Plasma Physics and Controlled Nuclear Fusion, Miyamoto

Springer Series on

atomic, optical, and plasma physics

38


Springer Series on

atomic, optical, and plasma physics
The Springer Series on Atomic, Optical, and Plasma Physics covers in a comprehensive manner theory and experiment in the entire f ield of atoms and molecules
and their interaction with electromagnetic radiation. Books in the series provide
a rich source of new ideas and techniques with wide applications in f ields such as
chemistry, materials science, astrophysics, surface science, plasma technology, advanced optics, aeronomy, and engineering. Laser physics is a particular connecting
theme that has provided much of the continuing impetus for new developments
in the f ield. The purpose of the series is to cover the gap between standard undergraduate textbooks and the research literature with emphasis on the fundamental
ideas, methods, techniques, and results in the f ield.

27 Quantum Squeezing
By P.D. Drumond and Z. Ficek
28 Atom, Molecule, and Cluster Beams I
Basic Theory, Production and Detection of Thermal Energy Beams
By H. Pauly
29 Polarization, Alignment and Orientation in Atomic Collisions
By N. Andersen and K. Bartschat
30 Physics of Solid-State Laser Physics
By R.C. Powell
(Published in the former Series on Atomic, Molecular, and Optical Physics)
31 Plasma Kinetics in Atmospheric Gases
By M. Capitelli, C.M. Ferreira, B.F. Gordiets, A.I. Osipov

32 Atom, Molecule, and Cluster Beams II
Cluster Beams, Fast and Slow Beams, Accessory Equipment and Applications
By H. Pauly
33 Atom Optics
By P. Meystre
34 Laser Physics at Relativistic Intensities
By A.V. Borovsky, A.L. Galkin, O.B. Shiryaev, T. Auguste
35 Many-Particle Quantum Dynamics in Atomic and Molecular Fragmentation
Editors: J. Ullrich and V.P. Shevelko
36 Atom Tunneling Phenomena in Physics, Chemistry and Biology
Editor: T. Miyazaki
37 Charged Particle Traps
Physics and Techniques of Charged Particle Field Confinement
By V.N. Gheorghe, F.G. Major, G. Werth
38 Plasma Physics and Controlled Nuclear Fusion
By K. Miyamoto

Vols. 1–26 of the former Springer Series on Atoms and Plasmas are listed at the end of the book


K. Miyamoto

Plasma Physics
and Controlled
Nuclear Fusion
With 117 Figures

123

Professor emer. Kenro Miyamoto
Univesity of Tokyo
E-mail:

Originally published in Japanese under the title "Plasma Physics and Controlled Nuclear Fusion"
by University of Tokyo Press, 2004

ISSN 1615-5653
ISBN 3-540-24217-1 Springer Berlin Heidelberg New York
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Preface

The primary objective of these lecture notes is to present the basic theories
and analytical methods of plasma physics and to provide the recent status
of fusion research for graduate and advanced undergraduate students. I also
hope that this text will be a useful reference for scientists and engineers
working in the relevant fields.
Chapters 1–4 describe the fundamentals of plasma physics. The basic
concept of the plasma and its characteristics are explained in Chaps. 1 and
2. The orbits of ions and electrons are described in several magnetic field
configurations in Chap. 3, while Chap. 4 formulates the Boltzmann equation
for the velocity space distribution function, which is the basic equation of
plasma physics.
Chapters 5–9 describe plasmas as magnetohydrodynamic (MHD) fluids.
The MHD equation of motion (Chap. 5), equilibrium (Chap. 6) and plasma
transport (Chap. 7) are described by the fluid model. Chapter 8 discusses
problems of MHD instabilities, i.e., whether a small perturbation will grow
to disrupt the plasma or damp to a stable state. Chapter 9 describes resistive
instabilities of plasmas with finite electrical resistivity.
In Chaps. 10–13, plasmas are treated by kinetic theory. The medium in
which waves and perturbations propagate is generally inhomogeneous and
anisotropic. It may absorb or even amplify the waves and perturbations. The
cold plasma model described in Chap. 10 is applicable when the thermal velocity of plasma particles is much smaller than the phase velocity of the wave.
Because of its simplicity, the dielectric tensor of cold plasma is easily derived
and the properties of various waves can be discussed in the case of cold plasmas. If the refractive index becomes large and the phase velocity of the wave
becomes comparable to the thermal velocity of the plasma particles, then the
particles and the wave interact with each other. Chapter 11 describes Landau

damping, which is the most characteristic collective phenomenon of plasmas,
and also cyclotron damping. Chapter 12 discusses wave heating (wave absorption) and velocity space instabilities (amplification of perturbations) in hot
plasmas, in which the thermal velocity of particles is comparable to the wave
phase velocity, using the dielectric tensor of hot plasmas. Chapter 13 discusses instabilities driven by energetic particles, i.e., the fishbone instability
and toroidal Alfv´en eigenmodes.


