Declan A. Diver

A Plasma Formularyfor Physics, Technology

and Astrophysics

'WILEY-VCHBerlin • Weinheim • New York • Chichester • Brisbane • Singapore • Toronto

dcd-wgC1.jpg

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Declan A. Diver

A Plasma Formularyfor Physics, Technology and Astrophysics

A Plasma Formulary for Physics, Technology and Astrophysics. Declan DiverCopyright ' 2001 WILEY-VCH Verlag Berlin GmbH, BerlinISBN: 3-527-40294-2

Declan A. Diver

A Plasma Formularyfor Physics, Technology

and Astrophysics

'WILEY-VCHBerlin • Weinheim • New York • Chichester • Brisbane • Singapore • Toronto

Author:Dr. Declan A. Diver, Department of Physics & Astronomy, University of Glasgow, U.K.e-mail: diver@astro.gla.ac.uk

This book was carefully produced. Nevertheless, author and publisher do not warrant the information con-tained therein to be free of errors. Readers are advised to keep in mind that statements, data, illustrations,procedural details or other items may inadvertently be inaccurate.

Cover:Solar image from the NASA TRACE satellite. With kind permission of NASA (background). A plasma plumecreated by laser ablation of a solid surface. With kind permission of Dr. K.W.D. Ledingham, Department ofPhysics & Astronomy, University of Glasgow, UK (left). Atmospheric glow discharge between glass elec-trodes. With kind permission of Prof. W. Graham and Dr. P. Steen, Queen’s University Belfast, UK (right).

1st edition

Library of Congress Card No: applied for

British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the BritishLibrary.

Die Deutsche Bibliothek - CIP Cataloguing-in-Publication-DataA catalogue record for this publication is available from Die Deutsche Bibliothek

' WILEY-VCH Verlag Berlin GmbH, Berlin (Federal Republic of Germany), 2001

ISBN 3-527-40294-2

Printed on non-acid paper.

Printing: StraussOffsetdruck GmbH, MorlenbachBookbinding: Wilhelm Osswald & Co.,Neustadt (WeinstraBe)

Printed in the Federal Republic of Germany.

WILEY-VCH Verlag Berlin GmbHBiihringstrasse 10D-13086 Berlin

To Anne, Caitlin and Ronan

Contents

Preface xv

1 Basic Physical Data 11.1 Basic Physical Units 2

1.1.1 SI Units 21.1.2 cgs-Gaussian Units 3

1.2 Maxwell 9s Electromagnetic Equations 31.3 Special Relativity 41.4 Physical Constants 51.5 Dimensional Analysis 71.6 lonization Energies of Gas-Phase Molecules 91.7 Characteristic Parameters for Typical Plasmas 10

2 Basic Plasma Parameters 132.1 Notation 142.2 Natural Timescales 15

2.2.1 Characteristic Frequencies 152.2.2 Characteristic Times 16

2.3 Natural Scalelengths 172.3.1 Debye Length 17

vi CONTENTS

2.3.2 Mean Free Path 172.3.3 Plasma Skin Depth 172.3.4 Larmor Radius 17

2.4 Natural Speeds 182.4.1 Alfven Speed 182.4.2 Sound Speed 18

2.5 Miscellaneous Parameters 192.5.1 Collision Cross-Section 192.5.2 Differential Scattering Cross-Section 192.5.3 Magnetic Moment 192.5.4 Mobility 19

2.6 ˝ on-Dimensional Parameters 202.6.1 Dielectric Constant 202.6.2 Hartmann Number 212.6.3 Knudsen Number 212.6.4 Lundquist Number 212.6.5 Mach Number 212.6.6 Magnetic Reynolds Number 212.6.7 Plasma Beta 22

3 Discharge Plasmas and Elementary Processes 233.1 Notation 243.2 Plasma Sheath 25

3.2.1 Planar Sheath Equation 253.2.2 Child-Langmuir Law 263.2.3 Collisional Sheaths 27

3.3 Double-Layer 283.4 Diffusion Parameters 29

3.4.1 Free Diffusion 293.4.2 Mobility 303.4.3 Ambipolar Diffusion 303.4-4 Ambipolar Diffusion in a Magnetic Field 32

3.5 lonization 323.5.1 Townsend Breakdown 323.5.2 Alfven lonization 373.5.3 Secondary Electron Emission 373.5.4 Townsend Breakdown Criterion 393.5.5 Paschen Curve 39

