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

But even though the spatial extension, the density, the ioni-zation degree and the plasma temperature may vary by many orders of mag-nitude, the physical similarities – or the plasma pro

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A Dinklage T Klinger G Marx L Schweikhard (Editors)

Plasma Physics

Confinement, Transport and Collective Effects

ABC

Priv-Doz Dr Andreas Dinklage

Professor Dr Thomas Klinger

Domstr 10a

17489 Greifswald, Germany

marx @physik.uni-greifswald.de lutz.schweikhard@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

ISBN -13 978-3-540-25274-0 Springer Berlin Heidelberg New York

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Plasma, sometimes called the fourth state of matter, is a multifaceted stance which poses a variety of challenges Plasma physics deals with thecomplex interaction of many charged particles with external or self-generatedelectromagnetic fields It is this unique entanglement which makes plasmaphysics a fascinating field for basic research At the same time, plasma plays

sub-an essential role in msub-any applications, rsub-anging, e.g., from advsub-anced ing devices and surface treatments for semiconductor applications or surfacelayer generation to the efforts to tame nuclear fusion as an energy source forour future harnessing the nuclear processes which fuel our sun

light-Modern plasma research is a multidisciplinary endeavor which includesaspects of electrodynamics, many-particle physics, quantum effects and non-linear dynamics But even though the spatial extension, the density, the ioni-zation degree and the plasma temperature may vary by many orders of mag-nitude, the physical similarities – or the plasma properties – of, e.g., thesolar corona, non-neutral plasmas in ion-traps, the electron gas of metals orplanetary interiors lead to similarities of these systems

Plasmas on earth are evanescent The confinement of plasmas for tended times is a very difficult task and one of the central keys for plasmaresearch and applications Consequently, transport phenomena which go farbeyond classical transport are highly relevant This also leads to the ulti-mate challenge of many-particle physics: the understanding of turbulence Inaddition, a variety of “ordered” collective effects can be studied in uniqueclarity, for example, phase transitions in “dusty” plasmas or the large vari-ety of plasma waves The corresponding investigations are at the forefront ofcurrent research and development

ex-This volume of Springer Lecture Notes in Physics provides an overview

of modern plasma research with a special focus on confinement and relatedissues Beginning with a broad introduction, the book leads graduate studentsand researchers – including those not specialized in plasma research – tothe state of the art of modern plasma physics The book also presents amethodological cross section ranging from plasma applications and plasmadiagnostics to numerical simulations, an important link between theory andexperiment which is gaining more and more importance The references arechosen to guide the reader from basic concepts to current research Exercises

in computational plasma physics are supplied on a Web site (see Chap 16 inPart III of this book).

The contributions are structured in three parts: After a broad

introduc-tion to Fundamental Plasma Physics, the focus of this volume on

Confine-ment, 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 respectivefields The book is based on a series of lectures for graduate students in theframework of a W.E.–Heraeus Summer School

We would like to thank the W.E.–Heraeus Foundation for funding andthe International Max Planck Research School “Bounded Plasmas” for sup-porting 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 benefittedfrom their encouragement and support

We thank Dr Angela Lahee from Springer Heidelberg for her friendlycollaboration throughout this project We also appreciate the professional andfriendly support from Ms Jaqueline Lenz, Ms Gabriele Hakuba, Ms ElkeSauer 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 AndreaPulss, for whom it must have been much more than a “challenging effort” to

