Basic Plasma Physics [1 ed.] 9781783320295, 9781842658567

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Basic Plasma Physics

Basic Plasma Physics

Basudev Ghosh

Alpha Science International Ltd. Oxford, U.K.

Basic Plasma Physics 266 pgs. | 67 figs. | 15 tbls.

Basudev Ghosh Department of Physics Jadavpur University Kolkata Copyright © 2014 ALPHA SCIENCE INTERNATIONAL LTD. 7200 The Quorum, Oxford Business Park North Garsington Road, Oxford OX4 2JZ, U.K. www.alphasci.com All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the publisher. ISBN 978-1-84265-856-7 E-ISBN 978-1-78332-029-5 Printed in India

To My Mother Manibala Ghosh

Preface

Plasma physics is a relatively new and fast growing branch of physics. Extensive interest in controlled fusion power over decades is now resulting in two huge projects ITER (International Thermonuclear Experimental Reactor) and HiPER (High Power laser Energy Research) which are expected to come on line in about a decade or so. Industrial applications of plasma science and technology are now rapidly expanding into diverse areas such as high efficiency lighting, microelectronics, nanoscience, polymer and textile processing, packaging, bioscience and medicine. All these will require a large number of scientists and engineers trained in plasma physics. To keep pace with this fast changing scenario in plasma research and applications most universities have introduced in recent years plasma physics as a part of the curriculum. While teaching this subject very often we refer to different reference books for different topics. In most cases students do not get the opportunity to consult so many books. Some students though lucky enough to get this opportunity, get confused with different styles of communication used by different authors and become unable to derive interest out of it. The purpose of the present book is to provide a coherent piece of text covering the current syllabus of plasma physics offered by most Indian universities and technical institutions. The book is designed to serve as an introductory compact text book for advanced undergraduate, post-graduate and research students taking plasma physics as one of their subject of study for the first time. It is primarily based on my long experience with the subject both as a teacher and a research worker. The book presents a basic introduction to the subject and requires no background in plasma physics but only elementary knowledge of basic physics and mathematics. The subject matter has been presented in an easy-tounderstand way. For brevity technical details and historical backgrounds have been omitted. Emphasis has been given on the analytical approach. I have tried to develop the topics from first principle so that the students can learn through selfstudy. In most cases apart from theoretical study students are to perform some experimental works as a part of their curriculum. With this in mind two chapters have been devoted to describe some practical aspects of plasma physics and some experiments suitable for class-room teaching.

viii

Preface

Each chapter contains a good number of solved problems some of which contain subtle points not mentioned in the text. This will help the readers to increase their level of understanding. To test the ability of the students in correct application of the basic principles a variety of unsolved problems with answers have also been given at the end of each chapter. It will surely help to grow self-confidence. A variety of subjective questions, mostly taken from recent examination papers, have been incorporated at the end of each chapter. A number of thought provoking multiple choice questions (MCQ) given in the Appendix F will surely help the students to check their level of understanding. In an introductory book like the present one it is not possible to cover all areas of plasma physics. Most of the topics covered in the book contain only the basic principles suitable for the beginners. For interested readers a list of some advanced text books on the subject has been given in the Appendix G as reference. It is hoped that the book will be able to meet up all the requirements of the students taking up plasma physics as their first course. However, any constructive criticism and suggestion for the improvement of the book will always be appreciated. In a text book of this type much originality cannot be claimed. I therefore take this opportunity to express my gratitude to the renowned authors whose writings have contributed to my understanding of the subject of this book. I take this opportunity to thank my students who inspired me to write this book and provided me with the opportunity to test my ideas. Finally it is pleasure for me to thank my wife, Mrs. Shyamali Ghosh, for sparing me from the regular household duties during the preparation of the manuscript.

Basudev Ghosh

Contents



Preface..................................................................................................... vii

1. INTRODUCTORY ............................................................................1.1—1.7 1.1 What is a plasma?................................................................................1.1 1.2 Definition of a Plasma.........................................................................1.2 1.3 Occurrence of Plasma in Nature..........................................................1.2 1.4 Brief History of Plasma Physics..........................................................1.2 1.5 Applications of Plasma Physics...........................................................1.4 1.6 Theoretical Study of Plasma................................................................1.6 Questions.............................................................................................1.7 2. BASIC PLASMA CHARACTERISTICS........................................2.1—2.18 2.1 Temperature and Density of Plasma....................................................2.1 2.2 Quasineutrality in Plasma....................................................................2.3 2.3 Debye Shielding..................................................................................2.4 2.4 Criteria for Ionized Gas to be a Plasma...............................................2.7 2.5 Plasma Oscillations.............................................................................2.8 2.6 Key Parameters of Some Plasmas.....................................................2.10 Solved Problems................................................................................2.10 Questions...........................................................................................2.15 Exercises............................................................................................2.16 3.

ORBIT THEORY OF PLASMA.......................................................3.1—3.33 3.1 Introduction.........................................................................................3.1 3.2 Particle Motion in a Static Uniform Magnetic Field...........................3.1 3.3 Particle Motion in Static Uniform Electric and Magnetic Fields........3.5 3.4 Particle Motion in Magnetic and Gravitational Fields........................3.9

x

Contents



3.5 Grad-B Drift......................................................................................3.10



3.6 Curvature Drift..................................................................................3.12



3.7 Polarization Drift...............................................................................3.14



3.8 Adiabatic Invariance of Magnetic Moment.......................................3.17



Solved Problems................................................................................3.23



Questions...........................................................................................3.27



Exercises............................................................................................3.29

4. FLUID DESCRIPTION OF PLASMAS..........................................4.1—4.14

4.1 Introduction.........................................................................................4.1



4.2 Fluid Equation of Motion....................................................................4.1



4.3 Equation of Continuity........................................................................4.2



4.4 Closed System of Fluid Equations......................................................4.3



4.5 Complete Set of Fluid Equations for a Simple Plasma ......................4.4



4.6 Fluid Drifts..........................................................................................4.5



4.7 Single Fluid Equations........................................................................4.6



4.8 Frozen-in Magnetic Fields...................................................................4.8



4.9 Magnetic Pressure.............................................................................4.10



4.10 Magnetic Tension.............................................................................. 4.11



Solved Problems................................................................................ 4.11



Questions...........................................................................................4.13



Exercises............................................................................................4.14

5. WAVES IN PLASMAS.....................................................................5.1—5.27

5.1 Introduction...........................................................................................5.1



5.2 Electron Plasma Wave in Cold Plasmas...............................................5.1



5.3 Electron Plasma Waves in a Warm Plasma...........................................5.4



5.4 Ion-acoustic Wave.................................................................................5.6



5.5 Electromagnetic Waves in Cold Field-free Plasma............................. 5.11

5.6 Electrostatic Electron Oscillation Perpendicular to Applied Magnetic Field...................................................................................5.14

5.7 MHD Waves in a Uniform Plasma.....................................................5.16



5.8 Mathematical Analysis of MHD Waves............................................5.18



Solved Problems................................................................................5.21



Questions...........................................................................................5.23



Exercises............................................................................................5.25



Contents

xi

6. KINETIC THEROY OF PLASMA..................................................6.1—6.20

6.1 Introduction.........................................................................................6.1



6.2 Distribution Function and Macroscopic Variables..............................6.1



6.3 Boltzmann Equation............................................................................6.2



6.4 Fluid Equations....................................................................................6.4

6.5 An Application of Vlasov Equation: Dispersion Relation of Longitudinal Electron Plasma Waves..................................................6.8

6.6 Landau Damping............................................................................... 6.11



6.7 Physical Mechanism of Landau Damping.........................................6.15



Solved problems.................................................................................6.16



Questions...........................................................................................6.18



Exercises............................................................................................6.19

7. TRANSPORT PROCESSES IN PLASMAS....................................7.1—7.17

7.1 Introduction.........................................................................................7.1



7.2 Mobility and Electrical Conductivity of Weakly Ionized Plasma.......7.1



7.3 Electrical Conductivity of Fully Ionized Plasma.................................7.3



7.4 Free Diffusion of Weakly Ionized Plasmas.........................................7.5



7.5 Ambipolar Diffusion in Plasma...........................................................7.6



7.6 Diffusion Equation..............................................................................7.8



7.7 Diffusion of a Weakly Ionized Plasma Across a Magnetic Field......7.10



7.8 Diffusion of a Fully Ionized Plasma Across a Magnetic Field..........7.13



Solved problems.................................................................................7.14



Questions...........................................................................................7.15



Exercises............................................................................................7.16

8. STABILITY OF FLUID PLASMA..................................................8.1—8.10

8.1 Introduction.........................................................................................8.1



8.2 Classification of Plasma Instabilities...................................................8.1



8.3 Methods of Stability Analysis.............................................................8.2



8.4 Two-stream Instability.........................................................................8.3

8.5 Instability of Fluid Plasma Supported Against Gravity by a Magnetic Field.....................................................................................8.5

Solved problems...................................................................................8.7



Questions...........................................................................................8.10



Exercises............................................................................................8.10

xii

Contents

9. NONLINER EFFECTS IN PLASMA..............................................9.1—9.28

9.1 Introduction.........................................................................................9.1



9.2 Plasma Sheath......................................................................................9.2



9.3 Langmuir Probe...................................................................................9.4



9.4 The Pondermotive Force.....................................................................9.6



9.5 Ion-acoustic Solitary Waves and the Korteweg-de Vries Equation.....9.9



9.6 Sagdeev Potential Approach..............................................................9.14

9.7 Nonlinear Schrödinger Equation, Modulational Instability and Envelope Soliton...............................................................................9.17

9.8 Wave-wave Interaction......................................................................9.21



Solved Problems................................................................................9.23



Questions...........................................................................................9.27



Exercises............................................................................................9.28

10. SOME PRACTICAL ASPECTS OF PLASMA PHYSICS..........10.1—10.22

10.1 Introduction.......................................................................................10.1



10.2 Plasma Production.............................................................................10.1



10.3 Plasma Diagnostic Techniques..........................................................10.5



10.4 Heating of the plasma......................................................................10.14



10.5 Confinement of the Plasma..............................................................10.15



Solved Problems..............................................................................10.19



Questions.........................................................................................10.21



Exercises..........................................................................................10.22

11. SOME ELEMENTARY CLASS-ROOM EXPERIMENTS ON PLASMA...................................................................................... 11.1—11.26

11.1 Introduction....................................................................................... 11.1

11.2 To Draw Paschen Curve and Study the Dependence of Breakdown Voltage on the Pressure —Interelectrode Gap Length Product........................................................................... 11.1 11.3 To Study the Conditions of Occurrence of Striations in Gas Discharge.................................................................................... 11.4 11.4 To Measure Plasma Parameters by Using Single Langmuir Probe.................................................................................................. 11.7 11.5 To Measure Plasma Parameters by Using Double Langmuir Probe................................................................................................ 11.10

11.6 Experimental Study of Ion-acoustic Waves..................................... 11.15



Question with Answers for Viva Voce.............................................. 11.18



Contents

xiii

APPENDICES........................................................................................ A.1—A.18 Appendix A : Some Useful Vector Relations..............................................A.1 Appendix B : Concept of Tensors...............................................................A.4 Appendix C : Some Important Physical Constants.....................................A.6 Appendix D : List of Some Symbols and Their Principal Meanings ........A.7 Appendix E : Some Important Formulae in Plasma Physics....................A.10 Appendix F : Some Typical Multiple Choice Questions.........................A.14 References........................................................................................................... R.1 Index..................................................................................................................... I.1

CHAPTER

1 Introductory

1.1 WHAT IS A PLASMA? The word ‘plasma’ comes from the Greek and means something molded or fabricated. The word ‘plasma’ was first used by two American Physicists Langmuir and Tonks, in 1929, for the positive column region in the discharge tube which contains ions and electrons in about equal numbers. They found that under certain circumstances some sort of cohesiveness exist among the particles of an ionized gas which is reminiscent of jelly or blood plasma. The way the particles of such ionized gas moves reminds one of the way the red and white corpuscles move through blood plasma. For this the interesting name ‘plasma’ was given to it. The plasma is regarded as the part of the sequence solid-liquid-gas-plasma and is sometimes referred to as the fourth state of matter. The state of matter depends on the relative magnitudes of the random thermal kinetic energy of its atoms or molecules and the interparticle binding potential energy. The interparticle binding energy is relatively strong in solid, weak in a liquid and almost zero in gaseous state. On heating a substance when thermal kinetic energy of particles exceeds the binding energy there results phase change which occurs at constant temperature for a given pressure. Thus when a solid is heated it first changes to liquid state and then to gaseous state. Now if heated further a molecular gas dissociates into atomic gas when thermal kinetic energy of the particles exceeds molecular binding energy. At sufficiently high temperature outermost orbital electrons can overcome their binding forces and an ionized gas or plasma results. However this transition from gas to plasma is not a phase transition in true thermodynamic sense because it occurs gradually with increasing temperature. However we call plasma a distinct phase because it has many distinctive properties as compared to gaseous phase. The basic difference between plasma and neutral gas arises due to different character of the interparticle forces in them. In neutral gas the interparticle force is of shortrange Van der Waals type whereas in plasma this force is of long-range Coulomb type. As a result of this each particle in a plasma can simultaneously interact with

1.2

Basic Plasma Physics

all other particles and gives rise to the characteristic collective behaviour which is not found in neutral gases. Natural occurrence of plasmas at high temperatures is also one of the reasons for its designation ‘the fourth state of matter’.

1.2 DEFINITION OF A PLASMA A plasma may be defined as a quasineutral collection of charged and neutral particles which shows a characteristic collective behaviour because of the fact that in plasma long-range electromagnetic interactions with a given charged particle due to all other charged particles in the collection are much larger than the interactions due to short-range binary collisions. The number of charged particles in the collection must not be very low so that the notion of collective motion fails. In fact a collection of charged and neutral particles will exhibit plasma behaviour provided it satisfies certain conditions or criteria. These criteria will be discussed in some detail in Section 2.4.

1.3 OCCURRENCE OF PLASMA IN NATURE Plasma is the most natural state of matter. It is estimated that about 99.9% of the matter in the known universe is in the plasma state. We are lucky enough to live in the rest 0.1% of the universe in which plasma does not occur naturally. In earlier epoch of the universe, everything was in plasma state. In the present epoch stars, nebulae, stellar interiors, much of interstellar space are filled with plasma. In our own sun one may observe spectacular plasma in the form of solar flares, solar prominences, and sun spots. Closer to home the ionosphere, solar wind, aurora, Van Allen radiation belts are the natural examples of plasmas. Plasmas can also be found in our own neighbourhood. For example, lightening and aurora borealis are natural plasmas whereas man-made plasmas can be found in fluorescent lamps and in many industrial processes. Apart from ionized gases, some liquids and solids can also exhibit plasma behaviour. Free electrons inside metals, free electrons and holes in semiconductors can exhibit collective effects that characterize plasma. Certain liquids such as mercury, solution of sodium in ammonia etc. exhibit many properties characteristics of conventional plasmas. It is assumed that the whole matter in the early universe immediately following the big bang was in extremely hot plasma state now known as quark-gluon plasma.

1.4 BRIEF HISTORY OF PLASMA PHYSICS Plasma physics is a relatively new and one of the advanced disciplines of physics. It draws its methodology from diverse fundamental areas of physics such as mechanics, electrodynamics, statistical mechanics, kinetic theory, fluid mechanics



Introductory

1.3

and atomic physics. During the early part of twentieth century the progress of physics was characterized by the remarkable advances in the knowledge of electrons, ions, atoms and molecules individually. At the same time scientists like J.J.Thomson and J.S.Townsend tried to understand the luminous and fascinating discharge phenomenon in gases. They tried to understand how the characteristics of individual electron, ion and atom cooperate and give rise to the phenomenon of electric discharge. The success came from two American Scientists I. Langmuir and L. Tonk who discovered that certain regions of a gas discharge tube exhibit periodic variations of electron density. It is now known as Langmuir oscillations. The name ‘plasma’ was first given to the positive column region in the discharge tube which contains ions and electrons in about equal number. The study of plasma is known as plasma physics. After Langmuir plasma research gradually spread in many other directions. Increased understanding of plasma physics has been stimulated and paced by the development of its many important applications. Some are discussed below:

(i) The development of radio broadcasting led to the discovery of the ionosphere which is in plasma state. In order to understand the radio wave propagation through ionosphere in 1920s scientists like Appleton, Budden and others systematically developed the theory of wave propagation through nonuniform magnetized plasma.



(ii) Astrophysicists quickly recognized that much of the universe is in plasma state and understanding of astrophysical phenomena requires a better grasp of plasma physics. H. Alfven, pioneer in this field, developed the theory of magnetohydrodynamics in 1940s and successfully employed it to understand many astrophysical phenomena such as sun spots, solar flares, solar wind, star formation etc. He was awarded Noble prize in physics in 1970 for this work.



(iii) The creation of hydrogen bomb in 1952 raised great interest in controlled thermonuclear fusion as a possible power source for the future. Thermonuclear fusion reactions require very high temperature (~ 107 K). The most practical laboratory system capable of operating at the necessary high temperature is the fully ionized plasma. Fundamental problem of thermonuclear reaction control is that of confining a high energy plasma long enough so that an appreciable number of fusions can take place. Solid walls make an ineffective container because they cool the plasma. Fusion physicists are trying to trap plasma by magnetic field and investigating the plasma instabilities which may allow it to escape. The problem is still unsolved and most of the active plasma research is centred about it. It is responsible for the rapid growth of plasma physics since 1952.

1.4

Basic Plasma Physics



(iv) Discovery of Van Allen radiation belts in 1958 through satellite observation marked the beginning of systematic exploration of the earth’s magnetosphere. It opened up the field of space plasma physics. Today it is an active area of plasma research contributing to the understanding of observations in space.



(v) The development of high power lasers in 1960s opened up a new field of research, called laser plasma physics. A major application of laser plasma physics is to explore the possibility of inertial confinement fusion. The strong electric field generated when a high power laser pulse passes through a plasma is expected to be used for particle acceleration. This technique can dramatically reduce the size and cost of conventional particle accelerators. The laser produced plasmas have extreme conditions of density and temperature not found in conventional plasmas. For high density low temperature plasmas quantum effects become important and we need to develop quantum plasma physics—a new area of plasma research.



(vi) Today plasma physics is a challenging field of research in which exist many problems and few satisfactory solutions. The present problems are indeed formidable, but the richness of the rewards for their solution warrants the interest and effort which is and will be devoted to plasma physics.

1.5 APPLICATIONS OF PLASMA PHYSICS ince plasma is the most abundant state in the universe it is only fitting and natural S that man should try to understand and benefit from plasma. The progress in plasma research has led to a wide range of plasma applications. Some of the applications of plasma physics are briefly discussed below: (i) Controlled thermonuclear fusion he most important application of man-made plasmas is in the control of T thermonuclear fusion reaction which promises a solution for future power problem. (ii) Astrophysics Most of the astrophysical objects are in the plasma state. Various plasma theories are being used to explain different astrophysical phenomena with the hope of clarifying about the origin of the universe. (iii) Space physics An important application of plasma physics is the study of earth’s environment in space including ionosphere and magnetosphere. The propagation of radio waves



Introductory

1.5

through ionosphere, formation of Van Allen radiation belts, aurora borealis etc. can be understood by using plasma physics. (iv) Magnetohydrodynamic conversion of energy Another possible application of plasma physics is the direct conversion of thermal energy into electrical energy with the help of magnetohydrodynamic (MHD) generators. Its basic principle is very simple. Suppose a plasma jet flows with a velocity vv along x-axis across an applied magnetic field Bv in the y-direction (Fig. 1.5-1). The Lorentz force q_vv # Bv i causes the ions of charge q to drift upward (in the z-direction) and the electrons downward. It charges the electrodes kept in contact with the plasma jet. If the electrodes are connected to external circuit then a current density vj = s Evind = s _vv # Bv i flows across the plasma jet in v z-direction where s is the plasma conductivity and Eind is the induced electric field. This current in turn generates a force density vj # Bv which decelerates the flowing plasma. The net result is the conversion of the plasma kinetic energy into electrical energy.

Fig. 1.5-1 : Basic arrangement of a MHD generator

(v) Plasma propulsion The plasma propulsion systems are based on a process that converts electrical energy into plasma kinetic energy, that is, the reverse of MHD generator process. It can be accomplished by applying both electric field and magnetic field perpendicular to each other across a plasma (Fig.1.5-2). The resulting current density vj flowing in the direction of applied Ev -field gives rise to a force per unit volume which shoots the plasma out of the system. The associated reaction force accelerates the system in the direction opposite to the plasma flow. This principle has been used to develop rocket engines for interplanetary mission.

1.6

Basic Plasma Physics

Fig. 1.5-2 : Principle of plasma propulsion

(vi) Industrial and technological applications Industrial and technological applications of plasma science are now rapidly expanding into diverse areas. Some are listed below: (a) Plasma processing, thin film deposition, IC fabrication (b) Plasma-based high efficiency lighting, display systems and TVs (c) Surface cleaning and surface treatment, polymer and textile processing (d) Material synthesis (e) Gas discharge devices such as mercury rectifiers, hydrators, ignitrons, neon and fluorescent light, gas lasers etc. (f) Bioscience and medicine

1.6 THEORETICAL STUDY OF PLASMA he charged particles in a plasma can move around and generate local T concentrations of charges which give rise to electric field. Motion of these charged particles is equivalent to electric current and it produces magnetic field. The dynamic behaviour of a plasma is governed by the interaction of plasma particles with these internally created electromagnetic fields and the externally applied fields. For theoretical description of plasma phenomena we draw methodology from diverse fundamental areas of physics such as mechanics, electrodynamics, kinetic theory, statistical mechanics, fluid mechanics and the atomic physics. Plasma is highly complex system and hence approximation is essential. In fact it is a challenge to simplify the description in a manner that is physically convincing. Depending on the density, temperature and other physical parameters there have been developed basically four principal approaches with appropriate choice of approximations. (i) Single particle (or orbit) theory his method consists in studying the motion of each charged particle in T presence of specified force fields. It is not really a plasma theory. However, it



Introductory

1.7

provides some physical insight for better understanding of the dynamic processes in plasma. This method is found to be useful for predicting the behavour of very low density plasmas such as can be found in Van Allen radiation belts, solar corona, cosmic rays, cathode ray tubes, etc. Orbit theory can be used with simplicity and success where collisions are very few and far between. (ii) Statistical approach s the plasma consists of a very large number of interacting particles for A macroscopic description one can use statistical approach. Here one introduces the distribution function for the system of particles under consideration and solve the appropriate kinetic equations that govern the evolution of the distribution function in phase space. One such kinetic equation is the Vlasov equation in which the effects of close collisions are neglected and the interaction between the charged particles is described by the self-consistent internal electromagnetic fields. (iii) Two-fluid theory hen the plasma density is high it becomes an impossible task to follow W the trajectory of each particle and to predict plasma behaviour. Fortunately when collisions between plasma particles become very frequent each species can be treated as a fluid described by local density, temperature and velocity. Here identity of individual particle is neglected and only the motion of fluid elements is considered. In this approach the plasma is treated as a mixture of two or more interpenetrating fluids depending on the number of plasma species. (iv) Single-fluid theory When the density of plasma is high enough the whole plasma may be considered as a single conducting fluid. A simplified form of this single fluid theory is known as magnetohydrodynamics (MHD) which is applicable to the study of very low frequency phenomena in highly conducting magnetized fluids.

QUESTIONS 1. What is a plasma? 2. Plasma is called fourth state of matter—why? 3. Plasma is the most natural state of matter—discuss. 4. In what way is plasma different from neutral gas? 5. Briefly discuss the historical developments of plasma physics. 6. Mention some of the important applications of plasma physics. 7. Give the basic principles of operation of a MHD generator. 8. Give the basic principles of operation of a plasma propulsion system for rocket engines. 9. Briefly mention different theoretical approaches for the study of plasma. qqq

CHAPTER

2 Basic Plasma Characteristics

2.1 TEMPERATURE AND DENSITY OF PLASMA he temperature and the charged particle number density are the two important T parameters of a plasma. Behaviour of a plasma can be distinguished by these two characteristic parameters. These parameters vary over wide ranges for naturally occurring and man-made plasmas. In conventional plasmas with low density and high temperature thermal de Broglie wavelength associated with plasma particles is small compared to average interparticle distance. The wave functions associated with neighbouring particles do not overlap and hence plasma can be assumed to behave classically. So plasma particles in thermal equilibrium may be assumed to follow classical Maxwellian distribution.

Concept of Temperature classical plasma in thermal equilibrium has particles of all velocities and the A velocity distribution of these particles is known to be Maxwellian. In the simplest one-dimensional case the Maxwell’s distribution is given by

1 f(u) = A exp c- 2 mu 2 /kB T m

...(2.1-1)

where f(u)du is the number of particles per unit volume with velocity between u 1 and u+du, 2 mu 2 is the kinetic energy and kB = 1.38 × 10-23 J⋅K–1 is the Boltzmann constant. The density n, number of particles per unit volume, is given by

n =

#-33 f_uidu

...(2.1-2)

This on integration yields

m 1/2 A = n c 2pk T m B

...(2.1-3)

2.2

Basic Plasma Physics

The width of the distribution is characterized by the constant T which we call the temperature. In thermal equilibrium average kinetic energy per particle is 1 y-33 2 mu2 f^uhdu Eav = ...(2.1-4) y + 3 f^uhdu -3

Using (2.1-1) it can be shown by simple integration that 1 Eav = 2 kB T

...(2.1-5)

Generalizing the above results to three-dimensions one can get 3 m 3/2 A = n c 2pk T m and Eav = 2 kB T B



...(2.1-6)

Thus the temperature T may be considered as a measure of the mean kinetic energy of the random thermal motion of particles. It is customary in plasma physics to give temperature in units of energy (i.e., in electron-volt units). The energy corresponding to kBT is used for this purpose. Thus the absolute temperature of 1 eV plasma is given by

kBT = 1 eV = 1.6 ×10–19 J

1.6 # 10-19 = 11600 K 1.38 # 10-23 Thus the conversion formula is T =

or,



1 eV = 11600 K

...(2.1-7)

ere it is interesting to point out that each component of a plasma can H have temperature of its own which may be different from each other. This is primarily because of the fact that the collision rate among electrons or among ions themselves is larger than the rate of collisions between an electron and an ion. As a result electrons and ions can have separate Maxwellian distribution with different temperatures Te and Ti. Plasma may not last long for the two temperatures to equalize. In presence of magnetic field even a single species, say electrons, can have two different temperatures. This is so because the force acting on electrons along the magnetic field is different from that acting normal to the magnetic field. As a result the components of velocity parallel and perpendicular to the magnetic field may then belong to different Maxwellian distributions with temperatures T­| | and T= .

Degree of Ionization plasma may contain electrons, ions as well as neutral particles. Plasma behaviour A is determined by the density of charge particles relative to neutral particles. This



Basic Plasma Characteristics

2.3

relative density of charged particles in a plasma is described by a parameter called the degree of ionization. The fractional ionization to be expected in a gas in thermal equilibrium is given by the Saha equation:

ni T3/2 # 1021 $ . 2.4 exp ^- Ui /kB T h nn ni

(2.1-8)

where ni and nn are the densities (number per m3) of ion and neutral respectively, T is the temperature in kelvin (K) and Ui is the ionization energy of the gas. The fractional ionization (ni/nn) as given by the Eq. (2.1-8) represents a balance between the rate of ionization and the rate of recombination. The rate of ionization increases with temperature T. An ionized atom may recombine with an electron to become a neutral again. Clearly the recombination rate will increase and hence the ion density will decrease with increase in the density of electrons, which we can take as equal to ni. This explains the appearance of the factor ni–1 on the right-hand side of Eq. (2.1-8). The degree of ionization thus depends not only on the temperature but also on the density of the gas. or ordinary air at room temperature kBT c it is not possible to transform away the electric field. However in this case it is possible to transform away the magnetic field. For an S l -frame observer to see only electric field and no magnetic field we must have

Bv l = 0

or

1 v Bv = c 2 `vvD # E j

or or

1 E2 Ev # Bv = 2 Ev # `vvD # Ev j = 2 vvD c c

8a vvD = Ev B

c2 vvD = 2 _ Ev # Bv i E

c 2 _ Ev # Bv i relative to S-frame will E2 see only an electric field and no magnetic field. Such an observer will find the charged particle to have accelerated motion along a straight line parallel to Ev l . A laboratory frame observer will find the particle moving under the combined action of the constant drift vvD and an accelerated motion in a straight line parallel to Ev l . So an observer moving with a velocity

PROBLEM 3.2 : Suppose the axial magnetic field in a magnetic bottle geometry formed by a pair of coaxial coils is given by z 2 B(z) = B0 1 - B ! z H m _ mi

z 2 B_! zmi = B0 =1 + d m n G , we get a 0

v ||2 _ z i =

2mB0 zm 2 z 2 m =d a0 n - c a0 m G

(b)

2B0 2B z 2z = a 02

\

2mB0 t zk Fv|| = a 0

So equation of motion of the particle is

2 2mB m d 2z = - 2 0 z dt a0

This represents a simple harmonic motion of angular frequency

w =

2mB0 ma 02

PROBLEM 3.3 : In a magnetic mirror configuration with a mirror ratio Rm show that the Larmor radius of the particle at the centre is Rm times the Larmor radius at the point of maximum magnetic field. Solution : Since magnetic moment (m) is invariant

1m v2 1m v2 ` =j ` =j m = 2 B 0 = 2 B m 0 m

...(1)

3.26

Basic Plasma Physics

where the subscripts 0 and m corresponds respectively to the centre and the point of maximum magnetic field. Mirror ratio Larmor radius

B Rm = Bm 0 v= v= m R = w = q B c

_v=i0 Bm _ Ri \ 0 = B $ v = _ =im _ Rim 0

Bm B0 Rm

= Thus

[Using (1)]

Rm _ Rim .

