Principles of Plasma Physics 0070353468, 9780070353466

Citation preview

systems to note that

for curvilinear

convenient

sometimes

(v

\342\200\242 V)v =

iV(v

-

\342\200\242 V v)

X v)

X (V

where the operation is reduced to gradient and curl operations. Many useful identities of them are listed in P. M, can be proved Many using these operational properties. New Morse and H. Feshbach, \"Methods of Theoretical Physics,\" p. 114, McGraw-Hill, York, 1953.

volume and

surface

(V

III

mechanics a very the Gauss theorem,

In continuum

THEOREMS

INTEGRAL

is

integrals

. A) rfx

useful

integral

theorem

relating

=

closed volume V

(n \342\226\240 A) ds

II

containing

OF

PRINCIPLES

>S

surface

V

PHYSICS

PLASMA where

ft

=

unit

normal

vector

from

outward

the surface,

or

IfJaf^^'' 8xit integral

may

also

coordinates,

=

V

The surface

cartesian

in

JJ^^\"^^*

S

be written = JJ(A-n)rfi

Similar

relations

are valid for

scalars

and

tensors.

V

and

or

JJJ(V.T,rfx^JJ(n.T)rf, for

cartesian

JJ^\342\200\236rfi

coordinates. \342\200\2428Tu 1/ V

'- \342\200\242> Sc

S

Adler,

AND APPLIED PHYSICS

and Schiffer Introduction to General Berlin and Statistical Thermodynamics Introduction to Theoretical Mechanics

Becker Bjorken and Drell Bjorken and Drell Chodorow and Susskind Clark

Applied

Collin

Field

Evans

The

Feynman and Hardy

PURE

Bazin,

Allls and

Hall

IN

SERIES

INTERNATIONAL

Relativistic

Quantum

Relativistic

Quantum Fundamentals

Relativity

Mechanics

Fields Mechanics of Microwave Electronics

X-rays Theory

Atomic

of Guided Waves Nucleus

Hibbs

Quantum

to Electron The Principles

Introduction andPerrin

Harnwell Electricity Harnwell and Livingood

and

and Path

Mechanics

Integrals

Microscopy

of Optics Electromagnetism Atomic Experimental

Physics

Conceptual Development of Quantum Kinetic Theory of Gases Kinnard and Trivelpiece Ktall Principles of Plasma Physics of Modern Principles Physics Liighlon

Jammer

The

Lindsay

Mechanical Radiation

LMngston and Midelkton

Motso Mors* and

Mont

and

Particle Accelerators Blewett An Introduction to Statistical Communication

NtWton

Methods of Theoretical Physics Theoretical Acoustics Theory of Waves and Particles

Feshbach

Kinetic

lUad

Theory of Gases in Crystals

Dislocations

Rkhtmyer, Kennard, and

Rossi and

Olbert

Sohlff

Quantum

Introduction

Cooper

Introduction

to Modern Physics

to the Physics of

Space

Mechanics

^hwartz

Introduction

Sehwei'tx

Principles

StttM

Theory

Sound

and

Vibration

Ingard Scattering

Prtsint

Mechanics

to Special Relativity of Electrodynamics

Modern Theory of Solids Quantum Theory of Atomic Structure, Vol. I Theory of Atomic Structure, Vol. II Quantum Quantum Theory of Matter of Molecules: Quantum Theory Electronic Structure Symmetry and Energy Bands in Crystals: Quantum

The

Slettr SIttlii' Slatil'

Slalir Slatir

of Molecules and Solids, Molecules and

Theory of

Vol.

1

Solids,

Vol. 2 Slater

Insulators, Vol.3

Semiconductors,

Slater and

Frank

Introduction

Slatir and Frank Smytht Stratton Ttnkham

Tbwnes Wane fVhfti

and Metals: Quantum

to Theoretical

Theory

of Molecules and

Solids,

Physics

Mechanics

Static and Dynamic Electricity Theory Electromagnetic Theory and Quantum Group

Mechanics

Microwave Spectroscopy and Schawlow Solid-state Electronics to Atomic Spectra Introduction

Editor of the series from its inception in 1929 to his was Consulting late F, K, Richtmyer from 1939 to 1946; and G. P. Harnwell dMth 111 J939, Lee A. DuBildge was Consulting Editor I, Schiff served as consultant from 1954 until his death in 1971. from 1947 to 1954. Leonard The

OF

PRINCIPLES

PHYSICS

PLASMA

Nicholas

A. Krall of

Professor

University

Alvin

W.

Trivelpiece Professor

University

McGRAW-HILL

New York

London

St.

Louis

Mexico

Physics

of Maryland

BOOK

of

Physics

of Maryland COMPANY

San Francisco Dilsseldorf Kuala Lumpur Johannesburg New Delhi Panama Rio de Janeiro Singapore Toronto Sydney

Montreal

OF PLASMA PHYSICS

PRINCIPLES

1973 by McGraw-Hill, Inc. All rights reserved. \302\251 Copyright No part of this Printed in the United States of America. publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, without mechanical, photocopying, recording, or otherwise, of the publisher. the prior written permission

This

book

798765

KPKP

4567890

was

set in Times New Roman. The editors were and M. E. Margolies; and the production Thomas J. Lo Pinto. The line drawings were done

Jack L. Farnsworth

supervisor was by James S. Kempton. The

printer

Library

and binder was

of Congress

physics.

series in pure & bibliographies.

