Plasma physics for controlled fusion second edition

Springer Series on Atomic, Optical, and Plasma Physics 92

Kenro Miyamoto

Plasma Physics
for Controlled
Fusion
Second Edition


Springer Series on Atomic, Optical,
and Plasma Physics
Volume 92

Editor-in-chief
Gordon W.F. Drake, Windsor, Canada
Series editors
James Babb, Cambridge, USA
Andre D. Bandrauk, Sherbrooke, Canada
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Charles J. Joachain, Bruxelles, Belgium
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Gerd Leuchs, Erlangen, Germany
Pierre Meystre, Tucson, USA


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Kenro Miyamoto

Plasma Physics
for Controlled Fusion
Second Edition

123


Kenro Miyamoto
Tokyo
Japan

First edition published with the title: Plasma Physics and Controlled Nuclear Fusion
ISSN 1615-5653
ISSN 2197-6791 (electronic)
Springer Series on Atomic, Optical, and Plasma Physics
ISBN 978-3-662-49780-7

ISBN 978-3-662-49781-4 (eBook)
DOI 10.1007/978-3-662-49781-4
Library of Congress Control Number: 2016936992
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Preface

The worldwide effort to develop the fusion process as a new energy source has been
going on for about a half century and has made remarkable progress. Now construction stage of “International Tokamak Experimental Reactor”, called ITER,
already started. Primary objective of this textbook is to present a basic knowledge
for the students to study plasma physics and controlled fusion researches and to
provide the recent aspect of new results.
Chapter 1 describes the basic concept of plasma and its characteristics. The
orbits of ion and electron are analyzed in various configurations of magnetic field in

Chap. 2.
From Chap. 3 to Chap. 7, plasmas are treated as magnetohydrodynamic
(MHD) fluid. MHD equation of motion (Chap. 3), equilibrium (Chap. 4), and
confinement of plasma in ideal cases (Chap. 5) are described by the fluid model.
Chapters 6 and 7 discuss problems of MHD instabilities whether a small perturbation will grow to disrupt the plasma or will damp to a stable state. The basic
MHD equation of motion can be derived by taking an appropriate average of
Boltzmann equation. This mathematical process is described in Appendix A. The
derivation of useful energy integral formula of axisymmetric toroidal system and
the analysis of high n ballooning mode are introduced in Appendix B.
From Chap. 8 to Chap. 13, plasmas are treated by kinetic theory. Boltzmann’s
equation is introduced in Chap. 8. This equation is the starting point of the kinetic
theory. Plasmas, as mediums in which waves and perturbations propagate, are
generally inhomogeneous and anisotropic. It may absorb or even amplify the wave
and perturbations.
Cold plasma model described in Chap. 9 is applicable when the thermal velocity
of plasma particles is much smaller than the phase velocity of wave. Because of its
simplicity, the dielectric tensor of cold plasma can be easily derived and the
properties of various waves can be discussed in the case of cold plasma.
If the refractive index of plasma becomes large and the phase velocity of the
wave becomes comparable to the thermal velocity of the plasma particles, then the
particles and the waves interact with each other. Chapter 10 describes Landau

v


vi

Preface

damping, which is the most important and characteristic collective phenomenon of

plasma. Waves in hot plasma, in which the wave phase velocity is comparable to
the thermal velocity of particles, are analyzed by use of dielectric tensor of hot
plasma. Wave heating (wave absorption) in hot plasmas and current drives are
described in Chap. 11. Non-inductive current drives combined with bootstrap
current are essential in order to operate tokamak in steady state condition.
Instabilities driven by energetic particles (fishbone instability and toroidal
Alfvén eigenmodes) are treated in Chap. 12. In order to minimize the loss of alpha
particle produced by fusion grade plasma, it is important to avoid the instabilities
driven by energetic particles and alpha particles.
Chapter 13 discusses the plasma transport by turbulence. Losses of plasmas with
drift turbulence become Bohm type or gyro Bohm type depending on different
magnetic configuration. Analysis of confinement by computer simulations is greatly
advanced. Gyrokinetic particle model and full orbit particle model are introduced.
Furthermore it is confirmed recently that the zonal flow is generated in plasmas by
drift turbulence. Understanding of the zonal flow drive and damping has suggested
several routes to improving confinement. Those new topics are included in Chap. 13.
In Chap. 14, confinement researches toward fusion plasmas are reviewed.
During the last two decades, tokamak experiments have made a remarkable progress. Chapter 15 introduces research works of critical subjects on tokamak plasmas
and the aims of ITER and its rationale are explained. Chapter 16 explains reversed
field pinch including PPCD (pulsed parallel current drive), and Chap. 17 introduces
the experimental results of advanced stellarator devices and several types of
quasi-symmetric stellarator. Boozer equation to formulate the drift motion of particles is explained in Appendix C. Chapter 18 describes open-end systems including
tandem mirrors. Elementary introduction of inertial confinement including the fast
ignition is added in Chap. 19.
Readers may have an impression that there is too much mathematics in this
book. However, it is one of motivation to write this text to save the time to struggle
with the mathematical deduction of theoretical results so that students could spend
more time to think physics of experimental results.
This textbook has been attempted to present the basic physics and analytical
methods comprehensively which are necessary for understanding and predicting

plasma behavior and to provide the recent status of fusion researches for graduate
and senior undergraduate students. I also hope that it will be a useful reference for
scientists and engineers working in the relevant fields.
Tokyo, Japan

