X. 1 1 vs First it can be shown that by the definition of U the integral 5‘ fwiw. dU3 vanishes if i 75 J so that vs J a Z —3 (V°\P)i = m S fa.)i dw e 8X. 1 vs 8 2 : —-—— < > 3X1 [mene mi ] (2.11) where < wiz > is the mean square of the i-th component of particle velocity in the reference coordinate system moving with velocity E . e Next a differential volume of cross section dA and length A d Xi is considered. Within this volume all particles with Xi directed velocity components (in the moving system) equal to wi dw. within :b-z—l— are considered as a group to carry, in time dt, a A momentum dp out of the volume in the Xi direction. l9 dp = m f(w.) dw. dAdX. w. e 1 1 1 1 This corresponds to a pressure of m f(w.) dw. dA dX. w. e 1 1 1 1 _ __1. £2 _ dP ‘ dA dt " dA dt‘w dX. But mi = Tit—1 so that this ensemble of particles contributes a pressure dP : m f(ca.)m:Z dw.. e 1 1 1 Integrating the contribution of all particles of all velocities in the Xi direction 2 - 2 P: dP= m f(w)w.dw=mn . e 1 e o 1 so that 01‘ V-up = VP (2.12) where P is the usual pressure in the moving system. It must be remembered that for this expression to be useful the gas considered must behave in a sufficiently ideal manner that the concept of a. pressure is justified. In the electron gas,collisims are assumed to be sufficiently frequent so that in the equilibrium state, i. e. , when no external fields are present, the distribution function feo is nearly Maxwell-Boltzmann, and the gas obeys to a good approximation the ideal gas laws. If feo is Maxwell-Boltzmann the concept of a pressure and a temperature is valid; the electron temperature 20 T8 of the electron gas is assumed to be constant so that P = K T n and VP = KT Vn (2.13) e e o for the unperturbed gas (n0 will henceforth be used to denote the unperturbed value of ne). In all of the work to follow all of the electron plasma variables (ne, Ee’ 15, etc.) are assumed to consist of their unperturbed values (those existing in the absence of external fields) plus a relatively small perturbation term due to external fields. All of the unperturbed terms will carry the subscript zero while the perturbation terms will carry the subscript one. The ion variables will be assumed to be unperturbed and will carry the usual symbolism. In so far as perturbations are concerned it is assumed that collisons are sufficiently frequent to allow description in terms of a perturbation pressure term but that the effect of the external fields is so great as to reduce the number of degrees of freedom in the perturbation system to one. Perturbation density variations are assumed to form adiabatically so that the ideal adiabatic gas law applies. Just how well all of these conditions are satisfied is open to question, but these assumptions are common ones evidently representing the best approximations available. The ideal adiabatic gas law states that P~ n‘1 (2.14) 21 C where a is the specific heat ratio EP— which is equal to L? v from the kinetic theory, and y is the number of degrees of freedom of the perturbation system. Here Y = 1 which implies that o. = 3 . Equation (2. 14) implies that P (11 )Cl __9. = P o o n Cl. _ e _ VPe—V P0 (11—) _v Pe(n1,no). (2.15) 0 film or that The chain rule can be used to write 6P 8P - VPe=a—n—e vno+anle an . (2.16) O Evaluat1ng a Pe/ a no and a Pe/ a n1 8P 8 n1 0. nl a-l ezKT —— 11 1+— =KTa(1+—) an e an o n e n 1 1 o 0 or 8Pe P6 8111 = (1 I—1_ (2.17) e and a ape z KT 3 (n1 +110) = KT :9. n°+(l-a)n1 (218) an e an a-l e P n +n ° ° 0 o n o 1 0 Assuming the perturbation density term n1 is much smaller than the unperturbed term n0 equations (2.17) and (2.18) become, respectively 22 BPe PO : G. —— = (1 KT (2.19) anl no e and BP ane = KTe . (2.20) 0 Substituting equations (2.19) and (2. 20) into equation (2.16) VPe take 3 the form VPe = KTeVnO+aKTean . (2.21) The assumption that the unperturbed electron gas obeys the ideal gas laws allows application of the equipartition principle. Since there are three degrees of freedom in the unperturbed gas the mean kinetic energy of the unperturbed electrons is my 3 K.E. 2 3° = EKT (2.22) where v: is the mean square unperturbed electron velmity. The constant a hasbeen given the value three; use of this value and equation (2. 22) in equation (2. 21) imples VP =KT Vn +mv2 Vn . (2.23) e e o e o 1 This is the common scalar pressure subsitution for the stress tensor; it can be no better as an approximation than the set of assumptions leading to its development. 23 2. 4. The Moment Equations The mean electron acceleration term -a: of equation (2. 3) can easily be shown to be a. : .- ‘m" e in the absence of gravity and other external potentials. If equations (2. 23) and (2. 10) are substituted in equation (2. 3) the moment equations for the electron gas with small perturbations from equilibrium take the form — anl 224 V-(neue) = - ‘52—'- (- ) and 351 _ _ e _ _ __ ——t-— +(ue°V)ue=-;;(E+uex) - 1 (KT Vn +mv7‘vn)-f€ nem e o 80 1 Cl (2.25) It is possible to separate equations for the unperturbed variables by dropping all perturbation terms, i. e. , V ' (nouo) = O and ... .— _ e _ _ __ (uO°V)uo---rH(EO+qu )-n m Vno. (2.26) e o e The moment equations for the perturbation terms can be simplified by retaining only first order terms in the perturbation terms, 1. e. , dropping all the unperturbed terms as well as all of those involving 24 the product of more than one perturbation term. Equation (2. 24) is reduced by replac1ng neue by noul + nluo so that an — - _ 1 V ~[noul +nluo] - --—at . (2.27) The first order terms corresponding to neh-J:e - V) Se are [ne(ue - V)ue]l = n0(uo - V)ul + no(ul - V)uo + n1(uo - V)uo; (2. 28) those corresponding to ne[E + 3e x a are [n (E+u xB)]l =nlEo+nO l+nO 0xBl +n0ulxfio+nluox§o (2.29) It follows that equation (2. 25) is reduced to aul _ _ _ eno _ _ _ 2 no -8t— +[ne(ue - V)ue]l + fcu1 = - -1_-rT--[E1 + ue x )]l - voan e (2.30) Equations (2. 26), (2. 27), and (2. 30) are general and complete for the case of small perturbation of the equilibrium plasma. In the special case to be considered in the bulk of this work where collisions, the Lorentz force, and any constant magnetic field are neglected the moment equations for the perturbation terms become _ _ anl V- [noul +n1uo] : - 3t and _ aul - - e —- — 2 no 3?— + [ne(ue - V)ue]l = - Eel: noEl + nlEo] - voan . (2. 31) In a region of uniform plasma where the densities of ions and electrons are constant and equal, :0 = O, collisions can be neglected, the perturbation Lorentz force term is neglected, and the unperturbed E field is zero equat'nls (2. 27) and (2. 30) take the relatively simple form 25 8n n V° E = - —1- o 1 3t and __ aul eno _ 2 no—at— = - m El - Voan . (2.32.) e Maxwell's equations can also be separated into equations in unperturbed and perturbation terms, respectively. _ -1 _ v. E0 _ 60 (ni no) (2.33) — - .53. V' E1“ "e n1 0 v x150 = o (2.35) VxE1= -—3-t- (2.36) V x30 = o (2.37) _ _ _ 315 V x B1 = - p.0e(noul + nluo) + 11060 —5t— (2.38) V°B0=V°B1=O (2.39) In the uniform plasma region just described equation (2. 38) can be simplified to __ 8E1 (2.34) V X B1 = - uoe noul + p.060 -5_t_ . (2. 40) Thus equations (2. 32), (2. 34), (2. 36), and (2. 39),together with equation (2. 40),comp1etely specify the dynamics of the uniform plasma. The solution of the perturbation equations for the uniform plasma can be simplified by separation of the plasma variables into two sets of variables corresponding to an electromagnetic and electroacoustic mode as described by Cohen. 8 In this mode separation technique the four perturbation plasma variables 26 are considered each to be the sum of an ”EM” and an "EA" component to be denoted with subscripts 1e and 1p, respectively. E1 B1 3: It can be shown that +E E1e 1p 1316MB,1p u 1e+u1p +n n1e 1p (2. 41) such a mode separation is indeed valid in the frequency domain and that the new modal variables are determined by the following equations. EMmode V XEle = -JwBl —- __S — - — Vx 1— LLOJ -|J.Oen0u1e+_)wp.oeoEle s - "' z 2.... V E:1e 6(a)) 2 1.) did) - 60(1 ~---%) 1.) 2 2 noe w = (2.42) p meeo EA mode V xEl = O P proeoElp-uoe non—1 = O V - E = £— - — n -V ° E 1p 6 e 1 le 0 .2 _ enO E1 jwn 1.11 = -——In—E' -V an (2.43) 27 The terms 78 and pS are source current density and source charge density terms which are independent of the plasma variables; 0) is a plasma parameter called the plasma frequency. It can be seen that .1311) = file = 0. Equations (2. 42) and (2. 43) indicate that the EM mode has an electric field, all of the magnetic field, but no charge accumulation. The EA mode consists of an electric field, no magnetic field, and all of the charge accumulation. It is a simple matter to manipulate equations (2. 43) into a wave equation in 111 2 s w )0 V2+—12-(w2-w2) n = -—P—— . (2.44) p 1 2 V0 evo This is the well known inhomogeneous Klein-Gordon equation. This equation has the particular solution 1 w ' s E-jfipr : ._ __L __._____ 111 411' 2 5 p r dv (2. 45) e v V01 0 where 2 2 b.) - (1) Bp - V0 . The EM equations can be solved considering that the sources exist in a region of constant permitivity 2 (.1) 6(6)) 2 e0 (1 -43). w This mode separation technique is unfortunately not valid when external magnetic fields are present or when the plasma 28 density is not uniform, i. e. , in the plasma sheath surrounding an antenna emersed in a compressive plasma, for example. For the case of the plasma sheath a wave equation for nl similar in form to equation (2. 44) is developed in chapter IV; its extremely complicated nature requires that its solution be numerical. CHAPTER III THE PLASMA SHEATH 3 .1. Sheath Formation Whenever a conducting object is submerged in a hot plasma it assumes a negative potential relative to the surrounding plasma. If it is assumed that initially both electrons and ions are individually in thermal equilibrium at temperatures T6 and Ti’ respectively, where Ti < < Te (generally in arcs and ionospheric plasmas, where the plasma density is low, Ti < < Te) the electrons because of their small relative mass have much greater thermal velocities than do the ions. Therefore, upon introduction of the conductor into the previously uniform plasma, the electrons initially impact upon it in relatively large numbers before the ions can reach it in appreciable numbers. As a result a negative potential is produced at the conductor surface which accelerates the ions to the conductor and repells the electrons from it. Equilibrium is achieved when electrons and ions impact the conductor in equal numbers per unit time. As a result of this process there exists in the plasma surrounding the conducting object a negative potential distribution which increases monotonically with distance from the conductor to the potential of the main body of the plasma. Also electron and ion densities as well as the corresponding drift velocities vary in the same manner in space as a function of the potential. If the conducting object is an antenna the nature of these spatial distributions of 29 30 of potential, densities, and drift velocities will influence the electro- magnetic and electroacoustic waves radiated into the plasma. This region of nonuniform plasma will henceforth be termed the "plasma sheath" or merely the "sheath". Specification of the plasma sheath parameters is a very difficult problem although approximate analyses of various geometries under restrictive assumptions have been carried out by numerous workers. In genera1,the character of the plasma sheath is determined by the effects of particle collisions as well as the effect of the fields due to charge separation. The solution of the problem considering the effects of collisions, diffusion, and drift due to charge separation is a very elusive one, and no one to date has adequately attacked this general situation. A large class of plasma problems involves the so-called "low density" or "low pressure" plasma where collision effects can be largely neglected relative to the dominant effect of drift due to the electric fields of charge separation. Once attention is limited to this class of plasmas a number of simplifying assumptions can be made allowing approximate solution for the major parameters describing the plasma sheath. Various sets of assumptions and corresponding analyses have been made by many authors. 3.2. Analyses of Low Density Plasmas In the case of low density plasmas contained in a finite volume where the mean free path of the ions through the neutral particles (and therefore of the electrons through the neutrals) is much greater 31 than any linear dimension of the plasma a simplified analysis (such as that given by Tonks and Langmuir10 and extended by Self“) for the sheath can be carried out. It is found that the magnitude of the conductor potential corresponds to several times the average electron thermal energy outside the sheath; therefore, nearly all electrons entering the sheath are reflected by the retarding sheath potential so that the distribution function for the electrons outside and inside (except perhaps very near the conductor surface) the sheath should be nearly Maxwell -Boltzmann. It is easily shown that the ion velocities cannot be Maxwellian distributed anywhere in the vicinity of the conductor since the conductor absorbs nearly every ion impacting it yielding a deficiency of ions traveling outward from the conductor. The ion density distribution can be related to the potential distribution by various orbital analyses; Poisson's equation can then be solved for the potential distribution. To illustrate the mathematical difficulty incurred if a Maxwellian ion velocity distribution is used (whether totally Maxwellian or Maxwellian for incident ions only) to calculate the ion drift current density to the conductor a Maxwellian distribution is assumed valid outside the sheath at a distance far enough from the sheath to allow the ion velocity distribution to become essentially Maxwellian with a drift term yet close enough that collisions between this point and the conductor can be neglected. To simplify the description the conductor is assumed spherical and the plasma surrounding the sphere has spherical symmetry. The ion distribution function is 32 M. 3/2 --2—I-<—:I.