The Physics of Plasmas
Richard Fitzpatrick1
Professor of Physics
The University of Texas at Austin
1In association with R.D. Hazeltine and F.L. Waelbroeck.

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Contents
1 Introduction
5
1.1
Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2
What is Plasma? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.3
Brief History of Plasma Physics . . . . . . . . . . . . . . . . . . . .
7
1.4
Basic Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5
Plasma Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6
Debye Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7
Plasma Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.8
Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.9
Magnetized Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.10 Plasma Beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Charged Particle Motion
21
2.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2
Motion in Uniform Fields . . . . . . . . . . . . . . . . . . . . . . . 22
2.3
Method of Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4
Guiding Centre Motion . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5
Magnetic Drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6
Invariance of Magnetic Moment . . . . . . . . . . . . . . . . . . . . 32
2.7
Poincar´
e Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.8
Adiabatic Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.9
Magnetic Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.10 Van Allen Radiation Belts . . . . . . . . . . . . . . . . . . . . . . . 38
2.11 Ring Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.12 Second Adiabatic Invariant . . . . . . . . . . . . . . . . . . . . . . 46
2.13 Third Adiabatic Invariant . . . . . . . . . . . . . . . . . . . . . . . 48
2.14 Motion in Oscillating Fields . . . . . . . . . . . . . . . . . . . . . . 50
3 Plasma Fluid Theory
53
3.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2
Moments of the Distribution Function . . . . . . . . . . . . . . . . 56
3.3
Moments of the Collision Operator . . . . . . . . . . . . . . . . . . 58
3.4
Moments of the Kinetic Equation . . . . . . . . . . . . . . . . . . . 61
2

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3.5
Fluid Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6
Entropy Production . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.7
Fluid Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.8
Braginskii Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.9
Normalization of the Braginskii Equations . . . . . . . . . . . . . . 85
3.10 Cold-Plasma Equations . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.11 MHD Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.12 Drift Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.13 Closure in Collisionless Magnetized Plasmas . . . . . . . . . . . . . 100
3.14 Langmuir Sheaths . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4 Waves in Cold Plasmas
111
4.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2
Plane Waves in a Homogeneous Plasma . . . . . . . . . . . . . . . 111
4.3
Cold-Plasma Dielectric Permittivity . . . . . . . . . . . . . . . . . . 113
4.4
Cold-Plasma Dispersion Relation . . . . . . . . . . . . . . . . . . . 116
4.5
Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.6
Cutoff and Resonance . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.7
Waves in an Unmagnetized Plasma . . . . . . . . . . . . . . . . . . 120
4.8
Low-Frequency Wave Propagation . . . . . . . . . . . . . . . . . . 122
4.9
Parallel Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . 125
4.10 Perpendicular Wave Propagation . . . . . . . . . . . . . . . . . . . 130
4.11 Wave Propagation Through Inhomogeneous Plasmas . . . . . . . . 133
4.12 Cutoffs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.13 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.14 Resonant Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.15 Collisional Damping . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.16 Pulse Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.17 Ray Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.18 Radio Wave Propagation Through the Ionosphere . . . . . . . . . . 155
5 Magnetohydrodynamic Fluids
158
5.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.2
Magnetic Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.3
Flux Freezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
3