_ Ri0 =

PROBLEM 3.4: Suppose the earth’s magnetic field in the equatorial plane at a distance r from the centre of the earth is given by R 3 B(r) = B0 c rE m



(r > RE)

where B0 is the magnetic field at the equator and RE is the radius of the earth. Assuming an isotropic population of electrons with temperature Te and density ne at r = 5RE in the equatorial plane show that the grad-B drift speed is given by vG =

75kB Te . What is the direction of this drift? How long does it take to drift eB0 RE

around the earth? Solution : Grad-B drift velocity

W v Bi vvG = =3 _ Bv # d qB

\

vG =

= Now, or,

...(1)

W= v v B # dB q B3 W= v dB qB 2

3 v B_ r i = 2B rt =- 3B0 R E rt d 4 2r r

vB d = 3 r B W= = 1 mv =2 = kB Te 2

v B = r Bv A 7a d ...(2)



Orbit Theory of Plasma \ At Therefore,

kB Te 3 $ vG = eB r

3.27 (3)

r = 5RE, B(r) = B0/125 75k T vG = eB BR e 0 E

(4)

v B points in the – rt We know that Bv is directed from south to north. Here d direction, i.e., towards the centre of the earth. As q for electron is negative, Eq. (1) indicates that the direction of vvG will be towards east. Time to drift once around the earth is

T =

2p # 5RE 2pr 2p eB0 R 2E = = . $ vG 75kB Te /eB0 RE 15 kB Te

QUESTIONS 1. Show that a steady uniform magnetic field cannot do any work on a charged particle. 2. Considering the non-relativistic motion of charged particle in a static uniform magnetic field Bv = B0 kt find expressions for cyclotron frequency and cyclotron radius. Show that the magnetic moment associated with the orbital motion can be expressed as W= t k B where W= is the kinetic energy associated with transverse component of motion. Hence explain the diamagnetic behaviour of plasma particles.

mv = -

3. Consider the non-relativistic motion of a charged particle in crossed static uniform electric _ Ev i and magnetic _ Bv i fields. Assuming that E/B < c (velocity of light in free space) find the velocity of a moving frame in which electric field is zero. Describe the motion of the particle in this frame. What would be the corresponding motion in rest frame? 4. Consider a uniform plasma consisting of equal number of electrons and protons placed in orthogonal uniform gravitational field gv = - gjt and magnetic field Bv = Bkt . Calculate the drift velocities of each species. 5. Show that a uniform plasma in a slowly varying electric field together with a perpendicular magnetic field _ Bv i acts as a dielectric of permittivity

e = ε0 =1 +

ρm G ε0 B 2

where rm is the mass density.

3.28

Basic Plasma Physics

6. Consider the motion of a charged particle in a crossed static uniform electric field Ev = Ejt and magnetic field Bv = Bkt . Assuming that initially (t = 0) the particle is at rest at the origin of a Cartesian coordinate system show that the particle trajectory will be a cycloid:

E x(t) = Bw _wc t - sin wc t i c



E y(t) = Bw _1 - cos wc t i c

where wc is the cyclotron frequency. 7. Consider the motion of a charged particle in a crossed static uniform magnetic field Bv = B0 kt and electric field Ev = E0 tj . Assuming that initially (t = 0) the particle is at the origin and has a velocity vv = v0 it show that the path of the particle will be a straight line. 8. Consider a system of two coaxial magnetic mirrors whose axis coincides with z-axis. Suppose the system is symmetrical about z = 0 plane and mirroring planes are at z = ± zm. Find the relation between B (z = 0), B (z = ± zm) and the particle pitch angle q0 at z = 0 such that the particle can be reflected at z = ± zm. 9. Consider the motion of a charged particle in a spatially uniform but slowly time varying magnetic field. Verify the adiabatic invariance of orbital magnetic moment. 10. Suppose a positively charged particle is moving in the magnetic field produced by a long straight steady current carrying conductor and a uniform electric field applied parallel to the wire. Indicate the directions of Ev # Bv , grad-B and curvature drifts. [Hints :

v v vvD = E # B , radially inward B2



v Bi, along Ev vvG = W= _ Bv # d qB 3



vvR =

2W|| v v _ R # Bi, along Ev] qR 2 B 2

Fig. 3.Q-10



Orbit Theory of Plasma

3.29

v B in 11. Consider the motion of a charged particle in a magnetic field Bv having d v v perpendicular direction. Show that there is a force F = - mdB on the guiding centre where m is the orbital magnetic moment of the particle. 12. Consider a charged particle moving with a speed v|| along a circular Bv -field line having radius of curvature R much larger than the Larmor radius RL of the particle in the field of magnitude B. Treating the centrifugal force on the particle as an external force find the resulting drift. 13. Enumerate different types of drifts that plasma particles suffer in electromagnetic fields. 14. Consider the motion of a charged particle in crossed uniform static electric (E) and magnetic (B) fields. Show that for E > Bc the magnetic field is transformed away in a coordinate system moving with a velocity vvD = c2 ( Ev / Bv ) / E2. Hence discuss the motion of the particle as observed from this moving frame and the rest frame. 15. Consider a uniform and slowly time varying electric field pointing in a direction perpendicular to the static uniform magnetic field. Find the current density due to drift motion of ions and electrons. Show that the work done by the electric field in the act of polarization is equal to the increase in kinetic energy of the particles due to their collective motion with a Ev # Bv /B2 drift velocity. 16. Considering both the grad-B and curvature drifts show that the drift speed of a charge q in a torodial magnetic field can be expressed as

vT = 2kBT/qBR

where R is the radius of curvature of the field. [Hints: v ||2 ~kB T/m and v =2 ~2kB T/m ]

EXERCISES 1. Calculate the Larmor radius of a solar wind proton streaming with a velocity 300 km/s in a magnetic field of 5 × 10–9 T. Assume that the velocity component parallel to Bv -field is negligible. [Hints:

v RL = w= c

=

mv= q B

5 -27 # # # = 1.6 10-19 3 10-9 m # 5 # 10 1.6 # 10

= 626 km]

3.30

Basic Plasma Physics

2. (a) Calculate the gyroradius and gyrofrequency of a 4 MeV alpha particle in earth’s magnetic field (10-4 T). Assume that parallel and perpendicular energies of the particle are equal. [Ans. 2 km, 5 × 103 rad/s] (b) Calculate the cyclotron radius of a 10 keV electron in the earth’s magnetic field at a point where magnetic field is 5 × 10–5 T. Assume that the velocity component parallel to Bv -field is negligible. [Ans. 6.75 m] 3. A 20 keV deuteron nucleus has a pitch angle of 45° at the mid plane of a large mirror fusion device, where magnetic field is 0.7 T. Calculate its Larmor radius. [Ans. 0.03 m] 4. A long straight wire is carrying a constant current I in the z-direction. At time t = 0 an electron is in the z = 0 plane and at a distance r = r0­from the wire. Assuming that the electron starts with equal parallel and perpendicular components of velocity, i.e., v|| = v= , calculate the magnitude and direction of grad-B drift. µ0 I v µ I , dB = 2B = 0 2 2 πr 2r 2πr

[Hints:

B(r) =

\

vB d = 1 r B

Now,

vvG =

W= v v _ B # dB i qB 3

or,

vG =

W= v dB qB 2

v B = Bv ad

1 mv 2 mv 2 $ 2π πmv 2 = 1 = 2 qB $ r = 2q =$ µ I = 2qµ =I 0 0 Grad-B is along negative z-direction.] 5. Assuming the earth’s magnetic field to be a simple symmetric dipole field show that the grad-B drift of a charged particle of charge q and energy U at a distance r from the centre of the earth is given by

vG =

3U qrB _ r i

; Hints : B _ r i = C3 E r 6. Calculate the Larmor radius of a doubly charged helium nucleus of energy 3.5 MeV in a magnetic field of 8T. [Ans. 3 cm] 7. Suppose near the equatorial plane the magnetic dipole field can be approximated as B _ z i . B0 + 1 az 2 2 where a is a small constant. Show that in this approximation the parallel motion of a charged particle is simple harmonic with frequency µα/m , where m is the orbital magnetic moment of the particle of mass m.



Orbit Theory of Plasma

3.31

8. Derive polarization drift from the following idea: In a time varying Ev -field with v# v crossed uniform Bv -field. The Ev # Bv drift vvD = E 2B takes place with acceleration B vD dv and hence there is an inertial force Fv = - m associated with the guiding centre. dt This force gives rise to an additional drift vvp , called polarization drift. v# v = Hints: vvp = F 2B G . qB 9. Calculate the current density generated due to gravitational drift at an altitude of 300 km where there exists equal number of electrons and protons each of density 2 × 1012 per m3. Assume that the earth’s magnetic field to be 10–5 T there and perpendicular to gravitational field gv . [Ans. 2.8 × 10-9 A/m2] 10. An electron moves along the axis of a tube of length 2L in which there is an axial magnetic field of the form

B(z) = B0 e a

z

where a is a small positive constant. Find the minimum value of the pitch angle at z = 0 plane for which the electron is trapped. 11. Consider a uniform plasma consisting of equal number of electrons and protons each of density n per metre3 at an attitude equal to the radius of the earth. Assuming magnetic field and gravitational fields to be normal to each other calculate the gravitational drift of each species. Take the magnetic field of the earth to be a dipolar field with a value 10–4 T on the earth’s surface.

R > Hints: g = gsurface d1 - 2h n, B_ r i = B0 d E n , r = RE + h = 2RE H r RE 3

12. Calculate the curvature drift of an electron on the earth’s surface. Assume that the magnetic field on the earth’s surface = 40 mT, parallel energy of the electron is equal to its thermal energy at a temperature of 300 K and radius of the earth = 6400 km. [Hints: vR =

2W|| ; W k T 1.38 # 10 -23 # 300J, q = 1.6 # 10 -19 C, qRB || = B =

R = 6400 × 103 m, B = 40 × 10–6 T. Thus vR ≈ 2 × 10–4 ms–1] 13. A charged particle moves in a magnetic field given by Bv = B _ x itj + B kt y

0

2By where By(x) and are very small quantities. 2x

Verify that the grad-B drift for the particle is almost zero. [Hints: vvG =

W= v v _ B # dB i qB 3

3.32

Basic Plasma Physics

Here

where \

tj kt it v B = 0 By B0 Bv # d 2B 2B 2B 2x 2y 2z B =

B 2y _ x i + B 02 , 2B = 2B = 0 2y 2z

v B = tjB 2B - kB t 2B Bv # d 0 2x y 2x

2By By _ x i 2 B 2x . 0 (Being product of two very small quantities) Now, B 2x = Thus vG - 0 .] 14. A collisionless plasma with an isotropic velocity distribution is suddenly placed at the equatorial plane of a magnetic bottle with mirror ratio Rm. Calculate the fraction of plasma particles that will be lost from the magnetic bottle. [Hints: Particles within loss cone escape and rest remain trapped. The fraction of particles that will be lost is given by FL = 1 2π

iC

y0

dΩ = 1 2π

iC

y0

2π sin θdθ =

1/2 1 - cos qc = 1 - d1 - 1 n .] Rm

15. Suppose the magnetic field along the axis of a magnetic bottle is given by Bz = B0 (1 + a2z2) where a is a positive constant and a2z2 0.

Thus grad-B drift separates electrons and ions vertically in the zt -direction. The resulting Ev -field is in zt -direction. Hence Ev # Bv drift pushes the plasma outwards.] 20. Considering both the grad-B and curvature drifts calculate the drift speed for a plasma particle at a temperature of 10 keV, a magnetic field of 2T and radius of curvature of 1 m. qqq

CHAPTER

4 Fluid Description of Plasmas

4.1 INTRODUCTION plasma generally consists of a large number of particles. It is practically A impossible to follow the complicated trajectory of each of these particles and predict the behaviour of plasma. Fortunately majority of plasma phenomena observed in real experiments can be explained by a rather crude model known as fluid model. Here identity of the individual particle is neglected and only the motion of fluid elements is considered as in fluid mechanics. In the fluid approximation it is assumed that each plasma species is able to maintain a local equilibrium. Then each species can be treated as a fluid described by local density, local macroscopic velocity and local temperature. Time evolution of these quantities is determined by means of fluid equations which are analogous to, but generally more complicated than, the equations of hydrodynamics. The macroscopic variables are related to the moments of particle distribution function f_rv, vv, t i . The equations satisfied by these macroscopic variables can be obtained by taking various moments of the Boltzmann equation which describes the time evolution of f_rv, vv, t i . It requires more mathematical calculations than is appropriate for the beginners. It will be considered in Chapter 6. In this chapter we shall consider plasma fluid equations in their simplest form by using hydrodynamic approach and some simple physical arguments.

4.2 FLUID EQUATION OF MOTION he equation of motion of a plasma particle of charge q and mass m in the selfT consistent electric _ Ev i and magnetic _ Bv i fields is

v m dv = q_ Ev + vv # Bv i dt

...(4.2-1)

4.2

Basic Plasma Physics

Suppose that all the particles in a fluid element feel essentially the same forces and they move in the same way so that the average velocity uv of the particles in the fluid element is the same as the individual particle velocity vv . It is true for collisionless cold plasma. Then multiplying Eqn. (4.2-1) by the particle number density n we get the simple fluid equation of motion: v mn du = qn_ Ev + uv # Bv i dt



...(4.2-2)

Here the time derivative is with respect to a reference frame moving with the mean velocity uv . Therefore, the time derivative operator, d ≡ 2 uv $ d v 2t + dt



...(4.2-3)

v corresponds to where 2 is the time derivative in a fixed frame and the term uv $ d 2t change as the observer moves with the fluid. Thus,

v v i uv E = qn_ Ev + uv # Bv i mn ; 2u + _uv $ d 2t

...(4.2-4)

When thermal motions are taken into account an additional force due to pressure gradient has to be added to the right-hand side of Eqn. (4.2-4). This force arises due to random motion of particles in and out of a fluid element. For a collisionless plasma with isotropic velocity distribution of plasma particles pressure gradient v p where p is the scalar kinetic force per unit volume may be calculated as - d pressure. Then we have the fluid equation of motion:

v v i uv E = qn_ Ev + uv # Bv i - d v p mn ; 2u + _uv $ d 2t

...(4.2-5)

In the general case of anisotropic distribution and the presence of shearing v p is to be replaced by - d v $ Pv where Pv is the forces and effects of viscosity - d pressure tensor. In a collisional plasma collisions between plasma particles will lead to an additional force term to be added in the equation of motion (4.2-5).

4.3 EQUATION OF CONTINUITY The equation of continuity follows from the principles of conservation of mass. It can be easily derived using the method of fluid dynamics. Conservation of mass requires that the rate of decrease of the number of particles in a volume V must be equal to that leave the volume V per unit time through the surface S bounding the volume V. Therefore,

-

2 2t

y ndV = y muv $ dsv v

s

where n is the number density of particles and uv is the fluid velocity.

...(4.3-1)



Fluid Description of Plasmas

4.3

Using Gauss’s divergence theorem we can write

-

or, y ; V

2 y ndV = 2t V

y dv $ ^nuvhdV V

2n v + d $ ^nuvhEdV = 0 2t

...(4.3-2)

Since this result must be valid for any arbitrary volume V the integrand must vanish identically. Thus we get the equation of continuity: 2n d v $ _nuvi = 0 2t + There is one such equation of continuity for each species.

...(4.3-3)

4.4 CLOSED SYSTEM OF FLUID EQUATIONS he continuity equation (4.3-3) relates two macroscopic variables, number density T n with flow velocity uv . To determine these two variables we need two independent macroscopic transport equations. The equation of motion or the momentum conservation equation (4.2-5) relates uv with n and pressure p [the variables Ev and Bv being provided by Maxwell’s equations]. Therefore, the two transport equations (4.3-3) and (4.2-5) relate three independent variables. We may go to the next higher order transport equation, namely the energy conservation equation. But it is found to relate four variables, uv , n, p and heat flow. Then one gets three independent equations for four variables. Thus, we find that transport equations do not form a closed set in the sense that number of equations is not sufficient to determine all the variables that appear in them. Thus one needs to introduce some simplifying assumptions. In order to get a complete set of fluid equations generally there are two widely used approximations – the so-called cold and warm plasma models.

Cold Plasma Model he cold plasma model uses only the continuity equation and the equation of T motion. The effects due to thermal motion of plasma particles are neglected and the force due to pressure gradient term is taken equal to zero. This cold plasma model has been successfully applied, for example, in the investigation of wave propagation in plasma with phase velocities much larger than the thermal velocity of the particles.

Warm Plasma Model I n warm plasma model the heat flux term that appears in the energy transport equation is taken to be zero. This means that the processes occurring in plasma are such that there is no heat flow. This approximation is also called adiabatic approximation. In addition neglecting the effects of viscosity and energy transfer

4.4

Basic Plasma Physics

due to collision it can be shown that the energy equation reduces to the following adiabatic equation of state relating pressure to density: prm- c = C



...(4.4-1)

where C is a constant, r­m is the mass density and g = C­P/CV­ is the ratio of specific heats at constant pressure and at constant volume. The parameter g is related to the number of degrees of freedom f of a gas by the relation

g = 1 + 2 f

...(4.4-2)

The equation of continuity, equation of motion and the adiabatic equation of state (4.4-1) form a closed set of equations. Compared to cold plasma model the warm plasma model gives a more precise description of various plasma phenomena.

4.5

COMPLETE SET OF FLUID EQUATIONS FOR A SIMPLE PLASMA

e consider a simple plasma consisting of electrons and ions. In the warm plasma W model the closed set of hydrodynamic equations are the following: 2n j v $ ^n uv h = 0 +d j j 2t 2uv j vp v j uv G = q j n j a Ev + uv j # Bv k - d m j n j = + `uv j $ d j j 2t



p j = C j n cj j

...(4.5-1) ...(4.5-2) ...(4.5-3)

where, n j, uv j, p j, m j and q j are respectively the number density, flow velocity, pressure, mass and charge of the jth species (j = e for electrons and j = i for ions); Cj is a constant and gj is the ratio of two specific heats; Ev and Bv are respectively the self-consistent electric and magnetic fields inside the plasma. As plasma particles are charged and move around they can generate charge density (r), current density ( vj ), electric and magnetic fields. These variables are provided by Maxwell’s equations: v $ Ev = ρ d ε0



v v # Ev = - 2B d 2t v $ Bv = 0 d v v # Bv = µ d vj + ε 2E n d 0 0 2t

where, r = ni qi + ne qe, vj = ni qi uvi + ne qe uve

...(4.5-4) ...(4.5-5) ...(4.5-6) ...(4.5-7) ...(4.5-8)



Fluid Description of Plasmas

4.5

The above two set of equations provide 16 independent scalar equations for the 16 scalar unknowns n j, uv j, p j, Ev and Bv . Simultaneous solution of this set of 16 equations gives the self-consistent fields and motions in warm plasma model.

4.6 FLUID DRIFTS fluid element contains many individual particles and it is expected that a fluid A element will exhibit drifts similar to the drifts of individual particles. However in a fluid plasma there is an additional drift associated with the pressure gradient that is not found in the single particle picture. To see how this happens so we consider the fluid equation of motion for each species:

v v i uv E = qn_ Ev + uv # Bv i - d v p mn ; 2u + _uv $ d 2t

...(4.6-1)

2uv For drifts slow compared to cyclotron frequency we may neglect the term mn 2t v i uv and then compared to the term qn_uv # Bv i . Neglecting the convective term _uv $ d taking the cross product of Eqn. (4.6-1) with Bv , we get Now,

v p # Bv 0 = qn 8 Ev # Bv + _uv # Bv i # Bv B - d

...(4.6-2)

_uv # Bv i # Bv = - Bv # _uv # Bv i

v 2 + Bv _ Bv .uvi = – uB = -`uv|| + uv=j B 2 + Bv ` Bv .uv||j = -`uv|| + uv=j B 2 + uv|| B 2 = - uv= B 2

8a Bv $ uv= = 0B 8a Bv || uv||B

...(4.6-3)

Therefore, from Eqn. (4.6-2),

v p # Bv v# v d uv= = E 2B B qnB 2

= uvE + uvD

...(4.6-4)

v p # Bv v# v d where, uvE = E 2B can be recognized as the Ev # Bv drift and uvD =is the qnB 2 B so-called diamagnetic drift – a fluid effect. Note that uvD depends on the sign of charge. So electrons and ions will drift in opposite directions and so gives rise to a diamagnetic current:

vj = n _uv - uv i e Di De D

4.6

Basic Plasma Physics = _ kB Ti + kB Tei

vn Bv # d 2 B

...(4.6-5)

The physical origin of the diamagnetic drift can be understood from Fig. 4.5-1 which shows the orbits of plasma ions gyrating in a magnetic field. For a pressure or density gradient towards left there are more ions moving downward than upward since the downward-moving ions come from a region of higher density. Thus a v n and Bv . As the guiding centres are stationary, fluid drift occurs perpendicular to d this drift does not occur in single particle theory.

Fig. 4.5-1 : Physical origin of diamagnetic drift

The curvature drift of single particle picture also exists in fluid picture because all the particles in a fluid element feel centrifugal force as they move around curved magnetic field. However, the grad-B drift of single particle picture does not exist in fluid picture. This is so because magnetic field cannot change energy and distribution of plasma particles. The particle drifts in any fixed fluid element cancel out.

4.7 SINGLE FLUID EQUATIONS nder certain conditions it is possible to consider plasma as a single conducting U fluid without specifying its various individual species (electrons, ions and neutral particles). The fluid equations for individual species are combined into one set of equations that describe the macroscopic behaviour of the plasma as a whole without considering the individual species present. The set of equations thus obtained are usually known as the magnetohydrodynamic (MHD) equations. Plasma then behaves like a simple conducting fluid. In this approach each macroscopic variable is combined by adding contributions of the various species in plasma.



Fluid Description of Plasmas

4.7

In order to describe the plasma according to one-fluid model we need to confine ourselves to low frequency (w wpe. For w < wpe, k would become imaginary. As the wave has spatial dependence as exp ikx, it will be exponentially damped as e–|k|x. Thus w = wpe is a cut-off frequency. As a consequence, a low frequency electromagnetic wave cannot pass through a plasma of very high density.



Waves in Plasmas

5.13

For w > wpe, the phase velocity of the electromagnetic wave is vp =



w =c k

1+

w2pe k2 c2



...(5.5-19)

and group velocity is

dw kc2 vg = dk = w =

c 1+

w2pe



...(5.5-20)

k2 c2 Note that here phase velocity is greater than the speed of light in vacuum. It does not violate the principle of relativity because the group velocity with which a signal propagates is always less than c. For w > wpe the dispersion relation (5.5-18) reduces to

w = kc and then vp = vg = c.

It corresponds to electromagnetic waves in free space. Physically this behaviour is expected because in the limiting case of infinite frequency even the electrons are unable to respond to the oscillating electric field. In this limit electromagnetic waves are not damped by the presence of plasma. The dispersion relation (5.5-18) is shown graphically in Fig. 5.5-1.

Fig. 5.5-1 : Dispersion curve for electromagnetic waves in cold unmagnetized plasma

Reflection of Radio Waves from Ionosphere ong distance radio communication around the earth is possible due to reflection L of radio waves from the ionosphere which is in plasma state. When a radio wave reaches at altitude in the ionosphere where w = wpe then the wave cannot go beyond that point. The wave is reflected making it possible to send signals around the earth. To communicate with space vehicles or artificial satellites it is necessary for the radio waves to penetrate the ionosphere and we must use radio

5.14

Basic Plasma Physics

waves with frequencies w > wpe, where wpe corresponds to maximum density in the ionosphere. For example, if the maximum density of ionospheric plasma is 1012 m–3 the critical or cut-off frequency will be of the order of 10 MHz. Hence frequency used for radio communication on the earth should be less than 10 MHz and for communication with space vehicle frequency should be greater than 10 MHz.

Measurement of Plasma Density he phenomenon of cut-off of electromagnetic waves by plasma suggests an T easy way to measure plasma density. For most laboratory plasmas wpe lies in the microwave region. A beam of microwaves is allowed to incident on plasma and the transmitted beam is detected by a crystal. The frequency (w) of the microwaves is varied and the frequency below which the waves are not transmitted by the plasma is noted. It gives plasma frequency w­pe and hence the plasma density.

5.6

ELECTROSTATIC ELECTRON OSCILLATION PERPENDICULAR TO APPLIED MAGNETIC FIELD

o far we have assumed that there is no external applied magnetic field. In presence S of an applied magnetic field it is found that many more modes of oscillations and waves are possible. However, we shall examine only certain simple cases. To begin with we consider electron oscillations perpendicular to the applied magnetic field Bv0 . We assume the plasma to be cold and ions are immobile. The plasma electron fluid dynamics are then governed by the following equations: 2ne v + d $ ^ne uveh = 0 2t 2 v m uv = - en 8 Ev + uv # Bv B me ne c + uve $ d e e e 2t v $ Ev = e (n – n ) e0 d i e

...(5.6-1) ...(5.6-2) ...(5.6-3)

where the symbols have their usual meaning. Assuming small perturbations in plasma parameters about their equilibrium values the Eqns. (5.6-1) – (5.6-3) can be written in terms of first order perturbed quantities (denoted by the subscript 1) as

2n1 v + d $ ^n0 uv1h = 0 2t 2uv m 1 = - e` Ev1 + uv1 # Bv0j 2t v $ Ev = – en e0 d 1 1

...(5.6-4) ...(5.6-5) ...(5.6-6)



Waves in Plasmas

5.15

Suppose we consider longitudinal waves with Ev1 parallel to wave vector kv assumed to be along x-axis and external magnetic field Bv along z-axis (Fig. 5.6-1). 0

Now assuming harmonic dependence of the perturbed quantities as exp i(kx – wt) we obtain from Eqns. (5.6-4) – (5.6-6).

Fig. 5.6-1 : Geometry of longitudinal wave propagating perpendicular to Bv0



–iwn1 + ikn0u1x = 0



...(5.6-7)

- iwmuv1 = - e` Ev1 + uv1 # Bv0j e0ikE1 = –en1

...(5.6-8) ...(5.6-9)

Equation (5.6-8) can be written in the following component forms:

– iwmu1x = - eE1 - eu1y B0



– iwmu1y = eu1x B0



– iwmu1z = 0

(5.6-10)

Solving (5.6-10) for u1x and then using it in Eqn. (5.6-7) we obtain

where wce =

eE1 im w u1x = w2ce 1- 2 w

(5.6-11)

eB0 is the electron cyclotron frequency. m

Now using Eqns. (5.6-7) and (5.6-10) we get from Eqn. (5.6-9)



ikE1 = -

ek n u = ωε0 0 1x

ikω2pe ω2 E ω2 1 1 - ce2 ω

5.16

or

Basic Plasma Physics

e1 -

w2pe w2ce o = E E w2 1 w2 1

...(5.6-12)

For non-trivial solution (E1 ≠ 0) we get the dispersion relation

w2 = w2pe + w2ce

...(5.6-13)

It represents a non-propagating fixed frequency oscillation. The frequency of oscillation w =

w2pe + w2ce = wh is called upper hybrid frequency. The frequency

of oscillation is higher than the electron plasma frequency wpe because here there are two restoring forces on the electrons – the electrostatic force and the Lorentz force.