(International

Includes

1. 1931-

Plasma

QC7I8,K72

Data

1932-

of plasma

Principles

Inc.

in Publication

Cataloging

A

Krall, N

Kingsport Press,

gasos)

(Ionized

joint author, II. 330.4'4

ISBN 0-07-033346.B

I,

applied

physics)

Trivelpiccc,

Tide,

72-369.1

A. W.,

CONTENTS

xi

Preface

CHAPTER Part 1.1

TO

INTRODUCTION

1 One

Equilibrium

Plasma

and

Concepts

PLASMA

PHYSICS

1

3

Terminology

3

and Metaequilibrium

1.2 Debye Length

3

1.3

4

Parameter

Plasma

5

1.4 Distribution Function

1.5

Temperature

and

Other

Moments of the Distribution Function

Magnetic Pressure 1.7 Particle Drifts 1.8 Plasma Frequency 1.9 Waves in Plasmas 1.10 Landau Damping and Controlled 1.11 Plasma Stability Thermonuclear 1.12 Shock Waves and Solitary Waves 1.13 Collisions 1.14 Diffusion and Bohm Diffusion Radiation 1.15 Plasma

7

1.6

Part

Two

Plasma

7 9

9 12

Fusion

Production

Part

Three

Measurement

of Plasma

Properties

Current and Voltage Measurements in Plasmas 1.25 Plasma Probes of Plasma Properties 1.26 Other Methods of Measurement References

12 16 18

20 23 25

1.16 The Low-pressure Cold-cathode Discharge 1.17 The Thermionic-arc Discharge 1.18 Plasma Guns Alkali Metal Vapor Plasma\342\200\224QMachines 1.19 1.20 RF-produced Plasmas 1.21 Dense-plasma Focus 1.22 The Solar Plasma 1.23 Laser-produced Plasmas 1.24

6

27 27 30

33

33 37

39 39 41

42

45 50

54

CONTENTS

vi

CHAPTER

2.1

2

AND THERMODYNAMICS EQUILIBRIUM PLASMAS

STATISTICAL

MECHANICS

OF

55

2.6

Parameter Gibbs Distribution and Correlation Functions in an Equilibrium Correlations Plasma Two-particle Free Energy of a Plasma Equation of State of a Plasma The as a Fluid Plasma

63 65 66

2.7

The Ideal

67

2.8

Potential of a Test

2.2 2.3

2.4 2.5

The

Plasma

57

58 60

Plasma

Other Examples of 2.10 Coulomb Energy of 2.11 Discussion 2.9

Plasma the Plasma as a Fluid in a

Particle

68 70

a Plasma

74 76

Tl

References

CHAPTER 3 MACROSCOPIC PROPERTIES OF PLASMAS 3.1 The Distribution Function and the Liouville Equation 3.2 Macroscopic Variables of a Plasma 3.3 Macroscopic Equations for a Plasma; Fluid Equations 3.4 Two-fluid Plasma Theory 3.5 One-fluid Plasma Theory; Magnetohydrodynamics 3.6 Approximations Commonly Used in One-fluid Theory 3.7 Simplified One-fluid Equations and the MHD Equation 3.8 Properties of the Plasma Described and MHD Models by the One-fluid of a Plasma Described 3.9 Dynamic Properties by the One-fluid and MHD

78 79

82 84 88

89 92 95

98 Plasma

Theories

3.10 Double-adiabatic 3.11 The Dynamic References

CHAPTER 4.1

4.2 4.3 4.4

4.5 4.6 4.7

4.8 4.9 4.10 4.11

4.12 4.13 4.14

4

128

WAVES

Dielectric

104 I]8 J23

Theory Pinch

FLUID

IN THE

Constant of a

Plasma

Oscillations

Plasma

Oscillations

PLASMA

Field-free

Fluid

129

Plasma

Bo

= 0)

One-dimensional Drifting Plasma Waves in a Warm Plasma Space-charge in a Cold Plasma Plane Waves Microwave Method of Measuring Plasma Properties Transmission Plasma-column Resonances Waves in Finite Plasmas Space-charge Constant Plasma Dielectric of a Cold Magnetized (Eo = 0, Bo = Bo 2) in a Cold Magnetized to the Magnetic Field Waves that Propagate Parallel Plasma (Eo = 0,Bo = Boz) to the Magnetic Field in a Cold Magnetized Waves that Propagate Perpendicular Plasma (Eo = 0, Bo = Bo z) Wave in Typical Plasmas Frequencies in a Finite Magnetic Field Waves in Cold Finite Plasmas Space-charge Low-frequency Drift Waves in Nonuniform Plasmas in a