VI

Preface

In order to understand the complex nonlinear behavior of plasmas, computer simulation becomes a dominant factor in the theoretical component of
plasma research, and this is briefly outlined in Chap. 14.
Chapter 15 reviews confinement research toward fusion grade plasmas.
During the last decade, tokamak research has made remarkable progress. Today, realistic designs for tokamak reactors such as ITER are being actively
pursued. Chapter 16 explains research work into critical features of tokamak plasmas and reactors. Non-tokamak confinement systems are also receiving great interest. The reversed field pinch and stellarators are described
in Chap. 17 and inertial confinement is introduced in Chap. 18.
The reader may have the impression that there is too much mathematics
in these lecture notes. However, there is a reason for this. If a graduate student
tries to read and understand, for example, frequently cited short papers on
the analysis of the high-n ballooning mode and fishbone instability [Phys.
Rev. Lett 40, 396 (1978); ibid. 52, 1122 (1984)] without some preparatory
knowledge, he must read and understand a few tens of cited references, and
references of references. I would guess that he would be obliged to work hard
for a few months. Therefore, one motivation for writing this monograph is to
save the student time struggling with the mathematical derivations, so that
he can spend more time thinking about the physics and experimental results.
This textbook was based on lectures given at the Institute of Plasma
Physics, Nagoya University, Department of Physics, University of Tokyo and
discussion notes from ITER Physics Expert Group Meetings. It would give me

great pleasure if the book were to help scientists make their own contributions
in the field of plasma physics and fusion research.
Tokyo, November 2004

Kenro Miyamoto


Contents

Part I Plasma Physics
1

Nature of Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Charge Neutrality and Landau Damping . . . . . . . . . . . . . . . . .
1.3
Fusion Core Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3
3
5
6

2

Plasma Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Velocity Space Distribution Function . . . . . . . . . . . . . . . . . . . .

2.2
Plasma Frequency. Debye Length . . . . . . . . . . . . . . . . . . . . . . .
2.3
Cyclotron Frequency. Larmor Radius . . . . . . . . . . . . . . . . . . . .
2.4
Drift Velocity of Guiding Center . . . . . . . . . . . . . . . . . . . . . . . .
2.5
Magnetic Moment. Mirror Confinement . . . . . . . . . . . . . . . . . .
2.6
Coulomb Collision. Fast Neutral Beam Injection . . . . . . . . . .
2.7
Runaway Electron. Dreicer Field . . . . . . . . . . . . . . . . . . . . . . . .
2.8
Electric Resistivity. Ohmic Heating . . . . . . . . . . . . . . . . . . . . .
2.9
Variety of Time and Space Scales in Plasmas . . . . . . . . . . . . .

13
13
14
15
16
19
21
27
28
28

3

Magnetic Configuration and Particle Orbit . . . . . . . . . . . . . . .
3.1
Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Magnetic Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
Equation of Motion of a Charged Particle . . . . . . . . . . . . . . . .
3.4
Particle Orbit in Axially Symmetric System . . . . . . . . . . . . . .
3.5
Drift of Guiding Center in Toroidal Field . . . . . . . . . . . . . . . .
3.5.1 Guiding Center of Circulating Particles . . . . . . . . . . .
3.5.2 Guiding Center of Banana Particles . . . . . . . . . . . . . .
3.6
Orbit of Guiding Center and Magnetic Surface . . . . . . . . . . .
3.7
Effect of Longitudinal Electric Field on Banana Orbit . . . . .
3.8
Polarization Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31
31
33
34
36
38
39
40
42
44

45

4

Velocity Space Distribution Function
and Boltzmann’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1
Phase Space and Distribution Function . . . . . . . . . . . . . . . . . . 47
4.2
Boltzmann’s Equation and Vlasov’s Equation . . . . . . . . . . . . 48


VIII

Contents

5

Plasma as MHD Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
Magnetohydrodynamic Equations for Two Fluids . . . . . . . . .
5.2
Magnetohydrodynamic Equations for One Fluid . . . . . . . . . .
5.3
Simplified Magnetohydrodynamic Equations . . . . . . . . . . . . . .
5.4
Magnetoacoustic Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51
51

53
55
58

6

Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1
Pressure Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
Equilibrium Equation for Axially Symmetric Systems . . . . . .
6.3
Tokamak Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4
Upper Limit of Beta Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5
Pfirsch–Schl¨
uter Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6
Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61
61
63
67
69
70
71

7

Plasma Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1
Collisional Diffusion (Classical Diffusion) . . . . . . . . . . . . . . . .
7.1.1 Magnetohydrodynamic Treatment . . . . . . . . . . . . . . . .
7.1.2 A Particle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
Neoclassical Diffusion of Electrons in a Tokamak . . . . . . . . . .
7.3
Fluctuation Loss. Bohm and Gyro-Bohm Diffusion.
Convective Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4
Loss by Magnetic Fluctuation . . . . . . . . . . . . . . . . . . . . . . . . . .

