Advanced course on thermal plasmas : technology and applications, Eindhoven University of Technology, Eindhoven, the Netherlands, June 26-28, 1985

applications, Eindhoven University of Technology, Eindhoven,

the Netherlands, June 26-28, 1985

Citation for published version (APA):

Boulos, M. I. (1985). Advanced course on thermal plasmas : technology and applications, Eindhoven University of Technology, Eindhoven, the Netherlands, June 26-28, 1985. Eindhoven University of Technology.

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THERMA

L P

LASM

AS

T

ECHNOLO

GY

AND

AP

PLIC

A

TI

ON

S

EINDHOVEN UNIVERSITY OF TECH

N

OL

O

GY

EINDHOVEN, THE NETHER

LAN

D

S

JUNE

26-28, 1985

LE

C

T

U

RE

NOT

E

S

VOLUME 1

EDITED BY M.I. BOULOS

THERMAL PLASMAS

TECHNOLOGY AND APPLICATIONS

EINDHOVEN UNIVERSITY OF TECHNOLOGY

EINDHOVEN, THE NETHERLANDS

JUNE 26-28, 1985

LECTURE NOTES

VOLUME 1

Or. Maher I. Boulos

Dr. Pierre Fauchais

Dr. W.H. Gauvin

Or. Emil Pfender

by

Department of Chemical Engineering

Universite de Sherbrooke

2500 Un;versite Boul.

Sherbrooke, Que., J1K 2Rl

CANADA

Laboratoire de Thermodynamique

Universite de Limoges

123 Ave Albert Thomas

87060 Limoges Cedex

FRANCE

Hydro Quebec Research Institute

1800 Montee Ste Julie

Varennes, Que., JOL 2PO

and Department of Chemical Engineering

Meti11 University, Montreal, Que.

CANADA

Department of Mechanical Engineering

University of Minnesota

111 Church St., Minneapolis

Minnesota 55455

U.S.A.

INTRODUCTION

With the continuous change in our available energy resources and the

increasing use of electricity in the chemical and metallurgical industries,

plasma processing is becoming more and more attractive and competitive with

conventional technologies. Even if at present there is only a limited

num-ber of large scale thermal plasma installations around the world, plasma

techno-logy is no longer viewed as an exotic technotechno-logy but rather as the technotechno-logy

of the 80 ies.

This two-day course will aim at introducting the partiCipants to the

fundamental aspects of plasma technology and its applications. It will be

res-tricted to thermal plasmas with emphasis on its characteristics, means of

gene-ration, diagnostic techniques and industrial applications. The course will

co-ver d.c. arcs, transfered arcs, induction radio-frequency plasmas and a.c. plasma

heaters.

While the participants are not expected to emerge from this course as

instant plasma experts, they will certainly be familiar with the fundamental

con-cepts involved and the current state of the art of this technology.

The course will be adressed to scientists and engineers, with a good

general background, who are involved in advanced material processing technologies.

No prior knowledge or experience with plasma processing is required.

THE LECTURERS

The course will be given by an interdisciplinary group of four lecturers

with an applied physics, chemical and mechanical engineering background. They

have all been activly engaged in fundamental and applied research and teaching

in the field of thermal plasmas for a number of years.

and director of the High temperature plasma Laboratory of the University

de Sherbrooke, Sherbrooke s

Qu~bec.

His principal areas of research

inclu-de the thermal treatment of powinclu-ders in induction and d.c. plasmas. He

pu-blished numerous papers in the area of fluid flow and heat transfers

induc-tion plasma modelling and laser-doppler-anemometry under plasma condiinduc-tions.

Dr. Pierre Fauchais:

Professor at the

Universit~

de Limoge and director of the Equipe de

de Thermodynamique et' Plasma of the Laboratoire CNRS, ceramique nouvelle.

He is also director of the Institut de Gestion des Energies, Limoges, France.

Professor Fauchais has been actively involved in research work in the areas

of plasma diagnostics, chemical synthesis and plasma spray-coating. He has

given numerous invited lectures and is responsable for a plasma chemistry

course that is given at the University of Limoges.

Dr. W.H. Gauvin:

Scientific Councillor of Hydro

Qu~bec

Research Institute, and Senior

Research Associate in the Department of Chemical Engineering of McGill

Univer-sity. Dr. Gauvin has a vast research and development experience on both the

academic and industrial levels. His principal areas of research are fluid

mechanics, heat transfer, chemical kinetics and particle dynamics.

Dr. E. Pfender:

Professor of Mechanical Engineering and director of the High

Tempe-rature Laboratory, University of Minnesota, U.S.A. is a physicist by training.

He has lectured at the University of Stuttgart, Germany before coming to the

U.S. His principal areas of research included plasma heat transfer, arc

technology, and plasma chemisty. He authored

num~rous

papers in theses fields,

and he also teaches part of a plasma chemistry courses at the University of

Minnesota.

1.

Fundamental concepts (P. Fauchai s) ••••••••••••••••••••••••• l •••••• 1.1

1.1 Definitions ••••.••.•.•••.••.••.••••••.•••.•.••.••••• l .•••.• 1.1

1 .1 .1 1 .1 .2

Electronic. vibrational and rotational

states of atoms and molecules

Col11c10ns, effective crossections.

kinetic constants, statistical

ap-proach, distribution function

1 .1

1

1.30

1.2

Equilibrium and non-equilibrium p1asmas ••••••••••••• 1 •••••• 1.51

1 .2.1

1.2.2

1 .2.3

1 .2.4

1

.2.5

Plasma in complete thermodynamic

equi-librium

Equilibrium concepts

Transport phenomena

Transport properties

Classification of plasma generating

devices

References

Tables

Fi gures

1 1 1 1 1 1

1.51

1.72

1.95 1.104

1 .108

1 .122

1.130 1 .136

2.

Plasma generation •••••••••••••••••••••••••••••••••••••••••• l •••••• 2.1

2.1

Thermal arcs (E. Pfender) ••••••••••••••••••••••••••• l •••••• 2.1

2.2

2.1

.1

2.1 .2

2.1 .3

Plasma

2.2.1

2.2.2

The arc column, the electrode regions

Electrical Stati1ity

Classification of arcs

2.4

2.44

2.45

torches and arc gas

heaters •••••••••••••••••• l •••••• 2.64

Components of arc gas heaters and p1as-

2.69

ma torches

Classification and performance of arc

1

2.78

heaters

2.3.1

Int roduct i on

2

2.108

2.3.2

Coupling Mechanism

2

2.109

2.3.3

Torch design

2

2.112

2.3.4

Energy balance

2

2.113

2.3.5

Principal characteristics of the induc-

2

2.114

tively coupled r.f. discharges

2.3.6

Mathematical mode11 i ng

2

2.120

References

2

2.151

Figures

3.