do the technical editorial work

Gerrit Marx Lutz Schweikhard

Part I Fundamental Plasma Physics

1 Basics of Plasma Physics

U Schumacher 3

1.1 Definition, Occurrence and Typical Parameters of Plasmas 3

1.2 Ideal Plasmas 5

1.3 Important Plasma Properties 7

1.3.1 Debye Shielding 7

1.3.2 The Plasma Parameter 8

1.3.3 Landau Length 8

1.3.4 Plasma Frequency 9

1.4 Single Particle Behavior in Plasmas 10

1.4.1 Coulomb Collisions, Collision Times and Lengths 11

1.4.2 Electrical Conductivity of Plasmas 14

1.4.3 Single Charged Particle Motion in Electric and Magnetic Fields 15

1.5 Kinetic Description 19

References 20

2 Waves in Plasmas A Piel 21

2.1 Introduction 21

2.2 Dispersion Relation for Waves in a Fluid Plasma 22

2.2.1 Maxwell’s Equations 22

2.2.2 The Equation of Motion 22

2.2.3 Normal Modes 23

2.2.4 The Dielectric Tensor 23

2.2.5 Phase and Group Velocity 24

2.3 Waves in Unmagnetized Plasmas 24

2.3.1 Transverse Waves 25

2.3.2 Longitudinal Waves 31

2.3.3 Electron Beam Driven Waves 35

2.4 Waves in Magnetized Plasmas 37

2.4.1 Propagation Along the Magnetic Field 39

2.4.2 Cut-Offs and Resonances 40

2.4.3 Propagation Across the Magnetic Field 43

2.5 Concluding Remarks 47

References 48

3 An Introduction to Magnetohydrodynamics (MHD), or Magnetic Fluid Dynamics B.D Scott 51

3.1 What MHD Is 51

3.2 The Ideas of Fluid Dynamics 52

3.2.1 The Density in a Changing Flow Field – Conservation of Particles 52

3.2.2 The Advective Derivative and the Co-moving Reference Frame 54

3.2.3 Forces on the Fluid – How the Velocity Changes 55

3.2.4 Thermodynamics of an Ideal Fluid – How the Temperature Changes 57

3.2.5 The Composite Fluid Plasma System 58

3.3 From Many to One – the MHD System 59

3.3.1 The MHD Force Equation 60

3.3.2 Treating Several Ion Species 60

3.3.3 The MHD Kinematic Equation 61

3.3.4 MHD at a Glance 62

3.4 The Flux Conservation Theorem of Ideal MHD 62

3.4.1 Proving Flux Conservation 62

3.4.2 Magnetic Flux Tubes 64

3.5 Dynamics, or the Wires-in-Molasses Picture of MHD 64

3.5.1 Magnetic Pressure Waves 65

3.5.2 Alfv´en Waves: Magnetic Tension Waves 67

3.6 The Validity of MHD 68

3.6.1 Characteristic Time Scales of MHD 68

3.6.2 Checking the Assumptions 69

3.6.3 A Comment on the Plasma Beta 70

3.7 Parallel Dynamics and Resistivity, or Relaxing the Ideal Assumption 71

3.8 Towards Multi-Fluid MHD 73

3.9 Further Reading 73

References 74

4 Physics of “Hot” Plasmas H Zohm 75

4.1 What is a Hot Plasma? 75

4.2 Kinetic Description of Plasmas 77

4.2.1 The Kinetic Equation 77

4.2.2 Landau Damping 78

4.3 Fluid Description of Plasmas 79

4.3.1 The MHD Equations 79

4.3.2 Consequences of the MHD Equations 82

4.4 MHD Instabilities 86

4.4.1 Classification of MHD Instabilities 86

4.4.2 Examples of MHD Instabilities 88

4.5 Summary 92

References 93

5 Low Temperature Plasmas J Meichsner 95

5.1 Introduction 95

5.2 Gas Discharges and Low Temperature Plasmas: Basic Mechanisms and Characteristics 98

5.2.1 Classical Townsend Mechanism and Electric Breakdown in Gases 98

5.2.2 Townsend and Glow Discharge 100

5.2.3 Arc Discharge 102

5.2.4 Streamer Mechanism and Micro-Discharges, Dielectric Barrier and Corona Discharge 102

5.2.5 Glow Discharge at Alternating Electric Field, RF and Microwave Discharge 104

5.3 Plasma Surface Transition 106

5.3.1 Plasma Boundary Sheath, Bohm Criterion 106

5.3.2 RF Plasma Sheath 108

5.3.3 Electric Probes 110

5.4 Reactive Plasmas and Plasma Surface Interaction 114

References 116

6 Strongly Coupled Plasmas R Redmer 117

6.1 Introduction 117

6.2 Many-Particle Effects and Plasma Properties 118

6.2.1 Green’s Function Technique: Spectral Function 119

6.2.2 Cluster Decomposition of the Self-Energy 122

6.3 Composition of Strongly Coupled Plasmas 125

6.4 Electrical Conductivity 127

6.5 Conclusion 130

References 131

Part II Confinement, Transport and Collective Effects

7 Magnetic Confinement

F Wagner and H Wobig 137

7.1 Conditions for Fusion 137

7.2 The Need for Magnetic Confinement 138

7.3 Particle Motion in Electro-Magnetic Fields 139

7.4 Constants of Motion 144

7.4.1 Exact Invariants 144

7.4.2 Adiabatic Invariants 144

7.5 Concepts of Magnetic Confinement 147

7.5.1 Introduction 147

7.5.2 The Mirror Machine 147

