.... 5..., . 1:; £3. .13 Swift: 1) a 5‘... x e. 2.2. ‘ git mflu,w.~ not" .. .24... .. a «drawififin; _ 9%??? 1w Illuuiflfi.’ r1 I l...:) VP. nas‘irQHL .VA 8.". ‘1 "U‘a.u( ’31 a . , (ol_ ~ I t A i ‘n B. N .l l . ' l «:13- u. .5. . v4 .4 1.17.. 1-2! 1.1.. I A I \ wk. 9 p v t . I , o. x 9.. . A I A I. l . uy y . I r . .o a n‘ . . . u z ‘ . I I. ll. 10‘ u. IA A ‘ . I l: $41.”! .,.A.n...\“ “u . r N..>{\...r..rlht . ._........ n .....|f....~...I-w.lnwdufl.,‘ ‘utiaufiiuuwu1. .Tlcanituty-Mac... . .. . f0 , . ..:.l.¥....vt..io ;!1. 3i .3. I - lullO¢ANL In . .LJ- 0‘ Ti 1 ‘___ ‘.,3 A :-, ‘37 3:1.- ng... ‘RJY’ “Kl ‘fl;r‘. 106.?! Cram 11> UM“ t--.,iJ-J.-"”y _?I - _ . uvw—P—r— ’. 'r..Y a. fit-5"“ This is to certify that the thesis entitled RADIATION AND RESONANCES OF ELECTROACOUSTIC AND IéNACOUSTIC WAVES IN COMPRESSIBLE PLASMAS presented by Kam-Chi Li has been accepted towards fulfillment of the requirements for Ph. D. degreein Electrical Engineering flack Major professor Date November 41 1974 0-7639 ABSTRACT RADIATION AND RESONANCES OF ELECTROACOUSTIC AND IONACOUSTIC WAVES IN COMPRESSIBLE PLASMA BY Kam-Chi Li The present study consists mainly of two major parts. The first part is the study on the basic properties of the electroacoustic and ionacoustic waves excited by an electro- magnetic source or field in an infinite, homogeneous, isotropic, compressible and lossy plasma. A two-fluid plasma model is employed and this leads to the formulation of the generalized electroacoustic and ionacoustic waves. The electron-ion compositions, as well as the propagation constants of the generalized electroacoustic and ionacoustic waves with various collision frequencies and under various electron and ion temperatures, are obtained. The radiation patterns of the generalized electro- acoustic and ionacoustic waves excited by simple antennas, such as Hertzian dipole, disk monopole, disk dipole and cylindrical antennas, are studied. They agree very closely Kam-Chi Li with the results of some recent experimental studies. The second part is the investigation of the excitation of an electroacoustic wave in the plasma sheath surrounding a cylindrical antenna, the excitation of electroacoustic resonances in various plasma geometries, and the reflection behavior of electroacoustic waves on various surfaces. A new diagnostic scheme for measuring the plasma density directly has been developed. In this scheme, a cylindrical antenna immersed in a compressible plasma is driven by a frequency— sweeping electromagnetic wave, and its d.c. bias voltage is varied. Based on the information on the electroacoustic wave excited in the plasma sheath surrounding the antenna, the plasma density can be read directly on the oscilloscope. The behaviors of electroacoustic resonances excited in the plasma sheaths at the boundaries of various plasma geometries which include cylindrical, rectangular and single- slope density profile plasma columns were studied. The technique of exciting electroacoustic resonances was then applied to study the reflection behavior of electroacoustic waves on dielectric and metallic surfaces. RADIATION AND RESONANCES. OF ELECTROACOUSTIC AND IONACOUSTIC WAVES IN COMPRESSIBLE PLASMAS By Kam—Chi Li A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1974 To my parents Mr. & Mrs. Chung-Wah Li ii ACKNOWLEDGMENTS The author is sincerely grateful to his major professor, Dr. K. M. Chen, for his guidance, wisdom and encouragement during the course of this research. He also wishes to thank his committee members, Dr. B. Ho, Dr. J. Asmussen, Dr. J. S. Frame and Dr. D. P. Nyquist, for their time and interest in this work. Finally, the author thanks his wife, Joyce Pui-Ping, for the typing of the manuscript, and especially for the love and understanding shown by her during the entire course of this study. iii TABLE OF CONTENTS ACKNOWLEDGMENTS . . . . . . . . . . LIST OF FIGURES . . . . . . . . . . LIST OF TABLES. . . . . . I. . . . . INTRODUCTION . . . . . . . . . . . ELECTROACOUSTIC WAVE AND IONACOUSTIC WAVE EXCITED IN AN INFINITE, HOMOGENEOUS, COMPRESSIBLE AND LOSSY PLASMA BY AN ELECTRO- MAGNETIC SOURCE . O O O O C C O O O 2.1 2.2 2.3 2 4 NM N O O O ”\l O RADIATION PATTERNS OF ELECTROACOUSTIC AND ION- Geometry and the Related Equations . . Equations for Electroacoustic and Ion- acoustic Waves . . . . . . . . Decoupling of ne and ni Waves . . . Electron- -ion Composition Ratios of the Generalized Ionacoustic Wave (n1 Wave) and the Generalized Electroacoustic Wave (n2 Wave). . . . . . Propagation Constants of the Generalized Ionacoustic Wave and the Generalized Electroacoustic Wave . . . Differential Equations of the Magnetic Field 0 O O 0 I O O O The Electric Field in the Plasma . . . Average Velocities of Electrons and Ions in the Plasma . . . . . . . . . ACOUSTIC WAVES EXCITED BY VARIOUS ANTENNAS . 3.1 Introduction. . . . . . . . . . 3.2 Hertzian Dipole Antenna . . . 3.2.1 Geometry and Statement of the Problem . . . . . 3.2.2 Radiation Patterns of the Generalized Ionacoustic Wave (n1 Wave). . . iv ISOTROPIC, Page .iii .vii . 3S . 38 . 41 3.2.3 Radiation Patterns of the Generalized Electroacoustic Wave (n2 Wave) 3.2.4 Radiation Patterns of the Electro— magnetic Wave . . . . . . 3.3 Disk Monopole Antenna. . . . 3. 3.1 Geometry and Statement of the Problem . . . . 3.3.2 Radiation Patterns of the Generalized Ionacoustic Wave (n1 Wave). . 3.3.3 Radiation Patterns of the Generalized Electroacoustic Wave (n2 Wave) 3.4 Disk Dipole Antenna . . . 3. 4.1 Geometry and Statement of the Problem . . . . . . 3.4.2 Radiation Patterns of the Generalized Ionacoustic Wave (n1 Wave). . 3.4.3 Radiation Patterns of the Generalized Electroacoustic Wave (n2 Wave) 3.5 Cylindrical Antenna . . . 3.5.1 Geometry and Statement of the Problem . . 3.5.2 Radiation Patterns of the Generalized Ionacoustic Wave (n1 Wave). . 3.5.3 Radiation Patterns of the Generalized Electroacoustic Wave (n2 Wave) 3.5.4 Radiation Patterns of the Electro- magnetic Wave . . . . . . EXCITATION OF AN ELECTROACOUSTIC WAVE IN THE PLASMA SHEATH SURROUNDING A CYLINDRICAL ANTENNA O O O O C C O O O O O 0 Introduction. . . . . . . . . Experimental Setup. . . . . . . ExPerimental Results . . . . Interpretation of the EXperimental Results . . . 4.4.1 The Case When the Cylindrical Antenna is Biased Positively . 4.4.2 The Case When the Cylindrical Antenna is Biased Negatively .5 Potential Application. . . 6 Analysis of the Coupling between the bhbcb I book)!" bub O Electromagnetic Mode and Electroacoustic Mode in the Plasma Sheath . .~ . . Page 47 50 52 52 57 63 66 66 70 76 79 79 81 82 88 97 97 97 101 107 110 112 115 116 EXCITATION OF ELECTROACOUSTIC RESONANCES IN VARIOUS PLASMA GEOMETRIES AND STUDY OF THE REFLECTION BEHAVIOR OF ELECTROACOUSTIC WAVES ON VARIOUS SURFACES . . . . . . . . . 5.1 Introduction. . . . . . . . . . 5.2 Experimental Setup. . . . . . 5.3 Electroacoustic Resonances and Dipole Resonance in a Cylindrical Plasma Column. . . . . . . . . . . . 5.4 Electroacoustic Resonances and Dipole Resonance in a Rectangular Plasma Column. . . . . . . 5.5 Resonances in Single-profile Plasma Column in the Rectangular Tube. . . . 5.5.1 Glass Reflector Region . . . . 5.5.2 Metallic Reflector Region . . . 5.6 Reflection Behavior of Electroacoustic Wave from Metallic and Non-metallic Surfaces . . . . . . . . . . . APPENDICES Appendix A. Numerical Calculation of R1, R2, the Electron-ion Composition Ratios of the n1 Wave and the n2 Wave . . . . . . B. Numerical Calculation of k1, k2, the Propagation Constants of the n1 Wave and the n2 Wave . . . . . . . . . . C. Tables of Data for the Calculation of Radiation Patterns of the n1 Wave and the n2 Wave. . . . . . . . . . . . REFERENCES . . . . . . . . . . . . vi 122 122 122 124 132 136 136 138 143 149 152 157 171 Figure 2.1 2.4 3.1 LIST OF FIGURES Electron-ion composition ratio of the generalized ionacoustic wave (n1 wave) as a function of (we/w)2 for various ratios of electron temperature to ion temperature . . Electron-ion composition ratio of the generalized electroacoustic wave (n2 wave) as a function of (we/w)2 for various ratios of electron temperature to ion temperature . Phase constant of the generalized ionacoustic wave (n1 wave) as a function of (we/w)2 for various ratios of electron temperature to ion temperature in a hydrogen plasma . . . Attenuation constant of the generalized ionacoustic wave (n1 wave) as a function of (we/w)2 for various collision frequencies in a hydrogen plasma. (Te/Ti = l) . . . . Attenuation constant of the generalized ionacoustic wave (n1 wave) as a function of (we/w)2 for various collision frequencies in a hydrogen plasma. (Te/Ti = 100) . . . Phase constant of the generalized electro- acoustic wave (n2 wave) as a function of (we/u) for various collision frequencies. (Te/Ti = 100) o o o o o o o o o o . Attenuation constant of the generalized electroacoustic wave (n2 wave) as a function of (we/w)2 for various collision frequencies. (Te/Ti = 100) o o o o o o o o o o o Geometry of a Hertzian dipole antenna . . . vii Page 19 20 28 29 30 31 32 42 Figure Page 3.2 Radiation patterns of the generalized ion- acoustic wave excited by a Hertzian dipole antenna for various electron temperatures. (f = 30 kHz, Te/Ti = 10, d1 = 1 cm). . . . . 44 3.3 Radiation patterns of the generalized ion- acoustic wave excited by a Hertzian dipole antenna for various ratios of electron temperature to ion temperature. (f = 30 kHz, Te = 6000°K, d1 = 1 cm). . . . . . . . . 45 3.4 Radiation patterns of the generalized ion- acoustic wave excited by a Hertzian dipole antenna for various antenna frequencies. (Te = 6000°K, Te/Ti = 10, d1 = 2.5 cm). . . . 46 3.5 Radiation patterns of the generalized electroacoustic wave excited by a Hertzian dipole antenna for various electron tempera- tures. (f = 1 GHz, Te/Ti = l to 10“, meZ/wz = 0.95, d1 = 1 mm) . . . . . . . . . . 48 3.6 Radiation patterns of the generalized electroacoustic wave excited by a Hertzian dipole antenna for various antenna fre- quencies. (Te = 4000°K, Te/Ti = l to 10“, meZ/mZ = 0.95, d1 = 1 mm) . . . . . .1 . . 49 3.7 Radiation pattern of the electromagnetic wave excited by a Hertzian dipole antenna in a plasma. . . . . . . . . . . . . 53 3.8 Geometry of a disk monopole antenna. . . . . 54 3.9 Radiation patterns of the generalized ion- acoustic wave excited by a disk monopole antenna for various electron temperatures. (f = 30 kHz, Te/Ti = 100, a = 2.25 cm). . . . 58 3.10 Radiation patterns of the generalized ion- acoustic wave excited by a disk monopole antenna for various ratios of electron temperature to ion temperature. (f = 30 kHz, Te = 2000°K, a = 2.25 cm) . . . . . . . . 59 viii Figure Page 3.11 Radiation patterns of the generalized ion- acoustic wave excited by a disk monOpole antenna. (f = 16.3 kHz, phase velocity VA = 1.05 x 103 m/sec, Te = Ti 2 1200°K, a = 2.25 cm) . . . . . . . . . . . . 60 3.12 Radiation patterns of the generalized ion- acoustic wave excited by a disk monopole antenna. (f = 23.3 kHz, phase velocity VA = 1.05 x 103 m/sec, Te = Ti 2 1200°K, a = 2.25 cm) . . . . . . . . . . . . 61 3.13 Radiation patterns of the generalized ion- acoustic wave excited by a disk monopole antenna. (f = 58.3 kHz, phase velocity VA = 1.05 x 103 m/sec, Te = Ti 2 1200°K, a = 2.25 cm) . . . . . . . . . . . . 62 3.14 Radiation patterns of the generalized electro- acoustic wave excited by a disk monOpole antenna for various electron temperatures. (f = 17.5 MHz, 1 = 13.1 cm, weZ/wz = 0.95, ye/w = 0, Te/Ti = l to 10“, 2a = 0.6 A) . . . 64 3.15 Radiation patterns of the generalized electro- acoustic wave excited by a disk monopole antenna for various electron temperatures.r (f = 17.5 MHz, 1 = 13.1 cm, weZ/wZ = 0.95, Ye/w = o, Te/Ti z 1 to 10“, 2a = 1.1 A) . . . 65 3.16 Geometry of a disk dipole antenna . . . . . 67 3.17 Radiation patterns of the generalized ion- acoustic wave excited by a disk dipole antenna for various electron temperatures. (f = 30 kHz, Te/Ti = 10, a = 2.25 cm) . . . . 71 3.18 Radiation patterns of the generalized ion- acoustic wave excited by a disk dipole antenna for various ratios of electron temperature to ion temperature. (f = 30 kHz, Te = 4000°K, a = 2.25 cm) . . . . . . . . 72 3.19 Radiation patterns of the generalized ion- acoustic wave excited by a disk dipole antenna. (f = 35 kHz, phase velocity VA = 1.05 x 103 m/sec, Te = Ti 2 1200°K, a = 2.25 cm) . . . . . . . . . . . . 73 ix Figure Page 3.20 Radiation patterns of the generalized ion- acoustic wave excited by a disk dipole antenna. (f = 46.6 kHz, phase velocity VA = 1.05 x 103 m/sec, Te 2 Ti 2 1200°K, a = 2.25 cm) . . . . . . . . . . . . 74 3.21 Radiation patterns of the generalized ion- acoustic wave excited by a disk dipole antenna. (f = 93.3 kHz, phase velocity VA = 1.05 x 103 m/sec, Te = Ti 2 1200°K, a = 2.25 cm) . . . . . . . . . . . . 75 3.22 Radiation patterns of the generalized electroacoustic wave excited by a disk dipole antenna for various electron tempera- tures. (f 17.5 MHz, 682/62 = 0.95, ye/w = 0, Te/Ti 1 to 10“, a = 7.2 cm) . . . . . 77 :2 II 3.23 Radiation patterns of the generalized electroacoustic wave excited by a disk dipole antenna for various antenna fre- quencies. (Te = 2000°K, wez/wz = 0.95, ye/w = 0, Te/Ti = l to 10“, a = 7.2 cm) . . . 78 3.24 Geometry of a cylindrical antenna . . . . . 80 3.25 Radiation patterns of the generalized ion- ,acoustic wave excited by a cylindrical ' antenna for various electron temperatures. (f = 30 kHz, Te/Ti = 10, h = 2.5 cm) . . . . 83 3.26 Radiation patterns of the generalized ion- acoustic wave excited by a cylindrical antenna for various electron temperatures. (f = 30 kHz, Te/Ti = 10, h = 5 cm) . . . . . 84 3.27 Radiation patterns of the generalized ion- acoustic wave excited by a cylindrical antenna for various ratios of electron temperature to ion temperature. (f = 30 kHz, 85 Te = 6000°K, h = 2.5 cm) . . . . . . . . 3.28 Radiation patterns of the generalized ion- acoustic wave excited by a cylindrical antenna for various ratios of electron temperature to ion temperature. (f = 30 kHz, Te = 6000°K, h = 5 cm) . . . . . . . . . 86 Figure Page 3.29 Radiation patterns of the generalized ion— acoustic wave excited by a cylindrical antenna for various antenna frequencies. (Te = 6000° K, Te/Ti = 10, h = 5 cm). . . . . 87 3.30 Radiation patterns of the generalized electroacoustic wave excited by a cylin- drical antenna for various electron tempera- tures. (f 5.5 MHz, = 4. 5 MHz, ye/w = 0, Te/Ti 1 to 10“, fgh/ = 0.7, h = 6 cm). . 89 II II 3.31 Radiation patterns of the generalized electroacoustic wave excited by a cylin- drical antenna for various antenna frequencies. = 4. 5 MHz, Te = 6000°K, Ye/w = o, Te/Ti 91 to 10“, = 8. S cm). . . . . . . 90 3.32 Radiation pattern of the generalized electro-' acoustic wave excited by a cylindrical antenna. (Te = 51500 K, fp = 4.5 MHz, f = 5. 5 MHz, h = 6 cm) . . . . . . . . . 91 3.33 Radiation pattern of the generalized electroacoustic wave excited by a cylin- drical antenna. (Te = 5150°K, fp = 4.5 MHz, f = 5.5 MHz, h = 8.5 cm) . . . . . . . . 92 3.34 Radiation pattern of the generalized electroacoustic wave excited by a cylindrical antenna. (Te = 5150° K, fp = 4. 5 MHz, f’= 7 MHz,l1=-8. 5 cm) . . . . . . . . . 93 3.35 Radiation patterns of the electromagnetic wave excited by a cylindrical antenna in a plasma for various antenna frequencies. fp = 4.5 MHz, h = 8.5 cm) . . . . . . . . 96 4.1 Experimental setup for the excitation of the electroacoustic wave in the plasma sheath surrounding a cylindrical antenna . . . 98 4.2 The plasma tube and accessories . . . . . . 100 4.3 A typical reflected wave versus sweeping frequency curve . . . . . . . . . . . 102 xi Figure Page 4.4 Oscillograms of the reflected wave versus sweeping frequency curves for various plasma currents. Frequency range from 0.5 to 1.0 GHz . . . . . . . . . . - . . . . 105 4.5 Oscillograms of the reflected wave versus sweeping frequency curves for various plasma currents. Frequency range from 0.4 to 1.4 GHz . . . . . . . . . . . . . . . 106 4.6 Affected frequency bands of the RW-SF curves for the cases of various plasma currents . . . 108 4.7 Plasma density profiles surrounding the antenna for various positive bias voltages . . 111 4.8 Plasma density profiles surrounding the antenna for various negative bias voltages . . 113 4.9 Geometry of a cylindrical antenna surrounded by a plasma sheath . . . . . . . . . . 117 5.1 Structure of the cylindrical plasma tube . . . 125 5.2 Structure of the rectangular plasma tube . . . 