CONTENTS vii

3.6 lonization Equilibrium 403.6.1 Local Thermodynamic Equilibrium 403.6.2 Saha Equation 41

4 Radiation 434.1 Notation 444*2 Radiation from a Moving Point Charge 4$

4-2.1 Lienard- Wiechert Potentials 4$4.2.2 Electric and Magnetic Fields of a Moving

Charge 4$4-2.3 Power Radiated by an Accelerating Point

Charge 464-2.4 Frequency Spectrum of Radiation from an

Accelerating Charge 504-3 Cyclotron and Synchrotron Radiation 50

4-3.1 Spectral Power Density 514-3.2 Power in Each Harmonic 524.3.3 Total Radiated Power 534-3.4 â í < 1-* Cyclotron Emission 534-3.5 â ı ~ 1: Synchrotron Emission 53

4.4 Bremsstrahlung 544-5 Radiation Scattering 55

4-5.1 Thomson Scattering 564.5.2 Incoherent Thomson Scattering from an

Unmagnetized Plasma 584-5.3 Coherent Thomson Scattering from an

Unmagnetized Plasma 604-5.4 Compton Scattering 614-5.5 Klein-Nishina Cross-Section 61

5 Kinetic Theory 635.1 Notation 645.2 Fundamentals 645.3 Boltzmann Equation 655.4 Maxwellian Distribution 655.5 Vlasov Description 67

5.5.1 Equilibrium Solutions 675.6 Collisional Modelling 68

5.6.1 Boltzmann Collision Term 68

viii CONTENTS

5.6.2 Simplified Boltzmann Collision Term 695.6.3 Fokker-Planck 695.6.4 Fokker-Planck Potentials 70

5.7 Driven Systems 715.7.1 Generalized Distribution 71

6 Plasma Transport 756.1 Notation 766.2 Basic Definitions 766.3 Binary Collisions 77

6.3.1 Elastic Collisions Between ChargedParticles 77

6.4 Particle Dynamics 806.4.1 Drifts 816.4-2 Adiabatic Invariants 836.4.3 Magnetic Mirror 84

6.5 Transport Coefficients 856.5.1 Fully Ionised Plasma, Zero Magnetic

Field, Krook Operator 856.5.2 Lorentzian and Spitzer Conductivity 856.5.3 Fully Ionized and Magnetized Plasma:

Braginskii Coefficients 866.5.4 Corrections to Braginskii Coefficients 906.5.5 Equal Mass Plasma Transport 91

7 Plasma Waves 937.1 Notation 947.2 Waves in Cold Plasmas 95

7.2.1 Model Equations 957.2.2 Cold Plasma Variable Dependencies 967.2.3 Dielectric Tensor for a Cold Magnetised

Plasma 967.2.4 General Dispersion Relation 977.2.5 Equal-Mass Cold Plasmas 103

7.3 Fluid Plasmas 1037.3.1 Hydromagnetic Equations 1047.3.2 Single Fluid MHD Plasma 1057.3.3 Variable Dependencies in Ideal MHD 1067.3.4 General Dispersion Relation: Ideal MHD 107

CONTENTS ιχ

7.4 Waves in Hot Plasmas 1097.4-1 Dielectric Function for an Unmagnetized

Plasma 1097.4-2 Langmuir Waves 1097.4-3 Ion-Acoustic Waves 1107.4-4 Dielectric Tensor for a Hot Plasma 111

8 Flows 1178.1 Notation 1188.2 Fundamental Results 118

8.2.1 Alfven’s Theorem 1188.2.2 Cowling’s Anti-Dynamo Theorem 1198.2.3 Ferraro ’s Law of Isorotation 1198.2.4 Kelvin’s Vorticity Theorem 119