-:[(vr-uoo)+ve+v¢] __1__ noo (21rKT.) 1 {(17, £7) = . (3.1) For the case of negligible collisions between r = d and r = a the continuity equation becomes _ 2 _ O = V ° (nu) or 411' r nu = constant (3.2) so that the ion current directed to the sphere at r = d equals that at r = a. The total ion current density at r = dis then . . M. 3/2 nu _( fl? ate?” (———1—-)- r=d ’ 0° ZTrKT. vel. space v. s. 1 —M-i—- [( u )2+ 2+ 2] '2KT. Vr' co V9 V4» __ _3 e 1 v dv or nur___d=nGO u,Jo . (3.3) This follows from the definition of the drift velocity and direct integration of f(;, V) V over the velocity space. No information on the value of u,Jo is available from this integration process; also, the assumed distribution f(F, _) vanishes only at v = infinity irrespective of the absorptive properties of the sphere. Generally, the sphere is assumed to absorb all ions impacting it, hence, in the absence of collisions in the sheath no ”outward going" ions should exist at r = a or r = d, i. e. , f(d, v) should vanish for vr > 0 contradicting the Maxwellian form of the distribution for positive vr . The standard Maxwellian distribution seems invalid and an alternative must be considered. 33 The next reasonable approach seems to be that of assuming that the incoming ions have a Maxwellian velocity distribution while the distribution function vanishes indentically for Vr > O . One might then attempt to calculate nco um as 3 f(r, v) V dv by writting this integral in terms of um therlfossoolving the resulting equation in uGO for u(D . This,at first,mlght seem plausible since in this case n u = ‘3 f(l: V) VdV3 as in the case of the complete Maxwellian distribution. This equation takes the form ,2 M 3/2 .0 .+00 - - — —-e- 1 nooucJo =3 f(r, v) v dv - C3 (ZTTKT) S ‘8 V S :-_ 1 so that variations in the values of nw and u00 from their assumed values have little effect upon the erf and hence upon the order of the approximation for 1 [ 1 + erf Kid: (vC ~um)] and um . 2 The approximate relation for um in terms of nw which is normally used has been developed. e W (3.11) It was assumed that collisions within the sheath could be neglected so that V - (nu) = O . This implie s that ~ .0 .0 (g) n > R , the largest plasma dimension without symmetry, so that only drift due to charge separation is considered. They solved by a series technique a "plasma equation" which is really the Poisson equation (3.16) with n. -n 1 e the left hand side set equal to zero in the plasma since . approximates zero everywhere except in the sheath. Theilr solution yields ion density and ion current density at the sheath edge which allows solution for the “wall" potential (conductor potential) ¢w by equating ion and electron currents at the conductor surface as demanded by continuity. Self11 following Langmuir's formulation obtains complete solutions for the sheath potential and density profiles. Since this formulation and the corresponding results. are so commonly used a brief sketch seems in order here. Three symmetric geometries are considered: planar, cylindrical, and spherical with the spherical case pictured in Figure 3.2. Ions are assumed generated at rest everywhere in the main plasma body at a rate C(r) ions/cm3/sec. Mi?) is taken as zero at the plasma center. The drift of ions to the sheath is considered 40 as a result of the presence of ';7(;) only, i. e. , thermal effects are neglected. Further all collisions of the ions once formed are neglected so that the ion drift and density can be determined from simple orbital analysis. To determine n(r) u(r) we consider the number 6N of ions generated in the differential volume pictured in Figure 3. Z of volume ,2 dv' : A(r') dr' : 4TT1‘ dr' at r = r' in time dt with zero velocity and accelerated to m. r') = if @(r) —§(r')) 1 at r = r. Now 6N = C(r’) dv' dt (3.18) and the contribution to n(r) u(r) in time dt when they arrive there (at r = r) is _ 6N * G(r') dv'dt _ 3;! , d(nu) — Am”) dt .- A(r)dt - G(r')(r ) dr (3.19) where I = 0,1, 2 for the planar, cylindrical, and spherical cases, respectively. n(r) u(r) is obtained by integrating the contributions of each differential volume over 0 _<_ r’ E r . Then .r 2 2 n(r) u(r) = 3 G 43—) dr‘ . (3.20) O This integral is generally transformed to an integral over n(r) and the variables are transformed to dimensionless ones to facilitate numerical integration, i. e. , the following definitions are applied: 41 . _ C(S') ___I_‘_ _ n“ g(s ) — C(O) , s — L L — (3(0) (3.21) Then (3.20) becomes n(r)u(r) = not, 1(n(r)) (3.22) where .n s’ I as’ 1M) = 50 g(S') (g‘) 5}]— d-‘l - (3-23) I(n) is obtained numerically and tabulated for I = 0,1, 2. Except for the planar case I(n) is a weak function of the form of the generation function. Two cases are considered: (1) ionization rate constant in space and (2) ionization rate prOportional to electron density. The electrons are assumed Boltzmann distributed so that g(s) can be written Yr; g(S) = 61 where Y1 = O for case (1) and Y1 = l for case (2). Self indicates the following values are valid for Iain“): TABLE 3.1. Self‘s Values for H71») [:0 leo 1(rLN)=O.3444 — __.-_.——-—-_ .3444 .2914 . 2703 . 2571 . 2136 NNt-‘l—‘O t—‘ov—‘or—I OOOOO The contribution to n(r) due to 6N is d(nu) (I I dn00 . Eventually)as r increases,collisions become more important, the analysis breaks down, and drift currents and densities are maintained finite. Bernstein and Rabinowitzl‘2 analyzed the external infinite plasma for the spherical and cylindrical cases. Electrons are assumed to be Boltzmann distributed while an attempt to solve the Boltzmann equation for the ion distribution function is made. Then the Poisson equation is solved for the potential distribution. The ion current density is formulated as a complicated integral. This can be solved in conjunction with Poissons equation only by tedious numerical computation. To simplify the problem ions are assumed to be monoenergetic. Solution for the potential and density distributicns still involves numerical calculation but Bernstein and Rabinowitz present the results of this computation. Although more dependence of the results on ion temperature was found than previously predicted by a majority of analyses indicating essentially no dependence of sheath parameters on ion temperature, the dependence is still not great (approximately 20% over the entire range of possible ion temperatures). In 1949 Bohm14 gave a Simplified treatment ignoring the distribution of energy of the p051tive ions entering the sheath. Bohm 44 showed that the positive ions have an energy given by when they arrive at the sheath edge. This result is obtained by considering the planar case with ions assumed entering the sheath region all at the same velocity uO, electrons are Boltzmann distributed, and collisions of ions in the sheath are neglected. The density distribution of the ions is computed assuming n(r) u(r) = nco u00 : constant where the drift velocity is computed from the potential distribution Lflr) = «fag +-3%q?iii (3.27) 1 so that the density distribution becomes 1"; ni : m . (3.28) V1 +C"'mj) where ZKT 0' '2' e_ . M. ui 1 t» Poisson's equation then becomes 2 d n 1 . 6‘” (3.29) 3 = “—“7: dx \1 l + 0' 3’) where x '5 r/XD and XD is the debye shielding length for the main plasma defined by 45 If the R. H. S. of equation (3. 29) is expanded in a power series in n for small 77, i. e. , in the plasma near the sheath edge, equation (3. 29), becomes —1-—GW,-u_n)+u.:=nu—%q. (i3m If 0' > 2 the solutions of equation (3. 30) in the plasma region will oscillate with wavelength x = O 2nJ%—1 which obviously is not allowable. Therefore cr 5 2 and X KT e M. 1 2 um: from a physical argument; unfortunately nothing can be said at this K T point about an upper bound for u: although M e is found to be i of the right order of magnitude from other analyses. Allen and Thonemann15 also considered the planar case and arrived at the same density expressions starting with the same physical model. They note,however, that at the sheath edge n. =n 5n 1 e s and dn dne a? : dx (3.31) to a high degree of accuracy. It is found as a result that 2 MET ~LI Me at the sheath edge and the conflict is resolved. 1 7 Additionally, Laframboise1 ' has performed an analysis and numerical integration similar to the approach of Bernstein and Rabinowitz but retaining a Boltzmann distribution for the ion density. 47 Other authors have attacked the problem with various approaches largely similar to those previously described. A general result of all these analyses is that the ion drift KT velocity at the sheath edge is approximately Me , The ion and i electron densities at the sheath edge are roughly half their values in the main plasma. The sheath potential and density profiles are nearly independent of ion temperature being dependent almost entirely on the electron density and temperature in the main plasma C a body. 3. 3. Conductor Potential The potential of the conducting object must be determined by equating the electron and ion currents to the conductor in accordance with continuity. It has been previously show that the ion current density is given by ZKT 1/2 e while the electron current density is KTe 'le ZTrm 6 e nooe 1(nW) is determined by geometry and generation function form as is shown in Table 3. l. Equating the current densities yields the potential at the conductor surface as M. _ 1 1 77w - 1n HUW) 41Tme o (3.33) 48 In the planar case Selfll obtains H77) 2 0.3444 W s 0 that n = 3.56+:— ln (3,34) W M M P where M/Mp is the molecular weight of the gas in question. Larson3 has collected typical profiles of potential and electron density from the analyses of various authors for the spherical case and for conductor potentials corresponding to nw= 3. 5. These data are presented in Figure 3. 3. In each case Larson has plotted the curves to correspond to nw = 3. 5 and a/XD = 19.6. 3. 4. Selection of a Sheath Model The value nw = 3. 5 was arbitrarily chosen by Larson because most theories predict nw near 3. 5 for hydrogen and because several of the investigators presented profile data for hydrogen. Since heavier gases are normally encountered a value of nW: 4. 5 was arbitrarily chosen for the following work as it better approximates the values expected for most gases. Larson's analysis indicates that EA wave propagation is not greatly affected by changing the sheath profiles, hence the arbitrary adoption of a set of profiles from any one of the sheath analyses reviewed is justified. As stated in the introduction the sheath thickness will commonly be many times less than the sphere radius; in this Hthin sheath" case the geometry within the sheath and that of the plasma adjacent to the sheath becomes nearly planar. The profiles 49 to be used correspond to the solution of equation (3. 29) deve10ped KT by Bohm for the planar case with um taken as _M.e , i. e. , . i .2 L; 2 __.._._l . _ E"? (3.35) dxd NJ 1 + 22’) where x 2 r/)\D with the boundary conditions .7) r:a '5 nw : 4.5 and (2"0 as xr"+00 . (3.36) Although the sheath potential profile is determined from the planar case for simplicity (because the potential profile description is not a function of the sphere radius a) the other sheath parameters are determined on the basis of spherically symmetric geometry. The equations governing the sheath parameters in terms of r) are repeated here for ease of reference. n(r) 2‘ no nus e (3.37) Us 6” (3.38) m. W (3.39) (3.40) 1 n. : n‘ —————-———-——- (3.41) i 30 *- 1 + 2:7 The solution of equation (3‘. 35) is given in Figure 3. 4. Also given in Figure 3. 4 are the slope of “73(X) from equation (3. 35) and the correSpondiiig electron density profile. 50 metallic \ sphe re sheath W edge Figure 3. l. Maxwellian Distribution at r=d a< d<< 1f the conducting sphere the sheath Figure 3. Z. Self's Plasma Geometry. C) 51 (A) Allen, Boyd, and Reynolds -- a/XD=14. 7, "W = 3. 5 (B) Bernstein and Rabinowitz -- a/XD=13. 0, nw :-. 3. 5, T1 7i.— 30.1 e (C) Laframboise -- a/xD = 10.0, "w = 3. 5 1.6 1.8 2.0 1.0 1.2 1.4 1.6 1.0 ' 2.0 Figure 3. 3. Comparison of Sheath Potential and Density Profiles. 52 .Eufi ottofi can asaaom 53.6 G H 60 co o. ofi . o.m c..~ in .v .m 93m?“ d . . . 4 a C HA PT ER IV DEVELOPMENT OF THE WAVE EQUATION 4.1. Wave Equation Form In this section a wave equation for the density perturbation 111 will be developed from the collisionless first-order moment equations deve10ped in chapter II; this equation will be of general validity inside as well as outside of the sheath. No consideration is given to any possible static magnetic field in the interest of maintaining symmetry in order to simplify the problem. When the moment equations are combined to form a. Klein-Gordon type equation for n1 it is found that the equation contains terms in electron drift velocity perturbation 111 as well as a term in the electric field perturbation E1 . It can be shown that the E term contributes negligibly to the solution for 1 nl once the drift velocity terms are eliminated. Elimination of the El term leaves one equation in n and u1 which cannot be I solved since 1.11 cannot be independently related to I11 . It is possible to rewrite this equation in conjunction with the moment equations to form a system of two coupled second order linear partial differential equations in n1 and ul . In the symmetrical geometry considered in this work this system can be simplified to a system of two coupled second order linear ordinary differential equations in n1 and 111 each of the basic form .2 i d . 