125 5.3 Experimental setup for the excitation and observation of electroacoustic resonances in different plasma geometries . . . . . . 126 5.4 Cross-sectional view of the cylindrical plasma tube. (a) without metallic backing (b) with metallic backing . . . . . . . . 128 5.5 Electroacoustic resonance in a cylindrical plasma column . . . . . . . . . . . . 128 5.6 Resonance curves observed in a cylindrical plasma column. (f = 2.4 GHz, Ip0 = 95 mA) . . 129 5.7 Resonance curves observed in a cylindrical plasma column. (f = 2.4 GHz, Ip0 = 115 mA) . . 130 5.8 Resonance curves observed in a cylindrical plasma column. (f = 2.45 GHz, Ip0 = 120 mA). . 131 xii Figure Page 5.9 Resonance curves observed in the uniform region of the rectangular plasma tube. 250 mA) . . . . . . . . . . . . . . 133 5.10 Resonance curves observed in the uniform region of a rectangular plasma tube. (f = 2.0 GHz, Ip0 = 150 mA) . . . . . . . 134 5.11 Resonance curves observed in the uniform region of a rectangular plasma tube. (f = 2.4 GHz, Ipo = 150 mA) . . . . . . . 135 5.12 Plasma density distribution in the glass reflector region . . . . . . . . . . . 137 5.13 Plasma density distribution in the metallic reflector region . . . . . . . . . . . 137 5.14 Resonance curves observed in the neck section of the glass reflector region of a rectangular plasma tube. (f = 2.4 GHz, Ip0 = 150 mA, 200 mA)...............l39 5.15 Resonance curves observed in the center sec- tion of the glass reflector region of a rectangular plasma tube. (f = 2.4 GHz, I = 150 mA) . . . . . . . . . . . . . 140 5.16 Resonance curves observed in the center sec- tion of the glass reflector region of a rectangular plasma tube. (f = 2.4 GHz, I = 200 mA) . . . . . . . . . . . . . 141 5.17 Resonance curves observed in the tail sec— tion of the glass reflector region of a rectangular plasma tube. (f = 2.4 GHz, Ip0 = 150 mA) . . . . . . . . . . . . . 142 5.18 Resonance curve observed in the neck section of the metallic reflector region of a rectangular plasma tube. (f = 2.4 GHz) . . . 144 5.19 Resonance curve observed in the center sec- tion of the metallic reflector region of a rectangular plasma tube. (f = 2.4 GHz) . . . 144 xiii Figure Page Resonance curve observed in the tail section of the metallic reflector region of a rec- tangular plasma tube. (f = 2.4 GHz) . . . . 144 Reflection curves observed in uniform, glass reflector and metallic reflector regions of a rectangular plasma tube. (f = 2.33 GHz, Ipo = 190 mA) . . . . . . . . . . . . 145 Reflection curves observed in the uniform region of a rectangular plasma tube. (f = 2.4 GHz, Ip0 = 150 mA) . . . . . . . 146 Reflection curves observed in a rectangular plasma tube with the inside and outside metallic backing. (f = 2.4 GHz, Ip0 = 150 mA) . . . . . . . . . . . . . 148 xiv LIST OF TABLES Table . _ Page 4.1 Ambient plasma frequency versus plasma current . . . . . . . . . . . . . . 107 c-1 k1d1 versus Te. (f = 30 kHz, Re[k1/(w/Vi)] = 0.3016 for (Te/Ti) = 10, d1 = 1 cm) . . . . 159 c-z kldl versus Te/Ti. (f = 30 kHz, Te = 6000°K, d]- = 1 CHI) 0 o o o o o o o o o o o o 159 c-3 kldl versus f. (Te = 6000°K, Re[k1/(w/Vi)] = 0.3016 for (Te/Ti) = 10, d1 = 2.5 cm). . . . 160 C-4 kzdl versus Te. (f = 1 GHz, Re[k2/(w/Ve)] = 0.2235 for (weZ/wz) = 0.95 and (Te/Ti) = 1 to 10“, d1 = 1 mm). . . . . . . . . . . 160 C-5 kzdl versus f. (Te = 4000°K, Re[k2/(w/Ve)] = 0.2235 for (692/62) = 0.95 and (Te/Ti) =-1 to 10“, d1 = 1 mm). . . . . . . . . . . 161 C-6 kla versus Te. (f = 30 kHz, Re[k1/(w/Vi)] = 0.0995 for (Te/Ti) = 100, a = 2.25 cm) . . . 161 c-7 kla versus Te/Ti. (f = 30 kHz, Te = 2000°K, a = 2025 cm). 0 o o o o o o o o o o o 162 C-8 kla versus f. (VA = 1.05 x 103 m/sec, Te = Ti 1200°K, Re[k1/(w/Vi)] = 0.7071 for (Te/Ti) 1, a = 2.25 cm) . . . . . . . . . . . 163 ll C-9 kga versus Te. (f = 17.5 MHz, 1 = 13.1 cm, Re[k2/(w/Ve)] = 0.2235 for (weZ/wZ) = 0.95 and (Te/Ti) = 1 to 10“) . . . . . . . . . 162 C-10 kla versus Te. (f = 30 kHz, Re[k1/(w/Vi)] = 0.3016 for (Te/Ti) = 10, a = 2.25 cm). . . . 163 C-ll kla versus Te/Ti~ (f = 30 kHz, Te = 4000°K, a = 2025 cm). 0 o o o o o o o o o o o 164 XV Table Page C-12 kla versus f. (VA = 1.05 x 103 m/sec, Te = Ti 2 1200°K, Re[k1/(w/Vi)] = 0.7071 for (Te/Ti) a 1, a = 2.25 cm) . . . . . . . . . . . 164 C-13 kza versus Te. (f = 17.5 MHz, Re[k2/(w/Ve)] = 0.2235 for (ueZ/uz) = 0.95 and (Te/Ti) = 1 to 10“, a = 7.2 cm) . . . . . . . . . . 165 C-14 kza versus f. (Te = 2000°K, Re[k2/(w/Ve)] = 0.2235 for (meZ/uZ) = 0.95 and (Te/Ti) = 1 to 10“, a = 7.2 cm) . . . . . . . . . . 165 C-lS klh versus Te- (f = 30 kHz, Re[k1/(w/Vi)] = 0.3016 for (Te/Ti) = 10) . . . . . . . . 166 C-16 klh versus Te/Ti. (f = 30 kHz, Te = 6000°K) . . 167 c-17 klh versus f. (Te = 6000°K, Re[k1/(w/Vi)] = 0.3016 for (Te/Ti) = 10, h = 5 cm). . . . . 166 4.5 MHz, (Te/Ti) = 1 to 10“ Re[k2/(w/ 0.57 and ke = 0.066 for (we /w2) = 0.6 , = 6 C111). 0 o o o o o o o o o o o o 168 C-18 kzh and ke/kz versus Te. (f = 5.5 MHz, 5p e)] 7 2h and ke/kz versus f. (Te = 6000°K, h k (Te/Ti) = l to 10“, fp = 4.5 MHz, h = 8.5 cm). . 169 C-20 ikzh and ke/kz versus f. (Te = 5150°K, (Te/Ti) = 1 to 10“, fp = 4.5 MHz). . . . . . 169 C-21 keh versus f. (fp = 4.5 MHz, h = 8.5 cm) . . . 168 xvi CHAPTER 1 INTRODUCTION The research described in this dissertation deals with the interaction of the electromagnetic radiation with a plasma. The first part of the dissertation studies the radiation of various antennas imbedded in an infinite, homogeneous, isotropic, compressible and lossy plasma. A two-fluid model is used to describe the plasma. The second part of the dissertation investigates the excitation of an electroacoustic wave in the plasma sheath surrounding a cylindrical antenna, the excitation of electroacouStic resonances in various plasma geometries, and the reflection behavior of electroacoustic waves on various surfaces. The excitation and radiation of the electroacoustic and ionacoustic waves from various simple antennas imbedded in a plasma medium is a subject that has received a great deal of attention from researchers. As to the excitation and prOpagation properties of the electroacoustic and ionacoustic waves, theoretical and experimental investigations have been done by the researchers such as Cohen [1], Hessel and Shmoys [2], Kuehl [3], Barrett and Little [4], Jones and Alexeff [5,6], an V¢ .A 'H ID' , I Mahmberg and Wharton [7], Chen and Lin [8], Doucet [9], Lonngren et al.[10] and Alexeff, Jones and Montgomery [11]. More recently, Nakamura et al.[12], Ishizone et al.[13] and Shen et al.[14] have detected the electroacoustic and ion— acoustic waves excited by some simple antennas, and their radiation patterns have also been measured. In treating the plasma, most of the workers, including Majumdar [15], Cohen [1], Hessel amd Shmoys [2] and Seshadri [16], have idealized the plasma to be a homogeneous, collisionless and compressive electron fluid with stationary ions that neutralize the electrons on the average. Recently, Kuehl [3] has studied the excitation of waves in a warm plasma by an electric dipole wherein the motion of the ion has been included. Seshadri [17] studied the radiation from electric current sources in a two- component finite temperature plasma and Maxam and Chen [18] decoupled electroacoustic and ionacoustic wave equations based on a two—fluid plasma model using macrosc0pic approach. It is the purpose of this research to apply the decoupled equations of electroacoustic and ionacoustic waves, with the consideration of various collision frequencies and under various electron and ion temperatures, to investigate in detail the electron—ion compositions and the prOpagation constants of the so—called generalized electroacoustic and ionacoustic waves. The radiation patterns of the generalized electroacoustic and ionacoustic waves excited by some simple antennas including Hertzian dipole, disk monopole, disk dipole and cylindrical antennas are calculated. Theoretical radiation patterns are then compared with recent experimental results by Nakamura et al.[12], Ishizone et al-[l3] and Shen et al-[l4]. A good agreement is obtained between the present theory and experimental results. The excitation of an electroacoustic wave in an inhomogeneous compressible plasma and the resonance of the electroacoustic wave in a plasma sheath leading to the so- called Tonks-Dattner's resonance, or thermal resonance, have been studied by numerous workers including Tonks [19], Dattner [20], Crowford [21], Parker et al.[22], Vandenplas [23], Tutter [24], Van Hoven [25], Derfler and Simonen [26] and Golddan and Yadlowsky [27]. Recently, Baldwin [28] and Parbhakar and Gregory [29], through their theoretical and experimental studies, proposed a new physical mechanism for the electroacoustic resonance in the plasma sheath of a cylindrical plasma column. The mechanism implies that in order to excite an electroacoustic wave, an electromagnetic wave is required to interact with the plasma at the critical density point where the plasma frequency is equal to the frequency of the electromagnetic wave. If no critical density point exists in the plasma, an electroacoustic wave may not be excited. In the second part of this research, experimental studies have been conducted to study (1) the excitation of the electroacoustic wave in the plasma sheath surrounding a cylindrical antenna imbedded in a compressible plasma, (2) the excitation of electroacoustic resonances in various plasma geometries which include cylindrical, rectangular and single- slope density profile plasma columns, and (3) the reflection behavior of electroacoustic waves on dielectric and metallic surfaces based on the technique of exciting electroacoustic resonances. Baldwin's mechanism [28] was used to explain some experimental results. In this part of the experimental study, a new diagnostic scheme for plasma density measurement was developed. A cylindrical antenna immersed in a compressible plasma is driven by a frequency-sweeping electromagnetic wave and a variable d.c. bias voltage is applied to the antenna. By observing the effect of the d.c. bias voltage on the excitation of the electroacoustic wave in the plasma sheath surrounding the antenna, the plasma density at the location of the antenna can be directly read on the oscillosc0pe. Throughout the study, the macroscopic approach is used. The problem was solved based on the hydrodynamic equations and Maxwell's equations. Chapter 2 studies the generalized electroacoustic and ionacoustic waves, their electron—ion compositions, propagation constants, the effects due to the collision frequency and electron and ion tempera— tures. Chapter 3 applies the results of Chapter 2 to calculate the radiation patterns of generalized electroacoustic and ionacoustic waves excited by four different types of antennas. Theoretical results are then compared with some recent experimental results. Chapter 4 studies the excitation of an electroacoustic wave in the plasma sheath surrounding a cylindrical antenna. A new diagnostic method for the plasma density measurement is described in this chapter. The excitation of electroacoustic waves in various plasma geome- tries and the reflection behavior of electroacoustic waves on various surfaces are investigated in Chapter 5. , CHAPTER 2 ELECTROACOUSTIC WAVE AND IONACOUSTIC WAVE EXCITED IN AN INFINITE, HOMOGENEOUS, ISOTROPIC, COMPRESSIBLE AND LOSSY PLASMA BY AN ELECTROMAGNETIC SOURCE 2.1 Geometry and the Related Equations We consider a system in which an electromagnetic source with current density 38 and charge density p8 is immersed in an infinite, homogeneous, isotrOpic, compressible and lossy plasma. The plasma is assumed to consist of two fluids, the electrons and the ions. The neutral particles of the plasma contribute to the dynamics of the plasma by collisions with the charged particles. The electromagnetic source excited a longitudinal electroacoustic wave and a longitudinal ionacoustic wave in addition to the usual electromagnetic wave. Since the excitation and propagation of the electromagnetic wave in the plasma are well known, only the electroacoustic wave and ionacoustic wave are investigated in detail in this study. A macroscopic approach is used to describe this system. It is assumed that the perturbation of the plasma due to the source is small, so that the linearized equations are applicable. Under these assumptions, the basic equations which govern this system are Maxwell's equations and the hydrodynamic equations. Maxwell's equations: + —_§.§ V x E - at (2.1.1) V x B = u JS + u e(n U - n + ) + U e E: (2 1 2) o o 01 oe e o 03 ° ° 8 VoE=L+§_(n.-n) (2.1.3) 6 e 1 e o o v-§=o (2.1.4) where noi and n0e are the unperturbed ion and electron densities which can be assumed to be equal and uniform throughout the system, that is, n . = n = n . (2.1.5) ni and ne are the perturbed ion and electron densities such tha . << n r n << n . n. and n are functi ns of th t n1 0’ e o 1 e 0 b0 position and time. 61 and fie are the average velocities of the ions and electrons induced by the external force. E and B are the electric and the magnetic fields. 38 and OS are the current and charge density of the source and are related by the equation of continuity as s v-35+§—f:—=0 . (2.1.6) “o and so are the permeability and permittivity of free space respectively. Hydrodynamic equations: The linearized equations of motion for the electrons are an _ e + 8?- + “6(V . Ue) _ 0 (2.1.7) 2 36 v at + YeUe me E no Vne . (2.1.8) The linearized equations of motion for the ions are ani 4 + 2 SU V. where Ye and Yi are the mean electron-neutral particle collision frequency and mean ion-neutral particle collision frequency respectively. V and Vi are the thermal velocities e of electrons and of ions, and are defined as 3kT v 2 = 8 (2.1.11) 8 m e 2 3kTi v. = (2.1.12) 1 mi where me and mi are the electron and ion masses. Te and Ti are the electron and ion temperatures. e is the magnitude of electron charge and k is the Boltzmann's constant. It is assumed that the electromagnetic source oscil- lates with a constant frequency w, consequently, all quantities vary with time as ejwt. The phaser analysis method is then applied in the following development. 