8.3 Hydromagnetic Flows 1208.3.1 Hartmann Flow 1218.3.2 Couette Flow 1238.3.3 Field-Aligned Flows 123

8.4 Solar Wind 1258.5 Neutral Gas/Magnetized Plasma Flows 1278.6 Beams 128

8.6.1 Beam Parameters 1288.6.2 Special Cases 131

8.7 Hydromagnetic Shocks 1348.7.1 Further Notation 1358.7.2 Shock Classification 1368.7.3 Shock Propagation Parallel to ´ º 1378.7.4 Shock Propagation Perpendicular to BI 1398.7.5 General Case: Fast Magnetic Shocks 1408.7.6 General Case: Slow Magnetic Shocks 1418.7.7 Further Reading 1\2

8.8 Ion-Acoustic Shock 142

9 Equilibria and Instabilities 1459.1 Notation 1\69.2 General Considerations 1479.3 Fluid Equilibria 147

9.3.1 Ideal MHD 1479.3.2 Cylindrical Equilibria 149

χ CONTENTS

9.4 Fluid Instabilities 1529.4-1 Firehose Instability 1529.4-2 Gravitational Instability 1539.4-3 Kelvin-Helmholtz Instability 1559.4.4 Cylindrical Pinch Instabilities 1559.4-5 Generalized Pinch Instabilities 1579.4-6 Resistive Drift Wave Instability 1619.4.1 MHD Resistive Wall Instability 1619.4.8 MHD Resistive Tearing Mode 1629.4-9 Streaming Instability 163

9.5 Kinetic Instabilities 1649.5.1 Bump-in- Tail Instability 1649.5.2 Electron Runaway 1659.5.3 Ion-Acoustic Instability 165

10 Mathematics 16710.1 Vector Algebra 16810.2 Vector Calculus 168

10.2.1 Cartesian Co-ordinates 16910.2.2 Cylindrical Co-ordinates 17010.2.3 Spherical Co-ordinates 172

10.3 Integral Theorems 17410.3.1 Stokes7 Theorems 17410.3.2 Gauss’ Theorems 17510.3.3 Green’s Theorems 175

10.4 Matrices 17510.4-1 Matrix Transpose 17610.4-2 Complex Conjugate 17610.4.3 Symmetric 17610.4.4 Orthogonal 17610.4.5 Nilpotent 17610.4.6 Idempotent 17610.4.7 Triangular 17710.4.8 Trace 17710.4-9 Determinant and Inverse 17710.4.10Partitioned Matrices 17810.4-11 Eigenvalues and Eigenvectors 17810.4.12Hermitian Matrix 179

CONTENTS xi

10.4.13 Unitary Matrix 17910.5 Eigenfunctions of the Curl Operator 17910.6 Wave Scattering 180

10.6.1 Simple Constant Barrier 18010.6.2 Phase Integral Method 18210.6.3 Mode Conversion 183

10.7 Plasma Dispersion Function 185

Appendix A Guide to Notation 187

References 193

Index 199

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List of Tables

1.1 Fundamental and supplementary SI units 2

1.2 Standard prefixes for SI units 21.3 Comparison of SI and cgs units 3

1.4 Maxwell’s equations 3

1.5 Lorentz transformations 4

1.6 Values of physical constants 5

1.7 Dimensions of common variables 7

1.8 lonization energies of gas-phase molecules 9

1.9 Operating parameters for plasma reactors 10

1.10 Ionospheric parameters 11

1.11 Solar plasma parameters 11

3.2 First Townsend ionization coefficients 34

3.3 First Townsend ionization coefficients for noblegases 35

6.2 Braginskii numerical transport coefficients 888.2 Average quiet sun conditions in the solar

equatorial plane 126

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Preface

Plasma physics has matured rapidly as a scientific and technological disciplinewith a vast span of relevant application in many different fields. As a con-sequence, no single textbook is able to address all aspects of plasma physicsrelevant to such a burgeoning community.

With this reference text I have attempted to bridge the gap between theexcellent variety of traditional, broadly-based plasma books, and more special-ist, device-oriented reference texts. David L Book’s NRL Plasma Formularywas an inspiration, as too was Andre Anders’ Formulary for Plasma Physics;however, I believe that this book offers a different perspective which makesit complementary to existing handbooks. I have tried to give the reader anoverview of the key aspects of plasma physics without being too specialist inany particular area. Since this book is not a textbook, there is more roomfor not just contemporary findings, but also many traditional established re-sults from the 1950’s and 60’s that are not often found in modern texts, andwhich are once more becoming important as imperfectly ionised and boundedplasmas enjoy a resurgence in relevance.