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5.4
MHD Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.5
The Solar Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.6
Parker Model of Solar Wind . . . . . . . . . . . . . . . . . . . . . . 170
5.7
Interplanetary Magnetic Field . . . . . . . . . . . . . . . . . . . . . 174
5.8
Mass and Angular Momentum Loss . . . . . . . . . . . . . . . . . . 180
5.9
MHD Dynamo Theory . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.10 Homopolar Generators . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.11 Slow and Fast Dynamos . . . . . . . . . . . . . . . . . . . . . . . . 189
5.12 Cowling Anti-Dynamo Theorem . . . . . . . . . . . . . . . . . . . . 191
5.13 Ponomarenko Dynamos . . . . . . . . . . . . . . . . . . . . . . . . 195
5.14 Magnetic Reconnection . . . . . . . . . . . . . . . . . . . . . . . . 200
5.15 Linear Tearing Mode Theory . . . . . . . . . . . . . . . . . . . . . . 202
5.16 Nonlinear Tearing Mode Theory . . . . . . . . . . . . . . . . . . . . 211
5.17 Fast Magnetic Reconnection . . . . . . . . . . . . . . . . . . . . . . 213
5.18 MHD Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
5.19 Parallel Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
5.20 Perpendicular Shocks
. . . . . . . . . . . . . . . . . . . . . . . . . 225
5.21 Oblique Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
6 Waves in Warm Plasmas
232
6.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
6.2
Landau Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
6.3
Physics of Landau Damping . . . . . . . . . . . . . . . . . . . . . . 242
6.4
Plasma Dispersion Function . . . . . . . . . . . . . . . . . . . . . . 244
6.5
Ion Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
6.6
Waves in Magnetized Plasmas . . . . . . . . . . . . . . . . . . . . . 248
6.7
Parallel Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . 254
6.8
Perpendicular Wave Propagation . . . . . . . . . . . . . . . . . . . 256
4

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1
INTRODUCTION
1
Introduction
1.1
Sources
The major sources for this course are:
The Theory of Plasma Waves: T.H. Stix, 1st Ed. (McGraw-Hill, New York NY, 1962).
Plasma Physics: R.A. Cairns (Blackie, Glasgow UK, 1985).
The Framework of Plasma Physics: R.D. Hazeltine, and F.L. Waelbroeck (Westview,
Boulder CO, 2004).
Other sources include:
The Mathematical Theory of Non-Uniform Gases: S. Chapman, and T.G. Cowling (Cam-
bridge University Press, Cambridge UK, 1953).
Physics of Fully Ionized Gases: L. Spitzer, Jr., 1st Ed. (Interscience, New York NY,
1956).
Radio Waves in the Ionosphere: K.G. Budden (Cambridge University Press, Cam-
bridge UK, 1961).
The Adiabatic Motion of Charged Particles: T.G. Northrop (Interscience, New York
NY, 1963).
Coronal Expansion and the Solar Wind: A.J. Hundhausen (Springer-Verlag, Berlin,
1972).
Solar System Magnetic Fields: E.R. Priest, Ed. (D. Reidel Publishing Co., Dordrecht,
Netherlands, 1985).
Lectures on Solar and Planetary Dynamos: M.R.E. Proctor, and A.D. Gilbert, Eds.
(Cambridge University Press, Cambridge UK, 1994).
Introduction to Plasma Physics: R.J. Goldston, and P.H. Rutherford (Institute of
Physics Publishing, Bristol UK, 1995).
5

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1.2
What is Plasma?
1
INTRODUCTION
Basic Space Plasma Physics: W. Baumjohann, and R. A. Treumann (Imperial Col-
lege Press, London UK, 1996).
1.2
What is Plasma?
The electromagnetic force is generally observed to create structure: e.g., stable
atoms and molecules, crystalline solids. In fact, the most widely studied conse-
quences of the electromagnetic force form the subject matter of Chemistry and
Solid-State Physics, which are both disciplines developed to understand essen-
tially static structures.
Structured systems have binding energies larger than the ambient thermal en-
ergy. Placed in a sufficiently hot environment, they decompose: e.g., crystals
melt, molecules disassociate. At temperatures near or exceeding atomic ioniza-
tion energies, atoms similarly decompose into negatively charged electrons and
positively charged ions. These charged particles are by no means free: in fact,
they are strongly affected by each others’ electromagnetic fields. Nevertheless,
because the charges are no longer bound, their assemblage becomes capable of
collective motions of great vigor and complexity. Such an assemblage is termed a
plasma.
Of course, bound systems can display extreme complexity of structure: e.g.,
a protein molecule. Complexity in a plasma is somewhat different, being ex-
pressed temporally as much as spatially. It is predominately characterized by the
excitation of an enormous variety of collective dynamical modes.
Since thermal decomposition breaks interatomic bonds before ionizing, most
terrestrial plasmas begin as gases. In fact, a plasma is sometimes defined as a gas
that is sufficiently ionized to exhibit plasma-like behaviour. Note that plasma-
like behaviour ensues after a remarkably small fraction of the gas has undergone
ionization. Thus, fractionally ionized gases exhibit most of the exotic phenomena
characteristic of fully ionized gases.
Plasmas resulting from ionization of neutral gases generally contain equal
numbers of positive and negative charge carriers. In this situation, the oppo-
6