2.2 Equations for Electroacoustic and Ionacoustic Waves To establish equations for the electroacoustic wave, ne, and the ionacoustic wave, ni, equation (2.1.8) is written as ' —-§_"-____ (3w + ye)U — m E Vn . (2.2.1) e 0 Taking the divergence of equation (2.2.1) yields 2 v (fun + yelV - fie = -fi- v . E - 33-— Vzne .' (2.2.2) 8 O V - U can be obtained from equation (2.1.7) as v-fi =-%9’-n . (2.2.3) 0 v - '13.: — Mn. . (2.2.4) n0 10 V - E can be obtained by taking the divergence of equation (2.1.2); +8 -> ~> . -> 0 = V - J + eno(V ° U- - V 0 U ) +.jweOV ° E . (2.2.5) Substituting equations (2.2.3) and (2.2.4) into equation (2.2.5) gives + ' +8 . v - E = 6%;[V - J - jew(ni - ne)] . (2.2.6) Substituting equations (2.2.6) and (2.2.3) into equation (2.2.2), we have 2 2 we2 Ye wez v. +-“-’--(1-—--- ...). ._.. e V2 2 w e V 1 e w e 2 - - j -f§—— V - 3S (2 2 7) V em e where the electron plasma frequency is 2 1 nee-2- we = (m e > . (2.2.8) e o . +8 . S . . . U51ng V - J = - jwp from the equat1on of continuity and defining B 2 = wZ <1 - wez - - :2) (2 2 9) e V2 2 3(1) I o. w ll equation (2.2.7) can be rewritten as 2 2 we we 08 V De + Be ne + 97 Hi = " W(g-) 0 (2.2.10) e e ’ Similarly, we can get an equation for n- as 1 2 2 ' w. m. S Vzn. + 8.2n. + —£— n = —£—(£-) (2.2.11) 1 1 1 2 e e V. V 1 1 where the ion plasma frequency is 2 l noe 3 1 o and 2 2 u)- y. 3.2 =L(1-J—-- '4) . (2.2.13) 1 2 w Vi w 2.3 Decoupling of ha and ni Waves Equations (2.2.10) and (2.2.11) can be decoupled mathematically into two independent wave equations [18] as S 2 0 (V 2 + k1 )n1 ((72 + 1622):.2 = 8 (2.3.2) (DID where n1 and n2 are linear combinations of ne and ni; namely, 12 V. V = .1 - .2 n1 (w- T12)ni (w T22)ne (2.3.3) 1 e n = (33 T )n - (Xi T )n (2 3 4) 2 we 21 e mi 11 i ' ° ° which represent two new waves of perturbation densities. On the other hand, ne and ni can be written in terms of n1 and n2 as w e ne V;(Tllnl + lenz) (2.3.5) “’1 ni = V:(T21n1 + T22n2) . (2.3.6) The propagation constants, k1 and k2, for the 111 wave and the n2 wave are given by (A) 2(1).2 J'- 2 2 2 2 2 e 1 2 e i x I N]l-‘ (2.3.7) 22_1_ (1)0) 2_(B'2_82)2+4e1]2 1 e V2V2 e i W M II any M ‘CD (‘0 N + ‘CD (2.3.8) The constants 81' 82, T11, T21 and T22 are expressed as functions of plasma parameters as 13 w w. _ e 1 ‘ e 1 w w. _ - .2 - _£ 52 — T21 v T11 v. _ (2.3.10) e 1 T s 1 (2 3 11) ll 2 2 l o o 1 Va V1 2 2 2 2 l + I 2 2(Be - B1 - A0) w w. e 1 2 2 T = - J; V V e - BL - A0 21 2 wewi V 2V.2 i l + 1 e 1 (B 2 _ 8.2 _ A )2 2 4 w 2w.2 e 1 o e 1 (2.3.12) T - l (2 3 13) 12 “ V 2V 2 g ' ' l e i 2 2 2 l + — (B — B + A ) [ 4 (0 200.2 e 1 0 ] e 1 2 2 T = _ 1 veVi 8e I 8i + A0 1 + 1 1 (B 2 _ B 2 + A )2 2 4 w 2w.2 e o e 1 (2.3.14) where 2 2 l _ 2 _ 2 2 we mi 7 e i Physically, ml and n2 represent two separate longi- tudinal plasma waves each consisting of electrons and ions and propagating with a particular velocity. For convenience, 14 we will call n1 the generalized ionacoustic wave and n2 the generalized electroacoustic wave. 2.4 Electron-ion Composition Ratios of the Generalized Ionacoustic Wave (n1 Wave) and the Generalized Electro- acoustic Wave (n2 Wave) The electron-ion composition ratios for the n1 wave and the n2 wave are studied for various collision frequencies and various ratios of electron temperature to ion temperature. From equation (2.3.3), we have ve vi 111 = - ('m—e- T22)ne + (a: T12)ni 0 (2.4.1) Let R1 be the electron-ion composition ratio for the n1 wave such that V w,T ( e 1 22) . ’ (2.4.2) ViweTlZ R1:- Using equations (2.3.13) and (2.3.14), equation (2.4.2) can be written as ) (B — B. + A ) . (2.4.3) Similarly, from equation (2.3.4), we calculate R2, the electron- ion composition ratio for the n2 wave, as V w.T ( e 1 21) R:- 2 ViweTll (2.4.4) 15 Using equations (2.3.11) and (2.3.14) in equation (2.4.4), we have v . _ 1 e 2 2 _ 2 _ R - 7‘52) (Be Bi A0) - (2.4.5) N R1 and R2 are numerically calculated for various collision frequencies, various Te/Ti and various source fre- quencies. A hydrogen gas plasma is assumed in the numerical example. The detail of this calculation is shown in Appendix A. The numerical calculation of R1 and R2 for various parameters was carried out on the CDC 6500 computer in five programs. In each program, we assign one of the Te/Ti ratios (1, 10, 100, 1000, 10000) and consider six different collision frequency ratios ye/w (0, 0.001, 0.01, 0.1, 1.0 and 10). Figures 2.1 and 2.2 plot the electron-ion composition ratios of the n1 wave and n2 wave reSpectively for ye/w = 0, 0.001, 0.01, 0.1, 1.0 and 10 with Te/Ti = 1, lo, 100, 1000, 10000 as a function of the plasma frequency square over the frequency square. The range of (082/032 considered in these 4tolxlo6 figures is from 1 x 10- which corresponds to a hgih frequency region and a low frequency region respectively. It can be seen in Figure 2.1 that at the high frequency limit, the n1 wave consists mainly of ions regardless of the Ye/w and Te/Ti values. At the low frequency limit, electron 16 composition is Te/Ti times higher than the ion composition; in the case of T8 = Ti' the nl wave consists of equal amount of ions and electrons. In Figure 2.2, at the high frequency limit, the n2 wave consists mainly of electrons; the higher the Te/Ti values, the higher is the composition of electrons. At the low frequency limit, the n2 wave consists of equal amount of electrons and ions regardless of the ye/w and Te/Ti values. It should be noted that the n2 wave is evanescent at the low frequency. In both figures, the effect due to the ye/w value is not very obvious. The numerical output of the computer can be checked analytically for the simple case where Te = Ti and ye/m = 0. In the low frequency limit, we have m < mi << we, and we can assume(we2/w2)+ m;(wiZ/w2)+ m. Under these conditions, equations (2.2.9), (2.2.13) and (2.3.15) are reduced to 2 w 2 82:0) (_e) e V 2 mi e 2 82=i(-(_D_L) 1 V.2 NZ 1 miz wez A= +_.., ° 3!.2 v2 1 e 2 2 (41:901.) . l 17 Using equations (2.1.11), (2.1.12), (2.2.8) and (2.2.12), we have = — ’ R1 (2.4.6) which is reduced to 1 when Te = Ti' This result is consistent with Figure 2.1. Similarly, equation (2.4.5) becomes R2 = -1. Since R1 and R2 are ratios of two waves, we are interested only in their absolute values, that is, the ratio of their magnitudes. Therefore we have 'R2| = 1. This result is consistent with Figure 2.2. In the high frequency limit, we have m > me >> wi' and we can assume (weZ/wz) + 0; (wiz/wz) + 0. Equations (2.2.9), (2.2.13) and (2.3.15) are reduced to 2 882 = 9—7 (2.4.7) V e 2 8.2 = 59—5- (2.4.8) 1 v. 1 2 2 1 2 2 2 u) m. _. A0 = (L. .. L) + 4 42—1.]2 . (2.4.9) v.2 v2 vzv.2 1 e wezwiz w Since w > me >> mi, 1t 15 true that 4 ;—§;—§ << 2 ;_§;—§. e i e i w 2w.2 After omitting the term 4 2 12 in equation (2.4.9), we have Ve Vi A =B.2- 82 . (2.4.10) 18 Substituting equations (2.4.7), (2.4.8) and (2.4.10) into equation (2.4.3), we have R1 = 0. This result implies that the n1 wave in the high frequency region consists mainly of positive ions. This phenomenon is shown in Figure 2.1. Substituting equation (2.4.10) into equation (2.4.5), we have V 2 32 = (:11) (e 2 - 8-2) . (2.4.11) 0) e 1 e Using equations (2.4.7) and (2.4.8) in equation (2.4.11), we have e V With Te/Ti = l and for hydrogen gas plasma model, Ve'Z/Vi2 =‘mi/me = 1836, thus, we have IRZI = w. This result indicates that the n2 wave in the,high frequency region consists mainly of electrons. This fact is shown in Figure 2.2. Furthermore, since R1 at the low frequency limit is equal to Te/Ti as given in equation (2.4.6), it can easily be seen that in the cases of T = 10 T., T = 100 T., T = 1000 T. e 1 e 1 e 1 and Te = 10000 Ti' the ratios of n to ni are 10, 100, 1000 e and 10000 respectively. These results are shown in Figure 2.1. l9 10‘- Te = TI 10000 Te _ TI _ 1000 102 - Te = 100 Ti- 89—" TI 10 1- 12:1 3 Ti ‘6 3 E‘ -2 m 10 P ‘5 H O ‘H ’7 -4_ 510 \ 3’ Y " —E-= 10 0.) 1o"6 . _ Y 10 8 r 53 = 0 ~ 1 1 L l J 1:, _ (L) 10‘4 10 2 1 102 1o4 106 (53) Figure 2.1 Electron-ion composition ratio of the generalized ionacoustic wave (n1 wave) as a function of (we/w)2 for various ratios of electron temperature to ion temperature. 2 20 1012 " Te . if = 10000 Te _ TI — 1000 10 10 33 = 100 Ti Te _ TI — 10 3 _ 10 22 = 1 g T1 3‘5 3 O 3' 0 o ‘5 H O ‘H 2‘104 * \ Q) 5 102 . l u— 1 _l L I J , (U 10’4 10"2 1 102 104 1o6 (52) Figure 2.2 Electron—ion composition ratio of the generalized electroacoustic wave (n2 wave) as a function of (we/m)2 for various ratios of electron temperature to ion temperature. 21 2.5 Propagation Constants of the Generalized Ionacoustic Wave and the Generalized Electroacoustic Wave The propagation constants of the generalized ionacoustic wave and the generalized electroacoustic wave, k1 and k2, for the cases of various collision frequencies and various ratios of electron temperature to ion temperature are studied in this section. k1 and k2 are given by equations (2.3.7), (2.3.8) and (2.3.15) as (Be + Bi + A) (2.5.1) a: Anya kflh‘ a: a: (8 +8. -A) . (2.5.2) It is shown in Appendix B that equations (2.5.1) and (2.5.2) are reduced to two sets of equations; one set for hydrogen gas and another set for xenon gas, such that kl/(w/Vi) and kz/(w/Ve) for each gas assumption can be cal- culated numerically by using CDC 6500 computer in five programs. In each program, we assign one of the Te/Ti ratios (1, 10, 100, 1000, 10000) and consider six different collision frequency ratios Ye/w (0, 0.001, 0.01, 0.1, l and 10). The numerical results for the hydrogen gas are drawn in Figures 2.3 to 2.7. The numerical results for the xenon gas are used in the electroacoustic wave and the ionacoustic wave radiation pattern calculations. Figure 2.3 plots the real part of kl/(w/Vi), or the phase constant of the n1 wave, for the cases of ye/m = 0, 22 0.001, 0.01, 0.1, 1.0 and 10 with Te/Ti = l, 10, 100, 1000 and 10000 as a function of the plasma frequency square over the frequency square. The range of w 2/w2 considered in these 4 6 e figures is from 1 x 10' to l x 10 which corresponds to a high frequency region and a low frequency region respectively. The effect due to the collision frequency is not very evident so that it is not shown in the figure. However, the tempera- ture ratio, Te/Ti' has a big effect in the low frequency region. It should be noted that kl/(w/Vi) does not vanish at any frequency range. This implies that the n1 wave propagates under all conditions. The phase velocity of the n1 wave, Vphl’ can also be observed in this figure, since it is given as w/[Re(k1)]. At the high frequency limit, we have Re[k1/(w/Vi)] = l, or Re(w/k1) = Vi' This implies that at the high frequency limit, or in the low plasma density region, the phase velocity of the n1 wave approaches to the thermal velocity of ions. Also, it can be seen in the figure that at the low frequency region or as the plasma density in- creases, the phase velocity becomes greater and then approaches to the value of Vi/(Te + Ti)/Ti' which is called VA’ the phase velocity of the pseudosonic wave. Figures 2.4 and 2.5 plot the negative imaginary part of kl/(w/Vi), the attenuation constant, of the n1 wave. In Figure 2.4, the cases for Te = Ti' ye/w = l and T8 = Ti' ye/w = 10 are plotted. In Figure 2.5, the cases for T8 = 100 Ti with ye/w = 0.001, 0.01, 0.1 and l are plotted. 23 It is noted that for the case of ye/w = 0, the attenuation constant is zero. From Figures 2.4 and 2.5, the most striking phenomenon is that the attenuation constant of the ni wave decreases (drastically once w becomes smaller than wi. It is also seen that the attenuation of the n1 wave is reduced as the colli- sion frequency becomes smaller or the temperature ratio Te/Ti becomes higher. It should be noted that the Landau damping is very high for the n1 wave at high frequency range where Vphl approaches to Vi‘ Figure 2.6 plots the real part of [ke/(w/Ve)], or the phase constant of the n2 wave, as a function of (we/w)2 for various collision frequencies. The effect due to the collision frequency is signifi- cant. For the collisionless case, it is seen that the real part of [ke/(w/Ve)] changes from one in the high frequency region to zero abruptly as w approaches m It is understood e' that as the phase constant of a wave goes to zero, the wave becomes evanescent. Therefore, it can be seen that the n2 wave is cut off when w < we. The phase velocity of the n2 wave, Vph2' in the high frequency region is Re(w/k2) which is equal to Ve' As the collision frequency becomes higher, the region in which the n2 wave propagates is extended further to the lower frequency region, though it can be seen in Figure 2.7 that the wave in this region will suffer a very high 24 attenuation. When w is around ”1’ a peak appears in the curve, this peak probably corresponds to the oscillation of ions at this frequency. The ratio Te/Ti affects the phase constant curve of the n2 wave only slightly on the low frequency region, therefore, it will not be plotted. Figure 2.7 plots the negative imaginary part of ke/(w/Ve), that is, the attenuation constant of the n2 wave. It is seen in the figure that the higher the collision fre- quency, the higher is the attenuation factor. Once w becomes smaller than w the attenuation constant becomes extremely e' large implying that the n2 wave is nearly cut off. It is noted that our theory based on the macrosc0pic approach does not predict the Landau damping which occurs at the high ph2 approaches to Ve' The numerical results for the propagation constants, frequency region where V k1 and k2, can also be checked analytically for the case where Te = Ti and ye/w = 0. In the low frequency limit, we have m < “i << we. Using equation (2.2.9) with ye/w : O, we have 2 82_w2_e (253) e“ 2 o o. e e < Using equations (2.2.13), (A-12) with ye/w = 0, we have 2 -2 U) l B.2=-—§--——2 . (2.5.4) 1 i 8 P < < Then (8.2 - B 2)2 = B 4 — 28 28_2 + 8.4 becomes 1 e e e 1 l w 2 w 2 2 w 2 w 2 + w 2 w 2 (B 2 _ 8 2)2 _ ( e _ 1 ) _ 2w2< e _ e 1 + 1 > 1 e V2 v2 v1 vzv2 v3 e i E e e i 1 + w4( 12 _ 12)2 . V V. e 1 Noting that w < mi << we, we drop the term containing w4 and neglect the term wiz in comparison with wez, thus,this equation reduces to 2 2 2 2 2 2 (B 2 _ 8 2’2 : (we _ “1 ) _ 2“(we _ me + ”1 ) 1 e v 2 v 2 v 4 v 2v.2 v I e 1 e 1 1 (2.5.5) Using equation (2.5.5) in equation (2.3.15), we have 1 wezviz + wizvez 2 2 A0 = (l — 2w A4) , (2.5.6) vzv2 e i where m 2V 4 - m 2V.2 + w.2V 4 A4 = e 1 2 2e 1 2 212 e . (a)e Vi + mi Ve ) A4 can be expressed in terms of Ti and Te by using equations (2.1.11), (2.1.12), (2.2.8) and (2.2.12); 1 T 2 - T.T + T.2(m /m,) 2 m- (Te + Ti) 26 Because T. = T and m. >> m , then T.2(m /m.) << T 2. After 1 e 1 e 1 e 1 e neglecting Ti2(me/mi) in the numerator of equation (2.5.7) and recognizing that (Te + Ti)2 >> (Te2 becomes A4 < (l/wiz), or szA - TiTe)' the 1nequa11ty 2 2~ . 4 < (2w /wi ). Since (oz/wiz) << 1, then 2(02A4 << 1. Using binomial expansion in equation (2.5.6) and keeping the first two terms, we have m 2V.2 + w.2V 2 2 m 2V.4 - m 2V 2V,2 + w,2V 4 A 2 e 1 1 e [i _ w e 1 e e 1 1 e 0 2 2 2 2 2 2 2 Ve Vi (we Vi + mi Ve ) + 0.9] (20508) Substituting equations (2.5.3), (2.5.4) and (2.5.8) into equation (2.5.1), we have 2 (.2 1 k1 = 2 L 2 2] Vi + (mi/we) (vs/Vi) 2 2 _ _ where (mi/we) (Ve/Vi) - l for Te - Ti' Thus, k l _ 757VEY - . (2.5.9) file This result is confirmed in Figure 2.3. Putting equations (2.5.3), (2.5.4) and (2.5.8) into equation (2.5.2), we drOp terms with w2 since w < wi << we, then 8 N E N k 2 = -(._e._. + .1.) (2.5.10) < 1"“ 27 which is a negative value. Since k2 is purely imaginary, the n2 wave will not propagate in the low frequency limit. In the high frequency limit, we have m > me >> mi. Using equation (2.2.9) with ye/w = 0, we have 2 E 2 w2 e B = ———(1 - ———) . (2.5.11) e Vez w2 Using equations (2.2.13), (A-lZ) with ye/m = 0 and wi (a m, we have 2 8.2 =9L— . (2.5.12) 1 2 i < Thus, equation (2.3.15) reduces to 2 2 we2 zw___-w 4' Ac V2 {72 ‘72 i e e and finally, k12 = (112/Vi2 or kl/(w/Vi) = 1. This result is shown in Figure 2.3. Similarly, 2 w2 we .2 2.39.4. The last term in the bracket can be drOpped, because w > we. 'Therefore, k22 = wz/Vez, or kZ/(w/Ve) = 1. This result is shown in Figure 2.6. 28 J. 0.9 - 0.8 .- Te _ Ff" 1 0.7 - G) > (U 3 20.5 r- 0) ,Q 4..) '44 00.5% 4.) 1: f6 4.) U) 80.4 .. U 3 g 33:: 10 04 T1 0.3 - Y 3- = 0.0 ~ 10 0.2 - (small effect due to collision) Te _ \ fi— 100 0.1 " T e “'7': 1000 353: 10000 t T1 T1. 1 0 l l l J _L 1 w 10‘4 10‘2 1 102 104 106 (ES) 2 Figure 2.3 Phase constant of the generalized ionacoustic (nave (n1 wave) as a function of (we/m)2 for various ratios of (electron temperature to ion temperature in a hydrogen plasma. 29 A-1[k1 ] 1 :_ m (OJ/Vi) 0 n > Y 32 39:10 H Clo-1:" !\ .2 h +’ i ‘H ,. O u . g 1. 4) 2 0 P 'Y 0 _e___1 s w " m I 5 b 5 1. «U .- .p (U .- J 1 1 1 1 __ (l) 10'2 1 102 104 106 (53)2 Figure 2.4 Attenuation constant of the generalized ionacoustic wave (n1 wave) as a function of (we/w)2 for various collision frequencies in a hydrogen plasma. (Te/Ti = l) ‘II I IL I1! F... C, .l)...‘ J. v‘lhh'h' I‘D-nil! 9 s 30 ’D l H a . P—‘I I w H |_—.l (w/Vi) 10.3— Y I- £—1 I w ' y- 10"4 —' y P e —-=0.].. 2 E m m L 3 . E‘ _ o. .. .G 4) m D O ‘5 .+110-5 :' Ye 2 ~ -— = 0.01 o r- (0 O I- c: f' 0 . -H 4..) m I a c m b .p 4.) m 10"6 c- e - -— = 0.001 Remark: k n For 8:0,- m[——1——]= m (w/Vi) 1 1 Lw=wil 1 3,. w 2 10"4 10 2 1 102 T 104 106 (53) Figure 2.5 (Attenuation constant of the generalized ionacoustic wave (n1 wave) as a function of (we/w)2 for various collision frequencies in a hydrogen plasma. (Te/Ti = 100) ‘ Re[——E3—T] 31 (cu/Ve 1 t L. 9 m ‘1 _. 310 _ N Z c - v 2 *' .2: 1 JJ )- (1) ‘H o F 4) c .. M {J m 5 010'2 :- 0) .- m - (U ‘3. 3 Y __ 59-: 0.1 _ r _3 _ 10 C )- r Y _ —9-= 0.01 Y Y m - f- = 0 e— 0.001 1 l l 1 A. 10'4 10"2 1 102 104 106 (.332 Figure 2.6 Phase constant of the generaliZed electro- acoustic wave (n2 wave) as a function of (we/w)2 for various collision frequencies. (Te/Ti = 100) ‘ k 32 - Im[:;7%;T] attenuation constant of the n2 wave Y i _e=10 00 1 L. C ' Y _ —e=1 w 10'1 -— I L— V Y - £=0.1 0.) 