7.5.3 Toroidal Confinement 148

7.5.4 Magnetic Surfaces and Toroidal Equilibrium 149

7.5.5 Confinement in Tokamaks 151

7.5.6 Coil System of Tokamaks 152

7.5.7 Theory of Tokamak Equilibria 153

7.5.8 Cylindrical Approximation 154

7.5.9 Confinement in Stellarators 156

7.6 Transport in Plasmas 163

7.6.1 Collisional Losses 164

7.6.2 Particle Picture of Classical Diffusion 165

7.6.3 Neoclassical Transport 166

7.6.4 Turbulent Transport 168

7.6.5 Empirical Scaling Laws 169

References 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

8.11 No Coupling – the Hydrodynamic Limit 201

8.12 The Effects of Dissipative Coupling 205

8.13 Summary 208

8.14 Further Reading 210

References 211

9 Transport in Toroidal Plasmas U Stroth 213

9.1 Experimental Confinement Times and Diffusion Coefficients 214

9.1.1 Global Confinement Times 214

9.1.2 Diffusion Coefficients 218

9.1.3 The Collisional Transport Matrix 221

9.1.4 Diffusion as Random-Walk 223

9.2 Particle Orbits in Toroidal Magnetic Fields 225

9.2.1 Particles in a Toroidal Magnetic Mirror 225

9.2.2 Passing Particles 226

9.2.3 Trapped Particles and Banana Orbits 227

9.2.4 Trajectories in Stellarator Fields 228

9.2.5 Influence of a Radial Electric Field 229

9.3 Collisional Transport 231

9.3.1 Classical Transport in the Particle Picture 231

9.3.2 Classical Transport in the Fluid Picture 232

9.3.3 Pfirsch–Schl¨uter Transport in the Particle Picture 234

9.3.4 Pfirsch–Schl¨uter Transport in the Fluid Picture 235

9.3.5 The Toroidal Resonance 236

9.3.6 Neoclassical Transport in the Particle Picture 237

9.3.7 Elements of Stellarator Transport 239

9.3.8 Neoclassical Transport in the Fluid Picture 240

9.3.9 The Ambipolar Electric Field 242

9.4 Turbulent Transport 245

9.4.1 Fluid Turbulence 245

9.4.2 Phenomenology of Turbulent Plasma Transport 249

9.4.3 Two Fundamental Linear Instabilities 252

9.4.4 Elements of a Drift Wave Model 255

9.4.5 Experimental Results 257

9.4.6 Transport Barriers 260

References 264

10 Non-Neutral Plasmas and Collective Phenomena in Ion Traps G Werth 269

10.1 Introduction 269

10.1.1 Basics of Ion Traps 269

10.2 Ion Cloud as Non-Neutral Plasma 278

10.3 Weakly Coupled Non-Neutral Plasmas 279

10.3.1 Plasma Oscillations 280

10.3.2 Rotating Walls 282

10.4 Collective Effects 284

10.4.1 Individual and Center-of-Mass Oscillations 284

10.4.2 Instabilities in the Ion Motion 287

10.5 Strongly Coupled Non-Neutral Plasmas 287

10.6 Summary 293

References 294

11 Collective Effects in Dusty Plasmas A Melzer 297

11.1 Introduction 297

11.2 Particle Charging 298

11.2.1 Orbital Motion Limit Currents 298

11.2.2 Other Charging Currents 300

11.2.3 Particles as Floating Probes 300

11.2.4 Charging in the RF Sheath 302

11.3 Forces on Particles 302

11.3.1 Gravity 302

11.3.2 Electric Field Force 302

11.3.3 Ion Drag Force 303

11.3.4 Neutral Drag Force 304

11.3.5 Thermophoresis 304

11.3.6 Dust Levitation and Trapping 305

11.3.7 Vertical Oscillations and Dust Charges 305

11.4 Particle–Particle Interaction 309

11.4.1 Strongly Coupled Systems and Plasma Crystals 309

11.4.2 Horizontal Interaction 311

11.4.3 Vertical Interaction 311

11.4.4 Phase Transitions 312

11.5 Waves in Weakly Coupled Dusty Plasmas 313

11.5.1 Dust-Acoustic Waves 313

11.5.2 Dust Ion-Acoustic Wave 316

11.6 Waves in Strongly Coupled Dusty Plasmas 316

11.6.1 Compressional Mode in 1D 317

11.6.2 Compressional Dust Lattice Waves 319

11.6.3 Shear Dust Lattice Waves 320

11.6.4 Mach Cones 320

11.6.5 Transverse Dust Lattice Waves 322

11.6.6 Normal Modes in Finite Clusters 324

11.7 Summary 327

References 327

12 Plasmas in Planetary Interiors

R Redmer 331

12.1 Introduction 331

12.2 Solar System 332

12.3 Extrasolar Planets 334

12.4 Equation of State for Partially Ionized Plasmas 337

12.4.1 Dense Hydrogen and Helium 337

12.4.2 Free Energy 337

12.4.3 Fluid Variational Theory 338

12.4.4 Plasma Component 339

12.4.5 Hugoniot Curves 341

12.5 Electrical and Thermal Conductivity 343

12.6 Conclusion 345

References 346

Part III Methods and Applications 13 Plasma Diagnostics H.-J Kunze 351

13.1 Introduction 351

13.2 Scattering of Laser Radiation by Plasma Electrons 352

13.2.1 Laser-aided Diagnostics 352

13.2.2 Incoherent Thomson Scattering 353