75
77
77
79
80

Magnetohydrodynamic Instabilities . . . . . . . . . . . . . . . . . . . . . . .
8.1
Interchange Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Interchange Instability . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.2 Stability Criterion for Interchange Instability.
Magnetic Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2
Formulation of Magnetohydrodynamic Instabilities . . . . . . . .
8.2.1 Linearization of Magnetohydrodynamic Equations . .
8.2.2 Energy Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.3
Instabilities of a Cylindrical Plasma . . . . . . . . . . . . . . . . . . . . .
8.3.1 Instabilities of Sharp-Boundary Configuration . . . . . .
8.3.2 Instabilities of Diffuse Boundary Configurations . . . .
8.3.3 Suydam’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.4 Tokamak Configuration . . . . . . . . . . . . . . . . . . . . . . . . .
8.4
Hain–L¨
ust Magnetohydrodynamic Equation . . . . . . . . . . . . . .
8.5
Energy Integral of Axisymmetric Toroidal System . . . . . . . . .
8.5.1 Energy Integral in Illuminating Form . . . . . . . . . . . . .
8.5.2 Energy Integral of Axisymmetric Toroidal System . .
8.5.3 Energy Integral of High-n Ballooning Mode . . . . . . . .
8.6
Ballooning Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.7
Eta-i Mode Due to Density and Temperature Gradient . . . .

91
92
92

8

83
89

95
99

99
102
104
104
109
113
115
117
119
119
121
126
128
133


Contents

9

IX

Resistive Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.1
Tearing Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.2
Resistive Drift Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

10 Plasma as Medium of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1 Dispersion Equation of Waves in a Cold Plasma . . . . . . . . . .

10.2 Properties of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.1 Polarization and Particle Motion . . . . . . . . . . . . . . . . .
10.2.2 Cutoff and Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Waves in a Two-Component Plasma . . . . . . . . . . . . . . . . . . . . .
10.4 Various Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.1 Alfven Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.2 Ion Cyclotron Wave and Fast Wave . . . . . . . . . . . . . . .
10.4.3 Lower Hybrid Resonance . . . . . . . . . . . . . . . . . . . . . . . .
10.4.4 Upper Hybrid Resonance . . . . . . . . . . . . . . . . . . . . . . . .
10.4.5 Electron Cyclotron Wave . . . . . . . . . . . . . . . . . . . . . . . .
10.5 Conditions for Electrostatic Waves . . . . . . . . . . . . . . . . . . . . . .

147
148
152
152
153
153
158
158
159
161
162
162
164

11 Landau Damping and Cyclotron Damping . . . . . . . . . . . . . . . .
11.1 Landau Damping (Amplification) . . . . . . . . . . . . . . . . . . . . . . .
11.2 Transit Time Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Cyclotron Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11.4 Quasi-Linear Theory of Evolution
in the Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167
167
171
171

12 Hot Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1 Energy Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Ray Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Dielectric Tensor of Hot Plasma . . . . . . . . . . . . . . . . . . . . . . . .
12.4 Wave Heating in the Ion Cyclotron Frequency Range . . . . . .
12.5 Lower Hybrid Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.6 Electron Cyclotron Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.7 Velocity Space Instabilities (Electrostatic Waves) . . . . . . . . .
12.7.1 Dispersion Equation of Electrostatic Wave . . . . . . . . .
12.7.2 Electron Beam Instability . . . . . . . . . . . . . . . . . . . . . . .
12.7.3 Various Velocity Space Instabilities . . . . . . . . . . . . . . .
12.8 Derivation of Dielectric Tensor in Hot Plasma . . . . . . . . . . . .
12.8.1 Formulation of Dispersion Relation in Hot Plasma . .
12.8.2 Solution of Linearized Vlasov Equation . . . . . . . . . . .
12.8.3 Dielectric Tensor of Hot Plasma . . . . . . . . . . . . . . . . . .
12.8.4 Dielectric Tensor of Bi-Maxwellian Plasma . . . . . . . .
12.8.5 Dispersion Relation of Electrostatic Wave . . . . . . . . .

177
178
182
183

189
192
196
199
199
201
202
202
202
204
206
209
210

174


X

Contents

13 Instabilities Driven by Energetic Particles . . . . . . . . . . . . . . . .
13.1 Fishbone Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.2 MHD potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.3 Kinetic Integral of Hot Component . . . . . . . . . . . . . . .
13.1.4 Growth Rate of Fishbone Instability . . . . . . . . . . . . . .
13.2 Toroidal Alfv´en Eigenmode . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2.1 Toroidicity-Induced Alfv´en Eigenmode . . . . . . . . . . . .
13.2.2 Instability of TAE Driven by Energetic Particles . . .

