Plasma Physics like-A. Dinklage, T. Klinger, G.Marx, L. Schweikhard

Lecture Notes in Physics
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A. Dinklage T. Klinger G. Marx L. Schweikhard
(Editors)

Plasma Physics
Confinement, Transport and Collective Effects

ABC


Editors
Priv-Doz. Dr. Andreas Dinklage
Professor Dr. Thomas Klinger
MPI Plasmaschutz
EURATOM Association
Wendelsteinstr. 1
17491 Greifswald, Germany



Dr. Gerrit Marx
Professor Dr. Lutz Schweikhard
Ernst-Moritz-Arndt Universität

Institut für Physik
Domstr. 10a
17489 Greifswald, Germany
marx @physik.uni-greifswald.de


Andreas Dinklage et al., Plasma Physics,
Lect. Notes Phys. 670 (Springer, Berlin Heidelberg 2005), DOI 10.1007/b103882

Library of Congress Control Number: 2005923687
ISSN 0075-8450
ISBN -10 3-540-25274-6 Springer Berlin Heidelberg New York
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Dedicated to Billa, Frauke and Johanna



Preface

Plasma, sometimes called the fourth state of matter, is a multifaceted substance which poses a variety of challenges. Plasma physics deals with the
complex interaction of many charged particles with external or self-generated
electromagnetic fields. It is this unique entanglement which makes plasma
physics a fascinating field for basic research. At the same time, plasma plays
an essential role in many applications, ranging, e.g., from advanced lighting devices and surface treatments for semiconductor applications or surface
layer generation to the efforts to tame nuclear fusion as an energy source for
our future harnessing the nuclear processes which fuel our sun.
Modern plasma research is a multidisciplinary endeavor which includes
aspects of electrodynamics, many-particle physics, quantum effects and nonlinear dynamics. But even though the spatial extension, the density, the ionization degree and the plasma temperature may vary by many orders of magnitude, the physical similarities – or the plasma properties – of, e.g., the
solar corona, non-neutral plasmas in ion-traps, the electron gas of metals or
planetary interiors lead to similarities of these systems.
Plasmas on earth are evanescent. The confinement of plasmas for extended times is a very difficult task and one of the central keys for plasma
research and applications. Consequently, transport phenomena which go far
beyond classical transport are highly relevant. This also leads to the ultimate challenge of many-particle physics: the understanding of turbulence. In
addition, a variety of “ordered” collective effects can be studied in unique
clarity, for example, phase transitions in “dusty” plasmas or the large variety of plasma waves. The corresponding investigations are at the forefront of
current research and development.

This volume of Springer Lecture Notes in Physics provides an overview
of modern plasma research with a special focus on confinement and related
issues. Beginning with a broad introduction, the book leads graduate students
and researchers – including those not specialized in plasma research – to
the state of the art of modern plasma physics. The book also presents a
methodological cross section ranging from plasma applications and plasma
diagnostics to numerical simulations, an important link between theory and
experiment which is gaining more and more importance. The references are
chosen to guide the reader from basic concepts to current research. Exercises


VIII

Preface

in computational plasma physics are supplied on a Web site (see Chap. 16 in
Part III of this book).
The contributions are structured in three parts: After a broad introduction to Fundamental Plasma Physics, the focus of this volume on Confinement, Transport and Collective Effects is covered. Modern plasma physics is
also applied science and has many methodological branches as described in
the third part on Methods and Applications.
The chapters have been written by prominent experts in their respective
fields. The book is based on a series of lectures for graduate students in the
framework of a W.E.–Heraeus Summer School.
We would like to thank the W.E.–Heraeus Foundation for funding and
the International Max Planck Research School “Bounded Plasmas” for supporting the 50th Heraeus Summer School “Plasma Physics: Confinement,
Transport and Collective Effects” held in Greifswald during October 2003.
We are indebted to those speakers who contributed; this book has benefitted
from their encouragement and support.
We thank Dr. Angela Lahee from Springer Heidelberg for her friendly
collaboration throughout this project. We also appreciate the professional and

friendly support from Ms. Jaqueline Lenz, Ms. Gabriele Hakuba, Ms. Elke
Sauer and Ms. Shanya Rehman during the editorial and technical realization
of this book.
And last – but certainly not least – we are deeply grateful to Ms. Andrea
Pulss, for whom it must have been much more than a “challenging effort” to
do the technical editorial work.