13.2.3 Collective Thomson Scattering 357

13.2.4 X-ray Scattering 361

13.3 Plasma Spectroscopy 361

13.3.1 Overview 361

13.3.2 Charge State Distribution 364

13.3.3 Line Emission 366

13.3.4 Line Profiles 370

13.3.5 Continuum Radiation 372

References 372

14 Observation of Plasma Fluctuations O Grulke and T Klinger 375

14.1 Introduction 375

14.2 Basics 376

14.3 Fluctuation Diagnostics 383

14.3.1 Invasive Fluctuation Diagnostics 384

14.3.2 Non-invasive Fluctuation Diagnostics 390

14.3.3 Electron Cyclotron Emission 392

14.3.4 Beam Emission Spectroscopy 393

14.3.5 Heavy Ion Beam Probe 394

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 399

15.1 Introduction to Light Sources 399

15.1.1 The Lighting Market 399

15.1.2 Overview of Discharge Lamps and Applications 401

15.1.3 Aspects of Lamp Research 404

15.2 High Intensity Discharge Lamps 404

15.2.1 Construction and Working Principle 404

15.2.2 Light Technical Properties 406

15.3 Modelling of High Intensity Discharge Lamps 407

15.3.1 Physical Modelling 407

15.3.2 Thermochemical Modelling 412

15.4 Thermochemical Experiments 414

15.4.1 Knudsen Effusion Mass Spectrometry (KEMS) 414

15.4.2 Corrosion Analysis 418

15.5 Conclusions 421

References 422

16 Computational Plasma Physics R Schneider and R Kleiber 425

16.1 Introduction 425

16.2 Plasma Edge Physics 426

16.2.1 Models 428

16.3 Turbulence 434

16.3.1 Gyro-kinetic Theory 435

16.3.2 The PIC Method 437

16.4 Outlook 441

16.5 Seminars 441

References 441

17 Nuclear Fusion H.-S Bosch 445

17.1 Introduction 445

17.2 Energy Production in the Sun 448

17.3 Fusion on Earth 449

17.4 Conditions for Nuclear Fusion 453

17.5 Power Balances 455

17.6 Development of a Fusion Power Plant 458

17.7 Muon-catalyzed Fusion 458

References 459

18 The Possible Role of Nuclear Fusion

in the 21st Century

T Hamacher 461

18.1 Introduction 461

18.2 The Challenges 462

18.2.1 Energy Demand and Lifestyle 463

18.2.2 Efficient Use of Energy and Energy Saving 464

18.2.3 Energy Resources 464

18.2.4 Geopolitical Frictions 465

18.2.5 Environmental Damages 465

18.2.6 Possible Supply Options 466

18.3 Characteristics of Nuclear Fusion as Power Source 467

18.3.1 Overall Design of a Fusion Power Plant 467

18.3.2 Resources 469

18.3.3 Environmental and Safety Characteristics, External Costs 470 18.3.4 Economic Consideration 473

18.4 The Possible Role of Fusion in a Future Energy System 475

18.4.1 The Global Dimension 475

18.4.2 Fusion in Western Europe 476

18.4.3 Fusion in India 477

18.5 Conclusion and Outlook 480

References 481

Abbreviations 483

Index 489

Fundamental Plasma Physics

U Schumacher

Institut f¨ur Plasmaforschung, Universit¨at Stuttgart, Pfaffenwaldring 31

70569 Stuttgart, Germany

schumach@ipf.uni-stuttgart.de

Abstract Basic properties of plasmas are introduced, which are valid for an

ex-tremely wide range of plasma parameters Plasmas are classified by different ical behaviour The motion of charged particles in electromagnetic fields is revisedwith respect to drift motions Adiabatic invariants are discussed and the kineticdescription of plasmas is briefly presented

phys-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 byelectric 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 physicalmechanisms and the same formulae, even if their parameters range e.g fromextremely 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 electrondensities 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 anextremely wide range of their parameters like temperatures, particle densitiesand plasma generated magnetic field strengths, as expressed in Table1.1.Examples of plasmas in nature are all stars like the Sun, which has acentral temperature of about 17 million degrees Its surface, the photosphere,radiates at a temperature of 5 700 K, and the corona has a temperature ofmore than one million degrees Also the outer parts of the Earth’s atmosphereconsist of plasmas as, e.g the ionosphere, the plasmosphere, and the radiationbelts in the magnetosphere Terrestric plasmas are found in gas discharges

as in lightnings, in sparks, arcs, fluorescent lamps, energy saving lamps, arclamps, plasma displays, plasma thrusters and plasma torches These plasmas