13.2.3 Various Alfv´en Modes . . . . . . . . . . . . . . . . . . . . . . . . . .

215
215
215
216
218
221
224
225
229
237

14 Computer Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1 MHD model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 Linearized Kinetic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3 Modeling Bulk Plasma and Energetic Particles . . . . . . . . . . .
14.4 Gyrofluid/Gyro-Landau-Fluid Models . . . . . . . . . . . . . . . . . . .
14.5 Gyrokinetic Particle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.6 Full Orbit Particle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

239
240
242
243
244
247
251

Part II Controlled Nuclear Fusion

15 Development of Fusion Research . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1 From Secrecy to International Collaboration . . . . . . . . . . . . . .
15.2 Artsimovich Era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.3 The Trek to Large Tokamaks Since the Oil Crisis . . . . . . . . .
15.4 Alternative Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259
260
262
263
266

16 Tokamaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.1 Tokamak Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.3 MHD Stability and Density Limit . . . . . . . . . . . . . . . . . . . . . . .
16.4 Beta Limit of Elongated Plasma . . . . . . . . . . . . . . . . . . . . . . . .
16.5 Impurity Control, Scrape-Off Layer and Divertor . . . . . . . . . .
16.6 Confinement Scaling of L Mode . . . . . . . . . . . . . . . . . . . . . . . . .
16.7 H Mode and Improved Confinement Modes . . . . . . . . . . . . . .
16.8 Non-Inductive Current Drive . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.8.1 Lower Hybrid Current Drive . . . . . . . . . . . . . . . . . . . . .
16.8.2 Electron Cyclotron Current Drive . . . . . . . . . . . . . . . .
16.8.3 Neutral Beam Current Drive . . . . . . . . . . . . . . . . . . . . .
16.8.4 Bootstrap Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.9 Neoclassical Tearing Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.10 Tokamak Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

269
269

272
274
277
278
284
286
293
293
297
300
302
304
311


Contents

XI

17 RFP Stellarator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1 Reversed Field Pinch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1.1 Reversed Field Pinch Configuration . . . . . . . . . . . . . . .
17.1.2 MHD Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1.3 Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1.4 Oscillating Field Current Drive . . . . . . . . . . . . . . . . . .
17.2 Stellarator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2.1 Helical Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2.2 Stellarator Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2.3 Neoclassical Diffusion in Helical Field . . . . . . . . . . . . .
17.2.4 Confinement of Stellarator System . . . . . . . . . . . . . . . .

319
319
319
320
323
325
325
325
329
331
334

18 Inertial Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.1 Pellet Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2 Implosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3 MHD Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4 Fast Ignition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

337
337
342
345
347

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367


Part I

Plasma Physics


1 Nature of Plasma

1.1 Introduction
As the temperature of a material is raised, its state changes from solid to
liquid and then to gas. If the temperature is elevated further, an appreciable
number of the gas atoms are ionized and a high temperature gaseous state is
achieved, in which the charge numbers of ions and electrons are almost the
same and charge neutrality is satisfied on a macroscopic scale.
When ions and electrons move, these charged particles interact with the
Coulomb force which is a long range force and decays only as the inverse
square of the distance r between the charged particles. The resulting current
flows due to the motion of the charged particles and Lorentz interaction takes
place. Therefore many charged particles interact with each other by long
range forces and various collective movements occur in the gaseous state. In
typical cases, there are many kinds of instabilities and wave phenomena. The
word ‘plasma’ is used in physics to designate this high temperature ionized
gaseous state with charge neutrality and collective interaction between the
charged particles and waves.
When the temperature of a gas is T (K), the average velocity of the thermal
motion of a particle with mass m, that is, thermal velocity vT is given by
2
mvT
/2 = κT /2 ,

(1.1)

where κ is the Boltzmann constant κ = 1.380 658(12) × 10−23 J/K and κT
denotes the thermal energy. Therefore the unit of κT is the joule (J) in MKSA
units. In many fields of physics, the electron volt (eV) is frequently used as
the unit of energy. This is the energy required to move an electron, charge
e = 1.602 177 33(49) × 10−19 coulomb, against a potential difference of 1 volt:
1 eV = 1.602 177 33(49) × 10−19 J .
The temperature corresponding to a thermal energy of 1 eV is 1.16 × 104 K
(= e/κ). The ionization energy of the hydrogen atom is 13.6 eV. Even if the
thermal energy (average energy) of hydrogen gas is 1 eV, that is T ∼ 104 K,
there exists a small number of electrons with energy higher than 13.6 eV,
which ionize the gas to a hydrogen plasma.