5.7 MHD WAVES IN A UNIFORM PLASMA I n the MHD description of plasma we have seen that the magnetic field lines behave as mass-loaded string under tension. The MHD fluid feels a magnetic B2 B2 tension along the field lines and an isotropic pressure . So by analogy 2m0 m0 with transverse vibration of elastic strings we can expect that if the conducting fluid is slightly disturbed from equilibrium conditions the magnetic field lines will perform transverse vibrations and the disturbance would propagate along the field lines with a velocity.

VA =

tension = density

B02 µ0 ρm

...(5.7-1)

This is known as Alfvén speed. In 1942 Alfvén first pointed out the existence of this type of low frequency wave motion in a conducting magnetized fluid. There is no density or pressure fluctuations associated with this wave and it is often called the torsional or shear Alfvén wave. The fluid motion and magnetic perturbations are all perpendicular to the field lines. Figure 5.7-1 gives a pictorial representation of torsional Alfvén waves in a compressible conducting MHD fluid.

Fig. 5.7-1 : Torsional Alfvén waves



Waves in Plasmas

5.17

Longitudinal oscillations are also possible in a compressible conducting MHD fluid. For motion of particles and propagation of waves along the magnetic field lines there will be no magnetic field perturbation along the magnetic field lines since the particles are forced to move in this direction. The corresponding wave is simply the ordinary longitudinal sound waves. In Fig. 5.7-2 we give a pictorial representation of longitudinal sound waves propagating along the magnetic field lines.

Fig. 5.7-2 : Longitudinal sound waves propagating along magnetic field lines

In a direction perpendicular to Bv -lines a new type of longitudinal wave motion is possible, which involves compression and rarefaction of both the lines of force and the conducting fluid (Fig. 5.7-3). It is known as magnetosonic, magnetoacoustic or simply compressional wave.

Fig. 5.7-3 : Magnetosonic wave propagating perpendicular to magnetic field lines

5.18

5.8

Basic Plasma Physics

MATHEMATICAL ANALYSIS OF MHD WAVES

o examine the wave motion in a MHD plasma we start from the following MHD T fluid equations:

2rm v + d $ ^rm uvh = 0 2t



rm c



c

2 v p = 1 ^d v # Bv h # Bv - d v p v m uv = vj # Bv - d + uv $ d m0 2t

2 v mc p m = 0 + uv $ d 2t rcm 2Bv v # ^uv # Bv h = d 2t

...(5.8-1) ...(5.8-2) ...(5.8-3) ...(5.8-4)

where the symbols have their usual meaning. We assume that in the equilibrium state the fluid is uniform with a constant density r0, the equilibrium velocity is zero and the magnetic field is constant ` Bv0j throughout. To deduce the dispersion relation for small amplitude waves we assume small amplitude departures from the equilibrium values:

rm ^rv, t h = r0 + r1 ^rv, t h



Bv ^rv, t h = Bv0 + Bv1 ^rv, t h



uv^rv, t h = 0 + uv1 ^rv, t h



p^rv, t h = p0 + p1 ^rv, t h

...(5.8-5)

Substituting (5.8-5) in Eqns. (5.8-1) – (5.8-4) and then linearizing, we get

2p1 v $ uv = 0 + r0 d 1 2t 2uv 1 v v vp v -d # B j# B r0 1 = `d 1 0 1 m0 2t 2p1 γp0 2ρ1 2t ρ0 2t = 0 2Bv1 v # `uv # Bv j d 1 0 2t =

...(5.8-6) ...(5.8-7) ...(5.8-8) ...(5.8-9)

Let us search for wave-like solutions of Eqns. (5.8-6) – (5.8-9) in which the perturbed quantities vary like exp i^kv $ rv - wt h . v by ikv and 2 by – iw. Thus we obtain Then we can replace d 2t



Waves in Plasmas

- ωρ1 + ρ0 kv $ uv1 = 0



- ωp1 +



.

...(5.8-10)

1 v - ωρ0 uv1 = ` kv # Bv1j # Bv0 - kp 1 m0



5.19

...(5.8-11)

γp0 ωρ1 = 0 ρ0 - wBv1 = kv # `uv # Bv0j

...(5.8-12) ...(5.8-13)

These equations can be combined to yield an equation for uv1 alone: - w2 uv1 - VvA # $kv # 8kv # `uv1# VvAjB. + v s2 ` kv $ uv1j kv = 0 where VvA = Bv0 / µ0 ρ0 is the Alfvin velocity vector and vs =

...(5.8-14) γp0 is the usual ρ0

sound speed. Using the rules of vector triple product we get finally - w2 uv1 + ^v s2 + V A2h` kv $ uv1j kv + ` kv $ VvAj$uv1 ` kv $ VvAj - VvA ` kv $ uv1j - `VvA $ uv1j kv. = 0 ...(5.8-15) Let us now consider two special cases of wave propagation:

Wave Propagation Parallel to Bv0 In this case kv $ Vv = kV and Eqn. (5.8-15) reduces to A

A

^k2 V A2 - w2h uv1 + e

v s2 - 1o k2 `VvA $ uv1j VvA = 0 V A2

...(5.8-16)

This relation supports two types of wave motion. For uv1 parallel to kv and v B0 , Eqn. (5.8-16) yields an ordinary longitudinal sound wave with phase velocity

vp =

w = vs k

...(5.8-17)

This wave is not associated with any electric field or magnetic field. For uv1 perpendicular to kv and Bv0, uv1 $ VvA = 0 , and Eqn. (5.8-16) yields a transverse wave with phase velocity

w vp = k = VA

(5.8-18)

This is the dispersion relation for torsional Alfvén wave. The magnetic field perturbation associated with the Alfvén wave can be obtained by using Eqn. (5.8-13):

- wBv1 = uw1 ` kv $ Bv0j - Bv0 ` kv $ uv1j

5.20

Basic Plasma Physics kv $ uv1 = 0,

Since

B Bv1 = - 0 uv1 w c m k



...(5.8-19)

So magnetic field perturbation is perpendicular to the original magnetostatic field Bv0 . Thus the Alfvén wave parallel to Bv0 causes the lines of force to oscillate back and forth laterally (Fig. 5.7-1).

Wave Propagation Perpendicular to Bv 0 In this case kv $ Bv = 0 . The last term in Eq. (5.8-15) vanishes and it simplifies to 0



uv1 =

1 2 ^v + V A2h` kv $ uv1j kv w2 s

...(5.8-20)

Therefore, uv1 is parallel to kv and the associated wave is longitudinal in nature with a phase velocity

w vp = k =

v s2 + V A2

...(5.8-21)

The magnetic field perturbation associated with the wave can be obtained from (5.8-13) as

u Bv1 = 1 Bv0 w c m k

...(5.8-22)

Thus the magnetic field perturbation is parallel to the background magnetostatic field. There are compressions and rarefactions in the lines of force without changing their directions (Fig. 5.7-3). The wave is longitudinal since the velocity of mass flow and the mass density fluctuations are both in the direction of wave propagation. For this the wave is called magnetosonic wave. In the limit Bv " 0, v " v , the 0

p

s

magnetosonic wave turns into an ordinary ion-acoustic wave. On the other hand if Bv is very strong then v " V (Alfvén velocity). The phase velocity of the 0

p

A

magnetosonic wave is always larger than VA and for this it is often called the fast hydromagnetic wave. In the above discussions we have neglected the effects of displacement current. This approximation is valid only for highly conducting fluids at relatively low frequencies well below the ion cyclotron frequency. We have assumed the fluid to be non-viscous and perfectly conducting. If viscous effects are present and the fluid has a finite conductivity the MHD waves are found to be damped.



Waves in Plasmas

5.21

SOLVED PROBLEMS PROBLEM 5.1: Show that for Alfvén wave motion the magnetic energy density of the wave motion is equal to the time-averaged kinetic energy density. Solution: The magnetic energy density of the wave motion is

B B B12 where Bv1 = 0 uv1 =- 0 uv1 2m0 VA w c m k B2u2 B12 1 = 0 12 = ρ0 u12 2m0 2 2µ0 V A

Now,

=` VA =

B0 G µ0 ρ0

PROBLEM 5.2: Assuming that the solar corona consists of ionized hydrogen of electron density 1012 m–3 and the magnetic field there to be 10–3 T calculate the speed of an Alfvén wave. Solution: Alfvén speed VA = Here

B µ0 ρm

B = 10-3 T, rm . n $ mi = 1012 # 1.67 # 10-27 kg.m-3 = 1.67 # 10-15 kg.m-3 VA =

\

10-3 ^4p # 10-7 # 1.67 # 10-15h1/2

= 2.2 # 107 m.s.-1

PROBLEM 5.3: In a microwave interferometer experiment a fringe shift of 1/10 is obtained when one of the interfering beams is passed through a plasma slab 8 cm thick. Assuming the wavelength of the microwave used as 8 mm calculate the density of plasma. Solution: The refractive index of plasma is

µ=

c w2 1/2 e1 - pe o = ω w2 c m k

[using (5.5 – 18)]

Extra path difference introduced is (1 – m)l where l = 8 mm, thickness of plasma. Therefore number of fringe shift or

DN = (1 – m) l/l0 m = 1 -

(l0 = free space wavelength)

∆N $ λ0 0.1 # 0.8 = 1= 1 - 0.01 l 8

5.22

or

Basic Plasma Physics

1-

w2pe w2

= 1 – 0.02 2

or

ne = e

3 # 108 o 8 # 10-3

#

0.02 = 3.5 # 10-17 m-3 . 92

PROBLEM 5.4: Suppose electron plasma waves of frequency 1 GHz are propagating in a uniform plasma with electron temperature 100 eV and density 1016 m–3. Find the wavelength. Solution: From Bohm-Gross dispersion relation

3 kB Te 2 w2 = w2pe + k 2 me

Here w = 2pf with f = 109 Hz. \

ω pe . 2π # 9 ne = 18p × 108 rad/s 2 (2p × 109)2 = ^18π # 108h +

3 100 # 1.6 # 10-19 2π 2 $ $c m λ 2 9.1 # 10-31

From this we get l = 1.18 × 10–2 m = 1.18 cm. PROBLEM 5.5: A uniform plasma has ions that are initially at rest, but its electrons are streaming through the ions with velocity u0. Using the two-fluid equations and neglecting thermal effects show that the dispersion relation for electrostatic oscillations involving both electrons and ions is



1 = w2pe

>

me mi 1 H + 2 2 w ^w ku0h

Solution: The linearized equations of continuity and motion for both electrons and ions are

2ni1 v $ uv = 0 + n0 d i1 2t

2ne1 v $ uv + uv $ d v n = 0 + n0 d e1 0 e1 2t 2uv mi n0 i1 = en0 Ev1 2t 2uv v j uv E = - en Ev me n0 ; e1 + `uv0 $ d 0 1 e1 2t



Waves in Plasmas

5.23

Poisson’s equation:

v $ Ev = e(n – n ) e0 d 1 i1 e1

Assuming one-dimensional plane wave solutions varying as exp i (kx – wt) and expressing ne1, ue1, ni1, ui1 in terms of E1 we get from the Poisson’s equation 1 1 + e0 ikE1 = ie2 n0 kE1 = G 2 m1w me ^w - ku0h2 Thus we get the desired dispersion relation, me 1 + G. 1 = w2pe = 2 mi w ^w - ku0h2 PROBLEM 5.6: Show that the propagation time of radio signals from pulsars over a large distance d through interstellar plasma containing free electrons may be expressed approximately as

t =

d e2 + c 2cme ε0 ω2

y0d ne ds

where ds is a path element over which plasma density is ne. Solution: Dispersion relation is w2 = w2pe + k2 c2 dw kc2 c = = \ Group velocity = vg = w2 - w2pe = c dk w w \

1-

w2pe w2

w2pe 1 1 ≈ =1 + 2 G vg c 2w

Now propagation time of the signal is

t =

n e2 Using ω2pe = e , we get ε0 me

ds

y0d v

g

=

1 d + c 2cw2

d e2 t = c + 2cε0 me ω2

y0d w2pe ds

y0d ni ds .

QUESTIONS 1. Show that when the motion of both electrons and ions is considered the natural frequency of oscillation of the net charge density in a cold homogeneous infinite plasma is given by

w =

w2pe + w2pi

5.24

Basic Plasma Physics

where ω pe =

n0 e 2 ε0 me

and ω pi =

n0 e 2 . The symbols have their usual ε0 mi

significance. OR Starting from the equations of continuity and momentum for each species of an electron-ion plasma and Poisson’s equation show that the natural frequency of oscillation of the net charge density in a cold homogeneous infinite plasma is given by w =

w2pe + w2pi where the symbols have their usual significance.

2. (a) Starting from the fluid equations for electrons and the Poisson’s equation derive the Bohm-Gross dispersion relation 2 w2 = w2pe + 3 k 2 V Th 2 for longitudinal electron plasma wave.



(b) Discuss the case w < wpe. (c) Show that for w >> wpe­, the wave group travels essentially at thermal velocity (VTh). 3. (a) Starting from the basic fluid equations obtain the dispersion relation for an ionacoustic wave. You can use the plasma approximation. (b) Give the physical picture of ion-acoustic wave propagation. 4. Using the equilibrium fluid equation of motion show that the particle number density distribution for Maxwellian electrons of temperature Te is described by the Boltzmann relation ef ne = n0 exp e k T o B e [Hints: Set the convective derivative to zero for equilibrium.] 5. Show that the dispersion relation for plane transverse electromagnetic waves propagating in a cold, unbounded unmagnetized stationary plasma with immobile ions is given by

w2 = w2pe + c 2 k 2

Sketch the dispersion relation. Comment on the physical significance of the dispersion relation near the region w = wpe­. 6. (a) Discuss how the frequency range for radio communication/satellite communication is related to the maximum density of ionospheric plasma. (b) Discuss how the phenomenon of cut-off of electromagnetic waves by plasma can be used to measure plasma density. 7. Consider electrostatic electron oscillation perpendicular to the applied magnetic field in a magnetized plasma and find the dispersion relation

w2 = w2pe + w2ce .



Waves in Plasmas

5.25

8. What are Alfvén waves? Give the physical picture of Alfvén wave propagation and calculate the value of Alfvén speed. 9. Describe the physical process by which magnetosonic waves can be generated. Deduce an expression for the velocity of magnetosonic waves in plasma. 10. (a) Considering the propagation of electromagnetic waves through a plasma show that the index of refraction is equal to

w2pe

. w2 (b) Phase velocity of electromagnetic waves through plasma is greater than c. Does it violate the principle of relativity? 1-

11. Obtain the dispersion relation for ion-acoustic waves without using the plasma approximation. Hence discuss the regions of validity of plasma approximation. 12. Derive the dispersion relation for electron plasma waves in a plasma consisting of cold electrons and ions moving with the relative velocity u0. 13. Consider Alfvén waves propagating in a uniform plasma in a uniform field with the wave vector kv parallel to the applied magnetic field Bv0 . Assuming that the displacement current is negligible find the dispersion relation. 14. Show that the Alfvén wave propagating along the magnetic field is circularly polarized.

EXERCISES 1. A plasma is immersed in an external magnetic field of 10 T. Calculate the magnetic pressure. [Ans. 4 × 107 Nm–2] 2. Find the Alfvén wave speed in the interstellar plasma. Assume that the electron density ne = 107 m–3, magnetic field B = 10–7 T and the positive charge carriers are protons. [Ans. 6.9 × 105 ms–1] 3. Show that for electron plasma oscillation (i.e., Langmuir oscillation) time-averaged electron kinetic energy per unit volume is equal to electric field energy density. 1 1 e2 [Hints: From Eqn. (5.2-10), n0 2 me u 12 = n0 2 me . 2 2 E 12 w me n e2 1 1 Using ω2 = ω2pe = ε 0 m we get, n0 2 me u 12 = 2 e0 E 12 ] 0 e 4. The maximum electron density in ionospheric plasma is 1012 m-3. (a) Find the minimum frequency of the radio signal that can be used for communication with space vehicle. [Ans. 10 MHz] (b) Find the maximum frequency of electromagnetic waves that can be used for world-wide radio communication through ionospheric waves. [Ans. 10 MHz] 5. A beam of microwaves from a variable frequency microwave source is incident on a uniform plasma and the transmitted beam is detected by a crystal. As the frequency

5.26

Basic Plasma Physics of the source is slowly decreased it is found that at the frequency 9 GHz the detected signal disappears. Find the density of plasma. [Ans. 1018 m–3]

6. The dispersion relation for electromagnetic waves in plasma is given as w2 = w2pe + k 2 c 2 . An electromagnetic wave is incident obliquely on a plasma slab. Find the critical angle for total internal reflection as a function of frequency w. c kc =Hints: Refractive index µ = ω = ω = ak k Critical angle θc = sin -1 1 = sin -1 µ

1-

1 1-

ω2pe ω2

=

ω2pe ω2

.

7. Find the speed of ion-acoustic waves in a uniform plasma with electron temperature 104K and ion mass mi = 65 × 10–27 kg. > Hints: vs =

kB Te mi =

1.38 # 10 -23 # 10 4 1.46 # 10 3 ms -1 .H = 65 # 10 -27

8. Consider electron plasma oscillations in a cold unbounded plasma with inhomogeneous density. Starting from the linearized equation of continuity and equation of motion for electron fluid show that the Poisson’s equation can be written in the following form where

v $ _eEv i = 0 d e = 1 -

w2pe _ x i

w2 [Hints: Derive Eq. (5.2-13) and use it.] 9. Considering the dispersion relation for electromagnetic waves in plasma show that the product of the phase velocity and group velocity is c2. 10. Consider simple electron plasma oscillation in a cold unbounded homogeneous plasma with immobile ions. Assuming space time dependence of fluctuating quantities as exp [i(kx – wt)] find the phase difference between electron fluid velocity and electric field. [Hints: Use Eq. (5.2-10).]

[Ans. 90°]

11. Suppose a two-component electron-ion plasma is moving with a uniform constant velocity u0 along x-axis with respect to a stationary observer. Assuming plasma approximation show that the dispersion relation for ion-acoustic waves in this case is given by _w - ku0i2 = v s2 k 2 , where vs is the usual ion-acoustic speed. [Hints: Use uv j = uv0 + uv j1 .] 12. An electron plasma wave of frequency 51 MHz is propagating in a plasma with electron number density 1.88 × 1013 m–3 and temperature 2640 K. Find the wavelength of the wave. [Ans. 0.75 cm]



Waves in Plasmas

5.27

13. Assuming that the plasma density in the ionosphere increases linearly with height the plasma frequency wpe can be expressed as

w2pe _ z i = K(z – h0)

for z > h0

where z = h0 corresponds to the base of the ionosphere. Calculate the height at which waves of frequency w transmitted vertically from a transmitter on the ground are reflected from the ionosphere. [Ans. w2/K + h0] qqq

CHAPTER

6 Kinetic Theory of Plasma

6.1

INTRODUCTION

plasma is a system containing large number of interacting charged particles. A The fluid theory provides a simple description of plasma. Though it can describe the majority of observed phenomena in plasma, it is found to be inadequate in some cases. More accurate description of plasma can be obtained from statistical kinetic approach. In this approach we need to introduce the concept of distribution function. From a knowledge of the distribution function the macroscopic plasma variables necessary for the macroscopic description of the plasma behaviour, can be systematically deduced. In fact all physically interesting information about the plasma is contained in the distribution function. The differential equation satisfied by the distribution function is generally known as the Boltzmann equation. In the kinetic description of plasma it plays an important role. As the present text is designed for the beginners the complex collision integral involved in the Boltzmann equation will not be considered here. Instead, we shall introduce the collisionless Boltzmann equation – the so-called Vlasov equation. As an example of its use we shall derive the dispersion relation for electron plasma waves in a warm plasma and discuss the phenomenon of Landau damping.

6.2 DISTRIBUTION FUNCTION AND MACROSCOPIC VARIABLES he one-particle distribution function f^rv, vv, t h is so defined that the number of T particles within the volume element d3 r d3 v of phase space about the phase space coordinates ^rv, vvh at time t is given by where

dn^rv, vv, t h = f^rv, vv, t h d3 r d3 v d3r d3v = dx dy dz dvx dvy dvz.

...(6.2-1)

6.2

Basic Plasma Physics

In a statistical sense the distribution function provides a complete description of the system under consideration. The macroscopic property of the system is due to the average behaviour of innumerable particles whereas the motion of each individual particle is controlled by the usual rules of mechanics. Knowing the distribution function f^rv, vv, t h we can deduce all the macroscopic variables of physical interest for the system. The average value of a physical quantity g ^rv, vv, t h will be defined by 1 y gf d3 v ...(6.2-2) n where n = n^rv, t h is the number of particles per unit volume about rv at time t having any velocity:

g =



n^rv, t h =

=

yv f^rv, vv, th d3 v

...(6.2-3)

y-+33 y-+33 y-+33 f^rv, vv, th dvx dvy dvz

...(6.2-4)

For example, the average flow velocity 1 v ^rv, vv, t h d3 v uv^rv, t h = y vf n v

...(6.2-5)

6.3 BOLTZMANN EQUATION I n order to describe a system from statistical kinetic approach the distribution function for the system under consideration must be known. The dependence of the distribution function f ^rv, vv, t h on rv, vv and t is governed by an equation known as Boltzmann equation. Let us consider the derivation of this equation in a simple way. The distribution function f ^rv, vv, t h is so defined that f^rv, vv, t h d3 r d3 v gives the number of particles in the volume element d3r d3v of phase space about the phase space coordinates ^rv, vvh at time t. Suppose that each particle is subjected to an external force Fv . In the absence of collisions a particle with coordinates about ^rv, vvh in phase space will be found after a time interval dt about the new coordinates ^rvl , vvlh such that

rvl = rv + vvdt

...(6.3-1)



vvl = vv + avdt

...(6.3-2)

where av = Fv/m is the field-induced acceleration of the particle of mass m. In the absence of collisions particles which were in the phase volume d3r d3v at time t will now occupy the phase volume d3 r l d3 vl d at time t + dt (Fig. 6.3-1), f^rvl , vvl , t + dt h d3 r l d3 vl = f^rv, vv, t h d3 r d3 v

...(6.3-3)



Kinetic Theory of Plasma

6.3

For the transformations defined by Eqns. (6.3-1) and (6.3-2) it can be shown that to first order in dt, d3 r l d3 vl = d3 rd3 v



...(6.3-4)

Thus from Eqn. (6.3-3), f^rvl , vvl , t + dt h = f^rv, vv, t h



or f^rv + vvdt, vv + avdt, t + dt h = f^rv, vv, t h

...(6.3-5)

Expanding left hand side in Taylor series and keeping terms of first order in dt only, we get v fdt + av $ d v fdt = f^rv, vv, t h f^rv, vv, t + dt h + vv $ d v f^rv, vv, t + dt h - f^rv, vv, t h v f - av $ d v f = - vv $ d ...(6.3-6) v dt v v corresponds to differentiations with respect to x, y, z and d where the operator d v with respect to vx, vy, vz. or

Fig. 6.3-1 : Particles in phase volume d3rd3v at time t occupy the volume d 3 r l d 3 v l at time t + dt, in the absence of collision.

In the limit dt → 0, we get 2f v f + av $ d v f = 0 ...(6.3-7) + vv $ d v 2t This is the Boltzmann equation in the absence of collision. If collision is taken into consideration the Boltzmann equation becomes



2f v f + av $ d v f = c 2f m + vv $ d v 2t 2t c

The collision term c

...(6.3-8)

2f m on the right-hand side of (6.3-8) should be properly 2t c

evaluated from the collision dynamics of the interacting particles. It depends on the type of particles and nature of collisions. Note that as a result of collision some

6.4

Basic Plasma Physics

of the particles that were initially within the volume element d3r d3v at time t may not be within the volume element d3 r l d3 vl at time t + dt. Also, some new particles which were initially outside d3rd3v may end up inside d3 r l d3 vl at time t+dt.

A Simple Model for the Collision Term When there are collisions with neutral atoms the complicated collision term c in the Boltzmann equation can be approximated as follows:

c

f -f 2f m = n tc 2t c

2f m 2t c

...(6.3-9)

where fn is the distribution function of neutral atoms and tc is the relaxation time constant. The model is particularly applicable to weakly ionized plasmas in which charge-neutral collisions are important. The model has many limitations. However, it is still in use because of its simplicity.

Vlasov Equation he Vlasov equation is a partial differential equation that describes the time T evolution of the distribution function f^rv, vv, t h in phase space. It can be obtained from the Boltzmann equation (6.3-8) with the collision term neglected but including the self-consistent internal electromagnetic fields produced by the presence and motion of all charged particles inside plasma: 2f v f + q ^ Ev + vv # Bv h $ d v f = 0 +v$d v 2t m

...(6.3-10)

Sometimes it is referred to as the ‘collisionless Boltzmann equation’. Because of its comparative simplicity this equation has been widely used to describe the dynamics of plasma, particularly sufficiently hot plasma where collisions can be neglected. The neglect of collision term limits the applicability of Vlasov equation to phenomenon with characteristic time t satisfying t > v m k

Let us now consider another important limiting case in which the wave phase w velocity is large compared to the velocity of almost all of the electrons. In this k



Kinetic Theory of Plasma

6.11

w >> v m we may expand the integral in (6.5-8) as k w2pe + 3 f0 y 1 = dv n0 w2 - 3 kv 2 c1 - m w

limit c

= =

w 2pe n0 w 2

+∞

∫ −∞

  2kv 3k 2 v 2 f0  1 + ... + +  dv w w2  

w 2pe   k 3k 2 2 1 + 2 v + v + ...  2  2 w w  w 

...(6.5-11)

Since the plasma is considered to be stationary v = u = 0. If we consider f0 to be Maxwellian, then, in one dimension,

1 1 m v2 = 2 kB Te 2 e

...(6.5-12)

where Te is the temperature of the electron gas at equilibrium. Therefore, the dispersion relation (6.5-11) becomes

w2 ≈ w2pe =1 + 3

k2 kB Te G $ w2 me

...(6.5-13)

Since the second term on the right-hand side of (6.5-13) arising due to thermal correction is small in the high phase velocity limit we can replace w by wpe in the second term. We then have

3kB Te w2 = w2pe + m k2 e

...(6.5-14)

The result is the same as obtained by using the warm plasma model fluid equations with g = 3. The result is known as the Bohm-Gross dispersion relation for electron plasma waves (or Langmuir waves).

6.6 LANDAU DAMPING andau damping is a phenomenon of wave damping without energy dissipation L by collision. It was first discovered by Landau (1946) purely mathematically in course of a careful analysis of a contour integral. However, it is a real effect and has been demonstrated in the laboratory. he evaluation of the integral in the dispersion relation (6.5-7) is not straight T forward because of the singularity at v = w/k. Landau first treated the problem as an initial value problem in which the plasma is given a sinusoidal perturbation and

6.12

Basic Plasma Physics

therefore k is real. In general w(k) is complex. Im (w) > 0 corresponds to unstable waves and Im (w) < 0 corresponds to damped waves. The integral requires to be evaluated by using contour integration in the complex v-plane.

Fig. 6.6-1 : Integration contour in the complex v-plane for small Im (w)

An exact analysis of the problem is complicated. However, a simplified analysis can be given by considering large phase velocity and weak damping [Re(w) >> Im(w)]. In this case the pole at w/k lies near the real axis (Fig. 6.6-1). The contour prescribed by Landau in this case is a straight line along the Re (v) axis with a small semicircle about the pole. The contribution to the integral in (6.5-7) in going around this small semicircle about the pole is 2pi times half the residue there. Therefore, Eqn. (6.5-7) becomes R V 2f0 W ω2pe SS + 3 2v 2f0 W + dv πi W 1 = 2 S P y- 3 ...(6.6-1) ω 2v k n0 S W vk T X where P denotes the Cauchy’s principal value of the integral obtained by removing w the small semicircle around the pole at v = from the range of integration. If k 2f0 we choose Re (w/k) large both f0 and 2v will be very small there and then the principal part of the integral in (6.6-1) can be approximated as an ordinary integral. On integrating by parts, we get



y-33

2f0 f0 2v dv = w > -w vv k k

=

y-+33

+3

H

+

y-+33

-3

f0 dv w 2 cv - m k

f0 dv w 2 cv - m k ...(6.6-2)



Kinetic Theory of Plasma

6.13

Since we have assumed w/k >> v we can expand the denominator and write

+∞

∫ −∞

∂ f0 2 ∂ v dv = k w w2 v− k

=

k2 w2

f0

+∞

∫ −∞

kv    1 −  w

y-+33 f0 ;1 +

2

dv

2kv 3k2 v2 + + ...Edv w w2

...(6.6-3)

If we consider f0 to be Maxwellian then the odd terms will vanish and in one dimension

1 1 me v2 = 2 kB Te 2

...(6.6-4)

Then for real w we can write from (6.6-1) or

1 = w2

=

w2pe k2 3k2 kB Te + + e o $ 1 0 $ w2 me k2 w2 w2pe +

w2pe 3kB Te $ k2 me w2

...(6.6-5)

Since the second term in the right-hand side is small in the high phase velocity limit we can replace w by wpe in it. Then

3kB Te w2 = w2pe + m k2 e

...(6.6-6)

It corresponds to the real part of w. o find the small imaginary term in Eqn. (6.6-1) we neglect the thermal T correction term and assume that Re(w) ≈ we. Then Eqn. (6.6-1) becomes or ω2 e1 - iπ or

1 =

ω2pe ω2

+ iπ

ω2pe 2f0 k2 n0 2v

ω2pe 2f0 o = w2pe k2 n0 2v 2 π ω pe 2f0 w ≈ ω pe e1 + i 2 $ 2 2v o k n0

Suppose f0 is one-dimensional Maxwellian:

f0 = n0 c

- me v2 me 1/2 o m exp e 2pkB Te 2kB Te

...(6.6-7)

6.14

Basic Plasma Physics - v2 n0 exp e 2 o VTh p VTh

=

2kB Te me .