130 133 136

143 147 153

157 167 178 182

196 200 202

206 213

References

CHAPTER

Part 5.1 5.2

(Eo =

5 One

STABILITY

OF

The Plasma

The Equilibrium Classification of

FLUID

THE

Stability

Problem

Problem Plasma

Instabilities

PLASMA

215 215

217 218

CONTENTS

5.3 5.4

Methods of Stability Regions of Stability Part Two

Two-stream

5.6

Fire-hose

Part 5.7 5.8 5.9

Instabilities

Instability Three

220

Analysis

221

Stability of

5.5

from Macroscopic Waves

Plasma

Unconfined

of Space-charge Wave of an Alfven

Stability

of Magnetically

Fluid

221

Equations

222 228

Plasma

Confined

from

Macroscopic

Fluid

Equations

229

Stability of the Fluid Plasma Supported against Gravity by a Magnetic Fluid Confined Fluid Plasma from Stability of Magnetically Thermodynamic Considerations; Interchange Instability of Macroscopic Equations for the Study of the Hydrodynamic Stability

230

Stability

5.11

Energy Principle

5.12

Stability of a Plane Plasma-Magnetic Field of Self-confined Plasma (B,, Only): Stability Influence of Line Tying Stabilizing

of the

Normal-mode

Part

CHAPTER

Plasma

Supported

against

Gravity by a

Magnetic-field:

246

Analysis

251

Stability Theory and

Four

5.15 Open 5.16 Closed 5.17 Other References

Fluid

233

241

Plasmas

Confined Magnetically

5.10

5.13 5.14

VU

Plasma-confinement

Controlled

Interface:

Principle Analysis Energy Principle Analysis Energy

263

Fusion Research

Thermonuclear

Experiments

Plasma-confinement

265 267 275 281

Experiments

Plasma-confinement

257 261

Experiments

285 6

TRANSPORT

PHENOMENA

IN PLASMA

287

Coulomb Collisions 6.2 Deflection of a Charged Particle by Multiple Coulomb Collisions in a Fully Ionized Plasma 6.3 Fokker-Planck Theory for Transport Times in a Fully Ionized Plasma 6.4 Relaxation 6.5 Transport Properties of a Fully Ionized Plasma 6.6 Boltzmann and the Lorentz Model for a Weakly Ionized Transport Equation Plasma 6.7 Modified Boltzmann Equation Coefiicients in a Weakly Ionized Plasma 6.8 Transport

289

6.9

321

6.1

Binary

Ambipolar

Diff\"usion

Weakly Ionized Plasma in a

6.10

Transport

6.11 6.12

Magnetic Field of a Weakly Ambipolar Diffusion MHD Power Generators

Properties

of a

and

295 301

307 311

315 318

Homogeneous 327

Ionized

Plasma

across

References

A PLASMA FOR CHAPTER 7 KINETIC EQUATIONS 7.1 The Microscopic for a Many-body System Equations for a Many-body System 7.2 The Statistical Equations 7.3 Statistical Equations for a Coulomb Plasma the Chain of Statistical Equations 7.4 Closing Kinetic Equation in Zero Order\342\200\224^TheVlasov Equation 7.5 in First Order 7.6 Kinetic Equation of the Vlasov Equation 7.7 Properties to Order g 7.8 Equation Properties of the Kinetic References

Steady

291

a Magnetic

Field

333 340

347

349 350 351

355 357

358 359 360 366

367

CONTENTS

VUI

CHAPTER

8

THE VLASOV THEORY

8.1 The Vlasov 8.2 The Linearized 8i3

OF

368 368

Equations Vlasov

369

Equations

of the Linearized Vlasov a l-'iold-free Plasma Equilibrium

Solution of

WAVES

PLASMA

Equations

for

Electrostatic

Perturbations 371

Solutions for (^^(0 375 Derivation for Electrostatic Waves in a Plasma 8.5 Simplified 381 TKe Vlasov Theory of Langmuir 8.6 Waves, Ion-sound Waves, and Landau Damping 383 (Eo - Bo - 0) Function for Plasma Waves 391 8.7 Perturbed Distribution for Waves in a General Plasma Equilibrium 8.8 The Dispersion Relation 395 Wavesina 8.9 The VlasovTheoryof Small-amplitude Field-free Plasma Equilibi-ium\342\200\224 and Electromagnetic Waves [Eo = Bo = 0, /o = foiv^)] Electrostatic 398 Waves in a Uniformly 8>10 The Vlasov Theory of Small-amplitude Magncli/xd = f.oivi\\ Plasma [Bo = Boz, Eo = 0,/\342\200\236o 402 v\342\200\236)] Plasma 407 8.1 i The Vlasov Theory of Waves in Cold Magnetized to the Equilibrium Field in a 8.12 Waves That Propagate Perpendicular Magnetic Plasma Hot Magnetized Waves and (he (Eo = 0, Bo = iBo)\342\200\224Electromagnetic 8.4