Diagnostic techniques •••••••••••••

~

•••••••••••••••••••••••• 2 •••••• 3.1

3.1

Temperature measurement (E.Pfender) ••••••••••••••••• 2 •••••• 3.1

3.1 .1

Emission spectroscopy

2

3.1

3.1 .2

Methods for emitting -absorbing pl as-

2

3.43

mas

3.1 .3

Enthalpy probes

2

3.55

3.2

Velocity measurements (M.I. Boulos) •••••••••••••••••

2 ••••••

3.60

Laser doppler anemometry

2 ••••••

3.61

4.

Applications of plasma chemistry to hemogenous ••••••••••••• 2 •••••• 4.116

reactions (P. Fauchais)

4.1

Chemical equilibrium •••••••••••••••••••••••••••••••• 2 •••••• 4.1

4.2

Reaction kinetics ••••••••••••••••••••••••••••••••••• 2 •••••• 4.4

4.3

Case study ego

NO

synthesis ••••••••••••••••••••••••• 2 •••••• 4.18

References

Figures

Tables

2 2 2

4.57

4.69

4.1

5.

Applications of plasma technology to metallurgical ••••••••• 3 •••••• 5.1

operations (W.H. Gauvin)

5.2.1

5.2.2

5.2.3

5.2.4

5.2.5

5.2.6

Momentum transfer

Pure heat transfer

Effect of mass transfer on heat

transfer

Chemical kinetics

Modell i n9

References

3 3 3 3 3 3

5.3

5.4

5.5

5.6

5.7

5.7

5.3

The quality of thermal energy ••••••••••••••••••••••• 3 •••••• 5.8

5.4

The electric furnace •••••••••••••••••••••••••••••••• 3 •••••• 5.l6

5.5

Plasma furnaces ••••••••••••••••••••••••••••••••••••• 3 •••••• 5.18

5.6

Research on transferred arc ••••••••••••••••••••••••• 3 •••••• 5.25

5.6.1

5.6.2

5.6.3

5.6.4

5.6.5

5.6.6

Apparatus

Measurement techniques, instrumentation

and procedures

Results

Conclusions

Nomenclature

References

3 3 3 3 3 3

5.26

5.32

5.37

5.37

5.51

5.53

5.7

Advanced design of transferred arc reactor •••••••••• 3 •••••• 5.54

5.7.1

5.7.2

5.7.3

5.7.4

5.7.5

Description of design

Typical applications

Engineering considerations

Features of transferred arc reactor

design

Types of transferred arc reactor design

3 3 3 3 3

5.54

5.56

5.57

5.58

5.60

furnaces

5.10 Techno-economics •••••••••••••••••••••••••••••••••••• 3 •••••• 5.71

5.10.1

5.10.2

5.10.3

5.10.4

Oesign and cost of a ferromolybdenum

plant (1982 CAN.

$)

basis

Alternative flow sheets

Sens it i vity analysis

Other factors

3

5.71

3

5.84

3

5.88

3

5.89

5.11

Chemical processing ••••••••••••••••••••••••••••••••• 3 •••••• 5.92

5.11.1

Superheated steam plasmas

5.11.2

The atomized suspension technique

5.11.3

AST applications

5.11.4

Plasmas as heat sources

3 3 3 3

5.92

5.97

5.99

5.100

5.12 Appendix 1 •••••••••••••••••••••••••••••••••••••••••• 3 •••••• 5.114 5.13 Appendix 11 ••••••••••••••••••••••••••••••••••••••••• 3 •••••• 5.121

5.13.1

Appendix II. The plasma gasification

of peat

5.13.2

Preliminary cost estimate of methanol

production from peat

3

3

5.122

5.127

6.

Plasma spray coating (P. Fauchais) ••••••••••••••••••••••••• 3 •••••• 6.1

6.1 Introductlon •••••••••••••••••••••••••••••••••••••••• 3 •••••• 6.1

6.2 Spraying devices •••••••••••••••••••••••••••••••••••• 3 •••••• 6.2

6.3

Plasma sprayed coatings ••••••••••••••••••••••••••••• 3 •••••• 6.9

6.4

Plasma sprayed materia1s •••••••••••••••••••••••••••• 3 •••••• 6.20

6.5

Industrial use of plasma spraying ••••••••••••••••••• 3 •••••• 6.24

6.6

Controls with plasma spraying ••••••••••••••••••••••• 3 •••••• 6.29

References

Figures

3 3

6.35

6.39

P.

Fauchais

Laboratoire de Thermodynamique

Universit~

de Limoges

1. FUNDAMENTAL CONCEPTS (P. Fauchais)

The plasmas used for chemical reactions contain usually : neutral atoms and molecules, ions, electrons and photons. The desired atomic or molecular species are obtained through the various interactions that the neutral species undergo with energetic electrons and/or photons. To get a detailed understanoing of these interactions, various concepts have been introduced to specify the laws that govern the elementary processes and to determine the various interactions. Collisions are the basis of most of the processes and many of the concepts are similar to those used in kinetic theory of gases. However this theory has to be modified to describe the experimental results observed and the exact treatment re-quires rigorous application of quantum mechanics that however, is generally too complicated to perform. Fortunately, a number of very general and simple principles that govern the collisions are helpfull:

- conservation of energy (kinetiC, internal or photon) - conservation of momentum

- conservation of angular-momentum - conservation of charge

- conservation of mass.

The macroscope properties of a plasma depend strongly on the mi-croscopic interactions beetween the particles present in the plasma, which particles have a distribution of velocity or energy. The macroscope properties are then rela-ted to the microscopic processes averaged over velocity distributions and to the distribution and exchanges of energy between the different species present in the plasma : equilibrium concept.

Thus we will deal first with the elementary phenomena and concepts before the discussion of the concept of thermodynamic equilibrium and his conditions.