Greifswald,
April 2005

Andreas Dinklage
Thomas Klinger
Gerrit Marx
Lutz Schweikhard


Contents

Part I Fundamental Plasma Physics
1 Basics of Plasma Physics
U. Schumacher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Definition, Occurrence and Typical Parameters
of Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Ideal Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Important Plasma Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Debye Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 The Plasma Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 Landau Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.4 Plasma Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Single Particle Behavior in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4.1 Coulomb Collisions, Collision Times and Lengths . . . . . . . . .
1.4.2 Electrical Conductivity of Plasmas . . . . . . . . . . . . . . . . . . . . .
1.4.3 Single Charged Particle Motion in Electric
and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Kinetic Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Waves in Plasmas
A. Piel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Dispersion Relation for Waves in a Fluid Plasma . . . . . . . . . . . . . . .
2.2.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 The Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 The Dielectric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Phase and Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Waves in Unmagnetized Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Transverse Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Longitudinal Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Electron Beam Driven Waves . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Waves in Magnetized Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Propagation Along the Magnetic Field . . . . . . . . . . . . . . . . . .

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2.4.2 Cut-Offs and Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Propagation Across the Magnetic Field . . . . . . . . . . . . . . . . .
2.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 An Introduction to Magnetohydrodynamics (MHD),
or Magnetic Fluid Dynamics
B.D. Scott . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 What MHD Is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Ideas of Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 The Density in a Changing Flow Field – Conservation
of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 The Advective Derivative
and the Co-moving Reference Frame . . . . . . . . . . . . . . . . . . . .
3.2.3 Forces on the Fluid – How the Velocity Changes . . . . . . . . .
3.2.4 Thermodynamics of an Ideal Fluid – How
the Temperature Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 The Composite Fluid Plasma System . . . . . . . . . . . . . . . . . . .
3.3 From Many to One – the MHD System . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 The MHD Force Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Treating Several Ion Species . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 The MHD Kinematic Equation . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 MHD at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 The Flux Conservation Theorem of Ideal MHD . . . . . . . . . . . . . . . .
3.4.1 Proving Flux Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Magnetic Flux Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Dynamics, or the Wires-in-Molasses Picture
of MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Magnetic Pressure Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Alfv´en Waves: Magnetic Tension Waves . . . . . . . . . . . . . . . . .
3.6 The Validity of MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Characteristic Time Scales of MHD . . . . . . . . . . . . . . . . . . . . .
3.6.2 Checking the Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.3 A Comment on the Plasma Beta . . . . . . . . . . . . . . . . . . . . . . .
3.7 Parallel Dynamics and Resistivity, or Relaxing

the Ideal Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Towards Multi-Fluid MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Physics of “Hot” Plasmas
H. Zohm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 What is a Hot Plasma? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Kinetic Description of Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 The Kinetic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.2.2 Landau Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fluid Description of Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 The MHD Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Consequences of the MHD Equations . . . . . . . . . . . . . . . . . . .
4.4 MHD Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Classification of MHD Instabilities . . . . . . . . . . . . . . . . . . . . .
4.4.2 Examples of MHD Instabilities . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3

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5 Low Temperature Plasmas
J. Meichsner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Gas Discharges and Low Temperature Plasmas: Basic
Mechanisms and Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Classical Townsend Mechanism
and Electric Breakdown in Gases . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Townsend and Glow Discharge . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Arc Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4 Streamer Mechanism
and Micro-Discharges, Dielectric Barrier and Corona
Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.5 Glow Discharge at Alternating Electric Field, RF
and Microwave Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Plasma Surface Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Plasma Boundary Sheath, Bohm Criterion . . . . . . . . . . . . . .
5.3.2 RF Plasma Sheath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.3 Electric Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Reactive Plasmas and Plasma Surface Interaction . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Strongly Coupled Plasmas
R. Redmer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Many-Particle Effects and Plasma Properties . . . . . . . . . . . . . . . . . .
6.2.1 Green’s Function Technique: Spectral Function . . . . . . . . . .
6.2.2 Cluster Decomposition of the Self-Energy . . . . . . . . . . . . . . .
6.3 Composition of Strongly Coupled Plasmas . . . . . . . . . . . . . . . . . . . . .
6.4 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part II Confinement, Transport and Collective Effects
7 Magnetic Confinement
F. Wagner and H. Wobig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Conditions for Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 The Need for Magnetic Confinement . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Particle Motion in Electro-Magnetic Fields . . . . . . . . . . . . . . . . . . . .
7.4 Constants of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Exact Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2 Adiabatic Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Concepts of Magnetic Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.2 The Mirror Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.3 Toroidal Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.5.4 Magnetic Surfaces and Toroidal Equilibrium . . . . . . . . . . . . .
7.5.5 Confinement in Tokamaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.6 Coil System of Tokamaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.7 Theory of Tokamak Equilibria . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.8 Cylindrical Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.9 Confinement in Stellarators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6 Transport in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.1 Collisional Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.2 Particle Picture of Classical Diffusion . . . . . . . . . . . . . . . . . . .
7.6.3 Neoclassical Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.4 Turbulent Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.5 Empirical Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137
137
138
139
144
144
144
147
147
147
148
149
151
152
153
154