Kenro Miyamoto


Contents

1

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

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Damping
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1
1
3
5
9

2

Orbit of Charged Particles in Various Magnetic Configuration
2.1
Orbit of Charged Particles . . . . . . . . . . . . . . . . . . . . . . .
2.1.1
Cyclotron Motion . . . . . . . . . . . . . . . . . . . . . .
2.1.2
Drift Velocity of Guiding Center . . . . . . . . . . .
2.1.3
Polarization Drift . . . . . . . . . . . . . . . . . . . . . .
2.1.4
Pondromotive Force . . . . . . . . . . . . . . . . . . . .
2.2
Scalar Potential and Vector Potential . . . . . . . . . . . . . . .
2.3
Magnetic Mirror. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
Toroidal System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1
Magnetic Flux Function. . . . . . . . . . . . . . . . . .

2.4.2
Hamiltonian Equation of Motion . . . . . . . . . . .
2.4.3
Particle Orbit in Axially Symmetric System . . . .
2.4.4
Drift of Guiding Center in Toroidal Field . . . . .
2.4.5
Effect of Longitudinal Electric Field
on Banana Orbit . . . . . . . . . . . . . . . . . . . . . . .
2.4.6
Precession of Trapped Particle . . . . . . . . . . . . .
2.4.7
Orbit of Guiding Center and Magnetic Surface . .
2.5
Coulomb Collision and Neutral Beam Injection . . . . . . . .
2.5.1
Coulomb Collision . . . . . . . . . . . . . . . . . . . . .
2.5.2
Neutral Beam Injection . . . . . . . . . . . . . . . . . .
2.5.3
Resistivity, Runaway Electron, Dreicer Field . . .
2.6
Variety of Time and Space Scales in Plasmas . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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28

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vii


viii

Contents

3

Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . .
3.1
Magnetohydrodynamic Equations for Two Fluids .
3.2
Magnetohydrodynamic Equations for One Fluid . .
3.3
Simplified Magnetohydrodynamic Equations . . . .
3.4
Magnetoacoustic Wave . . . . . . . . . . . . . . . . . . .

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51
51
54
56
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4

Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Pressure Equilibrium. . . . . . . . . . . . . . . . .
4.2
Grad–Shafranov Equilibrium Equation . . . .
4.3
Exact Solution of Grad–Shafranov Equation
4.4

Tokamak Equilibrium . . . . . . . . . . . . . . . .
4.5
Upper Limit of Beta Ratio. . . . . . . . . . . . .
4.6
Pfirsch Schluter Current . . . . . . . . . . . . . .
4.7
Virial Theorem . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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63

63
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5

Confinement of Plasma (Ideal Cases) . . . . . . . . . . . .
5.1
Collisional Diffusion (Classical Diffusion) . . . . .
5.1.1
Magnetohydrodynamic Treatment . . . .
5.1.2
A Particle Model . . . . . . . . . . . . . . .
5.2
Neoclassical Diffusion of Electrons in Tokamak .
5.3
Bootstrap Current . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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85
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6

Magnetohydrodynamic Instabilities. . . . . . . . . . . . . . . . . . . . .
6.1
Interchange Instabilities. . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1
Interchange Instability . . . . . . . . . . . . . . . . . . .
6.1.2

Stability Criterion for Interchange Instability . . .
6.2
Formulation of Magnetohydrodynamic Instabilities . . . . . .
6.2.1
Linearization of Magnetohydrodynamic Equations .
6.2.2
Rayleigh–Taylor (Interchange) Instability. . . . . .
6.3
Instabilities of Cylindrical Plasma with Sharp Boundary . .
6.4
Energy Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5
Instabilities of Diffuse Boundary Configurations . . . . . . . .
6.5.1
Energy Integral of Plasma with Diffuse Boundary .
6.5.2
Suydam’s Criterion . . . . . . . . . . . . . . . . . . . . .
6.5.3
Tokamak Configuration . . . . . . . . . . . . . . . . . .
6.6
Hain Lust Magnetohydrodynamic Equation . . . . . . . . . . .
6.7
Ballooning Instability . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8
ηi Mode Due to Density and Temperature Gradient . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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135

7

Resistive Instabilities . . . . . . . . .
7.1
Tearing Instability . . . . . .
7.2
Neoclassical Tearing Mode
7.3
Resistive Drift Instability. .