8.13

Time-asymptotic

Bernstein Modes Waves That Propagate Hot

8.14

to the

Equilibrium

Magnetic

Field in a Magnc(i/cO

9. Ih) and Electromagnetic Waves (Eo = 0, Bo at an Arbitrary with Respect to the Propagating Angle Field in a Magnetized Hot Plasma (Eo = 0, Bq i Bu) Magnetic an Inhomogeneous [Eo = 0, Bo Magnetized Hot Plasma 2/Jo(a-),

Plasma\342\200\224Electrostatic

Electromagnetic Equilibrium

8.13

407 Parallel

Waves

in

tio-iio^x)]

in an Inhomogeneous Magnetized Waves Low-frequency Electrostatic Waves Nonlinear Electrostatic (BGK Waves) Waves vs. Vlasov Waves Fluid of Wavelike Vlasov States 8.19 Summary

8.16 8.17 8.18

Plasma

9.5 9.6 9.7 9.8 9.9

9.10

THE

OF PLASMA

THEORY

VLASOV

439

STABILITY

442

Introduction of Monotone-decreasing Distribution; The Newcomb-Gardner of Multipeaked Distributions\342\200\224The Two-stream Instability

Theorem

of Multipeaked Distribution in Warm Plasmas\342\200\224Gentle-bump Mechanism of the Two-stream Instability The Nyquist Method and the Penrose Criterion for Stability Ion-acoustic Instability Theory Applications of Two-stream-instability Instabilities in Anisotropic Plasmas

Instability

Stability Stability Stability

Pinching

Electromagnetic

9.11 9.12

Discussion of

9.13 9.14

Loss-cone Instability Other Instability

9.15

Thermodynamic Bounds

Pinching of Anisotropic

Stability

463 476

478

482 483

Instabilities Instabilities

Magnetized

442 445 449 458 464

494 495

Plasma

497

505

Mechanisms on Field

Levels and Growth

Rates

in Unstable

Plasma

506 511

References

CHAPTER

10 THE AND

10.1

418 427

441

9

CHAPTER 9.2 9.3 9.4

417

432 437

References

9.1

414

Waves

The

Need

for

VLASOV

NONLINEAR

THEORY

INSTABILITIES a Nonlinear

Theory of Plasmas

OF PLASMA

WAVES

512 512

CONTENTS

10.2 10.3 10.4

10.5 10.6

10.7

10.8 10.9

in a Plasma Distribution Quasilinear Equations for Changes in Quasilinear Theory of Particles, Momentum, and Energy Landau Theory Damping in Quasilinear in Quasilinear Theory The Gentle-bump Instability Quasilinear Theory of the Two-stream Instability Electron Trapping in a Single Plasma Wave

514

518

Conservation

Wave

Plasma

Nonlinear

520 527

532

536 539 549

Echoes Interactions

Wave-particle

(Weak

Turbulence)

554

References

CHAPTER

11.1 11.2 11.3

11.4 11.5 11.6 11.7

11.8 11.9 11.10 11.11

11 FLUCTUATIONS, of a

Shielding

RADIATION

AND

CORRELATIONS,

556

Moving Test Charge

557

Field Fluctuations in a Plasma Plasma Electric Field Fluctuations in a Nonmaxwellian Waves Drag on a Test Particle; Emission of Electrostatic and Radiation Fluctuations Electromagnetic of Incoherent Radiation from a Plasma Density Fluctuations Scattering Emission of Radiation from a Plasma; Kirchhoff's Law from and in a Plasma Radiation Blackbody Radiation from a Plasma in a Magnetic Field Cyclotron (Synchrotron) from a Plasma Test Source Theory of Radiation of a Plasma Kinetic Equations Including Collisional Relaxation

563

Electric

568 570

573 575

588 590 592 595

599 605

References

APPENDIX 1.1 1.2

I

PARTICLE

MOTION

607

The Equations of Motion Particle Motion in Static Homogeneous

1.3

Particle Motion

1.4

Particle

Motion

in

608

Homogeneous

Homogeneous Magnetic Field [Bo i^ 0, E(0

Electric

and Magnetic

Electric

in a Homogeneous

Motion

Particle

Varying

Slowly

in a Static

Small-amplitude, 1.5

5^

608

Fields

and Magnetic Fields and a RapidlylVarying,

Electric

Field

Plane

Electromagnetic

618 Motion

Particle

in Static

Inhomogeneous Magnetic

Adiabatic Invariants 1.8 Plasma Properties from References

Orbit

APPENDIX II

OF

621

Fields

1.7

SUMMARY

627

628

Theory

630

III

SYSTEMS OF

UNITS,

FREQUENTLY

USED

OF

PROPERTIES

SOME

SOME INTEGRAL TENSORS, COORDINATES CURVILINEAR APPENDIX

VECTORS

CONVERSION

AND

AND

THEOREMS,

631

FACTORS,

AND 644

SYMBOLS

111.1 Systems of Units and Physical Constants 111.2 Conversion Factors III. 3 Frequently used Symbols

APPENDIX

613

,615

0]

