1 INTRODUCTION
1 Introduction
1.1 Sources
The major sources for this course are:
The theory of plasma waves: T.H. Stix, 1st edition (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 (Perseus,
Reading MA, 1998).
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 edition (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, Ger-
many, 1972).
Solar system magnetic fields: edited by E.R. Priest (D. Reidel Publishing Co., Dor-
drecht, Netherlands, 1985).
Lectures on solar and planetary dynamos: edited by M.R.E. Proctor, and A.D. Gilbert
(Cambridge University Press, Cambridge, UK, 1994).
5
1.2 What is plasma? 1 INTRODUCTION
Introduction to plasma physics: R.J. Goldston, and P.H. Rutherford (Institute of Physics
Publishing, Bristol, UK, 1995).
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, both disciplines d eveloped to understand essentially 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 simil arly 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 fie lds. 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.
6
1.3 A brief history of plasma physics 1 INTRODUCTION
Plasmas resulting from ionization of neutral gases generally contain equal
numbers of positive and negative charge carriers. In this situation, the oppo-
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 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 A 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
7
1.3 A brief history of plasma physics 1 INTRODUCTION
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-
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 theory 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 Lang-
muir waves. This was the genesis of plasma physics. Interestingly enough, Lang-
muir’s research nowadays forms the theoretical basis of most plasma processing
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 diff erent 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, s ome 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 a non-uniform magnetized plasma.
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
8
1.3 A brief history of plasma physics 1 INTRODUCTION
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
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 imme nsely 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
9
1.4 Basic parameters 1 INTRODUCTION
target, material is immediately ablated, and a plas ma forms at the boundary
between the beam and the target. Laser plasmas tend to have fairly extreme
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 m
e
and charge −e (here, e denotes the magnitude of the electron charge ),
and ions, with mass m
i
and charge +e. We do not necessarily demand that the
system has attained thermal equilibrium, but nevertheless use the symbol
T
s
≡
1
3
m
s
v
2
(1.1)
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 measur ed in
electron-volts (1 joule is equivalent to 6.24 ×10
18
eV).
Quasi-neutrality demands that
n
i
n
e
≡ n, (1.2)
where n
s
is the number density (i.e., the number of particles per cubic meter) of
species s.
10
1.5 The plasma frequency 1 INTRODUCTION
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
speeds via the so-called thermal speed,
v
ts
≡
2 T/m
s
. (1.3)
Note that the ion thermal speed is usually far smaller than the electron thermal
speed:
v
ti
∼
m
e
/m
i
v
te
. (1.4)
Of course, n and T are generally functions of position in a plasma.
1.5 The plasma frequency
The plasma frequency,
ω
2
p
=
n e
2
0
m
, (1.5)
is the most fundamental time-scale in plasma physics. Clearly, there is a different
plasma frequency for each species. However, the relatively fast electron frequency
is, by far, the most important, and references to “the plasma frequency” in text-
books invariably mean the electron plasma frequency.
It is easily seen that ω
p
corresponds to the typical electrostatic oscillation fre-
quency of a given species in response to a small charge separation. For instance,
consider a one-dimensional situation in which a slab consisting entirely of one
charge species is displaced from its quasi-neutral position by an infinitesimal dis-
tance δx. The resulting charge density which develops on the leading face of the
slab is σ = e n δx. An equal and opposite charge density develops on the oppo-
site face. The x-directed electric field generated inside the sl ab is of magnitude
E
x
= −σ/
0
= −e n δx/
0
. Thus, Newton’s law applied to an individual particle
inside the slab yields
m
d
2
δx
dt
2
= e E
x
= −m ω
2
p
δx, (1.6)
giving δx = (δx)
0
cos (ω
p
t).
11
1.6 Debye shielding 1 INTRODUCTION
Note that plasma oscillations will only be observed if the plasma system is
studied over time periods τ longer than the plasma period τ
p
≡ 1/ω
p
, and if
external actions change the system at a rate no faster than ω
p
. In the opposite
case, one is clearly studying something other than plasma physics (e.g., nuclear
reactions), and the system cannot not usefully be considered to be a plasma. Like-
wise, observations over length-scales L shorter than the distance v
t
τ
p
traveled by
a typical plasma particle during a plasma period will also not detect plasma be-
haviour. In this case, particles will exit the system before completing a plasma
oscillation. This distance, which is the spatial equivalent to τ
p
, is called the Debye
length, and takes the form
λ
D
≡
T/m ω
−1
p
. (1.7)
Note that
λ
D
=
0
T
n e
2
(1.8)
is independent of mass, and therefore generally comparable for diff erent species.