VTh =

where Therefore,

...(6.6-8)

- v2 2vn0 2f0 = exp e 2 o 3 VTh 2v p VTh

...(6.6-9)

From (6.6-7) and (6.6-9),

Im(w) = -

π ω3pe $ $ 2 k2

- v2 2v exp e 2 o 3 VTh π VTh

...(6.6-10)

Let us approximate v by wpe/k in the coefficient and in the exponent we put Then

v2 =

w2pe 3 w2 2 = + VTh 2 k2 k2

Im(w) = - π ω pe e

= -

ω pe kVTh

3

o exp =-

ω pe

...(6.6-11) ω2pe 2 k2 VTh

-

3 G ...(6.6-12) 2

1 3 π exp =- 2 2 - G $ 2 8 k2 λ3D 2k λD

...(6.6-13)

For a standing wave where k is real the waves are proportional to

exp i (kx – wt) = exp i(kx – wrt) $ exp wit

...(6.6-14)

where we have separated w into its real and imaginary parts as w = wr+ iwi. Since Im (w) = wi is negative there is temporal decay of the wave amplitude. This type of damping of longitudinal plasma waves was first pointed out by L.D. Landau and for this reason it is known as Landau damping. Note that this type of damping arises even in the absence of any dissipative mechanism such as collisions of the electrons with heavy ions. For this Landau damping is sometimes called collisionless damping. The damping is important for waves of wavelength smaller or comparable with the Debye length. For long wavelengths (kl­D 0 . Note that the Landau damping is essentially due to pole of the integrand 2v w in (6.5-7) at v = k . This is a mathematical manifestation of the fact that the waveparticle interaction is effective only when the velocity of the electrons is very close to the phase velocity of the wave. for



6.7

Kinetic Theory of Plasma

6.15

PHYSICAL MECHANISM OF LANDAU DAMPING

he physical mechanism responsible for collisionless Landau damping is the waveT particle interaction, i.e., the interaction of the electrons with the wave electric field. The electrons that initially have velocities very close to the phase velocity of the wave do not see a rapidly fluctuating electric field and can effectively exchange energy with the wave. These ‘resonant’ particles are trapped inside the potential wells of the wave and it results in a net interchange of energy between the electrons

Fig. 6.7-1 : Maxwellian distribution

and the wave. Particles with velocities slightly less than the wave phase velocity are accelerated by the wave electric field, gaining energy from the wave while particles with velocities slightly greater than the phase velocity of the wave are decelerated by the wave electric field, losing energy to the wave. In a plasma there are electrons both faster and slower than the wave. For Maxwellian distribution if the wave velocity

Fig. 6.7-2 : A double-humped distribution

6.16

Basic Plasma Physics

2f0 is negative then in the small velocity 2v range around w/k there will be more electrons initially moving slower than w/k than moving faster than w/k [Fig. 6.7-1]. Thus in this case there will be more electrons gaining energy from the wave than losing to the wave. It leads to wave damping. In some cases the wave phase velocity w/k may correspond to the region 2f of positive slope, i.e., 0 > 0 [Fig. 6.7-2]. In this case there will be more electrons 2v giving energy to the wave than gaining energy from the wave. It leads to an unstable situation with the wave amplitude growing with time. w/k corresponds to the region where

SOLVED PROBLEMS PROBLEM 6.1 : Consider a plasma with two cold electron streams with equal and opposite velocity (+ v0) in a background of fixed ions. The equilibrium distribution function for electron velocities is given by

f0(v) =

1 n 6d^v - v0h + d^v + v0h@ 2 0

1 n is the density of each stream. Using the following integral form of the 2 0 plasma dispersion relation derive the dispersion relation for the above case: where

e2 1 + me ε0 k

y- 3

+3

2f0 2v dv = 0. ω - kv

Solution : Integrating by parts the given dispersion relation can be written in the following convenient form f0 e2 + 3 y 1 dv = 0 me ε0 - 3 ^ω - kvh2 ow using the given distribution function and properties of delta function we N get the desired dispersion relation, 1 1 1 + G. 1 = 2 w2pe = 2 ^w - kv0h ^w + kv0h2



PROBLEM 6.2 : Taking into account the effects of both electron and ion motion the dispersion relation for electrostatic plasma waves can be expressed as e2 1+ m0 ε0 k

y- 3

+3

2f0e 2 2v dv + e mi ε0 k ω - kv

y- 3

+3

2f0i 2v dv = 0 ω - kv



Kinetic Theory of Plasma

6.17

Show that for a wave with phase velocity, w/k, much less than electron thermal velocity but much greater than ion thermal velocity the above dispersion relation, to a first approximation, can be written as

ω2pi 1 1 + 2 2 - 2 = 0 k λD ω

Identify the wave mode. Solution: Integrating the ion term by parts the given dispersion relation can be put in the following form e2 1+ me ε0 k

y- 3

+3

2f0e 2 2v dv - e mi ε0 k ω - kv

y- 3

+3

2f0i 2v dv = 0 ω - kv

For the electron term we may assume that w V

where V is a real positive constant. Find the dispersion relation. 2. Insert one-dimensional Maxwellian distribution function into the dispersion relation 2 1+ e $ 1 me ε0 k



y- 3

+3

2f0 2v dv = 0 ω - kv

and derive the Bohm-Gross relation for the Langmuir wave: 2 w2 . w2pe + 3 k 2 V Th 2 3. Consider one-dimensional motion of charged particles in an electric potential field

f(x). Show by direct substitution that a function of the form f = f b 1 mv 2 + qfl is a 2 solution of the Boltzmann equation under steady state condition. [Hints: In the steady state or, Here,

df =0 dt

2f 2f 2f + v + vo = 0 2t 2x 2v 2f 2f 2f 2w = 0, $ 2t 2x = 2w 2x

=

2f 2f $q , 2w 2x

2f 2f 2w 2f = $ $ mv 2v 2w 2v = 2w

where w = 1 mv 2 + qf 2

6.20

Basic Plasma Physics

and

q 2f E vo = F = - 1 2w = m m 2x m 2x

4. Suppose a system of particles with a constant particle number density n0 is characterized by a one-dimensional velocity distribution function f (v) such that

f(v) = A for |v| < v0

= 0

for |v| > v0

where v0 is a real positive constant. Determine A in terms of n0 and v0. 5. Consider the following two-dimensional Maxwellian distribution function:

f_vx, vyi = n0 b

- m`v 2x + v 2y j m > H exp 2pkB T l 2kB T

Verify that n0 represents the particle number density, that is, the number of particles per unit area. 6. One-dimensional electron distribution function in presence of an electric potential f(x) is

1 b 2 mv 2 - eφl 1/2 m > H f(v) = n0 b exp 2πkB T l kB Te

Integrating f (v) over v show that

n(f) = n0 exp e

ef . kB Te o

7. Since the Maxwell’s velocity distribution is isotropic (independent of velocity direction) it is of interest to define a distribution of speeds v = vv . Using Maxwell’s velocity distribution show that the desired distribution of speeds is given by

(v) = 4pn b

m 3/2 v 2 exp - mv 2 . e 2k T o 2pkB T l B qqq



CHAPTER

7 Transport Processes in Plasmas

7.1

INTRODUCTION

I n real plasmas there may have gradients of macroscopic parameters such as concentration, temperature and velocity. These gradients give rise to fluxes that finally equalize the macroscopic parameters over the plasma volume. These are known as transport phenomena. For example, in a nonuniform plasma with density gradient plasma particles tend to diffuse from a region of higher concentration towards the region of lower concentration. The presence of velocity gradient causes transport of momentum and gives rise to the phenomenon of viscosity. Similarly the presence of temperature gradient causes transfer of thermal energy and gives rise to the phenomenon of thermal conductivity. In this chapter we shall consider the phenomena of electrical conductivity and diffusion in plasmas.

7.2

MOBILITY AND ELECTRICAL CONDUCTIVITY OF WEAKLY IONIZED PLASMA

o begin with we consider the plasma to be weakly ionized and consist of nonT uniform distribution of ions and electrons in a dense uniform background of neutrals. The charge-neutral collisions are dominant here. he macroscopic equation of motion for electrons in a weakly ionized cold T plasma in presence of an applied constant (DC) uniform electric field, including collisions is duv = - ene Ev - ven me ne uve ...(7.2-1) dt where uve is the average electron velocity and ven is an effective constant collision frequency for momentum transfer between the electrons and neutral particles. Here we have neglected the average motion of neutral particles as they are much

me ne

7.2

Basic Plasma Physics

heavier than electrons. In the steady state the action of the applied electric field is balanced dynamically by electron-neutral collisions. Then

- eEv - ven me uve = 0

...(7.2-2)

The electron mobility me is defined as the average velocity per unit applied field. Therefore, e me = m v e en



...(7.2-3)

The electric current density associated with electron motion in response to an applied electric field is vj = - en uv e e e



...(7.2-4)

From (7.2-1) and (7.2-4), we get the Ohm’s law e2 ne v vj = E = se Ev e me ven

where se =

...(7.2-5)

e2 ne is the DC electrical conductivity of the electron gas. me ven

Similarly, for ions, we have ion mobility,

e mi = m v i in

...(7.2-6)

electrical conductivity, e2 ni si = mi vin



...(7.2-7)

For a plasma with contributions from both positive and negative charges vj = sEv

where with me >> mi,

ne ni s = e2 c m v + m v m e en i in

...(7.2-8)

e2 ne s ≈ m v e en

...(7.2-9)

In cases when electric field Ev and average electron velocity uve vary harmonically in time as exp (–iwt), Eqn. (7.2-1) can be linearized as or

- iwme ne uve = - ene Ev - ven me ne uve uve = -

eEv me ^ven - iwh

...(7.2-10)



Transport Processes in Plasmas

7.3

Therefore, in this case electron mobility

me =

e



...(7.2-11)

e2 ne me ^ven - iwh

...(7.2-12)

me ^ven - iwh

and electrical conductivity

se =

The complex conductivity means that there is a phase difference between the current density and the applied electric field.

7.3

ELECTRICAL CONDUCTIVITY OF FULLY IONIZED PLASMA

or a fully ionized plasma consisting of electrons and ions only current flow due to F applied electric field is mainly due to the motion of more mobile electrons. These electrons are scattered by Coulomb collisions with slow ions. Assuming that in the steady state the action of the applied electric field is balanced dynamically by electron-ion collisions we can write from the fluid equation of motion (7.2-1) for electrons

0 = - eEv - vei me uve

...(7.3-1)

where we have neglected the pressure-gradient force. The electric current density associated with the electron motion in response to applied electric field and decelerating effect due to electron-ion collision is vj = - neuv ...(7.3-2) e

Combining (7.3-1) and (7.3-2), we get where

vj =

e2 ne v $ E = sEv me vei

e2 ne s = m v e ei

...(7.3-3) ...(7.3-4)

is the electrical conductivity of fully ionized plasma (neglecting ionic contribution). o obtain an estimate of plasma conductivity we need to know vei. Suppose T an electron of velocity v approaches a fixed ion of charge e. Let in the absence of Coulomb forces the distance of closest approach (or impact parameter) is r0. The Coulomb force between the electron and ion is

F = -

1 e2 $ 2 4πε0 r0

...(7.3-5)

7.4

Basic Plasma Physics

This force is felt during the time the electron is in the vicinity of the ion. This time is roughly T ≈



r0 v

...(7.3-6)

The change in electron momentum is therefore approximately given by

e2 D(mev) ≈ FT . 4πε r v 0 0

...(7.3-7)

For a 90° scattering the change in momentum is of the order of momentum mev itself. So, or

e2 mev = 4πε r v 0 0 e2 r0 = 4πε m v2 0 e

...(7.3-8)

The collision cross-section for such large angle scattering is therefore,

s90° = πr02 =

e4 16πε02 m e2 v4

...(7.3-9)

Fig. 7.3-1 : Trajectory of an electron making a Coulomb collision with a massive ion

The collision frequency is, therefore,

vei = ne s90° v =

ne e4 16πε02 m e2 v3

...(7.3-10)

Assuming Maxwellian distribution for the electrons in the plasma we may replace v2 by

kB Te . Then, me

ne4 ...(7.3-11) vei = 1 3 2 16πε0 m e2 ^kB Teh2



Transport Processes in Plasmas

7.5

Because of long-range nature of Coulomb force small-angle collisions are much more frequent and an effective 90° scattering involves an accumulation of many small-angle scattering. Using Debye length as an upper limit of the impact parameter it has been shown by Spitzer that Eqn. (7.3-11) should be multiplied by a factor ln Λ, called Coulomb logarithm. It is a slowly varying function of electron density and temperature. For most plasmas of interest 5 < ln Λ < 20 and one usually takes ln Λ = 10 regardless of the type of plasma involved. Therefore, multiplying (7.3-11) by ln Λ and then using it in Eqn. (7.3-4) we can write for the conductivity as 1

3

3

16πε02 ^kB Teh2 e2 ne e2 ne 16πε02 m e2 ^kB Teh2 = = σ= $ 1 me vei me ne e4 ln Λ e2 m e2 ln Λ

...(7.3-12)

The plasma resistivity h is given by 1



1 η = = σ

e2 m e2 ln Λ 3

16πε02 ^kB Teh2



...(7.3-13)

Note that unlike a weakly ionized plasma the conductivity for a fully ionized plasma is independent of the number density of charge carriers. This is due to the fact that in fully ionized plasma an increase in electron density causes a proportional increase in ions that scatter the electrons. So the effect of increase in the number of charge carriers is counteracted by the corresponding increase in collisions. 3

Equation (7.3-12) shows that s is proportional to T e2 . As the plasma is heated Coulomb scattering cross-section decreases. It decreases the plasma resistivity and increases plasma conductivity. Plasmas at high temperatures (a few keV) are essentially collisionless. For this Ohmic heating of plasmas, that is heating of plasma by sending current through it, cannot be used to heat plasmas to thermonuclear temperatures.

7.4

FREE DIFFUSION OF WEAKLY IONIZED PLASMAS

I n real plasmas there may have density gradients. Then on an average plasma particles flow from regions of greater concentration to regions of lower concentration. This process is called diffusion. To study diffusion we consider a weakly ionized plasma consisting of a non-uniform distribution of electrons and ions in a dense uniform background of neutrals. The diffusion rates of electrons and ions are different. As a result space charge electric field is created which influence the diffusion process. The diffusion in which the effect of space charge electric field is neglected is known as free diffusion.

7.6

Basic Plasma Physics

Free diffusion of particles in a plasma results from the pressure gradient force. To deduce an expression for free electron diffusion coefficient for a warm weakly ionized plasma we start from electron fluid equation of motion including collision with a suitably defined collision frequency n: 2 v E uv = - d v p - m n nuv me ne ; + uve $ d ...(7.4-1) e e e e e 2t where we have neglected any space charge-electric field. We assume that the deviations from the equilibrium state due to density inhomogeneities are very small. Then uve is a small quantity of first order. Therefore, 2uv assuming a steady state in which e = 0 and linearizing (7.4-1) we can write for 2t an isothermal plasma (pe = nekBTe), or where

v n - m n nuv 0 = - kB Te d e e e e uve = De =

vn vn d kB Te d $ e =- De e me n ne ne

kB Te me n

...(7.4-2) ...(7.4-3)

is called the electron free diffusion coefficient. The flux of electrons is given by

v v = n uv =- D d Γ e e e ne e

...(7.4-4)

which is the Fick’s law of diffusion. The law expresses the fact that a net flux of particles occurs in diffusion from a denser region to less dense region simply because there are more randomly moving particles in the dense region. In plasma Fick’s law is not necessarily obeyed because of collective motion of plasma particles.

7.5

AMBIPOLAR DIFFUSION IN PLASMA

I n deriving the results of Section 7.4 we neglected the effect of space-charge electric field. Since diffusion coefficient is inversely proportional to the particle mass, electrons tend to diffuse faster than the ions. As a result a space-charge electric field is created which tend to slow down the diffusion of electrons and enhance that of ions. The combined diffusion of the electrons and the ions forced by the space charge Ev -field is known as ambipolar diffusion. In this case the diffusion rate has a value intermediate to electron and ion free diffusion rates.



Transport Processes in Plasmas

7.7

To find the ambipolar diffusion coefficient we consider a warm weakly ionized plasma and start from the electron and ion fluid equations of motion including collision with a suitably defined collision frequency n: mjnj ;

2u j v Euv = q n Ev - d v p - m n nuv + uv j $ d j j j j j j j 2t

...(7.5-1)

where j = e for electrons and j = i for ions. If we assume that the deviation from equilibrium due to density inhomogeneity is very small, uv j will be a small quantity of first order. Then assuming a steady 2uv j = 0 and linearizing (7.5-1) we can write for an isothermal (pj state in which 2t = njkBTj) plasma, w n - m n nuv 0 = q j n j Ev - kB Tj d j j j j



uv j = ! m j Ev - D j

or where, µ j = and D j =

qj is the mobility mjν

wn d j nj

...(7.5-2)

kB Tj is the diffusion coefficient of the j-th species; + signs correspond mjn

to ions and electrons respectively. Note that the diffusion coefficient and mobility are connected by the relation Dj kB Tj = mj qj



...(7.5-3)

It is known as Einstein relation. The particle flux is w v = n uv = ! n m Ev - D d Γ j j j j j nj j



...(7.5-4)

v In order that the plasma remains quasineutral we expect in the steady state Γ e v v v and Γ to be equal. Equating Γ and Γ and writing n = n = n we get from (7.5-4) i

or,

e

i

e

i

w n = nm Ev - D d w - nme Ev - De d i i n wn D - De d $ Ev = i mi + me n

Note that the ambipolar electric field is due to the density gradient.

...(7.5-5)

7.8

Basic Plasma Physics The common flux is then given by

w v = nm Ev - D d Γ i i n

= mi

Di - De w wn dn - Di d mi + me

= –

mi De + me Di w dn mi + me

w n = - Da d

...(7.5-6)

which is just Fick’s law but with a new diffusion coefficient

Da =

mi De + me Di mi + me

...(7.5-7)

called the ambipolar diffusion coefficient. As the electrons are much more mobile than the ions me >> mi we can approximate (7.5-7) as

mi Da ≈ Di + m De e

Te = Di e1 + T o i

...(7.5-8)

where we have used the relation (7.5-3). For Te = Ti we get the simple result

Da = 2Di

...(7.5-9)

The ambipolar electric field enhances the diffusion rate of ions and retards the diffusion of electrons. The combined diffusion rate has a value intermediate to their free diffusion rates. However, the diffusion rate is primarily controlled by the slower species ions, i.e., Di < Da < De. For the case Te = Ti we find that the ambipolar diffusion rate is just twice the free diffusion rate of ions.

7.6

DIFFUSION EQUATION

I n the previous section we have seen that for ambipolar diffusion the common particle flux can be expressed as

v v = - D d Γ a n

...(7.6-1)



Transport Processes in Plasmas

7.9

The equation of continuity can be expressed as or

2n v + d.^nuvh = 0 2t 2n v v + d.Γ = 0 2t

...(7.6-2)

From (7.6-1) and (7.6-2) we get the diffusion equation: 2n = Da d2 n 2t



...(7.6-3)

This equation can be used to study the temporal and spatial evolution of the density profile of a plasma. As an example, let us examine how a plasma created between two parallel planes decays by diffusion to the planes. In one dimension the Eqn. (7.6-3) can be written as 22 n 2n = Da 2 2x 2t



...(7.6-4)

It can be solved by the method of separation of variables. Substituting n (x, t) = X(x)T(t) in Eqn. (7.6-3) and dividing throughout by XT, we get

D d2 X 1 dT = a T dt X dx2

...(7.6-5)

Left-hand side is a function of t only and right-hand side is a function of x only. So for equality each side of (7.6-5) must be equal to some constant. Let us call it 1 - . Then we have t 1 dT = - T ...(7.6-6) t dt and

d2 X 1 = X Da t dx2

...(7.6-7)

t

Solution of (7.6-6) is T = T0 e - x

...(7.6-8)

where T0 is a constant. The solution of (7.6-7) is

X(x) = A cos

1 x + B sin Da t

1 x Da t

...(7.6-9)

where the constants A and B are to be found from the given boundary conditions. Suppose in the plasma slab geometry shown in Fig. 7.6-1, initially the plasma

7.10

Basic Plasma Physics

density is maximum in between the two walls and the density decreases to zero at the walls x = ! L . Symmetry of the problem indicates that B = 0. Then the condition X = 0 at x = ! L requires that or

L π = 2 Da t t = c

2L 2 1 m $ π Da

...(7.6-10)

Combining Eqns. (7.6-8) – (7.6-11), we get the complete solution as

n(x, t) = X(x) T(t)

...(7.6-11)

πx = n0 e-t/x cos 2L

Fig. 7.6-1 : Density profile of a plasma at different time instants as it decays by diffusion to the walls

It shows how the initial density distribution decays exponentially with time as a result of diffusion. From (7.6-10) we find that the time constant increases with L and decreases with Da.

7.7

DIFFUSION OF A WEAKLY IONIZED PLASMA ACROSS A MAGNETIC FIELD

s the motion of electrons and ions are affected by the presence of magnetic field A it is natural to expect that the diffusion of electrons and ions will be changed by the application of magnetic field. In fact the rate of plasma loss by diffusion can be decreased by applying a magnetic field. It is a central problem in controlled thermonuclear fusion research. To study the effect of magnetic field on diffusion we consider a weakly ionized magnetized plasma consisting of a nonuniformly distributed electrons and ions in a



Transport Processes in Plasmas

7.11

dense background of neutrals. Since the magnetic field does not affect the motion of plasma particles in the direction parallel to the magnetic field, the diffusion and mobility of plasma particles in parallel direction will be similar to those in magnetic field-free cases. In presence of collisions plasma particle can diffuse across Bv to the walls. To see how this comes about we start from the fluid equation for either species including collision with a suitably defined collision frequency n :

mn

duv v p - mnnuv = ! en^ Ev + uv # Bv h - d dt

...(7.7-1)

where + signs correspond to ions and electrons respectively. Assuming that the deviation from equilibrium due to density inhomogeneity duv is very small and considering steady state we can put = 0. Then for isothermal dt (p = nkBT) plasma,

v n mnnuv = ! en^ Ev + uv # Bv h - kB Td

...(7.7-2)

Assuming the steady magnetic field to be in the z-direction Eqn. (7.7-2) can be separated into component equations:

2n mnnux = ! enEx ! enuy B - kB T 2x

...(7.7-3)



2n mnnuy = ! enEy " enux B - kB T 2y

...(7.7-4)



2n mnnuz = ! enEz - kB T 2z

...(7.7-5)

Equation (7.7-5) indicates that the mobility and diffusion coefficient in the magnetic field direction remain unchanged from its zero magnetic field value. Equations (7.7-3) and (7.7-4) can be rewritten as

ωc D 2n ux = ! µEx ! ν uy - n 2x

...(7.7-6)



uy = ! µEy "

ωc D 2n ux ν n 2y

...(7.7-7)

k T e is the mobility D = B is the diffusion coefficient in fieldmn mν eB free plasma and wc = is the cyclotron frequency. Solving Eqns. (7.7-6) and m (7.7-7), we get where µ =

7.12

Basic Plasma Physics



ux ^1 + ω2c τ2h = ! µEx -

k T 1 2n D 2n ω2c τ2 Ey + " ω2c τ2 B n 2x B eB n 2y

...(7.7-8)



uy ^1 + ω2c τ2h = ! µEy -

k T 1 2n D 2n ω2c τ2 Ex ! ω2c τ2 B n 2y B eB n 2x

...(7.7-9)

1 is the time between successive collisions. ν An examination of the Eqns. (7.7-8) and (7.7-9) reveals that the normal velocity of either species is composed of two parts. The first two terms of these equations represent mobility and diffusion drifts parallel to the gradients in potential and density. The last two terms of these equations represent Ev # Bv and diamagnetic drifts perpendicular to the gradients in potential and density. Considering drift associated with mobility and diffusion only we can rewrite Eqns. (7.7-8) and (7.7-9) as 1 2n ux = ! m= Ex - D= n 2x ...(7.7-10) where τ =



uy = ! m= Ey - D=

where

m= =

µ 1 + ω2c τ2

and

D= =

D 1 + ω2c τ2

1 2n n 2y

...(7.7-11) ...(7.7-12) ...(7.7-13)

Thus it is evident that the mobility and diffusion perpendicular to the magnetic field are reduced by the factor and we have

D= =

1 . For a strong magnetic field ω2c τ2 >> 1 1 + ω2c τ2 k Tνm k T v2 D = B = B2 2 2 2 mν q 2 B 2 q B ωc τ 2 m

...(7.7-14)

Note that in case of diffusion perpendicular to the magnetic field the diffusion coefficient D= is proportional to the collision frequency because collision is needed for diffusion across the magnetic field. On the other hand, the diffusion coefficient D in the absence of magnetic field or parallel to magnetic field is inversely proportional to the collision frequency because collisions retard diffusion. ince n \ m-1/2 we find that and D= \ m1/2 . Thus in diffusion parallel to S Bv electrons diffuse more rapidly than the ions because of their higher thermal velocity. In perpendicular diffusion electrons diffuse more slowly than ions. This



Transport Processes in Plasmas

7.13

is because the charge particles walk one cyclotron radius per collision and the electron cyclotron radius is much smaller than that of the ions. Note that

D =

2 λ2 kB T VTh = m ~ mν ν τ

where lm is the distance between successive collisions. This expression indicates that the parallel diffusion can be considered as a random-walk process with a step length lm. Similarly, where RL =

D= =

kB Tvm R L2 ~ t q2 B2

qB is the cyclotron radius. mVTh

It shows that the perpendicular diffusion can be considered as a random-walk process with a step length RL. Thus D= is related to the cyclotron radius rather than the mean free path.

7.8

DIFFUSION OF A FULLY IONIZED PLASMA ACROSS A MAGNETIC FIELD

o describe the problem of diffusion in a fully ionized plasma we describe the T plasma, for simplicity, as a single conducting fluid. For a steady state plasma in presence of magnetic field and pressure gradient the equation of motion is

v p vj # Bv = d

..(7.8-1)

The generalized Ohm’s law in simplified form is

vj = s^ Ev + uv # Bv h

...(7.8-2)

where s is the longitudinal electrical conductivity. Eliminating vj from Eqs. (7.8-1) and (7.8-2), we get vp s 8 Ev # Bv + `uv= # Bv j # Bv B = d

vp Ev # Bv d ...(7.8-3) B2 sB 2 where uv= is the component of uv in a direction normal to the external magnetic field Bv . The first term on the right-hand side of (7.8-3) represents the Ev # Bv drift of or ­uv= =

both species together. The second term represents the diffusion velocity in the v p. direction of - d The flux associated with the diffusion only is given by

7.14

Basic Plasma Physics



v v = nu =- ndp Γ = sB 2

...(7.8-4)

where n is the number density of electrons (or ions). Now for a two-fluid plasma.

p = pe + pi = nkB (Te + Ti)

...(7.8-5)

Therefore,

+ v v v = nkB ^Te Tih dn =- D d Γ = n sB 2

...(7.8-6)

where

D= =

nkB ^Te + Tih sB 2

...(7.8-7)

is known as the classical diffusion coefficient for a fully ionized plasma. Note that the diffusion coefficient D= of fully ionized plasmas is proportional 1 to 2 , just as in the case of weakly ionized plasmas. Nevertheless, there are B some basic differences between D= as given by (7.8-7) and the corresponding diffusion coefficient for a weakly ionized plasma as given by (7.7-12). Firstly, in a fully ionized plasma D= is not constant, but is proportional to the number density n. Secondly, since s is proportional to T3/2 for a Maxwellian distribution, D= decreases with increase in temperature in a fully ionized plasma. Finally, for the diffusion coefficient in (7.8-7) there is no ambipolar electric field because both ions and electrons diffuse at the same rate.