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Contents

ix

7.4
Resistive Wall Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8

Boltzmann’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1
Phase Space and Distribution Function . . . . . . . . . . . . . .
8.2
Boltzmann’s Equation and Vlasov’s Equation . . . . . . . . .
8.3
Fokker–Planck Collision Term . . . . . . . . . . . . . . . . . . . .
8.4
Quasi Linear Theory of Evolution in Distribution Function
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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163
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9

Waves in Cold Plasmas . . . . . . . . . . . . . . . . . . . . . . . . .
9.1
Dispersion Equation of Waves in Cold Plasma. . . . .
9.2
Properties of Waves . . . . . . . . . . . . . . . . . . . . . . .
9.2.1
Polarization and Particle Motion . . . . . . . .
9.2.2

Cutoff and Resonance . . . . . . . . . . . . . . .
9.3
Waves in Two Components Plasma . . . . . . . . . . . .
9.4
Various Waves. . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.1
Alfvén Wave . . . . . . . . . . . . . . . . . . . . .
9.4.2
Ion Cyclotron Wave and Fast Wave . . . . .
9.4.3
Lower Hybrid Resonance . . . . . . . . . . . .
9.4.4
Upper Hybrid Resonance . . . . . . . . . . . . .
9.4.5
Electron Cyclotron Wave (Whistler Wave).
9.5
Conditions for Electrostatic Waves . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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194

10 Waves in Hot Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1 Landau Damping and Cyclotron Damping . . . . . . . . . . . . .
10.1.1 Landau Damping (Amplification) . . . . . . . . . . . .
10.1.2 Transit-Time Damping . . . . . . . . . . . . . . . . . . .
10.1.3 Cyclotron Damping. . . . . . . . . . . . . . . . . . . . . .
10.2 Formulation of Dispersion Relation in Hot Plasma . . . . . . .
10.3 Solution of Linearized Vlasov Equation . . . . . . . . . . . . . .
10.4 Dielectric Tensor of Hot Plasma. . . . . . . . . . . . . . . . . . . .
10.5 Dielectric Tensor of bi-Maxwellian Plasma . . . . . . . . . . . .
10.6 Plasma Dispersion Function. . . . . . . . . . . . . . . . . . . . . . .
10.7 Dispersion Relation of Electrostatic Wave . . . . . . . . . . . . .

10.8 Dispersion Relation of Electrostatic Wave in Inhomogeneous
Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.9 Velocity Space Instabilities . . . . . . . . . . . . . . . . . . . . . . .
10.9.1 Drift Instability (Collisionless) . . . . . . . . . . . . . .
10.9.2 Ion Temperature Gradient Instability . . . . . . . . . .
10.9.3 Various Velocity Space Instabilities . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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x

Contents

11 Wave Heatings and Non-Inductive Current Drives . . . . .
11.1 Energy Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Ray Tracing. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Dielectric Tensor of Hot Plasma, Wave Absorption .
11.4 Wave Heating in Ion Cyclotron Range of Frequency
11.5 Lower Hybrid Wave Heating . . . . . . . . . . . . . . . . .
11.6 Electron Cyclotron Heating . . . . . . . . . . . . . . . . . .
11.7 Lower Hybrid Current Drive . . . . . . . . . . . . . . . . .
11.8 Electron Cyclotron Current Drive . . . . . . . . . . . . . .
11.9 Neutral Beam Current Drive . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Instabilities Driven by Energetic Particles . . . . . . . . . . . . . . . .
12.1 Fishbone Instability . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.2 MHD Potential Energy . . . . . . . . . . . . . . . . . .
12.1.3 Kinetic Integral of Hot Component . . . . . . . . . .
12.1.4 Growth Rate of Fishbone Instability . . . . . . . . .
12.2 Toroidal Alfven Eigenmode. . . . . . . . . . . . . . . . . . . . . .
12.2.1 Toroidicity Induced Alfvén Eigenmode . . . . . . .
12.2.2 Instability of TAE Driven by Energetic Particles.
12.2.3 Various Alfvén Modes . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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259
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266
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282
283

13 Plasma Transport by Turbulence . . . . . . . . . . . . . . . . . . . . . .
13.1 Fluctuation Loss, Bohm, GyroBohm Diffusion . . . . . . . . .