1.1. DEFINITIONS AND ELEMENTARY PROCESSES

The purpose of this section is to introduce the main definitions used in plasma physics and chemistry. But before defining the particle

we have to summarize briefly the units used. The most commonly unit used by pla:"--3 physicjstsis the electron-volt: 1eV is the energy gained by an electron when passing through a potential difference of 1 Volt,it follows that

1eV 1.6 x 10-12 ergs 1.6 x 10- 19 Joule

An other energy is often used by the spectroscopists. The deexci-tat ion of an excited atom or molecule from an energy E2 to an energy El leads to the emission of a photon with the energy hv, h being the PlancK constant and v the frequency of the corresponding electromagnetic associated radiation. This corres-ponds to the wave length

A

of the emitted line, the energy of the photon being given by

hc

e = A hcO' ( 1 )

Most of the spectroscopy tables are expressed in 0 units

(inde-pendent of the accuracy with which the velocity of light is known)that is to say in cm- l • The corresponding energies are then

1 cm- 1 = 1.9855 x 10- 16 ergs = 1.23941 x 10-4 eV

1.1.1. Electronic, vibrational and rotational states of atoms and molecules

The microscopic quantities we are going to describe now are related to individual particles

1.1.1.1.1. Definitions

The internal energy states wich a particle can occupy result from wave mechanic : SchrBdinger equation. This statistical treatment tells us what is the probability to find the particle in a certain elementary volume of the space and describing the electrons in an atom corresponds to a certain charge distribution. The wave functions statisfies the Schradinger wave equation which is a partial differential equation.

Applied to the problem of an electron moving in the field of a proton, it turns out that physically acceptable solutions for the waves functions exist only for specific integer values of the three quantum numbers /1/

- the principal quantum number n = 1, 2, 3 ' 0 0

- the angular momentum quantum number 1 = 0, 1 "" (n - 1) - the magnetic quantum number m

l

=

1, (1 - 1) •• , - (1 - 1), - 1

• The principal quantum number fulfills the same role as in the classical quantum theory of Bohr

• The angular momentum quantum number 1 determines the angular momentum of the electron

• The magnetic quantum number m

l determines the component of the angular momentum along a specifiea direction : magnetic or electric field.

However the fine structure of the spectral lines emitted by atoms shove that an additional phenomena has to be taken into account : the electron angular momentum or spin that could stand parallel or anti-parallel to the angular momentum of the orbital motion, i.e., that the spin quantum number is s • ±

~.

The orbital angular momentum of an electron in an atom is asso-ciated with the angular quantum number 1. The actual orbital angular momentum

corresponding to 1 is / 1(1+1) 2: where h is the Planck'sconstant (6.62 10-34J.sec). Electrons having 1 values of 0, 1, 2, 3, 4, 5, etc are designated s, p, d, f, g, h -electrons respectively. Thus, a d-electron is one having

16

units of angular momentum,

h

and the shells are numbered 1, 2, 3, 4, etc •••• number 1 being the innermost shell. The quantum number n is used to designate these shells in general. An electron in an

atom is designated by the two integer nUr:1bers nand 1. A 5p-electron for example deno-tes an electron occupying the fifth shell and having

12

units of angular momentum.

Electrons having the same values of nand 1 are called equivalent electrons ; if there are r equivalent electrons in an atom, they are denoted by nlr. Thus, 6d 2 denotes 2 electrons in the 6th shell, each having

16

units of an-gular momentum.

The quantum number I can take all integral values from 0 to n-1. Therefore there can only be s-electrons in the first shell. The second shell can contain s- and p-electrons and so on.

The electron configuration of the simplest atom, hydrogen, in its ground state is deSignated 1s showing that the electron occupies the first shell and possesses zero orbital angular momentum. A much more complicated atom, copper. has the following ground state electron configuration : - 1s2 2s2 2p 6 3~2

3p6 3d10 4s denoting

2 s-electrons in the 1st shell 2 s-electrons in the 2nd shell 6 p-electrons in the 2nd shell 2 s-electrons in the 3rd shell

6 p-electrons in the 3rd shell

10 d-electrons in the 3rd shell 1 s-electron in the 4th shell

The maximum number of electrons in a given shell is limited by Pauli's exclusion principle. This principle is understood on introducing the quan-tum numbers m_l and s characterising the electron. The number m_l can take integral values only and only those values from -1 to +1. Thus. for a given 1. there are 21+1 possible values of mI' Hence. the total number of atomic orbital states cor-responding to a given value of n is equal to

n-1

2 E (21 + 1)

=

2 n2

1=0

The remaining quantum number associated with an electron is tr! number s denoting its spin angular momentum (not to be confused with the letter designation appropriate to 1=0!) The spin angular momentum associated with s is

IS(S+1)2~'

The quantum number s can take only the two values s

=

±

t.

Thus, an electron is designated by a quartet of numbers

=

n, 1, ml and s. Pauli's exclusion principle states that no two electrons in an atom can have all four numbers the same. Thus for an s-electron, there is only one value of ml' viz., ml

=

0 and the spin quantum number can take the two values ±

! .

Therefore, there can never be

L.

more than 2 s-electrons in a shell. For a p-electron, ml can take the 3 possible values -1, 0 and + 1 and there are 2 possible values of spin associated with each

value of ml, giving a maximum number of 6 p-electrons. Combining this scheme with the fact that in the nth shell 1 can take all integral values from 0 to n-1, we may 'fill up' each shell in turn to give 1s1 2s2 2p6 3s 2 3p6 3d 10 ••• etc.

In the argon atom, the shells are completed up to third and the electron configuration is therefore written 1s2 2s2 2p6 3s 2 3p6 3d 10 • The copper atom on the other hand has all shells completed up to the third and has one elec-tron in the fourth shell. This difference in the configurations of the argon and copper atoms radically affects the spectrum of each atom. Table 1 gives the elac-tron configuration of the first 36 elements.

One introduces also the total angular momentum quantum number j

=

I ±

i

associated with the total angUlar momentum Ij(j+1)

2~

• When a magnetic field is applied the total angular momentum vector of the electron orbital can have

(2j+1) possible orientations in the magnetic field, i.e., mj = j.(j-1) ••• -(j-1),-j.