SOLVED PROBLEMS PROBLEM 7.1: In a weakly ionized 2eV plasma electron mobility is 120 Find the electron diffusion coefficient.

m2/V-s.

Solution: From Einstein’s relation

De kB Te = me e

or

De =

me kB Te 120 # 2 # 1.6 # 10-19 = = 240 m 2 /s . e 1.6 # 10-19

PROBLEM 7.2: A weakly ionized plasma in a slab geometry has the following density distribution:

πx n(x) = n0 cos c 2L m, - L # x # L .



Transport Processes in Plasmas

7.15

Suppose the plasma decays both by diffusion and recombination. Assuming that rate of loss due to recombination is given by –an2 where a is a constant, find the value of n at which the rate of loss by diffusion will be equal to that by recombination. Solution: Rate of loss due to diffusion = Dd 2 n = D

Dc

\

π 2 m n = an2 2L n =

or

22 n π 2 πx π 2 =- Dn0 c m cos =- D c m n 2 2L 2L 2L 2x

D π 2 c m. α 2L

QUESTIONS 1. Consider a weakly ionized cold unmagnetized plasma in presence of an applied DC electric field. Assuming the ions to be relatively immobile and using the equilibrium force balance equation show that the plasma conductivity is given by 2 s = ne me n



where n is the number density of electrons and n is an effective collision frequency. 2. (a) Derive an expression for the electrical conductivity of a fully ionized plasma. Neglect ionic contribution and pressure-gradient force. (b) Discuss the dependence of the electrical conductivity on electron temperature and density in fully ionized plasmas. 3. Show that the electrical resistivity of a fully ionized plasma can be expressed in the form

h =

me vei ne e 2

where vei is the electron-ion collision frequency. Without deriving an expression for vei try to explain physically (i) why h is almost independent of ne even though ne appears in the expression for h (ii) why h decreases as the electron temperature Te increases. 4. (a) What do you mean by ambipolar and free diffusions in plasmas? Show that the free electron diffusion coefficient for a warm weakly ionized plasma is given by

De = kBTe/men

where n is the effective collision frequency.

(b) Is Fick’s law obeyed by plasma?

7.16

Basic Plasma Physics

(c) Identifying diffusion as a random walk process obtain the step length for free diffusion. 5. (a) What do you mean by ambipolar diffusion in plasma? (b) Considering a warm weakly ionized plasma and starting from electron and ion fluid equations of motion obtain an expression for ambipolar diffusion coefficient. (c) Show that for the case of equal electron and ion temperatures the ambipolar diffusion rate is just twice the free diffusion rate of ions. 6. Consider the diffusion equation 2n = D d 2 n a 2t



where n = n_rv, t i is the number density of each species and Da is the ambipolar diffusion coefficient in a plasma. Solve the equation by the method of separation of variables for a plasma slab geometry given by - L # x # L . Assume that the initial plasma density distribution is symmetric with a maximum at x=0. Hence show that as a result of diffusion plasma density decays as n (x, t) = n0 e -t/x cos πx 2L

where

2L 2 1 t = b π l D . a

7. (a) For a weakly ionized plasma in a magnetic field show that the perpendicular mobility and diffusion coefficient are reduced by a factor e1 +

ω2c o , where wc ν2

and n are respectively the cyclotron frequency and the collision frequency. (b) Justify that in parallel diffusion electrons move faster than ions while in perpendicular diffusion electrons escape more slowly than ions. (c) Identifying the diffusion as random-walk process, obtain the step lengths for cross field and parallel diffusions. v v =- D d 8. (a) Using the single-fluid MHD equations and Fick’s law `Γ = nj , where v = nuv is the particle flux, obtain an expression for the diffusion coefficient Γ _ D=i across a magnetic field in fully ionized plasmas.

(b) Point out some basic differences between diffusion across a magnetic field in weakly ionized plasma and fully ionized plasma.

EXERCISES 1. Consider a weakly ionized plasma in a slab defined by - L # x # L . Assuming that the plasma decays only by ambipolar diffusion show that the plasma decay time in



Transport Processes in Plasmas

7.17

2 the lowest diffusion mode may be taken as b 2L l $ 1 , where Da is the diffusion π Da coefficient.

2. Show that in a plasma with equal ion and electron temperatures the ambipolar diffusion coefficient is equal to twice the free diffusion coefficient of ions. 3. Show that the resistivity of a fully ionized plasma decreases by a factor of about 1/125 when its temperature is increased from 1 keV to 25 keV. [Hints: h ∝ T-3/2] 4. Consider the one-dimensional diffusion equation 2n = D 2 2 n , 2t 2x 2



-3 < x 1 -



2ef_ x i H mi u 02

1/2

where n0 is the density at x = 0. 2. Considering Boltzmann distributed electrons and using Poisson’s equation show that the potential distribution in the plasma sheath region is governed by the following nonlinear equation:

ε0

ef 2ef -1/2 d 2φ = en exp 1 e o > H e 0 kB Te mi u 02 o dx 2

3. Consider the following KdV equation:

2φ 2φ 1 2 3 φ + φ + 2 3 = 0 2τ 2ξ 2ξ

Making the coordinate transformation as h = x – Ut, where U is a constant, show that the equation has a solitary wave solution of the form: f = 3U sech2

U .h. 2

4. Consider the following nonlinear Schrödinger equation: 2 i 2α + p 2 α +q α 2τ 2ξ2

2

α=0

Assuming a = f(x) exp. iAt, where f (x) is real and A is an arbitrary constant, show that when both p and q are positive the equation has a solitary wave solution of the form:

a =

2A sech d q

A ξ n exp iAτ p

2A sech d q

A ξn exp iAτ p

5. Show by direct substitution that

α_ξ, τi =

is a solution to the nonlinear Schrödinger equation: 2 2 i 2α + p 2 α + q α α = 0. 2τ 2ξ2

qqq

CHAPTER

10

Some Practical Aspects of Plasma Physics

10.1 INTRODUCTION I n order to appreciate the results of the theoretical studies of plasma physics we require to perform experimental studies of plasma. Also, the outcome of actual experiments on plasma may result in the discovery of new phenomenon which requires new theoretical models. Since plasma does not exist as a normal state of matter on the earth it is necessary to create plasma in the laboratory. Once a plasma has been created it is desirable to measure different parameters of that plasma. In this chapter we shall consider plasma production, plasma diagnostics and some practical aspects of heating and confinement of plasmas.

10.2 PLASMA PRODUCTION here are different methods of producing plasmas in the laboratory. Plasma T parameters depend to a great extent on the method of plasma production. Thus, depending on the method of creation plasma may be of high- or low-density, highor low-temperature, steady-state or transient, stable or unstable and so on. As the gas discharge is the principal mechanism of plasma production in the laboratory we first briefly consider the theory of gas discharge.

Townsend’s Theory of Gas Discharge ased on the theory of ionization by collision of electrons with atoms Townsend B developed the theory of gas discharge. Let us consider the self-maintained gas discharge between two electrodes separated by a distance d. The rate of production of ions in the gas volume is described by the Townsend’s first coefficient a which is defined as the number of ions produced by the electron per unit length of its path. Production of electrons at the cathode surface due to incidence of ions is described by Townsend’s second coefficient g which is defined as the number of secondary

10.2

Basic Plasma Physics

electrons ejected from the cathode due to incidence of an ion on it. Let n0 be the number of electrons leaving the cathode and it becomes n after moving through a distance x from the cathode. Let these n electrons on moving through a distance of dx produce additional dn electrons due to collision. Therefore dn = andx



...(10.2-1)

Integrating Eqn. (10.2-1) across the gap, we get nd = n0ead



...(10.2-2)

So we can say that each electron produced at the cathode generates 7 nd - n0A/n0 or [exp (ad) – 1] pairs of charged particles in the discharge gap. Each of the produced positive ions moves towards the cathode and upon incidence on it generates g number of electrons. So that number of electrons generated at the cathode due to incidence of all the positive ions is g[exp. (ad) – 1]. For the discharge to be self-maintained all the ions produced by one electron in the discharge gap should generate one new electron at the cathode. Therefore, we have the Townsend’s criterion for the self-maintained discharge as

g [exp (ad) – 1] = 1 1 ad = ln c1 + m g

or

...(10.2-3)

According to Townsend the first coefficient a depends on the electric field strength E and gas pressure p as a = A exp [– B/(E/p)] p



...(10.2-4)

where A and B are constants that depend on the nature of the gas. For uniform field E = Vb/d where Vb is the breakdown voltage of the uniform field gap. Therefore,

exp =

B^ pd h A^ pd h G = Vb ad

...(10.2-5)

Using Eqn. (10.2-3), we get

exp =

B^ pd h G = Vb

A^ pd h 1 c m ln 1 + g

...(10.2-6)

Taking logarithm of both sides, we get

Vb =

B^ pd h C + ln ^ pd h

...(10.2-7)



Some Practical Aspects of Plasma Physics

10.3

A where C = ln 1 H > ln c1 + m g It shows that the breakdown voltage of a uniform field gap is a unique function of the gas pressure-gap length product. In Eqn. (10.2-7) Vb is in V, p in atmosphere, d in m, B in V/(atm-m) and C is a number. For air B = 43.6 × 106 V/(atm-m) and C = 12.8.

Paschen’s Law rom experimental observations Paschen showed that the breakdown voltage (Vb) F of a gas is a function of the product of gas pressure (p) and the distance (d) between the electrodes only, i.e.,

V­b = f (pd)

...(10.2-8)

Experiments show that initially the breakdown voltage decreases as the product pd increases and then increases gradually once again, Vb is minimum for a certain value of the product pd. Thus Vb plotted against pd shows a minimum (Fig. 10.2-1).

Fig. 10.2-1 : Typical Paschen Curve

The voltage at which the breakdown occurs is given by Eqn. (10.2-7) as

Vb =

B^ pd h C + ln ^ pd h

where B and C are constants that depend on the gas composition. The minimum value of V­b is obtained as

dVb

d^ pd h

= 0

This yields (pd)min = e1–C

...(10.2-9) ...(10.2-10)

10.4

Basic Plasma Physics

Therefore, from (10.2-7),

(Vb)min = B(pd)min = B.e1–C

...(10.2-11)

For pd values greater than (pd)min, for example, if keeping d fixed p is increased, electrons collide frequently as it travel from cathode to anode. Then an electron may not be able to acquire sufficient energy between two successive collisions and higher voltage may be required for breakdown. For pd values smaller than (pd)min, that is for lower p or closely spaced electrodes there is a low probability that any electron will collide with neutral atoms during its journey from the cathode to anode. Hence higher voltage may be required for breakdown.

Low Pressure Cold-Cathode Discharge ow density (n0 ~ 1016 m–3) and low-temperature (Te < 0.5 eV) plasmas can be L produced in a low-pressure cold-cathode gas discharge tube. The discharge tube contains a gas at low pressure and is fitted with electrodes. Plasma is produced by arc discharge in the tube by applying an external high voltage (Fig. 10.2-2).

Fig. 10.2-2 : Plasma produced by low-pressure cold cathode gas discharge

Radio Frequency Discharge I t is found that the breakdown voltage of a gas with radio frequency (RF) field is much smaller than that when produced by a DC field. Initial ionization for a thermonuclear reaction either in pinched discharge or in a stellarator system is usually carried on by a radio frequency field. The frequency of the field may be in the range from 100 kHz to a few megahertz. The discharge can be excited even without any electrode and this type of discharge is known as electrodeless discharge. The output coil from an oscillator may be wrapped around the discharge tube as shown in Fig. 10.2-3. Sometimes RF produced plasma is created in a magnetic field. When the RF field has the same frequency as the cyclotron frequency for the electrons plasmas with substantially higher energies can be produced. These are called cyclotron resonance plasmas.



Some Practical Aspects of Plasma Physics

10.5

Fig. 10.2-3 : Plasma production by RF discharge

Plasma Production by Laser he focused output of a high power Q-switched laser can be used to irradiate T solids and compressed gases to produce dense high-temperature plasmas. A typical shock wave accompanies the generation of this type of plasma. By this method it is possible to create a plasma in which thermonuclear reactions can occur.

10.3 PLASMA DIAGNOSTIC TECHNIQUES he measurement of plasma parameters is called plasma diagnostics. In order to T understand plasma one must be able to measure a variety of the properties of the plasma. Among the most important quantities to be known are the electron and ion densities, electron and ion temperatures, collision frequency, thermal conductivity, resistivity, diffusion coefficients etc. There are four main categories of diagnostic techniques:

(i) electrical measurements,



(ii) radiations (light and X-rays) emitted by plasma,



(iii) particles emitted by plasma,



(iv) probing plasma with waves and particle beams.

Many experimental methods based on it have been developed. Here only a few common methods will be considered.

Resistivity of Plasma uch information about plasma parameters can be gained by measuring its M electrical characteristics. Application of an electric field to a plasma causes an

10.6

Basic Plasma Physics

electric current flow through the plasma. The ratio of the electric field to current density gives the resistivity of plasma. The current in a plasma is usually measured by means of a current transformer which generates a voltage proportional to the derivative of the current. The voltage from the current transformer is integrated electronically and then recorded in an oscilloscope.

Langmuir Single Probe Method angmuir probe is one of the most widespread methods of plasma diagnostics. It L is simply a small bare wire or metal disk introduced into the plasma which can be biased to different voltages relative to the plasma while measuring the current collected by the probe (Fig. 10.3-1). Plasma physicists use it to measure electron and ion densities, electron temperature, plasma potential, plasma floating potential and random electron and ion current densities in low-temperature plasmas. A typical current-voltage characteristic of a Langmuir probe immersed in a plasma is shown in Fig. 10.3-2.

Fig. 10.3-1: Langmuir probe (P) with biasing arrangement

When the probe potential is strongly negative with respect to plasma the probe accepts ions only and the current in the region AB is entirely due to positive ions. Even the most energetic electrons cannot reach the probe. However if the probe potential is made less negative, in the region BC, faster electrons can overcome the retarding field and reach the probe. So the probe current is then reduced. Eventually at some point C the probe current becomes zero. It corresponds to the case when ions and electrons reach the probe at equal rate. The potential corresponding to zero probe current is usually known as ‘floating potential’ (Vf). It is the potential that an isolated probe would reach if immersed in plasma. When the probe potential is made more positive beyond C in the CD region the number of electrons collected exceeds the number of ions and the current direction is reversed. When the applied probe voltage equals the plasma potential Vp (average potential inside plasma with respect to the walls of the device) all the electrons irrespective of their velocities



Some Practical Aspects of Plasma Physics

10.7

are able to reach the probe, as if they are unaffected by its presence. So the region DE corresponds to electron saturation current.

Fig. 10.3-2 : The I-V characteristic of a Langmuir probe in a plasma

In the linear region CD, assuming Boltzmann distributed electrons, probe current is given by

I = Ise exp (–eV / kBTe)

...(10.3-1)

where Ise is the electron saturation current and V = Vp – Va is the potential of the surrounding plasma relative to the probe. Taking natural logarithm of both sides of Eqn. (10.3-1), we can write

log I = constant +

eVa kB Te

...(10.3-2)

So a plot of log I against Va (in the region between the floating potential and plasma potential) should be a straight line and from the slope (e / kBTe) of the line the electron temperature Te can be determined. Here it is important to mention that the above measurement is based on the assumption that the dimension of the probe is so small that the plasma remains practically undisturbed. Also, the underlying theory is not valid when there is secondary electron emission or photo-emission from the probe surface due to the impact of incident electrons. The theory also becomes invalid for non-Maxwellian electron velocity distribution. The current collected by the probe may be significant and can affect the very parameters to be measured (especially the electron density). The electron saturation current is shown to be given by

Ise =

8k T 1/2 1 n0 eA e B e o 4 me π

...(10.3-3)

10.8

Basic Plasma Physics

where A is the probe collecting area. Thus if Te is known, plasma density n0 can be determined from Eqn. (10.3-3). The ion saturation current obtained from AB portion of the curve is shown to be given by

Isi = 0.6n0 eA e

kB Te 1/2 o mi

...(10.3-4)

where Mi is the mass of the ion. Thus, Eqn. (10.3-4) can be used to find ion density n0 if electron temperature is known. Equation (10.3-4) is applicable for plasmas with Te >> Ti. When Ti is comparable to Te, Isi is given by

Isi =

8k T 1/2 1 n0 eA e B e o 4 mi π

...(10.3-5)

Double Probe Method he double probe method makes use of two identical Langmuir probes. In this T method a very small current is drawn from the plasma and hence the plasma remains almost unperturbed during measurement. The basic circuit arrangement of a double probe method is shown in Fig. 10.3-3. B is a variable voltage source whose polarity can be changed. Current Id flowing in the external circuit is measured by the current meter A and the probe difference voltage Vd (potential of probe 2 with respect to probe 1) is measured by the voltmeter V.

Fig. 10.3-3 : Basic double probe circuit

Typical current-voltage characteristic of the double probe method is shown in Fig. 10.3-4.



Some Practical Aspects of Plasma Physics

10.9

Fig. 10.3-4 : Current-voltage characteristic of the double probe Id = Ii1 – Ie1 = – (Ii2 – Ie2)

When Vd = 0, each probe will collect zero net current from the plasma and each probe will be at floating potential. Then Id = 0 which corresponds to the point O in the curve of Fig. 10.3-4. When Vd is a large negative voltage, i.e., probe P2 is made highly negative with respect to probe P1, probe P2 collects entire ion current and probe P1 collects entire electron current. It corresponds to the AB portion of the curve. If Vd is made less negative electrons are also collected by P2 and BO portion of the curve is obtained. As the polarity of Vd is changed the symmetry of the system results OCD portion of the curve. The conservation of charge requires

Isi1 – Ie1 = – (Isi2 – Ie2)

or

/ Isi = Ie1 + Ie2

...(10.3-6)

where Isi1 and Ie1 are respectively the ion and electron currents to probe P1 and Isi2 and Ie2 be the corresponding currents to probe P2. Here ion currents are assumed to be almost equal to their saturation values. If Ise1 and Ise2 represent the electron saturation currents to the probes P1 and

P2 respectively, then

Ie1 = Ise1 exp e-

eV1 o kB Te

10.10 Basic Plasma Physics Ie2 = Ise2 exp e-



eV2 o kB Te

...(10.3-7)

where V1 and V2 are the potentials of the surrounding plasmas with respect to the corresponding probes. Substituting (10.3-7) in (10.3-6) and rearranging we obtain

/ Isi



Ie2

- 1 =

Ise1 I e^V - V2h = se1 exp =- 1 G Ise2 Ise2 kB Te

or ln 8/ Isi /Ie2 - 1B = constant –

eVd kB Te

...(10.3-8)

where Vd = V1 – V2 is the difference voltage. The value of Ie2 which corresponds to a voltage Vd is illustrated in Fig. 10.3-4. Ie2 can be related to Id as Ie2 = Id + Isi2 .

Thus the plot of ln 8/ IBi /Ie2 - 1B against Vd will be a straight line and from

the slope of this line electron temperature Te can be determined.

It is also possible to find Te directly from the slope of the Id – Vd curve at Vd = 0. Since Vd = V1 – V2,

1 =

dV1 dV2 dVd dVd

...(10.3-9)

Now

Id = Ie2 – Isi2 = Isi1 – Ie1

...(10.3-10)

Using the relations (10.3-7) and then differentiating (10.3-10) with respect to Vd, we get

Ie1

dV1 dV + Ie2 2 = 0 dVd dVd

...(10.3-11)

From (10.3-9) and (10.3-11) one easily gets

Ie2 dV1 = dVd Ie1 + Ie2

...(10.3-12)

From Eqn. (10.3-10)

dId dI = - e1 dVd dVd

...(10.3-13)

Using (10.3-7) and (10.3-12) we may get

Ie2 e dId = Isi1 k T $ + I Ie2 dVd B e e1

...(10.3-14)

Some Practical Aspects of Plasma Physics 10.11



Vd = 0, Ie1 = Isi1 and Ie2 = Isi2

Now at Therefore,

dId dVd

Isi1Isi2 e = k T $ Vd = 0 B e Isi1 + Isi2

...(10.3-15)

Thus from the slope of Id – Vd graph at Vd = 0 it is also possible to find Te by using Eqn. (10.3-15).

Magnetic Probe magnetic probe usually consists of a few turns of wire arranged in a loop A which may be a millimetre in diameter. It is used to sample the magnetic fields in or around plasmas. A magnetic probe operates on the principle that a timevarying magnetic field induces a voltage in the loop; the magnetic field can be determined from a measurement of the induced voltage. Since the induced voltage is proportional to the rate of change of magnetic field, the probe signal is integrated for the measurement of the magnetic field.

Radio Frequency Probe radio frequency (RF) field, not sufficient enough to cause breakdown, can be A used as a probe. The conductivity of ionized gas is found to be a function of the frequency of the applied RF field. If the conductivity is measured for different frequency of the applied RF field, it shows a maximum at a certain frequency. From a knowledge of this maximum value of RF conductivity it is possible to determine the plasma parameters such as electron density, collision frequency and electron temperature. The advantage of this radio frequency technique over the probe methods lies in the fact that the process of measurement does not alter the quantities to be measured.

Microwave Method of Plasma Density Determination icrowaves have been used extensively to study plasma parameters. A microwave M signal can be used as a probe in the determination of some plasma parameters. A beam of microwaves generated by a klystron is launched towards the plasma by a horn antenna (Fig. 10.3-5). The transmitted beam is collected by another antenna and then detected by a crystal. As the frequency of the signal is varied the detected signal disappears when the signal frequency equals the electron plasma frequency, i.e., ω =

n0 e2 . Thus noting this cut-off of the transmitted signal me ε0

one can determine plasma density. Note that this procedure gives the maximum value of the density.

10.12 Basic Plasma Physics

Fig. 10.3-5 : Microwave measurement of plasma density by the cut-off of the transmitted signal

Microwave Interferometer for Plasma Density Measurement icrowave signal from a klystron is split up into two. One beam is sent through M the plasma with horn antennas and the other beam is passed through a calibrated attenuator and phase shifter (Fig. 10.3-6). The two beams finally combine and the resultant beam is received by an oscilloscope. In the absence of the plasma the attenuator and the phase shifter are so adjusted that the detector output is zero. When the plasma is turned on a phase shift occurs through plasma and then the detector gives a finite output signal. By adjusting the calibrated phase shifter again a null condition is obtained. Thus one obtains the phase shift introduced by the plasma. For a uniform density plasma slab of width l the change in phase is given by 1



ω2pe 2 H ω Df = >1 - e1 - 2 o .l c ω

...(10.3-16)

By setting this expression equal to the measured change in phase it is possible to find the average density of plasma.

Fig. 10.3-6 : A microwave interferometer

Spectroscopic Method lasma emits spectral lines from atomic transitions in visible, UV or soft X-ray P regions of electromagnetic spectrum depending on its density and temperature. By analyzing these spectral lines by a spectrometer it is possible to measure plasma

Some Practical Aspects of Plasma Physics 10.13



parameters. The advantage of spectroscopic method is that the plasma is not disturbed at all as in other methods. For example, assuming local thermal equilibrium the relative intensities of two spectral lines emitted from plasma can be related to the electron temperature through some standard spectroscopic constants. Hence the electron temperature can be obtained by measuring the relative intensities of two spectral lines. The plasma is placed in front of a constant-deviation spectrograph and the lines whose intensities are to be compared are focussed on the slit of the spectrograph by a lens. A photomultiplier tube with an amplifying device is fixed at the position of the slit. The output currents are taken to be proportional to the intensities of the spectral lines. The temperature measurement is also possible from Doppler broadening of spectral lines. Since the atoms and ions in plasmas are moving with different velocities they emit light of different wavelengths from a given electronic transition due to Doppler effect. Consequently each spectral line is broadened. The broadening depends on velocity distribution. Assuming Maxwellian distribution the Doppler broadening of spectral lines in plasma can be related to temperature. Hence from Doppler broadening of a spectral line in plasma the plasma temperature can be determined. If l0 be the wavelength of a spectral line emitted from a particle at rest then the wavelength of the same line emitted from a particle moving away with the velocity v is

l = l0 ^1 + v/ch

or

c v = l ^l - l0h 0

...(10.3-17)

Assuming Maxwellian distribution of velocities the probability that the velocity will lie between v and v + dv will be proportional to exp. (–mv2 / 2kBTe)dv where m is the mass of the emitting particle and Te is the electron temperature. So the spectral intensity as a function of l will be given by

I(l) = A × exp (–mv2 / 2kBTe)

...(10.3-18)

where A is a constant. Using Eqn. (10.3-17) we can write

mc2 l - l0 2 # = e o G A exp I(l) = 2kB Te l0

...(10.3-19)

Obviously the spectral intensity has a central maximum at l = l0 and the intensity falls to half of its maximum value when

10.14 Basic Plasma Physics exp =-

mc2 l - l0 2 1 e o G = 2kB Te l0 2

or

l - l0 =

l0 c

...(10.3-20) 2kB Te ln 2 m

So the Doppler half-intensity breadth of spectral line is

Dl = 2 l - l0 =

2l0 c

2kB Te ln 2 m

...(10.3-21)

This relation can be used to measure electron temperature.

10.4 HEATING OF THE PLASMA I n order to have controlled fusion reaction for the generation of power by using plasma in the startup of a fusion reactor one must be able to heat the plasma to a very high temperature of the order of 106 - 107 kelvin. In an operating fusion reactor part of the energy generated must be sufficient to maintain the plasma temperature. A number of processes for initial heating of plasma to the required operating temperature have been proposed. Here we discuss a few of these methods:

Ohmic Heating lasma is a good conductor of electricity. Hence it is possible to raise its temperature P by Joule heating when a current is induced through it. Electrons nearly absorb all the energy from the field. Ions are heated only by receiving energy from the electrons during collision. The amount of heat generated depends on the resistance of plasma and the amount of current passing through it. With the rise of electron temperature electron-ion collisions become less frequent, plasma resistivity decreases and Joule heating becomes ineffective in heating the ions. To obtain still higher temperatures additional heating methods must be used.

Radio Frequency Heating A plasma can absorb the energy of a high frequency electromagnetic wave at a particular resonant frequency of the plasma and is thereby heated. There are various such techniques including electron cyclotron resonance heating and ion cyclotron resonance heating.

Shock Heating A plasma can be compressed and heated by passing a shock wave through it.



Some Practical Aspects of Plasma Physics 10.15

Magnetic Heating nother method of heating the plasma (used in mirror machines) is that of A ‘adiabatic compression’. In this technique the magnetic field strength is made to increase with time. Thus plasma particles injected into the machine at times of low field strength will increase their transverse energy as the magnetic field is increased, while their axial energy will remain the same.

Neutral Beam Heating I n this method the particles are first accelerated as ions and then subsequently neutralized before introducing them into the ohmically heated and magnetically confined plasma. The neutral atoms get ionized as they pass through the plasma and are trapped by the magnetic field. The energetic ions then transfer part of their energy to the plasma particles through repeated collisions and thereby increase plasma temperature.

