# Fundamentals of Plasma Physics by J. A. Bittencourt

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• Author: J. A. Bittencourt
• Title: Fundamentals of Plasma Physics
• Prerequisities: Better part of an undergraduate (BS) physics program

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1 INTRODUCTION
1. General Properties of Plasmas                                           1
1.1 Definition of a Plasma
1.2 Plasma as the Fourth State of Matter
1.3 Plasma Production
1.4 Particle Interactions and Collective Effects
1.5 Some Basic Plasma Phenomena
2. Criteria for the Definition of a Plasma                                 6
2.1 Macroscopic Neutrality
2.2 Debye Shielding
2.3 The Plasma Frequency
3. The Occurrence of Plasmas in Nature                                    11
3.1 The Sun and its Atmosphere
3.2 The Solar Wind
3.3 The Magnetosphere and the Van Allen Radiation Belts
3.4 The Ionosphere
3.5 Plasmas Beyond the Solar System
4. Applications of Plasma Physics                                         17
4.1 Controlled Thermonuclear Fusion
4.2 The Magnetohydrodynamic Generator
4.3 Plasma Propulsion
4.4 Other Plasma Devices
5. Theoretical Description of Plasma Phenomena                            25
5.1 General Considerations on a Self-Consistent Formulation
5.2 Theoretical Approaches
Problems                                                                  28

2 CHARGED PARTICLE MOTION IN CONTSTANT AND UNIFORM
ELECTROMAGNETIC FIELDS
1. Introduction                                                           33
2. Energy Conservation                                                    34
3. Uniform Electrostatic Field                                            36
4. Uniform Magnetostatic Field                                            37
4.1 Formal Solution of the Equation of Motion
4.2 Solution in Cartesian Coordinates
4.3 Magnetic Moment
4.4 Magnetization Current
5. Uniform Electrostatic and Magnetostatic Fields                         49
5.1 Formal Solution of the Equation of Motion
5.2 Solution in Cartesian Coordinates
6. Drift Due to an External Force                                         54
Problems                                                                  56

3 CHARGED PARTICLE MOTION IN NONUNIFORM
MAGNETOSTATIC FIELDS
1. Introduction                                                           59
2. Spatial Variation of the Magnetic Field                                61
2.1 Divergence Terms
2.3 Shear Terms
3. Equation of Motion in the First-Order Approximation                    66
4. Average Force Over One Gyration Period                                 68
4.1 Parallel Force
4.2 Perpendicular Force
4.3 Total Average Force
6. Parallel Acceleration of the Guiding Center                            74
6.1 Invariance of the Orbital Magnetic Moment and of the Magnetic Flux
6.2 Magnetic Mirror Effect
7. Curvature Drift                                                        84
Problems                                                                  89

4 CHARGED PARTICLE MOTION IN
TIME-VARYING ELECTROMAGNETIC FIELDS
1. Introduction                                                           95
2. Slowly Time-Varying Electric Field                                     95
2.1 Equation of Motion and Polarization Drift
2.2 Plasma Dielectric Constant
3. Electric Field with Arbitrary Time Variation                          100
3.1 Solution of the Equation of Motion
3.2 Physical Interpretation
3.5 Cyclotron Resonance
4. Time-Varying Magnetic Field and Space-Varying Electric Field          108
4.1 Equation of Motion and Adiabatic Invariants
4.2 Magnetic Heating of a Plasma
5. Summary of Guiding Center Drifts and Current Densities                115
5.1 Guiding Center Drifts
5.2 Current Densities
Problems                                                                 116

5 ELEMENTS OF PLASMA KINETIC THEORY
1. Introduction                                                          122
2. Phase Space                                                           123
2.1 Single-Particle Phase Space
2.2 Many-Particle Phase Space
2.3 Volume Elements
3. Distribution Function                                                 126
4. Number Density and Average Velocity                                   128
5. The Boltzmann Equation                                                129
5.1 Collisionless Boltzmann Equation
5.2 Jacobian of the Transformation in Phase Space
5.3 Effects of Particle Interactions
6. Relaxation Model for the Collision Term                               135
7. The Vlasov Equation                                                   136
Problems                                                                 138