156
163
164
165
166
168
169
171

8 Introduction to Turbulence in Magnetized Plasmas
B.D. Scott . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.1 Part A – Statistical Nonlinearity
and Cascade Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.2 Eddy Mitosis and the Cascade Model . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.3 The Statistical Nature of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.4 Quadratic Nonlinearity and Three Wave Coupling
for Small Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.5 Incompressible Hydrodynamic Turbulence – Energy and Enstrophy182
8.6 MHD Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.7 Part B – Gradient Driven Turbulence
in Magnetized Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
8.8 Passive Scalar Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.9 Dissipative Coupling and the Adiabatic Response . . . . . . . . . . . . . . 193
8.10 Computations in the Dissipative Coupling Model for Drift Wave
Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198


Contents

8.11 No Coupling – the Hydrodynamic Limit . . . . . . . . . . . . . . . . . . . . . .

8.12 The Effects of Dissipative Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.14 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 Transport in Toroidal Plasmas
U. Stroth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1 Experimental Confinement Times
and Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.1 Global Confinement Times . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.2 Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.3 The Collisional Transport Matrix . . . . . . . . . . . . . . . . . . . . . .
9.1.4 Diffusion as Random-Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Particle Orbits in Toroidal Magnetic Fields . . . . . . . . . . . . . . . . . . . .
9.2.1 Particles in a Toroidal Magnetic Mirror . . . . . . . . . . . . . . . . .
9.2.2 Passing Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.3 Trapped Particles and Banana Orbits . . . . . . . . . . . . . . . . . . .
9.2.4 Trajectories in Stellarator Fields . . . . . . . . . . . . . . . . . . . . . . .
9.2.5 Influence of a Radial Electric Field . . . . . . . . . . . . . . . . . . . . .
9.3 Collisional Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.1 Classical Transport in the Particle Picture . . . . . . . . . . . . . . .
9.3.2 Classical Transport in the Fluid Picture . . . . . . . . . . . . . . . . .
9.3.3 Pfirsch–Schl¨
uter Transport in the Particle Picture . . . . . . . .
9.3.4 Pfirsch–Schl¨
uter Transport in the Fluid Picture . . . . . . . . . .
9.3.5 The Toroidal Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.6 Neoclassical Transport in the Particle Picture . . . . . . . . . . . .
9.3.7 Elements of Stellarator Transport . . . . . . . . . . . . . . . . . . . . . .
9.3.8 Neoclassical Transport in the Fluid Picture . . . . . . . . . . . . . .
9.3.9 The Ambipolar Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4 Turbulent Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.1 Fluid Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.2 Phenomenology of Turbulent Plasma Transport . . . . . . . . . .
9.4.3 Two Fundamental Linear Instabilities . . . . . . . . . . . . . . . . . . .
9.4.4 Elements of a Drift Wave Model . . . . . . . . . . . . . . . . . . . . . . .
9.4.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.6 Transport Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 Non-Neutral Plasmas
and Collective Phenomena in Ion Traps
G. Werth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.1 Basics of Ion Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Ion Cloud as Non-Neutral Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XIII

201
205
208
210
211
213
214
214
218
221
223
225
225

226
227
228
229
231
231
232
234
235
236
237
239
240
242
245
245
249
252
255
257
260
264

269
269
269
278


XIV

Contents

10.3 Weakly Coupled Non-Neutral Plasmas . . . . . . . . . . . . . . . . . . . . . . . .
10.3.1 Plasma Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.2 Rotating Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4 Collective Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.1 Individual and Center-of-Mass Oscillations . . . . . . . . . . . . . .
10.4.2 Instabilities in the Ion Motion . . . . . . . . . . . . . . . . . . . . . . . . .
10.5 Strongly Coupled Non-Neutral Plasmas . . . . . . . . . . . . . . . . . . . . . . .
10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279
280
282
284
284
287
287
293
294

11 Collective Effects in Dusty Plasmas
A. Melzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Particle Charging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.1 Orbital Motion Limit Currents . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.2 Other Charging Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.3 Particles as Floating Probes . . . . . . . . . . . . . . . . . . . . . . . . . . .