U Schumacher: Basics of Plasma Physics, Lect Notes Phys 670, 3–20 (2005)

Table 1.1 Parameter range of plasmas in the universe and on Earth

Temperature T (K) Density n(m −3 ) Magnetic Field B(T)

plasmas

supernovae

fusion reactor pinch discharges tokamaks

stellarators

plasma focus

solar center

white dwarfs

e −

gas in

solar wind

solar corona interplanetar

plasmas

interstellar

plasmas sphereiono− flames

gas discharges fluorescence light

chromosphere

photosphere semiconductor plasmas arcs

MHD generators

gas discharges fluorescence light

Fig 1.1 Temperature (T ) versus density (n) and plasma frequency (fpe),

respec-tively, 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/k B

1.2 Ideal Plasmas

Matter is in the plasma state, if it is ionized to a certain degree In dynamic equilibrium the ionization degree is given by the ratio of the particle

thermo-densities n z+1,1 and n z,1of ions in the ground states of the ionization stages

z + 1 and z, respectively, multiplied by the electron density n e, which isexpressed by Saha’s equation

where g z+1,1 and g z,1 are the statistical weights of the ground states of the

ionization stages z + 1 and z, respectively, T e is the electron temperature,

and χ z 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 ofplasmas 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 E th = 3/2 k B T

of the particles exceeds the mean electrostatic interaction energy, which for

non−idealplasmas

idealplasmas

weaklyionized plasmas

fullyionized plasmas

non−ideallydegenerated

hydrogen as an example of equal electron density n e and ion density n i (n e=

n i = n) is given by E e = 1/(4πε0) e2n 1/3 , where k Bis Boltzmann‘s constant

In this case the ratio of the mean electrostatic interaction energy and themean thermal energy, the coupling parameter

Non-relativistically degenerated and relativistically degenerated plasmas are

separated by the vertical line at the electron density of about 3.1 × 1036m−3,which is given by the equality of the electron rest energy and the Fermi energy

, T e = 2.4 × 10 −19 (n

1.3 Important Plasma Properties

1.3.1 Debye Shielding

One of the most important properties of a plasma is the shielding of everycharge in the plasma by a cloud of oppositely charged particles, the Debye

shielding Its typical spatial scale, the Debye length λ D, is estimated – in

one dimension (x) – by equating the potential energy of charge separation

E p = eφ(λ D ) over this distance λ D with the kinetic particle energy 1/2 k B T

In this approximation the electric field E(x) in a hydrogen plasma, e.g with

n e = n i = n, is obtained from divE = /ε0= ne/ε0 E(x)/x The potential

energy hence results in

E p = eφ (λ D ) = e

0



gives the total Debye length λ −2 D = λ −2 De + λ −2 Di, which consists of the Debye

lengths of electrons (index e) and ions (index i):

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

plas-mas are quasi-neutral, i.e., on the macroscopic scale of the plasma extension

L, with L  λ D, the plasma appears to be neutral So-called non-neutralplasmas – 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 acharged particle in vacuum

r

φ(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 N D describes the number of particles in the Debye

sphere For a plasma with singly charged ions like a hydrogen plasma (n e=

3

In the parameter diagram of typical plasmas (Fig.1.4) the lines of constant

values of the Debye radius λ D and of the plasma parameter N D, respectively,are plotted

1.3.3 Landau Length

The Landau length λ Lis 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 E p = (Z e2)/(4 π ε0λ L ) equals the kinetic energy E i = k B T

and is equivalent to the backscattering criterion of Rutherford scattering(central force scattering), resulting in

fusion reactor

solar center

white dwarfs

N D = 10 2

N D = 10 4

N D = 10 6

Fig 1.4 Lines of constant Debye length λ D and plasma parameter ND,

respec-tively, in the diagram of typical plasmas

m e

d2x

dt2 =−eE = − n e e2x

of a plane plasma sheath in linear approximation relates the space charge

electric field E to the separation x of the electrons from the ions and hence

describes non-damped Langmuir oscillations with their electron plasma quency

ω pi=

n i Z2e2ε0m i

1/2

which can be approximated by the electron plasma frequency because of thelarge mass ratio of ions to electrons Due to the mere square root dependence

on the electron density n e the plasma frequency ω p can replace the electron

density as abscissa in all the plasma diagrams, i.e f pe(s−1 ) = ω pe /2π = 8.98 (n e)1/2

With these expressions some important plasma properties can be given:– A plasma is quasi-neutral: n = n e = n i (for ion charge Z = 1); generally