4

1 Nature of Plasma

Fig. 1.1. Various plasma domains in the n–κT diagram

Plasmas are found in nature in various forms (see Fig. 1.1). One example is
the Earth’s ionosphere at altitudes of 70–500 km, with density n ∼ 1012 m−3
and κT ≈ 0.2 eV. Another is the solar wind, a plasma flow originating from
the sun, with n ∼ 106 –107 m−3 and κT ≈ 10 eV. The sun’s corona extending
around our star has density ∼ 1014 m−3 and electron temperature ∼ 100 eV,
although these values are position-dependent. The white dwarf, the final state
of stellar evolution, has an electron density of 1035 –1036 m−3 . Various plasma
domains in the diagram of electron density n(m−3 ) and electron temperature
κT (eV) are shown in Fig. 1.1.
Active research in plasma physics has been motivated by the aim to create
and confine hot plasmas in fusion research. In space physics and astrophysics,

plasmas play important roles in studies of pulsars radiating microwaves or
solar X-ray sources. Another application of plasma physics is the study of
the Earth’s environment in space.
Practical applications of plasma physics are MHD (magnetohydrodynamic) energy conversion for electric power generation and ion rocket engines
for spacecraft. Plasma processes for the manufacture of integrated circuits
have attracted much attention recently.


1.2 Charge Neutrality and Landau Damping

5

1.2 Charge Neutrality and Landau Damping
One fundamental property of plasmas is charge neutrality. Plasmas shield
electric potentials applied to the plasma. When a probe is inserted into a
plasma and a positive (negative) potential is applied, the probe attracts (repels) electrons and the plasma tends to shield the electric disturbance. Let us
estimate the shielding length. Assume that heavy ions have uniform density
(ni = n0 ) and that there is a small perturbation in the electron density ne
and potential φ. Since the electrons are in the Boltzmann distribution with
electron temperature Te , the electron density ne becomes
ne = n0 exp(eφ/κTe )

n0 (1 + eφ/κTe ) ,

where φ is the electrostatic potential and eφ/κTe
1 is assumed. The
equation for the electrostatic potential comes from Maxwell’s equations (see
Sect. 3.1),
E = −∇φ ,

∇·( 0 E) = − 0 ∇2 φ = ρ = −e(ne − n0 ) = −

e2 n0
φ
κTe

and
∇2 φ =

φ
,
λ2D

λD =

0 κTe
ne e2

1/2

1 κTe
ne e

= 7.45 × 103

1/2

(m) , (1.2)

where 0 is the dielectric constant of the vacuum and E is the electric intensity. ne is in m−3 and κTe /e is in eV. When ne ∼ 1020 cm−3 , κTe /e ∼ 10 keV,

then λD ∼ 75 µm. In the spherically symmetric case, the Laplacian ∇2 becomes
∂φ
1 ∂
r
,
∇2 φ =
r ∂r
∂r
and the solution is
φ=

q
4π

exp(−r/λD )
.
r
0

It is clear from the foregoing formula that the Coulomb potential q/4π 0 r of
a point charge is shielded out to a distance λD . This distance λD is called the
Debye length. When the plasma size is a and a
λD is satisfied, the plasma is
considered to be electrically neutral. If on the other hand a < λD , individual
particles are not shielded electrostatically and this state is no longer a plasma
but an assembly of independent charged particles.
The number of electrons included in a sphere of radius λD is called the
plasma parameter and is given by
ne λ3D =

κTe
e e
0

3/2

1
1/2
ne

.

(1.3)


6

1 Nature of Plasma

When the density is increased while keeping the temperature constant, this
value becomes small. If the plasma parameter is less than say ∼ 1, the concept
of Debye shielding is not applicable, since the continuity of charge density
breaks down on the Debye length scale. Plasmas in the region of ne λ3D > 1
are called classical plasmas or weakly coupled plasmas, since the ratio of
the electron thermal energy κTe and the Coulomb energy between electrons
−1/3
ECoulomb = e2 /4π 0 d, with d ne
the average distance between electrons
with density ne , is given by
κTe

= 4π(ne λ3D )2/3 ,
ECoulomb

(1.4)

and ne λ3D > 1 means that the Coulomb energy is smaller than the thermal
energy. The case ne λ3D < 1 corresponds to a strongly coupled plasma (see
Fig. 1.1).
The Fermi energy of a degenerate electron gas is given by
F