The knowledge concerning the electron configuration of atoms is very important because the chemical properties of atoms are determined mainly by the electron configuration, Thus, elements such as the alkali metals (Li, Na, K, Rb and Cs) which have one outer-electron, have all the same chemical behavio~r,

1.1,1,1,2. Energy transition

The transition of an electron from an energy level El to another level EZ is associated with the emission or absorption of electro-magnetic radiation of a frequency v such that :

The permitted transitions are those given by the selection rules account for obser-ved spectra of atoms. The lowest energy state of the atom is the ground state. The other energy states, although still bound to the particle are the electronically excited states. Their lifetime is generally short (10- 8 to 10- 6 s), A special type of short living excited state is the resonance excited state for which the probability of a radiative transition to the ground level is very high. Photon with the corresponding energy gap will be absorbed by the particles with a very high efficiency. When the electron~ after radiative deexcitation or electronic excitation, occupy an energy level for which the transition rules forbidden the radiative transition to the ground level, the lifetime of this state, called metastable state, is very large (greater than 1 s sometimes). At least when one or more elec-trons are removed completely from the particle, the particle is in an ionized state.

1.1.1.1.3. Designation

or

electrons conriguration

A nomenclature is used to identify atomic energy levels. The nomen-clature adopted describes, in most cases, the nature of the couplings between the electrons that are effective in producing the spectrum. The core of completed atomic shells has zero angular momentum (neglecting any angular momentum of the atomic nucleus) and hence only electrons in unfilled outer shells need to be considered. (We must include those situations in which an atom with completed shells, such as argon, is excited so that an electron from a completed shell now occupies an outer, incom-plete shell).

Electrons interact with each other through the coupling forces arising from :

(1) electrostatic repulsions between electrons.

(ii) magnetic fields arising from both the orbital motion of the electrons and spins of the electrons,

(iii) exchange forces between electron spins (these forces are understood only on a quantum-mechanical basis).

Both the energy levels of the atom and the strengths of transitions are dependent upon the nature and magnitude of these interactions.

The potential energy of an atom is determined by those of the various electron orbitals. To each individual electrons an orbital motion quantum number Ii and a spin quantum number si (si

=

1)

are assigned and then the various interactions are taken into account either with the L-S coupling or the J-J coupling.

In L-S coupling, the total orbital angular momentum is formed by

coupling together the orbital angular momenta quantum numbers of the separate elec-trons to form a total orbital angular momentum characterised by the quantum number L where L can take the possible values

L

and so on down to the smallest possible number, generally zero when there are more than two electrons. (For two electrons, the smallest possible number obtained by combining the angular momenta of the electrons is /11 - 121). The magnituce of the orbital angular momentum corresponding to L is {L(L+1)

2~'

The total spin momentum is formed by coupling together the spin momenta quantum numbers of the separate electrons to form a total spin momentum characterised by the quantum number S where S is given by

1 1

and each of s1, s2. s3 etc may take the value

:z

or 2' The possible values of S are S = / s11 +

I

s21 + / s31 + ••• I / s11 + / s21 +

I

s31 + . , . -1, ••• down to 0 or

1

2

according as there is an even number or an odd number of electrons. The magni-tude of the spin angular momentum corresponding to S is

IS(S+1)2~'

In L-S coupling the total orbital angular momentum and total spin momentum are coupled by weak magnetic forces to form a total angular momentum cha-racterised by a quantum number J where

J '" L + S

The magnitude of the angular momentum corresponding to the quantum number J is

J may taKe the values

J

=

L+S. L+S-1, L+S-2. '0' L-S when S<L and

J • L+S, L+S-1, L+S-2, '0' S-L when S>L

In the first case (S<L) there are 2S+1 possible values of J in the second ~na. there are 2L+1 possible values of J.

2S+1 is called the mulciplicity.

A given L together with the multiplicity defines a spectral term A given J of a given term defines a spectral level.

Spectral lines arise from transitions between levels. All possible levels of a given term is called a multiplet.

Designation of L-values

The designations of the total orbital angular momentum quantum number are analogous to the designations used for orbital angular momenta of sin-gle electrons except that capital letters are used instead of small letters. The total orbital angular momenta designations are the following

Values of L 0 1 2 3 4 5

S P D F G Capital letters then in alphabetical order

DeSignation of terms

A term 1s defined once the L-value and the multiplicity have been given.

The term designation is formed by placing the number denoting the multiplicity (2S+j) as a left-hand superscript on the L-value designation. For example, suppose S

=

~

and L=i2, The multipliCity 1s given by

2S+1 2 X 1 _{2 +} 1

'" 2

The designation appropriate to L=2 is O. Therefore, the term is designated 20 (pronounced 'doublet dee'),

Designation of levels

A level, coresponding to a given term and given J-value is desi-gnated by adding the J-value as a right-hand subscript to the term designation.

1

Taking the example above of S = ~ and L '" 2 there are two possible J-values given by J _'" 2 + 1 2

.1.

2 2 and J 2 + 1 1 = 1 1 ~ 2

Therefore the two levels of the 0 term are designated 202~ and 201±, This is shown diagrammatically in figure

1.1. We may think of the term as being split into

two energy levels by the action of electron spin. Levels can be f~rther split

into a number of states, the number of states of a given J being 2J+1. This split-ting into 2J+1 states occurs in the presence of a magnetic field. 2J+1 is often referred to as the statistical weight of the level ; this quantity is inportant in deriving theoretically the radiative power in a spectral line.

We now consider a fUrther example of level designation. Consider an F-term (i,e, L=3) with two electrons, The possible values of the total spin quantum number are given by

S 1 1 j

'2

+ -2

=

and S

₌

1 1 1 0

'2

+ 2 '" (i) S '" 0

The multiplicity is given by 2S+1 '" 2xO+1 '" 1. This results in a singlet term, designated IF. Only one value of J is possible, given by

J L+S

=

3+0

=

3 Therefore, there is only one level designated If3.

( i l l S = 1

The multiplicity is given by 2S+1

=

2x1+1

=

3. This results in a triplet term, designated 3F. Three J-values are possible, given by

J ~ L+S

=

3+1

=

4 J

=

L+S-1

=

3

and

J

=

L-S 2

Therefore, there are 3 levels designated 3F~. 3F3. 3F2. Provided S<L, then the number of levels of a term is equal to the multiplicity. If S>L, then the number of levels is equal to 2L+1.

Designation of parity

The parity is odd or even according as

!

Ii is odd or even where

r

Ii denotes the sum of the oroital angular momenta quantum numbers over all the electrons in the atom.