Large-amplitude

Wave 1.6

IX

IV

SELECTED

ADDITIONAL

STUDENT

IV. 1 Turbulent Heating 1V.2 Plasma Shock Waves IV.3 Transport Theory IV.4 Hydromagnetic and Fluid

READINGS

644

647 647

FOR

THE

ADVANCED

654 654

655 655 Models

656

CONTENTS

Computer Techniques in Plasma Physics 1V.6 Normal Modes of a Plasma and Landau Plasma Equilibrium IV.7 and Stability IV.8 Plasma Oscillations IV.9 Solid State Plasmas IV.IO Applications of Plasma Physics to Space

656

IV.5

Damping

657 657

Physics

658 659 659

Name Index

661

Index

666

Subject

PREFACE

to provide

in and graduate students physics a comprehensive background in plasma for a two-semester course, normally taken physics. It was designed in the second year of graduate study by students who have completed the usual of graduate courses in classical and statistical electromechanics, sequence and mathematical methods. It is a basic introduction to the field magnetism, and assumes no background in plasma physics. it was intended as a Although it could be used for an advanced course by proper text, graduate undergraduate This

textbook

astronomy

selection

was

at the

developed

University

of

Maryland

with

of material.

The basic plan of the book is to begin with the exact statistical description of a. many-body system and then to obtain various reduced of the deseri^jtions called fluid theory,' or macroscopic plasmai slate. The most reduced description, a wide range of plasma first and is used to explore theory, is developed phenomena and involving equilibrium, waves, and instabilities. problems Macroscopic 6), are typically theory, together with transport phenomena in plasmas (Chapter of the course. covered in the first semester or Next, the book takes up a less reduced description, the Vlasov based on a continuous distribution function of velocity and microscopic theory, and again considers problems of plasma waves, configuration space, equilibrium, and instabilities, in addition to nonlinear of waves and the plasma interactions \" distribution. TJus Vlasov-Maxwell\" state description of the plasma predicts that depend on the details of the velocity-space distribution function, phenomena for Landau including, example, damping, velocity-space instabilities, velocity^ etc. The Vlasov results for these plasma properties are compared space diffusion, fluid treatment of the earUer with the results of the somewhat simpler chapters. and that fluctuations, radiation, being Correlations, depend phenomena on discreteness, are not included in either of the above reduced descriptions. these quantities. A test-particle approach is used to calculate Further discrete

fR\302\273l\302\253ACK

XU

of

properties

Ihe exact

plasmas

statistical

are obtained

from

leading

equations,

and caJculations of equations The theory of the orbits

transport of

reduced

higher-order

to the

for

coefficients

plasmas.

in static and time-

particles

single-charged

of

descriptions Balescu-Lenard

or

Folclcer-Planclc

with consistent magnetic fields is included as an appendix, our objectives of trying to be complete, and at the same time not devote extensive in many text space to treatment of topics that are simple and adequately treated available reference sources. readily One feature of this book is an extensive unique introductory chapter \" intended to eliminate the \"jargon gap that we found to be a major stumbling first encountering have block for students plasma physics. Most of our students of in but the mathematics and classical an adequate background physics, many and

electric

varying

the

who ohnpter

chapter Al

the equiUbrium

on

a

to

is

many-body calculates

chapter

seta the

stage

lltuations in that grasp

the

for

which

to

unfamiliar

a short

course

physics.

the field. statistical mechanics of student with the proper and the plasma state

problem the thermodynamic properties the rest of the book, which the plasma is not in a state

to four lectures) is useful to anyone feature is a unique of thi s The purpose

(three

As such it Another plasmas.

of plasma

perspective

as

a statistical

of the

equilibrium

deals

with

of

thermodynamic

the

physics

system. This plasma,

and

many realistic equilibrium.

books and review With regard to references, we selected papers primarily to students. we found of particular value It is our opinion that a proper of of the physics is the overriding to which the historical aspects concern, field are subordinate. a plasma have in the use of digital computers to simulate advances Recent considerable

provided

are

the

provide

are

physics

difficulty.

introductory chapter is really and concepts terminology of plasma a qualitative introduction desires to This

II^

with plasma

associated

concepts and the terminology them, creating an artificial

into

insight

where

included

they

plasma processes. Some simulation results a particular analytical result, extend

or

ampHfy

or

to a better understanding of plasma properties. in difficulty from routine There are many problems in each chapter, varying to challenging to well-known results problems manipulations leading algebraic obtain a better in research in plasma physics interested that will help the student feeling for this diverse and intriguing field. use is their units are used throughout this book, since Gaussian-cgs A conversion table from in the widespread physics research literature. plasma contribute

Gaussian-cgs units to electric

capacitance,

a list of cyclotron

the

A project large

most

frequency,

number

standard

practical

units for

etc., is included occurring symbols

field strength,

frequently

quantities such as III, Appendix as plasma (such in

resistivity,

along with frequency,

etc.).

as extensive of people.

