Introduction to Plasma Physics:
A graduate level course
Richard Fitzpatrick
1
Associate Professor of Physics
The University of Texas at Austin
1
In association with R.D. Hazeltine and F.L. Waelbroeck.
Contents
1 Introduction 5
1.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 What is plasma? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 A brief history of plasma physics . . . . . . . . . . . . . . . . . . . 7
1.4 Basic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 The plasma frequency . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Debye shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 The plasma parameter . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.8 Collisionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.9 Magnetized plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.10 Plasma beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Charged particle motion 20
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Motion in uniform fields . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Method of averaging . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Guiding centre motion . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Magnetic drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Invariance of the magnetic moment . . . . . . . . . . . . . . . . . 31
2.7 Poincar
´
e invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.8 Adiabatic invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.9 Magnetic mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.10 The Van Allen radiation belts . . . . . . . . . . . . . . . . . . . . . 37
2.11 The ring current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.12 The second adiabatic invariant . . . . . . . . . . . . . . . . . . . . 46
2.13 The third adiabatic invariant . . . . . . . . . . . . . . . . . . . . . 48
2.14 Motion in oscillating fields . . . . . . . . . . . . . . . . . . . . . . . 49
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
3.5 Fluid equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6 Entropy production . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.7 Fluid closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.8 The Braginskii equations . . . . . . . . . . . . . . . . . . . . . . . . 72
3.9 Normalization of the Braginskii equations . . . . . . . . . . . . . . 85
3.10 The cold-plasma equations . . . . . . . . . . . . . . . . . . . . . . 93
3.11 The MHD equations . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.12 The drift equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.13 Closure in collisionless magnetized plasmas . . . . . . . . . . . . . 100
4 Waves in cold plasmas 105
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2 Plane waves in a homogeneous plasma . . . . . . . . . . . . . . . . 105
4.3 The cold-plasma dielectric permittivity . . . . . . . . . . . . . . . . 107
4.4 The cold-plasma dispersion relation . . . . . . . . . . . . . . . . . 110
4.5 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.6 Cutoff and resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.7 Waves in an unmagnetized plasma . . . . . . . . . . . . . . . . . . 114
4.8 Low-frequency wave propagation in a magnetized plasma . . . . Introduction to Atomic Physics Introduction to Atomic Physics Bởi: OpenStaxCollege Individual carbon atoms are visible in this image of a carbon nanotube made by a scanning tunneling electron microscope (credit: Taner Yildirim, National Institute of Standards and Technology, via Wikimedia Commons) From childhood on, we learn that atoms are a substructure of all things around us, from the air we breathe to the autumn leaves that blanket a forest trail Invisible to the eye, the existence and properties of atoms are used to explain many phenomena—a theme found throughout this text In this chapter, we discuss the discovery of atoms and their own substructures; we then apply quantum mechanics to the description of atoms, and their properties and interactions Along the way, we will find, much like the scientists who made the original discoveries, that new concepts emerge with applications far beyond the boundaries of atomic physics 1/1 An Introduction to
GEOMETRICAL PHYSICS
R. Aldrovandi & J.G. Pereira
Instituto de F´ısica Te´orica
State Univers ity of S˜ao Paulo – UNESP
S˜ao Paulo — Brazil
To our parents
Nice, Dina, Jos´e and Tito
i
ii
PREAMBLE: SPACE AND GEOMETRY
What stuff’tis made of, whereof it is born,
I am to learn.
Merchant of Venice
The simplest geometrical setting used — consciously or not — by physi-
cists in their everyday work is the 3-dimensional euclidean space E
3
. It con-
sists of the set R
3
of ordered triples of real numbers such as p = (p
1
, p
2
, p
3
), q
= (q
1
, q
2
, q
3
), etc, and is endowed with a very special characteristic, a metric
defined by the distance function
d(p, q) =
3
i=1
(p
i
− q
i
)
2
1/2
.
It is the space of ordinary human experience and the starting point of our
geometric intuition. Studied for two-and-a-half millenia, it has been the
object of celebrated controversies, the most famous concerning the minimum
number of properties necessary to define it completely.
From Aristotle to Newton, through Galileo and Descartes, the very word
space has been reserved to E
3
. Only in the 19-th century has it become clear
that other, different spaces could be thought of, and mathematicians have
since greatly amused themselves by inventing all kinds of them. For physi-
cists, the age-long debate shifted to another question: how can we recognize,
amongst such innumerable possible spaces, that real space chosen by Nature
as the stage-set of its processes? For example, suppose the space of our ev-
eryday experience consists of the same set R
3
of triples above, but with a
different distance function, such as
d(p, q) =
3
i=1
|p
i
− q
i
|.
