"\H%/ PLASMA PHYSICS BY J. G. LINHART EURATOM/CNEN Association Frascati (Rome) Italy 1969 Published by the European Atomic Energy Community — EURATOM Brussels (Belgium), OCTOBER 1969 EUR 4268 e NOTE The Commission of the European Com­ munities and its departments decline all responsibility with regard to the use of the information contained herein. PREFACE TO THE THIRD EDITION During the years preceding 1960 plasma physics developped rapidly and the pioneering mood of those days influenced many, including the author of this book, to come out with articles and books in which a fermenting, partial and somewhat hasty spirit prevails. The times have changed and nowadays it is possible to attempt to write a text­ book rather than a work demonstrating novel or revolutionary features of the subject. This edition is meant for graduate and post-graduate students; wherever possible the treatment insists on physical insight rather than on mathematical rigour. I have tried to treat the subject in all its aspects without indulging in details and specialities (for which exist now several excellent books) and nevertheless aiming at students having a good background in physics and even having some acquaint­ ance with the concepts of physics of ionized gases. This should put the present edition in a bracket between elementary books on plasma physics and those dealing thoroughly with special domains, some of which will be found among the recommended literature on p. 332. Frascati, spring 1967. Acknowledgment I would like to thank those of my colleagues from Laboratorio Gas Ionizzati who have helped me so generously with the manuscript of the 3rd edition. CONTENTS PREFACE TO THE THIRD EDITION V INTRODUCTION 1 CHAPTER 1 1.1. Plasma state 6 1.1.1. Degree of ionisation — Sana's equation 6 1.1.2. Electric fields in plasma 16 1.1.3. Radiation in plasma Hi 1.1.4. Classification of plasmas 21 1.2. Plasma in nature and in laboratory 22 1.2.1. Stars and interstellar space — The diagramme of Hertzsprung and Russell 22 1.2.2. Planets 24 1.2.3. Plasma produced by man 24 References to Chapter 1 26 List of symbols used in Chapter 1 27 CHAPTER 2: MOTION OF ELECTRONS AND IONS IN ELEC­ TRIC AND MAGNETIC FIELDS Introduction 28 2.1. Motion in an electrostatic field 28 2.2. Motion in a magnetostatic field 31 2.2.1. Motion of charged particles in a toroidal magnetic field 37 2.2.2. Motion of charged particles in the field of a magnetic lens 39 2.2.3. Motion of charged particles in a helical magnetic field 45 2.2.4. Superimposed toroidal magnetic field and betatron magnetic field 46 2.3. Motion of charged particles in crossed electric and magnetic fields 52 2.3.1. Electric vortex field and magnetic lens field 56 2.4. Motion in crossed R. F. electric field and a magnetostatic field 58 VIII CONTENTS 2.5. The movement of a charged particle in the field of an electromagnetic wave 62 2.6. Radiation from accelerated charges 66 2.6.1. Bremsstrahlung 68 2.6.2. Cyclotron (betatron, synchrotron) radiation 71 2.6.3. Cerenkov radiation 77 References to Chapter 2 85 List of symbols used in Chapter 2 85 CHAPTER 3 : FLUID DESCRIPTION OF PLASMA Introduction 87 3.1. Stationary distributions 91 3.2. The collisionless Boltzmann equation 92 3.2.1. Non-relativistic ensemble 94 3.2.2. Relativistic ensemble 97 3.3. Integrals of Boltzmann's equations over the velocity space 98 3.3.1. Non-relativistic case 98 3.3.2. Relativistic case 104 3.4. Fluid models 106 References to Chapter 3 110 List of symbols used in Chapter 3 110 CHAPTER 4: EQUILIBRIUM CONFIGURATIONS (PLASM ASTATICS) Introduction Ill 4.1. Plasma in an external magnetic field 112 4.4.1. Cylindrical and cusp geometry 120 4.2. Confinement by magnetic fields generated by currents in the plasma 126 4.2.1. Non-relativistic streams 126 4.2.2. Relativistic streams 131 4.3. Plasma equilibrium in external and self-fields .... 135 4.3.1. Stabilized Z-pinch 135 4.3.2. Toroidal plasma loop 137 4.3.3. Force-free magnetic fields 140 References to Chapter 4 142 List of symbols used in Chapter 4 142 CONTENTS IX CHAPTER 5: WAVES AND INSTABILITIES IN PLASMA Introduction 143 5.1. Electron oscillations in plasma 144 5.1.1. The longitudinal oscillations 147 5.1.2. The transversal oscillations 147 5.1.3. Hybrid transversal and longitudinal waves 151 5.1.4. Reflection of electromagnetic waves by plasma 152 5.1.5. Electron waves on a plasma cylinder 155 5.1.6. Effects of random velocities on waves in plasma 160 5.2. Positive ion oscillations 171 5.2.1. Electrostatic ion oscillations 172 5.2.2. Hydromagnctic oscillations in a stationary infinite plasma —· waves on a plasma cylinder 174 5.2.3. Hydromagnetic oscillations in plasma streams 180 5.3. Growing waves and instabilities 183 5.3.1. Conversion of kinetic energy of particle streams into the energy of longitudinal plasma oscillations 185 5.3.2. Rayleigh-Taylor instability 188 5.3.3. Magnetohydrodynamic instability 192 5.3.4. Hydrodynamic instability 208 References to Chapter 5 213 List of symbols used in Chapter 5 214 CHAPTER 6 : SHOCK WAVES IN PLASMA Introduction 216 6.1. Relations of Rankine-Hugoniot. Shock-speed . 