as

this

text

naturally

involves the contribution

of

a

xm

PREFACE

We

the

in plasma thank our friends and colleagues physics with whom we and learned about this subject. We the students who took thank course at Maryland during the this book was being physics period in the development Their conscientious of the course help and advice textbook was very valuable to us. We are particularly grateful to Ted

worked

have

plasma

written.

the

and

Tidman, and Allan Kaufman a graduate to develop and teach benefited experience greatly from their previous We the advice, encouragement, and appreciate Derek

Northrop,

on

advice

how

to our

concentrate

our

on

which

chapters

Hans

particularly

Maryland,

are grateful

of

the

Ron Davidson, and

Chairman,

Howard

on teaching and developing is based. We appreciate the

manuscript

by Ron

their

course

efforts

book

this

Griem,

Department

early suggestions and in plasma physics. We in teaching plasma physics. of our colleagues at support

for

the careful

Davidson, Alan

Bob

Pechacek.

We

for allowing us to plasma physics course of one or more reading

Laster,

Seishi

DeSilva,

Hamasaki,

Klein, Paulett Liewer, Don Spero, Derek Tidman, and Maria Zales-Caponi. Our Marvin Schwartz, thanks and Mrs. Clara Rodriguez for their special go to Mrs. Mary Ann Ferg work in typing the various drafts of and to excellent the manuscript, Mrs. Barbara for preparing the Indexes. Hornady We are grateful to our wives, Terry Krall and Shirley for their Trivelpiece, this project. during patience and understanding John

Hey,

Walt

Jones,

Chris

Kapetanakos,

Hank

Nicholas A. Krall Alvin

W.

Trivelpiece

1

TO PLASMA

INTRODUCTION

PHYSICS

so is the study of charged collected in sufficient number particles physics is a factor in determining their the long-range Coulomb force statistical so that the force due to a near-neighbor properties, yet low enough in density is much less than the Coulomb force exerted by the many particle long-range It is the study of low-density ionized distant particles. The term \"plasma\" gases. was first used to describe a collection of charged by Tonks and Langparticles muir/ in 1929, in their studies of oscillations in electric discharges. However, of the plasma the most characteristic aspect of the state, the fact that because of the Coulomb force the charged exhibit a collective long range particles much first described behavior, was known earher, and was probably by Lord in in his of electron in the oscillations Thomson model 1906, Rayleigh,^ analysis Plasma

that

of

atom.

the

The term was coined gas

by

Crookes^

The

discharge.

is added

to a

state

\"fourth

W.

soUd,

term fourth it undergoes

' L. Tonks and ^

Lord

' W.

state

Lls^gniujr,

Croo]ii^s,PhiU,Trans.,

11:117

ionized

follows

matter

of

(1906).

(1879).

in

Ionized

the

Gases,

state,

state,

plasma

medium

from the

transition to a new

Oscillations 1:135

the

describe

a phase

Rayleigh;,PAi7:.!Ma^.,

to describe

often used

of matter,\" in 1879 to

created in a

idea that as usually

heat

hquid.

Phys, Rev., 33:195

(1929).

2

OF PLASMA

PRINCIPUKS

If heat

the

atoms.

state;

exist

to a liquid, it

is added

of

addition

The

this

PHYSICS

still

more

at temperatures

lower than

gas, and if

the

ionizing

energy

to the gas

a temperature above ionized state of matter is At

the

density

transition

a phase

undergoes

to the

gaseous

state.

results in the ionization of some of most matter exists in an ionized

100,000\302\260K

the

called

100,000\302\260K

is

low

fourth provided

enough

state. there

so that

A plasma state can is a mechanism for is

recombination

not

rapid. exists in a plasma state, 99.9 percent of the apparent universe little in the way of natural the low plasma here on earth because and high density of the earth and its near atmosphere the preclude

Although tliere

is very

temperature existence of

must be created by experimental plasma. This means that plasma in its the means study properties. However, upper atmosphere (ionosphere), does exist, created by photoionization of the tenuous atmosphere. plasma in the earth's the earth, plasma is trapped Fartlier out from magnetic field in the Plasma streams toward the earth from the near vacuum of space. sun (the of interstellar space, forming the medium solar wind), and fills many regions is viewed. tiirougli, which outer space the well-known Plasma physics generally involves physics of classical and nonrelativistic statistical mechanics. The meciianics, electromagnetism, of plasma challenge physics comes from the fact that many plasma properties are collective result from the long-range Coulomb interaction, and therefore that involve interacting simultaneously. many particles properties is a collection In its simplest form, a plasma of protons and electrons at are negligible. low density so that binary interactions gulTlciejitly (short-range) is the study of the properties of problem, Many-body theory, or the many-body a collection of protons and electrons When coexist such a medium. in an of this state are described by equihbrium equilibrium state, the properties with the appropriate Gibbs ensemble. most of the statistical mechanics However, to

of plasmas occur for nonequilibrium situations. in the United interest in plasma physics States began in 1952with of a program, then classified, known as Project the Sherwood,^ to attempts thermonuclear fusion reactor. Similar a controlled were develop programs and the U.S.S.R. at about the same time. These started in England, France, since that time, and now there are have grown substantially programs many in this field. the development nations with major research programs Although fusion reactor is one of the more challenging of a controlled practical apphcait is only one of the many areas in which tions of plasma physics, plasma physics a major role in the development Plasma physics has played of much a role. plays interesting

features

Revived

' A.