10"2 :— P Y * -9 = 0.01 (L) 10-3:- p- L. Y " —E = 0.001 b (A) 1 1 1 1 ._ 10‘ 10-2 102 104 106 (29, Figure 2.7 Attenuation constant of the generalized electroacoustic wave (n2 wave) as a function of (we/w)2 for various collision frequencies. (Te/Ti = 100) 33 2.6 Differential Equations of the Magnetic Field The magnetic field excited by the electric source in the plasma can be found as follows: From equation (2.1.2), with the assumption of periodic time dependence and using the relation of equation (2.1.5), we have V x g = uo3s + poenofl)i - fie) + jwuoeofi . (2.6.1) Taking the curl of equation (2.6.1), we get V x V x § = 00V x 35 + uoeno(V x 6i — V x U ) + jwuoeov x E (2.6.2) where Vxfi=- jwfi (2.6.3) is given by equation (2.1.1), V x fie and V x fii can be obtained by taking the curl of equations (2.1.8)and (2.1.10) and using equation (2.6.3), thus, . + + JweB V x U = , (2.6.4) 9 mewe + 3w) v X at = "' Jweg o (2.605) 1 mi(Yi + 3w) 34 Substituting equations (2.6.3), (2.6.4) and (2.6.5) into equation (2.6.2), we have m 2. w 2 V x V x B = 00V x 35 - jwuoeo[ i + ]B e (Y1 + jw) (Ye + jw) + wzuoeoB . (2.6.6) 2 . + + + -> . Since V x V x B = V(V - B) - V B and V . B = 0, equation (2.6.6) can be rewritten as 2 2 2+ V B + w u e 1 + , e , + , , B o o 3m(ye + 3w) 3w(yi + 3w) +S = - ro x J . (2.6.7) Let ke2 = wzpoe, where 2 2 e = e 1 + . . + . . ‘3[ jw(Ye + 3w) Jw(Yi + Jw)] w 2 w.2 w 2Y w + y w + y. w(w + y ) e 1 e w.zy. + 21 1 2 ) (2.6.8) w(w + Yi ) the equivalent complex permittivity in the plasma, equation (2.6.7) can be written as 2 2 _ _ +8 (v + ke )fi _ 00v x J (2.6.9) which is an inhomogeneous wave equation and its solution can 35 be expressed as 156) II .111: O < x <‘w Q R‘ (D . .__§___ dv' (2.6.10) + -> where R = Ir - r' 2.7 The Electric Field in the Plasma The electric field in the plasma can be derived from the equation of magnetic fields, equations of the n1 wave and n2 wave. It will be shown later in this section that the electric field in the plasma consists of three components; one of which is electromagnetic in nature, while the other two components, which are due to the presence of the electro- acoustic and ionacoustic waves, are longitudinal in nature. Let us consider the source free Maxwell's equation in the plasma: + V x B + + ' E 2 7 1 poenomi - Ue) + jwuoeo ( . . ) .y .+ where Ue and Ui can be found from equations of momentum conservation for electrons and ions. That is,from equations (2.1.8) and (2.1.10), we have i + 3kT e e Ue = - m ( + . ) E - , Vne (2.7.2) e Ye 3w nome(Ye + 3w) 3kT . 6. — e E - 1 Vno . (2.7.3) 1 mi(Yi + jw’ nomi(Yi + 3w) 1 36 It is seen that these average velocities of electrons and ions are proportional to the electric field in the plasma and the pressure gradient of the particles. Using equations (2.7.2) and (2.7.3) in equation (2.7.1) and after rearrange- ment, we have u e w 2 u e w 2 _ . o o e o o i V x B - [jwuoeo + ( + ]B Ye + jw) TYi + jw) BuoekTeVne - BuoekTiVni me(Ye + jw) mi(Yi + jw) which yields Bu ekT Vn V x B = 0 e e P , B + . (Ye + jw71Yi + 3w) me(Ye + 3w) 3n ekT.Vn. o 1 i - mi(Yi + ij V(2.7.4) where . . . 2 . P = jwuoeo(ye + 3w)(yi + 3w) + “oeowe (yi + 3m) + poeowiz(ye + jw) pon{[Yi(wez - wz) + Ye(wiz - w2)] . 2 2 2 + jw[we + wi - w + yeyi]} . (2.7.5) 37 EXpress E in terms of B, ne and ni in equation (2.7.4), we have 3n ekT (Y + jw)Vn = 1 . - o i e i i F [(ye + 3(1))(Yi + 3w)(V x B) + mi 3n ekT (y. + jw)Vn - 0 e 1 e] . (2.7.6) me Using equations (2.3.5) and (2.3.6) which express ne, ni in terms of n1 and n2, we can obtain B in terms of B, nl and n2 as follows: +_1 _2 . E - F[(YeYi w ) + Jw(Ye + Yi)](V x g) + 311oek [(YewiTiTZI YiweTeT11)+ . (“iTiTzi __....._.. - 3(1) .__...__ P miVi meVe miVi w T T w T T w T T _ e e 11)]an + [Ce 1 i 22 _ Y1 e e 12) meVe miVi meVe w T_T w T T + jw( 1 1 22 - —————e e 12)]Vn2 . (2.7.7) miVi meVe It is seen in this equation that the electric field in the plasma has three components. The first term on the right hand side of the equation is electromagnetic in nature, because B field is entirely electromagnetic. The second and third terms, which are due to the presence of the electro- acoustic and ionacoustic waves, are longitudinal in nature. 38 2.8 Average Velocities of Electrons and Ions in the Plasma The average velocities of electrons and ions in the plasma can be obtained from equations (2.7.2) and (2.7.3) with equations (2.3.5) and (2.3.6) as + 1 eg 3kTewe U = ——————— - —— - , (T Vn + T Vn ) e Ye + 3w [ me menove(Ye + 3m) 11 1 12 2:] (2.8.1) a 1 eE 3kTiwi . = __ - . (T Vn + T Vn fl . 1 Y1 + 3‘“[’“i minoViTYi + 3‘“) 21 1 22 2 (2.8.2) n1 and n2 can be found by solving equations (2.3.1) and (2.3.2) and E is given by equation (2.7.7). It is observed that fie and 61 also possess both electromagnetic and longitudinal natures. CHAPTER 3 RADIATION PATTERNS OF ELECTROACOUSTIC AND IONACOUSTIC WAVES EXCITED BY VARIOUS ANTENNAS 3.1 Introduction Our objective in this chapter is to calculate the radiation patterns of electroacoustic and ionacoustic waves excited by four different types of antennas; namely, Hertzian dipole antenna, disk monopole antenna, disk dipole antenna and cylindrical antenna. The antennas are assumed to be im- mersed in an infinite, homogeneous, isotropic and compressible plasma. In Chapter 2, the generalized ionacoustic and electro- acoustic waves which are excited by an electromagnetic source are given in equations (2.3.1) and (2.3.2) as 2 2 S (v + kl )nl = 51 ep— (3.1.1) (V2+k2)n =5 9.8. (312) 2 2 2 e o o o The propagation constants of the nl and n2 waves, that is, 39 40 k1 and k2, are shown graphically in Figures 2.3 to 2.7 for various electron and ion temperatures and various collision frequencies. Since equations (3.1.1) and (3.1.2) are of the same form, only a common equation such as 2 2 S (v +1. )n=Sg-— (3.1.3) will be considered. Equation (3.1.3) is a scalar inhomogeneous Helmholtz equation whose solution is _+ s S ., -jkI-f - I") n(r) = - -— f p (r') e + + dv' (3.1.4) 4Ne v' [r - r'I where the primed coordinates refer to the source points and the unprimed coordinates represent the field point. We assume that the antenna dimensions are small compared with a free space electromagnetic wavelength and the observation point is in the far zone of the antenna that the far zone approximations of I? - E'I = r for the amplitude term and I; - ?'| = r - z' 0086 for the phase term can be used. The radiation patterns of the generalized electro- acoustic wave (n2 wave) and the generalized ionacoustic wave (n1 wave) excited by various antennas can be calculated from and k to 2 2 replace n, S and k in equation (3.1.4) while for the nl wave equation (3.1.4) For the 112 wave, we use n2, S we use n1, S1 and k1 instead. 41 3.2 Hertzian Dipole Antenna 3.2.1 Geometry and Statement of the Problem The geometrical configuration of a Hertzian dipole antenna is shown in Figure 3.1 using a Spherical coordinate system (r, 9, 0). A Hertzian dipole antenna, with the assump- tion that the radius of the wire is thin and its length, d1, is very short compared with the wavelength, is immersed in the plasma. The ends of the antenna are large enough that the charge distribution of the antenna can be given approxi- mately as s _ 06(2' - d1)6(x)0(y) ‘ 'QG(z' + dl)5(x)0(y) ‘3-2-1’ where Q is the charge in coulomb and 6 is the Dirac delta function. The generalized electroacoustic and ionacoustic waves excited by a Hertzian dipole antenna can be obtained from equation (3.1.4) after the substitution of OS from equation (3.2.1). Using the far zone approximations, the integral in equation (3.1.4) becomes + + ~jklr - r'l -jkr f ps(;') e + + dv' = sz e [sin(kdl c080)] v' Ir ' r'l r (3.2.2) where k is the propagation constant of a particular wave. For the generalized electroacoustic wave, we use k2 to replace 42 + z=d1 '- 1; w H+ 11+ =-d1 Figure 3.1 Geometry of a Hertzian dipole antenna. 43 k, while for the generalized ionacoustic wave we use k1 to replace k. The term in the bracket in equation (3.2.2) will be used to calculate the radiation patterns of these plasma waves excited by the Hertzian dipole antenna. 3.2.2 Radiation Patterns of the Generalized Ionacoustic Wave (n1 Wave) The radiation pattern function of the generalized ion- acoustic wave can be obtained from equation (3.2.2) after replacing k by kl. The pattern function can be expressed as Fl(6) = sin(k1d1 cose) . (3.2.3) Since kl/(w/Vi) has been calculated by using the computer for a xenon gas plasma, we can determine the value of k1 by assum- ing the values of w, T6 and Te/Ti' d1 is the antenna half length and is assigned for various values. The phase velocity of the generalized ionacoustic wave, Vphl' at the low frequency range is approximated by /3ETT€ + Ti)7mi and is called VA. The results of some typical cases are plotted in Figures 3.2, 3.3 and 3.4 and their numerical results are given in Tables 1, 2 and 3 of Appendix C. Figure 3.2 shows the radiation patterns of the generalized ionacoustic wave at various electron temperatures. Since the wave length in the plasma is A = Vphl/f' as Te P increases, 1p increaseszand consequently, the antenna becomes relatively smaller. Figure 3.3 shows the radiation patterns of the 44 “so H u do .oH u Hs\me .me om u my .mmusumummEmu someomam msoflum> u0m mccmucm maomflc sownuumm m an nonwoxm m>m3 owumsoomsow wmnwamumsmm man no mcumuumm downwacmm m.m onsmflm as «a .2. . s .36 s.oOOOH u as 0 soooom u a x.ooo~ u we 45 150 H u as .xoooom n we .me om n my .wusumummEmu 20H ou manpmummEmu couuomam mo moaumn msoflum> How mccmuam maomflo CMHNuHmm m an wmufloxm m>m3 oaumsoomcofl UTNHHMHmcwm mnu mo mcumuumm coflumflwmm m.m musmfim od 0 1 @011 wo1v col coca fl .MI 9 .H . .9 wo1 ooa fl .MI E . 1!.l1 OH " MW. . o; B I a obi 8 Hum) 46 1&0 m.~ u as .OH u a9\0e .soooom u was .mmflocwsqmum mccmuom msoanm> now mocmvcm maomfip smaupumm m v.m madman >9 Umufloxm m>m3 Uflpmsoomcofl UTNHHmuwcmm may mo mahwuumm coaumwwmm me on wax ON me 0H 47 generalized ionacoustic wave at various ratios of electron temperature to ion temperature. It is seen that the change in the ratio Te/Ti does not affect the radiation patterns significantly. Figure 3.4 shows the radiation patterns of the generalized ionacoustic wave at various antenna frequencies. It is seen that as the antenna frequency increases, the wave- length of the generalized ionacoustic wave decreases; as a result, the antenna becomes relatively larger. 3.2.3 Radiation Patterns of the Generalized Electroacoustic Wave (n, Wave) The radiation pattern function of the generalized electroacoustic wave can be obtained from equation (3.2.2) after replacing k by k2. The pattern function can be expressed as F2(0) = sin(k2dl cose) . (3.2.4) Since kZ/(w/Ve) has been calculated by the computer for a xenon gas plasma, we can determine values of k2 based on assumed values of w, Te and Te/Ti' Some typical cases are chosen and plotted in Figures 3.5 and 3.6 and the corresponding numerical results are given in Tables 4 and 5 of Appendix C. Figure 3.5 shows the radiation patterns of the excited generalized electroacoustic wave at various electron tempera- tures. It is seen that as Te decreases, the phase velocity and the generalized electroacoustic wave length in the plasma 48 Ass H u He .mm.o u N3\m63 ..OH on H n Hs\me .Nmo H n we .mmnsumuwmfimu couuomHm msoHum> MOm mccmucm mHome smenuumm m xn UmuHoxm m>m3 oHumsoomouuomHm pmumeumcmm on» NO mcumuumm COADMHomm m.m muqum o; . ad 0.0 V6 «.0 o 1 1 «d *6 0.0 ad o; .2- _ . 1-..)...1..- .11 L... _ . .2. \. I. ./ \\ \ \ \ 31.. / / . I/ \. . \.\. _\ /. III. / a \\\ \ / I], o ’Ix‘ ‘\ \ V.Ol_l / I, III.‘\ \ . x 8 o \ \\/o ’ oOOO o \ \ /. I V. 0.0-.1 / .v .\ o / o —/0\ . _ _ vHoOOOOH H... 0H. I/ 0.011 \~ / I \ 0:88 u me .o/« r/ 1111 \\ .8 \ Moooom 1|. 08 I O... \ 49 165 H u H6 .mm.o u m3\m03 .OOH on H u HB\me .eoooos u was .mmHocmsvmum macwacm msoflum> mom mccmucm THOQH© CMHNuHmm m >9 UmuHoxw m>m3 oaumsoomouuomam Umuflamnmcmm mnu wo mcumuumm COHDMHomm m.m musmHm o— no 0.0 v.0 «.0 .1. o 1 1N0 Iva 0.0 m_o o p 00 .2. . . \l“. . 1 .I. ”II . \ \ // / / \\ .\ \\\ . 6 . // . l/ \\ o\ \\ \\ N61... /. //// / / \ I / 1 \6. // ./ .%0.IIIIIT.II\\\\.T\ WI/tllll. I|\\\\%3. ~ 9011 I 1 1 o _ 1 go m.H u w. z 811 \ N30 0...” u m can // con NmO m.c 1|. M Our ooflm 50 decreases; and consequently, the antenna becomes relatively larger. Figure 3.6 shows the radiation patterns of the excited electroacoustic wave at various antenna frequencies. In both Figures 3.5 and 3.6, we choose the propagation constant, k2, at the frequency of (082/002 = 0.95. The reason for this choice is that the electroacoustic wave suffer less Landau damping when w is close to w and slightly higher e than we. 3.2.4 Radiation Patterns of the Electromagnetic Wave The magnetic field in the plasma has been determined in Section 6 of Chapter 2 and is given by equation (2.6.10) as 4. T: -jkeR 6(2) = _% v x f 35(E') E__§__ dv' (3.2.5) v. where k8 is the propagation constant of the electromagnetic wave in the plasma and is given as w 2 w 2 w 2y 2 _ 2 e _ i _ . e e ke - U) U05: 1 (D2 + 2 L02 + 2 J[m(w2 + 2) Ye Yi Ye w 2y + l l ; (3.2.6) w(w2 + -2) Y1 +8 + . . . . J (r') is the source current den31ty and is given as 35(2') = i z = Ioz for —d1 5 z 2 dl (3.2.7) for a Hertzian dipole antenna whose cross—sectional area is A. 51 R represents I? - §' . Substituting equations (3.2.6) and (3.2.7) into equation (3.2.5) and using the far zone approxi- mations, we have u I d1 -jke(r - z'cosO) fiat): 230fo e dz'; " -dl' r (3.2.8) After evaluating the integral, using 2 = 2 c058 - 6 sine, taking the curl of the integral, neglecting l/r2 terms and retaining only the l/r term because of far zone approximation, equation (3.2.8) can be reduced to u I k d1 + A o o e -jker{sin(kedl c059) . §(r) = ¢< San 2n r i (kedl cosO) (3.2.9) Considering d1 as a small number, we have [sin(kedl cosG)]/(kedl cose) 2 l, and thus, + + A uOIOkedl e-jker - B(r) 2 ¢( 2n X r >Sln9 . (3.2.10) The electromagnetic component of E field in the plasma can be obtained from equation (2.7.7); + 2 . + Eem = %[(YeYi - w ) + 3w How mccmucm maomocoe xmflp m an pmufloxm m>m3 oeumsoomcofl pmuflamumgmm mcu mo mcumuumm coeumepmm m.m mnsmflm «.01 . v.01 / n 0.01 s.ooooH we ad 1 II E4 0:88. can (I E-1 M.o°o~ 59 Ago mm.m n m .soooom n we .me om n we .musumummamu cofl ou musumndemu couucmam mo moflpmu msoflum> now mccmucm QHOQOCOE xmflp m we pwufloxm m>m3 oeumsoomcofl pmmflamumcmm mnu mo mcumupmm COHuMHpmm oa.m musmflm 3 . no so to go o 3 to so as S 000 . q _ _ V f. .\ A _ _ a 000 \ \ / \ . / \ «.011 / \ . / \ / / to-.. . / .bo _ . . a %3 ’ 31.. _ / \ f / \ 33 u wk. / . no.1) , \ we ,/ // .\ on 2: u (m can / ‘ \ o 08 II. o \\ H / .8 0O"@ H u M». 60 Ago mm.m n m .xooomH u He u we .omm\emoa x mo.H u <> sufloon> mmmnm .me m.wa u we .mccmucm cacaocofi xmflp m an pmufloxw m>m3 oeumsoomcofl Ummflamumcmm mcu mo mcumuumm COHpMHpmm Ha.m musmam 31.2.. no amgm >2 ucmfiwnmmxm o o o whomsu pcmmmud as . ms 90 so «a o as ed so as o; .bo. . _ 9g 2 z 10)( _ z %6 . . C 0 e011 :01: . . 000 000 o 9011 o C . ad 11 on .bn a Axon.o u a .1: o; .\ .oum 61 Ago mm.~ u m .sOOONH u He u we .omm\e .oa x mo.a u <> suHoon> amuse .me m.mm n we .mccmucm maomocoe xmflp m we pmufloxm m>m3 oeumsoomcofl pmuaamumcmm men mo mcuwuumd coflumflpmm ma.m musmflm Hva_.am um cmcm >3 ucmEHumdxm o o o >uom£u ucmmmum 0A . md 90 to ad 0 N. 1d ad ad 0; nbo . . s _ _ J) co 3011 se 901T @011 mon mom 62 A80 mm.m ma Umufloxm m>m3 oaumsoomcofi Ummwamuwcmm mcu mo mcumuumm COHuMHUmm oo 0 o; co m .xooomH u fie u md we .omm\s Moe x mo.H u 4> muHoon> mmmnm .me m.mm u be .mccmucm mHOQOCOE xmflp m ma.m musmflm avaa.am um cwnm mg ucmEHummxm o o o whomnp ucmmmum od vd Nd nd 06 ad 0; @611 on .bn o xevm.m q as 2| 63 We choose the phase velocity of the generalized ionacoustic wave, VA' as 1.05 x 103 meter/sec.; the diameter of the disk antenna, 2a, as 4.5 cm and the normalized antenna length, L, (antenna length with respect to the generalized ionacoustic wavelength, i.e., L = 2a/(VA/f)) as 0.7 (A), l (A), 2.5 (A). In these cases, our radiation patterns agree very closely with the experimental result of Shen et al.[14J. 3.3.3 Radiation Patterns of the Generalized Electroacoustic Wave (n7 Wave) The radiation pattern function of the generalized electroacoustic wave can be obtained from equation (3.3.8) by replacing k by k2. That is, J1(k2a sine) F2(0) (3.3.10) kza sine The numerical values of k2 are calculated and are given in Table 9 of Appendix C. Figures 3.14 and 3.15 are the radiation patterns of the generalized electroacoustic wave. We choose (1) ye/m = 0 for simplification, (2) wez/wz = 0.95 such that Landau damping is small, (3) Te/Ti 2 l to 104, (4) the antenna frequency f = 17.5 MHz, and (5) L = 0.6 (1) and 1.1 (A) respectively in these two figures. The radiation patterns for Te between 2000°K and 4000°K agree very closely with the experimental results of Nakamura et al.[12] who used the grid with the plate as an antenna. 64 x4 o.o n ma ..OH on H u fie\me .o u sxo» .mm.o u m3\~m3 .Eo H.mH u 4 .Nm: m.sa u my .mmusumumafimu couuomam msofium> now mccmucm maomocoe xmflp m we pmufloxm m>m3 oflumsoomouuomam pmuflamumcmm mnu mo mcumuMMQ coeumepmm va.m musmflm fima_.am um musamxmz an unmaflummxm o o o muomsu ucmmmud Ilnlllll 02.3 . As new . t” 3 o «a +6 a ’23“. mo 9...? . l. . . , . ~ 0 sour o ._ 23.0 n a . _ _ . _ _ . . 