This would define a different metric space, in principle as good as that
given above. Were it only a matter of principle, it would be as good as
iii
iv
any other space given by any distance function with R
3
as set point. It so
happens, however, that Nature has chosen the former and not the latter space
for us to live in. To know which one is the real space is not a simple question
of principle — something else is needed. What else? The answer may seem
rather trivial in the case of our home space, though less so in other spaces
singled out by Nature in the many different situations which are objects of
physical study. It was given by Riemann in his famous Inaugural Address
1
:
“ those properties which distinguish Space from other con-
ceivable triply extended quantities can only be deduced from expe-
rience.”
Thus, from experience! It is experiment which tells us in which space we
actually live in. When we measure distances we find them to be independent
of the direction of the straight lines joining the points. And this isotropy
property rules out the second proposed distance function, while admitting
the metric of the euclidean space.
In reality, Riemann’s statement implies an epistemological limitation: it
will never be possible to ascertain exactly which space is the real one. Other
isotropic distance functions are, in principle, admissible and more experi-
ments are necessary to decide between them. In Riemann’s time already
other geometries were known (those found by Lobachevsky and Boliyai) that
could be as similar to the euclidean geometry as we might wish in the re-
stricted regions experience is confined to. In honesty, all we can say is that
E
3
, as a model for our ambient space, is strongly favored by present day
experimental evidence in scales ranging from (say) human dimensions down
to about 10
−15
cm. Our knowledge on smaller scales is limited by our ca-
pacity to probe them. For larger scales, according to General Relativity, the
validity of this model depends on the presence and strength of gravitational
fields: E
3
is good only as long as gravitational fields are very weak.
“ These data are — like all data — not logically necessary,
but only of empirical [...]... programming language Programming languages can be divided into two major categories: low-level languages designed to work with the given hardware, and high-level languages that are not related to any specific hardware Simple machine languages and assembly languages were the only ones available before the development of high-level languages A machine language is typically in binary form and is designed to work... computing and global computing is elucidated in Koniges (2000), Foster and Kesselman (2003), and Abbas (2004) 1.1 Computation and science Modern societies are not the only ones to rely on computation Ancient societies also had to deal with quantifying their knowledge and events It is interesting to see how the ancient societies developed their knowledge of numbers and calculations with different means and tools... of programming and debugging They are more advanced than machine languages because they have adopted symbolic addresses But they are still related to a certain architecture and wiring of the system A translating device called an assembler is needed to convert an assembly code into a native machine code before a computer can recognize the instructions Machine languages and assembly languages do not... discussion on the Fortran language and its applications, see Edgar (1992) The newest version of Fortran, known as Fortran 90, has absorbed many important features for parallel computing Fortran 90 has many extensions over the standard Fortran 77 Most of these extensions are established based on the extensions already adopted by computer manufacturers to enhance their computer performance Efficient compilers... programming languages that are used in scientific computing The longest-running candidate is Fortran (Formula translation), which was introduced in 1957 as one of the earliest high-level languages and is still one of the primary languages in computational science Of course, the Fortran language has evolved from its very early version, known as Fortran 66, to Fortran 77, which has been the most popular language... replace human beings in this regard and the quest for a better understanding of Nature will go on no matter how difficult the journey is Computers will certainly help to make that journey more colorful and pleasant 1.3 Computer algorithms and languages Before we can use a computer to solve a specific problem, we must instruct the computer to follow certain procedures and to carry out the desired computational. .. protocols and environments under various software packages, which we will leave to the readers to discover and explore 1.3 Computer algorithms and languages The other popular programming language for scientific computing is the C programming language Most system programmers and software developers prefer to use C in developing system and application software because of its high flexibility (Kernighan and... (Kernighan and Pike, 1984) now used on almost all workstations and supercomputers was initially written in C In the last 50 years of computer history, many programming languages have appeared and then disappeared for one reason or another Several languages have made significant impact on how computing tasks are achieved today Examples include Cobol, Algol, Pascal, and Ada Another object-oriented language... speed of the motorcycle is 67 m/s, the air density is ρ = 1.2 kg/m3 , the combined mass of the motorcycle and the person is 250 kg, and the coefficient c is 1, find the tilting angle of the taking-off ramp that can produce the longest range 1.9 One way to calculate π is by randomly throwing a dart into the unit square defined by x ∈ [0, 1] and y ∈ [0, 1] in the x y plane The chance of the dart landing inside... efficient approach to learning computational physics is