S.

Bishop,

\"Project Sherwood,\"

Addison-Wesley,

Reading,

Mass.,

1958.

TO PLASMA

INTRODUCTION

physics, and it is important in the study

of contemporary

areas as

atomic

astrophysics,

physics,

chemistry,

Hfe

of

in such

problems

molecular

sciences,

3

PHYSICS

physics,

power generation, and atmospheric physics. Plasma has its own and set of ideas. The main purpose vocabulary physics review on of this is to an a plasma physics elementary level, provide chapter of the field, of the sketch of the familiar background concepts identify many in discussing the plasma state, review some of the schemes terms used repeatedly and review some of the methods which is produced in the laboratory, by plasma which are measured. by plasma properties magnetohydrodynamic

PART ONE: 1.1

AND METAEQUILIBRIUM

EQUILIBRIUM term

The

used in

is often loosely

\"equilibrium\"

each

by a

described the

spectrum

of

emitted

The

electrostatic

is in

medium

energy at the

absorbs

at the

are neither

electrons

situations that

DEBYE

1.2

and

is blackbody. theoretical and experimental with

equilibrium

to describe

the ions and the electrons characterized by the same

In this situation, the

radiates

small

investigating

by

a plasma,

electrostatic field

their surroundings. will

eventually

electrons

charged

electrons

of an

isolated

are attracted

the rest of

and attracts

particle.

single

rate.

same

with

The

situations

same

in plasma

interest

nor

temperature

The term

be altered

of

in is used

metaequilibrium

by binary collisions.

LENGTH potential

from

are

equiUbrium

particle

of

q is

charge

0=^ In

describe

radiation

In many of the physics, the ions and thermodynamic

distribution

maxwellian

temperature.

its surroundings, and it

that

means

equilibrium

Thermodynamic

a the plasma particles colUde to

physics

plasma

until only quasi-steady-state condition that persists with each other. Frequently, plasma studies are made perturbations about such a metaequilibrium state.

parameter,

TERMINOLOGY

AND

CONCEPTS

PLASMA

The

potential

(1.2.1)

ion and shield its at rest repels in the vicinity This effect alters the potential at rest in a plasma of a charge is given by

the

ions.

to the

r

plasma.

vicinity

Similarly,

of an

an electron

4>

=^-e-'\"-\"

other of

(1.2.2)

a

4

OF PLASMA

FRJNCtPtES

where

is the

Ajj

PHYSICS

For an

electrolytes.

length originally defined in the

Debye

plasma

electron-proton

1/2

1/2

/rp^

\"-'

^'^rn^HW n =

where

density

T =

The

with

the

general,

to

respect

the

The

of the

sphere

1.38 x

10\"^*

of influence

depends on the

length

Debye

g indicates

is defined

and

sphere,

constant (=

ergs/\302\260K)

a given

of speed

the

of

number

iSi

pftraiweter

to

mean

the

tnergy uad the

plasma

kinetic

the

particles.

between

can

and

for

is treated

plasma

have a charge

density

of a

the description

in a Debye

of particles

The assumption g

=

(1.9.2)

'^ at rest repels inandtheimmobile vicinity is given

=^-e-'\"-\"

e=l-^

other of a ions,

by (1.2.2) (1.9.3)

TO PLASMA

INTRODUCTION

Problem plasma

a

from

The

Derive the dielectric electrons and stationary displacement of electrons. 1.9.1

ions

in (1.9.3)

given

the

by computing

field

a

for

arising

//// for electromagnetic

relation

dispersion

corresponding

as

constant

of cold

11

PHYSICS

a

in

waves

is

plasma

\"'

=

k^

(1.9.4)

-/^' than the

plasma frequency co^, the wave number is imaginary and the waves are evanescent. Above the plasma frequency (co > co^), waves propagate, and at very high frequencies, the free electrons of the plasma such as finite Effects a wave. size, only slightly influence the electromagnetic If

the

wave

frequency

magnetic

steady

co

field,

is less

or

this picture

modify

inhomogeneities

plasma

considerably.

In

a

homogeneous For

possible.

steady magnetic field a slow dispersionless example,

nethydrodynamic

wave,

called

an Alfven

below the ion-cyclotron

frequencies

Plasmas

or Nonuniform

in Magnetized

Waves

1.9.3

wave

electromagnetic

through the

wave, propagates with

frequency

result gradients.

wave

nnti is the mass density. As a second example, drift waves propagate of particle drifts and plasma currents This extra freedom introduced by in which motion

-i=

(1.9.5)

in inhomogeneous plasmas as a with plasma density

associated the

density

gradient

kT 1

k is

the Boltzmann constant, and

Plasma

1.9.4 These

at

p^ =

coK

where

or magplasma

speed

K^ =

where

are

motions

wave

additional

many

waves

Langmuir velocities

T is

m

aUows a

Vrir.

