Table Of ContentPhysics of Collisional Plasmas
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Physics of Collisional Plasmas is an improved version of the original book
“Physiquedesplasmascollisonnels”byMichelMOISANandJacquesPELLETIER,
EDPSciences-GrenobleSciences’collection,2006,ISBN9782868838223.
TheReadingCommitteeoftheFrenchversionincludedthefollowingmembers:
• MichelAUBÈS,ProfessoratthePaulSabatierUniversity,Toulouse,France
• Jacques DEROUARD, Professor at the Joseph Fourier University, Grenoble,
France
• AnneLACOSTE,ProfessorattheJosephFourierUniversity,Grenoble,France
and
• CédricDEVAULXandDidierRIEU
TranslationfromoriginalFrenchversionperformedbyGreameLISTER,Scientist
atOSRAMSylvaniaInc.
Typesettedby:DanielleKEROACK,Ph.DPhysics
Coverillustration:AliceGIRAUD
(withextractsfrom:Plasmalight-glow_net,flickr)
Michel Moisan (cid:2) Jacques Pelletier
Physics of
Collisional Plasmas
Introduction to High-Frequency Discharges
Translation by Graeme Lister
Prof.MichelMoisan Dr.JacquesPelletier
DépartementdePhysique CentredeRecherche
UniversitédeMontréal Plasmas-Matériaux-Nanostructures
Montreal,Québec,Canada LPSC
Grenoble,France
Translator
Dr.GraemeLister
OSRAMSylvaniaInc.
Beverly,MA,USA
Originaltitle:“Physiquedesplasmascollisionnels,applicationauxdéchargeshautefréquence”,
MichelMoisanetJacquesPelletier,CollectionGrenobleSciences-EDPSciences,2006
ISBN978-94-007-4557-5 ISBN978-94-007-4558-2(eBook)
DOI10.1007/978-94-007-4558-2
SpringerDordrechtHeidelbergNewYorkLondon
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Preface
During the 1960s, the public face of plasma physics was almost exclusively
represented by plasma confinement, with the goal of developing a reactor
to produce electricity by thermonuclear fusion. Such a reactor is still being
developed, without any guarantee as to its successful achievement, but since
then the applications of plasma physics have increased and diversified: one
of the best known, besides lighting, is etching in the fabrication of micro-
electronic computer chips, for which plasma is indispensable. At present, the
use of plasmas continues to expand and, from recent research publications,
a seemingly limitless number of applications will eventually see the light of
day. In this development, plasmas created by radiofrequency and microwave
fields play a particularly important role.
The present text is basically concerned with plasma physics of interest
for laboratory research and industrial applications, with emphasis on the
understanding of the physical mechanisms involved, rather than on minute
details and high-level theoretical analysis. At the introductory level to this
discipline, it is very important to assimilate its characteristic physical phe-
nomena, before addressing the ultimate formalism of kinetic theory, with its
microscopic, statistical mechanics approach. In this textbook, the physical
phenomena have been translated into more tractable equations, using the
hydrodynamic model; this treats the plasma as a fluid, in which the macro-
scopic physical parameters are the statistical averages of the microscopic
(individual)parameters.Thistextbookisintendedforstudentsintheirearly
years at the graduate level, and for engineers who are interested in applica-
tions. Its level of difficulty lies somewhat below that of JL Delcroix and A
Bers (from Universit´e Paris XI, Orsay and Sup´elec, Gif-sur-Yvette, France,
and MIT, Cambridge, MA, USA, respectively), which provides a series of
complementary and interesting theoretical treatments.
This book is divided into four chapters.
Chapter 1 is the introductory part of the textbook. It begins with a de-
scriptionoftheplasma,anionisedgas,asacollectiveandelectricallyneutral
gaseous medium, followed, for illustrative purposes, by a few selected scien-
v
vi Preface
tific and industrial applications. Then, the fundamental concepts of plasma
physicsareintroduced,withprogressivelyincreasingdetail:thechapteraims
topresentthebasicparametersrequiredtoreachastartingknowledgeofthe
plasma medium, such as the Debye length, the electron plasma frequency,
the various types of collision between particles and their description through
specific cross-sections. The concepts presented in this introduction will be
developed as a first approach, i.e. as the first step in an iterative process, to
becompletedbythedetailedandquantitativepresentationsintheremaining
chapters.
Chapter 2isathoroughexaminationofthetrajectoryofasinglecharged
particle (assuming no interaction whatsoever with other particles), subject
toanelectricfieldE,amagneticfieldB orboth.Inthecaseofelectricfields
E, special attention will be paid to those at RF and microwave frequencies,
designated jointly as high-frequency (HF) fields, in preparation for the mod-
elling of HF discharges developed in Chap. 4. The presence of a magnetic
field B results in a cyclotron motion, encountered for example in electron
cyclotron-resonancedischarges(Chap.4).ThecombinationofE andB fields
in different spatial configurations, and then the inclusion of the spatial inho-
mogeneity of the B field, reveals the so-called drift velocities, which have to
be “tamed” for an efficient operation of Tokomaks, nowadays investigated as
possible controlled-fusion reactors.