=

h2
(3π 2 ne )2/3 ,
2me

where h = 6.626 075 5(40) × 10−34 J s is Planck’s constant. When the density
becomes very high, it is possible to have F ≥ κTe . In this case, quantum
effects dominate over thermal effects. This case is called a degenerate electron
plasma. One example is the electron plasma in a metal. Most plasmas in
magnetic confinement experiments are classical weakly coupled plasmas.
The other fundamental plasma process is collective phenomena involving the charged particles. Waves are associated with coherent motions of
charged particles. When the phase velocity vph of a wave or perturbation
is much larger than the thermal velocity vT of the charged particles, the
wave propagates through the plasma media without damping or amplification. However, when the refractive index N of the plasma medium becomes
large and the plasma becomes hot, the phase velocity vph = c/N (where c is
the light velocity) of the wave and the thermal velocity vT become comparable (vph = c/N ∼ vT ). Then energy exchange is possible between the wave
and the thermal energy of the plasma. The existence of a damping mechanism
for these waves was found by L.D. Landau. The process of Landau damping

involves a direct wave–particle interaction in a collisionless plasma without
the need to randomize collisions. This process is the fundamental mechanism
in wave heating of plasmas (wave damping) and instabilities (inverse damping
of perturbations). Landau damping is described in Chaps. 11 and 12.

1.3 Fusion Core Plasma
Progress in plasma physics has been motivated by the desire to realize a fusion
core plasma. The necessary condition for fusion core plasmas is discussed in


1.3 Fusion Core Plasma

7

Fig. 1.2. (a) Dependence of the fusion cross-section σ on the kinetic energy E
of colliding nucleons. σDD is the sum of the cross-sections of D–D reactions (1)
and (2). 1 barn = 10−24 cm2 . (b) Dependence of the fusion rate σv on the ion
temperature Ti

this section. Nuclear fusion reactions are the fusion reactions of light nuclides
to heavier ones. When the sum of the masses of nuclides after nuclear fusion
is smaller than the sum before the reaction by ∆m, we call this the mass
defect. According to the theory of relativity, the amount of energy (∆m)c2
(c is the speed of light) is released by the nuclear fusion.
Nuclear reactions of interest for fusion reactors are as follows (D deuteron,
T triton, He3 helium-3, Li lithium):
(1)

D + D −→ T (1.01 MeV) + p (3.03 MeV) ,

(2)

D + D −→ He3 (0.82 MeV) + n (2.45 MeV) ,

(3)

T + D −→ He4 (3.52 MeV) + n (14.06 MeV) ,

(4)

D + He3 −→ He4 (3.67 MeV) + p (14.67 MeV) ,

(5)

Li6 + n −→ T + He4 + 4.8 MeV ,

(6)

Li7 + n (2.5 MeV) −→ T + He4 + n ,

where p and n are the proton (hydrogen ion) and the neutron, respectively
(1 MeV = 106 eV). Since the energy released by the chemical reaction
1
H2 + O2 −→ H2 O
2
is 2.96 eV, the fusion energy released is about a million times as great as the
chemical energy. A binding energy per nucleon is smaller in very light or very


8

1 Nature of Plasma

heavy nuclides and largest in nuclides with atomic mass numbers around 60.
Therefore, large amounts of energy can be released when light nuclides are
fused. Deuterium is abundant in nature. For example, it comprises 0.015 atom
percent of the hydrogen in sea water, with a volume of about 1.35 × 109 km3 .
Although fusion energy was released in an explosive manner by the hydrogen bomb in 1951, controlled fusion is still at the research and development
stage. Nuclear fusion reactions were found in the 1920s. When proton or
deuteron beams collide with a light nuclide target, the beam loses its energy
by ionization or elastic collisions with target nuclides, and the probability of
nuclear fusion is negligible. Nuclear fusion research has been most actively
pursued in the context of hot plasmas.
In fully ionized hydrogen, deuterium and tritium plasmas, the process of
ionization does not occur. If the plasma is confined adiabatically in some
specified region, the average energy does not decrease by elastic collision processes. Therefore, if very hot D–T plasmas or D–D plasmas are confined, the
ions have large enough velocities to overcome their mutual Coulomb repulsion, so that collision and fusion take place.
Let us consider the nuclear reaction wherein D collides with T. The crosssection of T nucleons is denoted by σ. This cross-section is a function of the
kinetic energy E of D. The cross-section of the D–T reaction at E = 100 keV
is 5 × 10−24 cm2 . The cross-sections σ of D–T, D–D, D–He3 reactions versus
the kinetic energy of colliding nucleons are shown in Fig. 1.2a [1.1, 1.2]. The
probability of the fusion reaction per unit time in the case where a D ion
with velocity v collides with T ions with density of nT is given by nT σv.
(We discuss the collision probability in more detail in Sect. 2.7.) When a
plasma is Maxwellian with ion temperature Ti , one must calculate the average
value σv of σv over the velocity space. The dependence of σv on the ion
temperature Ti is shown in Fig. 1.2b [1.3]. A fitting equation for σv for the
D–T reaction as a function of κT in units of keV is [1.4]
σv (m−3 ) =
H(κT ) ≡