5% +a(x) 53V; +b:1-—(Zp-J’\/ZTTQ€T’W Thar +7( -€T’+p)nl olar D XDJ1+2n 1. 69 The final form of the density wave equation in the frequency domain is obtained from equation (4. 27) as Inspection of equations (4. 30) and (4. 32) make it evident that the wave equations in n1 and 111 are coupled as indicated previously. They form a system of two linear ordinary differential equations requiring simultaneous solution; the system may be written in the form L n + L u 1121 L3111 + L4111 H O H O (4.33) where the L's are second order linear operators of the general form 2 2 + b(r) g? + C(I‘) = a(r) d 2 +b(r) g? + C(r) 8r dr L =a(r) and the values corresponding to the set [ai(r), bi”), Ci(1’)] 1:1, 2’ 3, 4 are obvious from equations (4. 30) and (4. 32). 70 Some insight into the extent of the modification of the solution for n1 to be expected upon inclusion of the drift velocity effects can be gained by consideration of the amplitude of the drift terms added to equation (4. 12) to form equation (4. 32). For example, in 2(n-le) is extremely small the first term of equation (4. 32) 6_lr_ 6 compared to unity in almost all of the sheath; it increases rapidly near the metallic surface to 6%- or approximately 0. 05. This term should have very little effect on the solution for n1 . In the second term :- 62(77-7’lw) increases rapidly to Z/Tr as the metallic surface is approached; although appreciable at this surface its overall effect should be small. The same general argument can be given for the third term. Unfortunately,there is no accurate means of comparing the RHS to the other terms in the equation; only solution for 1.11 can yield this information. CHAPTER V B OUNDAR Y CONDITIONS 5.1. IntrchBction Before any of the equations for n1 and 111 developed in chapter IV can be solved sufficient boundary conditions must be described. This can be done only approximately and will be based on the assumption that no electrons take part in the r. f. (perturbation) motion at the surface of the metallic sphere, i. e. , E A _ 1 nr=a_0’ and upon the continuity of the potential perturbation at r = a . Boundary conditions must also be specified at the sheath-plasma boundary where, as will be detailed in chapter VI, solutions for both the electromagnetic fields and the hydrodynamic variables will be matched; this sheath-plasma boundary is assumed to be just inside of the region of uniform plasma so that all electro- magnetic and hydrodynamic variables are infinitely differentiable, and the matching process is trivial. The matching of the far zone electromagnetic and hydrodynamic variables to their corresponding forms in the "quasi-static" zone (Region II) is simple and straight- forward; it is covered in chapter VI and need not be further mentioned here. The basic boundary conditions will be used to derive the boundary conditions required for the solution of the specific equations or systems of equations solved in chapters VI and VII. 71 72 5. Z. Sheath-Plasma BoundarLConditions The sheath-plasma boundary is a surface chosen, at r = d, sufficiently far away from the metallic sphere's surface that the electron density has returned essentially to its uniform value nco , the unperturbed potential is essentially zero, and the unperturbed electron drift velocity is negligible, while still close enough to the sphere to be in the quasi-static region surrounding it. In chapter VI solutions for the electromagnetic and hydrodynamic variables valid in the sheath are matched to those valid in the uniform ”quasi- static" region at the sheath-plasma boundary . As stated before at the sheath -p1asma boundary all electromagnetic and hydrodynamic variables are infinitely differentiable so that matching is easily accomplished. The continuity relations for the variables of interest are given for completeness. lim n = lim n (5.1) r -’ d' l r —' d+ 1 anl anl lim 8r : lim + Br (5. 2) r —’ d‘ r -" (:1 lim <19 2 11m Cb (5.3) r -’ d' l r -' (1+ 1 8¢l 8491 11m 8 = lim 8 (5. 4) r —» d“ r r » d+ r lim u1 = lim 111 (5. 5) r -’ d' r -‘ d Bul aul 11m 8r 2 11m + 8r (5.6) 73 where (bl is the scalar potential associated with the perturbation E field, i.e., since E1 is assumed quasi-static E1 = -vq>l . (5.7) 5. 3. Boundary Conditions at the Surface of the Metal Sphere At the sphere's surface the following field relations are valid 91 x E1 = 0 (5.8) (>1 = V (5.10) where V is the perturbation potential applied to the sphere from an external source, and as is the perturbation surface charge density on the sphere. Equation (5. 9) will be used in the determination of the current to the sphere, while equation (5. 10) will be used directly as a boundary condition on $1 . The boundary conditions for the hydrodynamic variables are not so easily specified, in fact, this difficulty represents one of the major weaknesses of the theory. If only the second order equation for 111 is to be solved it can be shown that only one boundary condition for the hydrodynamic variables is required; in fact, it can be shown that only one relation, independent of the moment anl aul 84> equations, in the variables n1, ul, -a—r——, 317" and 3;— 13 required to specify the problem. Obviously the independent specification of any one of these variables would suffice. Various authors have used various relations here. Cohen8 suggests the following linear relation 74 3.3 =Yfi-E+Yn 1 a 1 b (5.11) l in analogy with acoustics describing the boundary in terms of a surface admittance. He argues that the acoustic analogy to this reflection problem would lead to a linear relation between ii - 11.1 and 111 at the boundary surface, since the admittance is the ratio of the perturbation in velocity to the perturbation in pressure (the so-called ”excess pressure"). ”The excess pressure is the body pressure in acoustics; but in the plasma the electric field also contributes to the body force, and we should include it in the admittance. " Linear inclusion of the electric field leads directly to equation (5.11). The admittance coefficients Ya and Yb may be functions of the configuration of the fields at the metallic surface and are, in general, functions of frequency. Cohen indicates that, ”in principle, these admittance coefficients can be measured, since for each incident wave (impinging on the metal) there are two reflected waves (one EM and one EA wave). A calculation of the coefficients themselves, however, would have to start from the opposite point of view. One would have to solve in detail the plasma -metal boundary problem, including the fields and electron motions inside the metal. " No one has,as yet,devised a method of evaluating these coefficients so that the general form of equation (5.11), while of theoretical interest, is not useful. There are two assumptions commonly used here which are justified on the basis of analogy with acoustics. The first is the analog of the so-called acoustic I'soft" boundary condition in which the metallic surface is assumed to be perfectly absorptive s 0 that 75 n 5 O . (5.12) surface This model cannot be justified rigorously, but it has the advantages of great simplicity and of representing one extreme of surface absorptivity. Another more commonly used surface model is the analog of the so-called acoustic ”hard" boundary condition in which the metallic surface is assumed to be perfectly reflective for electrons so that there is no net perturbation in drift velocity at the surface, i. e. , microscopically every electron involved in perturbation motion which strikes the surface is elastically reflected. This leads to 3. H = o . (5.13) surface A criticism of this model is that, while in the case of the acoustically hard surface the neutral molecules of the fluid cannot penetrate the surface and 9-H isindeed zero, in the plasma case the fluid consists 1 of electrons which can, by nature of their charge, penetrate the surface. The advantages of this model lie in its simplicity, its ease of application, and in the fact that it represents the opposite conceptual extreme of the soft model. Larson3 performing an analysis similar to the present one uses both hard and soft boundary conditions and achieves‘similar results in each case. This. might be quite surprising considering the extreme difference in the surface models if it were not recognized that the unperturbed density drops to very small values at the metallic surface so that the perturbation density should be small also regardless of the surface model used. 76 In any case this result is a consoling one since the proper model is in doubt. It can be noted that the hard and soft boundary conditions correspond to the following admittance coefficients, respectively: HQ. B.C. Y = b Soft B. C. Har Ya 0 Balmain20 suggested the following absorptive boundary condition a u1= -u EBm Pl at r=a (5.14) KB e where uKB is the velocity of sound in the electron gas, no is the unperturbed electron density in the uniform plasma, and GKB is a dimensionless constant depending on the nature of the surface. For a "completely collapsed sheath" (totally absorptive surface) O'KB approximates «[2/17 , whereas a = 0 for the "perfectly rigid“ KB (perfectly reflective) boundary. While this absorptive boundary condition obviously represents an oversimplified picture of the actual behavior at the metallic surface, it would seem to be an improvement over the hard or soft boundary conditions; the problem with its use is the determination of QKB . All of the boundary conditions discussed have been used by various authors in various applications; each has its advantages. For the purposes of the present work the more conventional hard boundary condition is arbitrarily adopted,primari1y for reasons of simplicity, however use of a conventional boundary condition facilitates comparison of results with those of others. 77 5. 4. Hybrid Boundary Condition for the Density Solution The solution of the density wave equation in chapter VI requires 84) an a linear relation in -—1— , n , and —— . The perturbation moment 81' 1 8r Bu equations contain these variables plus 111 and Sr in addition. Use Bu of the hard boundary condition 111 = 0 eliminates ul , and —81-- can be eliminated from the moment equations by subtraction yielding a relation of the proper type. Consider the perturbation moment equations (2. 31) reduced to the frequency domain. . — — - _ _e__ — — Z jtonou1 + [ne(ue°V)ue] — - me[ noE1 + nlEo] - v0 an (5.15) l and Vo[nou1 +n1u0] = - jwn1 . (5.16) In the case of spherical symmetry these equations become Buo 8u duo Jn(3““11‘Lno‘11 8r +nouo 8r +nluo 8r = an e Z l - EB: [nlEo+noEl] -vo Br and due anl Bul BnO 2 nl 337+uo 8r'+no 8r + 8r u1+ r(nluo+noul) : -anl ' Applying the conditions 111 = O and r : a while multiplying the second equation through by uO leads to 8 u Bu . 2 anl 8431 (5.17) 78 and Bu BuO 2 u: 2 anl 110110 337+[u0 81‘ +qu0] n1 +1.10 ar 3 0. (5.18) Subtracting equation (5.18) from equation (5. 17) a relation of the desired form is obtained as Z 2u an en 84) e o . 2 l _ o l (m I50- a -3wuo)nl Jr(Vonuo)"5?- me 6r (5'19) Dividing through by v: and substituting for uo, E0, vi, and no equation (5.19) becomes 2 Bales-MEL [ _1_(sl.:|an1+ 1 3r — -61ra 6r 3X ve D 5 d4 . 52 d2 - P ---(—) +1 —(-)—‘n (5.20) [ 1* a ma __Jl 2 where the parameters Lop, XD’ nw, v0, p, 52, d, and a have been previously defined, and 6 is defined as xD/a. Equation (5. 20) can be written in a more compact form for use in chapter VI. 8491 anl _B?_ _ D1 3r _ + B1111 — (5.21) r—a r—a r—a where v e 4 77w l C1 D1 — "—7—— e l: - a; (5)] Lo 6 p o and v2 e 4 2 T? l 6 d . d B1: 0 6w_3_)\_. pa~;(a)+3 fig.) «(211' 79 For the thin sheath case these coefficients become 2 V e ' _ 0 ”W l- l 1 to 6 p o — V28 o T) l . Q B = 6w [p +j—] . (5.22) 1 (3:20 3)D a «[2? 5. 5. Boundary Conditions for the Drift Equation If the wave equations for 111 and 111 are solved by the iterative method discussed in chapter IV the solution of the drift wave equation Bu (4.30) requires specification of the values of 111 and 3;- at the metal boundary. The specification of 111 = O at the boundary fulfills the first requirement. It is not difficult to apply this condition to Bu the first moment equation thus determining 5}— at the boundary in Bnl terms of the values of n and -— 1 8r The first perturbation moment equation in the frequency domain for spherical symmetry is l 8 2 . _2 B? r (noul +n1uo) : - anl r which can be written in the form an] Brio 2 n0 anl Buo Zuo nO——8r +( 8r + r )ul+uO—--—ar +(-—-—-ar+~--—r +_]11.))n1 = O . (5.23) Letting u1 = O and r = a equation (5. 23) becomes Bu 8n due 2 + ; 110 + jw)n1 (5.24) 8O n u Bu u2 an Bu —°° 1-—3 1+--1-—( °+— +1.) can 2 ar‘ 28r Zuoar aquuo)l v v v o o o Remembering that u _ KTe d: 71-le o — - 2am 2 E e r so that au 0 - in 2 8r — uo [8r - r] then 2 2 nouO Bul uO anl 110 ‘81)- wu ' 2 8r =‘Zar +[——2-8r+J 21ml (5°25) v v v v o o o 0 Substituting for uO equation (5. 25) can be written as ncnucn 8‘11 1 ((1)2 81‘11 + 1 pla (d)2 . $2 .— : _— -— .——-——- — _J — n V2 81‘ 611' a 3r 3RD ZTT a m la 0 a a (5.26) For the thin sheath case this degenerates to -nm OD Bul : _L 8n1 + 1 p13. -j —-- n (5 27) V: 3r a 6n 3]: a 3RD Zn NI-Z—TT 1a Equation (5. 26) is the desired form of the boundary condition for 8ul 8r ' The possibility of solving the wave equations for I11 and 111 simultaneously has been discussed in chapter IV. Without going into the details of such a solution, it can be observed that specification 81 anvl aul of a set of boundary conditions for n , u , —-—--—- , and will be 1 1 Br Br required at some common point r on the interval [a, d] . If the On boundary conditions for n1 and Br can be determined at r = d, then it can be shown that 1.11 and Bul /8r can be specified in terms of the values of n1 and anl/Br . This relationship can be developed as follows. Since the plasma is assumed uniform at r = d with negligible unperturbed electron drift velocity or electric field, the uniform plasma relations described in section 2. 