10.5 CONFINEMENT OF THE PLASMA he most important application of plasma study is in the controlled thermonuclear T fusion reaction which promises a solution for the future power crisis. The fusion of two nuclei requires that the particles must have sufficient energy to overcome the Coulomb barrier of each other. That is why such reactions take place at high temperatures. For deuterium-tritium (D-T) reaction the temperature should be greater than 10 keV. For deuterium-deuterium (D-D) reaction this temperature should be greater than 100 keV. The reaction produces large amount of energy and is explosive in nature. However, if we can control the reactions the output energy could be used for the benefit of mankind. The controlled thermonuclear fusion promises an almost unlimited source of energy. The advantage of this fusion energy is that the raw material hydrogen is plenty available on the earth and the power generation process is almost pollution free and is not associated with the radiation problem. The most practical system capable of operating at high temperature needed for fusion reaction is the fully ionized plasma. The fundamental problem of controlled thermonuclear reaction is that of containing a high temperature plasma long enough so that an appreciable number of fusions can take place. In addition to the problem of heating plasma to high temperatures and confining plasma for long there is a serious problem of energy loss by radiation and various other processes. These losses constitute a serious problem in maintaining a self-sustaining thermonuclear reactor. It has been shown by J.D. Lawson that for successful operation of a thermonuclear reactor and release of useful output power the product of the density n of charged particles and the plasma confinement time t must be higher than a minimum value. For example for D-T reaction

10.16 Basic Plasma Physics nt > 1020 m–3 s (with T > 107 K) and for D-D reaction nt > 1022 m–3 s (with T > 108 K). This condition is known as the Lawson criterion. Existing laboratory installations do not achieve these values. The problem is still unsolved and most of the active research in plasma physics is directed towards the solution of the problem of heating and confinement of thermonuclear plasma. It is one of the most important scientific challenges of the present age. Solid walls make an ineffective container because they cool the plasma. So the plasma should be confined in space away from the walls of the container. This can be made possible with the help of applied magnetic field. As the charged particles cannot move easily across the magnetic lines of force most of the confinement schemes proposed so far use some type of magnetic field. We discuss below some common confinement schemes:

Magnetic Mirror I t is a kind of linear confinement device along the axis of which is applied a magnetic field which is converging on both sides (Fig. 10.5-1). By using a pair of coaxial coils it is possible to generate such a magnetic field structure. This field acts as a double magnetic mirror and hence a magnetic trap. It has been used in laboratory for plasma confinement. This system, however, suffers from end loss problems and instability arising due to transverse field gradient.

Fig. 10.5-1 : Converging field lines at both ends of a magnetic mirror device

Stellarator tellarator is a closed magnetic confinement device formed by twisting a torous into S a ‘figure-eight’ shape. Here the plasma is confined mainly by the externally applied axial magnetic field. Because of its closed geometry the loss of plasma particles at the ends is eliminated and hence confinement time is increased. The simplest method of heating plasma in the stellarator is to use the ring of plasma as a one-



Some Practical Aspects of Plasma Physics 10.17

turn secondary of a transformer. The induced current heats the plasma by Ohmic heating from the plasma resistivity. The heating current generates a magnetic field that has the effect of weakening the stabilizing features of the stellarator fields. Stellarator configuration though promising still seems to be plagued by different types of instabilities.

Fig. 10.5-2 : Stellarator

Tokamak he name tokamak comes from Russian words which means ‘toroidal magnetic T chamber’. It is the most promising device to achieve controlled fusion. Here a strong toroidal magnetic field is created by sending current through coils wound around the toroidal chamber (Fig. 10.5-3). The charge particles move in helical orbit with the magnetic lines of force as axis. In this way they are confined at a distance from the walls of the toroidal chamber. But here the field is not uniform. Due to gradient and curvature of the magnetic field positive and negative particles move in opposite directions. It results in charge separation and creation of electric field. This electric field tends to drive the plasma out. In order to prevent this effect, a relatively weak poloidal magnetic field is created by sending a current in the toroidal direction (Fig. 10.5-4). Tokamak device is apparently simple, but it is associated with many problems that need to be taken care of. The first tokamak was developed in Russia in 1968. In it the plasma confinement time was about 10 ms. After that a number of tokamaks have been developed. The confinement time has increased from one generation tokamak to the next. At present work on the world’s largest and most advanced tokamak project named ITER (International Tokamak Experimental Reactor) is going on at Cadarache, France. The ITER project aims to bring fusion energy to the commercial market in the coming decades. Current European Union, India, Japan, China, Russia, South Korea and USA are participating in the ITER project.

10.18 Basic Plasma Physics

Fig. 10.5-3 : Tokamak

Fig. 10.5-4 : Poloidal magnetic field in tokamak

Inertial Confinement new ray of hope of achieving controlled thermonuclear fusion is provided by A the inertial confinement. In this method a small pellet of solid D-T is compressed by high power laser beams arranged with spherical symmetry. When the lasers are fired simultaneously the heat of the laser burns the surface of the pellet into a plasma which explodes off the surface. The remaining part of the target is driven inwards due to reaction force and is eventually collapsed into a small point of very high density. Also, the shock waves created due to rapid blow off travel towards the centre of the compressed target from all sides. The energy associated with the shock wave further heats and compresses the tiny target. Thus a superdense high temperature plasma suitable for fusion reaction is produced. At present a large project HiPER (High Power Laser Energy Research Facility) is going on in

Some Practical Aspects of Plasma Physics 10.19



the European Union for possible construction of laser-driven inertial confinement fusion (ICF) device.

SOLVED PROBLEMS PROBLEM 10.1: In a plasma diagnostic experiment with Langmuir probe it is found that the slope of the log I – Va curve is 0.1, wherre I is the probe current and Va is the applied voltage to the probe. Find the electron temperature Te. Solution: Slope of log I - Va curve is \

e = 0.1 kB Te Te =

e 1.6 # 10-19 = K . 105 K . 0.1kB 0.1 # 1.38 # 10-23

PROBLEM 10.2: In a plasma diagnostic experiment with Langmuir probe the electron saturation current to the probe is 6.5 mA. If the probe area is 4 mm2 and the plasma temperature is 105 K, find the plasma density. Solution: The electron saturation current is given by

Ise =

8k T 1/2 1 n0 eA e B e o 4 me π

\

n0 =

4Ise me π 1/2 c m eA 8kB Te 1/2

4 # 6.5 # 10-6 9.1 # 10-31 # 3.14 e o = 19 6 1.6 # 10 # 4 # 10 8 # 1.38 # 10-23 # 105 = 2 × 1013 m–3.

PROBLEM 10.3: Considering the Townsend’s breakdown criterion as ad = k and dependence of a on the electric field (E) to gas pressure (p) ratio as

E E 2 a = C ; - c m E p p 0 p

show that the breakdown voltage V­b can be obtained as

Vb = c

E m pd + p 0

k . pd C

10.20 Basic Plasma Physics Here a is the Townsend’s first coefficient, d is the gap length, C is a constant and (E/p)0 is some minimum value of E/p at which effective ionization begins. Solution: We have ad = k and

E E 2 a = C ; - c m E p p 0 p

Combining these two equations we can write or

E E 2 k = C ; - c m E p p 0 pd E E k 1/2 m = c m + c p 0 Cpd p

For a uniform field gap, E = Vb / d. Therefore,

Vb = c

E m $ pd + p 0

k $ C

pd .

PROBLEM 10.4: In a gas discharge tube with two plane electrodes separated by a distance of 5 cm it is found that a steady current 600 mA flows when a voltage of 10 kV is applied. The current is found to reduce to 60 mA when the separation between the electrodes is reduced to 1 cm and terminal voltage is reduced to 2 kV. Using these data determine Townsend’s first ionization coefficient a. Solution: As the electric field E = V/d is being kept constant we can use the formula n = neead or I = I0ead \

600 = I0ea×0.05

and

60 = I0ea×0.01

Dividing, we get or

10 = e0.04a 1 a = ln 10 = 57.5 ions/m. 0.04

PROBLEM 10.5: The breakdown voltage across an air gap is given by

Vb =

B^ pd h C + ln ^ pd h

where B = 43.6 × 106 V/(atm-m), C = 12.8, p is the pressure in atmosphere and d is the gap length in m. Find the minimum value of Vb. Also calculate the voltage needed to arc a 1m air gap at STP.

Some Practical Aspects of Plasma Physics 10.21

Solution: For minimum,

dVb = 0 d^ pd h

This yields

(pd)min = e1–C and



(Vb)min = B(pd)min

= B.e1–C = 43.6 × 106 × e1–12.8 = 327 V For 1m air gap at STP

Vb =

43.6 # 106 # 1 # 1 12.8 + ln 1

= 3.4 × 106 V. PROBLEM 10.6: The Doppler breadth of the spectral line l = 4250 Å from an argon plasma is found to be 0.048 Å. Calculate plasma temperature. Solution: From Eqn. (10.3-21)

Dl =

2l0 c

2kB Te ln 2 m

Putting Dl = 0.048 Å, l0 = 4259 Å, c = 3 × 108 m/s, kB = 1.38 × 10–23 J/K and m = 40 × 1.67 × 10–27 kg, we get Te = 104K.

QUESTIONS 1. Define Townsend’s first and second ionization coefficients. Explain Townsend’s criterion for the breakdown of a gas in the discharge tube. 2. What is a plasma? Describe a method that has been used in the laboratory to produce a plasma. 3. State and explain Paschen’s law. How do you account for the minimum breakdown voltage under a given pd condition? 4. Describe with necessary theory and experimental set-up Langmuir probe method for the determination of electron temperature and density of a plasma. What are the sources of error in this probe measurement? 5. Describe with necessary theory the microwave transmission method for the determination of plasma density.

10.22 Basic Plasma Physics 6. Describe the microwave interferometer method for plasma density determination. 7. Explain the principle of the measurement of plasma temperature from Doppler broadening of spectral lines emitted from plasma. 8. Show that the Doppler broadening of the spectral lines emitted from a glow discharge plasma is proportional to the square root of the plasma temperature. 9. Enumerate different methods which have been suggested for heating plasma to high temperatures. 10. Discuss the principles of plasma confinement in magnetic mirror device, stellarator and tokamak. What is ITER? 11. Explain why plasma confinement is not possible in a purely toroidal magnetic field. How is this problem minimised?

EXERCISES 1. A microwave signal is launched towards a plasma and the transmitted signal is detected by a crystal. As the signal frequency is varied the detected signal disappears when the signal frequency is 1 GHz. Find the plasma density. [Ans. 1.2×1016 m–3] 2. In a microwave interferometer experiment a fringe shift of 1/10 is obtained when one of the interfering beams is passed through a plasma slab 8 cm thick. Assuming the wavelength of the microwave used is 8 mm calculate the density of plasma. [Ans. 3.5 × 1017 m–3] 3. The breakdown voltage of a gas between two electrodes is given by

Vb =

a_ pd i b + ln _ pd i

where p = gas pressure, d = distance between the electrodes, a and b are constants. Show that the minimum value of the breakdown voltage is given by (Vb)min = a(pd)min. 4. Considering Townsend’s breakdown criterion as ad = k and expressing the coefficient a/p as a function of the electric field (E) to pressure (p) ratio [i.e., a p = f(E/p)] show that the breakdown potential is a unique function of pd where d is the distance between the two electrodes and a is the Townsend’s first ionization coefficient. 5. In the Langmuir probe measurement for argon plasma with Te >> Ti compare the electron and ion saturation currents. [Hints: From Eqns. (10.3-3) and (10.3-4);

Ise /Isi =

mi /me /_0.6 2π i = 271/1.5 = 180]. qqq

CHAPTER

11

Some Elementary Class-room Experiments on Plasma

11.1 INTRODUCTION I n this chapter we describe some plasma experiments that are suitable for an introductory course in plasma physics. These experiments will help to familiarize students with the plasma state and to provide an elementary idea of experimental plasma physics. The equipments necessary for these experiments were first developed in India by FCIPT (IPR) and are now commercially available in the market. These experiments have been successfully used in some Indian universities. Use of vacuum pumps and pressure gauges are essentially required in most plasma physics experiments. So in doing plasma physics experiments one must know how to use vacuum pumps and pressure gauges. In a text book of this type it is not possible to go into details of the construction and working of these instruments. However the section ‘Some Oral questions and Answers’ added at the end of this chapter includes some important information about pumps and gauges.

11.2 TO DRAW PASCHEN CURVE AND STUDY THE DEPENDENCE OF BREAKDOWN VOLTAGE ON THE PRESSURE — INTERELECTRODE GAP LENGTH PRODUCT Theory: According to Paschen’s law the DC breakdown voltage of a homogeneous gap at a constant temperature is a function of the product of the gas pressure (p) and the distance (d) between the electrodes. The voltage (Vb) at which the breakdown occurs is given by

Vb =

B^ pd h C + ln ^ pd h

where B and C are constants that depend on the gas composition.

...(11.2-1)

11.2

Basic Plasma Physics

For air B = 43.6 × 106 V/(atm-m), C = 12.8; Vb is in V when p is in atmosphere and d is in m. The minimum value of breakdown voltage is given by

Vb, min = B(pd)min = Be1–C

...(11.2-2)

The curve obtained by plotting Vb against pd is called Paschen’s curve (Fig. 10.2-1). Initially Vb­ decreases as pd increases and then increases gradually with increase in pd. The curve shows a minimum for a particular value of the product pd for a given gas and electrode geometry. It is called the Paschen minimum. The electrode spacing corresponding to minimum breakdown voltage at a constant pressure is called critical spark length. Apparatus required: A glow discharge system consisting of a glass chamber with variable distance (1 cm – 30 cm) between the electrodes; a variable DC power supply (0-1000 V, 1 A); Pirani gauge, needle valve, rotary pump and vacuum related components. Procedure: (i) Close the air inlet tap and pump down the system to a base pressure of about 2 × 10-2 m bar by using the rotary pump. (ii) By adjusting the needle valve-control fill the system with air or any desired gas to the required pressure (may be in the range 0.05-10 mbar). Note the pressure from the Pirani gauge connected to the plasma chamber through a suitable opening. (iii) Adjust the interelectrode distance to a suitable value (~ 1 cm). Measure the gap length (d) from the attached scale. Note that the position of the electrodes can be changed from outside without affecting the vacuum. (iv) Connect the electrodes (through insulated steel rods) to the DC power supply as indicated in Fig. 11.2-1. Switch on the power supply and increase the voltage across the gap gradually until there is a breakdown (glow discharge) across the gap. It can be seen by the presence of a glow across the electrodes. Note the corresponding voltage which is the breakdown voltage (Vb). Take three independent readings for V­b and find the mean value. (v) Keeping the pressure fixed, increase the interelectrode distance in steps of say 1.5 cm and in each step measure the mean breakdown voltage. (vi) After taking 10-15 readings draw a graph by plotting the interelectrode separation d in cm (or the product pd in mbar-cm) along x-axis and the corresponding breakdown voltage Vb in volt along y-axis. The graph will be similar to that shown in Fig. 10.2-1. (vii) Keep pressure fixed at least at 3 or 4 other values (may be 1, 2, 3 4 mbar) and for each pressure repeat the above procedure and draw more d vs Vb or pd vs Vb graphs.



Some Elementary Class-room Experiments on Plasma

11.3

Note that the pressure can be changed by allowing air (or any specified gas) to enter the evacuated system (in a controlled way) through a needle valve.

Fig. 11.2-1 : Schematic diagram for the experimental set-up for Paschen curve

Experimental Data (A) Data for Paschen curve with fixed gas pressure Gas pressure inside glow discharge system = ........ mbar TABLE 1 No. of Obs.

Distance between electrodes (d) in cm

Breakdown voltage (V­b) in V

Average V­b in V

Product pd in mbar-cm

...

...

...

...

etc.

etc.

... 1.

...

... ... ...

2.

...

...

etc.

etc.

etc.

...

For each value of gas pressure make tables similar to Table 1.

Precautions and Discussions

(i) As we are dealing with high voltage proper care should be taken to avoid any risk of electrical shock. All connections and necessary adjustments

11.4

Basic Plasma Physics must be made with the high voltage OFF. For safety touch only the knob of the high voltage power supply.



(ii) Current limiting resistance should be used to limit the discharge current.



(iii) Discharge current should be kept constant during a set of observations. If it varies continuously the looseness of the pump belt should be checked and tightened.



(iv) After finishing the experiment or whenever you want to switch off the pump the valve connected to the atmosphere should be made open. Otherwise oil from the pump will rush into the plasma chamber.



(v) The given experimental set-up can be used to study Paschen curve by using different gases like Argon, Helium, Hydrogen, etc.



(vi) Using an AC voltage source one can study AC breakdown voltage for different frequencies.



(vii) To make the discharge stable the two electrodes may be connected to each other through a capacitor and a suitable resistor so that the time constant of the circuit is of the order of a few milliseconds.

11.3 TO STUDY THE CONDITIONS OF OCCURRENCE OF STRIATIONS IN GAS DISCHARGE Theory: In a gas discharge tube at a pressure of about 1 mm of Hg a brilliant glow discharge occurs in which different distinctive regions can be observed. Proceeding from the cathode to anode one finds cathode glow, Crookes dark space, negative glow, Faraday dark space and then a long most brilliant region, called the positive column (Fig. 11.3-1). Depending on the pressure and the current density the positive column region may be continuous or striated. The striations survive only in a limited range of current density and gas pressure values. The distance between the striations increases as the gas pressure is decreased or the distance between the electrodes is increased at a particular pressure.

Fig. 11.3-1 : Appearance of striations in glow discharge



Some Elementary Class-room Experiments on Plasma

11.5

The dependence of the distance between the striations on the gas pressure and the distance between the electrodes can be studied. Apparatus required: A glow discharge system consisting of a glass chamber with variable distance between the electrodes; a variable DC power supply (10 V-1000 V, 1 A); Pirani gauge, needle valve, rotary pump and vacuum related components. Procedure: (i) Close the vent valve and needle valve and pump down the system to a base pressure of about 0.02 mbar by using rotary pump. (ii) By adjusting the needle valve fill the system with air or any desired gas (e.g., Argon) to the required pressure (~0.5 mbar). Note the pressure from the Pirani gauge. (iii) Adjust the distance (d) between the electrodes to a fixed value (say, 8 cm). (iv) Connect the electrodes to the DC power supply. Switch on the power supply and increase the voltage in small steps (say, 10 V). Note the voltage when you see striations in the positive column near the anode glow. Also, note down the number of striations and the distance between the striations. (v) Increase the gas pressure inside by adjusting the needle valve-control in small steps (say, 0.1 mbar) and in each step note down the voltage (V), number of striations (N) and distance between the striations (ds). Repeat the process for a fixed distance between the electrodes and for at least five different pressures (say, 0.5, 0.6, 0.7, 0.8, 0.9 mbar). (vi) Repeat the processes (iii) – (v) for two other fixed values of interelectrode distances (say, 10 cm and 12 cm). (vii) Draw graphs by plotting pressure in mbar along x-axis and the corresponding distance between striations in mm along y-axis for each fixed value of interelectrode distance. The nature of the curves will be as shown in Fig. 11.3-2.

Fig. 11.3-2 : Typical variation of distance between striations with gas pressure for different fixed values of interelectrode distance

11.6

Basic Plasma Physics

(viii) If time permits repeat the experiment keeping pressure fixed and changing the distance between the electrodes to other values. Draw graphs by plotting the distance between striations as a function of interelectrode distance for each value of fixed pressure.

Experimental Data: (A) Data for ‘pressure’ versus ‘distance between striations’ graph TABLE 2 : Interelectrode distance d = ... cm (fixed) No. of obs.

Pressure in mbar

Voltage in V

No. of striations

Distance between striations in mm

1. 2. 3.

Make two other tables (Table 3 and Table 4) for two other fixed values of d. (B) Data for ‘interelectrode distance’ versus ‘distance between striations’ graph TABLE 5 : Pressure = ....... m bar (fixed) No. of obs.

Interelectrode distance in cm

Voltage in V

No. of striations

Distance between striations in mm

1. 2. 3.

Make two other tables (Table 6 and Table 7) for two other fixed values of pressure.

Precautions and Discussions: (i)- (iv) Same as in Experiment 11.2.

(v) The striations are the manifestation of the ionization waves and oscillations.



(vi) Striations can be seen only in a limited range of current density and gas pressure values.



(vii) The distance between the striations and the number of striations increase with the increase in distance between the electrodes at a given pressure.



(viii) The distance between the striations decrease with increase in pressure for a given interelectrode distance.



Some Elementary Class-room Experiments on Plasma

11.7

11.4 TO MEASURE PLASMA PARAMETERS BY USING SINGLE LANGMUIR PROBE Theory: Inserting a Langmuir probe within a plasma one can apply different bias voltage to the probe and measure the corresponding current collected by the probe. One thus obtains the I-V characteristics of the probe (Fig. 9.3-1). In the linear region the probe current I and probe voltage V are related by the equation

ln I = constant + c

e mV kB Te

...(11.4-1)

So a plot of ln I against V should be a straight line of slope = e/kBTe. Therefore,

Te =

e kB

# slope



...(11.4-2)

This relation can be used to measure plasma electron temperature. Electron saturation current Ise, ion saturation current Isi and floating potential Vf can be obtained directly from the I-V characteristics. Ise corresponds to the highest plateau, Isi to the lowest plateau and Vf to the probe voltage for zero probe current. Electron saturation current

8kB Te 1/2 1 o Ise = ne eA e 4 me π

...(11.4-3)

where A is the probe collecting area. This relation (11.4-3) can be used to measure plasma electron density ne when Te is known. For a glow discharge plasma with Te >> Ti, the ion saturation current is given by

Isi ≈ 0.6 nieA (kBTe / mi)1/2

...(11.4-4)

where mi is the mass of ion. This relation can be used to measure ion density ni. For a quasineutral plasma ne = ni. The plasma potential Vp can be determined from the following relation

VP = Vf +

kB Te Ies ln e Iis

...(11.4-5)

Apparatus required: A glow discharge system to produce plasma, a variable DC power supply (0-1000 V, 1 A), Langmuir probe and its bias voltage supply; voltage and current meters, Pirani gauge, needle valve, rotary pump and vacuum related components.

11.8

Basic Plasma Physics

Fig. 11.4-1 : Schematic diagram for experiment with Langmuir probe.

Procedure: (i) Close the air inlet valve and pump down the system to a base pressure of about 10–3 mbar by using the rotary pump. (ii) By adjusting the needle valve any desired gas (may be Argon) is then introduced into the system to raise the pressure to the required level (~0.1 mbar). Note the pressure from the Pirani gauge. (iii) Connect the electrodes to the discharge power supply. Switch on the power supply and increase the voltage gradually until there is a glow discharge between the electrodes. The plasma is thus generated. Adjust the discharge current to a suitable value (~40 mA). It should be kept constant.

Fig. 11.4-2 : ln I versus V curve to find electron temperature



Some Elementary Class-room Experiments on Plasma

11.9

(iv) Place the Langmuir probe at a proper place in the plasma and bias it externally to collect current from the plasma. (v) Vary the bias voltage in small steps from –50 V to + 50 V and in each step note the probe voltage and probe current. Note that at one point you require to reverse the polarity of the bias supply and the polarity of the meters. (vi) Draw the I-V characteristic by plotting V in volt along x-axis and the corresponding current I in mA along y-axis (Fig. 9.3-1). From the graph note down the values of ion saturation current (Isi­), electron saturation current Ise and the floating potential Vf at which I = 0. (vii) Taking data corresponding to the linear region of I-V characteristic, plot ln I versus V graph which will be a straight line (Fig. 11.4-2). From the slope of the curve calculate electron temperature as e ...(11.4-6) kB # ^ab/bch (viii) From the value of Ise obtained from the I-V curve calculate ne from Eqn. (11.4-3). Similarly, knowing Isi­from I-V curve, calculate ni from Eqn. (11.4-4). Te =



(ix) If time permits you can repeat the whole experiment by placing the Langmuir probe at a different location between the electrodes or for different gas pressure. Experimental Data:

Gas used = ,



Gas pressure = mbar,



Discharge current = mA

(A) Data for I-V characteristic curve TABLE 8 No. of obs.

1

2

...

V in V

...

I in mA

...

Floating potential from I-V curve = ......... V (B) To draw ln I vs V graph

(Take data corresponding to the linear part of I-V curve) TABLE 9 V in V

1

2

...

I in V

...

...

...

ln I

...

...

...

11.10 Basic Plasma Physics (C)

Calculation of Te from ln I vs V graph



e = 1.6 × 10-19 C, kB = 1.38 × 10–23 J/K TABLE 10 From graph ab

From graph bc in V

...

...

Te =

e in K kB # (ab/bc)

...

(D) Calculation of ion density from Eqn. (11.4-4)

A = ............. (given)



mi = 40 × 1.67 × 10–27 (for Argon),



Isi = ............ (from I-V curve)

∴

ni =

Isi 0.6eA^kB Te /mih1/2

= ... Precautions and Discussions: (i)-(iv) Same as in Expt. 11.2.

(v) For better results the probe size must be smaller than electron mean free path for collisions.



(vi) The experiment gives better results for low pressure plasmas. To generate plasmas at very low pressure hot cathode filament method can be used.



(vii) To limit the discharge current a current limiting resistance should be used.



(viii) The discharge voltage and discharge current should be kept constant throughout the experiment.



(ix) The bias voltage to the probe can be varied by using a ramp circuit and then the I-V characteristic of the probe can be obtained directly on the screen of an oscilloscope.



(x) The presence of the probe in the plasma may to some extent alter the plasma properties in its vicinity. The problem can be overcome by using double probe technique.

11.5 TO MEASURE PLASMA PARAMETERS BY USING DOUBLE LANGMUIR PROBE Theory: The double probe method makes use of two identical Langmuir probes. The basic circuit arrangement of a double probe method is shown in

Some Elementary Class-room Experiments on Plasma 11.11



Fig. 11.5-1. By plotting the current Id (flowing in the external circuit) against the probe difference voltage Vd (potential of probe 2 with respect to probe 1). One can draw the current voltage characteristic of the double probe as shown in Fig. 10.3-4. Assuming the two probes to be identical it can be shown that

ln =

Isi1 + Isi2 e - 1G = - c mV kB Te d Id + Isi2

...(11.5-1)

where Isi1 is the ion saturation current to probe 1 and Isi2 is the ion saturation current to probe 2, as indicated in Fig. 10.3-4. So a plot of the logarithmic term versus the different voltage Vd yields a straight line of slope e/kBTe. Therefore, electron temperature, Te =



kB

e # slope



...(11.5-2)

For a glow discharge plasma with Te >> Ti, the ion saturation current, with identical probes Isi1 = Isi2 . 0.6n0 eA^kB Te /mih1/2



...(11.5-3)

where A is the probe collecting area and mi is the mass of ion. his relation (11.5-3) can be used to estimate the ion or electron density n0 T (since ni ≈ ne = n0 for a quasineutral plasma). I f the probes are identical then Isi1 = Isi2 = Isi (say) and Eqn. (11.5-1) can be expressed as Id = Isi tanh e

\

dId dVd

Vd

eVd o 2kB Te

e = Isi 2k T B e =0

...(11.5-4) (11.5-5)

Hence in this case from the slope of current-voltage characteristics near Vd = 0 the electron temperature Te can be obtained by using Eqn. (11.5-5). If Isi1 and Isi2 are found to be different then the following relation can be used to find Te

dId dVd

Isi1 Isi2 e = k T $ B e Isi1 + Isi2 Vd = 0

...(11.5-6)

Apparatus required: A glow discharge system to produce plasma, a variable DC power supply (0-1000 V, 1 A) : two identical Langmuir probes and its bias voltage supply (0-100 V, 1 A): voltage and current meters, Pirani gauge, needle valve, rotary pump and vacuum related components.

11.12 Basic Plasma Physics

Fig. 11.5-1 : Schematic diagram for experiment with double Langmuir probe

Procedure: (i)-(iii) Same as in Expt. 11.4. (iv) Place the double Langmuir probe at a place in between the electrodes. Apply DC voltage to the double probe with respect to each other. (v) Vary the difference voltage Vd in small steps from – 40 V to + 40 V and in each step note the difference voltage Vd and the current Id in the external circuit. You require to reverse the polarity of the bias supply and of the meters at an intermediate stage.

ln =

Isi1 + Isi2 - 1G Id + Isi2

Fig. 11.5-2 : Double probe plot for temperature determination

Some Elementary Class-room Experiments on Plasma 11.13



(vi) Draw the current-voltage characteristic by plotting Vd in volt along x-axis and Id in mA along y-axis [Fig. 10.3-4]. From graph note down the values of Isi1 and Isi2. (vii) Calculate the values of the function ln =

Isi1 + Isi2 Gfor a number of Id + Isi2

selected points (15-20) on the Id – Vd graph. Take more points on the linear part of the curve. (viii) Plot ln =

Isi1 + Isi2 - 1G as a function of Vd. It would be a straight line as Id + Isi2

shown in Fig. 11.5-2. From the slope of the curve calculate electron temperature as Te =



e kB

# ^ab/bch



...(11.5-6)

(ix) Also calculate Te from the slope of the current-voltage characteristic (Fig. 10.3-4) near Vd = 0 and using Eqn. (11.5-5) or Eqn. (11.5-6). (x) Calculate plasma density n0 by using Eqn. (11.5-3). (xi) If time permits repeat the whole experiment for a different discharge current. Experimental Data: (Data given in the tables are only for illustrations. Students are to take their own data from actual experiment) Gas used = ......... Gas pressure = ........... mbar, Discharge current = ........... mA (A) Data for current-voltage characteristic curve TABLE 11 No. of obs.