3.7 × 10−18
20
exp −
H(κT ) × (κT )2/3
(κT )1/3

,

(1.5)

κT
5.45
.
+
37
3 + κT (1 + κT /37.5)2.8

Figure 1.3 shows an example of an electric power plant based on a D–T fusion reactor. Fast neutrons produced in the fusion core plasma penetrate the
first wall and a lithium blanket surrounding the plasma moderates the fast
neutrons, converting their kinetic energy to heat. Furthermore, the lithium
blanket breeds tritium due to reactions (5) and (6) above. [Triton beta-decays
to He3 with a half-life of 12.3 yr, T → He3 + e (< 18.6 keV), and tritium does
not exist as a natural resource.] The lithium blanket gives up its heat to generate steam via a heat exchanger and a steam turbine generates electric power.
Part of the generated electric power is used to operate the plasma heating
system. As alpha particles (He ions) are charged particles, they can heat the


1.3 Fusion Core Plasma

9

Fig. 1.3. Electric power plant based on a D–T fusion reactor

plasma directly by Coulomb collisions (see Sect. 2.6). The total heating power
Pheat is the sum of the ' particle heating power P' and the heating power
Pext due to the external heating system. The total heating power needed to
sustain the plasma in a steady state must be equal to the energy loss rate of
the fusion core plasma. Consequently, good energy confinement (small energy
loss rate) in the hot plasma is the key issue.
The thermal energy of the plasma per unit volume is (3/2)nκ(Ti + Te ).
This thermal energy is lost by thermal conduction and convective losses. The
notation PL denotes these energy losses from the plasma per unit volume and
unit time (power loss per unit volume). In addition to PL , there is radiation
loss R due to electron bremsstrahlung and impurity ion radiation. The total
energy confinement time τE is defined by
τE ≡

(3/2)nκ(Te + Ti )
PL + R

3nκT
.
PL + R

(1.6)

The input heating power Pheat required to maintain the thermal energy of
the plasma is equal to PL + R.
For the D–T reaction, the sum of kinetic energies Q' = 3.52 MeV of alpha

particles and Qn = 14.06 MeV of neutrons is QNF = 17.58 MeV per reaction
(Qn : Q' = m' : mn = 0.8 : 0.2 due to momentum conservation). Since
the densities of D ions and T ions in an equally mixed plasma are n/2, the
number of D–T reactions per unit time and unit volume is (n/2)(n/2) σv
(refer to the discussion in Sect. 2.6), so that the fusion output power per unit
volume PNF is given by
PNF = (n/2)(n/2) σv QNF .

(1.7)

If the fusion powers due to the neutron and alpha particle are denoted by Pn
and P' respectively, then Pn = 0.8PNF and P' = 0.2PNF . Let the thermalto-electric conversion efficiency be ηel and the heating efficiency (ratio of the


10

1 Nature of Plasma

Fig. 1.4. Condition of D–T fusion core plasma in nτE –T diagram in the case
η = 0.3, critical condition η = 1, and ignition condition η = 0.2

power deposited in the plasma to the input electric power of the heating
device) by ηheat . When a part (γ < 1) of generated electric power is used to
operate the heating system, then the available heating power to plasma is
(0.8ηel γηheat + 0.2)PNF = ηPNF ,

η ≡ 0.8γηel ηheat + 0.2 .

The burning condition is
Pheat = PL + R =

3nκT
< ηPNF ,
τE

(1.8)

that is,
QNF 2
3nκT
n σv ,
<η
τE
4
and hence,
nτE >

12κT
.
ηQNF σv

(1.9)

The right-hand side of (1.9) is a function of temperature T alone. When
κT = 104 eV and η ∼ 0.3 (γ ∼ 0.4, ηel ∼ 0.4, ηheat ∼ 0.8), the necessary
condition is nτE > 1.7 × 1020 m−3 s. The burning condition of the D–T fusion
plasma in the case η ∼ 0.3 is shown in Fig. 1.4. In reality the plasma is hot
in the core and cold at the edges. For a more accurate discussion, we must
take this temperature and density profile effect into account, an analysis
undertaken in Sect. 16.10.

The ratio of the fusion output power due to ' particles to the total is
Q' /QNF = 0.2. If the total kinetic energy (output energy) of alpha particles


1.3 Fusion Core Plasma

11

contributes to heating the plasma and alpha particle heating power can sustain the necessary high temperature of the plasma without heating from the
outside, the plasma is in an ignited state. The condition P' = PL +R is called
the ignition condition, which corresponds to the case η = 0.2 in (1.8).
The condition Pheat = PNF is called the break-even condition. This corresponds to the case of η = 1 in (1.8). The ignition condition (η = 0.2) and
break-even condition (η = 1) are also shown in Fig. 1.4.