6 AVERAGE VALUES AND MACROSCOPIC VARIABLES
1. Average Value of a Physical Quantity                                  141
2. Average Velocity and Peculiar Velocity                                142
3. Flux                                                                  143
4. Particle Current Density                                              146
5. Momentum Flow Dyad or Tensor                                          147
6. Pressure Dyad or Tensor                                               148
6.1 Concept of Pressure
6.2 Force per Unit Area
6.3 Force per Unit Volume
6.4 Scalar Pressure and Absolute Temperature
7. Heat Flow Vector                                                      154
9. Total Energy Flux Triad                                               155
10. Higher Moments of the Distribution Function                          157
Problems                                                                 157

7 THE EQUILIBRIUM STATE
1. The Equilibrium State Distribution Function                           161
1.1 The General Principle of Detailed Balance and Binary Collisions
1.2 Summation Invariants
1.3 Maxwell-Boltzmann Distribution Function
1.4 Determination of the Constant Coefficients
1.5 Local Maxwell-Boltzmann Distribution Function
2. The Most Probable Distribution                                        169
3. Mixture of Various Particle Species                                   170
4. Properties of the Maxwell-Boltzmann Distribution Function             171
4.1 Distribution of a Velocity Component
4.2 Distribution of Speeds
4.3 Mean Values Related to the Molecular Speeds
4.4 Distribution of Thermal Kinetic Energy
4.5 Random Particle Flux
4.6 Kinetic Pressure and Heat Flux
5. Equilibrium in the Presence of an External Force                      181
6. Degree of Ionization in Equilibrium and the Saha Equation             184
Problems                                                                 187

8 MACROSCOPIC TRANSPORT EQUATIONS
1. Moments of the Boltzmann Equation                                     193
2. General Transport Equation                                            194
3. Conservation of Mass                                                  197
3.1 Derivation of the Continuity Equation
3.2 Derivation by the Method of Fluid Dynamics
3.3 The Collision Term
4. Conservation of Momentum                                              200
4.1 Derivation of the Equation of Motion
4.2 The Collision Term
5. Conservation of Energy                                                204
5.1 Derivation of the Energy Transport Equation
5.2 Physical Interpretation
5.3 Simplifying Approximations
6. The Cold Plasma Model                                                 210
7. The Warm Plasma Model                                                 211
Problems                                                                 212

9 MACROSCOPIC EQUATIONS FOR A CONDUCTING FLUID
1. Macroscopic Variables for a Plasma as a Conducting Fluid              219
2. Continuity Equation                                                   222
3. Equation of Motion                                                    223
4. Energy Equation                                                       224
5. Electrodynamic Equations for a Conducting Fluid                       227
5.1 Maxwell Curl Equations
5.2 Conservation of Electric Charge
5.3 Generalized Ohm’s Law
6. Simplified Magnetohydrodynamic Equations                              234
Problems                                                                 236

10 PLASMA CONDUCTIVITY AND DIFFUSION
1. Introduction                                                          238
2. The Langevin Equation                                                 238
3. Linearization of the Langevin Equation                                240
4. DC Conductivity and Electron Mobility                                 242
4.1 Isotropic Plasma
4.2 Anisotropic Magnetoplasma
5. AC Conductivity and Electron Mobility                                 247
6. Conductivity with Ion Motion                                          249
7. Plasma as a Dielectric Medium                                         250
8. Free Electron Diffusion                                               251
9. Electron Diffusion in a Magnetic Field                                254
10. Ambipolar Diffusion                                                  256
11. Diffusion in a Fully Ionized Plasma                                  260
Problems                                                                 262