– The product of Debye length λ D e (λ D i ) and ω pe (ω pi) approximately

equals the electron (ion) thermal speed v th,e ≈ √ 3λ D e ω pe or v th,i ≈

where L is the plasma dimension, and τ stands for the collision time of

the charged plasma particles with neutrals

The first condition reflects the plasma being a many-particle ensemble, theparticle charges of which are shielded outside their Debye sphere The lastcondition expresses the Coulomb interaction as the dominant interactionmechanism in a plasma, as compared to collisions with neutral particles

1.4 Single Particle Behavior in Plasmas

The understanding of plasmas is based on the properties of the single ticle behavior with respect to Coulomb collisions as well as of the chargedparticle behavior in magnetic fields The single particle collision times, theirmean free paths and the Coulomb logarithm determine the particle distribu-tion functions, which can be calculated from the kinetic theory (Vlasov andBoltzmann equations) Gyration around the magnetic field lines, particle driftmotions and adiabatic invariants characterize the charged particle behavior

par-in magnetic fields and thus form the basis for the magnetic confinement ofhigh temperature plasmas

1.4.1 Coulomb Collisions, Collision Times and Lengths

If the collision parameter p, the lateral distance of two charged particles

ap-proaching to each other (e.g an electron interacting with another electron or

a negative ion, see Fig.1.5), is equal to the Landau length λ L(see Sect.1.3.3),

it is called critical collision parameter, because then the electron is deflected

by 90◦

Fig 1.5 Scheme of Coulomb interaction

The cross section for deflections larger than 90◦(backscattering cross section)

in a single interaction is given by

and the corresponding mean free path length hence is λ90◦ = 1/(σ90◦ n),

which results in the following relation:

and thus gives the definition of the Coulomb logarithm ln Λ, which is used for

collision times calculation subsequently The Coulomb logarithm is a weakfunction of temperature and density, as can be seen in the diagram of tem-perature versus density of typical plasmas in Fig.1.6 Since Λ can be approx- imated by Λ ≈ 9N D the lines in Fig 1.6 are directly related to the dashedlines in Fig.1.4

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Fig 1.6 The Coulomb logarithm ln Λ in the diagram of typical plasmas

The total Coulomb collision cross section

the density n and the thermal speed v th = [(3k B T )/m] 1/2, result in the

typical collision times τ = 1/(σnv th), which lead to 90◦deflection by the mulative action of many small angle collisions They are the electron–electron

cu-collision time τ ee , which is about equal to the electron–ion collision time τ ei,

the ion–ion collision time τ ii , and the ion–electron collision time τ ie If weassume test particles (first index in collision times) penetrating into thermalensembles, we arrive with Maxwellian velocity distributions for the collisiontimes at:

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Fig 1.7 The lines of constant Coulomb collision times τ eefor electron–electron and

τ for ion–ion collisions, respectively, in the diagram of typical plasma parameters

Multiplying the collision times τ ee and τ ii with the corresponding mean

thermal speeds, respectively, we obtain the mean free paths λ e and λ i forelectrons and ions, respectively:

λ e ∼= 16π ε2(k B T e)2

n e e4ln Λ and λ i ∼= 16π ε2(k B T i)2

Z4n i e4ln Λ . (1.32)For singly charged ions (Z = 1) – due to quasi-neutrality – both mean

free paths are equal They are plotted in the diagram of Fig.1.8

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1.4.2 Electrical Conductivity of Plasmas

If an electric field E is applied to the plasma, it accelerates the charged plasma

particles, which transfer the gained energy by Coulomb collisions to the otherplasma particles From the balance of the energy gain in the electric field andthe Coulomb collisional energy loss of the electrons, that play the dominantrole, to the other charged particles we obtain the specific electric resistance

which expresses the strong dependence of the specific electrical conductivity

of the plasma mainly on the electron temperature T e due to the ity of the Coulomb logarithm (see Fig 1.6) on density In the diagram oftypical plasma parameters in Fig.1.9the lines of constant specific electricalconductivity are plotted

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1.4.3 Single Charged Particle Motion in Electric

and Magnetic Fields

Charged Particle Motion in Magnetic Fields

and Diamagnetism of Plasmas

Particles of charge q α and mass m α = γm α0 , where γ is the relativistic mass factor and m α0 is the rest mass, are accelerated in electric fields E and magnetic fields B according to the (relativistic) equation of motion

d(m αv)

In magnetic fields the Lorentz force balanced by the centrifugal force leads tothe spiral particle motion around the magnetic field lines with the gyration

radius of r g = γm α0 v/(q α B), also known as Larmor radius, and the cyclotron frequency ω c = v/r g = q α B/(γ m α0) including the relativistic mass factor

γ =

1− (v/c)2 −1/2

The charged particles hence spiral around the magneticfield lines Without collisions they can not move across the field lines Theyare free, however, to move along the field lines This is the basic propertythat leads to the magnetic confinement of plasmas in closed configurations(see Chap 7 on magnetic confinement in Part II)