In contrast to Chap. 2, collisions between particles are taken into account
inChapter 3,toestablishthehydrodynamicdescriptionoftheplasma,con-
sidered asa fluid. Such adescription isobtained fromthe macroscopicquan-
titiescalculatedfromthedistributionfunctionofthe(microscopic)velocities
of individual particles. The transport equations, i.e. the equations describing
the space-time evolution of these quantities, are obtained from integration
of the Boltzmann equation over the distribution function of velocities. The
concepts of mobility (of charged particles) and diffusion of particles are then
introduced, where free mobility and diffusion tensors are deduced from the
(momentum transport) Langevin equation. Further, it is shown that, under
sufficientlydenseplasmaconditions,thespace-chargeelectricfieldmakeselec-
trons and ions diffuse together in the so-called ambipolar diffusion regime.
Finally, toward the end of the chapter, a first example of a scaling law in
plasmas is developed. Then, in the last section, the formation of sheaths lo-
catedattheinterfacebetweentheplasmaandthewallsisdescribed,together
with a straightforward and original derivation of the Bohm Criterion, which
provides the velocity of the ions as they enter the sheath.
Chapter 4, the last chapter, is dedicated to the mechanisms involved in
HF sustained discharges, which are developed based on an entirely new and
original approach. The key element is θ , the average power lost by an elec-
l
tron through its collision with heavy particles, in this way supplying power
to the plasma. It is shown that θ , the power taken on average per electron
a
from the HF field, adjusts so that θ = θ = θ, i.e. to compensate for the
l a
loss of charged particles. This implies, for instance, that the intensity of the
Preface vii
E field in the plasma is not set by the operator, but by this balance re-
quirement. A further consequence is that, for given operating conditions and
HF power density, whatever the means of supplying the HF field to achieve
the discharge, the θ value should be the same in all cases. The parameter θ
is also instrumental in demonstrating that, contrary to common belief, the
E-field intensity goes through a minimum at electron cyclotron resonance.
The influence of varying the field frequency on the EEDF, and ultimately
on plasma properties, is documented both theoretically and experimentally,
in the case of low-pressure (< 10torr) plasmas. The case of high-pressure
plasmas (including atmospheric pressure) is centred on the phenomena of
discharge contraction and filamentation in rare gases with low thermal con-
ductivity, emphasising the role of molecular ions in these monoatomic gas
discharges. Interrupting the kinetic cycle leading to dissociative recombina-
tion(ofmolecularions)byintroducingtracesofraregaseswithanionisation
potential lower than that of the carrier gas leads to the disappearance of
discharge contraction and filamentation.
In addition to the content of the main text, there are a large number of
remarksandfootnotes,forclarification,ortoqualifycertainpointsmorepre-
cisely.Fortyfiveproblems,withdetailedsolutions,whichareanindispensable
complement to this book, are distributed at the end of the first three chap-
ters.AsetofAppendicesprovidesclarificationsofthesubjectstreatedinthe
main text, together with a number of mathematical developments, and use-
ful mathematical formulae. Finally, an alphabetic index of important terms
is supplied, with a page reference to their first appearance in the text given
in bold type.
Montr´eal and Grenoble, Michel Moisan
January 2012 Jacques Pelletier
Acknowledgements
The authors would first like to congratulate Graeme Lister on his excellent
contribution in translating the content of our textbook, originally written in
(non-basic) French, into (English) English. Born in Australia, Graeme Lister
(Ph.D. on instabilities in plasmas) has been involved since the beginning in
modelling plasmas, first for thermonuclear research, in prestigious institutes
(Princeton, Garching) and, afterward for RF operated lamps, as a senior
scientist in International companies (Thorn Lighting, Osram Sylvania). He
is currently working for CeravisionLimited in the development of microwave
HID lighting. Needless to say that he has an interest in HF discharges!
AnotherinvaluablecontributiontothisbookhasbeenprovidedbyDanielle
K´eroack (Ph.D. in solid state physics) who edited the English manuscript
(afterhavingdonethesamewiththeFrenchtext),whilealsocarryingouther
activities in the laboratory and administrative duties in the Plasma-Qu´ebec
strategic network. Her calculation from basic equations of many figures and
her patience with Springer editorial guidelines are particularly noteworthy.