11 SOME BASIC PLASMA PHENOMENON
1. Electron Plasma Oscillations                                          269
2. The Debye Shielding Problem                                           273
3. Debye Shielding Using the Vlasov Equation                             278
4. Plasma Sheath                                                         279
4.1 Physical Mechanism
4.2 Electric Potential on the Wall
4.3 Inner Structure of the Plasma Sheath
5. Plasma Probe                                                          288
Problems                                                                 291

12 SIMPLE APPLICATIONS OF MAGNETOHYDRODYNAMICS
1. Fundamental Equations of Magnetohydrodynamics                         299
1.1 Parker Modified Momentum Equation
1.2 The Double Adiabatic Equations of Chew, Goldberger, and Low (CGL)
1.3 Special Cases of the Double Adiabatic Equations
1.4 Energy Integral
2. Magnetic Viscosity and Reynolds Number                                309
3. Diffusion of Magnetic Field Lines                                     311
4. Freezing of Magnetic Field Lines to the Plasma                        312
5. Magnetic Pressure                                                     316
6. Isobaric Surfaces                                                     318
7. Plasma Confinement in a Magnetic Field                                319
Problems                                                                 322

13 THE PINCH EFFECT
1. Introduction                                                          325
2. The Equilibrium Pinch                                                 326
3. The Bennett Pinch                                                     332
4. Dynamic Model of the Pinch                                            335
5. Instabilities in a Pinched Plasma Column                              341
6. The Sausage Instability                                               342
7. The Kink Instability                                                  345
8. Convex Field Configurations                                           346
Problems                                                                 348

14 ELECTROMAGNETIC WAVES IN FREE SPACE
1. The Wave Equation                                                     351
2. Solution in Plane Waves                                               352
3. Harmonic Waves                                                        354
4. Polarization                                                          358
5. Energy Flow                                                           363
6. Wave Packets and Group Velocity                                       366
Problems                                                                 370

15 MAGNETOHYDRODYNAMIC WAVES
1. Introduction                                                          375
1.1 Alfv´en Waves
1.2 Magnetosonic Wave
2. MHD Equations for a Compressible Nonviscous Conducting Fluid          379
2.1 Basic Equations
2.2 Development of an Equation for the Fluid Velocity
3. Propagation Perpendicular to the Magnetic Field                       382
4. Propagation Parallel to the Magnetic Field                            383
5. Propagation at Arbitrary Directions                                   384
5.1 Pure Alfv´en Wave
5.2 Fast and Slow MHD Waves
5.3 Phase Velocities
5.4 Wave Normal Surfaces
6. Effect of Displacement Current                                        390
6.1 Basic Equations
6.2 Equation for the Fluid Velocity
6.3 Propagation Across the Magnetostatic Field
6.4 Propagation Along the Magnetostatic Field
7. Damping of MHD Waves                                                  394
7.1 Alfv´en Waves
7.2 Sound Waves
7.3 Magnetosonic Waves
Problems                                                                 397

16 WAVES IN COLD PLASMA
1. Introduction                                                          400
2. Basic Equations of Magnetoionic Theory                                401
3. Plane Wave Solutions and Linearization                                402
4. Wave Propagation in Isotropic Electron Plasmas                        403
4.1 Derivation of the Dispersion Relation
4.2 Collisionless Plasma
4.3 Time-Averaged Poynting Vector
4.4 The Effect of Collisions
5. Wave Propagation in Magnetized Cold Plasmas                           413
5.1 Derivation of the Dispersion Relation
5.2 The Appleton-Hartree Equation
6. Propagation Parallel to B0                                            419
7. Propagation Perpendicular to B0                                       423
8. Propagation at Arbitrary Directions                                   430
8.1 Resonances and Reflection Points
8.2 Wave Normal Surfaces
8.3 The CMA Diagram
9. Some Special Wave Phenomena in Cold Plasmas                           439
9.1 Atmospheric Whistlers
9.2 Helicons
Problems                                                                 447