1

2

Vd in V

–40

Id in V

...

(B) Data for ln =

...

...

...

–30

0

+40

...

...

...

Isi1 + Isi2 - 1G versus Vd graph Id + Isi2

Isi1 = 47 mA, Isi2 = 50 mA

11.14 Basic Plasma Physics TABLE 12 Id in mA

Isi1 + Isi2 -1 Id + Isi2

–20

–48

47.5

3.86

–15

–45

18.4

2.91

–12

–41.5

–10

–35.5

–8

–30

–6

–25

–4

–22

...

...

0

0

0.94

– 0.06

2

5

0.76

– 0.27

4

12

.

.

6

18

.

.

...

...

.

.

20

49

0.09

– 2.41

Vd in V

ln >

Isi1 + Isi2 - 1H Id + Isi2

(c) Calculation of T­e from Id vs Vd graph Isi1 = ... mA, Isi2 = ... mA, Average Isi = (Isi1 + Isi2) / 2 = ... mA e = 1.6 × 10–19C, kB = 1.38 × 10–23 J/K TABLE 13 Value of DVd in V

Value of DId in mA

...

...

Te =

eIsi DVd in K $ 2kB DId

(d) Calculation of plasma density n0 Average ion saturation current Isi = (Isi1 + Isi2)/2 = ...mA Te = ... K (as obtained above) mi = 40 × 1.67 × 10–27 kg (for Argon) A = ... (given) Now find n0 from Eqn. (11.5-3).

...

Some Elementary Class-room Experiments on Plasma 11.15



Precautions and Discussions: (i)- (ix) Same as in Expt. 11.4.

(x) The values of Isi1 and Isi2 should preferably be chosen at which the current-voltage curve breaks away from the saturation region.



(xi) The advantage of the double probe method is that the probe current is only a very small fraction of the main discharge current. Thus the perturbation introduced by the probes is very small.

11.6 EXPERIMENTAL STUDY OF ION-ACOUSTIC WAVES Theory: Ion-acoustic wave in plasma is a low frequency wave in which ions form regions of compression and rarefaction similar to that formed by ordinary sound waves in a gas. But unlike sound waves ion-acoustic waves are associated with an oscillating electric field which arises because of charge separation due to different masses of ions and electrons. At low frequencies and in the limit of cold ions the ion-acoustic speed is given by

w = vs = k

kB Te mi

...(11.6-1)

where kB is the Boltzmann constant, Te is the electron temperature and mi is the mass of an ion. At high frequencies the dispersion relation of ion-acoustic wave is

k T 1 w2 = B e $ 2 m + k 1 k2 l2D i

...(11.6-2)

where lD is the Debye length. As the frequency approaches the ion plasma frequency,

fpi =

1 2π

n0 e2 ε0 mi

...(11.6-3)

the wave slows down and is damped. Ion-acoustic waves may be excited in plasma by applying sine wave bursts to an exciter grid. If t be the time taken by the wave to reach a receiver plate kept at a distance d. Then velocity of the wave will be given by

vs =

d t

...(11.6-4)

Once vs is known electron temperature Te can be estimated by using Eqn. (11.6-1). For a given frequency w the wave number k can be calculated as

11.16 Basic Plasma Physics

k =

ω 2πf.t = vs d

...(11.6-5)

Calculating k for different values of the frequency f of the excited wave, one can draw the dispersion (w – k) curve for the ion-acoustic wave. Apparatus required: A low pressure glow discharge system to produce plasma consisting of vacuum chamber, tungsten filaments, water cooler, water pump, needle valve, vacuum pumps, filament power supply (typically, 10 V, 20 A); discharge power supply (typically, 100 V, 0.5 A); exciter grid, collector plate, digital oscilloscope, function generator, Pirani gauge, Penning gauge, and vacuum related components. Procedure: (i) Close the air inlet valve and pump down the system to a base pressure of about 10–5 mbar by using the vacuum pumps (may be a combination of diffusion and rotary pumps). (ii) By adjusting the needle valve any desired gas (usually Argon) is then introduced into the system to raise the pressure to about 10–3 mbar. Note the pressure from Pirani gauge. (iii) Switch on the filament power supply and the slowly increase the discharge voltage until there is a glow discharge in the system. Adjust the discharge current to a suitable value (~30 mA). (iv) Adjust the distance between the exciter grid and collector plate to a suitable value d (say, 1 cm). Apply low frequency (~50 kHz) sine wave burst of several cycles (3-5 cycles at 1 V peak-to-peak) from a function generator to the exciter grid. The waveforms of the exciter and the signal received by the collector plate are observed on an oscilloscope. Note the signal time delay (t).

Fig. 11.6-1 : Schematic diagram for experiment with ion-acoustic wave

Some Elementary Class-room Experiments on Plasma 11.17



(v) Repeat the process (iv) for three other values of d (say, 2 cm, 3 cm and 4 cm) by changing the position of the collector plate. (vi) Draw a d versus t graph and from the slope of the graph find the mean wave velocity as vs = Dd / Dt.



(vii) Calculate electron temperature Te from Eqn. (11.6-1). (viii) For a fixed value of d measure the signal time delay (t) as described in process (iv). Then calculate the wave number k using Eqn. (11.6-5). (ix) Increase the frequency (f) of the excited signals in small steps and in each step find the value of k. Then draw the dispersion curve for the ion-acoustic wave by plotting k in cm–1 along x-axis and f in kHz along y-axis. Experimental Data: Gas used = ... (Argon), Gas pressure = ... mbar, Discharge current = ... mA (A) Data for the velocity of the ion-acoustic wave

Frequency (f) of the exciter signal = ... (50) kHz TABLE 14 No. of obs

Distance between exciter and collector (d) in cm

Time difference From d-t graph between launched and detected signals from oscilloscope Dd in cm Dt in ms (t) in ms

1

1

...

2

2

...

3

3

...

4

4

...

...

...

Ion-acoustic velocity vs = Dd # 10 4 Dt m/s

...

(B) Calculation of electron temperature (Te) From (11.6-1),

Te =

=

mi v s2 kB 40 # 1.67 # 10-27 # ^vs in m/sh2 K (for Argon plasma) 1.38 # 10-23

11.18 Basic Plasma Physics (C) Data for dispersion curve Distance between the exciter and collector = 1 cm (say). TABLE 15 No. of obs.

Frequency of the exciter signal (f) in kHz

Time difference between launched and detected signals from oscilloscope (t) in ms

Wave number k = 2πft # 10 -3 d in cm–1

1 2 . . .

Precautions and Discussions:

(i) The experiment gives better results for low density plasmas. To generate plasma at very low pressure hot cathode filament has been used.



(ii) The discharge voltage and discharge current are kept constant throughout the experiment.



(iii) In this experimental situation ion temperature is too low to affect the wave properties.



(iv) Performing the experiment with different gases it is possible to study the dependency of ion-acoustic wave speed on ion mass.



(v) In this experiment the collecting probe is usually biased positively to collect the electron saturation current which is a linear function of plasma density at the probe position. Thus probe signal indicates the density fluctuations associated with the ion-acoustic wave.



(vi) Suitable filter network can be used to remove plasma and other noise outside the range of ion-acoustic wave.

Question with Answers for Viva Voce

1. What is a plasma?



2. Plasma is called’ fourth state of matter’—why?



3. What is the temperature of a 1 eV plasma in kelvin unit?



4. What do you mean by Debye length?



5. In a plasma usually Te » Ti. How it happens so?



6. What are the criteria for an ionized gas to be called plasma?



Some Elementary Class-room Experiments on Plasma 11.19



7. In what way is plasma different from neutral gas?



8. Is there any relationship between plasma frequency and Debye length?



9. What do you mean by plasma oscillation?



10. What do you mean by quasineutrality in plasma?



11. Between electron plasma oscillation frequency and ion plasma oscillation frequency which one is greater and why?



12. Give an example of astrophysical / space / laboratory plasma.



13. What are the important applications of plasma physics?



Ans. For answers to Questions 1-13 see Chapters 1 and 2.



14. How is low-density, low-temperature plasma produced in the laboratory?



Ans. Such plasma can be produced by gas discharge in a discharge tube containing gas at low pressure and fitted with electrodes by applying an external high voltage.



15. What do you mean by glow discharge?



Ans. Glow discharge is an electrical discharge through a gas at low pressure that causes the gas to become luminous. The glow is produced by the decay of excited atoms and molecules.



16. Is the gas temperature equal to electron temperature in a glow discharge plasma?



Ans. No. In a glow discharge plasma Te >> Ti and Ti ≈ Tgas. But in an arc plasma Te ≈ Ti ≈ Tgas.







17. Why is it difficult to produce gas discharge at atmospheric pressure?

Ans. In this case mean free path is very small and the electrons get little opportunity to gain sufficient energy from the applied field and produce ionization by collision. Abnormally high voltage will be needed to produce breakdown in a gas at atmospheric pressure. 18. What can be done to keep the plasma chamber moisture free? Ans. Some drying agents such as silicon gel or calcium chloride can be used. 19. Can you give a practical example of glow discharge? Ans. A glow discharge is a self-sustaining discharge through a gas which occurs at pressures well blow atmospheric pressure. It has obvious glowing regions and fills the full cross-section of the tube. The discharge in a fluorescent light is a glow discharge. 20. What is Paschen’s law? Ans. According to Paschen’s law the breakdown characteristics of a gap are a function of the product of gas pressure and the distance between the electrodes only.

11.20 Basic Plasma Physics

21. How do you adjust the discharge current through the glow discharge system? Ans. By using a variable current limiting resistance. 22. During breakdown of the gas across the electrodes in a discharge system we find a luminous glow. What is the source of this glow? Ans. The glow appears due to decay of excited atoms and molecules.



23. Is it possible to produce a discharge in a gas by using radio frequency field?



Ans. Yes. In fact the breakdown voltage of a gas with radio frequency field is much smaller than that when produced by a DC field.



24. What is the advantage of hot cathode discharge system over cold cathode discharge system?



Ans. In case of hot cathode discharge system lower discharge voltage is required.



25. From Paschen’s curve we find that higher voltage is needed for breakdown across a gap with pd < (pd)min. How do you explain it physically?



Ans. For smaller pd valves, i.e., for lower p or closely spaced electrodes there is a low probability that any electron will collide with neutral atoms during its journey from the cathode to the anode. Hence higher voltage may be needed for breakdown.



26. From Paschen’s curve we find that higher voltage is needed for breakdown across a gap with pd > (pd)min. How do you explain it physically?



Ans. For higher pd valves i.e., for higher p or larger distance between the electrodes, electrons make frequent collisions as it travel from cathode to anode. Then an electron may not be able to acquire enough energy between two successive collisions and higher voltage may be needed for breakdown.



27. What is meant by breakdown in a gas?



Ans. Normally a gas is a good insulator. However if a strong electric field is applied across a gas ionization by collision sets in and gas becomes a good conductor of electricity. Then a disruptive discharge or spark takes place. This phenomenon is called breakdown in a gas.



28. In which region of a discharge tube there are almost equal number of electrons and ions?



Ans. Positive column region.



Some Elementary Class-room Experiments on Plasma 11.21



29. On what factors does the colour of the positive column in a gas discharge tube depend?



Ans. It depends on the nature of the gas in the tube. For air this colour is reddish for neon it is bright red and for carbon dioxide it is bluish white.



30. Is Paschen’s law applicable for electrodes of arbitrary shape?



Ans. The law is applicable only for uniform field distribution. So the electrodes should be such as to produce a uniform field between them.



31. Is it possible to reduce the breakdown voltage arbitrarily by adjusting electrode gap and gas pressure?



Ans. No. For a particular gas there is a minimum voltage below which it is not possible to strike the discharge.



32. How are the striations formed in the positive column region of a gas discharge tube?



Ans. Electrons travelling from cathode towards anode may acquire sufficient energy at some point and excite or ionize gas molecules. The glow is produced there by the decay of these excited molecules. The electrons after ionizing or exciting molecules lose energy and become unable to produce further ionization or excitation of molecules. Then the electrons move a short distance without causing any emission of light and hence this path appears black. After a short distance the electrons again acquire enough energy from the applied field and cause ionization and excitation of molecules. So after the dark region again a bright region is formed. This phenomenon is repeated again and again before the electrons reach the anode. For this a number alternate bright and dark regions, i.e., striations are formed.



33. What are the factors on which the distance between successive striations depends?



Ans. It depends on gas pressure and interelectrode distance. The distance between successive striations increases if the gas pressure is decreased or the distance between the electrodes is increased.





34. What do you mean by arc discharge? Ans. In a glow discharge the discharge fills the cross-section of the tube. As pressure is increased the current increases and its cross-section constricts to a narrow channel. The discharge is then called an arc discharge. 35. What is bar? How is it related to torr? Ans. ‘bar’ and ‘torr” both are units of pressure.

1 torr = 1 mm of Hg 1 bar = 105 N/m2 ≈ 750 mm of Hg = 750 torr.

11.22 Basic Plasma Physics

36. What is a Langmuir probe? Ans. It is simply a small bare wire or metal disk used for plasma diagnostics. 37. What is floating potential? Ans. It is the potential corresponding to zero probe current. It is the potential that an isolated probe would reach if immersed in plasma. 38. What is plasma potential?



Ans. It is the average potential inside plasma with respect to the walls of container. Electrons because of their much higher thermal velocities than ions are lost to the walls faster and leave the plasma with a net positive charge. The plasma will then have a positive potential with respect to the walls.



39. Can you measure plasma potential with respect to any reference electrode by just connecting a voltmeter?



Ans. Plasma potential cannot act as a source of power and hence it cannot be measured by connecting a voltmeter. For the voltmeter to show reading current must pass through it and it requires some power. Also it can be argued that if any current passes it will change the potential to be measured.



40. Explain physically the nature of the Langmuir probe I-V characteristic.



Ans. See, Section 9.3 41. What are the limitations of the probe theory that you apply for plasma diagnostics? Ans. Underlying theory is not valid



(i) for non- Maxwellian electron velocity distribution



(ii) when there are secondary electron emission form the probe s urface due to impact of incident electrons.



42. What is the advantage of double probe method over single probe method?



Ans. In double probe method current drawn to the probe from plasma is very small compared to that in single probe method and hence the plasma remains almost unperturbed during measurement.



43. In double probe method is there any limitation on the separation between the two probes?



Ans. Yes. The distance between the probes must be greater than the width of the sheath region because the discharge between the probes is not permitted. Moreover the probes must not emit any electron.



Some Elementary Class-room Experiments on Plasma 11.23



44. In double probe method the two ion saturation currents are found not to be equal in practice even with identical probes. What may be the possible cause for this?



Ans. This is probably due to the fact that ion densities at different points inside the plasma chamber are not exactly equal.



45. Do you get average plasma density by the probe method? Ans. No, we get plasma density at the position of the probe. 46. What is the main disadvantage of single probe method?



Ans. The probe current may be a significant part of the main discharge current. As a result the presence of the probe may change the plasma parameters that we are going to measure.



47. In experiments with the glow discharge system why do you keep the discharge voltage, discharge current and gas pressure constant?



Ans. Otherwise the plasma parameters will change.



48. Why is the magnitude of ion saturation current different from that of electron saturation current?



Ans. Due to smaller mass of electrons their drift velocity is much higher than that of ions. For this though electron and ion densities are equal, electron saturation current is much higher than that of ions. It can be shown that Ies .



49. What are the factors on which the floating potential depends? Ans. The magnitude of the floating potential is mainly determined by the electron temperature. It can be shown that Vf =



mi /πme $ Iis .

kB Te 1 m $ ln c i m . e 2 2πme

50. Is there any relationship between floating potential and plasma potential? Ans. Yes, it can be shown that Vp = Vf +



kB Te Ies ln e Iis

Vf is negative and its magnitude is greater than Vp.



51. What is an ion-acoustic wave? How does it differ from ordinary sound waves in air?



Ans. It is a low frequency wave in which ions form regions of compression and rarefaction similar to that formed by ordinary sound waves in air.

11.24 Basic Plasma Physics But unlike sound waves in air, ion-acoustic waves in plasma have oscillating electric field arising due to charge separation of ions and electrons due to different masses.

52. Ion-acoustic speed depends on electron temperature — why?



Ans. In plasma ion-acoustic waves propagate through intermediary of electric field. This electric field depends on electron temperature.



53. In the ion-acoustic wave experiment how do you become sure that the wave excited is an ion-acoustic wave and not an electron plasma wave?



Ans. In this experiment we are using a low density (n0 ~ 1014 m–3) plasma in which ion plasma frequency is of the order of a few hundred kHz and electron plasma frequency is about 100 MHz. Therefore electron plasma waves can be excited for frequencies exceeding 100 MHz. Here to launch the wave we are using sine wave bursts of frequencies less than the ion plasma frequency. Hence it is most likely that ion -acoustic waves will be excited.







54. What is a stabilized power supply? How is it fabricated? Ans. A stabilized power supply is a compact electronic regulated power supply. It consists of a rectifier that converts AC line voltage into unidirectional voltage, a filter network that smoothens the rectifier output and a regulator that stabilizes the output. 55. What is a function generator? Ans. It is an electronic instrument capable of producing different types of non-sinusoidal signals (pulse, square wave, triangular wave, saw tooth wave) in addition to sine wave signal. 56. What is an oscilloscope?



Ans. It is a very useful versatile testing and measuring instrument for electrical and electronic circuits. It can display electrical signals as a function of time.



57. How can you measure phase difference or time delay between two AC signals by using oscilloscope?

Ans. The two signals are displayed simultaneously as two separate traces on a dual trace oscilloscope. If d0 divisions be the distance between two consecutive peaks of a signal wave and d be the distance between two neighbouring peaks of the two waveforms then the phase difference would be d degrees. ∆φ = 360 # d0

The phase diference can also be measured by forming Lissajous patterns.



Some Elementary Class-room Experiments on Plasma 11.25 58. Why do you use moderate brightness on the oscilloscope screen? Ans. Excessive brightness can cause permanent damage of the phosphor on the screen. 59. What is a voltage ramp? Ans. A voltage that varies linearly with time is known as a voltage ramp.



60. How can you display the probe characteristics directly on the oscilloscope screen?



Ans. The probe bias voltage is to be varied from say, – 40 V to + 40 V using a ramp circuit. This voltage is to be applied to the horizontal plates of the oscilloscope and the voltage drop across a resistor connected in series with the probe is to be applied to the vertical plates of the oscilloscope.





61. What is the pumping mechanism of a rotary pump? Ans. A rotary pump consists of a solid cylinder (the rotor) which rotates eccentrically inside a hollow cylindrical stator. As the rotor rotates the gas taken in the volume between it and the stator from an inlet port is trapped and compressed and finally the gas escapes through an exhaust valve. It can produce a pressure as low as 10–3 torr. 62. What type of vacuum pump is used to produce high vacuum? Ans. A combination of rotary and diffusion pumps. 63. What is the principle of operation of a diffusion pump?



Ans. Diffusion pumps are vapour jet pumps. It uses a high speed jet of vapour to drive out gas molecules. By this pump one can produce a pressure as low as 10–8 torr. The diffusion pump can operate only after a partial vacuum is obtained with the help of a rotary pump.



64. Can you use a manometer to measure pressure in the discharge chamber?



Ans. No. Here pressure is very low compared to atmospheric pressure. Use of manometer is limited to near atmospheric pressure.



65. What is the principle of operation of a Pirani gauge? Ans. It uses the variation of thermal conductivity of a gas with change of pressure. 66. What is a Pirani gauge? Ans. It is a pressure gauge used to measure low pressure accurately. It contains an electrically heated filament exposed to the gas whose pressure is to be measured. Heat conduction from the filament depends on the gas pressure. Hence the equilibrium temperature and resistance of the filament is related to the gas pressure. The filament is made a

11.26 Basic Plasma Physics part of Wheatstone bridge and the pressure is read from a micrometer calibrated in pressure units.





67. What is principle of operation of a Penning gauge? Ans. It is a cold cathode ionization gauge suitable for very low pressure. It senses pressure indirectly by measuring ions produced when the gas is bombarded with electrons. Number of ions produced is a function of the gas pressure in the system. Ions are collected by a suitably biased collector. This collector current is a measure of the gas pressure. 68. What is the function of the needle valve? Ans. It is used for a very precise control of gas flow at low pressure. It takes many turns of a fine threaded screw to cause a small change in the rate of flow and hence precise control is obtained 69. Pirani gauge is suitable for what pressure range? Ans. 1 – 10–4 torr.



70. Suppose the vacuum pump is running but desired vacuum is not being produced? What may be the possible causes? What preliminary checks will you do in this case?



Ans. Either the pump is not working properly or there may be some leak in the plasma chamber. Connect the pump directly to a pressure gauge. If the pressure reads low (< 0.1 mm) the pump is OK. If the reading is more the pump is not working. Then check the looseness of the belt and if necessary tighten it. Then check the oil level of the pump. If the level is below add suitable amount of oil.



71. What is a Wilson seal?



Ans. It is a special type of vacuum seal which helps to change the position of electrodes, probes etc. from outside of the plasma chamber without affecting the vacuum condition of the chamber.



72. What do you mean by ‘roughing pump’? When do we need to use it?



Ans. A roughing pump is any vacuum pump (typically mechanical) used to initially evacuate a vacuum system as a first step towards high vacuum. Pumps that operate in the high vacuum generally do not operate or operate inefficiently at atmospheric pressure. For this as a first step a roughing pump is used. qqq



APPENDIX

A Some Useful Vector Relations

v , Bv and Cv represent vector functions whereas f and Y In the following relations A represent scalar functions. it, tj and kt are three Cartesian unit vectors. v $ Bv = Bv $ A v = A B +A B +A B 1. A x x y y z z it v # Bv =- Bv # A v= A 2. A x Bx

tj Ay By

kt Az Bz

v $ ^ Bv # Cv h = Bv $ ^Cv # A vh = Cv $ ^ A v # Bv h 3. A v # ^ Bv # Cv h = Bv ^ A v $ Cv h - Cv^ A v $ Bv h 4. A v $ ^d v fh = d2 f 5. d v # ^d v fh = 0 6. d v $ ^d v #A vh = 0 7. d v # ^d v # Ah = d v ^d v $A v h - d2 A v 8. d v $ ^fA v h = fd v $A v +d vf $ A v 9. d v # ^fAvh = fd v # Av + d v f # Av 10. d v $ ^A v # Bv h = Bv $ ^d v #A vh - A v $ ^d v # Bv h 11. d v # ^A v # Bv h = A v ^d v $ Bv h - Bv ^d v $A vh + ^ Bv $ d v hA v - ^A v $d v h Bv 12. d v ^φΨh = φd v Ψ + Ψd vφ 13. d

A.2

Appendix

I f V is a volume bounded by the closed surface S and nt is a unit vector drawn outwardly to the surface S, then v $A vh dV = 14. y ^d V

t (Gauss’s divergence theorem) yS Av $ ndS

I f S is an open surface bounded by the closed contour C, of which dlv is a line element, then v #A vh $ ndS t = 15. y ^d S

yC Av $ dlv (Stokes’ theorem)

In Cartesian coordinates (x, y, z): v f = it 2f + tj 2f + kt 2f 16. d 2x 2y 2z 2A 2A v $A v = 2Ax + y + z 17. d 2x 2y 2z it v #A vh = 2 18. ^d 2x Ax 19. d2 f =

tj 2 2y Ay

kt 2 2z Az

22 f 22 f 22 f + 2+ 2 2x2 2y 2z

In cylindrical coordinates(r, q, z): v φ = rt 2φ + θt 1 2φ + zt 2φ 20. d 2r r 2θ 2z v $A v= 21. d

1 2 1 2Ai 2Az + ^rAr h + r 2r r 2q 2z

rt 1 2 v #A v= 22. d r 2r Ar 23. d2 φ =

rqt 2 2q rAi

zt 2 2z Az

1 2 2φ 1 22 φ 22 φ cr m + 2 2 + 2 r 2r 2r r 2θ 2z

In spherical coordinates (r, q, j): v φ = rt 24. d

2φ t 1 2φ 1 2φ +θ + ϕt 2r r 2θ r sin θ 2ϕ



Appendix

A.3

2 1 2Az v $A v = 1 2 ^ r2 A h + 1 + 25. d θ sin A ^ h r r sin θ 2θ i r sin θ 2ϕ r 2 2r rt 2 1 v #A v= 26. d r2 sin θ 2r Ar 27. d2 φ =

rθt r sin θϕt 2 2 2θ 2ϕ rAi r sin θ A{

1 2 2 2φ 1 1 2 2φ 22 φ cr m+ 2 csin θ m + 2 2 2 2r 2θ r 2r r sin θ 2θ r sin θ 2ϕ2

If rv be the radius vector, of magnitude r, from origin to the point (x, y, z), then v $ rv = 3 28. d v # rv = 0 29. d v v 1 =- r 30. d r r3 31. d2

1 = 0 for r ! 0 . r

APPENDIX

B Concept of Tensors

Most physical quantities can be classified as scalars and vectors. A scalar F is a quantity completely defined by one component which does not depend on the v is a coordinate system, i.e., it transforms according to the law F′ = F · A vector A set of three components that transform in the same manner as the components of the position vector. Thus Ail =

/

k = x, y, z

aik Ak

^i = x, y, zh

where Ail and Ai are components of the vector in S′ and S coordinate systems respectively; aik are the direction cosines relating to the axes of S′ and S systems. There are certain entities which cannot be properly represented either by scalars or by vectors. For example, in an anisotropic dielectric medium the electric v is not in general in the direction of the electric field Ev . An displacement vector D v having components electric field applied in one direction may result in a vector D v = eEv , the vectors D v and Ev differ from in other directions. So in the relation D one another in both magnitude and direction and hence e cannot be a scalar or a vector. Thus the scalar or vector concepts become inadequate in this case. In three v can be expressed as a linear function of all the dimensions each component of D v components of E . Thus

Dx = exx Ex + exy Ey + exz Ez



Dy = eyx Ex + eyy Ey + eyz Ez



Dz = ezx Ex + ezy Ey + ezz Ez

In compact notation,

Di =

/

k = x, y, z

eik Ek

^i = x, y, zh



Appendix

A.5

The quantity represented by the nine (32) components eik which carry the v is known as a tensor of rank two. In general a tensor of rank n has vector Ev into D n 3 components. In this hierarchy, a scalar with 30 components is a tensor of rank 0 and a vector with 31 components is a tensor of rank 1. Formally a tensor of rank 2 in three dimensions is a quantity uniquely defined by 32 components which transform under change of coordinate systems according to the law

Tijl =

/

k = x, y, z

/

l = x, y, z

aik a jl Tkl

where Tijl and Tij are components of the tensor in S′ and S systems respectively. The tensor Tij is sometimes written with a double arrow: Tx . One can perform dot product of a tensor with a vector and the result is itself a vector. The relation v can be expressed as between Ev and D v = ex $ Ev D where ex is the dielectric permittivity tensor. A familiar example of tensor from plasma physics is the pressure tensor x P . The pressure at any point inside plasma may be defined as the rate of transport of molecular momentum due to random motion per unit area. In real plasma it is possible to transfer jth component of momentum due to motion in the ith direction. It cannot be represented by a scalar pressure or a vector but must be given by a tensor Px whose components specify both direction of motion and component of momentum involved. Physically Pij is the force per unit area in the ith direction acting on an element of surface oriented in the jth direction. The diagonal elements Pxx, Pyy, Pzz represent pressures and off-diagonal elements Pxy, Pxz etc. are shears. The force dFv on a surface element dsv inside a plasma can be expressed as

dFv = Px $ dsv

The expression for pressure tensor in a real plasma is a complicated one. However for the special case of isotropic velocity distribution Px can be expressed as

p v P = f 0 0

0 p 0

0 0 p p

v $ Px is just d v p , where p is the scalar pressure. In this case d

APPENDIX

C Some Important Physical Constants

Constant Speed of light in free space Electronic charge Electronic rest mass Proton rest mass Proton-electron mass ratio Avogadro number Molar volume of a gas at STP Boltzmann constant Universal gas constant Acceleration due to gravity Planck’s constant Gravitational constant Bohr magneton Electron-volt (energy) Electron-volt (temperature) Permittivity of free space Permeability of free space