13.2 Loss by Magnetic Fluctuation . . . . . . . . . . . . . . . . . . . .
13.3 Dimensional Analysis of Transport . . . . . . . . . . . . . . . . .
13.4 Analysis by Computer Simulations . . . . . . . . . . . . . . . . .
13.4.1 Gyrokinetic Particle Model . . . . . . . . . . . . . . .
13.4.2 Full Orbit Particle Model. . . . . . . . . . . . . . . . .
13.5 Zonal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.5.1 Hasegawa–Mima Equation for Drift Turbulence .
13.5.2 Generation of Zonal Flow . . . . . . . . . . . . . . . .
13.5.3 Geodesic Acoustic Mode (GAM) . . . . . . . . . . .
13.5.4 Zonal Flow in ETG Turbulence . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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298
299

303
307
307
316
320
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324

14 Development of Fusion Researches . . . . . . . . . . . . . . . . . . . . . . . . 327
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
15 Tokamak. . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1 Tokamak Devices. . . . . . . . . . . . . . . . .
15.2 Stability of Equilibrium Plasma Position .
15.3 MHD Stability and Density Limit. . . . . .
15.4 Beta Limit of Elongated Plasma . . . . . . .

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337
337
341
346
348


Contents

xi

15.5
15.6
15.7

15.8
15.9

Impurity Control, Scrape-Off Layer and Divertor . . . .
Confinement Scaling of L Mode . . . . . . . . . . . . . . .
H Mode and Improved Confinement Modes . . . . . . .
Steady-State Operation . . . . . . . . . . . . . . . . . . . . . .
Design of ITER (International Tokamak Experimental
Reactor) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.10 Trials to Innovative Tokamaks . . . . . . . . . . . . . . . . .
15.10.1 Spherical Tokamak . . . . . . . . . . . . . . . . . .
15.10.2 Trials to Innovative Tokamak Reactors . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16 Reversed Field Pinch . . . . . . . . . . . . . . .
16.1 Reversed Field Pinch Configuration
16.2 Taylor’s Relaxation Theory . . . . . .
16.3 Relaxation Process . . . . . . . . . . . .
16.4 Confinement of RFP . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . .

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357
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367

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389
389
390
393
397
401

17 Stellarator . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1 Helical Field . . . . . . . . . . . . . . . . . . . .

17.2 Stellarator Devices . . . . . . . . . . . . . . . .
17.3 Neoclassical Diffusion in Helical Field . .
17.4 Confinement of Stellarator . . . . . . . . . . .
17.5 Quasi-symmetric Stellarator System . . . .
17.6 Conceptual Design of Stellarator Reactor.
References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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439
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450
453

End System . . . . . . . . . . . . . . . . . . . .
Confinement Times in Mirror and Cusp
Confinement Experiments with Mirrors .
Instabilities in Mirror Systems . . . . . . .
Tandem Mirrors . . . . . . . . . . . . . . . . .
18.4.1 Theory . . . . . . . . . . . . . . . .
18.4.2 Experiments . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .

18 Open
18.1
18.2
18.3
18.4

19 Inertial Confinement . . .
19.1 Pellet Gain . . . . .
19.2 Implosion . . . . . .
19.3 MHD Instabilities.
19.4 Fast Ignition . . . .
References . . . . . . . . . . .

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xii

Contents

Appendix A: Derivation of MHD Equations of Motion . . . . . . . . . . . . . 455
Appendix B: Energy Integral of Axisymmetric Toroidal System . . . . . . 461
Appendix C: Quasi-Symmetric Stellarators . . . . . . . . . . . . . . . . . . . . . 473
Appendix D: Physical Constants, Plasma Parameters
and Mathematical Formula . . . . . . . . . . . . . . . . . . . . . . . 483
Curriculum Vitae in Sentence of Kenro Miyamoto . . . . . . . . . . . . . . . . 489
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491


Chapter 1

Nature of Plasma

Abstract Charge neutrality is one of fundamental property of plasma. Section 1.2
explains Debye length λD in (1.2), a measure of shielding distance of electrostatic

potential, and electron plasma frequency Πe in (1.4), a measure of inverse time
scale of electron’s motion to cancel the electric perturbation. Both parameters are
related with λD Πe = vTe , vTe being electron thermal velocity. Section 1.3 describes
the condition of fusion core plasma. The necessary condition for the density, ion
temperature and energy confinement time is given by (1.9).