For closed shells the total number of electrons is even and there-fore the parity must be even for closed shells. Therethere-fore, we need only evaluate

r

Ii for electrons in incomplete shells.

Odd parity is designated by

a

superscript 0 to the right of the term symbol, e.g. 2 pq.

Selection Rules for dipole radiation in the case of L-S coupling

In a dipole transition. the following selection rules apply in strict L-S coupling :

1. The parity must change.

2. The multiplicity must remain unchanged, i.e. intercombination lines are forbidden.

3. J must change by ± 1 or 0 except that the transition J a 0 ~ J

=

0 is not allowed

4. L must change by ± 1 or 0 except that the transition L

=

0 ~ L = 0 is not allowed.

It should be noted that rules 1 and 3 hold rigorously. whatever the coupling scheme; in addition rules 2 and 4 will hold in strict L-S coupling.

Example taken from the copper spectrum

The copper spectrum in an important one that often occurs in electrical discharge plasma due to trace amounts of copper from electrodes (Miyachi et ale 1972). The copper spectrum arises from doublet and quartet terms.

The ground state of the copper atom is deSignated

1S2 2s2 2p6 3s 2 3p6 3d10 _{4s 2S1}

"'2"

Since the shells up to 3p do not play any significant part in the formation of the spectrum this designation is abbreviated to

Since the shells up to 3d are complete, only the outer e-electron contributes to

1

the total angular momentum of the atom. For this s-electron, 1=0 and s-2' Since 1

there is only one electron in this outer shell, LaO and S-2' giving multiplicity 2S+1

=

2, leading to 2_St in the ground level designation.

Consider now the outer 4s-electron excited to 4p and 5d orbits. Using the arguments developed above, this leads to 2pO and 20 terms respectiveij. The 2pO term is then split into the two levels 2pg/2 and 2P

t

and the 20 term is split into the two levels 205/2 and 203/2' Applying the selection rules gives rise to three possible transitions as shown in figure 2. If doublet transitions involve an S-term, only two transitions are possible as shown in figure

1.3. There are im·

portant when there is good, L-S coupling.

One of the electrons of the completed 3d shell can become exci-ted to give an electron configuration :

••• 3d9 4s nx

where n>4 and x represents s, p, d, etc •••

The core now has a residual spin of

~

which can couple with the spin

~

of each of the 4s and nx electrons to give a total resultant spin

1

+

-2

..

3/2 2S+1 4

or

..

1/2

.

.

2S+1

..

2

That is, we now get terms of multiplicity 4 and 2 (quartets and double~s).

The maximum number of transition between two quartet terms is 9 as shown in figure

1 .4.

In this coupling scheme the magnetic interaction between the or-bital angular momentum and the spin angular momentum for each electron becomes dominant compared with the electrostatic and exchange interactions between diffe-rent electrons, Therefore, for each electron. the orbital and spin angular momenta couple stron~ly, giving a total angular momentum quantum number,

J

i, for the ith electron given by

..

1

The total angular momentum quantum number J for all the electrons can take the possible values J

=

jl + j2 + j3 + ••• , J

=

jl + j2 + j3 + ••• - 1,

J

=

j} + j2 + j3 + ••• - 2, and so on down to the smallest possible number. Consider for example a ps electron configuration.

For the s-electron. I} :z 0 _{jl 11}

±.1.

1 J

..

2 2

12 = 1 J _j2 !" 12

±.1.

= 1

±.1.

= 1-1 or -1

2 2 2 2

For the p-electron,

The following example is for tin which in its unexcited state has an outer shell (n 5) configuration 5s25p2. One of the p-electron is excited to the 6s state and the excited electron configuration of tin Is abbreviated to 5p6s. The 5p-electron and 5s-electron exhibit strong j-j coupling. The following table gives the J-values together with the designation :

J-value Designation 3 1 2 [5 P¥2 6sV2 )

"2

+ -2 '= ₂ 3 1 1 (5P

%

6S%J

-

..

2 2 1 1 1 , - +

"2

..

1 [5P1/2 6S%) 2 ₁ 1 1

_a

[5PY2 6S%] 0

"2

= 2

The designation is self-evident. being of the general form (5 P_j2 6S j1)J'

No coupling is rigorously L-S or j-j but many atoms can be more, or less adequately described within the framework of one scheme or the other. HowEver. when we come to the noble gases, neither coupling scheme proves adequate and it becomes necessary to introduce intermediate coupling.

The argon spectrum can only be described adequately using an intermediate coupling scheme and therefore we shall focus attention on this atom. This is probably one of the most important spectra encountered in plasma research as many industrial-type plasmas are either formed from pure argon or centain argon mixed with other gases, The reason for this is that it is cheap and readily ioni-sed, Argon is treated theoretically on the basis of intermediate coupling but this cannot be used to give us a simple spectroscopic notation. In order to designate atomic levels the j-l coupling scheme is used which approximates to the intermediate coupling for argon.

The ground state of argon is designated

The level designation IS arises because we have completed shells, giving zero

°

orbital angular momentum (S-term). zero spin giving a singlet-type term and there-fore zero total angular momentum (J ~ 0). Since the shells are complete the

parity is even,

One of the outer p-electrons can be excited to 4s, 4p, e • • 5s,

1

Sp, etc, •• orbits, leaving a core with residual electron spin

2

and electron configuration 152 2s2 2p6 3s2 3pS, This core represents the ion formed when an

electron is removed frOm the argon atom and is referred to as the parent ion of the argon atom.

and

For the ground state of the parent ion :

(i) the total orbital angular momentum is 1, i.e. L = 1 J

1

(ii) the residual electron spin is

2'

leading to a doublet-type term

(iii) the parity is odd,

leading to Therefore. 2pO ₃ and 2 J L + S. L + S - 1, etc ••• 1 + 1 = 3/2 2 J = 1

_{• 2"}

1 = 1/2

there are two levels of the ground term of the pr~rent ion designated

'2 _{P l ' These two possible levels for the ground term of the. parent ion}

°

2

give rise to two possible series limits for the argon spectrum. This leads to a large number of possible transitions.

The j-l coupling scheme which proves to be the most appropriJte for designating the atomic levels in arranged as follows

(i) the total angular momentum of the parent ion is coupled with the orbital angular momentum 1 of the excited electron to give a quantum number K.