3 are applicable. If the B- field is neglected the following relations can be developed. -enO-ii1 +jweo-E-1 = O (5.28) SE - _ _1_3_. 2 _ l a _ _e_. V'1‘31 ’ zarrE1‘ar +rE1“e n1 (5'29) 1‘ O jwnomeul -.- -nOeE1-mev:Vn1 (5.30) Solving for E1 from equation (5. 28) en 0 ._. . 5.3 l jweo ul ( l) Differentiating this expression with respect to r 8E en Bu 1 _ o 1 8r — jweo 8r ' (5°32) Substituting equation (5. 31) into equation (5. 30) and simplifying yields V2 8n - s. o 1 u1 ‘ Jn 2 2 8r 0 t) «w P L.) V: anl .—. jif- 2 8r ' (5.33) 82 Combining equations (5. 29), (5. 31), and (5.32.) 2 Bu . V 8n 1 . Lo 2 o 1 —— : -J _— n + _' C (5.34). 8r n0 1 a. “’12)“? _1). 8r Equations (5. 33) and (5. 34) specify 111 and Bul/Br in terms of I11 and 8nl/8r in the uniform plasma region which includes r = (1. These equations then specify boundary conditions for 111 and Bur/8r in terms of those for I11 and anl/Br . CHAPTER VI ANALYTICAL FORMULATION 6.1. Introduction As indicated in chapter I,the plasma surrounding each sphere of the antenna can be divided into three separate concentric spherical regions. The first of these is the sheath region, designated Region 1. extending from the sphereis surface, r = a , to an arbitrarily chosen radius r = d sufficiently far out into the plasma so that the unperturbed electron and ion densities have returned very nearly to their values in the uniform body of the plasma; the unperturbed electron drift velocity and the unperturbed potential have very nearly vanished at r = d so that for r i d they can be taken as identically zero with little error. Region II extends from r = d to r = cl1 where :11 is chosen to be within the "quasi-static" zone surrounding the sphere; since Regions I and II are within the quasi-static sons the solution for the "electromagnetic" fields can be simplified accordingly there. Region III includes the radiation zone and the transition sons between the radiation zone and the quasi-static zone; it extends from r = (11 to + 0° . Solutions for the electromagnetic and hydrodynamic quantities in regions II and III are easily obtained since the mode separation discussed in chapter 11 is valid. The solution for these quantities in Region I is greatly complicated by the effects of the nonuniformity in the plasma density; the only practical approach seems to involve numerical methods; a numerical approach to this solution is outlined in chapter VII. 83 84 The goals of this chapter include indicating the basic forms of the solutions for the pertinent electromagnetic and hydrodynamic quantities in each of the three regions and apprOpriately matching the corresponding solutions at the interregional boundaries in order to obtain the complete solutions. This matching process makes use of the boundary conditions developed in chapter V. The input admittance will be approximated making use of the quasi-static solutions for the electron density and electric field perturbations. The effect of electo- magnetic radiation will be included by computing an effective radiation conductance term calculated, as usual, by use of Poynting's theorem; the electromagnetic energy radiated is determined by comparing the far field scalar potential to that of an electric dipole and writing the classical electric dipole expressions in terms of the appropriate electric dipole moment. The density perturbation at the sphere's surface (r=a) is of interest for resonance probe applications; this is derived in a form useful for comparison with other theories. 6. 2. The Hydrodynamical Wave Equations The basic equations describing the propagation of the hydro- dynamic quantities n1 and 111 were derived in chapter IV, i. e. , equations (4. 32) and (4. 30), respectively. In Region I,where these equations assume their most complicated form,both must be solved. In Regions II and III,they assume relatively simple forms, are decoupled, and only the density wave equation need be solved; the drift velocity term 111 can be determined from the density term n1 by use of equation (5. 33). In chapter VII,equations (4. 32) and (4. 30) for Region I will be transformed to forms suitable for solutionby iteration. Only equation (4. 32) in its simplified form is solved in Regions II and III; exact analytical solution is accomplished in these regions. The form of equations (4. 32) and (4. 30) is expressed in opera- tional form in equation (4. 33), i. e. , ll 0 Lln1 + LZu1 H O L3n1 + L4u1 where the L's are second order linear ordinary differential operators of the general form 82 8 L = a(r) 8—2- +b(r) 3—17 + C(r) . r In the iterative approach to the solution of this system,L2 is first assumed identically zero, the coefficients of L1 are simplified by retaining only major terms, and the resultant zeroth order decoupled homogeneous equation is solved for two linearly independent solutions. These solutions are appropriately joined to the solutions for n1 in Region II; the boundary conditions at r = a,developed in chapter V, for n in terms of anl/ar and mil/a: are applied completely determining 1 the zeroth order solution for n1. Next this zeroth order solution is used in conjunction with L3nl to determine a decoupled inhomogeneous equation in 111; this equation is solved in conjunction with boundary conditions deve10ped previously to obtain the first order iterated solution for ul. This solution is then substituted into the first equation of (4. 33)(i. e. , as Lzul), and the resultant inhomogeneous equation in n1 is solved as before to obtain the first order iterated solution for ml. The exact form of the zeroth and first order Operators are determined from equations 86 (4.12), (4. 30), and (4. 32). In any of the three regions the basic equation to be solved, then,is a second order linear differential equation of the form anl —a—r— +[if(r)nl = G2(r) (6-1) Vznl + Gl(r) where the coefficients are determined from equation (4. 32). In Regions II and III,where uo and r) are taken as identically zero, it is evident that equation (6.1) reduces to the particularly simple form V2n1+82n1 = o (6.2) since G1(r) and G2(r) vanish and 8%”) goes over to 2 2 2 - Z (a) -U B1(r) : fl :: 2 ' v o In Region I, for the zeroth order solution for n1 , Gz(r), which corresponds to the inclusion of the drift terms, is dropped and (31(1') is reduced to -lfl1___1 61m“ 3 8r ' 3). P ' (6'3) D and only the dominant terms of [3‘12(r) are retained so that 032 -w2f fife) = 21’ r = szm (6.4) O V where and equation (6.1) becomes 87 8n 2 l l 2 _ an+?x-Bp—a-;—+B(r)nl—O. (6.5) It is evident that equation (6. 5) goes over to equation (6. 2) in accord with the assumption previously made regarding plasma uniformity at r = d. The solutions to either should apply equally well at r = d; this concept will be used to match the solutions at r = d. One further comment on the solution of equation (6.1) is required so that no further reference to the form of the equation need be made in this chapter. It will be useful to consider that equation (6. 1) has two linearly independent solutions so that independent boundary conditions can be specified for each solution; i. e. , if Gz(r) = 0, equation (6.1) is homogeneous and has two linearly independent solutions which can easily be shown to be of theform y1=l+a1x§+a3x:+... y2=x2+b3x3+b4x:+... (6.6) where x2 '55 r-d/XD, so that the following relation is true at x2 = 0 (r=d): ryl(0) 3’2“” 1 0 = (6. 7) 6371(0) 63’2“» 0 1 ' 8x2 8x2 i l— _J L _i The form of equations (6. 6) follows from the fact that equation (6.1) is homogeneous with coefficients regular everywhere in the region of interest which guarantees the existence of two linearly independent solutions in the form of Frobenius series (convergence is open to 88 question of course). Equation (6. 7) will prove very useful in numerical computations as well as in setting up analytical solutions. The problem arises when the first order inhomogeneous equation for 111 is solved (i. e. , Gz(r) # 0 in equation (6.1)); in addition to the two linearly independent solutions,there is a particular solution determined entirely by the zeroth order solution for 111 . The question arises as to the usefulness of a relation such as equation (6. 7). This problem can, fortunately, be dodged in the following manner. The drift terms are considered zero at r = d, and they remain relatively insignificant except for the portion of the sheath nearest the spheres surface. Since the effect of the drift terms is negligible or zero at and about r = d, so also should be the particular solution of equation (6.1) in the vicinity of r = d. Thus the homogeneous solutions represent the complete solution at r = d, and the situation is the same as for the zeroth order density solution, 1. e. , equation (6. 7) holds for the complete solution; adding the particular solution does not remove the possibility of forming two linearly independent solutions. The complete solution thenis (btermined by superimposing the particular and homogeneous solutions. The point of this argument, the justification of the general use of equation (6.7), has been made. 6. 3. Transformation of the Density Wave Equation Equation (6. 2) can be written in a more useful form by noting that for spherical symmetry .23 Sr2 sz = '1'“ (If) so that 2 l 8 _ :8r2 (rnl)+(3 n1 - 0 or EBZN1 2 a2+13le0 (6.8) r where N1 '=’ rn1 and 2 2 2 w -w (3 5 __B. 2 v o This equation, valid in Regions II and III, has the general solution N C 6361. +C e-jfir l o l where _ 1 2 2 B — :- 1.) ~10 o if w>w ,andifw1.) and P l Cle'br (6.11) N for w< (up . It might be questioned at this point whether or not it is advisable to denote the density coefficients in equations (6.10) and (6. 11) by the same symbol C1; no trouble arises from its use however since these equations apply to different frequency ranges, and the results are analogous and nearly identical, jfi being merely replaced by b = I (3| in extending the results for w > (up to the range w < wp. It is noted that equation (6. 8) has solutions of the form of equation (6.6) satisfying equation (6. 7); these are given by e-jfir = cos (3r - j sin Br and 6-br = cosh (3r - sinh (3r where the sets y1 = cos (3r y2 = %- sin Br and cosh (3r y1 y2 %- sinh (3r satisfy equations (6. 6) and (6. 7). Equation (6.1), valid in Region I, can be similarly transformed using the same variable changes and noting that 1_a_(rn)__l_61\11_3111+le r 8r 1 - r 8r — 8r r Equation (6.1) becomes 2 8N N l 8 l l l 2 _ FEB-:Z-(rnl)+Gl(r) (F-EF -;'z)+fi (r)n1 — G2(r) (6.13) which is easily rewritten in the form BZN1 8Nl ?- +g1(r)—a—r_'+g3(r) N1 : 32(1‘) (6°14) where the coefficients are obvious from equations (6.1) and (6.13) although not of any particular interest here except in so far as their regularity is concerned; the transformation to equation (6.14) involves multiplication of certain coefficients by r and 1; which,as can easily be seen,cannot effect their regularity on the interval r = a to r = d. On this interval equation (6.14) has two linearly independent solutions of the form of equation (6. 6) which satisfy the relation in equation (6. 7). 6. 4 Matching Solutions at r = d In Region I,the general solution for N1(r) is a linear combination of the linearly independent solutions justified. Nl(r) = Ay1+ByZ ‘ (6.15) where A and B are constants to be determined. The continuity of N1 and anl/Br at r = d guarantees the continuity of N1 and N'1 there (primes will henceforth be used to designate derivatives with respect to r, except where noted) so that N1(d’) Nl(d+) N1'(d") Ni(d+) . (6.16) 92 In Region II, from equation (6.10), (316-de -iBC1€-j5d (6.17) .+ N1(d ) , + N1(d ) where the notation N(d+) is shorthand for 1li_1;nc.l_,l\l(r). In section 6. 3, it was indicated that the development for w< top is completely analogous to the w > top case presented, the steps being identical if jB is replaced by (Lil. That this is true can easily be seen from the similarity of equations (6.10) and (6.11); if use of equation (6.11) is visualized in lieu of equation (6.10) in the development of this section the analogy is obvious. From equations (6.15), (6. 6), and (6. 7) N1(d') A - l N'(d ) _ B (6.18) 1 >‘D so that the following relations are developed C1 6"de A B _ijDA _ (6.19) Now if the following definition is made Mr) E ,1 ~16 nyz . (6.20) Nl (r) can then be written as Region I: N1(r) A(yl - jfix Dyz) = A N(r) RegionII: Nl(r) cle‘mr = Ae'jmr—d) . (6.21) The perturbation potential (1)1 must be determined everywhere and matched to the driving potential perturbation V at r = a. The potential perturbation can be determined in general form by integrating Poisson's equation after transforming it in the manner of equation (6. 8). 93 Poisson's equation in the perturbation terms as derived in chapter II, equation (2. 34), for the present symmetric geometry takes the form en 2 2 l 1 3 V $1 = e = g 2 (NP) 0 Br 01‘ 82 e 23—291”) = ;— N1(1‘) (6-22) r 0 where §1(r) is defined as r451 . If equation (6. 22) is integrated twice with respect to r in indefinite form ¢1(r) is determined as a(r) = C2 + C3r + P(r) where P(r) E‘ E?— §S.N1(r)dr dr . (6.23) o . In Region II N1(r) = c 6.3;” which implie s P(r) = .. 62 cle’jFir . (6.24) 608 Since (>1 must go to zero at least as fast as :7 as r approaches infinity, §1(r) : r¢1 should remain finite as r approaches infinity; C3r is unbounded so that C3 must be taken as zero. 2 In Region II, §l(r) is then given by i1 = CZ+P(r) . (6.25) In Region I the situation is more complex. Evaluating P(r), the first integral of Nl(r) is H > \I‘: Z fl 9.. H 3.N1(r) dr 94 and the second integral becomes SKNIU) dr A S S N(r) dr dr or or A ‘Sd dN(r) dr dr + (ea -j(3>\DYa)(r-d)+ub -j(37\DYb (6. 