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1.3
Brief History of Plasma Physics
1
INTRODUCTION
sitely charged fluids are strongly coupled, and tend to electrically neutralize one
another on macroscopic length-scales. Such plasmas are termed quasi-neutral
(“quasi” because the small deviations from exact neutrality have important dy-
namical consequences for certain types of plasma mode). Strongly non-neutral
plasmas, which may even contain charges of only one sign, occur primarily in
laboratory experiments: their equilibrium depends on the existence of intense
magnetic fields, about which the charged fluid rotates.
It is sometimes remarked that 95% (or 99%, depending on whom you are
trying to impress) of the baryonic content of the Universe consists of plasma. This
statement has the double merit of being extremely flattering to Plasma Physics,
and quite impossible to disprove (or verify). Nevertheless, it is worth pointing out
the prevalence of the plasma state. In earlier epochs of the Universe, everything
was plasma. In the present epoch, stars, nebulae, and even interstellar space, are
filled with plasma. The Solar System is also permeated with plasma, in the form
of the solar wind, and the Earth is completely surrounded by plasma trapped
within its magnetic field.
Terrestrial plasmas are also not hard to find. They occur in lightning, fluores-
cent lamps, a variety of laboratory experiments, and a growing array of industrial
processes. In fact, the glow discharge has recently become the mainstay of the
micro-circuit fabrication industry. Liquid and even solid-state systems can oc-
casionally display the collective electromagnetic effects that characterize plasma:
e.g., liquid mercury exhibits many dynamical modes, such as Alfv´
en waves, which
occur in conventional plasmas.
1.3
Brief History of Plasma Physics
When blood is cleared of its various corpuscles there remains a transparent liquid,
which was named plasma (after the Greek word πλασµα, which means “mold-
able substance” or “jelly”) by the great Czech medical scientist, Johannes Purkinje
(1787-1869). The Nobel prize winning American chemist Irving Langmuir first
used this term to describe an ionized gas in 1927—Langmuir was reminded of
the way blood plasma carries red and white corpuscles by the way an electri-
7

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1.3
Brief History of Plasma Physics
1
INTRODUCTION
fied fluid carries electrons and ions. Langmuir, along with his colleague Lewi
Tonks, was investigating the physics and chemistry of tungsten-filament light-
bulbs, with a view to finding a way to greatly extend the lifetime of the filament
(a goal which he eventually achieved). In the process, he developed the the-
ory of plasma sheaths—the boundary layers which form between ionized plasmas
and solid surfaces. He also discovered that certain regions of a plasma discharge
tube exhibit periodic variations of the electron density, which we nowadays term
Langmuir waves. This was the genesis of Plasma Physics. Interestingly enough,
Langmuir’s research nowadays forms the theoretical basis of most plasma process-
ing techniques for fabricating integrated circuits. After Langmuir, plasma research
gradually spread in other directions, of which five are particularly significant.
Firstly, the development of radio broadcasting led to the discovery of the
Earth’s ionosphere, a layer of partially ionized gas in the upper atmosphere which
reflects radio waves, and is responsible for the fact that radio signals can be re-
ceived when the transmitter is over the horizon. Unfortunately, the ionosphere
also occasionally absorbs and distorts radio waves. For instance, the Earth’s mag-
netic field causes waves with different polarizations (relative to the orientation
of the magnetic field) to propagate at different velocities, an effect which can
give rise to “ghost signals” (i.e., signals which arrive a little before, or a little
after, the main signal). In order to understand, and possibly correct, some of
the deficiencies in radio communication, various scientists, such as E.V. Appleton
and K.G. Budden, systematically developed the theory of electromagnetic wave
propagation through non-uniform magnetized plasmas.
Secondly, astrophysicists quickly recognized that much of the Universe con-
sists of plasma, and, thus, that a better understanding of astrophysical phenom-
ena requires a better grasp of plasma physics. The pioneer in this field was
Hannes Alfv´
en, who around 1940 developed the theory of magnetohydrodyamics,
or MHD, in which plasma is treated essentially as a conducting fluid. This theory
has been both widely and successfully employed to investigate sunspots, solar
flares, the solar wind, star formation, and a host of other topics in astrophysics.
Two topics of particular interest in MHD theory are magnetic reconnection and
dynamo theory. Magnetic reconnection is a process by which magnetic field-lines
suddenly change their topology: it can give rise to the sudden conversion of a
8