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 become the high temperature gaseous state in which the charge
numbers of ions and electrons are almost the same and charge neutrality is satisfied
in a macroscopic scale.
When the ions and electrons move collectively, these charged particles interact
with Coulomb force which is long range force and decays only in inverse square
of the distance r between the charged particles. The resultant 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. The typical cases are many kinds of
instabilities and wave phenomena. The word “plasma” is used in physics to designate
the 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,
that is, thermal velocity vT is given by
mvT2 /2 = κT /2

(1.1)

where κ is Boltzmann constant κ = 1.380658(12) × 10−23 J/K and κT indicates
the thermal energy. Therefore the unit of κT is Joule (J) in SI unit. In many fields
© Springer-Verlag Berlin Heidelberg 2016

K. Miyamoto, Plasma Physics for Controlled Fusion, Springer Series on Atomic,
Optical, and Plasma Physics 92, DOI 10.1007/978-3-662-49781-4_1

1


2

1 Nature of Plasma

Fig. 1.1 Various plasma domain in n–T diagram

of physics, one electron volt (eV) is frequently used as a unit of energy. This is the
energy necessary to move an electron, charge e = 1.60217733(49)×10−19 Coulomb,
against a potential difference of 1 volt:
1eV = 1.60217733(49) × 10−19 J.
The temperature corresponding to the thermal energy of 1eV is 1.16×104 K (=e/κ).
From now on the thermal energy κT is denoted by just T for simplicity and new
T means the thermal energy. The ionization energy of hydrogen atom is 13.6 eV.
Even if the thermal energy (average energy) of hydrogen gas is 1 eV, small amount
of electrons with energy higher than 13.6 eV exist and ionize the gas to a hydrogen
plasma. Plasmas are found in nature in various forms (see Fig. 1.1). There exits the
ionosphere in the heights of 70–500 km (density n ∼ 1012 m−3 , T ∼ 0.2 eV). Solar
wind is the plasma flow originated from the sun with n ∼ 106∼7 m−3 , T ∼ 10 eV.
Corona extends around the sun and the density is ∼1014 m−3 and the electron temperature is ∼100 eV although these values depend on the different positions. White
dwarf, the final state of stellar evolution, has the electron density of 1035∼36 m−3 .
Various plasma domains in the diagram of electron density n(m−3 ) and electron
temperature T (eV) are shown in Fig. 1.1. Active researches in plasma physics have
been motivated by the aim to create and confine hot plasmas in fusion researches.
Plasmas play important roles in the studies of pulsars radiating microwave or solar

1.1 Introduction

3

X ray sources observed in space physics and astrophysics. The other 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, ion rocket engines for space crafts, and plasma processing which
attracts much attention recently.

1.2 Charge Neutrality and Landau Damping
One of the fundamental property of plasma is the shielding of the electric potential
applied to the plasma. When a probe is inserted into a plasma and positive (negative)
potential is applied, the probe attracts (repulses) electrons and the plasma tends to
shield the electric disturbance. Let us estimate the shielding length. Assume that the
ions are in uniform density (n i = n 0 ) and there is small perturbation in electron
density n e or potential φ. Since the electrons are in Boltzmann distribution usually,
the electron density n e becomes
n e = n 0 exp(eφ/Te )

n 0 (1 + eφ/Te ).

Poisson’s equation is
E = −∇φ,

∇( 0 E) = − 0 ∇ 2 φ = ρ = −e(n e − n 0 ) = −

e2 n 0

φ
Te

and
∇2φ =

φ
,
λ2D

λD =

0 Te
n e e2

1/2

= 7.45 × 103

1 Te
ne e

1/2

(m)

(1.2)

where n e is in m−3 and Te /e is in eV. When n e ∼ 1020 cm−3 , Te /e ∼ 10 keV, then
λD ∼ 75 µm.

In spherically symmetric case, Laplacian ∇ 2 becomes ∇ 2 φ = (1/r 2 )(∂/∂r )
2
(r ∂φ/∂r ) and the solution is
φ=

q exp(−r/λD )
.
4π 0
r

It is clear from the foregoing formula that Coulomb potential q/4π 0 r of 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, then plasma is considered neutral in
charge. If a < λD in contrary, individual particle is not shielded electrostatically and
this state is no longer plasma but an assembly of independent charged particles. The
number of electrons included in the sphere of radius λD is called plasma parameter
and is given by


4

1 Nature of Plasma

Te
e e
0

nλ3D =

3/2

1
1/2
ne

.

(1.3)

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 in the
scale of Debye length. Plasmas in the region of nλ3 > 1 are called classical plasma
or weakly coupled plasma, since the ratio of electron thermal energy Te and coulomb
n −1/3 is the average distance
energy between electrons E coulomb = e2 /4π 0 d (d
between electrons with the density n) is given by
Te
= 4π(nλ3D )2/3
E coulomb
and nλ3 > 1 means that coulomb energy is smaller than the thermal energy. The
case of nλ3D < 1 is called strongly coupled plasma (see Fig. 1.1). Fermi energy of
degenerated electron gas is given by F = (h 2 /2m e )(3π 2 n)2/3 . When the density
becomes very high, it is possible to become F ≥ Te . In this case quantum effect is
more dominant than thermal effect. This case is called degenerated electron plasma.
One of this example is the electron plasma in metal. Most of plasmas in experiments are classical weakly coupled plasma except the plasma compressed by inertial
confinement.
Let us consider the case where a small perturbation occurs in a uniform plasma
and the electrons in the plasma move by the perturbation. It is assumed that ions

do not move because the ion’s mass is much heavier than electron’s. Due to the
displacement of electrons, electric charges appear and an electric field is induced.
The electric field is given by Poisson’s equation:
0∇

· E = −e(n e − n 0 ).