11.2.4 Charging in the RF Sheath . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Forces on Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.1 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.2 Electric Field Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.3 Ion Drag Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.4 Neutral Drag Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.5 Thermophoresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.6 Dust Levitation and Trapping . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.7 Vertical Oscillations and Dust Charges . . . . . . . . . . . . . . . . . .
11.4 Particle–Particle Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.1 Strongly Coupled Systems and Plasma Crystals . . . . . . . . . .
11.4.2 Horizontal Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.3 Vertical Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.4 Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Waves in Weakly Coupled Dusty Plasmas . . . . . . . . . . . . . . . . . . . . .
11.5.1 Dust-Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5.2 Dust Ion-Acoustic Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6 Waves in Strongly Coupled Dusty Plasmas . . . . . . . . . . . . . . . . . . . .
11.6.1 Compressional Mode in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6.2 Compressional Dust Lattice Waves . . . . . . . . . . . . . . . . . . . . .
11.6.3 Shear Dust Lattice Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6.4 Mach Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6.5 Transverse Dust Lattice Waves . . . . . . . . . . . . . . . . . . . . . . . . .
11.6.6 Normal Modes in Finite Clusters . . . . . . . . . . . . . . . . . . . . . . .
11.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

297
297
298

298
300
300
302
302
302
302
303
304
304
305
305
309
309
311
311
312
313
313
316
316
317
319
320
320
322
324
327
327

Contents

12 Plasmas in Planetary Interiors
R. Redmer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Extrasolar Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4 Equation of State for Partially Ionized Plasmas . . . . . . . . . . . . . . . .
12.4.1 Dense Hydrogen and Helium . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.2 Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.3 Fluid Variational Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.4 Plasma Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.5 Hugoniot Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5 Electrical and Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . .
12.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XV

331
331
332
334
337
337
337
338
339
341

343
345
346

Part III Methods and Applications
13 Plasma Diagnostics
H.-J. Kunze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Scattering of Laser Radiation
by Plasma Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2.1 Laser-aided Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2.2 Incoherent Thomson Scattering . . . . . . . . . . . . . . . . . . . . . . . .
13.2.3 Collective Thomson Scattering . . . . . . . . . . . . . . . . . . . . . . . . .
13.2.4 X-ray Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3 Plasma Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3.2 Charge State Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3.3 Line Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3.4 Line Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3.5 Continuum Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 Observation of Plasma Fluctuations
O. Grulke and T. Klinger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3 Fluctuation Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3.1 Invasive Fluctuation Diagnostics . . . . . . . . . . . . . . . . . . . . . . .
14.3.2 Non-invasive Fluctuation Diagnostics . . . . . . . . . . . . . . . . . . .
14.3.3 Electron Cyclotron Emission . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3.4 Beam Emission Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . .

14.3.5 Heavy Ion Beam Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351
351
352
352
353
357
361
361
361
364
366
370
372
372
375
375
376
383
384
390
392
393
394


XVI

Contents

14.3.6 Laser-induced Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
14.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
15 Research on Modern Gas Discharge Light Sources
M. Born and T. Markus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1 Introduction to Light Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1.1 The Lighting Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1.2 Overview of Discharge Lamps and Applications . . . . . . . . . .
15.1.3 Aspects of Lamp Research . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.2 High Intensity Discharge Lamps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.2.1 Construction and Working Principle . . . . . . . . . . . . . . . . . . . .
15.2.2 Light Technical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.3 Modelling of High Intensity Discharge Lamps . . . . . . . . . . . . . . . . . .
15.3.1 Physical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.3.2 Thermochemical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.4 Thermochemical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.4.1 Knudsen Effusion Mass Spectrometry (KEMS) . . . . . . . . . . .
15.4.2 Corrosion Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

399
399
399
401
404
404
404
406

407
407
412
414
414
418
421
422

16 Computational Plasma Physics
R. Schneider and R. Kleiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.2 Plasma Edge Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.2.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.3 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.3.1 Gyro-kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.3.2 The PIC Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.5 Seminars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