(iil the quantum number K is then coupled with the spin of the excited electron to give the total angular momentum J.

The level is designated

where n is the main quantum number of the excited electron. The prime denotes ai level belonging to a series converging on the higher limit of the parent ion

(i.e. 3 p 5 2Pi).

2

In addition, if the level has odd parity an upper right-hand superscript 0 is appended to the level designation.

The different electronic configurations and the corresponding energies (in cm-I ) are summarized in the tables of C.E. Moore /3 to 5/.

If a particle consists of more than one atom, the problem is much more complex and we will limit our discussion to diatomic molecules XV. On the

figure 1.5 a one can see that each atom X and Yare surrounded

by

a cloud of

electrons and the two centers at X and Yare bound together by outer shared electron orbitals. The balance of electrostatic forces involved results in a finite separation, r, between the two atomic centers. They can thus vibrate in a r - dependent force field about an equilibrium internuclear separation. The nuclear "dumbbell" can also rotate about an axis orthogonal to the nuclei axis. Therefore, aside from electronically excited state of X and Y, vibrationally and rotationally excited energy states are possible. The resolution of the equa-tion of Schradinger shows, as for the atoms, that only discrete energy levels are allowed but this resolution is much more complex than for the atoms due the fact that the electrostatic forces vary with rand that, for each electronic orbi-tal, potential energy curve must be determined. That is why, as for the atoms, we will use the momentum description of the molecule.

The differences between the energy levels (see

figure 1.5 b) are

very significant: a few eV between electronically excited states, about 0.1 eV between vibrational levels, about 0,01 eV between rotational levels. If one of

the atoms of the molecule is excited and the resulting configuration is stable, then an electronically excited molecule with associated vibrational and rotational le-vels is obtained, The spectral emission of these excited molecules consists of spectral bands due to electronic transitions between vibrational levels associated with the various possible potential curves, each line corresponding to the rota-tional levels associated with vibrarota-tional levels.

1.1.1.2.1. Classification of the electronic states of a diatomic molecule

Each electronic state of a diatomic molecule is defined by a quantum number A and by the multiplicity.

(i) The value of the quantum number A is associated with the projection on the internuclear axis of the orbital electronic angular momentum~. (In the case when L is defined. A takes the values A

=

0, 1. 2 •••• L.l.

The state of a molecule is designated according to the val~e of

A

as indicated in the following table :

A 0 1 2 3

STATE

I

IT t:. 41

When different electronic states of the same molecule have the same value of A, then Greek letter

(I.

n.

t:., 41, ••• ) is preceeded by a Roman capital letter, the ground electronic state being designated by X. the others by A, B. C •••• in order of increasing energy,

(ii) The multiplicity is given by 2S+1 where S is the quantum number asso-ciated with the total spin electronic angular momentum. The projection of the total spin angular momentum on to the internuclear axis is denoted by a quantum II: _

number

l..

2

can thus take the values S. S-1. S-2 ••••• -So Therefore the

total number of possible values that

2-

can take is equal to 2S+1. which is called the multiplicity of the electronic state. The multiplicity is indicated by a super-script to the left of the Greek letter denoting the value of A. For example, B

3n

denotes the electronic state with A

=

1 and S = 1.

(i) Case (a)

In this case the interaction of nuclear rotation N with the electronic angular momenta (~and ~) is very weak 171. 18/.

Taking account of the important magnetic field associated with

A,

produced along the internuclear axis by the rapid precession of L. the magnetic

interaction of the vectors S and A is strong. ~ coupled very strongly with the internuclear axis precesses around it and makes with it a constant angle in such a way that the axial component of ~ is quantized.

by

Lilli

takes the 2S+1 values :

',.

L.

=

-S, .,., S-1, S

The total electronic angular momentum

n

along the axis is given

,III

n

= A + L with

For given values of A and S. n takes 2S+1 values

n

= (A - S) • • • • (A + S)

n 1s compounded vectorially with the angular momentum of rotation of the nuclei N (not nuclear spin) to form a resultant angular momentum for the molecule

J

= n

+ N

l.

of constant amplitude, has a fixed direction, around which ~ and

g

(as well as the internuclear axis itself) precess much more slowly than do

h

and ~ about the internuclear axis.

For a given

n,

J takes the values

J =

n, n

+ 1,

n

+ 2. • ••

Levels with J <

n

dot not exist.

(ii) Case (b)

As in case (a), ~ precesses rapidly about the internuclear axis of the molecule and! is quantized , but the magnetic field associated with A is so weak that the interaction between hand S is small compared with the effect of rotation of the molecule on the spin. ~ is no longer coupled with the axis,

therefore

I-

does not exist and Q is not defined.

A and the orbital nuclear moment ~, which are parallel and per-pendicular respectively to the internuclear axis, form a resultant K with

K

=

h + N

around which h and ~ precess,

The corresponding quantum number K can take the integral values

K

=

h, A + 1, A + 2,

.t.

K and the resultant of the spins ~ form the total angular momentum

2

in a fixed direction around which they precess slowly compared with the molecular rotation.

J K + S

For a given values of K, J can take the values

J

=

K + S, K + S - 1, t'" (K-- S) Thus, each level K is composed of 2S + 1 sub-levels.

See

figure 1.5 d for the position of all these vectors.

(iii) Intermediate case

All the

I

states (h

=

0' come under case (b), whil~t all other states (A > 0, S > 0) may be either in case (a) or case (b) according to whether the coupling of ~ with! is very strong or very weak compared with the coupling of S with the rest of the molecule.

If none of the couplings can be neglected. the states will be in intermediate coupling cases.

As J increases, the rotational speed of the molecule, which is small compared with the speed of precession of ~ around ~ (case (a)). becomE~ comparable with it. then the influence of molecular rotation becomes predominant

(case (b)) 16/. 19/.

S is decoupled from the internuclear axis in order to couple with

K "" A

+

N

as J increases,

The coupling force between the spin S and the angular momentum A

Ae may be characterised by the coupling constant Y =

-Bv where A =; h e 8'1f 2I AC (3) and B _"" h (4) v 8'1f2I BC

tA represents the moment of inertia of the electrons rotating freely about the internuclear axis (dumb-bell model 16/. p. 117)),

IB

represents the moment of inertia of the molecule about an axis perpendicular to the internuclear axis, i.e. :

h

B ""

-v 8n 2 Cll

(~)

(5)

where r is the mean distance of separation between the two nuclei, the molecule being in the vibrational state defined by the quantum number v,

Notethat _Bv may be expressed by 161

B

=

h - a (v +

1)

v

8nclJr2 e 2 e

where curve first

r is the distance between the two nuclei at the minimum of the potential

e

and a is a tabulated coefficient. (The expression for B is limitea to the

e v

order in vl.