220 6.2. Structure of the shock front in absence of magnetic field 222 6.3. Shocks in a gyrotropic plasma 225 6.4. Diverging and converging shocks 227 6.4.1. Diverging shocks 227 6.4.2. Converging shocks 229 References to Chapter 6 232 List of symbols used in Chapter 6 232 CHAPTER 7: PLASMA DYNAMICS Introduction 233 X CONTENTS 7.1. Plasmoids 233 7.2. Steady plasma flow 242 References to Chapter 7 245 List of symbols used in Chapter 7 245 CHAPTER 8: COLLISION AND RELAXATION PROCESSES Introduction 246 8.1. Dynamics of a collision of two charged particles . 247 8.2. Fokker-Planck equation 251 8.2.1. Conduction of electricity in plasma — conduction of elec­ tricity in a gyrotropic plasma 256 8.2.2. Stopping power — relaxation to Maxwellian distribution — equipartition of energy 264 8.3. Diffusion in configuration space 267 8.3.1. Flux of particles 269 8.3.2. Conduction of heat and electricity 273 8.3.3. Diffusion of momentum. Viscosity 276 References to Chapter 8 278 List of symbols used in Chapter 8 278 Applications 280 CHAPTER 9: RESEARCH ON CONTROLLED FUSION Introduction 281 9.1. Sources of nuclear energy 281 9.1.1. Elementary nuclear concepts 281 9.1.2. Binding energy 283 9.1.3. Nuclear fusion 287 9.1.4. Fission and fusion reactions as sources of energy 292 9.1.5. Uncontrolled fusion reactions 295 9.2. Controlled fusion reactors 298 9.2.1. Stationary fusion reactors 303 9.2.2. Pulsed fusion reactors 306 9.2.3. Experiments in the research on controlled fusion 309 References to Chapter 9 311 List of symbols used in Chapter 9 312 CONTENTS XI CHAPTER 10: OTHER APPLICATIONS 10.1. Generation of electromagnetic waves 313 10.2. Direct conversion of chemical energy into electrical energy 316 10.3. Applications to particle-accelerators 322 10.3.1. Plasma betatron 322 10.3.2 Collective ion-acceleration 324 10.4. Rocket propulsion 326 10.5. Energy storage 328 List of symbols used in Chapter 10 331 References to Chapter 10 331 GENERAL LITERATURE 332 BOOKS ON SPECIAL TOPICS 332 The numbering of the equations in each chapter starts from one. When reference is made to equations in other chapters, the number of the chapter is added first (e.g., Equation (3.62) means Equation (62) in Chapter 3). INTRODUCTION Plasma physics is concerned with the behaviour of systems of many free electrons and ionised atoms where the mutual Coulomb interactions cannot be disregarded. In a restricted sense, such systems of particles consist of nearly equal numbers of positive and negative charges. Systems of this type are examples of a medium known as plasma which in many respects behaves differently from the solid, liquid and gaseous state of matter. All states of matter represent different degrees of organization, to which there correspond certain values of binding energy. Thus, in the solid state the important quantity is the binding energy of molecules in a crystal; in fact, a crystal could be considered as a macro- or super- molecule. If the average kinetic energy per molecule W exceeds the binding energy U (a fraction of an eV) the crystal structure breaks up, either into a liquid or directly into a gas. A similar law operates in the case of liquids, and in order to change a liquid into a gas, a certain minimum kinetic energy per molecule is required to break the bonds of the van der Waals forces. Matter can exist as plasma, i.e., in its fourth state, when the kinetic energy W per plasma particle exceeds the ionising potential of atoms which is usually a few eV. Thus the average kinetic energy per particle determines the state in which matter exists. A precise mathematical statement of this theorem is an equation of the Saha type. However, a simple criterion can be written as Un < W < Un+1 (1) where U„, U„+\ are the respective binding energies, expressing that matter exists in the (n -\- l)st state. The plasma state will correspond to an order-of-magnitude relation­ ship 1 < WA < 10" (eV). Extrapolating this principle to higher states of matter, so far unex­ plored, one may define the fifth state of matter as one in which 1 < Wr, < 500 (MeV).