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1.3
Brief History of Plasma Physics
1
INTRODUCTION
great deal of magnetic energy into thermal energy, as well as the acceleration of
some charged particles to extremely high energies, and is generally thought to be
the basic mechanism behind solar flares. Dynamo theory studies how the motion
of an MHD fluid can give rise to the generation of a macroscopic magnetic field.
This process is important because both the terrestrial and solar magnetic fields
would decay away comparatively rapidly (in astrophysical terms) were they not
maintained by dynamo action. The Earth’s magnetic field is maintained by the
motion of its molten core, which can be treated as an MHD fluid to a reasonable
approximation.
Thirdly, the creation of the hydrogen bomb in 1952 generated a great deal
of interest in controlled thermonuclear fusion as a possible power source for the
future. At first, this research was carried out secretly, and independently, by the
United States, the Soviet Union, and Great Britain. However, in 1958 thermonu-
clear fusion research was declassified, leading to the publication of a number
of immensely important and influential papers in the late 1950’s and the early
1960’s. Broadly speaking, theoretical plasma physics first emerged as a math-
ematically rigorous discipline in these years. Not surprisingly, Fusion physicists
are mostly concerned with understanding how a thermonuclear plasma can be
trapped—in most cases by a magnetic field—and investigating the many plasma
instabilities which may allow it to escape.
Fourthly, James A. Van Allen’s discovery in 1958 of the Van Allen radiation
belts surrounding the Earth, using data transmitted by the U.S. Explorer satellite,
marked the start of the systematic exploration of the Earth’s magnetosphere via
satellite, and opened up the field of space plasma physics. Space scientists bor-
rowed the theory of plasma trapping by a magnetic field from fusion research,
the theory of plasma waves from ionospheric physics, and the notion of magnetic
reconnection as a mechanism for energy release and particle acceleration from
astrophysics.
Finally, the development of high powered lasers in the 1960’s opened up the
field of laser plasma physics. When a high powered laser beam strikes a solid
target, material is immediately ablated, and a plasma forms at the boundary
between the beam and the target. Laser plasmas tend to have fairly extreme
9

