Introduction to Plasma Physics

I.H.Hutchinson

© 2001

Contents

1  Introduction
    1.1  What is a Plasma?
        1.1.1  An ionized gas
        1.1.2  Plasmas are Quasi-Neutral
    1.2  Plasma Shielding
        1.2.1  Elementary Derivation of the Boltzmann Distribution
        1.2.2  Plasma Density in Electrostatic Potential
        1.2.3  Debye Shielding
        1.2.4  Plasma-Solid Boundaries  (Elementary)
        1.2.5  Thickness of the sheath
    1.3  The `Plasma Parameter'
    1.4  Summary
    1.5  Occurrence of Plasmas
    1.6  Different Descriptions of Plasma
        1.6.1  Equations of Plasma Physics
        1.6.2  Self Consistency
2  Motion of Charged Particles in Fields
    2.1  Uniform B field, E = 0.
        2.1.1  Qualitatively
        2.1.2  By Vector Algebra
    2.2  Uniform B and non-zero E
        2.2.1  Drift due to Gravity or other Forces
    2.3  Non-Uniform B Field
    2.4  Curvature Drift
        2.4.1  Vacuum Fields
    2.5  Interlude: Toroidal Confinement of Single Particles
        2.5.1  How to solve this problem?
        2.5.2  The Solution: Rotational Transform
    2.6  The Mirror Effect of Parallel Field Gradients: E = 0, ∇B ||B
        2.6.1  Force on an Elementary Magnetic Moment Circuit
        2.6.2  μ is a constant of the motion
        2.6.3  Mirror Trapping
        2.6.4  Pitch Angle    θ
        2.6.5  Other Features of Mirror Motions
    2.7  Time Varying B Field     (E inductive)
    2.8  Time Varying E-field     (E, B uniform)
        2.8.1  Direct Derivation of [(dE)/dt] effect: `Polarization Drift'
    2.9  Non Uniform E    (Finite Larmor Radius)
    2.10  Summary of Drifts
3  Collisions in Plasmas
    3.1  Binary collisions between charged particles
        3.1.1  Frames of Reference
        3.1.2  Scattering Angle
    3.2  Differential Cross-Section for Scattering by Angle
    3.3  Relaxation Processes
        3.3.1  Energy Loss
        3.3.2  Cut-offs Estimates
        3.3.3  Momentum Loss
        3.3.4  `Random Walk' in angle
        3.3.5  Summary of different types of collision
    3.4  Thermal Distribution Collisions
        3.4.1  e → i
        3.4.2  i → e
        3.4.3  i → i
        3.4.4  e → e
        3.4.5  Summary of Thermal Collision Frequencies
    3.5  Applications of Collision Analysis
        3.5.1  Energetic (`Runaway') Electrons
        3.5.2  Plasma Resistivity (DC)
        3.5.3  Diffusion
        3.5.4  Energy Equilibration
    3.6  Some Orders of Magnitude
4  Fluid Description of Plasma
    4.1  Particle Conservation (In 3-d Space)
    4.2  Fluid Motion
        4.2.1  Lagrangian & Eulerian Viewpoints
        4.2.2  Momentum (Conservation) Equation
        4.2.3  Pressure Force
        4.2.4  Momentum Equation: Eulerian Viewpoint
        4.2.5  Effect of Collisions
    4.3  The Key Question for Momentum Equation:
    4.4  Summary of Two-Fluid Equations
    4.5  Two-Fluid Equilibrium: Diamagnetic Current
    4.6  Reduction of Fluid Approach to the Single Fluid Equations
        4.6.1  Summary of Single Fluid Equations: M.H.D.
        4.6.2  Heuristic Derivation/Explanation
        4.6.3  Maxwell's Equations for MHD Use
    4.7  MHD Equilibria
        4.7.1  θ-pinch
        4.7.2  Z-pinch
        4.7.3  `Stabilized Z-pinch'
    4.8  Some General Properties of MHD Equilibria
        4.8.1  Pressure & Tension
        4.8.2  Magnetic Surfaces
        4.8.3  `Current Surfaces'
        4.8.4  Low β equilibria: Force-Free Plasmas
    4.9  Toroidal Equilibrium
    4.10  Plasma Dynamics (MHD)
    4.11  Flux Conservation
    4.12  Field Line Motion
    4.13  MHD Stability
    4.14  General Perturbations of Cylindrical Equil.
    4.15  General Principles Governing Instabilities
    4.16  Quick and Simple Analysis of Pinches
5  Electromagnetic Waves in Plasmas
    5.1  General Treatment of Linear Waves in Anisotropic Medium
        5.1.1  Simple Case. Isotropic Medium
        5.1.2  General Case     (k in z-direction)
    5.2  High Frequency Plasma Conductivity
        5.2.1  Zero B-field case
        5.2.2  Meaning of Negative N2: Cut Off
    5.3  Cold Plasma Waves      (Magnetized Plasma)
        5.3.1  Derivation of Dispersion Relation
        5.3.2  Hybrid Resonances    Perpendicular Propagation
        5.3.3  Whistlers
    5.4  Thermal Effects on Plasma Waves
        5.4.1  Refractive Index Plot
        5.4.2  Including the ion response
    5.5  Electrostatic Approximation for (Plasma) Waves
    5.6  Simple Example of MHD Dynamics: Alfven Waves
    5.7  Non-Uniform Plasmas and wave propagation
    5.8  Two Stream Instability
    5.9  Kinetic Theory of Plasma Waves
        5.9.1  Vlasov Equation
        5.9.2  Linearized Wave Solution of Vlasov Equation
        5.9.3  Landau's original approach.    (1946)
        5.9.4  Solution of Dispersion Relation
        5.9.5  Direct Calculation of Collisionless Particle Heating
        5.9.6  Physical Picture
        5.9.7  Damping Mechanisms
        5.9.8  Ion Acoustic Waves and Landau Damping
        5.9.9  Alternative expressions of Dielectric Tensor Elements
        5.9.10  Electromagnetic Waves in unmagnetized Vlasov Plasma
    5.10  Experimental Verification of Landau Damping

Preface

These are transcriptions of the notes from which I teach the single semester course "Introduction to Plasma Physics". Despite the heroic efforts of Valerie Censabella (for which I am very grateful) to translate my hand-written materials into LaTeX, and extensive editing on my part, I don't doubt that there are many typographical errors. Moreover, since they are only notes, don't look for limpid prose, and realize that the detailed explanations are in my mind and orally in class, not all here.
The course is intended only as a first plasma physics course, but includes what I take to be the critical concepts needed for a foundation for further study. A solid undergraduate background in classical physics, electromagnetic theory including Maxwell's equations, and mathematical familiarity with partial differential equations and complex analysis are prerequisites.
Ian Hutchinson.

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