2 Plasma Characteristics

2.1 Velocity Space Distribution Function
In a plasma, electrons and ions move with various velocities. The number
o electrons in a unit volume is the electron density ne and the number of
electrons dne (vx ) with the x component of velocity between vx and vx + dvx
is given by
dne (vx ) = fe (vx )dvx .
Then fe (vx ) is called the electron velocity space distribution function. When
electrons are in a thermal equilibrium state with electron temperature Te ,
the velocity space distribution function is the Maxwell distribution:
fe (vx ) = ne

1/2

β
2π

exp −

βvx2
2

,

β=

me
.
κTe

From the definition, the velocity space distribution function satisfies
∞

fe (vx )dvx = ne .

−∞

The Maxwell distribution function in the three-dimensional velocity space is
given by
fe (vx , vy , vz ) = ne

3/2

me

2πκTe

exp −

me (vx2 + vy2 + vz2 )
2κTe

.

(2.1)

The ion distribution function is defined in the same way as for the electron.
The mean of the squared velocity vx2 is given by
vT2 =

1
n

∞
−∞

vx2 f (vx )dvx =

κT
.
m

(2.2)

The pressure p is

p = nκT .
The particle flux in the x direction per unit area Γ+,x is given by
∞

Γ+,x =

vx f (vx )dvx = n
0

κT
2πm

1/2

.


14

2 Plasma Characteristics

2.2 Plasma Frequency. Debye Length
Let us consider the case where a small perturbation occurs in a uniform
plasma and the electrons in the plasma move due to the perturbation. It is
assumed that the ions do not move because they have much greater mass than
the electrons. Due to the displacement of electrons, electric charges appear
and an electric field is induced. The electric field is given by
0 ∇·E

= −e(ne − n0 ) .

Electrons are accelerated by the electric field:
me

dv
= −eE .
dt

Due to the movement of electrons, the electron density changes:
∂ne
+ ∇·(ne v) = 0 .
∂t
Writing ne − n0 = n1 and assuming |n1 |
0 ∇·E

= −en1 ,

me

n0 , we find
∂n1
+ n0 ∇·v = 0 .
∂t

∂v
= −eE ,
∂t

For simplicity, the displacement is assumed to be only in the x direction and
sinusoidal with angular frequency ω :

n1 (x, t) = n1 exp(ikx − iωt) .
The time derivative ∂/∂t is replaced by −iω and ∂/∂x is replaced by ik. The
electric field has only the x component E. Then
ik 0 E = −en1 ,

−iωme v = −eE ,

−iωn1 = −ikn0 v ,

so that we find

n0 e2
.
(2.3)
0 me
This wave is called the electron plasma wave or Langmuir wave and its frequency is called the electron plasma frequency Πe :
ω2 =

Πe =

ne e2
0 me

1/2

= 5.64 × 1011

ne
1020

1/2

rad/s .

The following relation holds between the plasma frequency and the Debye
length λD :
λD Πe =

κTe
me

1/2

= vTe = 4.19 × 105

κTe
e

1/2

m/s .


2.3 Cyclotron Frequency. Larmor Radius

15

Fig. 2.1. Larmor motion of charged particle in magnetic field

2.3 Cyclotron Frequency. Larmor Radius

The equation of motion of a charged particle with mass m and charge q in
electric and magnetic fields E, B is given by
m

dv
= q(E + v × B) .
dt

(2.4)

When the magnetic field is homogeneous and in the z direction and the
electric field is zero, the equation of motion becomes v˙ = (qB/m)(v × b),
where b = B/B, and
vx = −v⊥ sin(Ωt + δ) ,

vy = v⊥ cos(Ωt + δ) ,

vz = vz0 ,

qB
.
(2.5)
m
The solution of these equations is a spiral motion around the magnetic line
of force with angular velocity Ω (see Fig. 2.1). This motion is called Larmor
motion. The angular frequency Ω is called cyclotron (angular) frequency.
2
Denoting the radius of the orbit by ρΩ , the centrifugal force is mv⊥
/ρΩ and
the Lorentz force is qv⊥ B. Since the two forces must balance, we find

Ω=−

ρΩ =

mv⊥
.
|q|B

(2.6)

This radius is called the Larmor radius. The center of the Larmor motion
is called the guiding center. The Larmor motion of the electron is a righthanded rotation (Ωe > 0), while the Larmor motion of the ion is a left-handed
rotation (Ωi < 0). When B = 1 T, κT = 100 eV, the values of the Larmor
radius and cyclotron frequency are given in Table 2.1.