3-- . _ , \ ./ > o o .\ a m \ 583 I e 3% \\. m fees I .H. can ,, .\\ con m moooow u e z I e. abfl@ 65 AK H.H u am ..QH on H u fle\me .o u 3\m> .mm.o n m3\mm3 .50 H.mH n K .Nm: m.sH u we .mmusumummEmu couuomam mSOflHm> How mccmucm wHOrooE xmflp m xn pmufloxm m>m3 oeumsoomouuomam UmNHHmnmcwm may mo mcumuumm cofluMHpmm mH.m mnsqflm Hm: .Hm um sunfimxmz mm ucmEflumdxm o o o knows» pcmmmum o.— . «.0 06 V6 «.0 o «.0 v.0 06 ad 0..— . _ .. .( .bo .36 4 2 NW s 1]., .M- _ A A) \ / \ . / \ NOII / . \ / \ / . /. A4VH.H u a .%o / .ho _ _ soooow u me \o 5.88 u we \ can 383 n we \ 66 3.4 Disk Dipole Antenna 3.4.1 Geometry and Statement of the Problem The geometrical configuration of a disk dipole antenna is shown in Figure 3.16 using a spherical coordinate system (r, 6, ¢). The antenna consists of two half circular metallic disks of radius a is immersed in the plasma. The antenna is excited by a radio frequency signal source and the charge distribution on the antenna can be given as pS= Go for 0 '00 for W ¢l ¢l TT 2TT IAIA (3.4.1) IAIA where 00 is the surface charge density. The generalized electroacoustic and ionacoustic waves excited by this antenna can be calculated by substituting equation (3.4.1) into equation (3.1.4). Using the far zone approximations and with ds' = r'd¢'dr', the integral becomes 'jkr . Go8 [In fa ejkr'51n9COS(¢ - ¢')r'dr'd¢' r ¢'=0 r'=0 2n a . , . , _ f f ejkr s1n0cos(¢ -¢ )r'dr'd¢{] . ¢'=n r'=0 Assuming that the observation point is in the y-z plane (¢ = n/2), we have -jkr c e a n . , . . , o r [I r'dr'f ejkr 51n081n¢ d¢. 0 0 a 2n 'k ' ° Osin¢' - f r'dr'f e3 r 51“ d¢{] . (3.4.2) 0 fl 67 \ 0 + (r,6'¢) j¢ Figure 3.16 Geometry of a disk dipole antenna. 68 Zn . . . Let 11 = f ejkr'51n051n¢'d¢. n 8' = ¢' - n, the integral becomes and replace the variable ¢' with I1: TT_' 1' 'l f e jkr SinGSinB dB' . 0 Since 8' is an independant variable, we can replace 8' by ¢' again and arrive at -° . ° 9 ' ' Jkr Sin Sin¢ d¢' . (3.4.3) n I = f e l 0 Substituting equation (3.4.3) into equation (3.4.2), we have 0e 0 r f r'dr'f 0 0 -jkr -jkr a “ 'kr'sinOSin¢' -'kr'sin05in¢' [e] _ e 3 ]d¢| j20 e a H = 0r f r'dr'f sin(kr'sinesin¢')d¢' .(3.4.4) o o 1T 2 f sin(kr'sinesin¢')d¢'. After replacing_kr'sin0 by Z 0 and ¢' by (w + n/2) such that sin¢' = cosw, the integral Let I becomes n/Z 12 = f sin(Zcosw)dw -n/2 n/2 = 2f sin(Zcosw)dw, 0 because the integrand is an even function. The Struve function is defined by the equation 212V 2 (2) "/ — f sin(Z cos¢)sin /Fr(v + 1) o 2v¢d¢ . a, (Z) 69 For v = 0, we have n/Z H (Z) = f sin(Zcos¢)d¢ . ° 0 =l|N Thus, 12 can be expressed by the Struve function as H II 2 nHO(Z) or nHO(kr'sin0) . (3.4.5) Substituting equation (3.4.5) into equation (3.4.4), we have j20 e.jkr a 0r f nHO(kr'sin0)r'dr' . (3.4.6) 0 a Let I3 = f Ho(kr'sin6)r'dr' and replace the variable 0 kr'sine by Z again, we have ka sine HO(Z)ZdZ I = f 3 0 (k sine)2 1 ka sin0 = 2 f HO(Z)ZdZ . (3.4.7) (k sine) 0 The recurrent equation of the Struve function is d v _ v d _ _ d§(ZH1) - ZHO for v — l, or 2H1 = fZHOdZ . (3.4.8) 70 Substituting equation (3.4.8) into equation (3.4.7), we obtain a2H1(ka sine) 3 = ka sine ° Then equation (3.4.6) or the integral of equation (3.1.4) becomes j2na200e-Jkr[Hl(ka sin0)] ka sine r (3.4.9) The term in the bracket will be used to calculate the radia- tion patterns of the plasma waves excited by a disk dipole antenna. 3.4.2 Radiation Patterns of the Generalized Ionacoustic Wave (n1 Wave) The radiation pattern function of the generalized ion- acoustic wave can be obtained from equation (3.4.9) by replacing k by kl. That is, H1(kla sine) F1(0) = . (3.4.10) kla sine The results of some typical cases are plotted in Figures 3.17 to 3.21 and their numerical results are given in Tables 10, 11 and 12 of Appendix C. Figure 3.17 shows the radiation patterns of the generalized ionacoustic wave at various electron temperatures. Figure 3.18 shows the radiation patterns of the 71 “so m~.~ n m .0H u H9\09 .me om u we .mmuaumummfiwu conuomflm wnoflum> How mccmucm waomflp xmflp m an Umefloxm m>03 UMHmSOUMGOw pmuflamumcmm may uo mcumuumm scannepmm 5H.m mnsmflm I \ / \ \\ / 0:82: u we (\\ no: //(.1. 0:83 u we can con scooom u me e. 72 A20 m~.m u m .moooov u we .nmx on n my .mnsumummfimu GOw on muaumummsmu couuomaw mo moflumu msoHHm> How mccmucm maomwp xmwp m an omuwoxm m>m3 owumaoOMCOw vmuwamumcmm man no mcuwuuMQ coHuMepmm mH.m musmflm to «.o o «a no 3 3 a... O).— . Dd ow q — A — 4 q 08 3,7 \ \ \ \ / / I l \ \ \ .9 OOH " MI Q6111 .H. A $3 I bun). n OH I m .b B 0‘ a He aqua H I MI 9 73 H50 m~.~ u m .mcoomH u He u we .omm\e .0H x mo.H u 4> euHoon> mmmem .me mm u we .mccmucm maomflp xmflp m wn wmvwoxm m>m3 owumsoomcoa pmNHHmumcwm mgp mo mcumuumm COHMMflpmm md.m musmflm HqH_.Hm um swam an ucmeHummxm o o o whomnu aammmhm . . 3 3 3 3 o 3 3 3 3 3 0000.— q — — A _ _ _ 000 o = 0 ad 61) 000 08 0611 / 3.... I \ $3” can A: mtm u A o.— 74 HEo mm.~ u m .MsOONH u AB u 09 .omm\E moa x mo.a n H~> >vfloon> mmmnm .me m.mv u my .mccmucm maomflp xmflp m >3 pmuwoxm m>mB Uflumsoumcofl Umnaamumcmm may mo mcumuumm coHuMHpmm om.m musmflm H¢HH.HM um cmnm >3 ucmfiwuwmxm. o o o >Hom£u ucwmmum o..— . ad 0.0 Vd «.0 o «.0 V6 0.0 md 0H 00 H z _ . . H H . 02. o o o 3 o o 0 V; 000 000 o .611 O O 0.011 I, .\ n , .bn no 1‘ ‘ 33 m u .H o.— .oua 75 RED mm.m n w .MOCONH a HR u we .omm\E moa x mo.a u <> >ueoon> mmm3m .me m.mm n mv .mccmucm mHOQHG xmflp m >3 Umueoxm m>03 owumsoomcofl pmnaamumcmm 033 m0 mcumuumm cofluMAUmm Hm.m musmflm avHH.Hm um :w3m >3 ucmsflummxm o o o >Hom3u unencum 3 . ad ad to «d o «6 Yo 0.0 ad 3 .50. . 3 H _ . _ _ H _ %& ud1r o o O to 000 000 3 _ @611 $3” can A: v u .H 3 76 generalized ionacoustic wave at various ratios of Te/Ti' Figures 3.19 to 3.21 show the radiation patterns of the generalized ionacoustic wave at various antenna fre- quencies. We choose the phase velocity of the generalized ionacoustic wave, V , to be 1.05 x 103 meter/sec., the diameter of the disk antenna, 2a, to be 4.5 cm, and the normalized antenna length, L, as 1.5 (A), 2 (A), 4 (A). In these cases, our radiation patterns again agree very closely with the experimental results of Shen et al.[14]. 3.4.3 Radiation Patterns of the Generalized Electroacoustic Wave (n2 Wave) The radiation pattern function of the generalized electroacoustic wave can be obtained from equation (3.4.9) by replacing k by k2. That is, H1(k2a Sine) 0 = . . . 1 F2( ) kza sin0 (3 4 l ) The results of two typical cases are plotted in Figures 3.22 and 3.23 and their numerical results are given in Tables 13 and 14 of Appendix C. Figure 3.22 shows the radiation patterns of the generalized electroacoustic wave at various electron tempera- tures. Figure 3.23 shows the radiation patterns of the generalized electroacoustic wave at various antenna frequencies. 77 3&0 N.h n 0 .:OH on H u w9\ma .o u 3\m> .mm.o n m3\m03 .Nmz m.>a u my .mmunumummfimu couuomam mzoflum> Mom mccmucm maomap xmflp 0 >3 woufioxm m>03 oaumsoomouuomam pmuwamumcmm 03» mo mcnmuumm cOHuMflpmm ~m.m mnsmflm . , __ .3 r _ to-) 3 _ \\ // ~— .3 o r . . /(’ .\\HHV\ 9611 /(////I \\\ / o\\ \ //o{. \\ m .\\ II. moooom u .H. II..\ .611 /(| 0:83 u we own .8 moooom u we 9. 78 .20 N.» u m ..oH on H u Hexme .o u 3\m> .mm.o u maxmma .moooom u mes .mmwoamnqmum 0330330 mSOHHm> How mccwucm maomflp xmflp m >3 cmuwoxm m>m3 cflumsoomonuomam penaamumcmm m3» m0 mcumuumm 30fl30fivmm mm.m mnsmflm coco... . ow do do «w\ c As... no mo e... 302. \ / /_ \ e. / . \ a . / \ / °3 ll " ‘\ 1’ /’ Ng ON ’ t \ 8.11 / o . I . .\\\\\\\ [I . N32 m NH | m .361}; \\ II - .bn N32 mH u M o— 79 3.5 Cylindrical Antenna 3.5.1 Geometry and Statement of the Problem The geometrical configuration of a cylindrical antenna is shown in Figure 3.24 using a spherical coordinate system (r, 8, 0). A cylindrical antenna with a thin radius is immersed in the plasma. For this antenna, charge and current distributions can be given approximately as s _ pocos[k(h - z')] for 0 f z' E h p _ -pocos[k(h + z')] for -h 5 z' 5 0 (3°5‘1) ' _ 1 A 1 is Imsin[k(h z')]; for 0 g z. 5 h (3.5.2) ImSln[k(h + z )]z for -h g z 5 0 The propagation constant, k, of the antenna charge or current is still not well known. Some theoretical studies performed by Seshadri [30] and Wunsch [31] predict an electroacoustic component in the antenna current while eXperimental studies conducted by Chen et al. [32,33] and Ishizone et a1. [34] found the antenna current to be predominantly electromagnetic in nature. This justifies the approximation of k = ke where ke is the propagation constant of the electromagnetic wave in the plasma. In our numerical calculation, k is assumed to be ke which is given by equation (2.6.8). The generalized electroacoustic and ionacoustic waves excited by a cylindrical antenna can be obtained by substitu- ting equation (3.5.l) into equation (3.1.4). Using the far zone approximations, the integral becomes 80 ’N 0 s: // z=h (r,0,¢) \/// R \ + )’ r ,\ I \ ( \ | 1" fly is) \ \ \\ \ \ \ \ \\ 3 I \ z=-h I Figure 3.24 Geometry of a cylindrical antenna. 81 -ij Zp k f 95(3)e dv' = .0 v' R J cos(kh cose) - cos(keh) -jkr . cose kzcosze - ke2 Thus, n(i?) = e jSp cos(kh cose) - cos(k h) -jkr °[ ]cose e 2nek c0520 - (ke/k)2 (3.5.3) where k is the propagation constant of the particular wave. 3.5.2 Radiation Patterns of the Generalized Ionacoustic Wave (n1 Wave) The radiation pattern function of the generalized ion- acoustic wave can be obtained from equation (3.5.3) by replacing k by k1. That is, cos(klh cose) - cos(keh) F (8) = cose . (3.5.4) 1 c0320 - (ke/kl)2 In order to excite the ionacoustic wave which does not suffer excessive Landau damping, we need to operate the antenna at low frequency region where me >> m. In this region the electromagnetic wave is cut off and it implies that ke is a pure imaginary number. In the present consideration, we have keh << 1 and ke/kl 2 0. Consequently, equation (3.5.4) is reduced to cos(klh cose) — l F1(9) '3 COSB . (3.5.5) 82 The results of some typical cases are plotted in Figures 3.25 to 3.29 and their numerical results are given in Tables 15, 16 and 17 of Appendix C. Figures 3.25 and 3.26 show the radiation patterns of the generalized ionacoustic waves at various electron tempera- tures. Figures 3.27 and 3.28 show the radiation patterns of the generalized ionacoustic waves for the cases of various ratios of electron temperature to ion temperature. Figure 3.29 shows the radiation patterns of the generalized ionacoustic wave at various antenna frequencies. 3.5.3 Radiation Patterns of the Generalized Electroacoustic Wave (n, Wave) The radiation pattern function of the generalized electroacoustic wave can be obtained from equation (3.5.3) by replacing k by k2. That is, cos(kzh cose) - cos(keh) F (6) = cose . (3.5.6) 2 cosze - (ke/k2)2 To excite an electroacoustic wave without suffering substantial Landau damping, the antenna frequency, w, is chosen to be slightly higher than the plasma frequency, we. The results of some typical cases are plotted in Figures 3.30 to 3.34 and their numerical results are given in Tables l8, l9 and 20 of Appendix C. Figure 3.30 shows the radiation patterns of the 83 Ago m.~ u a .oa u Ha\me .me om u to .mmusumuomfimu couuowam macaum> How occmucm Havauccflamo m >9 pmufioxm m>m3 oaumzoomgow pwuwamumcmm on» no mcnmuumm cofluwwomm m~.m whamwm o... . 8 3 3 «a o «a 3 3 no 3 / 3-1 \ Q MOOOCN II... mm... / // 0.01.! \ \ I m / / \ I 3.82: n we [r o... 1 84 “so m u a .oa u Ha\ma .me om u my .mmuaumummsmu couuomam mnofinm> How mccmucm Hmowuvcaamo m ha wmufioxm m>m3 oaumzoowcofl wmnwamumcmm may we mcumuumm GOHUMmem m~.m musmflm \mal ../ / . \ \ 3.: / .. \\\\\\ .IlIl. II.II .bo ll.ll. \\\\\\ L IIII. Ilunlullu\. .ba Moooom H E adJ1 388 u we , no... .8 382: n we 2 85 Ago m.m u n .xoooow u we .me om u we .musumnmmEmu COw 0v mnsumummfiwu conuomam mo moflumu mSOaum> How mccmucm Hmoflnocflaxo m an vmufloxm m>m3 owumsoomcofl Uwuflamumcmm mnu mo mammuumm cOHuMHUmm o.— v.0 0.0 v0 «.0 o 000 - q q — — \ \ \ «our \ v.01! 00 061' 811 S H mm. 00” HR. /H..,,' O.—; WI! .H. nm.m mnsmflm «.0 V6 0.0 0.0 o; 86 “so m u a .xoooom n we .me om u to .musumummEmu now on musumummamu conuomam mo moflpmn msofium> How mccmucm Hmoflupcfla>o a ma omufioxm m>m3 owgmsoomcow vaHHmumcmm Gnu mo mcnmuumm coHuMHpmm mm.m musmflm o... . as 3 so «a! o «a so am mo 38 .b0 4 . \\\.\Hu ‘ LI , llIaFll.lll o \\\\. II \\\ \\ // III/ll \ \ 3..- / / \\ \ / / \ // r \ \ to]- // \\\ III.III .bo;lllulu||.\ illnll.\\D%o can. . _ @641 2 u w we .3» .bn H H mm 3 me .95 87 .50 m u s .0H u aa\me .xcoooo u may .mmwocmnvmum «campus msoflum> Mom mccmucm Hmowupcfiaho m an vmuwoxm m>m3 owumaoomcow @0Nflamnocmm man no maumuumm coauMHomm mm.m musmwm \ \\\\\. ‘61. //////. / . «at. x \ me om me ON a / \ ‘3 Max OH ll ‘H O p 88 generalized electroacoustic wave at various electron tempera- tures in comparison with an eXperimental pattern measured by Ishizone et al.[13]. Figure 3.31 shows the radiation patterns of the generalized electroacoustic wave at various antenna frequencies. Figures 3.32 to 3.34 show the radiation patterns of the generalized electroacoustic waves predicted by the present theory in comparison with the experimental patterns measured by Ishizone et al.[13]. 3.5.4 Radiation Patterns of the Electromagnetic Wave To determine §(?), the integral e-jkeR I R dv in equation (2.6.10) is to be evaluated. For a cylindrical antenna, we assume 8 ¢*H+ l m 3 (2') = >H4 where A is the cross-sectional area of the antenna and +5 _ Imsin[ke(h - 2')12 for z' > 0 — Imsin[ke(h + z')]z for z' < 0 ° H (3.5.7) ke is the prOpagation constant of the electromagnetic wave in the plasma and is given by equation (3.2.6). For the collisionless case, equation (3.2.6) is reduced to 89 :6 m n a K6 n <5 3.3 cu a u asxma .o u a}; .55 m4 u mm .25 m.m u m: .mmusumuogfimu couuomam maoflum> How mccoucm HMUfluccflaxo m >n nonfioxm m>m3 owumsoomouuomam wmuwdmumcmm man «0 maumuumm cofiumflcmm om.m munmflm and .HM um meowwan an ucmfiwummxm o o o @393 Hammond Ill. o..— . ad 3 1.0 «.0 - o «d ‘0 0.0 «.0 o.— \ .3. , o3 . r. / 3-, I x ooom n ma / \\ o I m / ndnu \\ . 588 u .H. con/fl \s can 583 n ma / 1 a... .. \ 90 25 ma u a 3.3 cu H u Mime .o u 33 .383 u we .5? mé u at .mowocmflvmum macmucm mnoanm> Mom mccmucm Hmowupswamo m an omufioxm m>m3 owumnoomouuomflm cmuflamumcmm man «0 mcumuumm coauMHpmm Hm.m mnsmflm 0a ad 00 V0 «6 0 «0 V0 06 00 0p 000 _ a \hll.\ ”NJ-II! ll ._/_ d 000 \\\\\. \\\ .\/. l/I/l/ o.I///// \ \\ \SLT / / o ‘ \\\\\. ‘dnr Ill/I II 'H 0 \ , \\ mm: a u m ,.I 3% \ .3 an: m.m eon / \ «m: m ,, N $4 I 2 l 91 Eomunfimzméumgfizménmm .oxomamumi .mccmucm Hondupcaamo m >3 omufloxm m>m3 oaumsoomouuomaw pmNHHmumcmm wsu mo cumuumm coflumflpmm mm. m musmfim :22. am no mcoNHrmH >3 usmeHmmxm o o o wuomnu ucmmmum 2 . 3 3 a «d o «a to so no 3 .bo. _ a . _ z a .30 «dlr a tour a 000 000 0.011: 0611 OOM/ \oOn / 0a .35 92 Aev m.m n a .Nms m.m u m .Nm: m.v u an .xoomam u may . .mccmucm Havaupcflamo m an powwoxw m>m3 oflumsoomoupowam UmNflHmumcmm mnu mo cumuumm COH¢MH©mm mm.m musmflm mmH_.Hm um mcoNflan an ucwsflummxm .- o o mucosa unmmmum o.— . 8 3 to «d o «d to 9o 3 o... .bo- _ a _ . I. _ _ 3 .b0 «div tan: 000 as 3.: / \ ad .bn .bn c.— 93 Ago m.m u n .Nmr a u m .Nm: m.v u an .moomam u we .mccmucm HMUHHpCHawo m >n cmufloxm m>m3 Uflumsoomonuomdm UmNHHMHmcmm may mo cumppmm cofluMmem vm.m musmflh HMHH.HM um meoNflan ha usmfiflummxm o o o whomnu Hammond 3 e. , C 0.01... C no}. .2 .8 S 94 2 _ 2 e k - m uo€o . (3.5.8) After using the far zone approximations and neglecting l/r2 + -) . . terms, B(r) 18 determined to be fi + N A -jImuo cos(keh cose) - cos(keh) . e-jker (r) - ¢ 2 A 2 Sine --- . n cos 9 - 1 r (3.5.9) It is evident in equation (2.7.7) that E field contains ionacoustic, electroacoustic as well as electromagnetic com- ponents. To calculate the radiation patterns of the electro- magnetic wave, only the electromagnetic component is considered. This component can be obtained as ~> _ —jw '+ E - V B . 3.5.1 em u e (w2 - w 2 - w 2) X ( O) o o e i Substituting equation (3.5.9) into equation (3.5.10) and neglecting l/r2 terms, we have E _ 5 -ijmke cos(keh cose) - cos(keh) em 2_2_2 2_ 2nA€o(w we mi ) cos 0 1 "jker sine S_____ . (3.5.11) r The corresponding radiation pattern function can be expressed 95 cos(keh cose) - cos(keh) F (6) = sine . (3.5.12) em c0526 - l Figure 3.35 shows the radiation patterns of the electro- magnetic component of the electric field in the plasma. In this example, the plasma frequency is assumed to be 4.5 MHz and the antenna frequency is assumed to be 5, 5.5 and 7 MHz. The numerical results are given in Table 21 of Appendix C. Over this range of antenna frequency, the radiation patterns of the excited electromagnetic wave largely remains circular as shown in Figure 3.35. 