Electrons are accelerated by the electric field:
me

dv
= −e E.
dt

Due to the movement of electrons, the electron density changes:
∂n e
+ ∇ · (n e v) = 0.
∂t
Denote n e − n 0 = n 1 and assume |n 1 |
0∇

· E = −en 1 ,

me

n 0 , then we find

∂v
= −e E,
∂t

∂n 1
+ n 0 ∇ · v = 0.
∂t


1.2 Charge Neutrality and Landau Damping

5

For simplicity the displacement is assumed only in the x direction and is sinusoidal:
n 1 (x, t) = n 1 exp(ikx − iωt).
Time differential ∂/∂t is replaced by −iω and ∂/∂x is replaced by ik, then
ik 0 E = −en 1 ,

− iωm e v = −eE,

so that we find
ω2 =

− iωn 1 = −ikn 0 v

n 0 e2
.
0me

(1.4)

This wave is called electron plasma wave or Langmuir wave and its frequency is
called electron plasma frequency Πe :

Πe =

n e e2
0me

1/2

= 5.64 × 1011

ne
1020

1/2

rad/s.

There is following relation between the plasma frequency and Debye length λD :
λ D Πe =

Te
me

1/2

= vTe .

The other fundamental process of plasma is collective phenomena of charged
particles. Waves are associated with coherent motions of charged particles. When the
phase velocity vph of wave or perturbation is much larger than the thermal velocity vT
of charged particles, the wave propagates through the plasma media without damping

or amplification. However when the refractive index N of plasma media becomes
large and plasma becomes hot, the phase velocity vph = c/N (c is light velocity) of the
wave and the thermal velocity vT become comparable (vph = ω/k = c/N ∼ vT ),
then the exchange of energy between the wave and the thermal energy of plasma
is possible. The existence of a damping mechanism of wave was found by L.D.
Landau. The process of Landau damping involves a direct wave-particle interaction
in collisionless plasma without necessity of randomizing collision. This process is
fundamental mechanism in wave heatings of plasma (wave damping) and instabilities
(inverse damping of perturbations). Landau damping will be described in Chaps. 10
and 11.

1.3 Fusion Core Plasma
Progress in plasma physics has been motivated by how to realize fusion core plasma.
Necessary condition for fusion core plasma is discussed in this section. Nuclear
fusion reactions are the fused reactions of light nuclides to heavier one. When the
sum of the masses of nuclides after a nuclear fusion is smaller than the sum before


6

1 Nature of Plasma

the reaction by Δm, we call it mass defect. According to theory of relativity, amount
of energy (Δm)c2 (c is light speed) 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)
(2)
(3)
(4)

(5)
(6)

D + D→T(1.01 MeV) + p(3.03 MeV)
D + D→ He3 (0.82 MeV) + n(2.45 MeV)
T + D→ He4 (3.52 MeV) + n(14.06 MeV)
D + He3 → He4 (3.67 MeV) + p(14.67 MeV)
Li6 + n→T + He4 + 4.8 MeV
Li7 + n(2.5 MeV)→T + He4 + n

where p and n are proton (hydrogen ion) and neutron respectively (1 MV = 106 eV).
Since the energy released by chemical reaction of H2 + (1/2)O2 → H2 O is 2.96 eV,
fusion energy released is about million times as large as chemical one. A binding
energy per nucleon is smaller in very light or very heavy nuclides and largest in the
nuclides with atomic mass numbers around 60. Therefore, large amount of the energy
can be released when the light nuclides are fused. Deuterium exists abundantly in
nature; for example, it comprises 0.015 atom percent of the hydrogen in sea water
with the 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 in the stage of research development. Nuclear
fusion reactions were found in 1920s. When proton or deuteron beams collide with
target of light nuclide, beam loses its energy by the ionization or elastic collisions
with target nuclides and the probability of nuclear fusion is negligible. Nuclear fusion
researches have been most actively pursued by use of hot plasma. In fully ionized
hydrogen, deuterium and tritium plasmas, the process of ionization does not occur.
If the plasma is confined in some specified region adiabatically, the average energy
does not decrease by the processes of elastic collisions. Therefore if the very hot
D–T plasmas or D–D plasmas are confined, the ions have velocities large enough to
overcome their mutual coulomb repulsion, so that collision and fusion take place.
Let us consider the nuclear reaction that D collides with T. The effective cross