Since ions and electrons gyrate around the magnetic field lines (in site directions, see Fig.1.10) such, that the magnetic field generated by thisrotation of charges is oppositely directed to the applied magnetic field, it isobvious, that plasmas are diamagnetic

oppo-Fig 1.10 The charged particle rotation around the magnetic field lines expresses

the diamagnetism of the plasma

Charged Particle Drifts

If a homogeneous electric field is superimposed to a homogeneous magnetic

field the charged particles move at a constant drift velocity, the E× B–drift

This drift velocity in crossed homogeneous electric and magnetic fields is thesame for all charged particles irrespective of their velocity, mass, charge quan-

tity and sign Therefore, the E× B–drift does not lead to charge separation.

These specific properties do not hold for all other drift motions

Replacing the electric field E by the corresponding force F divided by the

particle charge q we obtain the general drift velocity

ion electron

x y

Although being negligible for the Earth’s low gravitation, the tional drift with its opposite drift motions of electrons and ions directly

gravita-demonstrates, that a net current density j = n e (m i + m e)(g× B)/B2 isgenerated Moreover, it is a simple model for particle drift motions under theaction of other forces like the centrifugal force for charged particles in curvedmagnetic fields In a curved magnetic field in vacuum with a curvature ra-

dius R c due to∇ × B = 0 the curvature is inherently related to a magnetic

mag-is an example for the second adiabatic invariant J [see (1.41) in the followingsubsection], and the generation of the so called electron current around theequator of the Earth, which is related to the third adiabatic invariant [see(1.42) in the following subsection as well].

These drift motions, moreover, are the reason for pure toroidal magneticfields (configurations with magnetic field lines, that are closed just after onecircumference around the symmetry axis) not being suitable for plasma con-

finement, because in such magnetic fields the curvature and gradient–B drifts

lead to charge separation in axis-parallel direction and hence to an electric

field perpendicular to the magnetic field direction, which results in an E×B–

drift in radial direction for all charged particles (see Chap 7 on magneticconfinement in Part II) There are several additional drift velocities, that oc-cur, e.g in inhomogeneous electric fields and in time dependent electric fields(polarization drift)

Adiabatic Invariants

The description of the single particle motion is appreciably simplified for riodic or quasi-periodic motions by the constancy of the adiabatic invariants.These are based on the line integral

taken over one period for the canonically conjugated quantities p and q, being

a constant of motion, an adiabatic invariant

The first adiabatic invariant is related to the gyration motion of thecharged particles around the magnetic field lines This is the magnetic mo-ment

The second adiabatic invariant J is related to the periodic bounce motion

of the particle between two mirror regions in which the particle is captured

For the second (or longitudinal) adiabatic invariant J the line integral is

taken over the particle’s velocity component v parallel to the magnetic field

B over the co-ordinate s:

where F is the area enclosed by the particles’ drift orbit The third

adia-batic invariant, however, in contrast to the first one can easily be violated byfluctuations or other temporal changes of the plasma

1.5 Kinetic Description

Although the single particle behavior already gives much detailed tion, the plasma as a many-particle system is determined by collective mo-tions as well The characteristic properties of collective motions in a plasmadetermine the propagation, the damping and the absorption of waves in plas-mas and the wave–plasma interaction, too (see the following Chapter onwaves in plasmas) Their knowledge allows to develop efficient wave heatingsystems and specific new plasma diagnostics, too (see also Chap 2 on waves

informa-in plasmas informa-in Part I and the Chap.13 informa-in Part III)

The plasma is described by distribution functions f α (p, r, t) for each species α, which depend on the momentum p, the radius vector r and the

time t The distribution function is obtained from the solution of a kinetic

equation, which in the limit of negligible particle interaction represents –analogous to the Liouville equation – the continuity equation in phase space

From the equation of motion for the particle species α

dp

we obtain the Vlasov equation (collision-less Boltzmann equation), which

contains self-consistent fields E and B:

has to be solved to obtain the distribution function f α (p, r, t), which

con-tains the Boltzmann collision term (or collision integral) on the right side.Combined with Maxwell’s equations this system forms the basis of magneto-hydrodynamics (MHD, see also Chaps 3 and 4 in Part I)

For further reading see [1,2,3, 4,5,6,7]