17 WAVES IN WARM PLASMA
1. Introduction                                                          453
2. Waves in a Fully Ionized Isotropic Warm Plasma                        453
2.1 Derivation of the Equations for the Electron and Ion Velocities
2.2 Longitudinal Waves
2.3 Transverse Wave
3. Basic Equations for Waves in a Warm Magnetoplasma                     460
4. Waves in a Warm Electron Gas in a Magnetic Field                      462
4.1 Derivation of the Dispersion Relation
4.2 Wave Propagation Along the Magnetic Field
4.3 Wave Propagation Normal to the Magnetic Field
4.4 Wave Propagation at Arbitrary Directions
5. Waves in a Fully Ionized Warm Magnetoplasma                           470
5.1 Derivation of the Dispersion Relation
5.2 Wave Propagation Along the Magnetic Field
5.3 Wave Propagation Normal to the Magnetic Field
5.4 Wave Propagation at Arbitrary Directions
6. Summary                                                               479
Problems                                                                 481

18 WAVES IN HOT ISOTROPIC PLASMA
1. Introduction                                                          483
2. Basic Equations                                                       483
3. General Results for a Plane Wave in a Hot Isotropic Plasma            485
3.1 Perturbation Charge Density and Current Density
3.2 Solution of the Linearized Vlasov Equation
3.3 Expression for the Current Density
3.4 Separation into the Various Modes
4. Electrostatic Longitudinal Wave in a Hot Isotropic Plasma             491
4.1 Development of the Dispersion Relation
4.2 Limiting Case of a Cold Plasma
4.3 High Phase Velocity Limit
4.4 Dispersion Relation for Maxwellian Distribution Function
4.5 Landau Damping
5. Transverse Wave in a Hot Isotropic Plasma                             503
5.1 Development of the Dispersion Relation
5.2 Cold Plasma Result
5.3 Dispersion Relation for Maxwellian Distribution Function
5.4 Landau Damping of the Transverse Wave
6. The Two-Stream Instability                                            506
7. Summary                                                               508
7.1 Longitudinal Mode
7.2 Transverse Mode
Problems                                                                 510

19   WAVES IN HOT MAGNETIZED PLASMA
1. Introduction                                                          515
2. Wave Propagation Along the Magnetostatic Field in a Hot Plasma        516
2.1 Linearized Vlasov Equation
2.2 Solution of the Linearized Vlasov Equation
2.3 Perturbation Current Density
2.4 Separation into the Various Modes
2.5 Longitudinal Plasma Wave
2.6 Transverse Electromagnetic Waves
2.7 Temporal Damping of the Transverse Electromagnetic Waves
2.8 Cyclotron Damping of the RCP Transverse Wave
2.9 Instabilities in the RCP Transverse Wave
3. Wave Propagation Across the Magnetostatic Field in a Hot Plasma       534
3.1 Solution of the Linearized Vlasov Equation
3.2 Current Density and the Conductivity Tensor
3.3 Evaluation of the Integrals
3.4 Separation into the Various Modes
3.5 Dispersion Relations
3.6 The Quasistatic Mode
3.7 The TEM Mode
4. Summary                                                               552
4.1 Propagation Along B0 in Hot Magnetoplasmas
4.2 Propagation Across B0 in Hot Magnetoplasmas
Problems                                                                 554

20 PARTICLE INTERACTIONS IN PLASMAS
1. Introduction                                                          560
2. Binary Collisions                                                     561
3. Dynamics of Binary Collisions                                         566
4. Evaluation of the Scattering Angle                                    569
4.1 Two Perfectly Elastic Hard Spheres
4.2 Coulomb Interaction Potential
5. Cross Sections                                                        572
5.1 Differential Scattering Cross Section
5.2 Total Scattering Cross Section
5.3 Momentum Transfer Cross Section
6. Cross Sections for the Hard Sphere Model                              578
6.1 Differential Scattering Cross Section
6.2 Total Scattering Cross Section
6.3 Momentum Transfer Cross Section
7. Cross Sections for the Coulomb Potential                              580
7.1 Differential Scattering Cross Section
7.2 Total Scattering Cross Section
7.3 Momentum Transfer Cross Section
8. Screening of the Coulomb Potential                                    582
Problems 586