For example for nitrogen B ~ 1.826 em-I, e

Be from which :

a_e 100 and therefore B V· ~ B • e

However for certain molecules Bv can become negative, Naturally in general

or

A » B

Kovacs /10/ gives the general rule;

The electronic state of a molecule belongs to case (a) i f

y » J(J+1) and belongs to case (bl if Y « J(J+1).

However Shemansky and Jones /11/, in their study of the tran~i­ tion B 3IT 4 A 3

L

+ of N2' assert that the 3

rr

state in fact belongs to an

interme-g u

diate coupling case between Hund's cases (al and (bl, Their assertion agrees with the distinction between Hund's cases (a) and (bl established by Budo /12/. In effect. according to him, a state belongs to Hund's cases (a) for Y » J. The more J increases, the less this condition is fulfilled J this is the transition

from case (a) to case (b) and eventually the state belongs to Hund's case (b) for J » y,

Note

If Y is positive the electronic state is said to be normal or regular (this is the case for the state B

3rr

of N21.

(iv) Other coupling cases

Other cases (c) and (d) come between the cases (a) and (b). Case (cl, encountered mostly for very heavy molecules (Xe2 for example), corresponds to a very strong interaction between Land S, which leads to the formation of a

-

Case (d) represents the limiting case where the cou~ling between L and the internuclear axis is very weak or even non-existent. The angular mom8ntu~ of the nuclear rotation is then quantized separately and L is coupled to it by forces which are essentially gyroscopic in origin.

c) §~~~~~~_e~ge~~~!~~_~f_r~~~~!2Q~!~!~~~!~_~Q~_~:~~~e!~~~_l_~~~~~~9!~~~ ~~!~~~!~_l~~_~~!~~~!~l

When the coordinates (x_k' Yk' Zk) of all the particles (nuclei and electrons) of the molecule are transformed to (-x_k' -Yk' -Zk)' the complete wave function may either remain unchanged, in which case the rotational level is said to be positive, or else the wave function changes sign is which case the rotational level is said to be negative 116/.

In the case of a homonuclear molecule, when the coordinates of the two nuclei alone are interchanged, the complete wave function may either remain unchanged in which case the rotational level is said to be symmetric, or else the sign char.ges and the rotational level is said to be anti-symmetric 16/.

(i) Case when A ~ 0

l -

_state

In this case there is no internal magnetic field resulting from the motion of the electron.

I-

is not defined and the state is not degenerate.

The electronic wave function W of a non-degenerate state can

e

remain unchanged or can change sign by reflection in the plane passing through the two nuclei. In the first case a

Y·

state results J in the second case a [-state.

In the case of a

,+

_L. state, where ~ _e remains unchanged, the rota-tional levels are positive or negative according as J is even or odd.

The even levels are either symmetric or antisyrrmetric. I f the

IS'Yen levels are symmetric and the odd levels antisyrrmetr.ic

state,

Y;

(g denotes gerade in German).

If the odd levels are symmetric and the even levels anti3yrr::-~-:':-i:::. then one has an odd state.

I+

is a state for which the even rotational levs:3 ~re

u

positive and antisymmetric and the odd rotational levels are negative and sYrT,:7;::otric. In the same way, for a

I-

state, the rotational levels are ~~si­ tive or negatlve according as J is odd or even. The g and u states are defi~ed in the same way as for

I+.

(ii) Case when h 1 0

rr,

li., ••• states

In this case, there is an interaction between the rotation of the nuclei N which creates a magnetic field and the rotation of the electrons l. This produces a separation into two components of each level J. This phenomenon is called h-doubling.

The rotational levels are then degenerate and for each value of

J. one level is positive and the other negative. It is possible to define odd Dr even electronic states inthecase of homonuclear molecules as for

L

states. The symmetry rules concerning the exchange of nuclei are the same for an even electro-nic state, the positive rotational levels are symmetric and the negative levels are antisymmetric whilst for an odd electronic state. the positive rotational levels are the negative levels are symmetric,

B

3rr

state for NZ

g

This concerns a 3

rr

state and therefore

A:

1. When ~ is coupled with the internuclear axis. i.e. for low values of J, then 25 + 1

=

3 (therefore 5 = 1),

I-

is defined and Q

=

h +

I-

exists.

I-

and 0 can take the values:

~,- :;

l, 1, O. 1

and

The groups of J- levels are therefore defined, 3ITo, 3ITl and 3IT2 where the subsc~~~t corresponds to the 3 values of g. Each is composed of levels J = g, g + 1, g + 2, taking account of the rotational angular momentum of the nuclei ~ (figure 6 shows the relative positions of the rotational levels in Hund's case (a)) which adds vec-rorially to g.

When the rotational speed of the molecule increases. for the lar-gest values of J say, g is no longer defined (cf. b (iii)) and strictly the rota-tional levels J can no longer be considered grouped as 3ITo, 3ITl and 3IT2 •

In reality, each rotational level K, corresponding to K is devided into 3 sub-levels J

=

K - 1, J = K. J • K + 1.

A

+ ~,

One could postulate, taking account of the numerical value of Y

measuring the coupling strength between ~ and A (cf. b (iii)) that the electronic state 3IT strictly belongs to Hund's case (b) beyond J = 15 /9/.

Hund's case (b).

3

Figure

1.7 shows relative positions of levels of the state

~

in

The A-doubling is such that the energy interval between two com-ponents of a level J is of the order of 1 cm-1 _{/6/, /7/, /8/, /13/.}

Taking into account this doubling, the parity, and the symmetry (cf § c (ii)) there are then 2 series of levels, a series c and a series d, for each of the series 3IT_g• The ~eries denoted by c is a series of levels for which the smallest value of K corresponds to a positive level or a group of positive levels and series d that corresponding to negative levels.

In a c-series, the levels are alternately + - + - and in a d-series. they are alternately - + - +.