425
425
426
428
434
435
437
441
441

441

17 Nuclear Fusion
H.-S. Bosch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2 Energy Production in the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3 Fusion on Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.4 Conditions for Nuclear Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.5 Power Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.6 Development of a Fusion Power Plant . . . . . . . . . . . . . . . . . . . . . . . . .
17.7 Muon-catalyzed Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

445
445
448
449
453
455
458
458
459


Contents

XVII

18 The Possible Role of Nuclear Fusion
in the 21st Century

T. Hamacher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2 The Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2.1 Energy Demand and Lifestyle . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2.2 Efficient Use of Energy and Energy Saving . . . . . . . . . . . . . .
18.2.3 Energy Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2.4 Geopolitical Frictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2.5 Environmental Damages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2.6 Possible Supply Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3 Characteristics of Nuclear Fusion as Power Source . . . . . . . . . . . . . .
18.3.1 Overall Design of a Fusion Power Plant . . . . . . . . . . . . . . . . .
18.3.2 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3.3 Environmental and Safety Characteristics, External Costs .
18.3.4 Economic Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4 The Possible Role of Fusion
in a Future Energy System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4.1 The Global Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4.2 Fusion in Western Europe . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4.3 Fusion in India . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.5 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

461
461
462
463
464
464
465
465

466
467
467
469
470
473
475
475
476
477
480
481

Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489


Part I

Fundamental Plasma Physics



1 Basics of Plasma Physics
U. Schumacher
Institut f¨
ur Plasmaforschung, Universit¨
at Stuttgart, Pfaffenwaldring 31
70569 Stuttgart, Germany

Abstract. Basic properties of plasmas are introduced, which are valid for an extremely wide range of plasma parameters. Plasmas are classified by different physical behaviour. The motion of charged particles in electromagnetic fields is revised
with respect to drift motions. Adiabatic invariants are discussed and the kinetic
description of plasmas is briefly presented.
This Chapter is also a guideline connecting the subsequent Chapters.

1.1 Definition, Occurrence and Typical Parameters
of Plasmas
A plasma (Greek πλασµα) is an ionised gas, consisting of free electrons,
ions and atoms or molecules. It is characterised by its collective behaviour.
Plasmas are many-particles ensembles; the charged particles are coupled by
electric and magnetic self-generated and self-consistent fields.
The majority of the matter of our visible universe is in the plasma state.
The fascinating fact of these plasmas is their description by the same physical
mechanisms and the same formulae, even if their parameters range e.g. from
extremely low charged particle densities (a few particles per cubic meter) as
in intercluster gases, which are the plasmas between the clusters of galaxies
in the universe, up to plasmas with 45 orders of magnitude higher electron
densities as in neutron stars. Several examples of these plasmas will be treated
in the different Chapters throughout this book.
The plasmas in nature as well as the man-made plasmas on Earth cover an
extremely wide range of their parameters like temperatures, particle densities
and plasma generated magnetic field strengths, as expressed in Table 1.1.
Examples of plasmas in nature are all stars like the Sun, which has a
central temperature of about 17 million degrees. Its surface, the photosphere,
radiates at a temperature of 5 700 K, and the corona has a temperature of
more than one million degrees. Also the outer parts of the Earth’s atmosphere
consist of plasmas as, e.g. the ionosphere, the plasmosphere, and the radiation
belts in the magnetosphere. Terrestric plasmas are found in gas discharges
as in lightnings, in sparks, arcs, fluorescent lamps, energy saving lamps, arc

lamps, plasma displays, plasma thrusters and plasma torches. These plasmas

U. Schumacher: Basics of Plasma Physics, Lect. Notes Phys. 670, 3–20 (2005)
c Springer-Verlag Berlin Heidelberg 2005
www.springerlink.com


4

U. Schumacher
Table 1.1. Parameter range of plasmas in the universe and on Earth
Temperature T (K) Density n(m−3 ) Magnetic Field B(T)

Intergalactic gas
Interstellar medium
gas clouds in galaxies
fusion plasmas
technical plasmas
electron cloud in metals
surface of stars
star center
white dwarf
neutron star