References

1 F.F Chen: Introduction to Plasma Physics (Plenum Press, New York 1988) 20

2 K Miyamoto: Fundamentals of Plasma Physics and Nuclear Fusion (Iwanami

Book Service Center, Toyko 1997) 20

3 J Raeder: Kontrollierte Kernfusion (B.G Teubner, Stuttgart 1981)20

4 U Schumacher: Fusionsforschung – Eine Einf¨ uhrung (Wissenschaftliche

Buch-gesellschaft Darmstadt 1993) 20

5 R.J Goldston and P.H Rutherford: Introduction to Plasma Physics (Institute

of Physics Publishing, Bristol and Philadelphia 1995) 20

6 M Kaufmann: Plasmaphysik und Fusionsforschung (B.G Teubner, Stuttgart

2003) 20

7 G Franz: Oberfl¨ achentechnologie mit Niederdruckplasmen (Springer–Verlag,

Berlin Heidelberg New York 1994) 20

A Piel

Institut f¨ur Experimentelle und Angewandte Physik, Christian-Albrechts-Univ

zu Kiel, Olshausenstr 40, 24098 Kiel, Germany

piel@physik.uni-kiel.de

Abstract Waves are basic manifestation of collective effects in plasmas Wave

types occurring in the plasma state are introduced and discussed with experimentalapplications, e.g for the diagnostics of plasmas

This Chapter is fundamental to Part II and covers many aspect introductory

to Chapters on applications (see Part III), e.g on plasma diagnostics

2.1 Introduction

Wave phenomena are ubiquitous in nature and immediately affect human life.Sound waves in air let us hear, light waves give us vision, and vibrations ofsolids can form music To the physicist, the study of wave phenomena gives

an immediate insight into the elastic properties of matter, which define thewave speed Sound waves in air and light waves in vacuum have the propertythat perturbations of different frequencies propagate with a unique velocity,the sound speed or the speed of light, respectively This is quite differentfrom surface waves on a pond, where sinusoidal waves of different frequencyhave different propagation speed and are therefore called dispersive Theyresemble light in a glass prism, which is dispersed into its various colors.Light waves and sound waves differ in their way of oscillation, which istransverse to the direction of propagation in light and along the propaga-tion for sound In solid matter, transverse “shear” waves and longitudinal

“compressional” waves can even coexist at the same frequency However, thedifferent propagation speed of these two “modes” gives the two waves dif-ferent wavelength at the same frequency Ordinary gases can only supportlongitudinal compressional waves, because there is no restoring shear force.Plasmas have a much higher variety of wave modes than ordinary matterbecause plasmas combine the aspects of a gas with electromagnetic forces.Moreover, the entanglement of particle motion with magnetic fields leads

to wave types unknown in other fields of physics This chapter attempts togive a survey of the various wave types in unmagnetized and magnetizedplasmas The mathematical apparatus is kept as simple as possible Thechosen topics of this tutorial, which emphasize diagnostic applications (seealso Chap 13 in Part III), reflect the preferences of the author, who is adevoted experimentalist Important aspects, like Landau damping and Alfv´en

A Piel: Waves in Plasmas, Lect Notes Phys 670, 21–50 (2005)

waves (see Chap 4 in Part I) or nonlinear waves (see Chap 8 in Part II onturbulence), had to be omitted as a tribute to the limited space The newfield of dusty plasmas, which leads to many more modes, like dust-acousticwaves, lattice waves or Mach cones, is left to a companion article (Chapter

11 in Part II)

There is quite a number of excellent textbooks on plasma waves that give

a general survey [1,2, 3], focus on cold plasmas [4] or on kinetic effects [5],which are recommended to the reader for more thorough studies

2.2 Dispersion Relation for Waves in a Fluid Plasma2.2.1 Maxwell’s Equations

Electromagnetic waves are derived from Maxwell’s equations, which define

the relationship between the electric field E, the magnetic field B and the electric current density j.

2.2.2 The Equation of Motion

The properties of the plasma medium appear in the dynamic response of thevarious plasma species to these fields Because of their different mass andsign of charge, this response is quite different for electrons and (positive)ions For each species, the periodic motion of the charged particles represents

an alternating electric current The superposition of the alternating currents

of all species is the required information about the plasma medium to closeMaxwell’s equations

The simplest approach to calculate the alternating current is to start from

a single-particle model For a plasma consisting of electrons and one species

of singly charged positive ions the current density is

Here σ(ω) is the conductivity tensor As in optics of anisotropic crystals, the

orientation of the current vector needs not to be parallel to the electric fieldvector In magnetized plasmas, we will even find non-diagonal elements Forstudying the dielectric properties of a plasma, we assume the propagation ofplane waves

Here k is the wave vector which defines the propagation direction of the

wave and is related to the wavelength λ by |k| = 2π/λ We convert the set

of Maxwell’s equations to the wave differential equation by taking the curl inthe induction law (2.1) and using (2.2)

2.2.4 The Dielectric Tensor

Although we have introduced a conductivity σ this does not imply that field and current are in phase Rather, the complex quantity σ gives a smooth

transition from dielectric behavior of the plasma, where the current lags

be-hind the field by π/2, to a conductor In the first case, the plasma is better described by a dielectric tensor (ω), which has real components