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1.4
Basic Parameters
1
INTRODUCTION
properties (e.g., densities characteristic of solids) not found in more conventional
plasmas. A major application of laser plasma physics is the approach to fusion
energy known as inertial confinement fusion. In this approach, tightly focused
laser beams are used to implode a small solid target until the densities and tem-
peratures characteristic of nuclear fusion (i.e., the centre of a hydrogen bomb)
are achieved. Another interesting application of laser plasma physics is the use
of the extremely strong electric fields generated when a high intensity laser pulse
passes through a plasma to accelerate particles. High-energy physicists hope to
use plasma acceleration techniques to dramatically reduce the size and cost of
particle accelerators.
1.4
Basic Parameters
Consider an idealized plasma consisting of an equal number of electrons, with
mass me and charge −e (here, e denotes the magnitude of the electron charge),
and ions, with mass mi and charge +e. We do not necessarily demand that the
system has attained thermal equilibrium, but nevertheless use the symbol
1
Ts ≡ ms v2
(1.1)
3
s
to denote a kinetic temperature measured in energy units (i.e., joules). Here, v is a
particle speed, and the angular brackets denote an ensemble average. The kinetic
temperature of species s is essentially the average kinetic energy of particles of
this species. In plasma physics, kinetic temperature is invariably measured in
electron-volts (1 joule is equivalent to 6.24 × 1018 eV).
Quasi-neutrality demands that
ni ≃ ne ≡ n,
(1.2)
where ns is the number density (i.e., the number of particles per cubic meter) of
species s.
Assuming that both ions and electrons are characterized by the same T (which
is, by no means, always the case in plasmas), we can estimate typical particle
10

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Document Outline

• Introduction
  • Sources
  • What is Plasma?
  • Brief History of Plasma Physics
  • Basic Parameters
  • Plasma Frequency
  • Debye Shielding
  • Plasma Parameter
  • Collisions
  • Magnetized Plasmas
  • Plasma Beta
• Charged Particle Motion
  • Introduction
  • Motion in Uniform Fields
  • Method of Averaging
  • Guiding Centre Motion
  • Magnetic Drifts
  • Invariance of Magnetic Moment
  • Poincaré Invariants
  • Adiabatic Invariants
  • Magnetic Mirrors
  • Van Allen Radiation Belts
  • Ring Current
  • Second Adiabatic Invariant
  • Third Adiabatic Invariant
  • Motion in Oscillating Fields
• Plasma Fluid Theory
  • Introduction
  • Moments of the Distribution Function
  • Moments of the Collision Operator
  • Moments of the Kinetic Equation
  • Fluid Equations
  • Entropy Production
  • Fluid Closure
  • Braginskii Equations
  • Normalization of the Braginskii Equations
  • Cold-Plasma Equations
  • MHD Equations
  • Drift Equations
  • Closure in Collisionless Magnetized Plasmas
  • Langmuir Sheaths
• Waves in Cold Plasmas
  • Introduction
  • Plane Waves in a Homogeneous Plasma
  • Cold-Plasma Dielectric Permittivity
  • Cold-Plasma Dispersion Relation
  • Polarization
  • Cutoff and Resonance
  • Waves in an Unmagnetized Plasma
  • Low-Frequency Wave Propagation
  • Parallel Wave Propagation
  • Perpendicular Wave Propagation
  • Wave Propagation Through Inhomogeneous Plasmas
  • Cutoffs
  • Resonances
  • Resonant Layers
  • Collisional Damping
  • Pulse Propagation
  • Ray Tracing
  • Radio Wave Propagation Through the Ionosphere
• Magnetohydrodynamic Fluids
  • Introduction
  • Magnetic Pressure
  • Flux Freezing
  • MHD Waves
  • The Solar Wind
  • Parker Model of Solar Wind
  • Interplanetary Magnetic Field
  • Mass and Angular Momentum Loss
  • MHD Dynamo Theory
  • Homopolar Generators
  • Slow and Fast Dynamos
  • Cowling Anti-Dynamo Theorem
  • Ponomarenko Dynamos
  • Magnetic Reconnection
  • Linear Tearing Mode Theory
  • Nonlinear Tearing Mode Theory
  • Fast Magnetic Reconnection
  • MHD Shocks
  • Parallel Shocks
  • Perpendicular Shocks
  • Oblique Shocks
• Waves in Warm Plasmas
  • Introduction
  • Landau Damping
  • Physics of Landau Damping
  • Plasma Dispersion Function
  • Ion Sound Waves
  • Waves in Magnetized Plasmas
  • Parallel Wave Propagation
  • Perpendicular Wave Propagation

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