96 6:“? 1.0‘ 30° 0.3 r- 0-6 " f = 7 MH 2 ; 60° OA- f = 5.5 MHz 0.2 - \ 1 L l l 96’ 0 d2 6.4 016 038 "' f = 5 MHz / Figure 3.35 Radiation patterns of the electromagnetic wave excited by a cylindrical antenna in a plasma for various antenna frequencies. (fp = 4.5 MHz, h = 8.5 cm) CHAPTER 4 EXCITATION OF AN ELECTROACOUSTIC WAVE IN THE PLASMA SHEATH SURROUNDING A CYLINDRICAL ANTENNA 4.1 Introduction The excitation of an electroacoustic wave by an antenna in an infinite, homogeneous, isotrOpic, compressible and lossy plasma was studied in Chapter 2. In practice, when an antenna is in contact with a compressible plasma, a plasma sheath is created on the antenna surface. In this chapter, we like to study the excitation of the electroacoustic wave by an actual antenna surrounded by a plasma sheath and imbedded in a compressible plasma. Main objectives of this chapter are (l) to study the effect of the plasma sheath on the excitation of the electroacoustic wave and (2) to seek the evidence of the excitation of the electroacoustic wave by an actual antenna. 4.2 Experimental Setup The schematic diagram of the experimental setup is shown in Figure 4.1. A mercury arc discharge was employed to produce the large volume and high density plasma in a large plasma tube which is made of an open end pyrex bell 97 98 .macwucm accentsflamo m mcwwcsounsm nummnm mammam on» Ca m>m3 oaumsoUMOHHUMHm on» NO nodumuwoxm mnu How moumm Hmucmeflnmmxm H.v musmflm mmoom j IoHHflomo mammam Houomump .uam _ aflmmnm was: t\¥ HmHMHHmEm Houmaaflomo Hm3om .u.© .IY cowwmmmnw.+1t Hwamdoo .B.3.B 4AI,.vmum mmmzm .n Hmcowuowuww 99 jar with the dimensions of l4-inch diameter and 18-inch length. The upper end of the tube is the anode with a cylindrical monopole antenna feeding through its center. The lower end of the tube is the cathode which consists of a mercury pool. A floating metallic ring is placed at the middle of the mercury pool to fix the moving hot spots of the mercury arc. An ignition circuit is installed in the mercury pool for the purpose of starting the plasma. Between the anode and the cathode, a d.c. power supply circuit is connected. Under the normal operation, the discharge current can run from zero to 50 amperes. The pumping system consists of two mechanical pumps and a mercury diffusion pump. The tube is continuously pumped during the eXperiment, and the pressure of the plasma is kept around 1 micron (10"3 mm Hg). The structure of the large plasma tube is shown in Figure 4.2. The output of a sweep frequency oscillator covering the frequency band of 0.4 to 1.4 GHz is amplified by a travelling wave tube amplifier and then connected through a directional coupler. It then passes through a bias insertion unit before reaching the antenna. Through this bias insertion unit, the d.c. bias voltage of the antenna can be varied from negative 40 volts to positive 25 volts. When the antenna excites an electro- acoustic wave in the plasma sheath, this effect appears in the reflected wave from the antenna. The reflected wave containing this electroacoustic resonance information is taken out through the directional coupler, and then connected to the 100 mesa HMOflcmnom 0>Hm> new» umacfl Ham .mmfluommmuom can mndu mammam one > 00m _._J .mewm nomm mcwHMOHM\\\ mcoa .mH umumsmac gas onsu mammam xman mesa cosmnmmwo mm mscmucm mmamo Edsom> .msm omuo uHo> omno g WWW. t. ~.v musmam l muoumflmmu muw3 mfiounowc 101 vertical input of the oscillosc0pe after detection. The horizontal input of the oscilloscope is fed by the sweep voltage of the sweep frequency oscillator. The curve dis- played on the oscillosc0pe is the reflected wave versus the sweeping antenna frequency. The scheme of the experiment is to observe the change in the curve of reflected wave versus sweeping frequency (RW-SF curve) as the antenna d.c. bias voltage is varied. As the bias voltage is varied, the size of the plasma sheath surrounding the antenna is changed. The observed change in the RW-SF curve as the bias voltage is varied supports the conjecture that this change is due to the excited electro— acoustic wave, because the excited electromagnetic wave should not be affected by the change of the plasma sheath which is at least a magnitude of order smaller than the electromagnetic wavelength. 4.3 Experimental Results When the sweep frequency signal covering the frequency range of 0.4 to 1.4 GHz was fed to the cylindrical antenna which is immersed in a large volume of compressible plasma, the reflected wave versus sweeping frequency displayed a curve such as shown in Figure 4.3 on the oscilloscope. Dips and peaks in the curve were probably due to the reflection of the electromagnetic wave from the antenna tip and the resonances excited by the electroacoustic wave in the plasma sheath. It reflected wave 102 l i. 1.4 GHz frequency Figure 4.3 A typical reflected wave versus sweeping frequency curve. 103 is well known that at discrete numbers of frequencies, the excited electroacoustic wave can set up resonances in the plasma sheath. Whenever an electroacoustic resonance is set up, a dip in the RW-SF curve is expected. The antenna bias voltage was then varied to observe the change in the RW-SF curve. The antenna was first biased positively with respect to the plasma. As the bias voltage was varied from zero volt to positive 25 volts, the RW-SF curve was not changed at all. When the bias voltage reached beyond positive 25 volts, the antenna started to draw a heavy d.c. current from the plasma evidenced by a red glowing at the antenna tip. It was concluded that the variation of the antenna bias voltage, which was positive relative to the plasma, did not change the RW-SF curve. The next step was to bias the antenna negatively with respect to the plasma. When the antenna bias voltage was varied from zero volt to negative 40 volts, a significant change in the RW—SF curve was observed. As the negative antenna bias voltage was substantially varied, the alternation of the RW-SF curve stopped at a particular frequency for a particular plasma density (discharge current). As the plasma density was increased, this particular frequency moved up indicating that a longer frequency range of the RW-SF curve was changed. This phenomenon is demonstrated in Figure 4.4. Figure 4.4(a) shows the RW-SF curve for the plasma current of 104 10 amperes, subject to the variation of antenna bias voltage from zero volt to negative 40 volts. It is clearly seen in this oscillogram that the lower frequency part (0.5 to 0.68 GHz) is substantially changed. Figure 4.4(b) shows the RW-SF curve for the case of 15 amperes plasma current. The frequency band of 0.5 to 0.78 GHz is affected. Figure 4.4(c) shows the RW-SF curve for the case of 20 amperes plasma current. The frequency band of 0.5 to 0.92 GHz is affected. Three oscillograms in Figure 4.5 show the similar phenomena. In these oscillograms, the range of sweeping fre- quency is from 0.4 to 1.4 GHz which is wider than the case of Figure 4.4. To understand the physics behind the observed pheno- mena, the correlation, between the plasma density and the highest frequency beyond which the antenna bias voltage ceased to affect the RW-SF curve, was investigated. It was found that this highest frequency was very close to the ambient plasma frequency. This finding implied that as the antenna bias voltage was varied, the affected part of the RW—SF curve was in the frequency band lower than the ambient plasma frequency. This phenomenon also implied that every possible electroacoustic resonance was excited in the plasma sheath for the antenna frequency lower than the ambient plasma frequency. Figure 4.6 summarizes the affected frequency bands of the RW-SF curves due to the variation of negative antenna (a) plasma current: 10 amp. CH2 (b) plasma current: 15 amp. CH2 (C) plasma current: 20 amp. 0.5 0.6 0.7 0.8 0.9 1.0 CH2 I-‘iuurv 4.4 Oscillograms of tho rvflvctwl wave versus; swooninn franvncv curvos for various plasma currents. Fro— :mvnc'; runw' (rum 0.") (‘0 1.0 0112. 106 (a) plasma current: 10 amp. 6112 (b) plasma current: 15 amp. CH2 (c) plasma current: 20 amn. 0.4 0.6 0.8 1.0 1.2 1.4 CHz Figure 4.5 Oscillograms of the reflected wave versus sweepinq frequency curves for various plasma currents. Fre— nut-m‘v ran'w‘ from 0.4 tr) 1.4 GHZ. 107 bias voltage for various plasma densities (plasma currents). The ambient plasma frequency in each case is indicated in the figure showing it to be close to the upper bound of the affected frequency band. It is noted that the ambient plasma frequency was measured by the conventional Langmuir probe method. The ambient plasma frequencies in the central part of the plasma tube, correSponding to various plasma currents, are shown in Table 4.1. Table 4.1 Ambient plasma frequency versus plasma current. Plasma current Ambient plasma frequency 5 0.46 10 0.57 15 0.68 20 0.87 25 1.00 30 1.12 35 1.30 40 1.47 45 1.47 4.4 Interpretation of the Experimental Results The excitation of an electroacoustic wave in a compress- ible plasma, and the resonance of the electroacoustic wave in a plasma sheath leading to the so-called Tonks-Dattner's 108 A.conuwfi wnoum Hwnfimsmq %n omusmmmE hocmsvmum mEmmHm ucmHnEm 0:» ma 0 .055 hocwsvmnm pmuommwm ecu we I v .mucwuuso mEmmam msoHum> mo mmmmu on» How mm>nso mmlzm may no magma mocmsvmum pmuummm< w.v musmam mocmnvmum Nmo v.H m.H N.H «.0 am; 1 _ ., _ as... a _ _ .QEm mm _ _ .98 ma q _ www.mfim 0H ucmuuso mammam 109 resonance or the thermal resonance have been studied by numerous workers. Recently, Baldwin [28] and Parbhakar and Gregory [29], through their theoretical and experimental studies, proposed a new physical mechanism for the electroacoustic resonance in the plasma sheath of a cylindrical plasma column. This new physical mechanism is the following: When an electro- magnetic wave is incident upon a bounded non-uniform plasma, the electromagnetic field will excite an electroacoustic wave at the critical density point on the density profile where the local plasma density is equal to the frequency of the incident wave. The electromagnetic energy is coupled to the electroacoustic wave at this critical density point. The excited electroacoustic wave then propagates in both directions; one attenuates into the overdense plasma and the other prOpa- gates, and sets up a standing wave in the underdense plasma region or the plasma sheath. In this physical mechanism, it is implied that in order to excite an electroacoustic wave, an electromagnetic wave is required to interact with the plasma at the critical density point. If no critical density point exists in the plasma, an electroacoustic wave may not be excited. This new physical mechanism will be used to interpret our experimental results. 110 4.4.1 The Case When the Cylindrical Antenna is Biased Positively: When the antenna is biased positively with respect to the plasma, the electron density in the vicinity of the antenna is increased and it may create a density profile surrounding the antenna as shown in Figure 4.7. In our eXperiment, the antenna frequency was contin- uously swept over a band and, at the same time, the antenna bias voltage was varied. At a particular instant, the antenna frequency is assumed to be ml. If “1 is higher than the ambient plasma frequency, an electroacoustic wave is excited at the critical density point where w = ml somewhere on the P density profile in the antenna vicinity. The excited electroacoustic wave which prOpagates out- wardly in a large volume of underdense ambient plasma is essentially a travelling wave because of the large plasma volume. It appears that the amount of energy used to excite the electroacoustic wave remains rather constant even for various antenna frequencies and various density profile which is changed by the variation of antenna bias voltage. The excited electroacoustic wave which propagates in- wardly toward the antenna becomes evanescent because an overdense plasma surrounds the antenna. Thus, no standing electroacoustic wave can be set up in this situation and no electroacoustic resonance can be observed through the reflected wave of the antenna. 111 ‘ne(the electron density in plasma) density profiles —> '1‘ Figure 4.7 Plasma density profiles surrounding the antenna for various positive bias voltages. 112 If the antenna frequency band is lower than the ambient plasma frequency, neither electroacoustic wave can be excited nor propagates in the plasma because every point in the plasma volume is overdense with respect to this frequency band. Therefore, one would not expect to observe any signifi- cant effect on the RW-SF curve as the antenna bias voltage is varied positively. 4.4.2 The Case When the Cylindrical Antenna is Biased Negatively: When the antenna is biased negatively with respect to the plasma, electrons in the antenna vicinity are repelled. This will create an electron-deficient region surrounding the antenna, or a conventional plasma sheath with a density profile as shown in Figure 4.8. For this situation, the local plasma frequency in the plasma sheath region is lower than the ambient plasma fre~ quency. When the antenna frequency is lower than the ambient plasma frequency, an electroacoustic wave can be excited at a critical density point on the density profile of the plasma sheath. This excited electroacoustic wave attenuates out- wardly; but can propagate inwardly because the plasma sheath region is underdense with respect to this frequency. The inward electroacoustic wave is essentially trapped in the finite plasma sheath region, so that it will set up a standing pattern. Furthermore, when the width of the plasma sheath is roughly in the order of an integral multiple of the half 113 ne(the electron density in plasma) standing electroacoustic wave J/ (mp)ambient ‘4 “ r ‘Vll ”" = ”2 w = w density V ‘V p 1 profiles A Figure 4.8 Plasma density profiles surrounding the antenna for various negative bias voltages. «V 114 electroacoustic wavelength, the electroacoustic wave will reach a resonance condition. Whenever the electroacoustic resonance is reached at a particular antenna frequency and at a particular antenna bias voltage, more power is transfered from the antenna to the plasma resulting a dip in the reflected wave from the antenna. Thus, as the antenna bias voltage is varied, while the antenna frequency is being swept, the electro- acoustic resonance is reached at some discrete frequencies. Since these discrete frequencies are dependent on the density profile of the plasma sheath, which are controlled by the antenna bias voltage, the low frequency part of the RW-SF curve will be altered when the antenna bias voltage is varied. When the antenna frequency is higher than the ambient plasma frequency, no critical density point can be found at any point of the plasma volume. Thus, according to Baldwin's [28] theory, no electroacoustic wave can be excited in the plasma. If no electroacoustic wave is excited for the fre- quency band higher than the ambient plasma frequency, no significant change on the Rw-SF curve can be observed when the antenna bias voltage is varied. Therefore, when the negative antenna bias voltage is varied, only the part of the RW-SF curve where the antenna frequency is lower than the ambient plasma frequency is affected. 115 4.5 Potential Application The result of study described in this chapter may lead to a convenient technique for plasma diagnostics: especially for the measurement of the local plasma density. A feasible scheme can be the following: A small movable monOpole can be built to probe the density of a plasma volume. The exciting frequency of the monopole is swept over an appro- priate frequency range. The bias voltage of the monopole is made variable from zero volt to a certain negative volt. The reflected wave versus sweeping frequency curve is displayed on the scope. As the bias voltage is varied (usually manually), the lower frequency part of the RW-SF curve will be altered. The highest frequency of this altered frequency band is the local plasma frequency at the location of this monopole probe. The advantage of this diagnostic technique is the direct reading of the local plasma frequency and the quick- ness of obtaining results. Unlike the conventional Langmuir probe method, this method does not require any graphical or computational intermediate steps. The disadvantage of this method is the requirement of a sweep frequency generator and a variable bias voltage setup. The commercially available sweep frequency generators usually have limited sweeping frequency bands so that the measurable range of the plasma density may also be limited. 