108
104
104 . . . 106
108
103 . . . 105
105

104
107 . . . 108
104
104

10−10
10−10
10−8 . . . 10−7
10−3 . . . 10
10−2

1
106
1012
1020 . . . 1032
1015 . . . 1025
8 × 1028
1022
1030
1036
1045

10−4 . . . 10−1
1
104
108

T (eV)
pulsar
magnetosphere

relativistic
plasmas

6

2

3/2 k T = m c (Eq. 1.4)

10

B

0e

supernovae

intercluster
gas
fusion
reactor

104

tokamaks
stellarators

pinch
discharges

solar
center

plasma
focus

2

10

solar
wind

chromosphere

interplanetar
plasmas

−

0

gas discharges
arcs
fluorescence light
photosphere
semiconductor plasmas
flames
MHD generators

10

interstellar
plasmas

white
dwarfs

solar
corona

iono−
sphere

e gas in
metals

−2

10

105

1010
4

10

6

10

1015
8

10

1020
10

10

1025
12

10

1030
14

10

16

10

1035 n (m−3)
18

10

fpe (Hz)

Fig. 1.1. Temperature (T ) versus density (n) and plasma frequency (fpe ), respectively, diagram of typical natural and man-made plasmas. fpe will be defined in
Sect. 1.3.4

can be presented in a plot of their temperatures versus their particle densities,
as given in Fig. 1.1. The temperatures are usually given in electron volts
(eV) with 1 eV ≡ 11 605.4 K representing the one degree of freedom energy
equivalent eV/kB .


1 Basics of Plasma Physics

5

1.2 Ideal Plasmas
Matter is in the plasma state, if it is ionized to a certain degree. In thermodynamic equilibrium the ionization degree is given by the ratio of the particle
densities nz+1,1 and nz,1 of ions in the ground states of the ionization stages
z + 1 and z, respectively, multiplied by the electron density ne , which is
expressed by Saha’s equation
3/2

nz+1,1 ne
gz+1,1 2 (2πme kB Te )
=
nz,1
gz,1
h3

exp −

χz
kB Te

,

(1.1)

where gz+1,1 and gz,1 are the statistical weights of the ground states of the
ionization stages z + 1 and z, respectively, Te is the electron temperature,
and χz is the ionization energy of the ion in stage z. The transition from
neutral gases to plasmas may be represented by the lower boundary line of
plasmas given by about 50% ionization of hydrogen and plotted in Fig. 1.2,
the plasma boundary diagram.
The majority of the plasmas, which occur in nature, are in the ideal
state. A plasma is called ideal, if the mean thermal energy Eth = 3/2 kB T
of the particles exceeds the mean electrostatic interaction energy, which for

T (eV)

relativistic
plasmas

2

3/2 k T = m c (Eq. 1.4)

106

0e

2)

H−ionization limit (Eq.

q.

1.1)
=
ND

weakly
ionized plasmas
10−2
105
104

1010
106

1015
108

E
1(

1020
1010

1012

1.1

non−ideally
degenerated
non−ideal
plasmas

1025
1014

ted
2

0

10

EF = me0c (Eq. 1.5)

=
Γc

1.2

E
1(

1030

1016

Fig. 1.2. Plasma boundary diagram

relativistically degenerated

F

E

ge

=

)

q.

th

E

fully
ionized plasmas

2

10

de

k T
B =
3/2

ideal
plasmas

ne

4

10

ra

(E

q.

1.6

)

B

1035 n (m−3)
1018

f

pe

(Hz)


6

U. Schumacher

hydrogen as an example of equal electron density ne and ion density ni (ne =
ni = n) is given by Ee = 1/(4πε0 ) e2 n1/3 , where kB is Boltzmann‘s constant.
In this case the ratio of the mean electrostatic interaction energy and the
mean thermal energy, the coupling parameter
1
e2
4πε0 a kB T

Γc =

(1.2)

with the mean particle distance (Wigner–Seitz radius) a = (4πn/3)−1/3 is
smaller than 1.
The boundary line between ideal and non-ideal plasmas hence is given by
e2 1/3
3
kB T ≈
n
2

4πε0

or

Γc ≈ 1 .