2.6) The continuity of 431 and W1 at r = d guarantees that of §l(r) and Qi there; this,in conjunction with equation (6. 25),guarantees the continuity of P(r) and P‘(r) there also. Potential matching at r = 6. will be carried out on the basis of the matching P(r) and P'(r); this process leads to a specialization of the integration constants ea, ob, Ya' and Vb . At r = d, the following relations are true: P(d+) = - 82 cle‘jfidz - ez A (6.27) 605 6013 P' + = ' e ‘jfid : ' _§_ .2 (d) 360 C16 jeoflA (6 8) mm = f— A[ab ~jsxDvb] (6.29) 0 Mel“) = g“?- AEaa ~1‘MDYa] (6.30) 0 Applying the conditions P(d+) = P(d‘) P'(d+) = P’(d‘) (6.31) the following relationships between the integration constants are obtained. - - _1.. ab_Jfi)\DYb ‘ " (6.32) l {—4. "531*“ 'U) Cla — Jfi‘XDYa 95 The set of integration constants satisfying equations (6. 32) is not unique,but if that set is considered real,then the set is uniquely determined and potential matching is accomplished if a = O a a _ l b ‘ "'2‘ (3 Y _ l a — '- 2 [3413 Vb = 0 . (6.33) Then,in Region I, P(r) becomes or s P(r) = i MS Srmr) dr dr + j l(r-d) - i] (6.34) 60 e d s d a B2 so that the potential perturbation is given by: C2 e A 2 .r 'r Region I: (bl = —r—— - -——2- ;-[l - j(3(r-d) - B ‘8 S N(r) dr dr] 600 d‘ d (6.35) C . Region II: (bl = 7.3- - :2 A? e-er-d) . (6.36) 6 o 6. 5. Matchinggat r = a In chapter V,a hybrid boundary condition was deve10ped from the hard boundary condition. This condition,(5. 21),and the condition that 491 = V at r = a (where V is the applied potential perturbation) can be used to determine the constants A and C2 in terms of V ; this process completely specifies the solutions for (PI and n1 , as well as the radiation fields, in terms of V as will be shown subsequently. The hybrid boundary condition to be applied has the form 96 where the coefficients are given in section 5. 4. At r = a, 8¢1/8r determined from equation (6. 35) is 69¢ (3 ,a ,r 81 = 'i " eA {“Lll " jB(a-d)-fizlS N(r) dr dr] + r 2 2 2 . r=a a €05 a d d -a 5 as 452) N(r) er} ‘ d from which 34> C 1 2 e E __r- ' ‘2 + “—77 A (6.37) r23. a, 606 a where 2 .a .r 2 ea E=l+de-B 8.8 N(r)drdr+(3a5 N(r)dr. ' d d d From the definition _ Nl _ A N(r) n _ —-—- _ 1 r r so that 8n 1 l l “5‘;- = A[-;—Z-N(r)+-1-: N'(1‘)] and 8n 1 __ A F 8r — - 2 (6.38) r=a a where F E N(a) - a Ni(a) If equations (6. 37) and (6.38) are used to substitute for anl/Br and 8431/ Gr in the hybrid boundary condition the following relation between C2 and A is obtained 2+ eEA _ DlFA+BlN(a)A - 2 2 2 _ - 2 a a E (3 a a O 9'7 01‘ e E2 - B1 N(a) a] . (6.39) 608 c‘2 = A[D1F+ Matching cpl at r = a to the driving potential perturbation V allows the determination of A in terms of V . From equation (6. 35) C -a -r V = 4’1 : .35. __27 §-[1-jfi(a-d)- (328 S N(r)drdr] . (6.40) rza €013 ' d. d Substituting for C from equation (6. 39) equation (6. 40) can be written 2 Ia er aV=A{D1F+ 6L; -B1N(a)a - ez[1 -js(a-d) -1325 S N(r)drdr]} 60(3 608 d d 01' aV = A[D1F+£—H2- -B1N(a) a] (6.41) 600 where ,a H '5 j8a+02a8 N(r)dr. ‘d A and C2 are completely determined in terms of V as eH '1 A = a.V[D1F+ 2 .131 N(a)a] (6.42) 6 P O and 131 F+9E2 -B1N(a)a 6 (3 C2 = aV OH (6-43) D1F+ e 2 -B1N(a)a 608 6.6. Matching at r 2 d1 Equations (2. 42) describe the EM mode in the uniform plasma without a constant magnetic field. They can be written in the same form as Maxwell's equations ina linear isotropic lossless medium of 98 permitivity 2 This can be seen if equation (2. 43) is subtracted from the frequency domain version of equation (2. 32) yielding ° _ — __3.— jwule - meEle . (6.44) If equation (6. 44) is substituted into the second equation of (2. 42) then z _ — —s “06 nOEle -- VxBlqu + . +jwueE o jmew o 0 le or (02 __ _ -s . - .2 _. 'V x13 — HOJ +jtsuoeo U. wZ)'Ele (6.45) This being the case,it is a simple matter to derive wave equations for the EM field quantities. For the purposes of this analysis the usual wave equation for the scalar potential in a homogeneous medium is sufficient, i. e. , 2 2 V ¢le+fle ¢le — O (6.46) where 2 2 2 to ~63 (3e : (,0 (1,05 : 7.2. and 2 l c = e “o 0 Equation (6. 46) is applicable everywhere in Regions II and III and has there the general solution (tiger. ¢1=——-; e I“ 99 only the forward prepagating term is retained since the geometry does not justify a reflected term so that E‘jfier (bl = a --—-—- . (6.47) e 4 r In the quasi-static region (I and II) the term 6 -3691. is nearly constant so that ¢le can be considered as being included in the Cz/r term; that this is true is the heart of the "quasi-static" approximation. The EA mode has a conservative field (V x Elp = 0 from equation (2. 43)) so that Elp = -V¢ 1p (6.48) There being no independent source charge in regions II and III equation (2. 43) yields V-E' :- 1, (6.49) and V ¢lp = 6 . (6.50) Equation (6. 2) is easily shown to apply here for the EA mode since this equation involves the total density perturbation n1 (the equation's derivation did not involve the concept of mode separation), and the EA mode contains all the charge accumulationfi. e. , n1p = n1). Equations (6. 2) and (6. 50) can be combined to form p 2 l e t3 0 Equation (6. 51) can be transformed to 2 2 8 e a (r4> ) = - (In) ; (6- 52) ar2 1p 6 62 ar2 l O 100 integrating twice with respect to r yields -en1 C16 ¢1p = Z+<15+ T (6.53) 6 B 0 so that 'jfier 0‘ _ .e. €_.___ .6. o tend to zero as r approaches infinity and dropping Demanding that ((91 the nonphysical nonpropagating terms o6/r equation (6. 54) becomes en -j(3 r _ l e e The potential expression (6. 55) can now be used to match the far field scalar potential perturbation to the quasi-static potential perturbation given in equation (6. 36). Demanding that both equations r = d leads to (6. 55) and (6. 36) hold at l t - __enl . . __u _ . . 1 _ ‘ ‘” 2 4 d ‘ d ' 2 1 _ r—d1 e 08 r=d1 l l 600 r--d1 (6.56) which implies immediately that (6. 57) C2 eJfiedl 6.7. The Far Zone Field As an immediate consequence of this matching process the radiation zone scalar potential is determined in terms of C2 , which has been determined in terms of the driving potential perturbation V, as (6.58) _ €"jl3e(1‘-dl) ¢le — C2 r V lOl Consider an oscillating point charge Q at r = 0 in a region of uniform permitivity 3 2 10 6:6 (1-—-§-) w the general particular solution to equation (6. 46) in terms of the source charge density ps is 1 ‘ F’s —js r : — e ¢le 47r€ l r 6 dv ’ and for the point charge 93 = Q 5(r) so that _ Q e'Jfier 4)le — 411' e r (6' 59) Comparing equations (6. 58) and (6. 59) it is seen that a single sphere of the antenna looks like a point charge Q located at the sphere's center where Q is related to C2 (and hence to V) by the relation 0 = 4n 6 c2 emedl . (6.60) Since d1 is arbitrary the phase relationship between the source and the far field is lost, but the amplitude and relative phase variations are maintained. By symmetry this description is valid for each sphere of the antenna and the resultant field is the superposition of those due the individual spheres; in performing this superposition the antenna-space geometry must be considered, i. e. , the relative phasing of the fields at any given point in space will depend upon 1 the orientation of that point relative to the antenna as well as upon the separation between the spheres. The geometry of the antenna, as detailed in the introduction, 102 was chosen so as to meet the requirements of an electric dipole; it is electrically small, its magnetic dipole moment is zero, and it is modeled as far as the radiation field is considered as two point charges Q oscillating at frequency 0) in antiphase and separated by a distance D . Since the phase relation between the source and a radius r in the radiation zone is lost in the expression for Q and only the relative phase of the fields from each sphere is important, Q can be taken as Q = 4nelc21 . (6.61) Neither EA or EM prOpagation is possible when w < Lop since both (32 and (3: are negative indicating attenuation so that this radiation model breaks down in this case. The radiation zone EM fields can be determined from the scalar potential given in equation (6. 58) and the vector potential due to the current in the antenna feed wires. A more simple method consists of merely writing down the radiation zone field expressions for an electric dipole in terms of the dipole moment QD; these classic dipole field expressions are given in many texts. 21’ 22 Ramo21 gives the following expressions in terms of the feed current assuming the dipole to be oriented with both sphere centers on the vertical axis in spherical coordinates. 391D jwu I D _' E9 2 ' “4:3“ Sin9 6 Jfier . (6°62) 103 The feed current is related to the dipole charge by IO : ij (6.63) so that equations (6. 62) become BerD HCb : - 41T1‘ 2 D . 0 “0 Q ‘Jfier . (6.64) E9 = --—-a-;T——1:-—- Slneé "jfier sine E The field expressions can be linearly related to V through equations (6.61) and (6. 43) replacing QD by DF+ 8E -13 N(a)a 1 (37‘ 1 QD=41TEICZI D=41reaDV 0H (6.65) DF+ e -B N(a)a 1 2 1 . 608 The radiated power is computed by integrating the time averaged outwardly directed Poynting vector over a large sphere of radius R The time averaged Poynting vector concentric with the antenna center. is given by __ _afc s __ ExI-I __ E9 HY); av — 2 _ 2 2 w2 2 “-0 fie 02 D 2 : --— 2 2 sin 9 . (6066) E 32w R The average EM power radiated is then (320) 2 ()2 D2 2:2] 12w (6.67) 01' 4 102 5/2 2 ._..1_ __ .2' 2 PaV—90( ) (1 —w2) lczl D . (6.68) 104 6. 8. Radiation Conductance Any solution for the driving point admittance (or impedance) using the quasi-static solutions cannot include the effect of electro- magnetic radiation. Because of the short electrical length of the antenna, its driving point admittance is largely capacitive so that a radiation conductance computed from the radiated EM power can very nicely be added to the quasi-static input admittance to yield a good approximation to the total input admittance. The total potential applied to the antenna is 2 V so that the effective radiation conductance is related to the EM power by Pav Ge = 2V2 (6.69) so that Ge can be written in terms of C2 in the form 4 L02 5/2 c 2 _ _l. w. .2. __3. 2 Ge _ 180 (It) (1 ‘1.)2) V D (6°70) This expression can be written in a form more useful for numerical calculation by recalling the definition of the normalized frequency variable $2 55)-- and by noting that equation (6. 43) for C2 can be P written in terms of a dimensionless quantity KC , i. e. , D F+ 6E .13 aN(a) 1 E (32 1 K S OH , (6.71) C D F+ e -13 aN(a) 1 £32 1 60 sothat C2 = aVKC . (6-72) 105 With these definitions equation (6. 70) can be written as __1 “:24 Ge 2180(C) 2 5/2 1 2 2 2 5(5) -1) DaIKCI or if (op/c is defined as kpe this equation takes the form _ 1 2 2 _1_ 2 5/2 2 C'e _180 (kpe D) (kpea) [$2 (9 -1) chl _ 1 2 2 - _180 (kpe D) (kpea) K.3 (6.73) where _ _1_ 2 5/2 2 Ke-Qm -1) (KC! 6. 9. Quasi-Static Input Admittance The perturbation current to each sphere can be calculated in terms of the quasi-static solutions indicated previously. The current supplied to the antenna is related to the sphere's surface charge and the plasma current as follows: I=J4Tra2 80's .1: 8t +Jp and Jp = -e(nu)l (6.74) where 0's is the surface charge density on the upper sphere, J is the magnitude of the plasma current density perturbation directed away from it, and (nu)l is the magnitude of the perturbation component of the electron drift current there. The (nu) term 1 can be reduced to 106 (nu)1 = nou1 +n11uO 01‘ u (6.75) (nu)1 2: 11 since 111 is taken as being identically zero at the sphere's surface. The surface charge density 0'S is related to the field parameters by the relation rza 8¢1 oar (6.76) II I m Combining equations (6. 74), (6. 75), and (6. 76) while transforming the first to frequency domain description,the current I can be written in the following te rm 3 2 . 84) I 2 417a -Jw€o ‘5';- -e(uon1) (6.77) r: 1'33. Use can once again be made of the hybrid boundary condition given in equation (5. 21) to reduce this current expression to one in n1 and 8n1/ 8r only. 2 8“1 I = 41Ta -Jw€O(D1 8r +Blnl )-e(uon1) _ r=a r=a r-a (6. 78) an Replacing 73-11!- using equation (6. 38) and noting that 107 and r=a from equation (3. 38), equation (6. 78) becomes KT 2 1:41: e 2am: ( ) N(a)a+jweO(D1F-B1N(a)a) A. (6.79) DJICL Replacing A , using equation (6. 42), a complete expression for I in terms of V is obtained as KT 2 e ane (g) N(a)a +3-on (D117 - B1N(a) a) I=4TTaV e H ' (6'80) D F+ e -B N(a)a 1 2 1 600 The quasi-static input admittance to the antenna is given by Yp = “zit? (6.81) since the potential applied to the antenna is twice the applied potential perturbation of the individual spheres. From equation (6. 80), Yp is KTe d 2 e ane (E) N(a)a + jweo(DlF - B1N(a) a) Y = 2n a H (6. 82) P D F + e — B N(a) a 1 2 l 6 l3 o It will be shown in chapter VII that Yp can be written in the form Y : 2 n a e 0.) K 6. 83 p 0 p y ( ) where the dimensionless quantity Ky is a function of frequency and plasma parameters; the definition of Ky is obvious from equations (6. 82) and (6. 83). 108 6.10.) EA-EM Power Ratio It is of interest to compare the ratio of the power radiated in the form of electroacoustic waves to that radiated in the form of electromagnetic waves. No loss mechanism has been included in the quasi-static analysis so that GP : Re[Yp] : Z'rr aeowae[Ky] (6.84) accounts for the power radiated in the form of electroacoustic waves. Similarly, no loss mechanism is included in the EM mode assumed here, and the radiation conductance computed in section 6. 8 from the power radiated in electromagnetic form represents a valid radiation correction to the quasi-static admittance. It follows that the ratio of the radiated electroacoustic power to the radiated electromagnetic power is given by the ratio of the conductances GP and Ge , i. e. , 5; EB Pe - Ge . (6.85) P 360 1r aeo w R [K ] 152 z 4 2 .21J 2 Y 5/2 2 (6°86) e k D a — (9 - 1) l K I pe 9 c Applying the definition of kpe’ equation (6. 