section of T nucleus is denoted by σ. This cross section is a function of the kinetic
energy E of D. The cross section of D–T reaction at E = 100 keV is 5 × 10−24 cm2 .
The cross sections σ of D–T, D–D, D–He3 reaction versus the kinetic energy of
colliding nucleus are shown in Fig. 1.2a [1, 2]. The probability of fusion reaction
per unit time in the case that a D ion with the velocity v collides with T ions with
the density of n T is given by n T σv (we will discuss the collision probability in more
details in Sect. 2.5). When a plasma is Maxwellian with the ion temperature of Ti , it
is necessary to calculate the average value σv of σv over the velocity space. The
dependence of σv on ion temperature Ti is shown in Fig. 1.2b [3]. A fitting equation
of σv of D–T reaction as a function of T in unit of keV is [4]
σv (m3 /s) =

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

,


1.3 Fusion Core Plasma

7

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

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

H (T ) ≡

5.45
T
+
37 3 + T (1 + T /37.5)2.8

(1.5)

Figure 1.3 shows an example of electric power plant based on D–T fusion reactor.
Fast neutrons produced in 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 reaction (5), (6).
Lithium blanket gives up its heat to generate the steam by a heat exchanger; steam


8

1 Nature of Plasma

turbine generates electric power. A part of the generated electric power is used to
operate heating system of plasma to compensate the energy losses from the plasma
to keep the plasma hot. The fusion output power must be larger than the necessary
heating input power taking account the conversion efficiency. Since the necessary
heating input power is equal to the energy loss rate of fusion core plasma, good
energy confinement of hot plasma is key issue.
The thermal energy of plasma per unit volume is given by (3/2)n(Ti + Te ). This
thermal energy is lost by thermal conduction and convective losses. The notation
PL denotes these energy losses of the plasma per unit volume per unit time (power

loss per unit volume). There is radiation loss R due to bremsstrahlung of electrons
and impurity ion radiation in addition to PL . The total energy confinement time τE
is defined by
3nT
(3/2)n(Te + Ti )
.
(1.6)
τE ≡
PL + R
PL + R
The necessary heating input power Pheat is equal to PL + R. In the case of D–T
reaction, the sum of kinetic energies Q α = 3.52 MeV of α particle (He4 ion) and
Q n = 14.06 MeV of neutron is Q fus =17.58 MeV per 1 reaction. Since the densities
of D ions and T ions of equally mixed plasma are n/2, number of D–T reaction per
unit time per unit volume is (n/2)(n/2) σv , so that fusion output power per unit
volume Pfus is given by
(1.7)
Pfus = (n/2)(n/2) σv Q fus .
Denote the thermal-to-electric conversion efficiency by ηel and heating efficiency
(ratio of the deposit power into the plasma to the electric input power of heating
device) by ηheat . Then the condition of power generation is
Pheat = PL + R =
that is

3nT
< (ηel )(ηheat )Pfus
τE

(1.8)

3nT
Q fus 2
n σv ,
< (ηheat )(ηel )
τE
4
nτE >

12T
η Q fus σv

(1.9)

where η is the product of two efficiencies. The right-hand side of the last foregoing equation is the function of temperature T only. When T = 104 eV and η ∼
0.3 (ηel ∼ 0.4, ηheat ∼ 0.75), the necessary condition is nτE > 1.7 × 1020 ms−3 · s.
The condition of D–T fusion plasma in the case of η ∼ 0.3 is shown in Fig. 1.4. In
reality the plasma is hot in the core and is cold in the edge. For the more accurate
discussion, we must take account of the profile effect of temperature and density and
will be analyzed in Sect. 15.9.


1.3 Fusion Core Plasma

9

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

condition (η = 0.2)

The condition Pheat = Pfus is called break even condition. This corresponds to
the case of η = 1 in the condition of fusion core plasma. The ratio of the fusion
output power due to α particles to the total is Q α /Q fus = 0.2. Since α particles
are charged particles, α particles can heat the plasma by coulomb collision (see
Sect. 2.5). If the total kinetic energy (output energy) of α particles contributes to
heat the plasma, the condition Pheat = 0.2Pfus can sustain the necessary high temperature of the plasma without heating from outside. This condition is called ignition
condition, which corresponds the case of η = 0.2.

References
1.
2.
3.
4.