116 4.6 Analysis of the Coupling between the Electromagnetic Mode and Electroacoustic Mode in the Plasma Sheath In this section, we aim to show that the electric field set up by the charge on the antenna will excite an electroacoustic wave in the plasma sheath surrounding the antenna. The excitation of an electroacoustic wave is poss- ible because the gradient of the electron density in the plasma sheath surrounding the antenna and the electric field on the antenna surface are both in the same direction--the radial direction. Thus, a strong coupling between the electroacoustic mode and the electromagnetic mode can exist. The theory presented in this section is to confirm the experimental observation that an antenna can excite an electroacoustic wave in the plasma sheath surrounding the antenna. Since we are concerned only with the electroacoustic wave in this section, the motion of positive ions is ignored in the analysis. Consider the geometry of Figure 4.9 where a cylindrical antenna is located along the z—axis. The electron density profile in the plasma sheath surrounding the antenna is also shown in this figure. Starting from the basic equations which govern the system, Maxwell's equations in the plasma sheath are v x E = - jwuofi (4.6.1) V x R = - enofie + jweofi . (4.6.2) 117 12 antenna 7//////////////////////% Wit/M. 9;7/////////////////////A’/////////////"//x;:7I - , Figure 4.9 Geometry of a cylindrical antenna surrounded by a plasma sheath. 118 The equation of mass conservation of electrons is 8N 3E2 + v . (Nefie) = 0 (4.6.3) where Ne = no(§) + ne(r,t) . (4.6.4) Thus, equation (4.6.3) becomes jwne + v . (nofie) = o ‘ (4.6.5) << . when n6 no The equation of momentum conservation of electrons is 36 V e + e e at e e me :10 e where g = fidc + g . (4.6.7) For the d.c. component of equation (4.6.6), o = - E— a - —E-Vn . (4.6.8) m 0 Equation (4.6.8) shows that the plasma density profile no(?) is maintained by the d.c. component of the electric field. For the a.c. component of equation (4.6.6) 3— E - —— Vn . (4.6.9) m e e no . + (3w + Ye)Ue = - 119 Taking the divergence of equation (4.6.2) and using equation (4.6.5), we have 0 = jwene + jweoV - E . (4.6.10) 01’." + ene V . E = - -— a (406.11) 80 Taking the curl of equation (4.6.1) and using equation (4.6.2), we have _ 2 - V x V x E - w poeoE + jwuoenofie . (4.6.12) Using equation (4.6.9), equation (4.6.12) reduces to 2 2 w Y m VxVx§=w2uoeo[-—-2-—e—-—2--j 2662]): w + Ye e(w + Ye ) w2+jwye) 2 + + < 2 2 Uoeove V(V . E) w + y e _ 2+ .«r BemE + aV(V E) (4.6.13) where 2 2 2 2 we . Yewe ] B = w u e [l - ———————— - 3 (4.6.14) em 0 O w2 + Yez w(w2 + Yez) propagation constant of the electromagnetic wave in the plasma sheath. 120 w + jwy V a = ( 2 e) 92 (4.6.15) 0.) C “080 Let us assume that E = fie + 8p where fie corresponds to the electric field of the electromagnetic wave such that V - fie = O and Ep corresponds to the electric field of the longitudinal electroacoustic wave such that V x B = 0. P Equation (4.6.13) then reduces to 2 2 + 2 2 = (v + Bem)Ee + (aV + eem)§ o . (4.6.16) P Taking the curl of equation (4.6.16), we have 2 (v + e 2)(v x E ) = - (vs 2) x (E + E ) .(4.6.l7) em e em e P Taking the divergence of equation (4.6.16), we have 2 B 2 em . = _ 1 2 . (v + —37)(v Ep) (; veem) (fie + Ep) ene Using V ° Ep = - _E_ from equation (4.6.11), we got 0 2 Be: 60 1 2 E g (V 4' Th'le = e—(E VBem) ° (6 + p) . (4.6.18) Equation (4.6.18) is the inhomogeneous wave equation for the electroacoustic wave. In this equation, Be: is in the r O O + c a O I direction. Be 15 also 1n the r d1rect1on on the antenna 121 surface because the electric field on the conductor surface is perpendicular to the surface. Therefore, there is a strong coupling between the electromagnetic mode and the plasma mode. In other words, the radial component of‘Ee field on the antenna surface can excite an electroacoustic wave, through the gradient of the density profile, in the plasma sheath. CHAPTER 5 EXCITATION OF ELECTROACOUSTIC RESONANCES IN VARIOUS PLASMA GEOMETRIES AND STUDY OF THE REFLECTION BEHAVIOR OF ELECTROACOUSTIC WAVES ON VARIOUS SURFACES 5.1 Introduction Electroacoustic resonances are excited in (l) a cylin- drical plasma column, (2) a rectangular plasma column and (3) a single-slope density profile plasma column. The nature of the electroacoustic resonances in different plasma geome- tries is studied. The techniques of exciting electroacoustic resonances are applied to study the reflection behavior of electroacoustic wave on (1) dielectric surface and (2) metallic surface. 5.2 Experimental Setup For the experiments in this chapter, two types of mercury-vapor plasma tubes have been constructed. One type was the cylindrical glass tube with a length of about 30 cm, outside diameter of 8 mm, inside diameter of 6 mm, and mercury pressure of about 1 micron. The structure of this tube is shown in Figure 5.1. The other type was the rectangular glass tube with a length of about 30 cm, outside cross 122 123 sectional dimensions of 12 mm by 8 mm with a wall thickness of 1 mm, and the mercury pressure of about 1 micron. This rectangular tube was divided into 3 sections; an uniform density section, a single-slope density profile (single- profile) section with a metallic reflector and a single- profile section with a glass reflector. A single-profile can be created in this tube by squeezing the plasma current flow at a gap close to the wall by means of a built-in glass plate. The structure of this tube is shown in Figure 5.2. The single-profile plasma column was constructed primarily for the purpose of studying the reflection behavior of an electroacoustic wave on various surfaces. It was hoped that the electromagnetic field of the electroacoustic probe can excite an electroacoustic wave in the region between the reflector and a point on the plasma density profile and not in the plasma sheath at the glass wall next to the electro- acoustic probe. Assuming that an electroacoustic wave can be excited in the region mentioned above by the electromagnetic field of the electroacoustic probe, a standing electroacoustic wave will be set up between the critical density point and the reflector if a sufficient amount of electroacoustic wave is reflected from the reflector surface. This standing electroacoustic wave will appear as resonances in the reflected electromagnetic wave which is picked up by the electroacoustic probe when the plasma current is varied. If the reflector surface absorbs the incident electroacoustic wave, no standing 124 electroacoustic wave will be set up and no resonances will be observed. From the patterns of resonances observed with different reflectors, the reflection behavior of the electro- acoustic wave on various reflector surfaces can be studied. The schematic diagram of the experimental setup is shown in Figure 5.3. The incident c. w. electromagnetic wave which excites an electroacoustic wave in the plasma column is fed to the electroacoustic probe which is essentially an open-ended coaxial line with a protruding center conductor with a disk tip. The reflected electromagnetic wave from the plasma column is picked up by the same electroacoustic probe. This reflected electromagnetic wave is passed through a directional coupler and a detector before reaching the vertical input terminal of the oscilloscope. The horizontal input of the oscilloscope synchronizes with 60 Hz sweeping of the plasma discharge current. The display of the reflected electromagnetic wave on the oscillOSCOpe contains all the information on the electroacoustic and dipole resonances and is called the reflection curve in the later sections of this chapter. 5.3 Electroacoustic Resonances and Dipole Resonance in a Cylindrical Plasma Column In this eXperiment, a cylindrical plasma tube was used in the setup as shown in Figure 5.3. The electromagnetic source was set at 2.4 GHz and the tube discharge current was 125 metallic baCking ammmmmwwwwnwns coaxial electroacoustic l : probe Figure 5.1 Structure of the cylindrical plasma tube. metallic glass uniform reflector reflector section section (#sectign.l m < \ 9— ' coaxial ' electroacoustic probe Figure 5.2 Structure of the rectangular plasma tube. 126 microwave signal generator "1detectorJ‘ 7 directional coupler electroacoustic probe 'géi: plasma tube H.._...__. O ._.., oscillo- A I sweep scope l tranSformer Figure 5.3 Experimental setup for the excitation and observation of electroacoustic resonances in different plasma geometries. 127 swept 60 Hz in the experiment. The reflected electromagnetic wave picked up by the electroacoustic probe went through the directional coupler (or a matched coaxial hybrid), detector and then was displayed on the oscilloscope. Resonance peaks were observed at various discharge currents. When a metallic backing was placed on the back side of the tube as shown in Figure 5.4, one of the resonance peaks was affected. Three sets of oscillograms were taken in this experiment and they are shown in Figures 5.6, 5.7 and 5.8. Figure 5.6 shows the resonance curves in the lower discharge current region. The Operating frequency was set at 2.4 GHz and the plasma current was swept around 95 mA. No effect on this part of the resonance curve was observed with a metallic backing to the tube. It is evident that these peaks are electroacoustic resonances which are excited in the plasma sheath directly near the probe. A metallic backing in the back side of the tube has little effect on this locally excited electroacoustic standing wave. This phenomenon is shown in Figure 5.5. Figures 5.7 and 5.8 show the resonance curves in the higher discharge current region observed in two different plasma tubes of same dimensions. When the tube was placed with a metallic backing on the back side, some effect was observed on the first highest peak of the resonance curve. This first highest peak is recognized as the dipole resonance which is physically different from the remaining electroacoustic 128 metallic EH -3—( electro- electro- acoustic acoustic probe probe (a) (b) Figure 5.4 Cross-sectional view of the cylindrical plasma tube. (a) without metallic backing (b) with metallic backing plasma tube) density profile .331 electroacoustic probe electroacoustic standing wave Figure 5.5 Electroacoustic resonance in a cylindrical plasma column. I29 m > P p without metallic 3 backing U m r-G ‘H m u Ipo = 95 mA plasma current m > P U with metallic w . u backlng o w H U: m u I r 95 mA plasma current Figure 5.6 Resonance curves observed in a cvlindrical plasma column. (f = 2.4 GHz, Ip0 — 95 mA) I30 w > m 3 8 without metallic u backing o m H 'H w u 11-_. _____ , [no , 115 mA plasma current 3’ V T 5 U with metallic 3 backing o w H 04 w u A Ip0 = 115 mA plasma current Figure 5.7 Resonance curves observed in a cylindrical plasma column. (f = 2.4 GHz, Ip0 = 115 mA) I31 without metallic backing reflected wave .——-».# , 71,__-_ ..».__.v-.._.——-—. IUD , 120 mA plasma current w > S g with metallic u backing o m r-4 U. m H I = 120 mA plasma current po Figure 5.8 Resonance curves observed in a cylindrical plasma column. (f = 2.45 GHz, Tpo : 120 mA) 132 resonances. Since a dipole resonance is an electromagnetic resonance and is excited over the whole column, a metallic backing will alter drastically the boundary condition and lead to a change in the dipole resonance peak. 5.4 Electroacoustic Resonances and Dipole Resonance in a Rectangular Plasma Column In this eXperiment, a rectangular plasma tube was used in the setup as shown in Figure 5.3. The electro- acoustic probe was placed at the uniform plasma section. The resonance curves were observed in both the low and the high discharge current regions and a complete series of electroacoustic and dipole resonances can be reconstructed in four oscillograms in Figure 5.9. To our best knowledge, the electroacoustic and dipole resonances have not been studied in this rectangular geometry. In Figure 5.9, it is observed that the resonance curve consists mainly of four distinct peaks; the highest peak occurs at the high discharge current end and the rest with descending order of magnitude toward the low discharge current end. This curve looks similar to the resonance curve observed in the cylindrical plasma tube. Figures 5.10 and 5.11 show the effect of a metallic backing on the resonance peaks. When the metallic backing was placed on the tube, the second highest peak was affected; but not the first highest peak as in the case of the cylindrical 133 on A45 0mm .uwmno mm>uso mUCMCOmwm m. ucwuuso ucmuuso . . om . I 00 mEmmam <8 oma u H mEmmHo «C maa I H A ucmuuso mfimmam 0E omm n 01H w m V .mnsu mEmmHe musmam A uflwhhfib mmeHo HE com u (r ‘1 I34 w > P 5 8 without metallic u backing U m H ‘H m u Ipo = 150 mA plasma current m Ix W U with metallic B backing U m H U: m p l s 150 mA plasma current po Figure 5.10 Resonance curves observed in the uniform region of a rectangular plasma tube. (f 2 GHz, 1”” a 150 mA) I35 m > m 3 8 without metallic 3 backing w H '44 m u 11 ,1 ,-., A -,,,_, , In” — 150 mA plasma current 0 > m 3 p with metallic (y . u backing o w r—4 M w u I = 150 mA plasma current DO Figure 5.11 Resonance curves observed in the uniform region of a rectangular plasma tube. (f = 2.4 CH2, Ip0 = 150 mA) 136 plasma tube. This may imply that the second highest peak in the resonance curve observed in a rectangular plasma tube is the dipole resonance. 5.5 Resonances in Singleéprofile Plasma Column in the Rectangular Tube As stated before, a single-profile column was fabricated in order to study the reflection behavior of an electro- acoustic wave on various boundary surfaces. Before this study was conducted, the plasma density profile of this plasma column was examined by observing the resonance curves created by the electroacoustic probe at different parts of the plasma column. 5.5.1 Glass Reflector Region We first examined the density profile at three differ- ent points in the glass reflector region as shown in Figure 5.12. A large density profile difference was expected to exist between the front and back sides at the neck section of this region. The density profile should become more uniform away from the neck section, so that a small density difference was expected to exist between the front and back sides at the center and tail sections of this region. The electroacoustic probe was placed at different positions along this glass reflector region and the reflection curves were studied. Figure 5.14 shows that at the neck section, the 137 back side g1:ss reflector "# neck center tail front side Figure 5.12 Plasma density distribution in the glass reflector region. back side metallic reflector \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\‘ plasma flow neck nter tail front Side Figure 5.13 Plasma density distribution in the metallic reflector region. 