86) can be written as 32 3 {2 Re [ KY] 2 (6. 87) Fe (k D)2k a(QZ-1)5/ZIKIZ e pe c P In chapter VII, KY and l KC) 2 are reduced to a form suitable for numerical evaluation. These dimensionless quantities, as well as others, are plotted in chapter VIII. Equation (6. 87) can be written 109 more compactly in terms of the dimensionless quantity Kp defined as 3 {2 R [K ] K E 2 5/2 8 ya P (52 -1) IKCI sothat 32 1 = K . (6.88) Fe (k D)Z(k a) P pe pe 6.11. Application to Resonance Probes In as much as one of the goals of this study is to check out the relatively simple analytical results of Fejer6 who studied essentially the same geometry but neglected the sheath, the application of his theory to resonance probles will be checked with the present theory. Without going into the theory of resonance probes in detail,it can be said that resonance probes operate on the principle that when a radio frequency voltage is applied to a Langmuir probe the perturbation in the current as wellas the direct current collected is a function of the applied frequency. It is normally accepted as an experimental fact that the collected DC current shows a sharp increase at the plasma frequency. Fejer disputes this claiming that (l) the change in the DC current is due to rectification caused by the nonlinear characteristic of the Langmuir probe, (2) the amplitude of the radio frequency perturbation in the collected current will be proportional to the density perturbation 111 at the probes surface, and (3) the peak in n1 in his theory does not occur at no but at a frequency appreciably P r=a lower than wp. To the extent that his arguments are valid, the 110 conventional theory of resonance probes leads to very serious errors. Fejer's expression for n in the notation used here is l r=a 2 a 3?. 92 -1 :2_wE ) V0 (6 39) n : V - ; o lr=a e v: (1 - €22)1/2 this expression is relatively small for $2 > 1 , has a zero at 52 = 1, and is entirely real for 82 < l with a pole at 2 2 2 1/2 v0 a 1.) 1/2 9 : ———-2 2 (1+4——B-2) -1 . (6.90) p 2a wp v0 It might be noted that as the factor amp/v0, which can be written a/Nf3 XD, approaches infinity the singular point 52p approaches zero so that for the case of the thin sheath very appreciable errors could be made assuming that 52p = 1 . The present study indicates a some- what different density-frequency profile; this is plotted in chapter VIII. The density perturbation can be written in terms of V and the plasma parameters in the form 2 (1) € : —E—-—2 O V Ka N(a) (6°91) r=a V e 0 where the dimensionless factor Ka is a function of frequency and plasma parameters; the form of Ka is determined from equation (6. 42.) and is developed in chapter VII. The quantity A is related to Ka by the relation A _ _P__° a—V— K . (6.92) I'll Iii.l11| I! CHAPTER VII a FORMULATION FOR NUMERICAL SOLUTION 7.1. The Mathematical Sheath Model The purpose of this chapter is to manipulate the results of chapters IV, V, and VI into a form amenable to numerical solution. It will be noted that the separation of the plasma into the sheath region and the uniform region (Regions II and III) as described in chapter VI is an artificial idealization of the actual situation described in chapter III. If the point r = d is chosen sufficiently far from r = a the conditions of uniformity at r = d assumed in the model will be very closely achieved; however, if d is chosen too far from r = a practical difficulties arise in the numerical solution of the wave equations on the interval [ a, d] . The second order differential equations involved are solved by numerical integration on this interval, and since the solutions are either oscillatory, at times involving many cycles of sinusoidal like functions on the interval, or exponential like functions with large growth rates, the stability of the method over the entire interval may be a problem. Thus, if d is chosen too far from r = a stability requirements may force the use of extremely fine subdivisions of the interval for calculation, and uneconomical amounts of machine time may be required. The choice of the arbitrary value of d represents a compromise between the extremes indicated. The sheath model chosen in chapter III for this study is not dependent upon the sphere radius and is shown in Figure 3. 4; the normalized potential 11 drops to less than 2. 5% of its value at the spheres lll 112 surface in 10 debye lengths from the sphere, and the electron density returns to within 10% of its uniform value. The sheath thickness for the mathematical model is arbitrarily chosen as 10 debye lengths regardless of sphere radius; a better approximation could be had if this thickness were extended to 20 debye lengths at the cost of appreciably more machine time. The various coefficients in the differential equations will be evaluated from an arbitrarily chosen mathematical model used in lieu of the physical model of Figure 3. 4; this mathematical model is chosen so that the normalized potential 1') (1) is described by a low order polynomial in r for simplicity of evaluation, (2) assumes the same value as the physical model at the spheres surface, chosen in chapter III to be 4. 50, and (3) goes to zero with zero slope at r = d maintaining uniformity (i. e. , continuity of derivatives of plasma variables) there. Plots of the physical model potential determined in chapter IIIJas well as the approximate mathematical model for n, and the coefficients of the zeroth order wave equation for n will be given in Figures 7.1 and 7. 2. The l analytical forms of the polynomials used are indicated in the figures. 7. 2. Transformation of the Wave Equations First consider the complete wave equation for n1 derived in chapter IV, namely equation (4. 32). If the variable substitution Nl(r) 5 rn1 is applied, and if it is noted that 113 equation (4. 32) becomes 2 4 ‘l a N 4 2(n-n ) x [1+ .1. (£1) 62(77’le)|1; 1+ _1_... [:p + ($1.) §__._w_ (2p _ 3%J 611' r arZ 3RD r 11’ (7.1) r-__| HI... Q’ .929 H 1—- I NI} L___l + .0) r—N *1 HI 2 p—J u 1 N :3 N o o: N NJ: If this equation is multiplied through by rkgA and the common terms are collected it becomes 4 2 2(n-n ) 4 x 1 d 2 - 8 N( l W l) 8N X 6w r 8 '1?) 1 L k a(r)—.77) 4 2 k 2 k 2 2 p__D e W' 9 8n 2 _P. __D — {8(r)>.D-3 r+3Tr (r) axz+2p +b(r) .6pr }N(r)- L l 2 2 nouO Ar 8 ul 2 2 (7.2) v 3x 0 1 where 82(r) is defined in equation (4.12) and r - a x 5 . (7.3) 1 RD Once n is defined all the terms in the coefficients of equation (7.. 2) are easily determined; this equation is in the proper dimensionless form for computation except for the right hand side which will be modified by the defintion of the dimensionless counterpart to N(r) for u1 . Since boundary conditions for the linearly independent solutions for n are given at r = d, a different normalized independent variable 1 x d-r: x RD 2 114 will be used for the density wave equations of zeroth and first order for convenience; x will be retained for the drift wave equation 1 since its boundary conditions have been given at r = a (x1 = O). In the development of the normalized drift wave equation it is convenient to start with equation (4. 15) transforming it in terms of N1 as in equation (7.1). 2 Z a2u+28r1o §E+(8 n0 £— ) - u 8 N1 + (2 fine +10) 8Nl + no 2 Br Br 2 - 2 no u_ - o 2 8r J 5r Br Br r 8r azuo Z w _| ( 2 “'17 o‘J?)Nli (7'4) 81‘ _- where u '5 r ul (7.5) If equation (7. 4) is multiplied through by 2 qui, and if the coefficients are written as explicit functions of r) the following equation is obtained. 2 ncouoo (g) azu 4P nocuoo (9)2 £1}- + Z noouoo(g)21 2 _ V? r 81-2 " 1D V2 r 8r 2" r 1'2 P o 0 V0 D 2 2 X 2 2(n-n ) 4 8 N 9—1-3- - 2(3)]11 ‘ - E W E 21 + 8 r 311 r 8 x1 1 r E. 62(77-77w) Q4 _ z x_D.) ,J i (21);: n'nw-I .8311. + 3 11 r r m r __1 8x1 201 n ) 4 2 k2 X e ' W q a g 2 _2 D 311 r) (8x2 +p +4r2 '4r p)+ l 2 $2 - d-2 )‘D J 3' —— 6?? 77w (1:) (T) N1 } - (7-6) VZW 5 11 Further,if the following definition is made anumAru - l U .. - 2 , (7.7) V0 . . . . . p Z Z 2 and 1f equation (7. 6) is multiplied through by kDr /d then the complete normalized drift wave equation is determined as 2 2 x 2 2(n-nw) 2 2 fl-zpfl_+pz_§_fl__z(_2) Uzi—__(i) .a_Nzl£1.+ 2 8 Z r 31r 3x1 1 8x1 3x1 L. I. 2(n-nw) 2 X g): 1%)1p-2r—D)-1——” e"'"w 44331“ + ‘_ V211 1 2(n-nw) 2 2 X X _ K -§-.-.-——<%> <—%+p2+4—-- wig-261% ~52 um- 3r r 3 211 The definition equation (7. 7) suggests the way to proceed in reducing the right hand side of equation (7. 2). From the definition of u it is easily shown that 22 - r 3111 + u 8r — 8r 1 and Z 2 8 u Bu 8 u 1 1 2 = r 2 + Z —a-— Br Br r from which azu - azu 3.. .82. + 2.1.1... 1' 2 ‘ 2 ‘ r 8r 2 Br Br r Multiplying equation (7. 9) through by 2nOuOA - ZnQuOOA (Si—)2 - 2 7 - 2 r v v o o (7.8) (7.9) 116 it becomes 2 - 2nouO Ar. 3 ul : (d)2(82U -£ 8U + 2 U) (7 10) 2 2 r 2 r 8r 2 ° ' 0 8r r Substituting from equation (7.10) in equation (7. 2) the complete normalized density wave equation is obtained after transformation of the independent variable x 1 4 2 2(n-n) 4 X 14.51.. (51) t::(n‘nw)]_uaNr E+€______W (51.) (213-33.) _LlaNl‘ + 11 r 3 311 r r 3x _) 8x2 1. 201- -n) 4 2 1.2 x rD W d an 2 D D [_sz frl-E Jr“3‘17"'-"---"(-1:)(altzp +6 2-6TPJNU‘) x r . z 2 k XD 2 (4, __3 U ‘2'}2 __aU +2U(—D—) (7.11) r 2 8X1 8X1 where fr is defined in equation (4.11). This equation will be used as the first order density wave equation in the iterative solution. A word about notation seems in order here. The forms of the functional coefficients of these wave equations are not changed as the independent variables are changed, although, strictly speaking, they are different functions of the new variables, and this fact should be recognized, by using different symbols, for example. The approach used is simple and probably less confusing. Such terms as d/r, d/a, xD/r, and XD/a can be written in terms of a single parameter, the debye length-sphere radius ratio defined as 6 ; this description will be given later. The zeroth order density equation is obtained from equation (7.11) by dropping the drift terms on the right hand side and retaining 117 only the leading terms in the coefficients (these are considered dominant), i.e., 2 2.32%11 -132 39.1311} +;_ [92-fr]N(r) = o; (7.12) X it is expected that equation (7. 12) will yield a good approximate solution for N(r) which can be used to start the iterative solution. 7. 3. The Iterative Solution The iterative approach to the solution for n1 is simple in concept and has been previously described in general in section 6. 2, however there are a few procedural details which should be mentioned. It is clear that equation (7.12) has two linearly independent solutions of the form given in equation (6. 6) satisfying the boundary conditions given in equation (6. 7); the boundary conditions will be used also for solution of the first order equation (7. 11) although this must be justified since equation (7.11) is inhomogeneous and has a particular solution depending on the drift terms on the right hand side in addition to the two linearly independent solutions. It is obvious that the two linearly independent solutions can be made to satisfy equation (6. 7); but what of the particular solution? The mathematical model calls for uniformity at r =d (x = 0) so that the drift terms become inconsequential in the vicinity of x = 0 where the boundary conditions are specified. To the extent that the drift terms on the right hand side of equation (7.11) are negligible the equation becomes homogeneous, and the particular solution vanishes in the vicinity of x = 0 . Boundary conditions suitable for particular solution of equation (7.11) are then given by 118 Np(0) = N'p(0) = 0 (7.13) where Np represents the particular solution to equation (7.12), and the complete solutions (as well as the homogeneous solutions) satisfy P110) yaw) _ 1 o (714) Jim) v'zw) o -1 — the same set of boundary conditions satisfied by solutions to the zeroth order equation (7.12). (Note: primes here indicate derivatives with respect to x -- here only.) The results of the solution of the zeroth order density wave equation are substituted into the drift wave equation (7. 8), and this equation is solved for its particular solution using the boundary conditions developed,in the next section, from the basic form given in section 5. 5. The results of this solution are then substituted into the right hand side of the first order density wave equation (7. 11) which now becomes an inhomogeneous equation in N(r) . Although the boundary conditions given by equation (7. 14) could be used to solve equation (7.11) for the complete solution for N(r) , it is more convenient to solve independently for the particular and homogeneous solutions. The homogeneous solution for N(r) is then NH(I‘) : Y1 ‘jfixDYZ (7°15) in accord with the definition in equation (6. 20). The first order iterated solution for N(r) is obtained by addition-of Np(r) and NH(r) . 119 7. 4. Boundary Conditions for the Drift Equation The boundary conditions for equation (7. 8) are obtained starting from equation (5. 26); making the usual variables substitutions and noting that 0.1 = 0 it becomes r=a - nmum aul _ 1 (2 2 l 3111— - :1- v 2 8r — 6? a> r 8r r o r=a r=a 2 l 1 d 0 + 5x" 2'; ('5') P 'J NIT n1 D r=a " r=a or ‘ 2 - nmum Bul _. 1 (2) BN1 2 ' - 61-1. a 8r v 8r 0 r=a r=a 1 1 d 2 "D o + T 7,," (1;) (P ”T )-1 N1 ”-161 D _ \[211 r—a r=a Multiplying equation (7. 16) through by ZAXD while noting the definition of "U yields 2 an _ _1_ 51 3N1 8561 _ .. ' 7 311 (a) 81’ch x1=0 xl='0 1 d2 "D 252 I (7..) (p - )‘j —"__— N o (7017) 371- a a 3 ("T17 l , _ r=a xl - 0 Completing the picture, the fact that 111 vanishes at r = a implies that 120 U = 0 . (7.18) Equations (7. 