W.R. Arnold, J.A. Phillips, G.A. Sawyer, E.J. Stovall Jr., J.C. Tuck, Phys. Rev. 93, 483 (1954)
C.F. Wandel, T.H. Jensen, O. Kofoed-Hansen, Nucl. Instr. Methods 4, 249 (1959)
J.L. Tuck, Nucl. Fusion 1, 201 (1961)
T. Takizuka, M. Yamagiwa, Japan Atomic Energy Research Institute JAERI-M 87-066 (1987)


Chapter 2

Orbit of Charged Particles in Various
Magnetic Configuration

Abstract Section 2.1 describes the drift motion of guiding center of cyclotron
motion, polarization drift which is important to study the zonal flow in Sect. 13.5.
Section 2.3 treats the drift motion in mirror configuration and Sect. 2.4 treats the drift

motion in toroidal configuration, the effect of longitudinal electric field on banana
orbit (Ware’s pinch) and the precession of banana orbit center which is important
topics for fishbone instability in Sect. 2.1. Coulomb collision and the heating rates of
ions and electrons by high energy neutral beam injection are described in Sect. 2.5.

2.1 Orbit of Charged Particles
2.1.1 Cyclotron Motion
The equation of motion of charged particle with the mass m and the charge q in an
electric and magnetic field E, B is given by
m

dv
= q(E + v × B).
dt

(2.1)

When the magnetic field is homogenous and is in the z direction and the electric field
is zero, the equation of motion becomes v˙ = (qB/m)(v × b) (b = B/B) and
vx = −v⊥ sin(Ωt + δ),
vy = v⊥ cos(Ωt + δ),
vz = vz0 ,
qB
Ω=− .
m

(2.2)

The solution of these equation is a spiral motion around the magnetic line of force
with the angular velocity of Ω (see Fig. 2.1). This motion is called Larmor motion.

The angular frequency Ω is called cyclotron (angular) frequency. Denote the radius
2
of the orbit by ρΩ , then the centrifugal force is mv⊥
/ρΩ and Lorentz force is qv⊥ B.
© Springer-Verlag Berlin Heidelberg 2016
K. Miyamoto, Plasma Physics for Controlled Fusion, Springer Series on Atomic,
Optical, and Plasma Physics 92, DOI 10.1007/978-3-662-49781-4_2

11


12

2 Orbit of Charged Particles in Various Magnetic Configuration

Fig. 2.1 Larmor motion of
charged particle in magnetic
field

Fig. 2.2 Drift motion of
guiding center in electric and
gravitational field
(conceptional drawing)

Table 2.1 Larmor radius and cyclotron frequency
B = 1T, T = 100 eV
Electron
(2T /m)1/2

Thermal velocity v⊥T =

Larmor radius ρΩ
(Angular) Cyclotron frequency Ω
Cyclotron frequency Ω/2π

5.9 × 106 m/s
33.7 µm
1.76 × 1011 /s
28 GHz

Proton
1.39 × 105 m/s
1.44 mm
−9.58 × 107 /s
−15.2 MHz

Since both forces must be balanced, we find
ρΩ =

mv⊥
.
|q|B

(2.3)

This radius is called Larmor radius. The center of Larmor motion is called guiding
center. Electron’s Larmor motion is right-hand sense (Ωe > 0), and ion’s Larmor
motion is left-hand sense (Ωi < 0) (see Fig. 2.2). When B = 1 T, T = 100 eV, the
values of Larmor radius and cyclotron frequencies are given in Table 2.1.

2.1.2 Drift Velocity of Guiding Center

When a uniform electric field E perpendicular to the uniform magnetic field is superposed, the equation of motion is reduced to


2.1 Orbit of Charged Particles

13

m

du
= q(u × B)
dt

by use of
v = ue + u,

ue =

E×b
.
B

(2.4)

Therefore the motion of charged particle is superposition of Larmor motion and drift
motion ue of its guiding center. The direction of guiding center drift by E is the same
for both ion and electron (Fig. 2.2). When a gravitational field g is superposed, the
force is mg, which corresponds to qE in the case of electric field. Therefore the drift
velocity of the guiding center due to the gravitation is given by
ug =

m
g×b
(g × b) = −
.
qB
Ω

(2.5)

The directions of ion’s drift and electron’s drift due to the gravitation are opposite
with each other and the drift velocity of ion guiding center is much larger than
electron’s one (see Fig. 2.2). When the magnetic and electric fields change slowly
1), the formulas of drift
and gradually in time and in space (|ω/Ω|
1, ρΩ /R
velocity are valid as they are. However because of the curvature of field line of
magnetic force, centrifugal force acts on the particle which runs along a field line
with the velocity of v . The acceleration of centrifugal force is
g curv =

v2
R

n

where R is the radius of curvature of field line and n is the unit vector with the
direction from the center of the curvature to the field line (Fig. 2.3).
Furthermore, as is described later, the resultant effect of Larmor motion in an
inhomogeneous magnetic field is reduced to the acceleration of

g ∇B = −

Fig. 2.3 Radius of curvature
of line of magnetic force

2
/2
v⊥
∇B.
B