Symbol c e me mp mp/me NA Vm kB R g h G mB eV eV e0 m0

Approximate Value 3 × 108 m/s 1.6 × 10–19 C 9.1 × 10–31 kg 1.67 × 10–27 kg 1836 6.02 × 1023 /mol 22.4 litre/mol 1.38 × 10–23 J/K 8.31 J/K-mol 9.8 m/s2 6.62 × 10–34 J-s 6.67 × 10–11 N-m2/kg2 9.27 × 10–24 J/T 1.6 × 10–19 J 11600 K 8.854 × 10–12 F/m 4p × 10–7 H/m

APPENDIX

D

List of Some Symbols and their Principal Meanings

Symbol

Meaning

Bv

Magnetic induction

c

Speed of light in free space

C

Capacitance

D

Diffusion coefficient

v D

Electric displacement

dlv, drv

Length element

dsv

Surface element

dV

Volume element

e

Electronic charge

f

Frequency, distribution function

Fv g

Force

Hv

Magnetic field intensity

I, i

Electric current

^it, tj, kth

j

Acceleration due to gravity

Cartesian unit vectors -1

vj

Volume current density

k

Wave number

kv

Wave vector

K

Dielectric constant

m

Mass

A.8

Appendix Symbol me mp mi N, n NA nt Pv

Meaning Electron mass Proton mass Ion mass Number density Avogadro number Unit normal to a surface element Polarization

p Px

Pressure (scalar) Pressure tensor

q

Charge Position vector

rv (r, q, z) (r, q, j) R RL Rm S t T uv

Cylindrical coordinates Spherical coordinates Radius of curvature, resistance Larmor radius Mirror ratio Surface area Time Absolute temperature, time period Fluid velocity

vv vs

Particle velocity, drift velocity

vv||

Parallel component of vv

vv= VA Vb Vth W

Perpendicular component of zv Alfvén speed Breakdown voltage Thermal speed Kinetic energy K E associated with parallel motion

W|| W= (x, y, z) e er

Ion-acoustic speed

K E associated with perpendicular motion Cartesian coordinates Permittivity Relative permittivity



Appendix Symbol e0 l lD m m0 n r rm s t f w

Meaning Permittivity of free space Wavelength Debye length Magnetic moment, mobility, refractive index Permeability of free space Collision frequency Volume charge density Mass density Electrical conductivity Relaxation time, time constant Electric potential Angular frequency, angular velocity

A.9

APPENDIX

E



Some Important Formulae in Plasma Physics

1. Debye Length



lD = ^e0 kB Te /n0 e2h1/2

= 69 Te /n0 m when Te in K, n0 in m–3

= 7430



2. Electron Plasma Frequency





ω pe . 9 n0 Hz with n0 in m–3 2π

wpi = ^n0 qi2 /mi e0h1/2 wpi = eB/me

5. Ion Cyclotron Frequency



fpe =

4. Electron Cyclotron Frequency



wpe = ^n0 e2 /me e0h1/2

3. Ion Plasma Frequency



Te /n0 m when Te in eV, n0 in m–3

wci = qiB/mi

6. Electron Cyclotron Radius / Larmor Radius



R= =

mv v= = e = eB wce

7. Magnetic Moment of Orbital Particle 1 2 mv= W= = 2 mv = B B



Appendix



2W|| ^ Rv # Bv h qR2 B2

vvp =

m 2Ev qB2 2t

uvD = –

v p # Bv d qnB2

w2 = w2pe +

3 2 2 k VTh 2

15. Ion-acoustic Speed



vvR =

14. Bohm-Gross Dispersion Relation



W= v v ^ B # dBh qB3

13. Diamagnetic Drift



vvG =

12. Polarization Drift



mgv # Bv qB2

11. Curvature Drift



Ev # Bv B2

vvg =

10. Grad-B Drift



vvD =

9. Gravitational / gv # Bv Drift



W= Bv e o B B

8. Ev # Bv Drift Speed



mv = –

vs = ^kB Te /mih1/2 with Ti=0

16. Dispersion Relation for Ion-acoustic Wave



k T γi kB Ti 1/2 1 w + = = B e G mi 1 + k2 λ2d mi k



w ≈ vs (approximately) k





17. Dispersion Relation for EM Waves in Cold Field—free Plasma



w2 = w2pe + k2 c2

A.11

A.12

Appendix 18. Refractive Index for Plasma



m = e1 -

19. Upper Hybrid Frequency 1/2



wh = `w2pe + w2cej

20. Alfvén Speed



w2 1/2 o w2pe

VA = (B2 / m0rm)

21. Vlasov Equation 2f v f + q ^ Ev + vv # Bv h $ d v f=0 + vv $ d v 2t m



22. Continuity Equation



23. Fluid Equation of Motion

mn ;

kB Te me v

Da =

mi De + me Di mi + me

D= =

kB Tvm q2 B2

28. Ion Saturation Current to Langmuir Probe



De =

27. Diffusion Coefficient Perpendicular to a Strong magnetic Field



e2 ne me v

26. Ambipolar Diffusion Coefficient



s =

25. Electron Free Diffusion Coefficient



2 v E uv = qn^ Ev + uv # Bv h - d vp + uv $ d 2t

24. DC Conductivity of Cold Plasma



2n v + d $ ^nuvh = 0 2t

Isi = 0.6n0 eA^kB Te /mih1/2

29. Electron Saturation Current to Langmuir Probe



Ise =

8k T 1/2 1 n0 eA e B e o 4 me p



Appendix 30. Pondermotive Force





ω2pe ω2

v 1 < ε E2 > d 0 2

ad = ln (1 + 1/g)

(d = gap length)

32. Breakdown Voltage of a Uniform Field Gap



FvNL ≈ –

31. Townsend’s Criterion for Self-maintained Discharge



A.13

33. Paschen’s Law

Vb =

B^ pd h (p = gas pressure, d = gap length) C + ln ^ pd h

Vb = f (pd)

APPENDIX

F Some Typical Multiple Choice Questions



1. Percentage of matter in the known universe, which is in the plasma state, is about

(a) 0.1%

(b) 1%

(b) 4×109

(c) 1.4×102

(d) 1.4×104

(b) 1 mm

(c) 2 m

(d) 2 mm

5. The electron plasma frequency in a plasma with density 1012m-3

(a) 9 MHz

(d) 11600 K

4. Debye length for ionospheric plasma with electron density 1012m-3 and temperature 100 K is

(a) 1 m

(c) 1160 K

3. The number of electrons in a Debye sphere for interstellar plasma with density n0 = 106 m-3 and temperature 0.1 eV is about

(a) 4×106

(d) 99%

2. Absolute temperature of a 0.1 eV plasma is about

(a) 1.6 ×10-20 K (b) 0.1 K

(c) 9%

(b) 90 MHz

(c) 9 GHz

(d) 90 GHz

6. For an ionized gas to behave like a plasma the number of particles in a Debye sphere must be

(a) much smaller than 1

(b) much greater than 1

(c) nearly equal to 1

(d) exactly equal to 1



7. The grad-B drift is

(a) always in the same direction as the curvature drift (b) always opposite to the curvature drift (c) in the same direction for ions and electrons (d) is sometimes parallel to Bv



Appendix

8. In a magnetic mirror device with mirror ratio Rm, the semi vertical angle ^qch of the loss cone is given by

(a) qc = tan-1 (c) qc = tan-1

A.15

1 Rm

1 Rm

1 Rm

(b) qc = sin-1 (d) qc = sin-1

1 Rm

9. Fluid equation of continuity is a consequence of the principle of conservation of

(a) momentum

(b) energy

(c) mass

(d) angular momentum

10. MHD equations can be used to describe (a) low frequency phenomenon

(b) high frequency phenomenon

(c) only steady state phenomena

(d) kinetic behaviour of plasma

11. The magnetic flux through any open surface moving with any highly conducting fluid remains constant. This statement comes from (a) Langmuir (c) Bohm and Gross

(b) Alfvén (d) Landau

12. The maximum value of Larmor radius of a solar wind proton streaming with a velocity 300 km/s is (a) 626 km

(b) 62.6 cm

(c) 6.26 m

(d) 6.26 mm

13. Electron plasma oscillations cannot propagate through a homogeneous plasma which is (a) cold and of infinite extent

(b) cold and of finite extent

(c) warm and of infinite extent

(d) warm and of finite extent

14. In a plasma “cut off” and “resonance” respectively refer to the situation when refractive index is (a) 0 and 1

(b) 1 and 0

(c) ∞ and 0

(d) 0 and ∞

15. Which of the following waves cannot propagate in unmagnetized plasma? (a) electron plasma wave

(b) ion-acoustic wave

(c) electromagnetic wave

(d) Alfvén wave

16. Ion-acoustic speed in a plasma is given by (a)

kB Ti /mi

(b)

kB Te /mi

(c)

kB Ti /me

(d)

kB Ti /mi + kB Te /me

A.16

Appendix

17. Maximum density in ionosphereric plasma is 1012m-3. Then for radio wave communication with space vehicles the frequency can be (a) 100 MHz

(b) 5 MHz

(c) 1 MHz

(d) 1 kHz

18. Regarding the phase velocity vp and group velocity vg of electromagnetic waves in a plasma which of the following statements is true? (a) both vp and vg always < c (b) both vp and vg are sometimes > c (c) sometimes vp > c but vg < c always (d) sometimes vg > c but vp < c always 19. Alfvén speed in a hydrogen plasma with electron density 1012 m-3 and magnetic field 10-3 T is about (a) 2.2 × 107m/s

(b) 4.4 × 107m/s

(c) 1.0 × 107m/s

(d) 2.2 × 107 cm/s

20. Vlasov equation can be well applied to describe the dynamics of a plasma which is (a) cold

(b) sufficiently hot

(c) cold and dense

(d) hot and superdense

21. First velocity moment of the distribution function f^rv, vv, t h gives (a) number density

(b) average velocity

(c) pressure tensor

(d) temperature

22. First velocity moment of Boltzmann equation gives the (a) equation of continuity

(b) equation of motion

(c) energy equation

(d) Vlasov equation

23. Landau damping (a) is associated with energy dissipation by collision (b) does not occur in collisionless plasma (c) is purely mathematical and cannot be demonstrated experimentally (d) is caused by the interaction of particles with wave electric field 24. The conductivity of a fully ionized plasma is found to be proportional to (a) T e3/2 (c) T e1/2

(b) T e-3/2 (d) T e-1/2



Appendix

A.17

25. If De, Di and Da are respectively the free diffusion coefficient of electrons, free diffusion coefficient of ions and the ambipolar diffusion coefficient then (a) Di < De < Da

(b) De < Di < Da

(c) Da < Di < De

(d) Di < Da < De



26. If D= is the diffusion coefficient of a plasma across a magnetic field B then D= is proportional to

(a) 1/B

(b) B

(c) 1/B2

(d) B2

27. Envelope soliton is described by

(a) KdV equation

(b) Nonlinear Schrodinger equation

(c) Vlasov equation

(d) Boltzmann equation

28. Korteweg de Vries equation (a) describes nonlinear evolution of long wavelength waves (b) describes nonlinear evolution of very short wavelength waves (c) admits envelop soliton solution (d) admits a solitary wave solution with sech profile 29. Plasma potential (a) is a few volt negative

(b) is a few volt positive

(c) can have any value

(d) is always zero volt

30. Floating potential of Langmuir probe corresponds to probe potential with current equal to (a) zero

(b) ion saturation current

(c) electron saturation current

(d) 1 micro ampere

31. Paschen law is related to (a) diffusion of plasma across a magnetic field (b) propagation of waves through ionosphere (c) breakdown voltage across a gap (d) controlled thermonuclear fusion 32. Which one of the following is not a plasma confinement device (a) Magnetic mirror

(b) Stellarator

(c) Tokamak

(d) MHD generator

A.18

Appendix

33. Which one of the following is not a possible way to heat plasma? (a) Adiabatic compression

(b) Ion cyclotron resonance

(c) Ambipolar diffusion

(d) Ohmic heating

34. Most promising fusion reaction for controlled thermonuclear reactor is (a) D-D

(b) D-T

(c) T-T

(d) D-He

ANSWERS 1. (d)

2. (c)

3. (a)

4. (d)

5. (a)

6. (b)

7. (a)

8. (b)

9. (c)

10. (a)

11. (b)

12. (a)

13. (a)

14. (d)

15. (d)

16. (b)

17. (a)

18. (c)

19. (a)

20. (b)

21. (b)

22. (b)

23. (d)

24. (a)

25. (d)

26. (c)

27. (b)

28. (a)

29. (b)

30. (a)

31. (c)

32. (d)

33. (c)

34. (b)

References

1. N.A. Krall and A.W. Trivelpiece, Principles of Plasma Physics, McGraw Hill Book Company, 1973

2. R.J. Goldstone and P.H. Rutherford, Introduction to Plasma Physics, IOP Publishing Ltd., 1995



3. F.F. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol.-1, Plenum Press, New York, 1984



4. B.M. Smirnov, Introduction to Plasma Physics, Mir Publishers, 1977



5. P.A. Sturrock, Plasma Physics, Cambridge University Press, 1994



6. J.A. Bittencourt, Fundamentals of Plasma Physics, Springer Verlag, New York, 2004



7. R.D. Hazeltine and F.L. Waelbroeck, The Framework of Plasma Physics, Perseus Books, Reading, Massachusetts, 1998



8. J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1962



9. D.J. Rose and M. Clark, Plasmas and Controlled Fusion, John Wiley and Sons, New York, 1961

10. L.A. Arzimovich, Elementary Plasma Physics, Blaisdell, New York, 1965 11. T.J.M. Boyd and J.J. Sanderson, Plasma Dynamics, Barnes & Noble, Inc. New York, 1969 12. R.A. Cairns, Plasma Physics, Blackie & Sons, Glasgow, 1985 13. L. Spitzer, Jr., Physics of Fully Ionized Gases, Interscience Publishers, 1962 14. D.R. Nicholson, Introduction to Plasma Theory, Wiley, New York, 1983 15. T.G. Cowling, Magnetohydrodynamics , Interscience, New York, 1957 16. T.H. Stix, The Theory of Plasma Waves, McGraw Hill, New York, 1962 17. V.I. Karpman, Nonlinear Waves in Dispersive Media, Pergamon Press, 1974 18. R.C. Davidson, Methods in Nonlinear Plasma Theory, Academic Press, New York, 1972

R.2

References

19. A.B. Mikhalovskii, Theory of Plasma Instabilities, Consultants Bureau, 1974 20. S.C. Brown, Introduction to Electrical Discharges in Gases, Wiley, 1966 21. I.H. Hutchinson, Principles of Plasma Diagnostics, Cambridge University Press, 2002 22. R.H. Huddlestone and S.L. Leonard, Plasma Diagnostics, Academic Press Inc. New York, 1965 23. J.D. Swift and M.J.R. Schwar, Electrical Probes for Plasma Diagnostics, IIiffe Books, 1970 24. H.V. Boenig, Plasma Science and Technology, Cornell Univ. Press, Ithaca, New York, 1982 25. K. Miyamoto, Plasma Physics for Nuclear Fusion , MIT Press, 1980.

qqq

Index

A Adiabatic approximation, 4.3 Adiabatic compression, 10.15 Adiabatic equation of state, 4.4, 6.8, 6.19

Cold plasma, 4.2, 4.3, 4.4 Cold plasma model, 4.3, 4.4, 6.8 Collisionless damping, 6.14 Confinement of plasma, 10.1

inertial, 10.18, 10.19

Alfven speed, 5.16, 5.21, 5.25



magnetic mirror, 3.18, 3.19, 3.20

Alfven theorem, 4.5, 4.13



stellarator, 10.4, 10.16, 10.17

Alfven waves, 5.16, 5.25



tokamak, 2.6, 3.14, 3.33



compressional, 5.17

Continuity equation, 4.3, 4.4, 5.4



shear, 5.16

Controlled thermonuclear reaction, 10.15

Adiabatic invariants, 3.17, 3.28, 4.3

Ambipolar diffusion, 7.6, 7.7, 7.8

Curvature drift, 3.12, 3.13, 3.14

Appleton-Hartree formula, 2.2, 2.7, 2.9

Cut-off frequency, 5.12, 5.14

Avogadro number, A.6, A.8

Cycloidal trajectory, 3.8

B Beam-plasma instability, 5.14, 5.21, 5.25 Bohm criterion, 5.5, 5.22 Bohm-Gross dispersion, 5.5, 5.22, 5.24 Boltzmann constant, 2.5, 11.15

Cyclotron frequency, 3.2, 3.6, 3.17 Cyclotron heating, 7.11, 7.13, 7.16 Cyclotron resonance, 10.4, 10.14 Cylindrical coordinates, 3.19

D

Boltzmann equation, 4.1, 6.1, 6.2

Debye length, 2.6, 2.7, 2.8

Boltzmann relation, 4.13, 5.24

Debye potential, 2.6, 2.11, 2.14

Bunemann instability, 8.1, 8.2, 8.3

Debye shielding, 2.4, 2.8, 2.11

C Carrier wave, 5.25, 7.5 Charge neutrality, 2.3, 2.7, 2.8

Degree of ionization, 2.2, 2.3, 2.16 Diamagnetic current, 4.5, 4.12 Diamagnetic drift, 4.5, 4.6, 4.11 Dielectric constant, 3.16, 3.17, 3.33

I.2

Index

Diffusion, 7.1, 7.5, 7.6 across Bv , 7.11

ambipolar, 7.6, 7.7, 7.8

Diffusion coefficient, 7.6, .7.7, 7.8

ambipolar, 7.6, 7.7, 7.8



fully ionized, 1.3, 7.3, 7.5



partially ionized, 1.1, 1.2

Diffusion equation, 7.8, 7.9, 7.16 Diffusion pump, 11.25

E

G Gas discharge, 1.3, 1.6, 2.10 Gauss’s theorem, 6.5 Glow discharge, 2.18, 10.22, 11.2 Grad-B drift, 3.11, 3.12, 3.13 Gradient drift, 3.10, 3.13, 3.14 Gravitational constant, A.6 Group dispersion, 5.1, 5.5 Group velocity, 5.3, 5.5, 5.6 Guiding centre, 3.3, 3.8, 3.11

Earth’s magnetic field, 3.22, 3.26, 3.30 Ev # Bv drift, 3.14, 3.31, 3.33

Gyrofrequency, 3.2, 3.15, 3.30

Einstein relation, 7.7

Gyroradius, 3.3, 3.18, 3.19

Gyroperiod, 2.8, 2.9

Electric permittivity of vacuum, A.5

H

Electron plasma frequency, 2.9, 5.1, 5.3 Electron plasma oscillation, 2.16, 5.25, 5.26

Harmonics, 9.21

Electron plasma wave, 5.1, 5.4, 5.5

Heat flow equation, 6.8

Electron thermal velocity, 6.17

Heating of plasma, 7.5, 10.14

Electron volt, 2.2



cyclotron, 3.2, 3.3, 3.6



magnetic, 1.2, 1.3, 1.5



neutral beam, 10.15



Ohmic, 7.5, 10.15, 10.17



shock, 10.5, 10.14, 10.18

Electromagnetic waves, 2.18, 3.4, 3.17 Electrodeless discharge, 10.4 Electrostatic oscillation, 5.2, 5.7, 5.22 Electrostatic probe, 2.3, 2.4, 2.5 Electrostatic shielding, 2.4, 2.6, 2.8 Electrostatic wave, 8.10 Energy conservation equation, 4.3 Envelope soliton, 9.17, 9.19, 9.20

Hot plasma, 1.2, 6.4 Hydromagnetic equations, 1.7, 3.8, 3.9 Hydromagnetic waves, 5.20

Alfven waves, 5.16, 5.25

Equation of motion, 3.1, 3.2, 3.5



fast wave, 5.7, 5.20



for a fluid, 4.13



magnetosonic, 5.17, 5.20, 5.25



for an ideal MHD system, 4.10, 4.13



slow wave, 3.14, 3.18

Equation of continuity, 4.2, 4.3, 4.4

Equation of state, 1.4, 4.8, 5.4 Escape cone, 3.22

I

Extraordinary mode, 5.14, 6.17, 7.17

Impact parameter, 7.3, 7.5

Evanescent wave, 1.3, 1.4

Index of refraction, 5.25

F Floating potential, 9.4, 9.5, 9.27 Fluid equations, 3.34, 4.3, 4.6 Frozen-in magnetic field, 4.13

Inertial confinement, 1.4, 2.10, 10.18 Instabilities, 1.3, 7.17, 8.1

configuration space, 2.17, 2.18



velocity space, 6.5, 6.6, 6.7



Index



kinetic, 8.2, 1.1

Larmor radius, 10.16



streaming, 3.29, 5.22, 8.2

Lawson criterion, 10.16



universal, 8.2

Loss cone, 3.22, 3.32, A.15



Bunemann, 2.18, 3.1, 3.9

Lower hybrid frequency, 5.16



gravitational, 3.1, 3.9, 3.10



Rayleigh-Taylor, 8.2, 8.5, 8.6



two-stream, 8.7

Interferometer, microwave, 5.21, 10.22, 2.9 Ion-acoustic wave, 5.6, 5.8, 5.9 Ion-acoustic velocity, 11.17 Ion-cyclotron resonance, A.18 Ion plasma frequency, 11.15, 11.24, A.10 Ion propulsion, 1.5, 1.6 Ion sound speed, 5.7, 5.8, 11.6 Ion wave, 5.4, 5.6 Ionization, 2.2, 2.3, 2.16 Ionization energy, 2.3, 2.16 ionosphere, 1.2, 1.3, 1.4

J

M Mach number, 9.14, 9.27 Magnetic bottle, 3.20, 3.21, 3.22 Magnetic compression, 3.17, 3.18 Magnetic confinement, 10.16 Magnetic heating, 10.15 Magnetic mirror effect, 3.2 Magnetic mirror geometry, 3.18 Magnetic moment, 3.4, 3.17, 3.18 Magnetic probe, 10.11 Magnetic pressure, 4.10, 4.11, 4.14 Magnetic pumping, 11.25 Magnetic tension , 4.11, 4.14 Magnetohydrodynamics, 1.3, 1.7, R.1 Magnetosonic wave, 5.17, 5.20, 525 Mass of electron, 11.23

Joule heating, 10.14

K

Mass of proton, 3.9, 3.16 Maxwell’s distribution, 2.1

KdV soliton, 9.24

Maxwell’s equations, 3.13, 4.3, 4.4

Kinetic description of plasma, 5.27, 6.1

Mean free path, 7.13, 11.10, 11.19

Kinetic pressure, 4.2, 4.10, 4.11

MHD equation, 4.6, 4.8, 4.10

Kinetic temperature, 3.32, 6.1

Microwave, 2.9, 5.14, 5.21

Krook collision model, 2.8, 3.1

MHD generation, 1.5, 1.7

Klystron, 10.11, 10.12

MHD waves, 5.16, 5.18, 5.20

Korteweg Devries equation, 9.9

Mirror ratio, 3.22, 3.25, 3.26

L Landau damping, 5.28, 6.10, 6.11 Landau damping constant Langmuir oscillation, 1.3, 5.25 Langmuir probe, 9.4, 9.5, 9.24

Mobility, 7.1, 7.2, 7.3 Modulational instabilitiy, 8.1, 8.2 Moment equations, 6.18, 6.19 Moments of Boltzmann equation, 6.19

O

Laplace equation, 4.1, 4.2

Ohmic heating, 7.5, 10.14, 10.17

Larmor frequency, 3.2, 3.3, 3.11

Ohm’s law, generalized, 4.8, 4.13

I.3

I.4

Index

One-fluid theory, 1.7, 6.1, 6.10

Positive column, 1, 1.3, 11.4

Orbit theory, 1.6, 1.7, 3.1

Presssure, 10.4, 11.6, A.8

Orbital magnetic moment, 3.18, 3.20, 3.24

Pressure force, 6.7

P Parallel conductivity, 5.19, A.8 Parameter b, 4.11 Parametric instability, 9.1 Particle drifts, 3.20, 3.21, 4.6 Particle trapping, 3.20, 3.22 Paschen’s law, 10.3, 10.21, 11.2 Penning gauge, 11.16, 11.26

Pressure tensor, 4.2, 6.5, 6.6 Propagation vector, 1.3, 1.4 Presheath, 9.4 Probes, 10.8, 10.10, 10.11

double, 6.15, 9.16, 10.8



single, 1.6, 2.2, 4.5

Pseudopotential, 9.15, 9.16 Pump wave, 11.26

Q

Perfect gas, 8.1 Permeabiliy of free space, A.6, A.9

Quasineutrality, 2.3, 2.4, 2.15

Permittivity of free space, 2.5, A.5, A.9

Quasistatic mode, 5.13, 5.14, 5.24

Perpendicular conductivity, 5.20, A.8 Phase space, 1.7, 6.1, 6.2 Phase velocity, 5.5, 5.9, 5.10 Physical constants, A.5 Pitch angle, 3.3, 3.28, 3.30 Planck’s constant, 9.8, A.5, 9.21 Plane waves, 4.14, 6.18 Plasma, 1.1, 1.2, 1.3

R Radio communication, 5.13, 10.4 Radius of gyration, 3.6, 3.11 Random-walk, 7.13, 7.16, 7.17 Ratio of specific heats, 4.4 Rayleigh-Taylor instability, 8.6, 8.7, 8.10 Refractive index of plasma, 5.21 Relative permittivity, A.8



approximation, 1.6, 3.30, 4.1



confinement, 1.4, 2.10, 3.14



cut-off, 5.6, 5.14, 5.24



density measurement, 10.12



diagnostics, 9.4, 10.1, 10.5



dielectric constant, 3.16, 3.17, 3.33



heating, 7.5, 9.28, 10.14



naturally occurring, 2.1, 2.10

Sagdeev potential, 9.14, 9.15, 9.16



parameters, 1.6, 2.1

Saha equation, 2.3, 2.16

Relaxation time, 6.4, A.9 Resistivity of plasma, 10.5, 10.6 RF discharge, 10.5 RF probe, 10.6, 10.7 Rotary pump, 11.2, 11.5, 11.7

S

Poisson’s equation, 2.5, 2.11, 2.13

Scalar potential, 4.2, 4.5

Polarization current, A.8, A.11

Scalar pressure, 4.13, 4.14, 5.4

Polarization drift, 3.14, 3.16, 3.31

Scattering angle, 7.4, 7.5

Polarization vector, 2.14, 3.5, 3.6

Scattering cross-section, 7.5

Pondermotive force, 9.6, 9.8, 9.18

Self-focussing, 9.8



Index

I.5

Sheath, 9.1, 9.2, 9.4

Townsend’s coefficient

Sheath criterion, 9.3, 9.4



first, 1.1, 1.3, 2.6

Shear stress, 5.16, A.5



second, 3.11, 4.9, 5.9

Shock wave, 10.5, 10.14, 10.18

Townsend’s theory, 10.1

Single fluid equation, 4.6, 4.7

Two-fluid equation for plasma, 1.7

Skin depth, 9.18

Two-humped velocity distribution, 2.1, 2.15

Solar wind, 1.2, 1.3, 3.29

Two-stream instability, 8.3, 8.7

Solid state plasma, 7.13 Solitary wave, 9.9, 9.12, 9.13 Soliton, 9.9, 9.12, 9.13

U Universal instability, 8.2

Sound wave, 5.6, 5.9, 5.17

Upper hybrid frequency, 5.16, A.12

Space physics, 1.4

Upper hybrid resonance, 9.21, 9.22

Speed of light, 3.7, 5.12, 5.13 Spitzer resistivity, 7.5, R.1 Stellarator, 10.4, 10.16, 10.17 Stimulated Brillouine scattering, 9.22 Stimulated Raman scattering, 9.21, 9.22, 9.23 Stokes’ theorem, 3.17, A.2

T Temperature, 1.1, 1.2, 1.3 Test charge, 2.4, 2.5, 2.6 Thermal conductivity, 6.8, 7.1, 10.5 Thermal energy, 1.5, 2.3, 2.14 Thermal velocity, 2.10, 4.3, 5.6 Thermonuclear fusion, 1.3, 1.4, 7.10 Tokamak, 2.10, 3.33, 10.17 Toroidal magnetic field, 3.33, 10.17, 10.22

V Van Allen radiation belts, 1.2, 1.4, 1.5 Vector relations, A.1 Velocity distribution function, 6.20 Veocity space, 6.10, 6.11 Vlasov equation, 1.7, 5.27, 6.4

W Warm plasma model, 4.3, 4.4, 4.5 Wavelength, 2.1, 2.18, 5.5 Wavenumber, 11.18, A.7 Waves in cold plasma, 4.3, 6.10 Wave-particle interaction, 9.1 Wave-wave interaction, 9.1, 9.21, 9.27 Weakly ionized plasma, 6.4, 7.1, 7.5