Furthermore, for a given level J, if the posit~ve sub-level is symmetric, the negative sub-level is anti symmetric and vice-versa (cf. § c (ii)).

We consider here the 3IT state, i.e. a state for which the negative rotational

g

levels are antisymmetric and the positive rotational levels are symmetriC (cf. § d (ii)).

Figure

1.8 shows the relative positions of rotational levels taking

account of the symmetry and sign in Hund's case (a) and figure

1.9 b in Hund's

case (b). Transition from case (a) to case (b) in so far as it affects the symme-tries is made clear in figure 1.9

a.

1.1.1.2.2, Energy of a diatomic molecUle in an electronic state

The total energy is given in the Born-Oppenheimer approximation by /6/

where E is the electronic energy, e

and

E is the vibrational energy v

E is the rotational energy.

r

(7)

Ek

If one introduces spectral terms Tk ~ expressed in cm-1 , hc

equation (7) is written

T

=

T + G(v) + F (J)

e v (8)

where V and J are respectively the quantum numbers of vibration and rotation.

This can be written /6/

(9)

where a term A

I!

has been added to the non-1~teracting term To' taking into account the coupling between total spin ~ and the internuclear axis, A being a constant depending upon the electronic state.

1 1 2 1 3 1 It

G(v) '" w (v+-) - w x (v+-) + w y (v+-) - w Z

(v+-4) .,. (10) e 2 e e 2 e e 2 e e

where w • w x • wyand w z are constants for a given electronic state _{e e e e e e e} (w »w x »w y »w Z , •• ), Note that the first term of equation (10)

e e e e e e e

represents the vibrational term when the vibrations are assumed to be simple harmonic.

If the symmetric vibrating top model is assumed. which takes account of the rotation-vibration interaction of the nuclei. the rotational spec-tral term may be written :

F (J) = B J(J+1) + (A'-B )A 2 - 0 J2(J+1)2 (11)

v v v v

B ,Dare constant functions of the vibrational state for a given electronic _{v v} state.

The quantlty A'A2. which is constant for a given electronic state. may be included in the electronic energy.

Furthermore. there are approximate expression for Bv and 0

1 v

retaining only the first order in (v+

_2).

Pekeris (1934) has shown that

B 1 B = _{- a} (v+-) v e e 2 (12 ) '" 0 1 0 +

a

e (v+

2

)

v e (13 )

In the tables one generally only finds

B = h (14 )

e

BTr2c\.lr2 e

where u is the reduced mass

and r is the average internuclear distance during an interaction. e

There are approximate expressions for a_e, ee' De as a function of tabulated data

0

₌

4 B~ (15 ) e w2 _e 6 Be (/w x B - B ) a = e _we e e e e (16 )

Be De [8 wexe _ 5ae _ _we

a~~~

_24B ] e

(1n

dJ Correction terms

The preceding expressions are, in fact, only simplified expressions. For a rigorous treatment, one must refer to Kovacs' excellent book /10/ which gives not only general formulae which are applicable to any type of electronic state but also all the correction terms which take into account

- the spin-orbit interaction between the magnetic moment associated with the electronic orbital angular momentum and the magnetic moments associa-ted with spin J

- the spin-spin interaction between the dipole magnetic moments of the elec-tron spins J

- the spin-rotation interaction between the magnetic field created by - the rotation of the nuclei and the spin of the electrons J

- the change in the coupling mode as a function of the speed of rotation of the molecule (intermediate case),

1.1.1.2.3. Electronic transition rules

The transition rules /6/ between two electronic states are very complex since they depend on the coupling case of each of the two electronic

states, and the type of coupling often depends on the value of J.

whatever the coupling case for the two states, the rule

tlJ =

o.

± 1

is strictly obeyed except that the transition J' :: 0 to JH ::: 0 is not allowed. Furthermore. the transition rules concerning symmetry properties are generally obeyed :

+ +-+ - +

-+

+. -

+

-and when the molecule is homonuclear.

s +-+ S J a +-+ a J s

-+

a.

and finally when the two nuclei have the same charge

g +-+ u g -+1+ g u

-+

u.

(1) Transition rules when the two electronic states belong to case (a) or to case (b)

It is now necessary to include the rules

till ::: 0, ± 1

tiS :: 0

(ii) Transition rules which are only obeyed when both electronic states belong to case (al

In this case

I-

is defined for each electronic state and

The two rules

M :: O. ± 1

are equivalent to the single rule

60

=

0, ± 1

If 0 0 for the two electronic states, the rule

AJ : 0, ± 1

reduces to

AJ ~ ± 1 for 0

=

0 ~

n •

0

(iii) Transition rules which are only obeyed when both electronic states belong to case (b)

In this case, K is defined for each electronic state and

AK • 0, ± 1

with the restriction 6K = ± 1 for the transition

l

~

l

(iv) Transition rules when the two electronic states do not belong to the same coupling case and when the coupling case varies with the value of J.

In this case the transition rules, and therefore the number of transitions, change with J.

1.1.2. Collisions, effectwe crossections, elementary processes, distrIbution function, mobility and diffusion of charged particles (see ref. /14 to 29/)

1.1.2.1. Collisions

In the plasma as in the gas the distances between particles are large compared to their sizes and there will be a collision, or more exactly the particles will influence each other, only when two or more happen to approach close-ly. The result of this mutual influence or collision is that the particles will sensibly deflect each other paths, a collision beingtaken into account only if this deflection is greater than a minimum value. As a consequence of a collision the kinetic and/or potential energy of the participating particles changes and one distinguishes :

elastic collisions during which the total kinetic energy is conserved. T~ese

collisions are practically the only kind that occur in neutral gases at ambient temperatures. Inthe elastic collision the fraction of energy. K, transferred from one particle of mass, m, to another of mass, M, averaged over all angles is

K :: 2mM ::: 2m

M when m « M

(m + M)2

that is why in the case of electron-atom collisions, this form of energy transfer is of little importance,

in elastic collisions during which the total kinetic energy of the particles is changed with a modification of the internal energy of the particles. Processes such as excitation, ionization, recombination, charge transfer, attachment, deta-chment and dissociation are brought about the type of collision

6Etotal < 0 : part of the kinetic energy of one of the particles is used to increase the internal energy of the other particle

~Etotal > 0 : part of the internal energy of one of the particle is used to increase the kinetic energy of the other : super elactic collision.