Clearly, our idealized system can only usefully be considered to be a plasma
provided that
λ
D
L
1, (1.9)
and
τ
p
τ
1. (1.10)
Here, τ and L represent the typical time-scale and length-scale of the process
under investigation.
It should be noted that, despite the conventional requirement (1.9), plasma
physics is capable of considering structures on the Debye scale. The most impor-
tant example of this is the Debye sheath: i.e., the boundary layer which surrounds
a plasma confined by a material surface.
1.6 Debye shielding
Plasmas generally do not contain strong electric fields in their rest frames. The
shielding of an external electric field from the interior of a plasma can be viewed
12
1.6 Debye shielding 1 INTRODUCTION
as a result of high plasma conductivity: plasma current generally flows freely
enough to short out interior electric fields. However, it is more useful to consider
the shielding as a dielectric phenomena: i.e., it is the polarization of the plasma
medium, and the associated redistribution of space charge, which prevents pen-
etration by an external electric field. Not surprisingly, the length-scale associated
with such shielding is the Debye length.
Let us consider the simplest possible example. Suppose that a quasi-neutral
plasma is sufficiently c lose to thermal equilibrium that its particle densities are
distributed according to the Maxwell-Boltzmann law,
n
s
= n
0
e
−e
s
Φ/T
, (1.11)
where Φ(r) is the electrostatic potential, and n
0
and T are constant. From e
i
=
−e
e
= e, it is clear that quasi-neutrality requires the equilibrium potential to be a
constant. Suppose that this equilibrium potential is perturbed, by an amount δΦ,
by a small, localized charge density δρ
ext
. The total perturbed charge density is
written
δρ = δρ
ext
+ e (δn
i
− δn
e
) = δρ
ext
− 2 e
2
n
0
δΦ/T. (1.12)
Thus, Poisson’s equation yields
∇
2
δΦ = −
δρ
0
= −
δρ
ext
− 2 e
2
n
0
δΦ/T
0
, (1.13)
which reduces to
∇
2
−
2
λ
2
D
δΦ = −
δρ
ext
0
. (1.14)
If the perturbing charge density actually consists of a point charge q, located
at the origin, so that δρ
ext
= q δ(r), then the solution to the above equation is
written
δΦ(r) =
q
4π
0
r
e
−
√
2 r/λ
D
. (1.15)
Clearly, the Coulomb potential of the perturbing point charge q is shielded on
distance scales longer than the Debye length by a shielding cloud of approximate
radius λ
D
consisting of charge of the opposite sign.
13
1.7 The plasma parameter 1 INTRODUCTION
Note that the above argument, by treating n as a continuous function, implic-
itly assumes that there are many particles in the shielding cloud. Actually, Debye
shielding remains statistically significant, and physical, in the opposite limit in
which the cl oud is barely populated. In the latter case, it is the probability of ob-
serving charged particles within a Debye length of the perturbing charge which
is modified.
1.7 The plasma parameter
Let us define the average distance between particles,
r
d
≡ n
−1/3
, (1.16)
and the distance of closest approach,
r
c
≡
e
2
4π
0
T
. (1.17)
Recall that r
c
is the distance at which the Coulomb energy
U(r, v) =
1
2
mv
2
−
e
2
4π
0
r
(1.18)
of one charged particle in the electrostatic field of another vanishes. Thus, U(r
c
, v
t
) =
0.
The significance of the ratio r
d
/r
c
is readily understood. When this ratio is
small, charged particles are dominated by one another’s electrostatic influence
more or less continuously, and their kinetic energies are small compared to the
interaction potential energies. Such plasmas are termed strongly coupled. On the
other hand, when the ratio is large, strong electrostatic interactions between in-
dividual particles are occasional and relatively rare events. A typical particle is
electrostatically influenced by all of the other particles within its Debye sphere,
but this interaction very rarely causes any sudden change in its motion. Such plas-
mas are termed weakly coupled. It is possible to describe a weakly coupled plasma
using a s tandard Fokker-Planck equation (i.e., the same type of equation as is con-
ventionally used to describe a neutral gas). Understanding the strongly coupled
14
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