17) and (7.18) are the desired boundary conditions for the dimensionless drift wave equation. 7. 5. Reduction of the K Forms The evaluation of the major functions derived in chapter VI depend upon the calculation of the dimensionless forms Ka" Kc? K9, Ke’ and Kp since it has been indicated that 2 w 6 V e 0 C2 = aVKE: Y = 211a€ to K p 0 P Y _ 1 2 _p 2 Ge " '1'8'6 (kpeD’ (kpea) Kg. 5 1 = . K o (7'19) 2 9 Pa (13551)) 1332a First solving for 1%, from equation (6.42) -l A e H —aV = DlF - B1N(a)a +—€o 73 where D1 and B are defined in equation (5. 21) while F and H are given 1 in equations (6. 38) and (6. 41). Substitution for these terms leads to __ 121 V 2e v 2e A _ o , o 1 TV- — —-2—- 01(N(a) - N (3)3.) +——z—- “x"— 0'2N(a)a w e w e D P 0 P 0 -1 .3 + ,—9—z-(j[5a+[52a 5 N(r)dr) (7.20) 6 13 d o where n 4 _ l d 0'1: 6 W 1-3-1;('a-) and 0' ._._€_:’: P -.1_.)\_R(é)4+J Q (2)2 2 3 r=a 11' a a V217 a Factoring common terms and writing the derivative and the integral in ': terms of x 2 .. A z 322:1 a N(a)+_a__ 3 N(a) 1W v e 1‘ in 1x 1 O 10 '1 a a l . 2 2 . ' + —— a" N(a)+—- --z—- (Jm - s x S N(r)dx) (7.21) If the 6 parameter is defined as x 6 E .32 (7.22) then factoring 6 from equation (7. 21) Ka is determined as 122 1 «r8752: K: 6 61 (6N(a) + 53x— N(a)) + -02N(a) +3 1 .0 -1 S N(r)dx . (7. 23) o L 3 o It should be noted that,as mentioned in chapter VI. all of the forms derived are valid for 52> 1 ; for fl< l the same forms are valid where applicable if j[3 is replaced by b E 18' or equivalently if jmz-l is replaced by m2. In equation (6.43),C2 is given as l 606 C2 = aV . eH DIF +1? - B1N(a)a O 1 N(a) a Substitution from equations (5. 21), (6. 37), (6. 38), and (6. 41) yields the following: 5.2. aV .a . .r 01(N(a)-aN'(a))+UZ—:'— N(a)+—l——[l+j 5d+flzaS'N(r)dr-flz‘8 N(r)drdr] D 812-1 'd d‘d a «1(N(a)-aN'. D'Sd N(r)dr) (7. 24) 123 K C 10 1010 8N(a) 6 0' 1(6N(a)+——-—-—-')Ir.1';,‘l\1'(a)+ + N(r)dx-— N(r)dxdx T J—_—2",-3-1_T"'3‘ 751,310.10 7 .17) BN a) . l 1 0'1 (6N(a)+-a—§‘—)+ 02N(a) +3 3 2: - -3- .80 N(r)dx (7. 25) Ky can be determined from equation (6. 82) which gives Yp in the form Z KTe d 2 e m:— (z) N(a)a+jw€o(DlF - B1N(a)a) Y =211a P eH DlF +——z- - B1N(a)a €08 Substituting for D F, Bl , and H l 9 Yp = 211a KTe d2 28 8V0 2 0'2 ° 211m (5' “MW Ham o[‘£2';e_°' (N(a1- -aN'(a))+-—T -- 2N(a>a] . e “p 60 66816 D . V0 e evo 2 0-2 a 106 70—01(N(a)- 2111\11(a))4—0h7°--)k — DN(a)a+——e—-2-(jpa+8 was N(r)dr) 9 ° ‘09 €°B d (7.26) Further reduction,proceeding as before,leads to Y =211a P 6 w (i 2 o[ )] "' Na '1‘ (1)6 016Na +38 xNa +0 2N3 A 1 10 47sz i N(r)dx .. ' O lIH 618(6N(a)+axN(a)) + O'ZN(a) +1 124 from which KY becomes 1 _ 3V 211 Y .10 ° 8 N(r)dx ’o 2 1%) N(a) +1016116N1a1+ ~53,- N(a))+ 62mm] K (7. 28) ooh- a . 1 0' (6N(a)+ -—- N(a))+0' N(a) +3 - 1 8X 2 ~73er Next K8 is determined from equation (6. 73) where it was defined as _ 1 2 5/2 2 K6 “Ti-(9 -1) (Kcl . Ke is determined once KC is obtained. Kp is determined from Ky and KC” as indicated in equation (6. 88), i. e. , 352118 [K2] (”2 _1)5/2 chi2 A Earlier reduction of %, _i-p’ and; was promised. These terms can be written in terms of the parameter 6 defined in equation (7. 22) noting that the sheath thickness has been taken as a constant 10 debye lengths, i. e. , d-a 2101‘ D O Dividing by a yields d '5- : 1'1‘106 . (7.29) )1 The factor can be written A D 1 r _ ———-1— (7. 30) 125 01‘ D __. 1 1 (7.31) 104-3-}: since, as it is easily shown, x =10-Xl= -X2 . (7.32) Finally 5:- Can be reduced to 10+— 9 5 (7.33) r +_l__ X1 6 7. 6. Comparison of Dege_nerate Forms with Feier's Results It might be noticed that the major differences between this analysis and that of Fejer stem from the inclusion of the plasma sheath. At first glance there appears to be little similarity in the results obtained,or in the analytical forms of YP, Pe’ n(a), etc. obtained,in this chapter and those of Fejer. Fejer ignores the existence of an unperturbed electric field, an electron drift velocity, and of the static potential of the spheres; these terms effect the results of this theory to a great extent. If the analytical forms for Yp’ Pe’ n(a), etc., (i. e., equations (6. 82), (6. 68), (6. 92), and (7. 23)), are modified by the deletion of the terms which Fejer ignores it is easily shown that the forms become identical to those derived by Fejer. To accomplish this modification, first let the sheath parameters, which Fejer ignores, take on their uniform plasma values, 1. e. , let 126 and n = n (7.34) in the sheath region; also let the static potential of each sphere vanish. Second let d-> a, i. e. , set d equal to a so that d - a x (d) = = 0 (7.35) l )‘D This is equivalent to shrinking the sheath to zero which is consistent with Fejer's uniform plasma assumption. As a result of this shrinking process all of the integrals in the equations mentioned vanish and the following degeneration occurs. N(a) —>1 8Na . .1 2— 8x *JfiXD-Tjfim-l 0'l ->l 02 -» 0 0'3 ->l -a 8 ~ 0 d '3, ex H - 0 'd d n e‘” -»1 (7.36) If a parameter 6f is defined by 127 6f 3 ———1— (7.37) «[3 6 and if substitutions are made using equations (7. 36) and (7. 37) into the expressions for Yp’ Pe’ n(a), etc. , it is seen that all of these expressions go over exactly to those deve10ped by Fejer if it is noted that 6f has the same definition as Fejer's 6 parameter. 128 x ~22; .o + x 38.0 13321.2 - m N 2 \ so a .3 3386.83... ax 1 a a 6“ Enos» 16235335 .I I I. II Lem 1 \ \ a no“ 3.08 233.322. . Mk 1 ohm o .4 d1h I A l ax .. S . . N x x 1 u aRTE 3m N + «RTE mi N as m CCNM >3 pondewxoummd c no.“ Hobos» 13398293982 1 11 1 I1 c HOW fivpoE Howmlfia m.~1 o.N o.m ow o.m 129 dxx. . um mom 33602 anon—massed: pom Hommlfna .N .N. ounwfim M v .01 I‘ll, T m 6D! /. . r ~61 / / Is- I: on. o.» /...~. o8 e.m or. as o.~ o; .1 v /- 11 n w v v n / L. 2 .o / / 1N .o 00.3.35 / H305 / rm .0 I .o , . m .o m / 1 o .o 6x61026635 + Nx 83 .o - 2 1 as a / w m r N. .o .2 33 h >3 poumegoummo w no“ HopoE dooSmEoLumE I1 1| 1| 1| // T w .o .H / / .1 ® .0 m HOW Epoch Hmowmcfim / I 1T o A CHAPTER VIII NUMERICAL R ESULTS In sections 6. 2 and 7. 3 the iterative solution of equations (7. 8) and (7.11) for the normalized versions of 111 and ul has been described. The first step in the procedure is to solve the simplified homogeneous equation (7. 12) for its two linearly independent solutions yl and y2 . Examples of these solutions are plotted for the case of 5 = O in Figures 8. 8 and 8.9. They exhibit the expected decaying oscillatory nature in that portion of the interval [0 f x _<_ 10] where w exceeds the effective local plasma frequency pr-fr ; on the rest of that interval they experience a growth with increasing X . This character is expected because if f1. and p are replaced by their "uniform plasma" values (i. e. , 1. 0 and 0, respectively) equation (7. 12) becomes 2 $11 + pa N(r) = o (8.1) which has solutions of the form of sines and cosines if (32 is positive and of the form of hyperbolic sines and cosines if (32 is negative; the solutions to equation (7.12.) bare a strong resemblance to these functions in the related regions. The increase of oscillation rate with X is due to the decrease of f1. with X; the damping of the oscillatory solutions is due to the presence of the first derivative term involving p. If (.0 > wp then yl and y2 with the imposed boundary conditions given by equation (7. 14) should approximate cosine and negative sine solutions, respectively, 130 131 which they appear to do. The solutions for N(r) and its appropriate derivatives and integrals are substituted into the K relations given in chapter VII; the results are plotted or tabulated in this section and represent the results of this study in that all of the quantities of interest are related by constants to the K's. Yp is related to Ky by equation (6. 83), and KY is defined by equation (7. 28). Zeroth order solutions for Ky are plotted in Figure 8.1 for 6 = 0, 0. 01, and 0. 05‘. A geometrical limitation has been placed on this analysis and the results presented herein by the use of a fixed sheath configuration. Reference to the work of Bernstein and Rabinowitz12 indicates that sheath thickness and potential profile change negligibly on the range 6 = 0 to 6 = 0. 05 but for larger 6 values the sheath shrinks. Extrapolation from their work indicates that the appropriate sheath thicknesses for this study were approximately 9 )‘D for 6 = 0.10 and 5 x for 6 = 0. 50. The sheath was shortened by D changing the variable X to allow the mathematical sheath potential model to assume its terminal value at X = 9 and X = 5, respectively. Values of the zeroth order solution for Kz determined by this process are tabulated in Tables 8. 7 and 8. 8. It is seen that the imaginary portion of Ky is a very weak function of 6 (at least over the range of 6 values considered); further, with the exception of a zero at 52 = 1, it corresponds fairly closely to a capacitor of capacitance 21r€ 0a which would be the capacitance of the antenna if only the capacitance of each sphere 132 with respect to infinity were considered with the sphere to sphere capacitance ignored and with a free space medium. The real part of K corresponds to EA wave propagation since no loss mechanism was included in the “quasi-static" analysis from which it was derived; it is negligible with respect to the imaginary portion for Q greater than about 1. 5 regardless of 6 but increases with decreasing Q as Q = l is approached from above until it reaches a large sharp peak very near 52 = 1 beyond which it drops rapidly to zero at Q = l ; it vanishes identically for (2 less than 1. 0 since EA propagation in the surrounding plasma is not possible. Figure 8. 1 would tend to indicate a general increase in ReKy with 6 for a given 9 although when the sheath model is appropriately shortened to accommodate larger 6's it is found that somewhere between 6 = O. 05 and 0.10 a maximum is achieved and ReKy decreases monotomically with 6 thereafter. Figure 8. 2. is a plot of the zeroth order solution for Kz; it is simply the reciprocal of Ky’ but it is plotted to facilitate comparison with other analytical and experimental results. The results of Fejer6 for the case of 6 = 0. 01 are plotted in the same figure for $2 > 1 for comparison purposes. It might be noted that Fejer's result is generally of much greater magnitude although it has similar trends and has the same limiting values as $2 - + 00 and as Q ‘14-. As in the case of Ky’ ReKz increases with 6, for fixed 9, until a peak is reached between 6 = 0. 05 and 0.10 thereafter decreasing with 6 . 133 The form I N(a) Kal is proportional to the density perturbation at the sphere's surface, i. e. , n1(a); the specific relation to n1(a) is given in equation (6. 91); the zeroth order solution for this quantity is plotted in Figure 8. 3. It can be seen that there are three major peaks in this term. The first peak occurs between 52 = 0 and Q = 1, its amplitude increasing and its position migrating upward with increasing 6; the second peak occurs very near but below (2 = l; the third occurs very near but above {2 = l . The second and third peaks appear to be essentially part of the same local maximum with the very sharp notch caused by a zero at 52 = 1 (it is noted that if a loss mechanism such as collision damping were considered this zero would not occur;there would most likely appear a. more shallow but wider notch). These multiple peaks contrast with Fejers results; his analysis yields a single pole between 52 = 0 and fl = 1 although his solutions do increase in magnitude with 6 and his pole does migrate upward with increasing 6 , and he does show a zero at Q = 1; there are then several major similarities in the two sets of results. Between the first and second peaks there is a local minimum of very small magnitude; it appears not to migrate appreciably with 5. Finally it is noted that [N(a) Kal = o for all a if 5 = 0; this is true also in Fejer's analysis. Ka is defined by equation (7. 23); zeroth order solutions for Kai are plotted in Figure'8. 4. Ka is identically zero for all 82 if 6 = 0. It is seen that Ka increases in magnitude with increasing 6; both the real and imaginary parts approximate slowly decaying sinusoids for Q5 1 except near (2 = l where each assumes 134 relatively large and sharp peaks before going to zero at Q = l . The value of the far field static potential is related to the parameter C2 by equation (6. 58); (32 ,in turn,is related to Kc by equation (6. 72.); Kc is defined by equation (6. 71) and more explicitly by equation (7. 25). Plots of zeroth order Kc are given in Figure 8. 5. It is seen that KC : 1+j0 (8.2) for 6 = 0; it assumes this value very nearly for all 6 except very near (2 = l where a minor deviation can be noted. The radiation conductance term Ge is related to Ke by equations (6. 73); Ke is defined in terms of Kc also in equations (6. 73) as 5/2 2. 2 - 1) IKCI . K 1: e ((2 3|“ Zeroth order solutions for Ke are given in Figure 8. 6. Since Kc is very nearly equal to l + jO except near 52 = 1 where its deviation is not great and since in this region of deviation ((22 - 1) is very small, Ke can be represented to a good approximation by . 5 2. KB = (:22 -11/ (8.3) SDI“ for all 6 . The ratio of power radiated in electroacoustic form to that radiated in electromagnetic form- Pr/Pe '.is related to' Kp by equation (7.19): Kp is defined by equation (6. 88). Zeroth order solutions for Kp are plotted in Figure 8. 7. Kp appears to be a weak function 135 of 6 with greatest dependence near 9 = 1; it increases without bound as Q = l is approached from above and decreases monotonically with 52 for all 52 > 1 . At 82 = 1.1 , PP is many times larger than Pa for all antenna geometries satisfying the "short antenna“ requirement imposed by the adopted model, but for 0 > 3. O the situation is reversed for all antenna dimensions except the most minute; the point of power equality falls at the value of Q where K = (k D)zk Pe P ea . (8. 