(1.3)

Numerically this line can be expressed by T ≈ 10−9 (n) , T in eV and n in
m−3 .
The boundaries of the ideal plasmas, which mainly occur in nature, are
given by the ionization limit and by the coupling parameter, which gives the
boundary to the non-ideal plasmas, that recently – as well as the ultra-cold
plasmas – gained increasing interest (see also Chap. 6 in Part I and Chap.
10 in Part II).
The boundary line between non-relativistic and relativistic plasmas is
given by the upper horizontal line in Fig. 1.2, for which the mean thermal
energy Eth = 3/2 kB T equals the rest energy m0e c2 of the electron:
1/3

3
kB T = m0e c2 = 511 keV .
2

(1.4)

Non-relativistically degenerated and relativistically degenerated plasmas are
separated by the vertical line at the electron density of about 3.1 × 1036 m−3 ,
which is given by the equality of the electron rest energy and the Fermi energy
EF :

m0e c2 = EF =

p2F
=
2me

2

3π 2 ne
2me

2/3

,

(1.5)

1/3

where pF = 3π 2 ne
is the Fermi momentum. The dividing line between
non-degenerated and non-relativistically degenerated plasmas results from
the equation of the mean thermal electron energy Eth = 3/2 kB Te and the
Fermi energy EF :
3
kB Te =
2

2

3π 2 ne
2me

2/3

,

Te = 2.4 × 10−19 (ne )

2/3

.

(1.6)


1 Basics of Plasma Physics

7

1.3 Important Plasma Properties
1.3.1 Debye Shielding
One of the most important properties of a plasma is the shielding of every
charge in the plasma by a cloud of oppositely charged particles, the Debye
shielding. Its typical spatial scale, the Debye length λD , is estimated – in
one dimension (x) – by equating the potential energy of charge separation
Ep = eφ(λD ) over this distance λD with the kinetic particle energy 1/2 kB T .
In this approximation the electric field E(x) in a hydrogen plasma, e.g. with
ne = ni = n, is obtained from divE = /ε0 = ne/ε0 E(x)/x. The potential
energy hence results in

λD

E(x) dx = ne2 λ2D /(2ε0 ) ,

Ep = eφ (λD ) = e
0

which gives
λD =

ε0 kB T
ne2

1/2

.

(1.7)

The solution of Poisson’s equation
φ=

1 d
r2 dr

r2

dφ
dr

= λ−2
D φ

(1.8)

−2
−2
gives the total Debye length λ−2
D = λDe + λDi , which consists of the Debye
lengths of electrons (index e) and ions (index i):

λDe,i =

ε0 kB Te, i
ne,i e2

1/2

(1.9)

and the potential distribution
φ(r) =

r
q 1
exp −
4πε0 r
λD

,

(1.10)

which is plotted in Fig. 1.3 as dotted line depicting the screening of the
Coulomb potential of every charge q of a particle in a plasma. Hence plasmas are quasi-neutral, i.e., on the macroscopic scale of the plasma extension
L, with L
λD , the plasma appears to be neutral. So-called non-neutral
plasmas – charged particle ensembles confined by electromagnetic fields –
lack of quasi-neutrality. Non-neutral plasmas will be addressed in Chap. 10
of Part II. The solid line represents the familiar potential distribution of a
charged particle in vacuum.


8

U. Schumacher

φ(r)

vacuum
plasma
r

λD

0

Fig. 1.3. Potential distribution of a charged particle in vacuum (solid line) and in
a plasma (dotted line)

1.3.2 The Plasma Parameter
The plasma parameter ND describes the number of particles in the Debye
sphere. For a plasma with singly charged ions like a hydrogen plasma (ne =
ni = n) ND is given by
4
ND = n πλ3D
3

with

ε0 kB Te, i
ne2

λDe,i =

1/2

.

(1.11)

With the mean particle distance (Wigner–Seitz radius) a = (4πn/3)−1/3 the
plasma parameter takes the simple form
ND =

λD
a

3

.

(1.12)

In the parameter diagram of typical plasmas (Fig. 1.4) the lines of constant
values of the Debye radius λD and of the plasma parameter ND , respectively,
are plotted.
1.3.3 Landau Length
The Landau length λL is a typical scale for Coulomb collisions to be addressed
in Sect. 1.4.1. It is the critical distance of two charged particles, for which the
potential energy Ep = (Z e2 )/(4 π ε0 λL ) equals the kinetic energy Ei = kB T
and is equivalent to the backscattering criterion of Rutherford scattering
(central force scattering), resulting in
λL =

Ze2
.
4πε0 kB T

(1.13)

For the ion charge Z = 1 the plasma parameter can also be written as
ND =

λD
.
3λL

(1.14)