21 THE BOLTZMANN AND THE FOKKER-PLANCK EQUATIONS
1. Introduction                                                          589
2. The Boltzmann Equation                                                590
2.1 Derivation of the Boltzmann Collision Integral
2.2 Jacobian of the Transformation
2.3 Assumptions in the Derivation of the Boltzmann Collision Integral
2.4 Rate of Change of a Physical Quantity as a Result of Collisions
3. The Boltzmann’s H Function                                            598
3.1 Boltzmann’s H Theorem
3.2 Analysis of Boltzmann’s H Theorem
3.3 Maximum Entropy or Minimum H Approach for Deriving the Equilibrium
Distribution Function
3.4 Mixture of Various Particle Species
4. Boltzmann Collision Term for a Weakly Ionized Plasma                  607
4.1 Spherical Harmonic Expansion of the Distribution Function
4.2 Approximate Expression for the Boltzmann Collision Term
4.3 Rate of Change of Momentum Due to Collisions
5. The Fokker-Planck Equation                                            612
5.1 Derivation of the Fokker-Planck Collision Term
5.2 The Fokker-Planck Coefficients for Coulomb Interactions
5.3 Application to Electron-Ion Collisions
Problems                                                                 621

22 TRANSPORT PROCESSES IN PLASMAS
1. Introduction                                                          628
2. Electric Conductivity in a Nonmagnetized Plasma                       629
2.1 Solution of the Boltzmann Equation
2.2 Electric Current Density and Conductivity
2.3 Conductivity for Maxwellian Distribution Function
3. Electric Conductivity in a Magnetized Plasma                          634
3.1 Solution of Boltzmann Equation
3.2 Electric Current Density and Conductivity
4. Free Diffusion                                                        640
4.1 Perturbation Distribution Function
4.2 Particle Flux
4.3 Free Diffusion Coefficient
5. Diffusion in a Magnetic Field                                         643
5.1 Solution of Boltzmann Equation
5.2 Particle Flux and Diffusion Coefficients
6. Heat Flow                                                             647
6.1 General Expression for the Heat Flow Vector
6.2 Thermal Conductivity for a Constant Kinetic Pressure
6.3 Thermal Conductivity for the Adiabatic Case
Problems                                                                 650

APPENDIX A Useful Vector Relations                                       655
APPENDIX B Useful Relations in Cartesian and in Curvilinear Coordinates  658
APPENDIX C Physical Constants (MKSA)                                     662
APPENDIX D Conversion Factors for Physical Units                         663
APPENDIX E Some Important Plasma Parameters                              664
APPENDIX F Approximate Magnitudes in Some Typical Plasmas                667
INDEX                                                                    669
Publisher said:
Fundamentals of Plasma Physics is a comprehensive textbook designed to present a logical and unified treatment of the fundamentals of plasma physics based on statistical kinetic theory, with applications to a variety of important plasma phenomena. The clarity and completeness of the text makes it suitable for self-learning.

Throughout the text the emphasis is on clarity, rather than formality. The various derivations are explained in detail and, wherever possible, the physical interpretations are emphasized. The mathematical treatment is set out in great detail, carrying out steps that are usually left to the reader. The problems form an integral part of the text and most of them were designed in such a way as to provide a guideline for the student, stating intermediate steps with answers.

The book is intended primarily for advanced undergraduate and first year graduate students meeting the subject of plasma physics for the first time and is suitable for those who have taken classical mechanics, electrodynamics and mathematics beyond sophomore level.

It is a valuable compendium for any serious student of plasma physics at the level of research student or research worker and it is also of interest to researchers in other related fields, such as space physics and applied electromagnetism.
Publisher's webpage - http://www.springer.com/physics/atomic,+molecular,+optical+&+plasma+physics/book/978-0-387-20975-3

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