4) P If,for example,the antenna geometry is such that D=0.5 a = 0.1 (3.5) k pe k pe then at the point of equality Kp = O. 025 which corresponds approximately to $2 = l. 35 essentially independent of 6 . In order that greater accuracy may be conveyed than that possible with curves Kz, Ka’ [N(a) Kal , and KP are tabulated for 6 = 0. 01 in Tables 8.1 through 8. 4. In order that a quantitative picture of the effect of the plasma on the input admittance may be obtained values of Yp and Ge were calculated for 6 = 0. 01 and the dimensions given in equations (8. 5); the results are tabulated in Table 8. 5. In this case it is easily shown that G = 1.39x10'5K e e P / P = 40 Kp 3 Y =1.667x10‘ K . (8.6) p v 136 For the same case Pp/Pe was calculated and tabulated in Table 8. 6. The total input admittance is YT = Ge + Yp (8.7) so that G T Ge + Rer . (8. 8) Zeroth order solutions for GT are also tabulated in Table 8. 6 for 6 = 0. 01 and the dimensions of equations (8. 5). It is seen that CT takes on relatively large values near 52 = 1 (in fact, GT increases without bound as $2 = 1 is approached from above) due to the excessive amount of EA power radiated there (EM power is negligible here). At first ((2 > 1) GT decreases with 52 as the dominant EA power decreases; as the increasing EA power becomes appreciable a minimum is approached near 52 = 1. 4, and beyond this point GT increases along with the then dominant EM power. For reasons of economy in the use of computer time,first order iterative solutions were sparingly computed. First order solutions for Kz, Ka’ [N(a) Kal , and Kp for 6 = 0.01 are tabulated in Tables 8. 9 through 8.12. *s'M r{.'_‘.'!I-'.l 137 > . Va HoUuO guonoN 4 .m 3ng vm 138 . ex sopuo fonoN .N .m ousmwm m 1 o 3\3H Din . J .4 o.N J H... l o.m . // 3! . n w AHOHOWV .mo 0 mO.OH O /‘ Sens ’ _ Ill/II. _ I III -_ .O ~O.O.OH© l\ U _ vcammo _ modno u. one . ) I); 2?\1 Edna I/l _ 5 ... . / _ d_~ 2‘ // ”z... I // _ 15 I, . .. .o x z, ..o N S x . u / moo / / z z I , , Cohort 3.0 .1. o / , / . .o 1 #111111. , . fomnmn Ema .CoEddE. / / NMEHH1 Nvmmm puma AMQH an 3E 385 fiouom 51w Samoa 139 a. 3 3 m e o.N OJ 0 wed .1. c 1m.o W i lees.“ ~06 .11. 0 mod n a L LmJ mo.o n w 10.~ m Lmd v2.37: n a ROM Munoz g01 x I 2)! MNI 140 .m cm & one .Sm "302 undo .Cdcwmmfifl. all. 1 1| 1| undo amok m¢.o u o . mm Honcho fouoN .v .w muawwh re; 10 .N vo.m to :v x €01 )I . 6M “250 foHoN .m .w enough 141 a 3\3m d w.~ cam ¢.~ Nd o.~ w; elk-lull) Sans U aim ououomnoaoz moans . mvm uoUHO nuOHoN .o .w onsmwm 142 @3\3m~m ow 3.8. ..~ wk. mum ohm . mm . 8.; 4.4 N; o; .1O~ ..o~ 110M #8. - p n 8 ism NE: NE # . x $23 Om so .0... 05:04 m or Nunez. 143 a M nopuO fiouoN .>.w ousmfm a 3\3m d o.m w.~ o.~ v.~ ~.~ o.~ m.“ o4 v4 N; o4 - d 1 J 1 4) q d 1 i L ,2 . c u a l\ //,... Hodno 1 mod n o 4 o .V1 o.M1 144 . 5» now mcofisfiom 1393.5:on .w .w Samoa 145 adnG . N> pom mcofisaom >Hm5ccflounw .o .w magmas 110 .mi... . IEoMII' is-.. 1.0.7. J-o.~.1 LLIO .N 41 r I om1 om_ ON A: ofi ON 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 1.01 1.05 1.10 1.20 1.30 1.40 1.50 1.70 1.90 2.10 2.50 3.00 146 R 1( 0.6503 0.0836 0.0314 0.0128 0.0057 0.0018 0.0006 0.0002 0.0002 0.0001 0.0000 0.0000 TABLE 8.1. Zeroth Order Kz for 6 = 0. 01. INIKE - 9. 565 - 4.770 -3.156 -2.328 -l.790 - 1.315 - 1.757 -1.398 - 1.230 - 1.211 - 1.499 -.0.7708 0.8642 - 0.8827 — 0.8360 . 0.7781 - 0.7228 - 0.6735 0.5928 0.5297 - 0.4788 0.4015 0.3342 147 TABLE 8.2. Zeroth Order Ka for 6 = 0. 01. _52_ 102 REKa 107' IMKa 0.10 -0.024 -0.002 0.20 0.001 0.000 0.30 0.001 0.000 0.40 0.002 0.000 0.50 0.004 0.001 0.60 0.009 0.013 0.70 0.014 0.005 0.80 0.011 0.000 0.90 0.017 -0.002 0.95 0.029 -0.007 0.99 0.065 -0.028 1.01 0.146 -0.107 1.05 0.113 0.020 1.10 0.082 0.000 1.20 0.054 -0.040 1.30 0.012 -0.060 1.40 0.028 - 0.046 1.50 0.044 -0.017 1.70 0.026 0.033 1.90 0.022 0.029 2.10 0.031 - 0.010 2.50 0.025 -0.010 3.00 0.019 0.011 148 TABLE 8.3. Zeroth Order [N(a) Kal for 6 = 0.01 0 lo2 |N(a) Kal 0.10 O. 026 0. 20 0. 027 0. 30 0. 029 0. 40 0. 034 0. 5O 0. 046 0. 60 0.094 0. 70 0. 033 0. 80 0. 002 0. 90 0. 010 O. 95 0. 017 0. 99 0. 027 1. 01 0. 043 1. 05 0. 029 1.10 0. 023 l. 20 0. 023 1. 30 0. 024 1. 40 0. 021 1. 50 0. 019 1. 70 0. 018 1. 90 0. 016 2.10 0. 014 2. 50 0. 012 3. 00 0. 010 149 TABLE 8. 4. Zeroth Order Kp for 6 = 0. 01. 52 K ._ __P__ 1.01 2.8031104 1.05 1.0451102 1.10 6. 5911100 1.20 5.151110"1 1.30 9.28::10‘2 1.40 1.641110”2 1.50 3.11::10'3 1.70 6.251110"4 1.90 3.45x10'4 2.10 2.10::10'4 2.50 5.101110"5 6 3.00 8.24x10' 1.30 1.40 1.70 1.90 2.10 2.50 3.00 150 Zeroth Order Y1) and Ge for 6 = 0. 01. RY _e_p_ 1. 0657 0.1848 0. 0671 0. 0306 0. 0157 0. 0059 0. 0020 0. 0010 0. 0008 0. 0007 0. 0005 0. 0003 Note: All values expressed in millimhos. k D=0.50andk a: 0.10. pe pe G e 0.0000 0.0000 0.0003 0.0015 0.0042 0.0091 0.0162 0.0402 0.0805 0.1422 0.3511 0.8387 For the case presented 151 TABLE 8. 6.: Zeroth Order PP/Pe and CT for 6 = 0. 01. .9... P /1=>e GT (millimhos) 1.01 1.12031106 1.0657 1.05 0.416x104 0.1848 1.10 0.264 x 103 0.0674 1.20 0.206 x 102 0.0321 1.30 0.371 x 101 0.0199 1. 40 0. 683 0.0148 1.50 0.124 0.0182 1.70 0.250 x10."1 0.0412 1.90 0.1381110'l 0.0813 2.10 0. 840 x10.2 0.1429 2. 50 0.204 x 10'“2 0.3516 3 3.00 0.330 x 10" 0.8390 0.10 0.30 0.50 0.60 0.70 0.80 0.90 0.95 0.99 1.01 1.05 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.50 3.00 152 TABLE 8.7. Zeroth Order Kz for 6 = 0. 5. .01300 .00245 .00146 .00095 00075 00063 .00054 .00047 .00042 0.00038 0.00034 0.00031 0.00019 0.00013 OOOCOOOOO - 0.400 - 0.333 153 TABLE 8. 8. Zeroth Order Kz for 6 = 0.1. fl REK «IK __ 2 M 2 0.10 -.- 8.786 0.30 --— 3.007 0.40 --- 2.260 0.50 --- 1.807 0.60 --- 1.498 0.70 —-- 1.285 0.80 --- 1.721 0.85 --- 1.667 0.90 --- 1.290 0.95 --— 1.205 0.99 --- 1.206 1.01 0.20374 1.296 1.05 0 05216 1.041 1.10 0.02649 0.933 1. 20 0 01662 '0.825 1.30 0 01127 0.759 1.40 0.00861 0.704 1.50 0 00595 0.658 1.70 0.00394 0.584 1.90 0 00257 0.523 2.10 0.00195 0.473 2.50 0.00111 0.398 3.00 0 00053 0.332 wNNHHHHHr—nr—IHH .01 .05 .10 .20 .30 .40 .50 .70 .90 .10 .50 .00 154 TABLE 8. 9. First Order Kz for 6 = 0. 01. o o o e o o e o e e 155 2 10RK ea TABLE 8.10. First Order Ka for 6 = 0. 01 10 2 I m K a 0. 011 0. 006 0. 007 0. 008 0. 010 0. 009 0. 018 0. 006 0. 007 0. 017 0. 018 0. 005 OOOOOOOOOOOO . 059 . 072 . 065 . 045 . 035 . 039 . 090 . 037 . 045 . 029 . 028 . 037 156 TABLE 8.11. First Order |N(a) Kal for 6 = 0. 01 s2 102 l N(a) Kal 1.01 0.017 1.05 0.024 1.10 0.024 1.20 0.014 1.30 0.007 1.40 0.013 1.50 0.043 1.70 0.012 1.90 0.018 2.10 0.014 2.50 0.018 3.00 0.017 157 TABLE 8.12. First Order Kp for 6 = 0. 01 52 K _ _E_ 1.01 2,243.104 1.05 9.8431101 1.10 6.30x100 1.20 4.78x10" 1.30 8.95x10- 1.40 1.55x10- 1.50 2.60x10" 1.70 5.71 x10‘ 1.90 3.23x10' 2.10 1.98x10' 2.50 4.51 x10" 3.00 7.32x10' CHAPTER IX CONCLUSION The major conclusion to be drawn from the results of this analysis is that Larson3 was seemingly quite justified in concluding, after his analysis of the spherical aperture antenna including sheath effects, that previous analyses ignoring the sheath had predicted excessively large effects due to EA radiation. His analysis involved the same geometry as and an approach similar to that of Waitl but included the plasma sheath which Wait did not do; this made it quite possible to compare directly his results with those of Wait to determine the effect of the sheath. The present geometry is the same as that of Fejer6 although he did not include the sheath; it is apparent here also that consideration of the sheath leads to the prediction of considerably reduced effects of EA radiation over those predicted by the corresponding "sheath-less" analysis. Larson noticed a considerably increased input susceptance (more nearly that for a free space environment) relative to Wait's result; the present study shows considerably increased input susceptance over Fejer's result. Larson noticed a considerably reduced amount of radiated EA power (hence a much reduced input conductance) in the region near 52 = 1; in chapter VIII,the same trend is observed when the results there are compared with Fejer's results. Figure 8. 2 shows Fejer's input resistance in the vicinity of 52 = l to be greater by nearly an order of magnitude for 6 = 0. 01; comparison for large 6 values shows an even greater disparity. 158 159 It is easily seen that except for the very close proximity of Q = 1 the input reactance follows that of a capacitor with capacitance 217 60a which would be the capacitance of the antenna if it were placed in a vacuum and the capacitance between the spheres were neglected (i. e. , only the self capacitance or the capacitance to infinity of each sphere were considered). Since the antenna is small electrically the input admittance is essentially capacitive. The fact that the radiated EA power is greatly reduced (as evidenced by a greatly reduced input conductance) and the input susceptance is greatly increased by the inclusion of the sheath makes it apparent that the sheath greatly decouples the antenna from the plasma. This is not surprising because the sheath represents a near void of plasma at the surface of the antenna; that coupling to the plasma is reduced by such a void seems reasonable. If a conventional radiation resistance term is derived from the total input admittance expression and compared with Fejer's result it is seen that each expression vanishes at $2 = 1 and increases monotonically with $2 to achieve the same limiting values for large 52; it is seen also that Fejer's term increases much faster initially although the ratio of the two is always less than 2. 0 if 52 is greater than about 1.10 so that the two analyses yield very similar results in the range where the radiation resistance term is appreciable. Larson considered in his sheath model the unperturbed electron density and the unperturbed electric field due to charge separation; he did not consider the unperturbed drift of electrons 160 toward the spheres. This study has included in an approximate manner the effect of electron drift; it is found to be inconsequential as far as the input susceptance is concerned but to have considerable effect upon the magnitude of the input conductance and upon the electron density perturbation. Both analyses dropped the effect of the perturbation in the electric field; it was indicated in chapter IV that its effect was small although there is reason to believe that its effect might not be entirely negligible upon the relatively sensitive solutions for the input conductance and the electron density pertur- bation. The inclusion of this electric field term would require the solution of a system of third order linear ordinary differential equations if drift effects were simultaneously included; this could be done in theory but at present a lack of suitable and sufficient boundary conditions makes it impossible. Boundary conditions represent a major weakness of this type of analysis and work should be done in the area of developing more reasonable boundary conditions. The “hard" boundary condition used here is arbitrary and somewhat unbelievable, but it is conventional. Other boundary conditions have been suggested by various peoPle some of which offer greater flexibility although all are arbitrary and open to question. lo. 11. 12. LIST OF REFERENCES Wait, J. R. , "Radiation from a Spherical Aperture Antenna Immersed in a Compressible Plasma, " IEEE Trans. on Antennas and Propagation, Vol. AP-14, No. 3, pp. 360-368, May 1966. Landau, L. , ”On the Vibrations of the Electronic Plasma, " J. Phys. U.S.S.R., 10, pp. 25-34, 1946. Larson, R. W. , "A Study of the Inhomogeneously Sheathed Spherical Dipole Antenna in a Compressible Plasma, " Dept. of Elect. Engr., University of Michigan, Ph. D. Thesis, 1966. Chen, K. M. , “Interaction of a Radiating Source with a Plasma, " Proc. 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A. , ”Exact Solution of the Collisionless Plasma -Sheath Equation," Phys. of Fluids, Vol. 6, No. 12, pp. 1762-1768, December 1963. ‘ Bernstein, I. B. and I. N. Rabinowitz, ”Theory of Electrostatic Probes in a Low-Density Plasma, " Phys. of Fluids, Vol. 2, No. 2, pp. 112-121, March-April 1959. 161 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. Bohm, D., E. H. S. Burhop, H. S. W. Massey, “The Use of Probes for Plasma Exploration in Strong Magnetic Fields, " Chapter II, Characteristics of Electrical Discharges in Magnetic Fields, A. Guthrie, Ed.: McGraw-Hill, New York, 1949. Bohm, D., E. H. S. Burhop, H. S. W. Massey, Characteristics of Electrical Discharges in Magnetic Fields, Chapter III, A. Guthrie, Ed., McGraw-Hill, New York, 1949. Allen, J. E., and P. C. Thonemann, Proc. Phys. Soc., B70, p. 297, 1957. Allen, J. E., R. L. F. Boyd, and P. Reynolds, "The Collection of Positive Ions by a Probe Immersed in a Plasma, " Proc. Phys. Soc.Lcndon,Vol. 70, No. 447B, Pt. 3, pp. 297-304, March 1957. Laframboise, J. , “Theory of Electrostatic Probes in a Collision- lee Plasma at Rest, " Fourth International Symposium on Rarified Gases, Toronto, 1964. Crawford, F. W. , "The Mechanism of Tonks-Dattner Plasma Resonances," Physics Letters, Vol. 5, No. 4, pp. 244-247, July 15, 1963. Cohen, M. H., "Radiation in Plasma. III. Metal Boundaries," Phys. Rev., Vol. 126, No. 2, pp. 398-404, April 15, 1962. Balmain, K. , "Impedance of a Radio Frequency Plasma Probe with an Absorptive Surface, " Radio Sci. , Vol. 1 (new series), pp. 1-12, January 1966. R‘amo, S. and J. R. Whinnery, Fields and Waves in Modern Radio, Second Edition, Wiley, New York, pp. 496-500, 1960. Stratton, J. A., Electromggnetic Theory, McGraw-Hill, New York, pp. 434-437, 1941. 162 M'Tlll'llllllllillllllll111114)1111111111)ES 1293 03146 11