We are happy to acknowledge the contribution made by several people
to the content of this book. We would first like to thank Professor Ana La-
coste(MasterofplasmaphysicsatUniversit´eJoseph-FourierinGrenoble)at
the origin of a number of developments, particularly in Chapter 4, and for
providing some problems with their solutions. The content of Chapter 4 was
originallydevelopedbyyoungresearchersinplasmaphysics,KremenaMaka-
sheva,YassineKabouziandEduardoCastan˜osMart´ınezaspost-doctoralfel-
lows and Ph.D. student in plasma physics respectively, particularly concern-
ing the atmospheric-pressure HF discharges. The new results on this topic
appearing in the English edition result from the Ph.D. research of Eduardo
Castan˜os Mart´ınez. The content of this book has also greatly benefited from
the many questions and comments from students in the classroom and in
the laboratory. Finally, we were lucky to have a group of kind consultants
and readers. We need to mention the long-running advice provided by Dr.
Antoine Royer, in commenting on the physics of the text and re-writing part
ofit.Finally,wearegratefultoJohn MichaelPearsonandRichardMacKen-
zie, both professors of physics at Universit´e de Montr´eal, for answering our
(almost daily) queries on the English language.
ix
Contents
1 The Plasma State: Definition and Orders of Magnitude
of Principal Quantities.................................... 1
1.1 Definition and essential nature of plasma .................. 1
1.1.1 A plasma behaves as a collective medium ........... 1
1.1.2 A plasma is a macroscopically neutral medium ....... 2
1.1.3 First examples of plasmas ......................... 3
1.2 Areas of research and applications (examples).............. 5
1.2.1 Controlled thermonuclear fusion.................... 5
1.2.2 Astrophysics and environmental physics ............. 7
1.2.3 Laser pumping................................... 8
1.2.4 Plasma chemistry ................................ 9
1.2.5 Surface treatment ................................ 10
1.2.6 Sterilisation of medical devices ..................... 11
1.2.7 Elemental analysis (analytical chemistry)............ 12
1.2.8 Lighting......................................... 13
1.2.9 Plasma display panels............................. 13
1.2.10 Ion sources ...................................... 14
1.2.11 Ion propulsion thrusters........................... 14
1.2.12 Further applications .............................. 15
1.3 Different types of laboratory plasmas ..................... 15
1.3.1 Discharges with continuous current or alternative
current at low frequency .......................... 15
1.3.2 High frequency (HF) discharges .................... 16
1.3.3 Laser induced discharges .......................... 16
1.4 Electron density and temperature of a plasma.............. 17
1.4.1 Range of electron density values in a plasma ......... 17
1.4.2 Definition of plasma “temperature” and the concept
of thermodynamic equilibrium (TE) ................ 17
1.4.3 Different levels of departure from complete
thermodynamic equilibrium........................ 21
1.5 Natural oscillation frequency of electrons in a plasma ....... 23
xi
xii Contents
1.5.1 Origin and description of the phenomenon........... 23
1.5.2 Calculation of the electron plasma frequency......... 24
1.6 Debye length: effect of screening in the plasma ............. 27
1.6.1 Description of the phenomenon..................... 27
1.6.2 Calculation of the potential exerted by an ion in a
two-temperature plasma: definition of the Debye length 28
1.7 Collision phenomena in plasmas .......................... 32
1.7.1 Types of collision................................. 33
1.7.2 Momentum exchange and energy transfer during a
collision between two particles ..................... 36
1.7.3 Microscopic differential cross-section ................ 44
1.7.4 Total (integrated) microscopic cross-section.......... 48
1.7.5 Total macroscopic cross-section .................... 49
1.7.6 Expression for the temperature of a plasma in
electron-volt ..................................... 53
1.7.7 Collision frequency and mean free path between two
collisions ........................................ 54
1.7.8 Average collision frequency and mean free path ...... 56
1.7.9 Examples of collision cross-sections ................. 58
1.8 Mechanisms for creation and loss of charged particles in a
plasma and their conservation equation.................... 64
1.8.1 Loss mechanisms ................................. 64
1.8.2 Creation mechanisms ............................. 66
1.8.3 Conservation equation for charged particles.......... 67
Problems .................................................. 68
2 Individual Motion of a Charged Particle in Electric and
Magnetic Fields .......................................... 101
2.1 The general equation of motion of a charged particle in E
and B fields and properties of that equation ............... 103
2.1.1 The equation of motion ........................... 103
2.1.2 The kinetic energy equation ....................... 104
2.2 Analysis of particular cases of E and B ................... 104
2.2.1 Electric field only (B =0)......................... 105
2.2.2 Uniform static magnetic field ...................... 113
2.2.3 Magnetic field either (slightly) non uniform or
(slightly) varying in time .......................... 135
Problems .................................................. 155
3 Hydrodynamic Description of a Plasma................... 203
3.1 Fundamental aspects of the Boltzmann equation............ 205
3.1.1 Formal derivation of the Boltzmann equation ........ 205
3.1.2 Approximation to the Boltzmann elastic collision
term: relaxation of the distribution function towards
an isotropic state................................. 208