138 reflection curve from the front side is significantly different from that of the back side implying the existence of a drastic density profile difference between the front and back sides of the tube at this section. Figures 5.15 and 5.16 show that a relatively small difference exists between the reflection curves from the front and back sides at the central section of the glass reflector region. This will imply the existence of only a small difference in the densities between the front and back sides at this section of the tube. Also in Figure 5.15, the effect due to the metallic backing is indicated. We can see that the second highest peak was altered when the metallic backing was placed on the tube. Figure 5.17 shows the existence of a small difference in density profile between the front and back sides at the tail section of the glass reflector region. The similar effect due to the metallic backing was also observed in the experiment. Experimental results shown in Figures 5.14, 5.15, 5.16 and 5.17 confirm that a single-profile was created in this rectangular plasma tube. 5.5.2 Metallic Reflector Region The density profile in the metallic reflector region of the same tube as shown in Figure 5.13 was studied. Three sections, the neck, center and tail sections, of this region were examined. A series of oscillograms of the reflection I39 09 om Aumwno mw>uso mocmcowwm va.m musqfim “coupon ucmuuso Q I 0Q mamas as com n H Bengal <2 ems spam some ouam iom acoppdo . because . mfimoam GE oom n 00H acmcan mu omH n eye I) A )3 spam ucoum back side I 2 150 mA plasma current front side —»———-—-. [P0 s 150 mA plasma current Front side with metallic backing IpO = 150 mA plasma current Figure 5.15 Resonance curves observed in the center section of the glass reflector region of a rectangular plasma tube. (F = 2.4 GHz, Ip0 = 150 mA) w > m 3 3 front side 4.) o m H '44 m u IDC 2 200 mA plasma current w > s 3 8 back side 4.) o w H ‘44 o u L Ipo = 200 mA plasma current Figure 5.16 Resonance curves observed in the center section of the glass reflector region of a rectangular plasma tube. (f = 2.4 GHz, IDo = 200 mA) I42 back side Ip0 = 150 mA plasma current front side In” : 150 mA plasma current front side with metallic backing IDO = 150 mA plasma current Figure 5.17 Resonance curves observed in the tail section of the glass reflector region of a rectangular plasma tube. (f a 2.4 GHz, Ipo = 150 mA) 143 curves were taken during the experiment at the neck, center and tail sections of the metal reflector region under various discharge currents. By grouping these oscillograms together, we obtained three complete curves of the resonance. Figure 5.18 shows a complete curve of resonance at the neck section. At this section no distinct electroacoustic resonance was observed. It was probably due to the turbulent plasma flow and irregular density distribution in this posi- tion. Figure 5.19 shows a complete curVe of resonance at the center section. The electroacoustic and dipole resonances were observed. Figure 5.20 shows a complete curve of resonance at the tail section. The electroacoustic and dipole resonances were clearly observed at this section. 5.6 Reflection Behavior of Electroacoustic Wave from Metallic and Non-metallic Surfaces Figure 5.21 shows the reflection curves observed in the uniform, glass reflector and metallic reflector regions of the tube. The reflection curve from the metallic reflector region is different from the other two cases. This appears to imply different reflection behaviors of an electroacoustic wave on metallic and non-metallic surfaces. However, Figure 5.22 shows that the reflection curve observed in the uniform column is affected by an external metallic backing and, 144 1 1 1 J ,, I 100 150 200 250 mA p Figure 5.18 Resonance curve observed in the neck section of the metallic reflector region of a rectangular plasma tube. (f = 2.4 GHz) 1 1 L l ,__1 100 150 200 250 mA p Figure 5.19 Resonance curve observed in the center section of the metallic reflector region of a rectangular plasma tube. (f = 2.4 GHz) l J l l I 100 150 200 250 mA” P Figure 5.20 Resonance curve observed in the tail section of the metallic reflector region of a rectangular plasma tube. (f = 2.4 GHz) I45 uniform region IPO = 190 mA plasma current glass reflector region I = 190 mA plasma current metallic reflector reg ion ___A. - _ L A *7 In” t 190 mA plasma current Figure 5.21 Reflection curves observed in uniform, glass reflector and metallic reflector regions of a rectangular plasma tube. (f 1 2.33 "Hz, Ip0 s 190 mA) 146 without metallic backing reflected wave ——~——> I = 150 mA plasma current wa V8 with metallic Iiackiivi reflected ,, - __, I - 150 mA plasma current Figure 5.22 Reflection curves observed in the uniform region of a rectangular plasma tube. (f - 2.4 GHz, In” 150 mA) ‘ 147 furthermore, Figure 5.23 shows that inside and outside metallic backing do not give different reflector curves. Based on the results observed in Figures 5.22 and 5.23, the different reflection curves observed in Figure 5.21 may not be due to the reflecting surface. This may imply that all the electro- acoustic resonances were still excited at the front side of the tube directly near the probe. The different reflection curves observed in the glass reflector and metallic reflector regions may be due to the electromagnetic effect of the metallic reflector to the reflected wave. Our attempt to study the reflection behavior of an electroacoustic wave on metallic and non-metallic surfaces using a single-profile plasma column was proved to be incon- clusive. A major disruption in the vacuum system prevented the continuation of this study. It is recommended that with some modifications on the tube construction, but based on the same idea of a single-profile plasma column, the reflection behavior of the electroacoustic wave can be successfully studied. m f, P ,5 8 inside metallic u backing U m r—I lu w u Ino : 150 mA plasma current w :, P a 3 outside metallic u backing o o H U 4 o )4 ,- 1 _ ,_ 1, n. In” " 150 mA plasma current Figure 5.23 Reflection curVes obserVed in a rectangular plasma tube with the inside and outside metallic backing. (f r 2.4 Cllz, Ipo = 150 mA) APPENDICES APPENDIX A NUMERICAL CALCULATION OF R1, R2, THE ELECTRON-ION COMPOSITION RATIOS OF THE n1 WAVE AND THE n2 WAVE To determine R1 and R2, the electron-ion composition ratios of the n1 wave and the n2 wave, for various source frequencies, various collision frequencies and various Te/Ti by using a computer, we write the following equations in terms of X, Y and Z where X = (we/w)2 , (A-l) Y = Ye/w (A-2) z = Te/Ti . (A-3) Equation (2.2.9) 2 w 2 y Be2 = E”__2_<1 - .2. .. j .2) (A-4) ve can be written as 82=K2 (A-S) 149 150 where x = 9—. (A-6) 2 Ve A1 = (1 - x - jY) . ' (A-7) For equation (2.2.13), 2 2 w y. 2 w i . i B, = —< - —— 'I' J —) (A‘s) l V12 042 w ' we use equations (2.2.8), (2.2.12), (2.1.11) and (2.1.12) with hydrogen gas plasma assumption, then we have wiz = (me/mime2 : or wiz = (1/1836)we2 ; { (A-9) vi2 = (me/mi)(Ti/Te)ve2 . 0r V12 = VeZ/(l8362) ; (A-lO) assuming (vi/Ye) = (Vi/Ve) (A-11) and using equation (A-lO), we have vi = Ye/(1836)1/2 . (A-12) 151 Equation (A-8) can be written as 312 = K22A2 (A-13) where A2 a 18362 - xz - jy(13362)1/2 . (A-14) Similarly, equation (2.3.15) 1 2 2 ‘5 4 w m 2 2 2 w e i . = - . , + 4(—)(—>(—) ° [ 1 e v 2v.2 (1)2 4:2 e 1 can be written as A = K 2A (A-lS) o 2 3 where l _ _ 2 2 2 _ Substituting equations (A-S), (A-13), (A-lS) into equations (2.4.3), (2.4.5) and using equations (A-l), (A-S) yield R1 = 52(A1 - A2 + A3) (A-17) R = l—(A - A - A ) (A-18) 2 2x 1 2 3 ° Equations for R1 and R2 are functions of X, Y and Z, i.e., functions of (we/w)2, ye/w and Te/Ti. Therefore, the electron- ion composition ratios of the n1 wave and the n2 wave can be determined by assuming various source frequencies, various collision frequencies and various Te/Ti' APPENDIX B NUMERICAL CALCULATION OF k1, k2, THE PROPAGATION CONSTANTS OF THE n WAVE AND THE n2 WAVE 1 To determine k1, k2, the propagation constants of the n1 wave and the n2 wave, for various source frequencies, various collision frequencies and various Te/Ti by using a computer, we rewrite equations (2.5.1) and (2.5.2). For hydrogen gas plasma, we use equation (A-l4) as well as equations (A-S), (A-7), (A-l3), (A-lS), (A-16) and (A-6). Then equations for k1 and k2 can be written as L— = 154 where A1 = 1 - x - jY (B-16) A2 = 991442 - x2 - jY(99144Z)1/2 (3-17) A.3 = BA? - Al)2 + 42x2]1/2 (B-18) x = (we/w)2 (3-19) Y = ye/w (B-ZO) z = Te/Ti . (B-21) Therefore, k1 and k2 for various source frequencies, various collision frequencies and various Te/Ti can be determined. For hydrogen gas plasma, we use equations (B-l) through (B-B). For xenon gas plasma, we use equations (B-l4) through (B-Zl). 155 Sample Program O(3()O(3()O(UF)O ()0(3()O ()O(3()O 10 PROGRAM PLASMA (OUTPUT) ********************************fk'k*********************** THIS PROGRAM CALCULATES (1) THE WAVE NUMBERS AKlK AND AK2K (2) THE RATIOS RNlEI AND RN2EI AS A FUNCTION OF x (x=(WE/W)**2) FOR AN ASSIGNED A (A=COLLISION FREQUENCY/W). THIS CASE (HYDROGEN GAS IS ASSUMED, TE=B*TI WHERE B=100) ********************************************************* REAL MOD1,MOD2 COMPLEX C,D,AM,AN,ANMX,AK1K,AK2K,RNlEI,RNZEI,P DIMENSION A(7),X(15),AM(7,15),AN(7,15),ANMX(7,15), lP(7,15) ,Q(15) ,AK1K(7,15) ,AK2K(7,15) , 2RN1EI(7,15),RNZEI(7,15),MOD1(7,15),MOD2(7,15) A(1)=o.o A(2)=lO.E-4 DO 1 I=3,6 A(I)=A(I-1)*10. CONTINUE x(1)=10.E-S DO 2 J=2,13 X(J)=X(J-1)*10. CONTINUE B=100. C=CMPLX(0.0,1.0) E=SQRT(1836.*B) D=CMPLX(0.0,E) DO 3 I=l,6 DO 4 J=1,13 AM(I,J)=1.—x(J)-C*A(I) AN(I,J)=1836.*B-B*X(J)-D*A(I) P(IIJ)=(AN(IIJ)-AM(IIJ))**2 Q(J)=4.*B*X(J)**2 ANMx(I,J)=CSQRT(P(I,J)+Q(J)) NOTING AT LARGE x, 4BX**2 IS MUCH SMALLER THAN THE RE(AN-AM), SO ANMx SHOULD HAVE THE SIGNS OF (AN-AM). SINCE IM(AN—AM) IS NEGATIVE. WE DEMAND IM(ANMx) NEGATIVE. IF(AIMAG(ANMx(I,J))-o.0) 20,20,10 ANMX(IIJ)=-(ANMX(IIJ)) AK1K MIGHT HAVE 2 SOLUTIONS,ONE IS THE NEGATIVE OF THE OTHER.SINCE IT IS A WAVE NUMBER WHICH HAS TO HAVE POSITIVE REAL PART AND NEGATIVE IMAGINARY PART.SO WE COULD PICK THE REQUIRED SOLUTION BY DOING THE FOLLOWING 20 30 4O 50 60 11 9 100 200 201 202 300 400 156 STATEMENTS.(SAME FOR AK2K) AK1K(I,J)=CSORT((AM(I,J)+AN(I,J)+ANMX(I,J))/(3672.*B)) IF (REAL(AK1K(I,J))-0.0) 30,30,40 AK1K(I,J)=-(AK1K(I,J)) AK2K(I,J)=CSQRT((AM(I,J)+AN(I,J)-ANMX(I,J))/2.) IF (REAL(AK2K(I,J))-0.0) 50,50,60 AK2K(I,J)=-(AK2K(I,J)) RNlEI(I,J)=(AM(I,J)-AN(I,J)+ANMX(I,J))/(2.*X(J)) RN2EI(I,J)=(AM(I,J)-AN(I,J)-ANMx(I,J))/(2.*X(J)) MOD1(I,J)=CABS(RN1EI(I,J)) MOD2(I,J)=CABS(RN2EI(I,J)) CONTINUE CONTINUE PRINT 100 D0 5 I=1,6 PRINT 200,I,A(I) DO 6 J=1,13 PRINT 201,J,X(J),AM(I,J),AN(I,J),P(I,J).Q(J),ANMX(I,J) CONTINUE CONTINUE PRINT 100 D0 7 I=1,6 PRINT 200,1,A(I) D0 8 J=1,13 PRINT 300,J,X(J),AK1K(I,J),AK2K(I,J) CONTINUE CONTINUE DO 9 I=1,6 PRINT 202,1,A(I) D0 11 J=1,13 PRINT 400,J,X(J),RN1EI(I,J),RN2EI(I,J),MOD1(I,J),MOD2(I,J) CONTINUE CONTINUE FORMAT(*1RESULTS*) FORMAT(1H0,*A(*,Il,*)=*,F7.3) FORMAT(3X,*X(*,I2,*)=*,E10.2,3X,*AM=*,E15.7,2X,E15.7,6X, 1*AN=*,E15.7,2X,E15.7/ 214X,11H(AN-AM)**2=,E15.7,2X,E15.7,2X,7H4BX**2=,E15.7/ 320X,*ANMX=*,E15.7,2X,E15.7) FORMAT(1H1,*A(*,I1,*)=*,F7.3) FORMAT(1H ,10X,*X(*,12,*)=*,E10.2,10X, 1*AK1K=*,E15.7,2X,E15.7,10X, 2*AK2K=*,E15.7,2X,E15.7) FORMAT(1H0,10X,*X(*,I2,*)=*,E10.2,10X, 1*RN1EI=*,E15.7,2X,E15.7,10X, 2*RN2EI=*,E15.7,2X,E15.7,/, 337X,*MOD1 =*,E15.7,27X,*MOD2 =*,E15.7) END ’APPENDIX c TABLES OF DATA FOR THE CALCULATION OF RADIATION PATTERNS OF THE n1 WAVE AND THE n2 WAVE Notations and constants used in this appendix: 1 Ve = (3kTe/me)2 , thermal velocity of electrons. .1. Vi = (3kTi/mi)2 , thermal velocity of ions. 1. VA = [3k(Te + Ti)/mi]2 , phase velocity of the HI wave at low frequency range. Re[kl/(w/Vi)] , numerical output of the computer. kl = {Re[k1/(w/Vi)]}(w/Vi) , phase constant of the n1 wave. Re[k2/(w/Ve)] , numerical output of the computer. k2 = {Re[k2/(w/Ve)]}(w/Ve) , phase constant of the n2 wave. .1. ke = w/uoeo(l - weZ/wz)2 , propagation constant of the electromagnetic wave in the plasma. f = antenna frequency. 157 158 fp = electron plasma frequency. me = anp , circular electron plasma frequency. L = 2a/(Vbh/f) , normalized antenna length. k = 1.38 x 10"23 joules/°K , Boltzmann's constant. m.e = 9.109 x 10’31 kg , electron mass. mi = 9.031 x 10'26 kg , xenon ion mass. Remark: A xenon gas plasma with (ye/w) = 0 is assumed for all cases in this appendix. 159 Table C-l kldl versus Te° (f = 30 kHz, Re[k1/(w/Vi)] = 0.3016 for (Te/Ti) = 10, d1 = 1 cm) w . 1 2000°K 623 188 1.88 6000°K 359 108 1.08 10000°K 278. 84 0.84 Table C-2 kldl versus Te/Ti. (f = 30 kHz, Te = 6000°K, d1 = 1 cm) 32 Re[——El——] 2— k k d1 Ti (w/Vi) vi 1 l 1 0.7071 114 80 0.80 10 0.3016 359 108 1.08 100 0.0995 1136 113 1.13 1000 0.0316 3594 114 1.14 160 Table C—3 kldl versus f. 6000°K, Re[k1/(w/Vi)] = 0.3016 for (Te/Ti) = 10, d1 (Te = = 2.5 cm) (.0 l 10 kHz 120 36 0.9 20 kHz 240 72 1.8 30 kHz 360 108 2.7 Table C-4 kzdl versus Te- (f = 1 GHz, Re[k2/(w/Ve)] = 0.2235 for (weZ/wz) = 0.95 and (Te/Ti) = 1 to 10“, d1 = 1 mm) w e 2000°K 2.08 x 10“ 4650 4.65 6000°K 1.20 x 10“ 2690 2.69 10000°K 9.32 x 103 2080 2.08 161 Table C- -S kzdl versus f. (T6 = 4000° K, Re[k2/(w/Ve)] = o. 2235 for (682/02) = 0.95 and (Te/Ti) = 1 to 10 , d1 = 1mm) m f v— k2 kzdl e 0.5 GHZ 7.37 x 103 1650 1.65 1.0 GHz 1.47 x 10“ 3290 3.29 1.5 GHz 2.21 x 10“ 4940 4.94 Table C- 6 kla versus Te (f = 30 kHz, Re[k1/(w/Vi)] = 0. 0995 for (Te/Ti) = 100, a = 2. 25 cm) (1) Te VA VT kl kla l 2000°K 962 2000 199 4.48 6000°K 1666 1136 113 2.55 10000°K 2150 880 88 1.97 162 mm.m «m.H a.hv HAN x.ooom ma.q m~.~ m.nm 5mm soooov hm.m ma.m m.Hm mam moooom Umx ONx . 0 mx I» we AKVH.H u A Axvm.o u A 1.0H 00 H u AHA\ma. ecu mm.o u A~3\~msc you mm-.o u HA6>\svxmx_mm .so H.ma u A .Nmz m.SH u me .me msmum> aux muo manna mw.¢ had mNNm mmm mHmc.o coca o¢.v mmH mmmH Nam mmmo.o OOH MH.m and hmH vaH finchoo H a H a . .> 3 . MHM HX I!» ‘1? HEH 0% W19 as a Ago m~.~ u m .xoooom u we .me on we .Aa\ma mamum> max RIO manna Table C-8 163 kla versus f. (VA = 1.05 x 103 m/sec, Te = Ti 2 1200°K, Re[k1/(w/Vi)] = 0.7071 for (Te/Ti) = 1, a 2.25 cm) m f L V— k1 kla 1 16.3 kHz 0.7 138 98 2.2 23.3 kHz 1.0 198 152 3.4 58.3 kHz 2.5 494 349 7.8 Table C-10 kla versus Te- 0.3016 for (Te/Ti) = 10, a (f = 30 kHz, Re[k1/(w/Vi)] cm) w 2000°K 1004 623 188 4.22 6000°K 1739 359 108 2.44 10000°K 2245 278 84 1.89 2.25 164 Table C-ll kla versus Te/Ti. (f = 30 kHz, Te = 4000°K, a = 2.25 cm) Te k1 w —— Re[————-{] —— k k a 1 0.7071 139 98.4 2.21 10 0.3016 440 133.0 2.99 100 0.0995 139 139.0 3.12 Table C-12 kla versus f. (V = 1.05 x 103 m/sec, Te = Ti 2 1200°K, Re[k1/(m/Vi)] = 0.7071 for (Te/Ti) = 1, a = 2.25 cm) OJ f L ‘7— k1 kla 1 35.0 kHz 1.5 297 210 4.72 46.6 kHz 2.0 395 279 6.28 93.3 kHz 4.0 790 559 12.60 165 Table C-13 kza versus Te- (f = 17.5 MHz, Relkz/(w/Vé)] = 0.2235 for (882/82) = 0.95 and (Te/Ti) = 1 to 10 , a = 7.2 cm) (J e 2000°K 365 81.5 5.87 4000°K 257 57.6 4.15 8000°K 182 40.8 2.94 Table C-14 kza versus f. (Te = 2000°K, Re[k2/(w/Ve)] = 0.2235 for (892/02) = 0.95 and (Te/Ti) = 1 to 10“, a = 7.2 cm) f E_ k k a Ve 2 2 15.0 MHz 313 69.8 5.03 17.5 MHz 365 81.5 5.87 20.0 MHz 417 93.2 6.71 Table C-lS klh versus Te° 166 (f = 30 kHz, Re[k1/(w/Vi)] - 0.3016 for (Te/Ti) = 10) T Q_ k1 klh 3 Vi h = 2.5 cm h = 5 cm 2000°x 623 188 4.7 9.4 6000°K 359 108 2.7 5.4 10000°x 278 84 2.1 4.2 Table C-17 k h versus f. h = 5 cm) w/Vi)] = 0.3016 for (Te/Ti) = 10, w 1 10 kHz 120 36 1.8 20 kHz 240 72 3.6 30 kHz 360 108 5.4 167 ¢.m h.N QOH amm mHom.o OH o.v O.N Om QHH HBCF.O H EU m .I... 3 EU m.N H H> AH>\3V HR. A A 4.. Jul. 9. m. n x x a Amooooo .me om mv .HB\OB msmum> sax mano magma 168 Table C-18 kzh and ke/kz versus Te (f= 5. 5 MHz, fp = 4. 5 MHz, (Te/Ti ) = 1 to 10“, Re[k2/(w/Ve)] = 0. S7 and ke — 0. 066 for (weZ/mz) = 0. 67, h = 6 cm) k T 2— k —2 k h e Ve 2 k2 2 4000°K 81.0 46.1 0.00143 2.70 6000°K 66.2 37.7 0.00176 2.26 8000°K 57.3 32.6 0.00203 1.95 Table C-21 keh versus f. (f = 4.5 MHz, h = 8.5 cm) P we2 f —§—. ke keh w 5.0 MHz 0.81 0.046 0.0039 5.5 MHz 0.67 0.066 0.0056 7.0 MHz 0.41 0.113 0.0096 169 Sm.m .II. mmaoo.o H.mo oo~aa.o m.om mn.o um: o.h mq.m qv.~ mmaoo.o S.oq hammo.o ¢.HS sm.o um: m.m 86 m.m u P so 6 u a N9 N 6 6> im>\3c m: x x um Imllmm a nwx x x . Ann: m.v u no ..oa op H u AHE\meV .moomam u 691 .m mamum> ~x\mx new nmx omuo wanna «v.m mnaoo.o m.mm oemaa.o «.vm mn.o H6.o mm: 0.5 o~.m msaoo.o 5.5m Sammo.o ~.om nm.o no.o um: m.m ma.~ Shaoo.o m.m~ qwmeo.o H.om mq.o Hm.o um: o.m N m m 3 3 .4. .. .. 1 I. a 11.. A. . 9 mg a m A80 m.m u n .Nmz m.v H mm ..OH on H u AAB\maV .xoooom u use .m msmum> ~x\mx can amx mauo manna 170 Sample Program 000000000 1 2 100 101 200 300 PROGRAM PLASMA (OUTPUT) ***************************************************** THIS PROGRAM CALCULATES THE RADIATION PATTERNS OF THE N2 WAVE EXCITED BY A CYLINDRICAL ANTENNA. RADPAT=COS(THETAR)*(COS(K2*H*COS(THETAR))-COS(KE*H))/ ((COS(THETAR))**2-(KE/K2)**2) WE LET P=K2*H, B=KE*H, C=KE/K2. *******it*******************************************t DIMENSION THETA(20),THETAR(20),P(5),A(5),B(5),C(5) P(1)=1.95 P(2)=2.26 P(3)=2.70 P(4)=3.20 B(1)=0.00396 B(2)=0.00388 B(3)=B(1) B(4)=0.00562 c(1)=0.00203 c(2)=0.00176 C(3)=0.00143 c(4)=c(2) PI=3.14159265 DO 2 J=1,4 PRINT 100 PRINT 101,J,P(J) D0 1 I=1,19 THETA(I)=-100.+10.*I PRINT 200,I,THETA(I) THETAR(I)=(PI/180.)*THETA(I) A(J)=P(J)*COS(THETAR(I)) RADPAT=COS(THETAR(I))*(COS(A(J))-COS(B(J)))/ 1((COS(THETAR(I)))**2-C(J)**2) PRINT 300,RADPAT CONTINUE CONTINUE FORMAT(*1RESULTS*) FORMAT(*0*,5X,*CASE(*,IZ,*)*,2X,6HK2*H =,ElS.7) FORMAT(* *,SX,*THETA(*,I2,*)=*,F6.1,* DEGREE*) FORMAT(*+*,33X,*RADPAT=*,E15.7) END REFERENCES 10. 11. REFERENCES M. H. Cohen, "Radiation in a plasma, III, metal boundaries," Phys. Rev., Vol. 126, 2, 398, (1962). A. Hessel and J. 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