The diverse nature of the plasma science community is matched by a con-fusing miscellany of physical units. Throughout this handbook, all formulaeare quoted in both SI and cgs-Guassian units where it is relevant. I hope thiswill maximise this book’s practicality and utility, and perhaps even assist thewhole community in the smooth transition to using SI units only....

It has been a guiding principle to reference the source (or sources) of anyformula quoted in this book, together with whatever caveats or restrictions

xvi PREFACE

tha t apply to its use. Where practica l I have reference d the original articles,subject to the importan t constrain t that verifiable sources are accessible tothe general reader . Please accept my apologies in advance for any misquotesor omissions, and please do let me know about them . As for the formula ethemselves , I am indebte d to Prof ¯ W Laing for his patien t and exactingexaminatio n of the manuscript , which led to the eliminatio n of a very largenumbe r of errors. Thank s are also due to my colleagues Brenda n Dowds, HughPotts , Richar d Barrett , Graha m Woan, Norma n Gra y and Graem e Stewart,for answering endless question s on WI^.2 £ formattin g and graphics, andpointin g out still more howlers in the iih iterat e of the book. Despit e all thisinvaluable and talente d assistance, I have no doubt that there remain , lurkingin dark corner s of the text, or even brazenl y displayed in large, open areas,error s in physics and formatting . I have no excuse; please let me know, and Ishall make good these mistakes.

I am also grateful to Prof Ken Ledingha m for lettin g me use his wonderfu limage of a laser-produce d plasma plume ; likewise, to Prof Bill Graha m forthe beautifu l high-pressur e discharge picture .

It is appropriat e to acknowledge the kind suppor t offered by David Hughe sin guiding me initially on this project , and latterly Vera Dederich s for patientl yendurin g one delay after anothe r in its prosecution . Thank s are also dueto Prof A E Roy for wise advice at the outset . Finally , I am grateful tomy Institut e for grantin g me the sabbatica l leave which was instrumenta l inallowing me to complet e the book.

DECLA N ANDRE W DIVERGlasgow, July 2001

Basic Physical Data

A Plasma Formulary for Physics, Technology and Astrophysics.DQclan DiverCopyright ' 2001 WILEY-VCH Verlag Berlin GmbH, BerlinISBN: 3-527-40294-2

1.1 BASIC PHYSICAL UNITS

1.1.1 SI Units

Table 1.1: Fundamenta l and supplementar y SI unit s

QUANTIT Y UNI T ABBREVIATIO N

Fundamental Units

masslengthtimetemperatur eelectrica l curren tluminou s intensit yamoun t of substanceplane anglesolid angle

Selected

frequenc yforceenergypowerelectrica l chargeelectri c potentia lelectrica l resistancecapacitanc einductanc emagneti c fluxmagneti c flux density

kilogrammetr esecondKelvinamper ecandel amoleradiansteradia n

derived units

hert znewtonjoulewattcoulom bvoltohmfaradhenr ywebertesla

kgms˚Acd

molradsr

Hz˝J

WCVÙF˙

WbÔ

Table 1.2: Standar d prefixes for SI unit s

PREFI X

yottazettaexapeta

SYMBOL

Õ˘¯Ñ

FACTO R

ɡ 24

ɡ 21

ɡ 18

ɡ 15

PREFI X

decicent imillimicro

SYMBOL

dcm

ì

FACTO R

ßï- 1io- 210~3

HT 6

A Plasma Formulary for Physics, Technology and Astrophysics.Declan DiverCopyrigh t ' 2001 WILEY-VCH Verlag Berlin GmbH , BerlinISBN : 3-527-40294- 2

MAXWELL'S ELECTROMAGNETIC EQUATIONS

Table 1.2: continued

PREFI X

teragiga

megakilo

hect odeca

SYMBOL

ÔGÌkhda

FACTO R

1012

109

106

103

102

101

PREFI X

nan opico

femtoatto

zeptoyacto

SYMBOL

çÑfaæ

y

FACTO R

io- 9io- 12io- 15io- 18io- 21io- 24

1.1.2 cgs-Gaussian Units

For a useful overview of non-S i unit s see [15].