\"-k oic

new

(1.9.6)

tiQ

the temperature.

Waves

are

waves.

or if the

also known as They electrons

propagate have

waves,

space-charge

only an

if there

average

is a

electrostatic

velocity in the

waves, of

distribution observer's

electron frame.

or

12

by an CO

PHYSICS

described

are the previously

They

is

OF PLASMA

PRINCIPLES

initially displaced

=

CO,

+

The dispersion

of charge.

clump

at frequency

oscillations,

plasma

waves

these

for

relatiojQ

aroused

cOp,

icoi.

CO/ = C(j/

+

k^

m

(1-9.7)

r

The The

finite

is evidenced

dispersion

second

the

evidenced

by

valid only

when

the

to

the medium

causes

temperature

by the equation,

and

is weak,

absorption

be

both

at

vanishes

The absorption is

T-^Q.

X^^ =

.'j'~r'-''^'-v^;.:v\302\253^?i^c:^::.:h\342\200\236 =YB\342\200\236r-''sin

The values of

-> 00 /\342\200\242

(4.7.2)

nG +

sin E,\342\200\236,r

G

(4.7.3)

at by the boundary condition dielectric-vacuum interface

at a

the are

160

PRINCIPLES

that

the normal

PHYSICS

component

electric

the

of

component

OF PLASMA

the

These

#2

This

= 4>o{a)

determines

condition

of the

incident

bE, continuous at r

dr

dr ^j(a)

terms

sE) and tangential conditions are satisfied if

(with

a)

co'j

\\

(D =

displacement

continuous.

be

field

surface at r =

the plasma

of

Eg continuous

and A\342\200\236 and

amplitude

the potential

gives

=a

(4.7.4)

atr = a

(4.7.5)

inside the

in

plasma

frequency:

Ai =

1+

' (4.7.6)

=

{1+b)A\342\200\236

Thus the

field

that

s =

1

+

in the

n>2

0

becomes large (resonant) when the

plasma

is such

frequency

0, or co

=

(4.7.7)

^

The fact that the plasma resonates at s = \342\200\224 1 and not at s = 0 is a consequence of the cylindrical geometry 4.7.2). (see Prob. Since the plasma column is resonant at this frequency, the electrons in the column oscillate in response to the driving field. This motion reradiates, electric or scatters, Since the amplitude the incident field in cylindrical waves. of the at resonance, the scattered power motion of the electrons in the plasma is largest will at resonance. If the plasma column extends across a be a maximum is a maximum at resonance. coefficient waveguide as in Fig. 4.7.1, the reflection 4,7.1

Problem

Eg be continuous

Show that (in that

implies

Problem 4.7.2 Show that as above has a resonant mode Figure 4.7.3 shows there is more peaks

than one

in terms

the

also

a sphere at

resonances at

ca

that

condition

////

of plasma with the

co^^^

=

same

assumptions

////

cojy/s.

Many

of multiple resonances

However, the multiple time. The resonance

the

coordinates)

is continuous.

of this type of experiment. As can be seen, workers tried to explain the additional

results

resonance.

cylindrical

cj)

\302\253C0pl^j2

with

associated

remained (called

as an the

the

unsolved main

dielectric problem

resonance)

constant. for

a long

is correctly

WAVES

resonance

Main

O U

IN THE FLXJID

PLASMA

161

ui-^ui^/-/Z

>s

due to thermal

Scattering ^

waves

plasma

\342\226\240o

3 c

ldr interface. This requirement leads to the result that continuous at the plasma-vacuum the amplitude of the nth mode can be nonzero only if Again

are

interface

kaJ'\342\200\236{ka)

ka =-^

where This gives the warm

of the

Because

plasma.

are no other

that there

can

Equation (4.7.17)

main

at

frequencies

resonance

arbitrary of the

modes

be solved at n = 0,

modes

which

{-

1

=

0 can

there

V^,

plasma which also

(graphically or numerically) at the same frequency

-

^

= 0, V^,

satisfying

restriction

)

\\co/

co/

for

be excited in a no

is

a/Ap.

large

resonance

as the

guarantee

excited.

be

could

(4.7.18)

There is a in the cold

plasma,

(4.7.19)

/ y/

//V'/V*

)/

y

y y yi

i

^x'PA

a

FIGURE

r

4.8.4

Electric-field

there

is a net

a real

therefore

The maximum phase and is given by

for space-charge

magnetic E,

means

in turn

which

current,

perturbation

magnetic field, and these disturbances.

waves in a plasmafield for lowest azimuthally have a velocity p are zero, and electrons

distribution

charge-density

axial waveguide in an infinite symmetric mode. One-quarter cycle later to E^. distribution proportional

filled

power

electromagnetic of the

velocity

there flow,

waves

space-charge

a perturbed

is

with

associated

occurs

for

low

frequency,

lim

=

Uphase

.,