Table 1.3: Compariso n of SI and cgs unit s

QUANTIT Y UNI T ABBREV. SI EQUIVALEN T

lengthmasstimeforceenergypowerelectrica l chargecurren telectri c potentia lmagneti c flux density

centimetr egramm eseconddyneergerg per secondstatcoulom bstatam pstatvoltgauss

cm

gsdynergergs"1

statcou lstatam pstatvoltG

10-2m10-3kgIs10~5N10~7Jio- 7w(3 ÷ ɡ 9)-^(3 ÷ ɡ 9)-^300V10-4T

1.2 MAXWELL'S ELECTROMAGNETIC EQUATIONS

Table 1.4: Maxwell’s equation s

V ÷ ¯

SI

dBdt

cgs-Gaussia n

IdB~ ~~c~dt

Faraday’s law

ð rr 9D r l 9D 4ð ô `V ÷ ˙ = h J = - -7; I J Ampere s lawdt c dt c

continued on next page

BASIC PHYSICAL DATA

Table 1.4: continued

SI

V • D =pc

V - B =0

D r()E

´ = ì ˆ ìïÀ.

cgs-Gaussia n

= 4ðæ ó Poisson equatio n

-0

= erE

É = ì ˆ ˙

Boundary Conditions The boundar y condition s at an interfac e for Maxwell’selectromagneti c equation s are that the tangentia l componen t of J£, and thenorma l componen t of B, must each be continuous , where norma l mean s par-allel to the local norma l vector to the interface , and tangentia l mean s in theplane perpendicula r to the local normal .

1.3 SPECIAL RELATIVITY

Assume standar d inertia l frames S and S", with respective cartesian co-ordinate s (or, ?/ , 2), ( x ’ , y f , z ’ ) aligned such that the origins 0, Of are co-inciden tat time t = t1 = 0, with S1 moving with velocity í with respect to S. Subscript|| will denot e the directio n of this mutua l motion , and subscript J_ denote sthe orthogona l plane . The Lorent z transformation s of various physically sig-nifican t quantitie s are given in the following table [61]:

Table 1.5: Lorent z transformation s

QUANTIT Y TRANSFORMATIO N

space-time : r = 7v(rji + vt1) + ˆ’–

invariant : r2 c2t2

velocity: u (w|| + í + «÷/7«)/( 1 + u’

momentum-mass : æ = % (pi, + m’v) + p’–

me = jv (m’c + i;p|| /c)

invariant : p2 m2c2

curren t & charge densities: J = 7v(J|| + vpc] + J’–

continued on next page

PHYSICAL CONSTANTS 5

Table 1.5: continued

QUANTIT Y TRANSFORMATIO N

invariant :

electri c & magnetic fields:

1.4 PHYSICAL CONSTANTS

The values of the constant s quoted here are the 1998 CODAT A recommende dvalues [66], reproduce d with permission .

Table 1.6: Values of physical constant s

QUANTIT Y

speed of light in vac-uumvacuum permeabilit y

vacuum permittivit y

vacuum impedanc e

gravitationa l constan t

Planc k constan t

Planc k mass

Planc k length

Planc k time

Avogadro constan t

Bohr magneto n

Bohr radius

Boltzman n constan t

SYMBOL

c

ì ï

eo

ZQ

G

h

ra-p

h

*7>

NA

ìå

Æ0

kB

VALUE

299 792 458

4ð ÷ 1CT7

8.854 187817•• - ÷ ˙Ô 12

376.730313461...

6.673(10) ÷ 10-11

6.62606876(52) ÷ 10~34

2.1767(16) ÷ 10-8

1.6160(12) xlO~ 3 6

5.3906(40) xlO- 44

6.022 141 99(47) ÷ 1023

927.400899(37) ÷ 10~26

0.5291772083(19) ÷ 10~10

1.380 650 3(24) ÷ ˙Ô 23

continued on

UNIT S

m s"1

Hm- 1

Fm- 1

Ù

m3 kg"1 s~2

Js- 1

kg

m

s

mol"1

JT- 1

m

JK- 1

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